4. The model file¶
4.1. Conventions¶
A model file contains a list of commands and of blocks. Each command
and each element of a block is terminated by a semicolon (;). Blocks
are terminated by end;.
If Dynare encounters an unknown expression at the beginning of a line or after a semicolon, it will parse the rest of that line as native MATLAB code, even if there are more statements separated by semicolons present. To prevent cryptic error messages, it is strongly recommended to always only put one statement/command into each line and start a new line after each semicolon. [1]
Lines of codes can be commented out line by line or as a block. Single-line
comments begin with // and stop at the end of the line. Multiline comments
are introduced by /* and terminated by */.
Examples
// This is a single line commentvar x; // This is a comment about x/* This is another inline comment about alpha */ alpha = 0.3;/* This comment is spanning two lines. */
Note that these comment marks should not be used in native MATLAB code regions
where the % should be preferred instead to introduce a comment. In a
verbatim block, see Verbatim inclusion, this would result in a crash since
// is not a valid MATLAB statement).
Most Dynare commands have arguments and several accept options, indicated in parentheses after the command keyword. Several options are separated by commas.
In the description of Dynare commands, the following conventions are observed:
Optional arguments or options are indicated between square brackets: ‘[]’;
Repeated arguments are indicated by ellipses: “…”;
Mutually exclusive arguments are separated by vertical bars: ‘|’;
INTEGER indicates an integer number;
INTEGER_VECTOR indicates a vector of integer numbers separated by spaces, enclosed by square brackets;
DOUBLE indicates a double precision number. The following syntaxes are valid:
1.1e3,1.1E3,1.1d3,1.1D3. In some places, infinite ValuesInfand-Infare also allowed;NUMERICAL_VECTOR indicates a vector of numbers separated by spaces, enclosed by square brackets;
EXPRESSION indicates a mathematical expression valid outside the model description (see Expressions);
MODEL_EXPRESSION (sometimes MODEL_EXP) indicates a mathematical expression valid in the model description (see Expressions and Model declaration);
MACRO_EXPRESSION designates an expression of the macro processor (see Macro expressions);
VARIABLE_NAME (sometimes VAR_NAME) indicates a variable name starting with an alphabetical character and can’t contain:
()+-\*/^=!;:@#.or accentuated characters;PARAMETER_NAME (sometimes PARAM_NAME) indicates a parameter name starting with an alphabetical character and can’t contain:
()+-\*/^=!;:@#.or accentuated characters;LATEX_NAME (sometimes TEX_NAME) indicates a valid LaTeX expression in math mode (not including the dollar signs);
FUNCTION_NAME indicates a valid MATLAB function name;
FILENAME indicates a filename valid in the underlying operating system; it is necessary to put it between quotes when specifying the extension or if the filename contains a non-alphanumeric character;
QUOTED_STRING indicates an arbitrary string enclosed between (single) quotes;
DATE indicates a time period which can be either a year (e.g.
2024Yor2024A), a half-year (2024S1or2024H1), a quarter (2024Q2) or a month (2024M3) (see Dates in a mod file). Optionally, the time period can be followed by a plus sign and a number of periods, in which case the date is shifted accordingly (e.g.2023Q1+6is accepted and is equivalent to2024Q3).
4.2. Variable declarations¶
While Dynare allows the user to choose their own variable names, there
are some restrictions to be kept in mind. First, variables and
parameters must not have the same name as Dynare commands or built-in
functions. In this respect, Dynare is not case-sensitive. For example,
do not use Ln or shocks to name your variable. Not conforming
to this rule might yield hard-to-debug error messages or
crashes. Second, when employing user-defined steady state files it is
recommended to avoid using the name of MATLAB functions as this may cause
conflicts. In particular, when working with user-defined steady state files, do not
use correctly-spelled greek names like alpha, because there are
MATLAB functions of the same name. Rather go for alppha or
alph. Lastly, please do not name a variable or parameter
i. This may interfere with the imaginary number i and the index in
many loops. Rather, name investment invest. Using inv is also
not recommended as it already denotes the inverse operator. Commands
for declaring variables and parameters are described below.
- Command: var VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)]...;¶
- Command: var(log) VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)]...;
- Command: var(deflator=MODEL_EXPR) VAR_NAME (... same options apply)
- Command: var(log, deflator=MODEL_EXPR) VAR_NAME (... same options apply)
- Command: var(log_deflator=MODEL_EXPR) VAR_NAME (... same options apply)
This required command declares the endogenous variables in the model. See Conventions for the syntax of VAR_NAME and MODEL_EXPR. Optionally it is possible to give a LaTeX name to the variable or, if it is nonstationary, provide information regarding its deflator. The variables in the list can be separated by spaces or by commas.varcommands can appear several times in the file and Dynare will concatenate them. Dynare stores the list of declared parameters, in the order of declaration, in a column cell arrayM_.endo_names.If the model is nonstationary and is to be written as such in the
modelblock, Dynare will need the trend deflator for the appropriate endogenous variables in order to stationarize the model. The trend deflator must be provided alongside the variables that follow this trend.Options
- log¶
In addition to the endogenous variable(s) thus declared, this option also triggers the creation of auxiliary variable(s) equal to the log of the corresponding endogenous variable(s). For example, given a
var(log) ystatement, two endogenous will be created (yandLOG_y), and an auxiliary equation linking the two will also be added (equal toLOG_y = log(y)). Moreover, every occurence ofyin the model will be replaced byexp(LOG_y). This option is for example useful when one wants to perform a loglinear approximation of some variable(s) in the context of a first-order stochastic approximation; or when one wants to ensure the variable(s) stay(s) in the definition domain of the function defining the steady state or the dynamic residuals when the nonlinear solver is used.
- deflator = MODEL_EXPR¶
The expression used to detrend an endogenous variable. All trend variables, endogenous variables and parameters referenced in MODEL_EXPR must already have been declared by the
trend_var, log_trend_var, varandparameterscommands. The deflator is assumed to be multiplicative; for an additive deflator, uselog_deflator. This option can be used together with thelogoption (the latter must come first).
- log_deflator = MODEL_EXPR¶
Same as
deflator, except that the deflator is assumed to be additive instead of multiplicative (or, to put it otherwise, the declared variable is equal to the log of a variable with a multiplicative trend). This option cannot be used together with thelogoption, because it would not make much sense from an economic point of view (the corresponding auxiliary variable would correspond to the log taken two times on a variable with a multiplicative trend).
- long_name = QUOTED_STRING¶
This is the long version of the variable name. Its value is stored in
M_.endo_names_long(a column cell array, in the same order asM_.endo_names). In case multiplelong_nameoptions are provided, the last one will be used. Default:VAR_NAME.
- NAME = QUOTED_STRING¶
This is used to create a partitioning of variables. It results in the direct output in the
.mfile analogous to:M_.endo_partitions.NAME = QUOTED_STRING;.
Example (variable partitioning)
var c gnp cva (country='US', state='VA') cca (country='US', state='CA', long_name='Consumption CA'); var(deflator=A) i b; var c $C$ (long_name=`Consumption');
- Command: varexo VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];¶
This optional command declares the exogenous variables in the model. See Conventions for the syntax ofVAR_NAME. Optionally it is possible to give a LaTeX name to the variable. Exogenous variables are required if the user wants to be able to apply shocks to her model. The variables in the list can be separated by spaces or by commas.varexocommands can appear several times in the file and Dynare will concatenate them.Options
- long_name = QUOTED_STRING
Like long_name but value stored in
M_.exo_names_long.
- NAME = QUOTED_STRING
Like partitioning but QUOTED_STRING stored in
M_.exo_partitions.NAME.
Example
varexo m gov;
Remarks
An exogenous variable is an innovation, in the sense that this variable cannot be predicted from the knowledge of the current state of the economy. For instance, if logged TFP is a first order autoregressive process:
\[a_t = \rho a_{t-1} + \varepsilon_t\]then logged TFP \(a_t\) is an endogenous variable to be declared with
var, its best prediction is \(\rho a_{t-1}\), while the innovation \(\varepsilon_t\) is to be declared withvarexo.
- Command: varexo_det VAR_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];¶
This optional command declares exogenous deterministic variables in a stochastic model. See Conventions for the syntax of VARIABLE_NAME. Optionally it is possible to give a LaTeX name to the variable. The variables in the list can be separated by spaces or by commas.varexo_detcommands can appear several times in the file and Dynare will concatenate them.It is possible to mix deterministic and stochastic shocks to build models where agents know from the start of the simulation about future exogenous changes. In that case
stoch_simulwill compute the rational expectation solution adding future information to the state space (nothing is shown in the output ofstoch_simul) and forecast will compute a simulation conditional on initial conditions and future information.Note that exogenous deterministic variables cannot appear with a lead or a lag in the model.
Options
- long_name = QUOTED_STRING
Like long_name but value stored in
M_.exo_det_names_long.
- NAME = QUOTED_STRING
Like partitioning but QUOTED_STRING stored in
M_.exo_det_partitions.NAME.
Example
varexo m gov; varexo_det tau;
- Command: parameters PARAM_NAME [$TEX_NAME$] [(long_name=QUOTED_STRING|NAME=QUOTED_STRING)...];¶
This command declares parameters used in the model, in variable initialization or in shocks declarations. See Conventions for the syntax ofPARAM_NAME. Optionally it is possible to give a LaTeX name to the parameter.The parameters must subsequently be assigned values (see Parameter initialization).
The parameters in the list can be separated by spaces or by commas.
parameterscommands can appear several times in the file and Dynare will concatenate them.Options
- long_name = QUOTED_STRING
Like long_name but value stored in
M_.param_names_long.
- NAME = QUOTED_STRING
Like partitioning but QUOTED_STRING stored in
M_.param_partitions.NAME.
Example
parameters alpha, bet;
- Command: change_type(var|varexo|varexo_det|parameters) VAR_NAME | PARAM_NAME...;¶
Changes the types of the specified variables/parameters to another type: endogenous, exogenous, exogenous deterministic or parameter. It is important to understand that this command has a global effect on the
.modfile: the type change is effective after, but also before, thechange_typecommand. This command is typically used when flipping some variables for steady state calibration: typically a separate model file is used for calibration, which includes the list of variable declarations with the macro processor, and flips some variable.Example
var y, w; parameters alpha, beta; ... change_type(var) alpha, beta; change_type(parameters) y, w;
Here, in the whole model file,
alphaandbetawill be endogenous andyandwwill be parameters.
- Command: var_remove VAR_NAME | PARAM_NAME...;¶
Removes the listed variables (or parameters) from the model. Removing a variable that has already been used in a model equation or elsewhere will lead to an error.
- Command: predetermined_variables VAR_NAME...;¶
In Dynare, the default convention is that the timing of a variable reflects when this variable is decided. The typical example is for capital stock: since the capital stock used at current period is actually decided at the previous period, then the capital stock entering the production function isk(-1), and the law of motion of capital must be written:k = i + (1-delta)*k(-1)
Put another way, for stock variables, the default in Dynare is to use a “stock at the end of the period” concept, instead of a “stock at the beginning of the period” convention.
The
predetermined_variablesis used to change that convention. The endogenous variables declared as predetermined variables are supposed to be decided one period ahead of all other endogenous variables. For stock variables, they are supposed to follow a “stock at the beginning of the period” convention.Note that Dynare internally always uses the “stock at the end of the period” concept, even when the model has been entered using the
predetermined_variablescommand. Thus, when plotting, computing or simulating variables, Dynare will follow the convention to use variables that are decided in the current period. For example, when generating impulse response functions for capital, Dynare will plotk, which is the capital stock decided upon by investment today (and which will be used in tomorrow’s production function). This is the reason that capital is shown to be moving on impact, because it iskand not the predeterminedk(-1)that is displayed. It is important to remember that this also affects simulated time series and output from smoother routines for predetermined variables. Compared to non-predetermined variables they might otherwise appear to be falsely shifted to the future by one period.Example
The following two program snippets are strictly equivalent.
Using default Dynare timing convention:
var y, k, i; ... model; y = k(-1)^alpha; k = i + (1-delta)*k(-1); ... end;
Using the alternative timing convention:
var y, k, i; predetermined_variables k; ... model; y = k^alpha; k(+1) = i + (1-delta)*k; ... end;
- Command: trend_var(growth_factor = MODEL_EXPR) VAR_NAME [$LATEX_NAME$]...;¶
This optional command declares the trend variables in the model. See Conventions for the syntax of MODEL_EXPR and VAR_NAME. Optionally it is possible to give a LaTeX name to the variable.The variable is assumed to have a multiplicative growth trend. For an additive growth trend, use
log_trend_varinstead.Trend variables are required if the user wants to be able to write a nonstationary model in the
modelblock. Thetrend_varcommand must appear before the var command that references the trend variable.trend_varcommands can appear several times in the file and Dynare will concatenate them.If the model is nonstationary and is to be written as such in the
modelblock, Dynare will need the growth factor of every trend variable in order to stationarize the model. The growth factor must be provided within the declaration of the trend variable, using thegrowth_factorkeyword. All endogenous variables and parameters referenced in MODEL_EXPR must already have been declared by the var and parameters commands.Example
trend_var (growth_factor=gA) A;
- Command: log_trend_var(log_growth_factor = MODEL_EXPR) VAR_NAME [$LATEX_NAME$]...;¶
Same astrend_var, except that the variable is supposed to have an additive trend (or, to put it otherwise, to be equal to the log of a variable with a multiplicative trend).
- Command: model_local_variable VARIABLE_NAME [LATEX_NAME]... ;¶
This optional command declares a model local variable. See Conventions for the syntax of VARIABLE_NAME. As you can create model local variables on the fly in the model block (see Model declaration), the interest of this command is primarily to assign a LATEX_NAME to the model local variable.Example
model_local_variable GDP_US $GDPUS$;
4.2.1. On-the-fly Model Variable Declaration¶
Endogenous variables, exogenous variables, and parameters can also be declared inside the model block. You can do this in two different ways: either via the equation tag (only for endogenous variables) or directly in an equation (for endogenous, exogenous or parameters).
To declare an endogenous variable on-the-fly in an equation tag, simply write
endogenous followed by an equal sign and the variable name in single
quotes. Hence, to declare a variable c as endogenous in an equation tag,
you can type [endogenous='c'].
To perform on-the-fly variable declaration in an equation, simply follow the
symbol name with a vertical line (|, pipe character) and either an e
(for endogenous), an x (for exogenous), or a p (for parameter). For
example, to declare a parameter named alphaa in the model block, you could
write alphaa|p directly in an equation where it appears. Similarly, to
declare an endogenous variable c in the model block you could write
c|e. Note that in-equation on-the-fly variable declarations must be made on
contemporaneous variables.
On-the-fly variable declarations do not have to appear in the first place where this variable is encountered.
Example
The following two snippets are equivalent:
model; [endogenous='k',name='law of motion of capital'] k(+1) = i|e + (1-delta|p)*k; y|e = k^alpha|p; ... end; delta = 0.025; alpha = 0.36;var k, i, y; parameters delta, alpha; delta = 0.025; alpha = 0.36; ... model; [name='law of motion of capital'] k(1) = i|e + (1-delta|p)*k; y|e = k|e^alpha|p; ... end;
4.3. Expressions¶
Dynare distinguishes between two types of mathematical expressions: those that are used to describe the model, and those that are used outside the model block (e.g. for initializing parameters or variables, or as command options). In this manual, those two types of expressions are respectively denoted by MODEL_EXPRESSION and EXPRESSION.
Unlike MATLAB or Octave expressions, Dynare expressions are necessarily scalar ones: they cannot contain matrices or evaluate to matrices. [2]
Expressions can be constructed using integers (INTEGER), floating point numbers (DOUBLE), parameter names (PARAMETER_NAME), variable names (VARIABLE_NAME), operators and functions.
The following special constants are also accepted in some contexts:
- Constant: inf¶
Represents infinity.
- Constant: nan¶
“Not a number”: represents an undefined or unrepresentable value.
4.3.1. Parameters and variables¶
Parameters and variables can be introduced in expressions by simply typing their names. The semantics of parameters and variables is quite different whether they are used inside or outside the model block.
4.3.1.1. Inside the model¶
Parameters used inside the model refer to the value given through
parameter initialization (see Parameter initialization) or homotopy_setup
when doing a simulation, or are the estimated variables when doing an
estimation.
Variables used in a MODEL_EXPRESSION denote current period values when
neither a lead or a lag is given. A lead or a lag can be given by
enclosing an integer between parenthesis just after the variable name:
a positive integer means a lead, a negative one means a lag. Leads or
lags of more than one period are allowed. For example, if c is an
endogenous variable, then c(+1) is the variable one period ahead,
and c(-2) is the variable two periods before.
When specifying the leads and lags of endogenous variables, it is important to respect the following convention: in Dynare, the timing of a variable reflects when that variable is decided. A control variable — which by definition is decided in the current period — must have no lead. A predetermined variable — which by definition has been decided in a previous period — must have a lag. A consequence of this is that all stock variables must use the “stock at the end of the period” convention.
Leads and lags are primarily used for endogenous variables, but can be used for exogenous variables. They have no effect on parameters and are forbidden for local model variables (see Model declaration).
4.3.1.2. Outside the model¶
When used in an expression outside the model block, a parameter or a
variable simply refers to the last value given to that variable. More
precisely, for a parameter it refers to the value given in the
corresponding parameter initialization (see Parameter initialization); for an
endogenous or exogenous variable, it refers to the value given in the
most recent initval or endval block.
4.3.2. Operators¶
The following operators are allowed in both MODEL_EXPRESSION and EXPRESSION:
Binary arithmetic operators:
+,-,*,/,^Unary arithmetic operators:
+,-Binary comparison operators (which evaluate to either 0 or 1):
<,>,<=,>=,==,!=
Note the binary comparison operators are differentiable everywhere except on a line of the 2-dimensional real plane. However for facilitating convergence of Newton-type methods, Dynare assumes that, at the points of non-differentiability, the partial derivatives of these operators with respect to both arguments is equal to 0 (since this is the value of the partial derivatives everywhere else).
The following special operators are accepted in MODEL_EXPRESSION (but not in EXPRESSION):
- Operator: STEADY_STATE (MODEL_EXPRESSION)¶
This operator is used to take the value of the enclosed expression at the steady state. A typical usage is in the Taylor rule, where you may want to use the value of GDP at steady state to compute the output gap.
Exogenous and exogenous deterministic variables may not appear in MODEL_EXPRESSION.
Warning
The concept of a steady state is ambiguous in a perfect foresight context with permament and potentially anticipated shocks occuring. Dynare will use the contents of
oo_.steady_stateas its reference for calls to theSTEADY_STATE()operator. In the presence ofendval, this implies that the terminal state provided by the user is used. This may be a steady state computed by Dynare (ifendvalis followed bysteady) or simply the terminal state provided by the user (ifendvalis not followed bysteady). Put differently, Dynare will not automatically compute the steady state conditional on the specificed value of the exogenous variables in the respective periods.
- Operator: EXPECTATION (INTEGER) (MODEL_EXPRESSION)¶
This operator is used to take the expectation of some expression using a different information set than the information available at current period. For example,
EXPECTATION(-1)(x(+1))is equal to the expected value of variable x at next period, using the information set available at the previous period. See Auxiliary variables for an explanation of how this operator is handled internally and how this affects the output.
4.3.3. Functions¶
4.3.3.1. Built-in functions¶
The following standard functions are supported internally for both MODEL_EXPRESSION and EXPRESSION:
- Function: exp(x)¶
Natural exponential.
- Function: log(x)
- Function: ln(x)¶
Natural logarithm.
- Function: log10(x)¶
Base 10 logarithm.
- Function: sqrt(x)¶
Square root.
- Function: cbrt(x)¶
Cube root.
- Function: sign(x)¶
Signum function, defined as:
\[\begin{split}\textrm{sign}(x) = \begin{cases} -1 &\quad\text{if }x<0\\ 0 &\quad\text{if }x=0\\ 1 &\quad\text{if }x>0 \end{cases}\end{split}\]Note that this function is not continuous, hence not differentiable, at \(x=0\). However, for facilitating convergence of Newton-type methods, Dynare assumes that the derivative at \(x=0\) is equal to \(0\). This assumption comes from the observation that both the right- and left-derivatives at this point exist and are equal to \(0\), so we can remove the singularity by postulating that the derivative at \(x=0\) is \(0\).
- Function: abs(x)¶
Absolute value.
Note that this continuous function is not differentiable at \(x=0\). However, for facilitating convergence of Newton-type methods, Dynare assumes that the derivative at \(x=0\) is equal to \(0\) (even if the derivative does not exist). The rational for this mathematically unfounded definition, rely on the observation that the derivative of \(\mathrm{abs}(x)\) is equal to \(\mathrm{sign}(x)\) for any \(x\neq 0\) in \(\mathbb R\) and from the convention for the value of \(\mathrm{sign}(x)\) at \(x=0\)).
- Function: sin(x)¶
- Function: cos(x)¶
- Function: tan(x)¶
- Function: asin(x)¶
- Function: acos(x)¶
- Function: atan(x)¶
Trigonometric functions.
- Function: sinh(x)¶
- Function: cosh(x)¶
- Function: tanh(x)¶
- Function: asinh(x)¶
- Function: acosh(x)¶
- Function: atanh(x)¶
Hyperbolic functions.
- Function: max(a, b)¶
- Function: min(a, b)¶
Maximum and minimum of two reals.
Note that these functions are differentiable everywhere except on a line of the 2-dimensional real plane defined by \(a=b\). However for facilitating convergence of Newton-type methods, Dynare assumes that, at the points of non-differentiability, the partial derivative of these functions with respect to the first (resp. the second) argument is equal to \(1\) (resp. to \(0\)) (i.e. the derivatives at the kink are equal to the derivatives observed on the half-plane where the function is equal to its first argument).
- Function: normcdf(x)¶
- Function: normcdf(x, mu, sigma)
Gaussian cumulative density function, with mean mu and standard deviation sigma. Note that
normcdf(x)is equivalent tonormcdf(x,0,1).
- Function: normpdf(x)¶
- Function: normpdf(x, mu, sigma)
Gaussian probability density function, with mean mu and standard deviation sigma. Note that
normpdf(x)is equivalent tonormpdf(x,0,1).
- Function: erf(x)¶
Gauss error function.
- Function: erfc(x)¶
Complementary error function, i.e. \(\mathrm{erfc}(x) = 1-\mathrm{erf}(x)\).
4.3.3.2. External functions¶
Any other user-defined (or built-in) MATLAB or Octave function may be used in both a MODEL_EXPRESSION and an EXPRESSION, provided that this function has a scalar argument as a return value.
To use an external function in a MODEL_EXPRESSION, one must declare
the function using the external_function statement. This is not
required for external functions used in an EXPRESSION outside of a
model block or steady_state_model block.
- Command: external_function(OPTIONS...);¶
This command declares the external functions used in the model block. It is required for every unique function used in the model block.
external_functioncommands can appear several times in the file and must come before the model block.Options
- name = NAME¶
The name of the function, which must also be the name of the M-/MEX file implementing it. This option is mandatory.
- nargs = INTEGER¶
The number of arguments of the function. If this option is not provided, Dynare assumes
nargs = 1.
- first_deriv_provided [= NAME]¶
If NAME is provided, this tells Dynare that the Jacobian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to NAME returns the Jacobian as its second output argument. When this option is not provided, Dynare will use finite difference approximations for computing the derivatives of the function, whenever needed.
- second_deriv_provided [= NAME]¶
If NAME is provided, this tells Dynare that the Hessian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to NAME returns the Hessian as its third output argument. NB: This option can only be used if the
first_deriv_providedoption is used in the sameexternal_functioncommand. When this option is not provided, Dynare will use finite difference approximations for computing the Hessian derivatives of the function, whenever needed.
Example
external_function(name = funcname); external_function(name = otherfuncname, nargs = 2, first_deriv_provided, second_deriv_provided); external_function(name = yetotherfuncname, nargs = 3, first_deriv_provided = funcname_deriv);
4.3.4. A few words of warning in stochastic context¶
The use of the following functions and operators is strongly
discouraged in a stochastic context: max, min, abs,
sign, <, >, <=, >=, ==, !=.
The reason is that the local approximation used by stoch_simul or
estimation will by nature ignore the non-linearities introduced by
these functions if the steady state is away from the kink. And, if the
steady state is exactly at the kink, then the approximation will be
bogus because the derivative of these functions at the kink is bogus
(as explained in the respective documentations of these functions and
operators).
Note that extended_path is not affected by this problem, because
it does not rely on a local approximation of the mode.
4.4. Parameter initialization¶
When using Dynare for computing simulations, it is necessary to calibrate the parameters of the model. This is done through parameter initialization.
The syntax is the following:
PARAMETER_NAME = EXPRESSION;
Here is an example of calibration:
parameters alpha, beta;
beta = 0.99;
alpha = 0.36;
A = 1-alpha*beta;
Internally, the parameter values are stored in M_.params:
- MATLAB/Octave variable: M_.params¶
Contains the values of model parameters. The parameters are in the order that was used in the
parameterscommand, hence ordered as inM_.param_names.
The parameter names are stored in M_.param_names:
- MATLAB/Octave variable: M_.param_names¶
Cell array containing the names of the model parameters.
- MATLAB/Octave command: get_param_by_name('PARAMETER_NAME');¶
Given the name of a parameter, returns its calibrated value as it is stored in
M_.params.
- MATLAB/Octave command: set_param_value('PARAMETER_NAME', MATLAB_EXPRESSION);¶
Sets the calibrated value of a parameter to the provided expression. This does essentially the same as the parameter initialization syntax described above, except that it accepts arbitrary MATLAB/Octave expressions, and that it works from MATLAB/Octave scripts.
4.5. Model declaration¶
The model is declared inside a model block:
- Block: model ;¶
- Block: model(OPTIONS...);
The equations of the model are written in a block delimited bymodelandendkeywords.There must be as many equations as there are endogenous variables in the model, except when computing the unconstrained optimal policy with
ramsey_model,ramsey_policyordiscretionary_policy.The syntax of equations must follow the conventions for MODEL_EXPRESSION as described in Expressions. Each equation must be terminated by a semicolon (‘;’). A normal equation looks like:
MODEL_EXPRESSION = MODEL_EXPRESSION;
When the equations are written in homogenous form, it is possible to omit the ‘=0’ part and write only the left hand side of the equation. A homogenous equation looks like:MODEL_EXPRESSION;
Warning
In Dynare, only equality signs can delineate the left and right-hand side of an equation. If Dynare encounters an expression like
a>=b, this will therefore not define an inequality constraint. Rather, it is interpreted as the homogenous equation (a>=b)=0;, i.e., the Boolean (a>=b) must evaluate to 0. Inequality constraints in Dynare instead need to be set up either via OccBin or as mixed complementarity problems.
Inside the model block, Dynare allows the creation of model-local variables, which constitute a simple way to share a common expression between several equations. The syntax consists of a pound sign (#) followed by the name of the new model local variable (which must not be declared as in Variable declarations, but may have been declared bymodel_local_variable), an equal sign, and the expression for which this new variable will stand. Later on, every time this variable appears in the model, Dynare will substitute it by the expression assigned to the variable (if the model-local variable appears with a lead or a lag attached to it between parenthesis, the substitution will be done by shifting the expression accordingly). Note that the scope of this variable is restricted to the model block; it cannot be used outside. To assign a LaTeX name to the model local variable, use the declaration syntax outlined bymodel_local_variable. A model local variable declaration looks like:#VARIABLE_NAME = MODEL_EXPRESSION;
It is possible to tag equations written in the model block. A tag can serve different purposes by allowing the user to attach arbitrary informations to each equation and to recover them at runtime. For instance, it is possible to name the equations with anametag, using a syntax like:model; [name = 'Budget constraint']; c + k = k^theta*A; end;
Here,
nameis the keyword indicating that the tag names the equation. If an equation of the model is tagged with a name, theresidcommand will display the name of the equations (which may be more informative than the equation numbers) in addition to the equation number. Several tags for one equation can be separated using a comma:model; [name='Taylor rule', endogenous='r'] r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e; end;
More information on tags is available at https://git.dynare.org/Dynare/dynare/-/wikis/Equations-Tags.
There can be several
modelblocks, in which case they are simply concatenated. The set of effective options is also the concatenation of the options declared in all the blocks, but in that case you may rather want to use themodel_optionscommand.Options
- linear¶
Declares the model as being linear. It spares oneself from having to declare initial values for computing the steady state of a stationary linear model. This option can’t be used with non-linear models, it will NOT trigger linearization of the model.
- use_dll¶
Instructs the preprocessor to create dynamic loadable libraries (DLL) containing the model equations and derivatives, instead of writing those in M-files. You need a working compilation environment, i.e. a working
mexcommand (see Compiler installation for more details). Using this option can result in faster simulations or estimations, at the expense of some initial compilation time. Alternatively, this option can be given to thedynarecommand (see Dynare invocation). [3]
- block¶
Perform the block decomposition of the model, and exploit it in computations (steady-state, deterministic simulation, stochastic simulation with first order approximation and estimation). See https://archives.dynare.org/DynareWiki/FastDeterministicSimulationAndSteadyStateComputation for details on the algorithms used in deterministic simulation and steady-state computation.
- bytecode¶
Instead of M-files, use a bytecode representation of the model, i.e. a binary file containing a compact representation of all the equations.
- cutoff = DOUBLE¶
Threshold under which a jacobian element is considered as null during the model normalization. Only available with option
block. Default:1e-15
- mfs = INTEGER¶
Controls the handling of minimum feedback set of endogenous variables for the dynamic model. Only available with option
block. Possible values:0All the endogenous variables are considered as feedback variables.
1The endogenous variables assigned to equation naturally normalized (i.e. of the form \(x=f(Y)\) where \(x\) does not appear in \(Y\)) are potentially recursive variables. All the other variables are forced to belong to the set of feedback variables.
2In addition of variables with
mfs = 1the endogenous variables related to linear equations which could be normalized are potential recursive variables. All the other variables are forced to belong to the set of feedback variables.3In addition of variables with
mfs = 2the endogenous variables related to non-linear equations which could be normalized are potential recursive variables. All the other variables are forced to belong to the set of feedback variables.
Default value is1.
- static_mfs¶
Controls the handling of minimum feedback set of endogenous variables for the static model. Only available with option
block. See themfsoption for the possible values. Default value is0.
- no_static¶
Don’t create the static model file. This can be useful for models which don’t have a steady state.
- differentiate_forward_vars¶
- differentiate_forward_vars = ( VARIABLE_NAME [VARIABLE_NAME ...] )¶
Tells Dynare to create a new auxiliary variable for each endogenous variable that appears with a lead, such that the new variable is the time differentiate of the original one. More precisely, if the model contains
x(+1), then a variableAUX_DIFF_VARwill be created such thatAUX_DIFF_VAR=x-x(-1), andx(+1)will be replaced withx+AUX_DIFF_VAR(+1).The transformation is applied to all endogenous variables with a lead if the option is given without a list of variables. If there is a list, the transformation is restricted to endogenous with a lead that also appear in the list.
This option can useful for some deterministic simulations where convergence is hard to obtain. Bad values for terminal conditions in the case of very persistent dynamics or permanent shocks can hinder correct solutions or any convergence. The new differentiated variables have obvious zero terminal conditions (if the terminal condition is a steady state) and this in many cases helps convergence of simulations.
- parallel_local_files = ( FILENAME [, FILENAME]... )¶
Declares a list of extra files that should be transferred to follower nodes when doing a parallel computation (see Parallel Configuration).
- balanced_growth_test_tol = DOUBLE¶
Tolerance used for determining whether cross-derivatives are zero in the test for balanced growth path (the latter is documented on https://archives.dynare.org/DynareWiki/RemovingTrends). Default:
1e-6
Example (Elementary RBC model)
var c k; varexo x; parameters aa alph bet delt gam; model; c = - k + aa*x*k(-1)^alph + (1-delt)*k(-1); c^(-gam) = (aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam)/(1+bet); end;
Example (Use of model local variables)
The following program:
model; # gamma = 1 - 1/sigma; u1 = c1^gamma/gamma; u2 = c2^gamma/gamma; end;
…is formally equivalent to:
model; u1 = c1^(1-1/sigma)/(1-1/sigma); u2 = c2^(1-1/sigma)/(1-1/sigma); end;
Example (A linear model)
model(linear); x = a*x(-1)+b*y(+1)+e_x; y = d*y(-1)+e_y; end;
- Command: model_options(OPTIONS...);¶
This command accepts the same options as the
modelblock.The purpose of this statement is to specify the options that apply to the whole model, when there are several
modelblocks, so as to restore the symmetry between those blocks (since otherwise onemodelblock would typically bear the options, while the other ones would typically have no option).
- Command: model_remove(TAGS...);¶
This command removes equations that appeared in a previous
modelblock.The equations must be specified by a list of tag values, separated by commas. Each element of the list is either a simple quoted string, in which case it designates an equation by its
nametag; or a tag name (without quotes), followed by an equal sign, then by the tag value (within quotes); or a list of tag-equals-value pairs separated by commas and enclosed within brackets, in which case this element removes the equation(s) that has all these tags with the corresponding values.Each removed equation must either have an
endogenoustag, or have a left hand side containing a single endogenous variable. The corresponding endogenous variable will be either turned into an exogenous (if it is still used in somewhere in the model at that point), otherwise it will be removed from the model.Example
var c k dummy1 dummy2 dummy3; model; c + k - aa*x*k(-1)^alph - (1-delt)*k(-1) + dummy1; c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); [ name = 'eq:dummy1', endogenous = 'dummy1' ] c*k = dummy1; [ foo = 'eq:dummy2' ] log(dummy2) = k + 2; [ name = 'eq:dummy3', bar = 'baz' ] dummy3 = c + 3; end; model_remove('eq:dummy1', foo = 'eq:dummy2', [ name = 'eq:dummy3', bar = 'baz' ]);
In the above example, the last three equations will be removed,
dummy1will be turned into an exogenous, anddummy2anddummy3will be removed.
- Block: model_replace(TAGS...);¶
This block replaces several equations in the model. It removes the equations given by the tags list (with the same syntax as in
model_remove), and it adds equations given within the block (with the same syntax asmodel).No variable is removed or has its type changed in the process.
Example
var c k; model; c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); [ name = 'dummy' ] c*k = 1; end; model_replace('dummy'); c^(-gam) = (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); end;
In the above example, the dummy equation is replaced by a proper Euler equation.
Dynare has the ability to output the original list of model equations
to a LaTeX file, using the write_latex_original_model
command, the list of transformed model equations using the
write_latex_dynamic_model command, and the list of static model
equations using the write_latex_static_model command.
- Command: write_latex_original_model ;¶
- Command: write_latex_original_model(OPTIONS);
This command creates two LaTeX files: one containing the model as defined in the model block and one containing the LaTeX document header information.If your
.modfile isFILENAME.mod, then Dynare will create a file calledFILENAME/latex/original.tex, which includes a file calledFILENAME/latex/original_content.tex(also created by Dynare) containing the list of all the original model equations.If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.
Time subscripts (
t,t+1,t-1, …) will be appended to the variable names, as LaTeX subscripts.Compiling the TeX file requires the following LaTeX packages:
geometry, fullpage, breqn.Options
- write_equation_tags¶
Write the equation tags in the LaTeX output. The equation tags will be interpreted with LaTeX markups.
- Command: write_latex_dynamic_model ;¶
- Command: write_latex_dynamic_model(OPTIONS);
This command creates two LaTeX files: one containing the dynamic model and one containing the LaTeX document header information.If your
.modfile isFILENAME.mod, then Dynare will create a file calledFILENAME/latex/dynamic.tex, which includes a file calledFILENAME/latex/dynamic_content.tex(also created by Dynare) containing the list of all the dynamic model equations.If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.
Time subscripts (
t,t+1,t-1, …) will be appended to the variable names, as LaTeX subscripts.Note that the model written in the TeX file will differ from the model declared by the user in the following dimensions:
The timing convention of predetermined variables (see
predetermined_variables) will have been changed to the default Dynare timing convention; in other words, variables declared as predetermined will be lagged on period back,The
EXPECTATIONoperators will have been removed, replaced by auxiliary variables and new equations (as explained in the documentation ofEXPECTATION),Endogenous variables with leads or lags greater or equal than two will have been removed, replaced by new auxiliary variables and equations,
Exogenous variables with leads or lags will also have been replaced by new auxiliary variables and equations.
For the required LaTeX packages, see
write_latex_original_model.Options
- write_equation_tags
- Command: write_latex_static_model ;¶
- Command: write_latex_static_model(OPTIONS);
This command creates two LaTeX files: one containing the static model and one containing the LaTeX document header information.If your
.modfile isFILENAME.mod, then Dynare will create a file calledFILENAME/latex/static.tex, which includes a file calledFILENAME/latex/static_content.tex(also created by Dynare) containing the list of all the steady state model equations.If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.
Note that the model written in the TeX file will differ from the model declared by the user in the some dimensions (see
write_latex_dynamic_modelfor details).Also note that this command will not output the contents of the optional
steady_state_modelblock (seesteady_state_model); it will rather output a static version (i.e. without leads and lags) of the dynamicmodeldeclared in the model block. To write the LaTeX contents of thesteady_state_modelseewrite_latex_steady_state_model.For the required LaTeX packages, see
write_latex_original_model.Options
- write_equation_tags
See
write_equation_tags.
- Command: write_latex_steady_state_model ;¶
This command creates two LaTeX files: one containing the steady state model and one containing the LaTeX document header information.If your
.modfile isFILENAME.mod, then Dynare will create a file calledFILENAME/latex/steady_state.tex, which includes a file calledFILENAME/latex/steady_state_content.tex(also created by Dynare) containing the list of all the steady state model equations.If LaTeX names were given for variables and parameters (see Variable declarations), then those will be used; otherwise, the plain text names will be used.
Note that the model written in the
.texfile will differ from the model declared by the user in some dimensions (seewrite_latex_dynamic_modelfor details).For the required LaTeX packages, see
write_latex_original_model.
4.6. Auxiliary variables¶
The model which is solved internally by Dynare is not exactly the model declared by the user. In some cases, Dynare will introduce auxiliary endogenous variables—along with corresponding auxiliary equations—which will appear in the final output.
The main transformation concerns leads and lags. Dynare will perform a transformation of the model so that there is only one lead and one lag on endogenous variables and no leads/lags on exogenous variables.
This transformation is achieved by the creation of auxiliary variables
and corresponding equations. For example, if x(+2) exists in the
model, Dynare will create one auxiliary variable AUX_ENDO_LEAD =
x(+1), and replace x(+2) by AUX_ENDO_LEAD(+1).
A similar transformation is done for lags greater than 2 on endogenous
(auxiliary variables will have a name beginning with
AUX_ENDO_LAG), and for exogenous with leads and lags (auxiliary
variables will have a name beginning with AUX_EXO_LEAD or
AUX_EXO_LAG respectively).
Another transformation is done for the EXPECTATION operator. For
each occurrence of this operator, Dynare creates an auxiliary variable
defined by a new equation, and replaces the expectation operator by a
reference to the new auxiliary variable. For example, the expression
EXPECTATION(-1)(x(+1)) is replaced by AUX_EXPECT_LAG_1(-1),
and the new auxiliary variable is declared as AUX_EXPECT_LAG_1 =
x(+2).
Auxiliary variables are also introduced by the preprocessor for the
ramsey_model and ramsey_policy commands. In this case, they
are used to represent the Lagrange multipliers when first order
conditions of the Ramsey problem are computed. The new variables take
the form MULT_i, where i represents the constraint with which
the multiplier is associated (counted from the order of declaration in
the model block).
Auxiliary variables are also introduced by the
differentiate_forward_vars option of the model block. The new
variables take the form AUX_DIFF_FWRD_i, and are equal to
x-x(-1) for some endogenous variable x.
Finally, auxiliary variables will arise in the context of employing the
diff operator.
Once created, all auxiliary variables are included in the set of endogenous variables. The output of decision rules (see below) is such that auxiliary variable names are replaced by the original variables they refer to.
The number of endogenous variables before the creation of auxiliary
variables is stored in M_.orig_endo_nbr, and the number of
endogenous variables after the creation of auxiliary variables is
stored in M_.endo_nbr.
See https://git.dynare.org/Dynare/dynare/-/wikis/Auxiliary-variables for more technical details on auxiliary variables.
4.7. Initial and terminal conditions¶
For most simulation exercises, it is necessary to provide initial (and possibly terminal) conditions. It is also necessary to provide initial guess values for non-linear solvers. This section describes the statements used for those purposes.
In many contexts (deterministic or stochastic), it is necessary to
compute the steady state of a non-linear model: initval then
specifies numerical initial values for the non-linear solver. The
command resid can be used to compute the equation residuals for
the given initial values.
Used in perfect foresight mode, the types of forward-looking models for which Dynare was designed require both initial and terminal conditions. Most often these initial and terminal conditions are static equilibria, but not necessarily.
One typical application is to consider an economy at the equilibrium
at time 0, trigger a shock in first period, and study the trajectory
of return to the initial equilibrium. To do that, one needs
initval and shocks (see Shocks on exogenous variables).
Another one is to study how an economy, starting from arbitrary
initial conditions at time 0 converges towards equilibrium. In this
case models, the command histval permits to specify different
historical initial values for variables with lags for the periods
before the beginning of the simulation. Due to the design of Dynare,
in this case initval is used to specify the terminal conditions.
- Block: initval ;¶
- Block: initval(OPTIONS...);
Theinitvalblock has two main purposes: providing guess values for non-linear solvers in the context of perfect foresight simulations and providing guess values for steady state computations in both perfect foresight and stochastic simulations. Depending on the presence ofhistvalandendvalblocks it is also used for declaring the initial and terminal conditions in a perfect foresight simulation exercise. Because of this interaction of the meaning of aninitvalblock with the presence ofhistvalandendvalblocks in perfect foresight simulations, it is strongly recommended to check that the constructedoo_.endo_simulandoo_.exo_simulvariables contain the desired values after runningperfect_foresight_setupand before runningperfect_foresight_solver. In the presence of leads and lags, these subfields of the results structure will store the historical values for the lags in the first column/row and the terminal values for the leads in the last column/row.The
initvalblock is terminated byend;and contains lines of the form:VARIABLE_NAME = EXPRESSION;
In a deterministic (i.e. perfect foresight) modelFirst, both the
oo_.endo_simulandoo_.exo_simulvariables storing the endogenous and exogenous variables will be filled with the values provided by this block. If there are no other blocks present, it will therefore provide the initial and terminal conditions for all the endogenous and exogenous variables, because it will also fill the last column/row of these matrices. For the intermediate simulation periods it thereby provides the starting values for the solver. In the presence of ahistvalblock (and therefore absence of anendvalblock), thishistvalblock will provide/overwrite the historical values for the state variables (lags) by setting the first column/row ofoo_.endo_simulandoo_.exo_simul. This implies that theinitvalblock in the presence ofhistvalonly sets the terminal values for the variables with leads and provides initial values for the perfect foresight solver.Because of these various functions of
initvalit is often necessary to provide values for all the endogenous variables in aninitvalblock. Initial and terminal conditions are strictly necessary for lagged/leaded variables, while feasible starting values are required for the solver. It is important to be aware that if some variables, endogenous or exogenous, are not mentioned in theinitvalblock, a zero value is assumed. It is particularly important to keep this in mind when specifying exogenous variables usingvarexothat are not allowed to take on the value of zero, like e.g. TFP.Note that if the
initvalblock is immediately followed by asteadycommand, its semantics are slightly changed. Thesteadycommand will compute the steady state of the model for all the endogenous variables, assuming that exogenous variables are kept constant at the value declared in theinitvalblock. These steady state values conditional on the declared exogenous variables are then written intooo_.endo_simuland take up the potential roles as historical and terminal conditions as well as starting values for the solver. Aninitvalblock followed bysteadyis therefore formally equivalent to aninitvalblock with the specified values for the exogenous variables, and the endogenous variables set to the associated steady state values conditional on the exogenous variables.
In a stochastic modelThe main purpose of
initvalis to provide initial guess values for the non-linear solver in the steady state computation. Note that if theinitvalblock is not followed bysteady, the steady state computation will still be triggered by subsequent commands (stoch_simul,estimation…).As such,
initvalallows specifying the initial instrument value for steady state finding when providing an analytical conditional steady state file forramsey_model-computations.It is not necessary to declare 0 as initial value for exogenous stochastic variables, since it is the only possible value.
The subsequently computed steady state (not the initial values, use histval for this) will be used as the initial condition at all the periods preceeding the first simulation period for the three possible types of simulations in stochastic mode:
stoch_simul, if theperiodsoption is specified.forecastas the initial point at which the forecasts are computed.conditional_forecastas the initial point at which the conditional forecasts are computed.
To start simulations at a particular set of starting values that are not a computed steady state, use
histval.Options
- all_values_required¶
Issues an error and stops processing the .mod file if there is at least one endogenous or exogenous variable that has not been set in the initval block.
- Example
initval; c = 1.2; k = 12; x = 1; end; steady;
- Block: endval ;¶
- Block: endval(OPTIONS...);
This block is terminated byend;and contains lines of the form:VARIABLE_NAME = EXPRESSION;
Theendvalblock makes only sense in a deterministic model and cannot be used together withhistval. Similar to theinitvalcommand, it will fill both theoo_.endo_simulandoo_.exo_simulvariables storing the endogenous and exogenous variables with the values provided by this block. If noinitvalblock is present, it will fill the whole matrices, therefore providing the initial and terminal conditions for all the endogenous and exogenous variables, because it will also fill the first and last column/row of these matrices. Due to also filling the intermediate simulation periods it will provide the starting values for the solver as well.If an
initvalblock is present,initvalwill provide the historical values for the variables (if there are states/lags), whileendvalwill fill the remainder of the matrices, thereby still providing i) the terminal conditions for variables entering the model with a lead and ii) the initial guess values for all endogenous variables at all the simulation dates for the perfect foresight solver.Note that if some variables, endogenous or exogenous, are NOT mentioned in the
endvalblock, the value assumed is that of the lastinitvalblock orsteadycommand (if present). Therefore, in contrast toinitval, omitted variables are not automatically assumed to be 0 in this case. Again, it is strongly recommended to check the constructedoo_.endo_simulandoo_.exo_simulvariables after runningperfect_foresight_setupand before runningperfect_foresight_solverto see whether the desired outcome has been achieved.Like
initval, if theendvalblock is immediately followed by asteadycommand, its semantics are slightly changed. Thesteadycommand will compute the steady state of the model for all the endogenous variables, assuming that exogenous variables are kept constant to the value declared in theendvalblock. These steady state values conditional on the declared exogenous variables are then written intooo_.endo_simuland therefore take up the potential roles as historical and terminal conditions as well as starting values for the solver. Anendvalblock followed bysteadyis therefore formally equivalent to anendvalblock with the specified values for the exogenous variables, and the endogenous variables set to the associated steady state values.Options
- all_values_required
See
all_values_required.
Example
var c k; varexo x; model; c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); end; initval; c = 1.2; k = 12; x = 1; end; steady; endval; c = 2; k = 20; x = 2; end; steady; perfect_foresight_setup(periods=200); perfect_foresight_solver;
In this example, the problem is finding the optimal path for consumption and capital for the periods \(t=1\) to \(T=200\), given the path of the exogenous technology level
x.cis a forward-looking variable and the exogenous variablexappears with a lead in the expected return of physical capital, whilekis a purely backward-looking (state) variable.The initial equilibrium is computed by
steadyconditional onx=1, and the terminal one conditional onx=2. Theinitvalblock sets the initial condition fork(since it is the only backward-looking variable), while theendvalblock sets the terminal condition forc(since it is the only forward-looking endogenous variable). The starting values for the perfect foresight solver are given by theendvalblock. See below for more details.Example
var c k; varexo x; model; c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); end; initval; k = 12; end; endval; c = 2; x = 1.1; end; perfect_foresight_setup(periods=200); perfect_foresight_solver;
In this example, there is no steady command, hence the conditions are exactly those specified in the initval and endval blocks. We need terminal conditions for
candx, since both appear with a lead, and an initial condition fork, since it appears with a lag.Setting
x=1.1in theendvalblock without ashocksblock implies that technology is at \(1.1\) in \(t=1\) and stays there forever, becauseendvalis filling all entries ofoo_.endo_simulandoo_.exo_simulexcept for the very first one, which stores the initial conditions and was set to \(0\) by theinitvalblock when not explicitly specifying a value for it.Because the law of motion for capital is backward-looking, we need an initial condition for
kat time \(0\). Due to the presence ofendval, this cannot be done via ahistvalblock, but rather must be specified in theinitvalblock. Similarly, because the Euler equation is forward-looking, we need a terminal condition forcat \(t=201\), which is specified in theendvalblock.As can be seen, it is not necessary to specify
candxin theinitvalblock andkin theendvalblock, because they have no impact on the results. Due to the optimization problem in the first period being to choosec,kat \(t=1\) given the predetermined capital stockkinherited from \(t=0\) as well as the current and future values for technologyx, the values forcandxat time \(t=0\) play no role. The same applies to the choice ofc,kat time \(t=200\), which does not depend onkat \(t=201\). As the Euler equation shows, that choice only depends on current capital as well as future consumptioncand technologyx, but not on future capitalk. The intuitive reason is that those variables are the consequence of optimization problems taking place in at periods \(t=0\) and \(t=201\), respectively, which are not modeled here.Example
initval; c = 1.2; k = 12; x = 1; end; endval; c = 2; k = 20; x = 1.1; end;
In this example, initial conditions for the forward-looking variables
xandcare provided, together with a terminal condition for the backward-looking variablek. As shown in the previous example, these values will not affect the simulation results. Dynare simply takes them as given and basically assumes that there were realizations of exogenous variables and states that make those choices equilibrium values (basically initial/terminal conditions at the unspecified time periods \(t<0\) and \(t>201\)).The above example suggests another way of looking at the use of
steadyafterinitvalandendval. Instead of saying that the implicit unspecified conditions before and after the simulation range have to fit the initial/terminal conditions of the endogenous variables in those blocks, steady specifies that those conditions at \(t<0\) and \(t>201\) are equal to being at the steady state given the exogenous variables in theinitvalandendvalblocks. The endogenous variables at \(t=0\) and \(t=201\) are then set to the corresponding steady state equilibrium values.The fact that
cat \(t=0\) andkat \(t=201\) specified ininitvalandendvalare taken as given has an important implication for plotting the simulated vector for the endogenous variables, i.e. the rows ofoo_.endo_simul: this vector will also contain the initial and terminal conditions and thus is 202 periods long in the example. When you specify arbitrary values for the initial and terminal conditions for forward- and backward-looking variables, respectively, these values can be very far away from the endogenously determined values at \(t=1\) and \(t=200\). While the values at \(t=0\) and \(t=201\) are unrelated to the dynamics for \(0<t<201\), they may result in strange-looking large jumps. In the example above, consumption will display a large jump from \(t=0\) to \(t=1\) and capital will jump from \(t=200\) to \(t=201\) when usingrplotor manually plottingoo_.endo_simul.
- Block: histval ;¶
- Block: histval(OPTIONS...);
In a deterministic perfect foresight contextIn models with lags on more than one period, the
histvalblock permits to specify different historical initial values for different periods of the state variables. In this case, theinitvalblock takes over the role of specifying terminal conditions and starting values for the solver. Note that thehistvalblock does not take non-state variables.This block is terminated by
end;and contains lines of the form:VARIABLE_NAME(INTEGER) = EXPRESSION;
EXPRESSION is any valid expression returning a numerical value and can contain already initialized variable names.By convention in Dynare, period 1 is the first period of the simulation. Going backward in time, the first period before the start of the simulation is period 0, then period -1, and so on.
State variables not initialized in the
histvalblock are assumed to have a value of zero at period 0 and before. Note thathistvalcannot be followed bysteady.Example
model; x=1.5*x(-1)-0.6*x(-2)+epsilon; log(c)=0.5*x+0.5*log(c(+1)); end; histval; x(0)=-1; x(-1)=0.2; end; initval; c=1; x=1; end;
In this example,
histvalis used to set the historical conditions for the two lags of the endogenous variablex, stored in the first column ofoo_.endo_simul. Theinitvalblock is used to set the terminal condition for the forward looking variablec, stored in the last column ofoo_.endo_simul. Moreover, theinitvalblock defines the starting values for the perfect foresight solver for both endogenous variablescandx.In a stochastic simulation context
In the context of stochastic simulations,
histvalallows setting the starting point of those simulations in the state space. As for the case of perfect foresight simulations, all not explicitly specified variables are set to 0. Moreover, as only states enter the recursive policy functions, all values specified for control variables will be ignored. This can be usedIn
stoch_simul, if theperiodsoption is specified. Note that this only affects the starting point for the simulation, but not for the impulse response functions. When using the loglinear option, thehistvalblock nevertheless takes the unlogged starting values.In
forecastas the initial point at which the forecasts are computed. When using the loglinear option, thehistvalblock nevertheless takes the unlogged starting values.In
conditional_forecastfor a calibrated model as the initial point at which the conditional forecasts are computed. When using the loglinear option, thehistvalblock nevertheless takes the unlogged starting values.In
Ramsey policy, where it also specifies the values of the endogenous states (including lagged exogenous) at which the objective function of the planner is computed. Note that the initial values of the Lagrange multipliers associated with the planner’s problem cannot be set (seeevaluate_planner_objective).
Options
- all_values_required
See
all_values_required.
Example
var x y; varexo e; model; x = y(-1)^alpha*y(-2)^(1-alpha)+e; end; initval; x = 1; y = 1; e = 0.5; end; steady; histval; y(0) = 1.1; y(-1) = 0.9; end; stoch_simul(periods=100);
- Command: resid ;¶
This command will display the residuals of the static equations of the model, using the values given for the endogenous in the lastinitvalorendvalblock (or the steady state file if you provided one, see Steady state).Options
- non_zero¶
Only display non-zero residuals.
- Command: initval_file(OPTIONS...);¶
In a deterministic setup, this command is used to specify a path for all endogenous and exogenous variables. The length of these paths must be equal to the number of simulation periods, plus the number of leads and the number of lags of the model (for example, with 50 simulation periods, in a model with 2 lags and 1 lead, the paths must have a length of 53). Note that these paths cover two different things:The constraints of the problem, which are given by the path for exogenous and the initial and terminal values for endogenous
The initial guess for the non-linear solver, which is given by the path for endogenous variables for the simulation periods (excluding initial and terminal conditions)
In perfect foresight and stochastic contexts,
steadyuses the first observation loaded byinitval_fileas guess value to solve for the steady state of the model. This first observation is determined by thefirst_obsoption when it is used.Don’t mix
initval_filewithinitvalstatements. However, afterinitval_file, you can modify the historical initial values withhistvalorhistval_filestatement.There can be several
initval_filestatements in a model file. Each statement resetsoo_.initval_series.Options
- datafile = FILENAME¶
- filename = FILENAME (deprecated)¶
The name of the file containing the data. It must be included in quotes if the filename contains a path or an extension. The command accepts the following file formats:
M-file (extension
.m): for each endogenous and exogenous variable, the file must contain a row or column vector of the same name.MAT-file (extension
.mat): same as for M-files.Excel file (extension
.xlsor.xlsx): for each endogenous and exogenous variable, the file must contain a column of the same name. NB: Octave only supports the.xlsxfile extension and must have the io package installed (easily done via octave by typing ‘pkg install -forge io’). The first column may contain the date of each observation.CSV files (extension
.csv): for each endogenous and exogenous variable, the file must contain a column of the same name. The first column may contain the date of each observation.
- first_obs = {INTEGER | DATE}¶
The observation number or the date (see The dates class) of the first observation to be used in the file
- first_simulation_period = {INTEGER | DATE}¶
The observation number in the file or the date (see dates) at which the simulation (or the forecast) is starting. This option avoids to have to compute the maximum number of lags in the model. The observation corresponding to the first period of simulation doesn’t need to exist in the file as the only dates necessary for initialization are before that date.
- last_simulation_period = {INTEGER | DATE}¶
The observation number in the file or the date (see dates) at which the simulation (or the forecast) is ending. This option avoids to have to compute the maximum number of leads in the model.
- last_obs = {INTEGER | DATE}¶
The observation number or the date (see The dates class) of the last observation to be used in the file.
- nobs = INTEGER¶
The number of observations to be used in the file (starting with first of
first_obsobservation).
- series = DSERIES NAME¶
The name of a DSERIES containing the data (see The dseries class)
Example 1
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv); perfect_foresight_setup(periods=200); perfect_foresight_solver;
The initial and terminal values are taken from file
mydata.csv(nothing guarantees that these vales are the steady state of the model). The guess value for the trajectories are also taken from the file. The file must contain at least 203 observations of variablesc,xande. If there are more than 203 observations available in the file, the first 203 are used byperfect_foresight_setup(periods=200). Note that the values for the auxiliary variable corresponding tox(-2)are automatically computed byinitval_file.Example 2
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, first_obs=10); perfect_foresight_setup(periods=200); perfect_foresight_solver;
The initial and terminal values are taken from file
mydata.csvstarting with the 10th observation in the file. There must be at least 212 observations in the file.Example 3
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; ds = dseries(mydata.csv); lds = log(ds); initval_file(series=lds, first_obs=2010Q1); perfect_foresight_setup(periods=200); perfect_foresight_solver;
The initial and terminal values are taken from dseries
lds. All observations are loaded starting with the 1st quarter of 2010 until the end of the file. There must be data available at least until 2050Q3.Example 4
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, first_simulation_period=2010Q1); perfect_foresight_setup(periods=200); perfect_foresight_solver;
The initial and terminal values are taken from file
mydata.csv. The observations in the file must have dates. All observations are loaded from the 3rd quarter of 2009 until the end of the file. There must be data available in the file at least until 2050Q1.Example 5
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, last_obs = 212); perfect_foresight_setup(periods=200); perfect_foresight_solver;
The initial and terminal values are taken from file
mydata.csv. The first 212 observations are loaded and the first 203 observations will be used byperfect_foresight_setup(periods=200).Example 6
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, first_obs = 10, nobs = 203); perfect_foresight_setup(periods=200); perfect_foresight_solver;
The initial and terminal values are taken from file
mydata.csv. Observations 10 to 212 are loaded.Example 7
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, first_obs = 10); steady;
The values of the 10th observation of
mydata.csvare used as guess value to compute the steady state. The exogenous variables are set to values found in the file or zero if these variables aren’t present.
- Command: histval_file(OPTIONS...);¶
This command is equivalent tohistval, except that it reads its input from a file, and is typically used in conjunction withsmoother2histval.Options
- datafile = FILENAME
- filename = FILENAME (deprecated)
The name of the file containing the data. The command accepts the following file formats:
M-file (extension
.m): for each endogenous and exogenous variable, the file must contain a row or column vector of the same name.MAT-file (extension
.mat): same as for M-files.Excel file (extension
.xlsor.xlsx): for each endogenous and exogenous variable, the file must contain a column of the same name. NB: Octave only supports the.xlsxfile extension and must have the io package installed (easily done via octave by typing ‘pkg install -forge io’). The first column may contain the date of each observation.CSV files (extension
.csv): for each endogenous and exogenous variable, the file must contain a column of the same name. The first column may contain the date of each observation.
- first_obs = {INTEGER | DATE}
The observation number or the date (see The dates class) of the first observation to be used in the file
- first_simulation_period = {INTEGER | DATE}
The observation number in the file or the date (see The dates class) at which the simulation (or the forecast) is starting. This option avoids to have to compute the maximum number of lags in the model. The observation corresponding to the first period of simulation doesn’t need to exist in the file as the only dates necessary for initialization are before that date.
- last_simulation_period = {INTEGER | DATE}
The observation number in the file or the date (see dates) at which the simulation (or the forecast) is ending. This option avoids to have to compute the maximum number of leads in the model.
- last_obs = {INTEGER | DATE}
The observation number or the date (see The dates class) of the last observation to be used in the file.
- nobs = INTEGER
The number of observations to be used in the file (starting with first of
first_obsobservation).
- series = DSERIES NAME
The name of a DSERIES containing the data (see The dseries class)
Example 1
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; steady_state_model; x = 0; c = exp(c*x/(1 - d)); end; histval_file(datafile=mydata.csv); stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken from the two first rows of file
mydata.csv.Example 2
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; histval_file(datafile=mydata.csv, first_obs=10); stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken from rows 10 and 11 of file
mydata.csv.Example 3
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; histval_file(datafile=mydata.csv, first_obs=2010Q1); stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken from observations 2010Q1 and 2010Q2 of file
mydata.csv.Example 4
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; histval_file(datafile=mydata.csv, first_simulation_period=2010Q1) stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken from observations 2009Q3 and 2009Q4 of file
mydata.csv.Example 5
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; histval_file(datafile=mydata.csv, last_obs = 4); stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken from the two first rows of file
mydata.csv.Example 6
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, first_obs = 10, nobs = 4); stoch_simul(order=1,periods=100);
The initial values for the stochastic simulation are taken from rows 10 and 11 of file
mydata.csv.Example 7
var c x; varexo e; parameters a b c d; a = 1.5; b = -0,6; c = 0.5; d = 0.5; model; x = a*x(-1) + b*x(-2) + e; log(c) = c*x + d*log(c(+1)); end; initval_file(datafile=mydata.csv, first_obs=10); histval_file(datafile=myotherdata.csv); perfect_foresight_setup(periods=200); perfect_foresight_solver;
Historical initial values for the simulation are taken from the two first rows of file
myotherdata.csv.Terminal values and guess values for the simulation are taken from file
mydata.csvstarting with the 12th observation in the file. There must be at least 212 observations in the file.
4.8. Shocks on exogenous variables¶
In a deterministic context, when one wants to study the transition of
one equilibrium position to another, it is equivalent to analyze the
consequences of a permanent shock and this in done in Dynare through
the proper use of initval and endval.
Another typical experiment is to study the effects of a temporary
shock after which the system goes back to the original equilibrium (if
the model is stable…). A temporary shock is a temporary change of
value of one or several exogenous variables in the model. Temporary
shocks are specified with the command shocks.
In a stochastic framework, the exogenous variables take random values
in each period. In Dynare, these random values follow a normal
distribution with zero mean, but it belongs to the user to specify the
variability of these shocks. The non-zero elements of the matrix of
variance-covariance of the shocks can be entered with the shocks
command.
If the variance of an exogenous variable is set to zero, this variable will appear in the report on policy and transition functions, but isn’t used in the computation of moments and of Impulse Response Functions. Setting a variance to zero is an easy way of removing an exogenous shock.
Note that, by default, if there are several shocks or mshocks
blocks in the same .mod file, then they are cumulative: all the
shocks declared in all the blocks are considered; however, if a
shocks or mshocks block is declared with the overwrite
option, then it replaces all the previous shocks and mshocks
blocks.
- Block: shocks ;¶
- Block: shocks(overwrite);
See above for the meaning of theoverwriteoption.In deterministic context
For deterministic simulations, the
shocksblock specifies temporary changes in the value of exogenous variables. For permanent shocks, use anendvalblock.The block should contain one or more occurrences of the following group of three lines:
var VARIABLE_NAME; periods INTEGER[:INTEGER] | DATE[:DATE] [[,] INTEGER[:INTEGER] | DATE[:DATE]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
It is possible to specify shocks which last several periods and which can vary over time. The
periodskeyword accepts a list of several dates or date ranges, which must be matched by as many shock values in thevalueskeyword. Note that a range in theperiodskeyword can be matched by only one value in thevalueskeyword. Ifvaluesrepresents a scalar, the same value applies to the whole range. Ifvaluesrepresents a vector, it must have as many elements as there are periods in the range.Note that shock values are not restricted to numerical constants: arbitrary expressions are also allowed, but you have to enclose them inside parentheses.
The feasible range of
periods, when specified as integers, is from 0 to the number ofperiodsspecified inperfect_foresight_setuporperfect_foresight_with_expectation_errors_setup. Alternatively, it is possible to use real dates if thefirst_simulation_periodand/orlast_simulation_periodoption has been passed to the aforementioned commands.Warning
Note that the first endogenous simulation period is period 1 (when specified as an integer). Thus, a shock value specified for the initial period 0 may conflict with (i.e. may overwrite or be overwritten by) values for the initial period specified with
initvalorendval(depending on the exact context). Users should always verify the correct setting ofoo_.exo_simulafterperfect_foresight_setuporperfect_foresight_with_expectation_errors_setup.Example (with scalar values)
shocks; var e; periods 1; values 0.5; var u; periods 4:5; values 0; var v; periods 4:5 6 7:9; values 1 1.1 0.9; var w; periods 1 2; values (1+p) (exp(z)); end;
Example (with vector values)
xx = [1.2; 1.3; 1]; shocks; var e; periods 1:3; values (xx); end;
Example (with dates)
shocks; var e; periods 2023Q1; values 0.5; var u; periods 2023Q2:2023Q4 2024Q1; values 0 0.1; end;
In stochastic contextFor stochastic simulations, the
shocksblock specifies the non zero elements of the covariance matrix of the shocks of exogenous variables.You can use the following types of entries in the block:
Specification of the standard error of an exogenous variable.
var VARIABLE_NAME; stderr EXPRESSION;
Specification of the variance of an exogenous variable.
var VARIABLE_NAME = EXPRESSION;
Specification the covariance of two exogenous variables.
var VARIABLE_NAME, VARIABLE_NAME = EXPRESSION;
Specification of the correlation of two exogenous variables.
corr VARIABLE_NAME, VARIABLE_NAME = EXPRESSION;
In an estimation context, it is also possible to specify variances and covariances on endogenous variables: in that case, these values are interpreted as the calibration of the measurement errors on these variables. This requires the
varobscommand to be specified before theshocksblock.Example
shocks; var e = 0.000081; var u; stderr 0.009; corr e, u = 0.8; var v, w = 2; end;
In stochastic optimal policy contextWhen computing conditional welfare in a
ramsey_modelordiscretionary_policycontext, welfare is conditional on the state values inherited by planner when making choices in the first period. The information set of the first period includes the respective exogenous shock realizations. Thus, their known value can be specified using the perfect foresight syntax. Note that i) all other values specified for periods than period 1 will be ignored and ii) the value of lagged shocks (e.g. in the case of news shocks) is specified withhistval.Example
shocks; var u; stderr 0.008; var u; periods 1; values 1; end;
Mixing deterministic and stochastic shocks
It is possible to mix deterministic and stochastic shocks to build models where agents know from the start of the simulation about future exogenous changes. In that case
stoch_simulwill compute the rational expectation solution adding future information to the state space (nothing is shown in the output ofstoch_simul) andforecastwill compute a simulation conditional on initial conditions and future information.Example
varexo_det tau; varexo e; ... shocks; var e; stderr 0.01; var tau; periods 1:9; values -0.15; end; stoch_simul(irf=0); forecast;
- Block: mshocks ;¶
- Block: mshocks(OPTIONS...);
The purpose of this block is similar to that of theshocksblock for deterministic shocks, except that the numeric values given will be interpreted in a multiplicative way. For example, if a value of1.05is given as shock value for some exogenous at some date, it means 5% above its steady state value.If no
endvalblock is present, the steady state as specified in theinitvalblock is used as the basis for the multiplication. If anendvalblock is present, the terminal steady state as specified in theendvalblock will be used as the basis for the multiplication (unless therelative_to_initvaloption is passed).The syntax is the same as
shocksin a deterministic context.This command is only meaningful in two situations:
on exogenous variables with a non-zero steady state, in a deterministic setup,
on deterministic exogenous variables with a non-zero steady state, in a stochastic setup.
Options
- relative_to_initval¶
If an
endvalblock is present, the initial steady state as specified in theinitvalblock will be used as the basis for multiplication (instead of the terminal steady state).
- Block: heteroskedastic_shocks ;¶
- Block: heteroskedastic_shocks(overwrite);
In estimation context, it implements heteroskedastic filters, where the standard error of shocks may unexpectedly change in every period. The standard deviation of shocks may be either provided directly or set/modified in each observed period by a scale factor. Ifstd0is the usual standard error forshock1, then:using a scale factor in period
timplies:std(shock1|t)=std0(shock1)*scale(t)using a provided value in period
timplies:std(shock1|t)=value(t).
The block has a similar syntax as the
shocksblock in a perfect foresight context. It should contain one or more occurrences of the following group of three lines (for setting values):var VARIABLE_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
OR (for setting scale factors):
var VARIABLE_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; scales DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
NOTE:
scalesandvaluescannot be simultaneously set for the same shock in the same period, but it is possible to setvaluesfor some periods andscalesfor other periods for the same shock. There can be only onescalesandvaluesdirective each for a given shock, so all affected periods must be set in one statement.Example
heteroskedastic_shocks; var e1; periods 86:87, 89:97; scales 0.5, 0; var e1; periods 88; values 0.1; var e2; periods 86:87 88:97; values 0.04 0.01; end;
- MATLAB/Octave command: get_shock_stderr_by_name('EXOGENOUS_NAME');¶
Given the name of an exogenous variable, returns its standard deviation, as set by a previousshocksblock.
- MATLAB/Octave command: set_shock_stderr_value('EXOGENOUS_NAME', MATLAB_EXPRESSION);¶
Sets the standard deviation of an exgonous variable. This does essentially the same as setting the standard error via ashocksblock, except that it accepts arbitrary MATLAB/Octave expressions, and that it works from MATLAB/Octave scripts.
4.9. Other general declarations¶
- Command: dsample INTEGER [INTEGER];¶
Reduces the number of periods considered in subsequent output commands.
4.10. Steady state¶
There are two ways of computing the steady state (i.e. the static equilibrium) of a model. The first way is to let Dynare compute the steady state using a nonlinear Newton-type solver; this should work for most models, and is relatively simple to use. The second way is to give more guidance to Dynare, using your knowledge of the model, by providing it with a method to compute the steady state, either using a steady_state_model block or writing matlab routine.
4.10.1. Finding the steady state with Dynare nonlinear solver¶
- Command: steady ;¶
- Command: steady(OPTIONS...);
This command computes the steady state of a model using a nonlinear Newton-type solver and displays it. When a steady state file is usedsteadydisplays the steady state and checks that it is a solution of the static model.More precisely, it computes the equilibrium value of the endogenous variables for the value of the exogenous variables specified in the previous
initvalorendvalblock.steadyuses an iterative procedure and takes as initial guess the value of the endogenous variables set in the previousinitvalorendvalblock.For complicated models, finding good numerical initial values for the endogenous variables is the trickiest part of finding the equilibrium of that model. Often, it is better to start with a smaller model and add new variables one by one.
Options
- maxit = INTEGER¶
Determines the maximum number of iterations used in the non-linear solver. The default value of
maxitis 50.
- tolf = DOUBLE¶
Convergence criterion for termination based on the function value. Iteration will cease when the residuals are smaller than
tolf. Default:eps^(1/3)
- tolx = DOUBLE¶
Convergence criterion for termination based on the step tolerance along. Iteration will cease when the attempted step size is smaller than
tolx. Default:eps^(2/3)
- non_zero
See
non_zero.
- solve_algo = INTEGER¶
Determines the non-linear solver to use. Possible values for the option are:
0Use
fsolve(under MATLAB, only available if you have the Optimization Toolbox; always available under Octave).1Use a Newton-like algorithm with line-search.
2Splits the model into recursive blocks and solves each block in turn using the same solver as value
1.3Use Chris Sims’ solver.
4Splits the model into recursive blocks and solves each block in turn using a trust-region solver with autoscaling.
56Newton algorithm with a sparse LU solver at each iteration.
7Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration.
8Newton algorithm with a Stabilized Bi-Conjugate Gradient (BiCGStab) solver at each iteration.
9Trust-region algorithm with autoscaling (same as value
4, but applied to the entire model, without splitting).10Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra (2004)). The complementarity conditions are specified using the perpendicular symbol, see
lmmcp.11PATH mixed complementarity problem solver of Ferris and Munson (1999). The complementarity conditions are specified using the perpendicular symbol, see
lmmcp. Dynare only provides the interface for using the solver. Due to licence restrictions, you have to download the solver’s most current version yourself from http://pages.cs.wisc.edu/~ferris/path.html and place it in MATLAB’s search path.12Computes a block decomposition and then applies a Newton-type solver on those smaller blocks rather than on the full nonlinear system. This is similar to
2, but is typically more efficient. The block decomposition is done at the preprocessor level, which brings two benefits: it identifies blocks that can be evaluated rather than solved; and evaluations of the residual and Jacobian of the model are more efficient because only the relevant elements are recomputed at every iteration. This option is typically used with theperfect_foresight_solvercommand with purely backward, forward or static models, or with routines for semi-structural models, and it must not be combined with optionblockof themodelblock ormodel_optionscommand. Also note that for those models, the block decomposition is performed as ifmfs=3had been passed to themodelblock ormodel_optionscommand, and the decomposition is slightly different because it is computed in a time-recursive fashion (i.e. in such a way that the simulation is meant to be done with the outer loop on periods and the inner loop on blocks; while for models with both leads and lags, the outer loop is on blocks and the inner loop is on periods).14Same as
12, except that it applies a trust region solver (similar to4) to the blocks.
Default value is4.
- homotopy_mode = INTEGER¶
Use a homotopy (or divide-and-conquer) technique to solve for the steady state. If you use this option, you must specify a
homotopy_setupblock. This option can take three possible values:0Do not use homotopy.
1In this mode, all the parameters are changed simultaneously, and the distance between the boundaries for each parameter is divided in as many intervals as there are steps (as defined by the
homotopy_stepsoption); the problem is solved as many times as there are steps.2Same as mode
1, except that only one parameter is changed at a time; the problem is solved as many times as steps times number of parameters.3Dynare tries first the most extreme values. If it fails to compute the steady state, the interval between initial and desired values is divided by two for all parameters. Every time that it is impossible to find a steady state, the previous interval is divided by two. When it succeeds to find a steady state, the previous interval is multiplied by two. In that last case
homotopy_stepscontains the maximum number of computations attempted before giving up.
Default value is0.
- homotopy_steps = INTEGER¶
Defines the number of steps when performing a homotopy. See
homotopy_modeoption for more details. Default is10.
- homotopy_force_continue = INTEGER¶
This option controls what happens when homotopy fails.
0steadyfails with an error message1steadykeeps the values of the last homotopy step that was successful and continues. BE CAREFUL: parameters and/or exogenous variables are NOT at the value expected by the user
Default is0.
- nocheck¶
Don’t check the steady state values when they are provided explicitly either by a steady state file or a
steady_state_modelblock. This is useful for models with unit roots as, in this case, the steady state is not unique or doesn’t exist.
- markowitz = DOUBLE¶
Value of the Markowitz criterion (in the interval \((0,\infty)\)) used to select the pivot with sparse Gaussian elimination (
solve_algo = 5). This criterion governs the tradeoff between selecting the pivot resulting in the most accurate solution (lowmarkowitzvalues) and the one that preserves maximum sparsity (highmarkowitzvalues). Default: 0.5.
- fsolve_options = (NAME, VALUE, ...)¶
A list of NAME and VALUE pairs. Can be used to set options for the
fsolveroutine, which is selected whensolve_algo = 0(this option has no effect for other values ofsolve_algo). For the list of available name/value pairs, see the documentation offsolvein the MATLAB or Octave manual. Note that Dynare already uses the values of themaxit,tolfandtolxoptions of thesteadycommand for initializing the corresponding options passed tofsolve, so you should not need to override those. Also note that you should not try to override the value of theJacobianorSpecifyObjectiveGradientoption.
Example
After computation, the steady state is available in the following variable:
- MATLAB/Octave variable: oo_.steady_state¶
Contains the computed steady state. Endogenous variables are ordered in the order of declaration used in the
varcommand (which is also the order used inM_.endo_names).
- MATLAB/Octave variable: oo_.exo_steady_state¶
Contains the steady state of the exogenous variables, as declared by the previous
initvalorendvalblock. Exogenous variables are ordered in the order of declaration used in thevarexocommand (which is also the order used inM_.exo_names).
- MATLAB/Octave command: get_mean('ENDOGENOUS_NAME' [, 'ENDOGENOUS_NAME']... );¶
Returns the steady of state of the given endogenous variable(s), as it is stored in
oo_.steady_state. Note that, if the steady state has not yet been computed withsteady, it will first try to compute it.
- Block: homotopy_setup ;¶
- Block: homotopy_setup(from_initval_to_endval) ;
This block is used to declare initial and final values when using a homotopy method. It is used in conjunction with the option
homotopy_modeof the steady command.The idea of homotopy (also called divide-and-conquer by some authors) is to subdivide the problem of finding the steady state into smaller problems. It assumes that you know how to compute the steady state for a given set of parameters, and it helps you finding the steady state for another set of parameters, by incrementally moving from one to another set of parameters.
The purpose of the
homotopy_setupblock is to declare the final (and possibly also the initial) values for the parameters or exogenous that will be changed during the homotopy. It should contain lines of the form:VARIABLE_NAME, EXPRESSION, EXPRESSION;
This syntax specifies the initial and final values of a given parameter/exogenous.
There is an alternative syntax:
VARIABLE_NAME, EXPRESSION;
Here only the final value is specified for a given parameter/exogenous; the initial value is taken from the preceeding
initvalblock (or from the preceedingendvalblock if there is one before thehomotopy_setupblock).A necessary condition for a successful homotopy is that Dynare must be able to solve the steady state for the initial parameters/exogenous without additional help (using the guess values given in the
initvalorendvalblock).The
from_initval_to_endvaloption can be used in the context of a permanent shock, when the initial steady state has already been computed. This option can be used following theendvalblock that describes the terminal steady state. In that case, in the subsequentsteadycommand, Dynare will perform a homotopy from the initial to the terminal steady state (technically, using this option is equivalent to writing ahomotopy_setupblock where all exogenous variables are asked to transition from their values in theinitvalto their values in theendvalblock). When this option is used, thehomotopy_setupblock is typically empty (but it’s nevertheless possible to add explicit directives for moving exogenous or parameters; these will be added on top of those implicitly generated by thefrom_initval_to_endvaloption).If the homotopy fails, a possible solution is to increase the number of steps (given in
homotopy_stepsoption ofsteady).Example
In the following example, Dynare will first compute the steady state for the initial values (
gam=0.5andx=1), and then subdivide the problem into 50 smaller problems to find the steady state for the final values (gam=2andx=2):var c k; varexo x; parameters alph gam delt bet aa; alph=0.5; delt=0.02; aa=0.5; bet=0.05; model; c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); end; initval; x = 1; k = ((delt+bet)/(aa*x*alph))^(1/(alph-1)); c = aa*x*k^alph-delt*k; end; homotopy_setup; gam, 0.5, 2; x, 2; end; steady(homotopy_mode = 1, homotopy_steps = 50);
4.10.2. Providing the steady state to Dynare¶
If you know how to compute the steady state for your model, you can
provide a MATLAB/Octave function doing the computation instead of
using steady. Again, there are two options for doing that:
The easiest way is to write a
steady_state_modelblock, which is described below in more details. See alsofs2000.modin theexamplesdirectory for an example. The steady state file generated by Dynare will be called+FILENAME/steadystate.m.You can write the corresponding MATLAB function by hand. If your
.modfile is calledFILENAME.mod, the steady state file must be calledFILENAME_steadystate.m. SeeNK_baseline_steadystate.min the examples directory for an example. This option gives a bit more flexibility (loops and conditional structures can be used), at the expense of a heavier programming burden and a lesser efficiency.
Note that both files allow to update parameters in each call of the
function. This allows for example to calibrate a model to a labor
supply of 0.2 in steady state by setting the labor disutility
parameter to a corresponding value (see NK_baseline_steadystate.m
in the examples directory). They can also be used in estimation
where some parameter may be a function of an estimated parameter and
needs to be updated for every parameter draw. For example, one might
want to set the capital utilization cost parameter as a function of
the discount rate to ensure that capacity utilization is 1 in steady
state. Treating both parameters as independent or not updating one as
a function of the other would lead to wrong results. But this also
means that care is required. Do not accidentally overwrite your
parameters with new values as it will lead to wrong results.
- Block: steady_state_model ;¶
When the analytical solution of the model is known, this command can be used to help Dynare find the steady state in a more efficient and reliable way, especially during estimation where the steady state has to be recomputed for every point in the parameter space.Each line of this block consists of a variable (either an endogenous, a temporary variable or a parameter) which is assigned an expression (which can contain parameters, exogenous at the steady state, or any endogenous or temporary variable already declared above). Each line therefore looks like:
VARIABLE_NAME = EXPRESSION;
Note that it is also possible to assign several variables at the same time, if the main function in the right hand side is a MATLAB/Octave function returning several arguments:
[ VARIABLE_NAME, VARIABLE_NAME... ] = EXPRESSION;
Dynare will automatically generate a steady state file (of the form
+FILENAME/steadystate.m) using the information provided in this block.Steady state file for deterministic models
The
steady_state_modelblock also works with deterministic models. Aninitvalblock and, when necessary, anendvalblock, is used to set the value of the exogenous variables. Eachinitvalorendvalblock must be followed bysteadyto execute the function created bysteady_state_modeland set the initial, respectively terminal, steady state.Example
var m P c e W R k d n l gy_obs gp_obs y dA; varexo e_a e_m; parameters alp bet gam mst rho psi del; ... // parameter calibration, (dynamic) model declaration, shock calibration... ... steady_state_model; dA = exp(gam); gst = 1/dA; // A temporary variable m = mst; // Three other temporary variables khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1)); xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1); nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp ); n = xist/(nust+xist); P = xist + nust; k = khst*n; l = psi*mst*n/( (1-psi)*(1-n) ); c = mst/P; d = l - mst + 1; y = k^alp*n^(1-alp)*gst^alp; R = mst/bet; // You can use MATLAB functions which return several arguments [W, e] = my_function(l, n); gp_obs = m/dA; gy_obs = dA; end; steady;
4.10.3. Replace some equations during steady state computations¶
When there is no steady state file, Dynare computes the steady state
by solving the static model, i.e. the model from the .mod file
from which leads and lags have been removed.
In some specific cases, one may want to have more control over the way this static model is created. Dynare therefore offers the possibility to explicitly give the form of equations that should be in the static model.
More precisely, if an equation is prepended by a [static] tag,
then it will appear in the static model used for steady state
computation, but that equation will not be used for other
computations. For every equation tagged in this way, you must tag
another equation with [dynamic]: that equation will not be used
for steady state computation, but will be used for other computations.
This functionality can be useful on models with a unit root, where
there is an infinity of steady states. An equation (tagged
[dynamic]) would give the law of motion of the nonstationary
variable (like a random walk). To pin down one specific steady state,
an equation tagged [static] would affect a constant value to the
nonstationary variable. Another situation where the [static] tag
can be useful is when one has only a partial closed form solution for
the steady state.
Example
This is a trivial example with two endogenous variables. The second equation takes a different form in the static model:
var c k;
varexo x;
...
model;
c + k - aa*x*k(-1)^alph - (1-delt)*k(-1);
[dynamic] c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam);
[static] k = ((delt+bet)/(x*aa*alph))^(1/(alph-1));
end;
4.11. Getting information about the model¶
- Command: check ;¶
- Command: check(OPTIONS...);
Computes the eigenvalues of the model linearized around the values specified by the lastinitval,endvalorsteadystatement. Generally, the eigenvalues are only meaningful if the linearization is done around a steady state of the model. It is a device for local analysis in the neighborhood of this steady state.A necessary condition for the uniqueness of a stable equilibrium in the neighborhood of the steady state is that there are as many eigenvalues larger than one in modulus as there are forward looking variables in the system. An additional rank condition requires that the square submatrix of the right Schur vectors corresponding to the forward looking variables (jumpers) and to the explosive eigenvalues must have full rank.
Note that the outcome may be different from what would be suggested by
sum(abs(oo_.dr.eigval))when eigenvalues are very close toqz_criterium.Options
- solve_algo = INTEGER
See solve_algo, for the possible values and their meaning.
- qz_zero_threshold = DOUBLE¶
Value used to test if a generalized eigenvalue is \(0/0\) in the generalized Schur decomposition (in which case the model does not admit a unique solution). Default:
1e-6.
Output
checkreturns the eigenvalues in the global variableoo_.dr.eigval.
- MATLAB/Octave variable: oo_.dr.eigval¶
Contains the eigenvalues of the model, as computed by the
checkcommand.
- Command: model_diagnostics ;¶
This command performs various sanity checks on the model, and prints a message if a problem is detected (missing variables at current period, invalid steady state, singular Jacobian of static model).
- Command: model_info ;¶
- Command: model_info(OPTIONS...);
This command provides information about the model. By default, it will provide a list of predetermined state variables, forward-looking variables, and purely static variables.The command also allows to display information on the dynamic and static versions of the block decomposition of the model:
The normalization of the model: an endogenous variable is attributed to each equation of the model (the dependent variable);
The block structure of the model: for each block
model_infoindicates its type, size as well as the equation number(s) or name tags and endogenous variables belonging to this block.
There are five different types of blocks depending on the simulation method used:
EVALUATE FORWARDIn this case the block contains only equations where the dependent variable \(j\) attributed to the equation appears contemporaneously on the left hand side and where no forward looking endogenous variables appear. The block has the form: \(y_{j,t} = f_j(y_t, y_{t-1}, \ldots, y_{t-k})\).
EVALUATE BACKWARDThe block contains only equations where the dependent variable \(j\) attributed to the equation appears contemporaneously on the left hand side and where no backward looking endogenous variables appear. The block has the form: \(y_{j,t} = f_j(y_t, y_{t+1}, \ldots, y_{t+k})\).
SOLVE BACKWARD xThe block contains only equations where the dependent variable \(j\) attributed to the equation does not appear contemporaneously on the left hand side and where no forward looking endogenous variables appear. The block has the form: \(g_j(y_{j,t}, y_t, y_{t-1}, \ldots, y_{t-k})=0\). Here,
xdenotes the subtype of the block.xis equal toSIMPLEif the block has only one equation. If several equations appear in the block,xis equal toCOMPLETE.SOLVE FORWARD xThe block contains only equations where the dependent variable \(j\) attributed to the equation does not appear contemporaneously on the left hand side and where no backward looking endogenous variables appear. The block has the form: \(g_j(y_{j,t}, y_t, y_{t+1}, \ldots, y_{t+k})=0\). Here,
xdenotes the subtype of the block.xis equal toSIMPLEif the block has only one equation. If several equations appear in the block,xis equal toCOMPLETE.SOLVE TWO BOUNDARIES xThe block contains equations depending on both forward and backward variables. The block looks like: \(g_j(y_{j,t}, y_t, y_{t-1}, \ldots, y_{t-k} ,y_t, y_{t+1}, \ldots, y_{t+k})=0\). Here,
xdenotes the subtype of the block.xis equal toSIMPLEif the block has only one equation. If several equations appear in the block,xis equal toCOMPLETE.
Options
- block_static¶
Prints out the block decomposition of the static model.
- block_dynamic¶
Prints out the block decomposition of the dynamic model.
- incidence¶
Displays the gross incidence matrix and the reordered incidence matrix of the block decomposed model for the
block_dynamicorblock_staticoptions.
- Command: print_bytecode_dynamic_model ;¶
Prints the equations and the Jacobian matrix of the dynamic model stored in the bytecode binary format file. Can only be used in conjunction with thebytecodeoption of themodelblock ormodel_optionscommand.
- Command: print_bytecode_static_model ;¶
Prints the equations and the Jacobian matrix of the static model stored in the bytecode binary format file. Can only be used in conjunction with thebytecodeoption of themodelblock ormodel_optionscommand.
4.12. Deterministic simulation¶
4.12.1. Perfect foresight¶
When the framework is deterministic, Dynare can be used for models
with the assumption of perfect foresight. Typically, the system is
supposed to be in a state of equilibrium before a period 1 when the
news of a contemporaneous or of a future shock is learned by the
agents in the model. The purpose of the simulation is to describe the
reaction in anticipation of, then in reaction to the shock, until the
system returns to the old or to a new state of equilibrium. In most
models, this return to equilibrium is only an asymptotic phenomenon,
which one must approximate by an horizon of simulation far enough in
the future. Another exercise for which Dynare is well suited is to
study the transition path to a new equilibrium following a permanent
shock. For deterministic simulations, the numerical problem consists
of solving a nonlinear system of simultaneous equations in n endogenous
variables in T periods. Dynare offers several algorithms for solving
this problem, which can be chosen via the stack_solve_algo
option. By default (stack_solve_algo=0), Dynare uses a Newton-type
method to solve the simultaneous equation system. Because the
resulting Jacobian is in the order of n by T and hence will be
very large for long simulations with many variables, Dynare makes use
of the sparse matrix capacities of MATLAB/Octave. A slower but
potentially less memory consuming alternative (stack_solve_algo=1)
is based on a Newton-type algorithm first proposed by Laffargue
(1990) and Boucekkine (1995), which avoids ever storing the full
Jacobian. The details of the algorithm can be found in Juillard
(1996). The third type of algorithms makes use of block decomposition
techniques (divide-and-conquer methods) that exploit the structure of
the model. The principle is to identify recursive and simultaneous
blocks in the model structure and use this information to aid the
solution process. These solution algorithms can provide a significant
speed-up on large models.
Warning
Be careful when employing auxiliary variables in the context
of perfect foresight computations. The same model may work for stochastic
simulations, but fail for perfect foresight simulations. The issue arises
when an equation suddenly only contains variables dated t+1 (or t-1
for that matter). In this case, the derivative in the last (first) period
with respect to all variables will be 0, rendering the stacked Jacobian singular.
Example
Consider the following specification of an Euler equation with log utility:
Lambda = beta*C(-1)/C; Lambda(+1)*R(+1)= 1;Clearly, the derivative of the second equation with respect to all endogenous variables at time
tis zero, causingperfect_foresight_solverto generally fail. This is due to the use of the Lagrange multiplierLambdaas an auxiliary variable. Instead, employing the identicalbeta*C/C(+1)*R(+1)= 1;will work.
- Command: perfect_foresight_setup ;¶
- Command: perfect_foresight_setup(OPTIONS...);
Prepares a perfect foresight simulation, by extracting the information in theinitval,endvalandshocksblocks and converting them into simulation paths for exogenous and endogenous variables.This command must always be called before running the simulation with
perfect_foresight_solver.Options
- periods = INTEGER¶
Number of periods of the simulation. This option is mandatory, unless both
first_simulation_periodandlast_simulation_periodoptions are given.
- first_simulation_period = DATE¶
Assign a date to the first simulation period, i.e. the first period in which endogenous variables are solved for. When this option is set, it becomes possible to declare shocks using dates, and
perfect_foresight_solverreturns the result of the simulation as a time series object inSimulated_time_series.
- last_simulation_period = DATE¶
Assign a date to the last simulation period, i.e. the last period in which endogenous variables are solved for. When this option is set, it becomes possible to declare shocks using dates, and
perfect_foresight_solverreturns the result of the simulation as a time series object inSimulated_time_series.
- datafile = FILENAME
Used to specify path for all endogenous and exogenous variables. Strictly equivalent to
initval_file.
- endval_steady¶
In scenarios with a permanent shock, specifies that the terminal condition is a steady state, even if the
steadycommand has not been called after theendvalblock. As a consequence, the subsequentperfect_foresight_solvercommand will compute the terminal steady state itself (given the value of the exogenous variables given in theendvalblock). In practice, this option is useful when the permanent shock is very large, in which case the homotopy procedure insideperfect_foresight_solverwill find both the terminal steady state and the transitional dynamics within the same loop (which is less costly than first computing the terminal steady state by homotopy, then computing the transitional dynamics by homotopy).
Output
The paths for the exogenous variables are stored into
oo_.exo_simul.The initial and terminal conditions for the endogenous variables and the initial guess for the path of endogenous variables are stored into
oo_.endo_simul.
- Command: perfect_foresight_solver ;¶
- Command: perfect_foresight_solver(OPTIONS...);
Computes the perfect foresight (or deterministic) simulation of the model.Note that
perfect_foresight_setupmust be called before this command, in order to setup the environment for the simulation.If the perfect foresight solver cannot directly find the solution of the problem, it subsequently tries a homotopy technique (unless the
no_homotopyoption is given). Concretely, this technique consists in dividing the problem into smaller steps by diminishing the size of shocks and increasing them progressively until the problem converges.Options
- maxit = INTEGER
Determines the maximum number of iterations used in the non-linear solver. The default value of
maxitis50.
- tolf = DOUBLE
Convergence criterion for termination based on the function value. Iteration will cease when it proves impossible to improve the function value by more than
tolf. Default:1e-5
- tolx = DOUBLE
Convergence criterion for termination based on the change in the function argument. Iteration will cease when the solver attempts to take a step that is smaller than
tolx. Default:1e-5
- noprint
Don’t print anything. Useful for loops.
- print¶
Print results (opposite of
noprint).
- stack_solve_algo = INTEGER¶
Algorithm used for computing the solution. Possible values are:
0Use a Newton algorithm with a direct sparse LU solver at each iteration, applied to the stacked system of all equations in all periods (Default).
1Use the Laffargue-Boucekkine-Juillard (LBJ) algorithm proposed in Juillard (1996) on top of a LU solver. It is slower than
stack_solve_algo=0, but may be less memory consuming on big models. Note that if theblockoption is used (see Model declaration), a simple Newton algorithm with sparse matrices, applied to the stacked system of all block equations in all periods, is used for blocks which are purely backward or forward (of typeSOLVE BACKWARDorSOLVE FORWARD, seemodel_info), since LBJ only makes sense on blocks with both leads and lags (of typeSOLVE TWO BOUNDARIES).2Use a Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration, applied on the stacked system of all equations in all periods. The following options can be used to control the behaviour of the algorithm:
preconditioner,iter_tol,iter_maxit,gmres_restart.3Use a Newton algorithm with a Stabilized Bi-Conjugate Gradient (BiCGStab) solver at each iteration, applied on the stacked system of all equations in all periods. The following options can be used to control the behaviour of the algorithm:
preconditioner,iter_tol,iter_maxit.4Use a Newton algorithm with a direct sparse LU solver and an optimal path length at each iteration, applied on the stacked system of all equations in all periods (requires
bytecodeand/orblockoption, see Model declaration).5Use the Laffargue-Boucekkine-Juillard (LBJ) algorithm proposed in Juillard (1996) on top of a sparse Gaussian elimination (SPE) solver. The latter takes advantage of the similarity of the Jacobian across periods when searching for the pivots. This algorithm requires the
bytecodeoption. The following options can be used to control the behaviour of the algorithm: markowitz,minimal_solving_periods.6Synonymous for
stack_solve_algo=1. Kept for backward compatibility.7Allows the user to solve the perfect foresight model with the solvers available through option
solve_algo, applied on the stacked system of all equations in all periods (See solve_algo for a list of possible values, note that values5,6,7and8, which requirebytecodeand/orblockoptions, are not allowed). For instance, the following commands:perfect_foresight_setup(periods=400); perfect_foresight_solver(stack_solve_algo=7, solve_algo=9)
trigger the computation of the solution with a trust region algorithm.
- preconditioner = OPTION¶
When
stack_solve_algois equal to2or3, this option specifies which preconditioner will be used in combination with the iterative sparse linear solver (either GMRES or BiCGStab). Possible values for OPTION are:umfiterAt the first iteration of the nonlinear Newton solver, compute the full LU decomposition with complete pivoting of the linear system (and use it to solve that first iteration rather than using the iterative solver). This LU decomposition is then used as the preconditioner in further Newton iterations. Inspired from TROLL’s option with the same name.
iterstackCompute the LU decomposition with complete pivoting for only a few simulation periods within the stacked Jacobian (which is a block tridiagonal matrix). Then repeat that LU decomposition over the block diagonal to construct a preconditioner for the linear system with all simulation periods. If the total number of simulations periods is not a multiple of the number of periods used for the small LU, then an additional LU is computed for the remainder. The following options can be used to control the construction of this preconditioner:
iterstack_maxlu,iterstack_nperiods,iterstack_nlu,iterstack_relu. Inspired from TROLL’s solver with the same name.iluUse an incomple LU decomposition as the preconditioner, recomputed at every iteration of the nonlinear Newton solver.
Default value isumfiter.
- iter_tol = DOUBLE¶
When
stack_solve_algois equal to2or3, this option controls the relative tolerance of the iterative linear solver (either GMRES or BiCGStab). It corresponds to thetoloption of thegmresandbicgstabMATLAB/Octave functions. Note that the perfect foresight solver uses an absolute tolerance for determining convergence, so this option should be used with care, and the default is meant to suit most situations. Default: the value of the :opt:tolf <tolf = DOUBLE>option, divided by 10 times the infinite norm of the right-hand side of the linear system.
- iter_maxit = INTEGER¶
When
stack_solve_algois equal to2or3, this option controls the maximum number of iterations of the iterative linear solver (either GMRES or BiCGStab). It corresponds to themaxitoption of thegmresandbicgstabMATLAB/Octave functions. It should not be confused with themaxitoption, which controls the (outer) nonlinear Newton loop, whileiter_maxitcontrols the (inner) linear loop. Default:500.
- gmres_restart = INTEGER¶
When
stack_solve_algois equal to2, this option controls the number of iterations before restart of the GMRES algorithm. It corresponds to therestartoption of thegmresMATLAB/Octave function. Default:100.
- iterstack_maxlu = INTEGER¶
When
preconditioneris equal toiterstack, controls the maximum size of the matrix for which the small LU will computed. The actual size of the matrix will be determined by the largest number of periods that, multiplied by the number of equations, is less than the value of the option. Note that when combined with block decomposition (seeblock), blocks for which the whole stacked system is less than this option will be solved using a regular LU decomposition instead of the iterative linear solver. Default:20000.
- iterstack_nperiods = INTEGER¶
When
preconditioneris equal toiterstack, controls the number of periods used for the small LU. If nonzero, this option overrides theiterstack_maxluanditerstack_nluoptions. Default:0.
- iterstack_nlu = INTEGER¶
When
preconditioneris equal toiterstack, specifies the number of times that the small LU should be repeated in the large preconditioner. If nonzero, this option overrides theiterstack_maxluoption. Default:0.
- iterstack_relu = DOUBLE¶
When
preconditioneris equal toiterstack, controls the relative position of the small LU within the whole stacked system. Must be a number between0and1. Default:0.5.
- robust_lin_solve¶
Triggers the use of a robust linear solver for the default
stack_solve_algo=0.
- solve_algo¶
See solve_algo. Allows selecting the solver used with
stack_solve_algo=7. Also used for purely backward, forward and static models (when neither theblocknor thebytecodeoption of themodelblock ormodel_optionscommand is specified); for those models, the values12and14are especially relevant.
- no_homotopy¶
This option tells Dynare to not try a homotopy technique (as described above) if the problem cannot be solved directly.
- homotopy_initial_step_size = DOUBLE¶
Specifies which share of the shock should be applied in the first iteration of the homotopy procedure. This option is useful when it is known that immediately trying 100% of the shock will fail, so as to save computing time. Must be between
0and1. Default:1.
- homotopy_min_step_size = DOUBLE¶
The homotopy procedure halves the size of the step whenever there is a failure. This option specifies the minimum step size under which the homotopy procedure is considered to have failed. Default:
0.001.
- homotopy_step_size_increase_success_count = INTEGER¶
Specifies after how many consecutive successful iterations the homotopy procedure should double the size of the step. A zero value means that the step size should never be increased. Default:
3.
- homotopy_linearization_fallback¶
Whenever the homotopy procedure is not able to find a solution for 100% of the shock, but is able to find one for a smaller share, instructs Dynare to compute an approximate solution by rescaling the solution obtained for a fraction of the shock, as if the reaction of the model to the shock was a linear function of the size of that shock. More formally, if \(s\) is the share of the shock applied (between \(0\) and \(1\)), \(y(s)\) is the value of a given endogenous variable at a given period as a function of \(s\) (in particular, \(y(1)\) corresponds to the exact solution of the problem), and \(s^*\) is the greatest share of the shock for which the homotopy procedure has been able to find a solution, then the approximate solution returned is \(\frac{y(s^*)-y(0)}{s^*}\).
If linearization is triggered, the variable
oo_.deterministic_simulation.homotopy_linearizationis set, and the simulation corresponding to share \(s^*\) is stored inoo_.deterministic_simulation.sim1.
- homotopy_marginal_linearization_fallback [= DOUBLE]¶
Whenever the homotopy procedure is not able to find a solution for 100% of the shock, but is able to find one for a smaller share, instructs Dynare to compute an approximate solution obtained by rescaling the solution obtained for a fraction of the shock, obtained as if the reaction of the model to the shock was, at the margin, a linear function of the size of that shock. More formally, if \(s\) is the share of the shock applied (between \(0\) and \(1\)), \(y(s)\) is the value of a given endogenous variable at a given period as a function of \(s\) (in particular, \(y(1)\) corresponds to the exact solution of the problem), \(s^*\) is the greatest share of the shock for which the homotopy procedure has been able to find a solution, and \(\epsilon\) is a small step size, then the approximate solution returned is \(y(s^*)+(1-s^*)\frac{y(s^*)-y(s^*-\epsilon)}{\epsilon}\). The value of \(\epsilon\) is
0.01by default, but can be modified by passing some other value to the option.If marginal linearization is triggered, the variable
oo_.deterministic_simulation.homotopy_marginal_linearizationis set. Moreover, the simulation corresponding to share \(s^*\) is stored inoo_.deterministic_simulation.sim1, and the one corresponding to share \(s^*-\epsilon\) is stored inoo_.deterministic_simulation.sim2.
Instructs Dynare, within the homotopy procedure, to not try to compute the solution for a greater share than the one given as the option value. This option only makes sense when used in conjunction with either the
homotopy_linearization_fallbackor thehomotopy_marginal_linearization_fallbackoption. It is typically used in situations where it is known that homotopy will fail to go beyond a certain point, so as to save computing time, while at the same time getting an approximate solution. Default:1.
- homotopy_exclude_varexo = (VARIABLE_NAME...)¶
A list of exogenous variables which are to be excluded from the homotopy procedure, i.e. which must be kept at their value corresponding to 100% of the shock during all homotopy iterations.
- markowitz = DOUBLE
Value of the Markowitz criterion, used to select the pivot (see markowitz for more details). Only used when
stack_solve_algo = 5. Default:0.5.
- minimal_solving_periods = INTEGER¶
Specify the minimal number of periods where the model has to be solved, before using a constant set of operations for the remaining periods. Only used when
stack_solve_algo = 5. Default:1.
- lmmcp¶
Solves the perfect foresight model with a Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra, 2004), which allows to consider inequality constraints on the endogenous variables (such as a zero lower bound, henceforth ZLB, on the nominal interest rate or a model with irreversible investment). This option is equivalent to
stack_solve_algo=7andsolve_algo=10. Using the LMMCP solver avoids the need for min/max operators and explicit complementary slackness conditions in the model as they will typically introduce a singularity into the Jacobian. This is done by setting the problem up as a mixed complementarity problem (MCP) of the form:\[ \begin{align}\begin{aligned}\begin{split}LB = X &\Rightarrow F(X)>0\\\end{split}\\\begin{split}LB < X < UB &\Rightarrow F(X)=0\\\end{split}\\X =UB &\Rightarrow F(X)<0.\end{aligned}\end{align} \]where \(X\) denotes the vector of endogenous variables, \(F(X)\) the equations of the model, \(LB\) denotes a lower bound, and \(UB\) an upper bound. Such a setup is implemented by specifying the complementarity condition after the equation to which it is attached, the two being separated by the perpendicular symbol (the latter can be input either in UTF-8, as ⟂, corresponding to Unicode codepoint U+27C2; or alternatively as pure ASCII, as _|_, i.e. a vertical bar enclosed within two underscores).
For instance, a ZLB on the nominal interest rate would be specified as follows in the model block:
model; ... r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e ⟂ r > -1.94478; ... end;
where
1.94478is the steady state level of the nominal interest rate andris the nominal interest rate in deviation from the steady state. This construct implies that the Taylor rule is operative, unless the implied interest rater<=-1.94478, in which case theris fixed at-1.94478(thereby being equivalent to a complementary slackness condition). By restricting the value ofrcoming out of this equation, the complementarity condition also avoids usingmax(r,-1.94478)for other occurrences ofrin the rest of the model. Two things are important to keep in mind. First, the complementary slackness condition cannot be simply attached to any equation; it must be attached to the correct affected equation as otherwise the solver will solve a different problem than originally intended. Second, the sign of the residual of the dynamic equation must conform to the MCP setup outlined above. In case of the ZLB, we are dealing with a lower bound. Consequently, the dynamic equation needs to return a positive residual. Dynare by default computes the residual of an equationLHS=RHSasresidual=LHS-RHS, while an implicit equationLHSis interpreted asLHS=0. For the above equation this impliesresidual= r - (rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e);which is correct, since it will be positive if the implied interest rate
rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + eis belowr=-1.94478. In contrast, specifying the equation asrho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e = r;would be wrong.
Note that both the lower and the upper bounds can be specified at the same time in a given complementarity condition. Moreover, arbitrary functions of parameters can appear in the bounds. As an example, the following complementarity condition is syntactically correct (assuming that
alphais a parameter):… ⟂ -1.94478 < r < 1+2*alpha;
- endogenous_terminal_period¶
The number of periods is not constant across Newton iterations when solving the perfect foresight model. The size of the nonlinear system of equations is reduced by removing the portion of the paths (and associated equations) for which the solution has already been identified (up to the tolerance parameter). This strategy can be interpreted as a mix of the shooting and relaxation approaches. Note that round off errors are more important with this mixed strategy (user should check the reported value of the maximum absolute error). Only available with
stack_solve_algooption equal to0.
- linear_approximation¶
Solves the linearized version of the perfect foresight model. The model must be stationary and a steady state needs to be provided. Linearization is conducted about the last defined steady state, which can derive from
initval,endvalor a subsequentsteady. Only available withstack_solve_algooption equal to0or7.
- steady_solve_algo = INTEGER¶
See solve_algo. Used when computing the terminal steady state when option
endval_steadyhas been specified to theperfect_foresight_setupcommand.
- steady_tolf = DOUBLE¶
See tolf. Used when computing the terminal steady state when option
endval_steadyhas been specified to theperfect_foresight_setupcommand.
- steady_tolx = DOUBLE¶
See tolx. Used when computing the terminal steady state when option
endval_steadyhas been specified to theperfect_foresight_setupcommand.
- steady_maxit = INTEGER¶
See maxit. Used when computing the terminal steady state when option
endval_steadyhas been specified to theperfect_foresight_setupcommand.
- steady_markowitz = DOUBLE¶
See markowitz. Used when computing the terminal steady state when option
endval_steadyhas been specified to theperfect_foresight_setupcommand.
Output
The simulated endogenous variables are available in global matrix
oo_.endo_simul.If any of the
first_simulation_periodorlast_simulation_periodoption was passed to the precedingperfect_foresight_setupcommand, a time series object containing both endogenous and exogenous variables is stored in the workspace variableSimulated_time_series.The variable
oo_.deterministic_simulation.statusindicates whether the simulation was successful or not.
- Command: simul ;¶
- Command: simul(OPTIONS...);
This command is deprecated. It is strictly equivalent to a call toperfect_foresight_setupfollowed by a call toperfect_foresight_solver.Options
Accepts all the options of
perfect_foresight_setupandperfect_foresight_solver.
- MATLAB/Octave variable: oo_.endo_simul¶
This variable stores the result of a deterministic simulation (computed byperfect_foresight_solverorperfect_foresight_with_expectation_errors_solver) or of a stochastic simulation (computed bystoch_simulwith theperiodsoption or byextended_path). The variables are arranged row by row, in order of declaration (as inM_.endo_names). Note that this variable also contains initial and terminal conditions, so it has more columns than the value of theperiodsoption: the first simulation period is in column1+M_.maximum_lag, and the total number of columns isM_.maximum_lag+periods+M_.maximum_lead.
- MATLAB/Octave variable: oo_.exo_simul¶
This variable stores the path of exogenous variables during a simulation (computed byperfect_foresight_solver,perfect_foresight_with_expectation_errors_solver,stoch_simulorextended_path). The variables are arranged in columns, in order of declaration (as inM_.exo_names). Periods are in rows. Note that this convention regarding columns and rows is the opposite of the convention foroo_.endo_simul! Also note that this variable also contains initial and terminal conditions, so it has more rows than the value of theperiodsoption: the first simulation period is in row1+M_.maximum_lag, and the total number of rows isM_.maximum_lag+periods+M_.maximum_lead.
- MATLAB/Octave variable: Simulated_time_series¶
This variable stores, as adseriesobject, the path of both endogenous and exogenous variables after a deterministic simulation (computed byperfect_foresight_solverorperfect_foresight_with_expectation_errors_solverusing thefirst_simulation_periodand/orlast_simulation_periodoption).
- MATLAB/Octave variable: oo_.initial_steady_state¶
If a permanent shock is simulated through the use of both
initvalandendvalblocks, this variable contains the initial steady state, as determined by theinitvalblock (when followed by asteadycommand). This variable has the same structure asoo_.steady_state(and this latter variable contains the terminal steady state, if theendvalblock is followed by asteadycommand).
- MATLAB/Octave variable: oo_.initial_exo_steady_state¶
If a permanent shock is simulated through the use of both
initvalandendvalblocks, this variable contains the initial steady state of the exogenous variables, as specified in theinitvalblock. This variable has the same structure asoo_.exo_steady_state(and this latter variable contains the terminal steady state of the exogenous variables).
- MATLAB/Octave variable: M_.maximum_lag¶
The maximum number of lags in the model. Note that this value is computed on the model after the transformations related to auxiliary variables, so in practice it is either 1 or 0 (the latter value corresponds to a purely forward or static model).
- MATLAB/Octave variable: M_.maximum_lead¶
The maximum number of leads in the model. Note that this value is computed on the model after the transformations related to auxiliary variables, so in practice it is either 1 or 0 (the latter value corresponds to a purely backward or static model).
- MATLAB/Octave variable: oo_.deterministic_simulation.status¶
Set totrueby theperfect_foresight_solvercommand if the simulation succeeded, otherwise set tofalse.
- MATLAB/Octave variable: oo_.deterministic_simulation.homotopy_linearization¶
Set totrueby theperfect_foresight_solvercommand if linearization has been used to compute an approximate solution.
- MATLAB/Octave variable: oo_.deterministic_simulation.homotopy_marginal_linearization¶
Set totrueby theperfect_foresight_solvercommand if marginal linearization has been used to compute an approximate solution.
- MATLAB/Octave variable: oo_.deterministic_simulation.sim1¶
Set by theperfect_foresight_solvercommand if either linearization or marginal linearization has been used to compute an approximate solution. This structure contains the simulation corresponding to the greatest share of the shocks for which an exact solution could be computed. The subfieldhomotopy_completion_sharecontains that share. The subfieldsendo_simul,exo_simul,steady_stateandexo_steady_staterespectively contain the path of endogenous, the path of exogenous, the steady state of endogenous and the steady state of exogenous for that simulation (with the same conventions as the fields of of the same name inoo_).
- MATLAB/Octave variable: oo_.deterministic_simulation.sim2¶
Set by theperfect_foresight_solvercommand if marginal linearization has been used to compute an approximate solution. This structure contains the simulation corresponding to a share marginally smaller that the one inoo_.deterministic_simulation.sim1. The subfields are the same as inoo_.deterministic_simulation.sim1.
4.12.2. Perfect foresight with expectation errors¶
The solution under perfect foresight that was presented in the previous section makes the assumption that agents learn the complete path of future shocks in period 1, without making any expectation errors.
One may however want to study a scenario where it turns out that agents make expectation errors, in the sense that the path they had anticipated in period 1 does not realize exactly. More precisely, in some simulation periods, they may receive new information that makes them revise their anticipation for the path of future shocks. Also, under this scenario, it is assumed that agents behave as under perfect foresight, i.e. they take their decisions as if there was no uncertainty and they knew exactly the path of future shocks; the new information that they may receive comes as a total surprise to them.
Such a scenario can be solved by Dynare using the
perfect_foresight_with_expectation_errors_setup and
perfect_foresight_with_expectation_errors_solver commands, alongside shocks
and endval blocks which are given a special learnt_in option.
- Block: shocks(learnt_in=INTEGER|DATE) ;
- Block: shocks(learnt_in=INTEGER|DATE, overwrite) ;
Theshocks(learnt_in=INTEGER|DATE)syntax can be used to specify temporary shocks that are learnt in a specific period. It should contain one or more occurences of the following group of three lines, with the same semantics as a regularshocksblock:var VARIABLE_NAME; periods INTEGER[:INTEGER] | DATE[:DATE] [[,] INTEGER[:INTEGER] | DATE[:DATE]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
If the period in which information is learnt is greater or equal than 2, then it is possible to specify the shock values in deviation with respect to the values that were expected from the perspective of the previous period. If the new information consists of an addition to the previously-anticipated value, the
valueskeyword can be replaced by theaddkeyword; similarly, if the new information consists of a multiplication of the previously-anticipated value, thevalueskeyword can be replaced by themultiplykeyword.The
overwriteoption says that this block cancels and replaces previousshocksandmshocksblocks that have the samelearnt_inoption. Note that a block with an integer-valuedlearnt_inoption never overwrites a block with a date-valuedlearnt_inoption, even if they correspond to the same period.Also note that a
shocks(learnt_in=1)block is equivalent to a regularshocksblock.Example
shocks(learnt_in=1); var x; periods 1:2 3:4 5; values 1 1.2 1.4; end; shocks(learnt_in=2); var x; periods 3:4; add 0.1; end; shocks(learnt_in=4); var x; periods 5; multiply 2; end;
This syntax means that:
from the perspective of period 1,
xis expected to be equal to 1 in periods 1 and 2, to 1.2 in periods 3 and 4, and to 1.4 in period 5;from the perspective of periods 2 (and 3),
xis expected to be equal to 1 in period 2, to 1.3 in periods 3 and 4, and to 1.4 in period 5;from the perspective of periods 4 (and following),
xis expected to be equal to 1.3 in period 4, and to 2.8 in period 5.
Example (with dates)
shocks(learnt_in=2023Q1); var x; periods 2023Q1:2023Q2 2023Q3:2023Q4 2024Q1; values 1 1.2 1.4; end;
- Block: endval(learnt_in=INTEGER|DATE) ;
Theendval(learnt_in=INTEGER|DATE)can be used to specify terminal conditions that are learnt in a specific period.Note that an
endval(learnt_in=1)block is equivalent to a regularendvalblock.Also note that, similarly to the regular
endvalblock, any variable specified in this block will jump to its new value in the same period as the one in which the information is learnt; and, from the perspective of that period, the variable is expected by agents to remain to that value until the end of the simulation. In particular, this means that any temporary shock that may have been anticipated on that variable (as specified through ashocks(learnt_in=...)block for a previous informational period) will be overridden; if this is not the desired behaviour, then the temporary shock will have to be reinstated through anothershocks(learnt_in=...)block.It is possible to express the terminal condition by specifying the level of the exogenous variable (using an equal symbol, as in a regular
endvalblocks without thelearnt_inoption). But it is also possible to express the terminal condition as an addition to the value expected from the perspective of the previous previous period (using the+=operator), or as a multiplicative factor over that previously expected value (using the*=operator).Example
endval(learnt_in = 3); x = 1.1; y += 0.1; z *= 2; end;
This syntax means that, in period 3, the agents learn that:
the terminal condition for
xwill be 1.1;the terminal condition for
ywill be 0.1 above the terminal condition forythat was expected from the perspective of period 2;the terminal condition for
zwill be 2 times the terminal condition forzthat was expected from the perspective of period 2.
Those values will be the realized ones, unless there is another
endval(learnt_in=p)block withp>3.The three variables will jump to their new value in period 3 and, from the perspective of period 3, they are expected by agents to remain there until the end of the simulation. In particular, any temporary shock on either
x,yorzspecified through a regularshocksblock or through ashocks(learnt_in=2)block will be overridden. If this is not the desired behaviour, ashocks(learnt_in=3)block will have to be added to reinstate the temporary shock.Example (with a date)
endval(learnt_in = 2024Q1); x = 1.1; end;
- Block: mshocks(learnt_in=INTEGER|DATE) ;
- Block: mshocks(learnt_in=INTEGER|DATE,OPTIONS...) ;
Themshocks(learnt_in=INTEGER|DATE)syntax can be used to specify temporary shocks that are learnt in a specific period, specified in a multiplicative way. It should contain one or more occurences of the following group of three lines, with the same semantics as a regularmshocksblock:var VARIABLE_NAME; periods INTEGER[:INTEGER] | DATE[:DATE] [[,] INTEGER[:INTEGER] | DATE[:DATE]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
As in the regular
mshocksblock (without thelearnt_inoption), the values are interpreted as a multiplicative factor over the steady state value of the exogenous variable (the latter being taken either from theinitvalorendval, seemshocksfor the details).If the terminal steady state as specified in the
endvalblock is used as a basis for the multiplication, its value as anticipated from the period given in thelearnt_inoption will be used.Note that a
mshocks(learnt_in=1)block is equivalent to a regularmshocksblock.Options
- overwrite
This block cancels and replaces previous
shocksandmshocksblocks that have the samelearnt_inoption. Note that a block with an integer-valuedlearnt_inoption never overwrites a block with a date-valuedlearnt_inoption, even if they correspond to the same period.
- relative_to_initval
Same meaning as in the regular
mshocksblock.
Example
mshocks(learnt_in=2); var x; periods 3:4; values 1.1; end;
This syntax means that from the perspective of period 2,
xin periods 3 and 4 is expected to be equal to 1.1 times its steady state. If there is noendvalblock, the initial steady state as given byinitvalis used; if there is anendvalblock, the terminal steady state as anticipated from the perspective of period 2 is used (as specified in the relevantendval(learnt_in=…block)).Example (with dates)
mshocks(learnt_in=2024Q2); var x; periods 2024Q3:2024Q4; values 1.1; end;
- Command: perfect_foresight_with_expectation_errors_setup ;¶
- Command: perfect_foresight_with_expectation_errors_setup(OPTIONS...);
Prepares a perfect foresight simulation with expectation errors, by extracting the contents of theinitval,endvalandshocksblocks (the latter two types of blocks typically used with thelearnt_inoption); alternatively, the information about future shocks can be given in a CSV file using thedatafileoption.This command must always be called before running the simulation with
perfect_foresight_with_expectation_errors_solver.Note that this command makes the assumption that the terminal condition is always a steady state. Hence, it will recompute the terminal steady state as many times as the anticipation about the terminal condition changes. In particular, the information about endogenous variables that may be given in the
endvalblock is ignored. Said otherwise, the equivalent of optionendval_steadyof theperfect_foresight_setupcommand is always implicitly enabled.Options
- periods = INTEGER
Number of periods of the simulation.
- first_simulation_period = DATE
Same meaning as the eponymous option of
perfect_foresight_setup.
- last_simulation_period = DATE
Same meaning as the eponymous option of
perfect_foresight_setup.
- datafile = FILENAME
Used to specify the information about future shocks and their anticipation through a CSV file, as an alternative to
shocksandendvalblocks.The file has the following format:
the first column is ignored (can be used to add descriptive labels)
the first line contains names of exogenous variables
the second line contains, in columns, indices of periods at which expectations are formed; the information set used in a given period is described by all the columns for which that line is equal to the period index
the subsequent lines correspond to the periods for which expectations are formed, one period per line; each line gives the values of present and future exogenous variables, as seen from the period given in the second line
the last line corresponds to the terminal condition for exogenous variables, as anticipated in the various informational periods
If
pis the value of theperiodsoption andkis the number of exogenous variables, then the CSV file hasp+3lines andk×p+1columns.Concretely, the value of a given exogenous in period
t, as anticipated from periods, is given in linet+2, and in the column which has the name of the variable on the first line andson the second line. Of course, values in cells corresponding tot<sare ignored.
Output
oo_.exo_simulandoo_.endo_simulare initialized before the simulation. Temporary shocks are stored inoo_.pfwee.shocks_info, terminal conditions for exogenous variables are stored inoo_.pfwee.terminal_info.Example
Here is a CSV file example that could be given to the
datafileoption (adding some extra padding space for clarity):Exogenous , x, x, x, x, x, x, x Period (info), 1, 2, 3, 4, 5, 6, 7 Period 1 (real), 1.2, , , , , , Period 2 (real), 1, 1.3, , , , , Period 3 (real), 1, 1, 1.4, , , , Period 4 (real), 1, 1, 1, 1, , , Period 5 (real), 1, 1, 1, 1, 1, , Period 6 (real), 1, 1, 1, 1, 1, 1.1, Period 7 (real), 1, 1, 1, 1, 1, 1.1, 1.1 Terminal (real), 1, 1.1, 1.2, 1.2, 1.2, 1.1, 1.1
In this example, there is only one exogenous variable (
x), and 7 simulation periods. In the first period, agents learn a contemporary shock (1.2), but anticipate no further shock. In period 2, they learn an unexpected contemporary shock (1.3), and also a change in the terminal condition (1.1). In period 3 again there is an unexpected contemporary shock and a change in the terminal condition. No new information comes in period 4 and 5. In period 6, an unexpected permanent shock is learnt. No new information comes in period 7.Alternatively, instead of using a CSV file, the same sequence of information sets could be described using the following blocks:
initval; x = 1; end; steady; shocks(learnt_in = 1); var x; periods 1; values 1.2; end; shocks(learnt_in = 2); var x; periods 2; values 1.3; end; endval(learnt_in = 2); x = 1.1; end; shocks(learnt_in = 3); var x; periods 3; values 1.4; end; endval(learnt_in = 3); x = 1.2; end; shocks(learnt_in = 6); var x; periods 6:7; values 1.1; end; endval(learnt_in = 6); x = 1.1; end;
- Command: perfect_foresight_with_expectation_errors_solver ;¶
- Command: perfect_foresight_with_expectation_errors_solver(OPTIONS...);
Computes the perfect foresight simulation with expectation errors of the model.Note that
perfect_foresight_with_expectation_errors_setupmust be called before this command, in order to setup the environment for the simulation.Options
This command accepts all the options of
perfect_foresight_solver, with the same semantics, plus the following one:- constant_simulation_length¶
By default, every time the information set changes, the simulation with the new information set is shorter than the previous one (because the terminal date is getting closer). When this option is set, every new simulation has the same length (as specified by the
periodsoption ofperfect_foresight_with_expectation_errors_setup); as a consequence, the simulated paths as stored inoo_.endo_simulwill be longer when this option is set (if s is the last period in which the information set is modified, then they will contain s+periods-1 periods, excluding initial and terminal conditions). Note that this option is not available iflast_simulation_periodoption has been passed toperfect_foresight_with_expectation_errors_setup.
Output
The simulated paths of endogenous variables are available in
oo_.endo_simul. The terminal steady state values corresponding to the last period of the information set are available inoo_.steady_stateandoo_.exo_steady_state.If any of the
first_simulation_periodorlast_simulation_periodoption was passed to the precedingperfect_foresight_with_expectation_errors_setupcommand, a time series object containing both endogenous and exogenous variables is stored in the workspace variableSimulated_time_series.
- MATLAB/Octave variable: oo_.pfwee.shocks_info¶
This variable stores the temporary shocks used during perfect foresight simulations with expectation errors, afterperfect_foresight_with_expectation_errors_setuphas been run. It is a three-dimensional matrix: first dimension correspond to exogenous variables (in declaration order); second dimension corresponds to real time; third dimension corresponds to informational time. In other words, the value of exogenous indexedkin periodt, as anticipated from periods, is stored inoo_.pfwee.shocks_info(k,t,s).
- MATLAB/Octave variable: oo_.pfwee.terminal_info¶
This variable stores the terminal conditions for exogenous variables used during perfect foresight simulations with expectation errors, afterperfect_foresight_with_expectation_errors_setuphas been run. It is a matrix, whose lines correspond to exogenous variables (in declaration order), and whose columns correspond to informational time. In other words, the terminal condition for exogenous indexedk, as anticipated from periods, is stored inoo_.pfwee.terminal_info(k,s).
4.12.3. Controlling the path of endogenous variables¶
In the usual perfect foresight problem, the user controls the path of exogenous variables for the simulation periods and the initial and terminal conditions for endogenous variables, while Dynare solves for the path of endogenous variables for the simulation periods.
However, Dynare offers the possibility of controlling the value of some
endogenous variables for some simulation periods (in which case some exogenous
variables must be left free and are thus solved for by Dynare, to avoid
over-determination of the problem). This exercise is called “conditional
forecasting” in some contexts (even though one may argue that this is not
really forecasting, since perfect foresight by the agents is assumed; for the
stochastic case, see the conditional_forecast command).
The description of controlled endogenous variables is done using the
perfect_foresight_controlled_paths block. The information given therein is
then processed by the perfect_foresight_setup (or
perfect_foresight_with_expectation_errors_setup) command, so
that the next perfect_foresight_solver (or
perfect_foresight_with_expectation_errors_solver) command
computes the simulation with controlled paths. In particular,
oo_.exo_simul will contain the computed value of exogenous variables
that have been left free.
- Block: perfect_foresight_controlled_paths ;¶
- Block: perfect_foresight_controlled_paths(OPTIONS...);
This block is used to tell the perfect foresight solver that the value of some endogenous variables will be controlled (in other words, they will be exogenized). It also gives the period(s) for which this control applies, the value(s) imposed to the endogenous variable(s), and the exogenous variable(s) that are left free at the same period(s) (in other words, those exogenous are endogenized).The block should contain one or more occurrences of the following group of four lines:
exogenize ENDOGENOUS_NAME; periods INTEGER[:INTEGER] | DATE[:DATE] [[,] INTEGER[:INTEGER] | DATE[:DATE]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...; endogenize EXOGENOUS_NAME;
Note that it is possible to have both
perfect_foresight_controlled_pathsand regularshocksblocks in the same.modfile (assuming of course that taken together they do not impose inconsistent constraints).The
perfect_foresight_controlled_pathsblock requires that thestack_solve_algooption be equal to either0,1,2,3,6or7, and is incompatible with theblockandbytecodeoptions of themodelblock andmodel_optionscommand.Options
- learnt_in = INTEGER | DATE¶
Used in conjunction with
perfect_foresight_with_expectation_errors_setupandperfect_foresight_with_expectation_errors_solvercommands, specifies the period or date at which this controlled paths block is learnt by agents.
Example (perfect foresight)
var c k; varexo x z; ... shocks; var x; periods 1; values 1.2; end; perfect_foresight_controlled_paths; exogenize c; periods 2 4:5; values 1.6 1.7; endogenize x; exogenize k; periods 7:9; values 13; endogenize z; end; perfect_foresight_setup(periods = 100); perfect_foresight_solver;
In this example, the exogenous variable
xis equal to 1.2 in period 1, but in periods 2, 4 and 5 it will be endogenized so that endogenous variable c is equal to 1.6 in period 2 and then 1.7 in periods 4 and 5. Similarly, the exogenous variablezwill be endogenized in periods 7 to 9 so that the endogenous variablekis equal to 13 over the same periods.Example (perfect foresight with expectation errors)
var c; varexo x; ... perfect_foresight_controlled_paths; exogenize c; periods 2002Y 2003Y:2005Y; values 1.6 1.7; endogenize x; end; perfect_foresight_controlled_paths(learnt_in=2004Y); exogenize c; periods 2004Y:2005Y; values 1.8; endogenize x; end; perfect_foresight_with_expectation_errors_setup(periods = 30, first_simulation_period = 2001Y); perfect_foresight_with_expectation_errors_solver;
In this example, agents in year 2001 (at beginning of the simulation) compute their plan under the assumption that the endogenous variable
cwill be equal to 1.6 in year 2002 and 1.7 from years 2003 to 2005, and that exogenous variablexwill behave so as to fulfill that constraint. Then, when 2004 arrives, they recompute their plan under the assumption thatcwill be equal to 1.8 in years 2004 and 2005 (and again thatxwill be endogenized accordingly).
4.13. Stochastic solution and simulation¶
In a stochastic context, Dynare computes one or several simulations corresponding to a random draw of the shocks.
The main algorithm for solving stochastic models relies on a Taylor
approximation, up to third order, of the expectation functions (see
Judd (1996), Collard and Juillard (2001a, 2001b), and
Schmitt-Grohé and Uríbe (2004)). The details of the
Dynare implementation of the first order solution are given in
Villemot (2011). Such a solution is computed using the
stoch_simul command.
As an alternative, it is possible to compute a simulation to a
stochastic model using the extended path method presented by Fair
and Taylor (1983). This method is especially useful when there are
strong nonlinearities or binding constraints. Such a solution is
computed using the extended_path command.
4.13.1. Computing the stochastic solution¶
- Command: stoch_simul [VARIABLE_NAME...];¶
- Command: stoch_simul(OPTIONS...) [VARIABLE_NAME...];
Solves a stochastic (i.e. rational expectations) model, using perturbation techniques.More precisely,
stoch_simulcomputes a Taylor approximation of the model around the deterministic steady state and solves of the the decision and transition functions for the approximated model. Using this, it computes impulse response functions and various descriptive statistics (moments, variance decomposition, correlation and autocorrelation coefficients). For correlated shocks, the variance decomposition is computed as in the VAR literature through a Cholesky decomposition of the covariance matrix of the exogenous variables. When the shocks are correlated, the variance decomposition depends upon the order of the variables in thevarexocommand.The Taylor approximation is computed around the steady state (see Steady state).
The IRFs are computed as the difference between the trajectory of a variable following a shock at the beginning of period
1and its steady state value. More details on the computation of IRFs can be found at https://archives.dynare.org/DynareWiki/IrFs.Variance decomposition, correlation, autocorrelation are only displayed for variables with strictly positive variance. Impulse response functions are only plotted for variables with response larger than \(10^{-10}\).
Variance decomposition is computed relative to the sum of the contribution of each shock. Normally, this is of course equal to aggregate variance, but if a model generates very large variances, it may happen that, due to numerical error, the two differ by a significant amount. Dynare issues a warning if the maximum relative difference between the sum of the contribution of each shock and aggregate variance is larger than
0.01%.The covariance matrix of the shocks is specified with the
shockscommand (see Shocks on exogenous variables).When a list of
VARIABLE_NAMEis specified, results are displayed only for these variables.Options
- ar = INTEGER¶
Order of autocorrelation coefficients to compute and to print. Default:
5.
- drop = INTEGER¶
Number of points (burnin) dropped at the beginning of simulation before computing the summary statistics. Note that this option does not affect the simulated series stored in
oo_.endo_simuland the workspace. Here, no periods are dropped. Default:100.
- hp_filter = DOUBLE¶
Uses HP filter with \(\lambda =\)
DOUBLEbefore computing moments. If theoretical moments are requested, the spectrum of the model solution is filtered following the approach outlined in Uhlig (2001). Default: no filter.
- one_sided_hp_filter = DOUBLE¶
Uses the one-sided HP filter with \(\lambda =\)
DOUBLEdescribed in Stock and Watson (1999) before computing moments. This option is only available with simulated moments. Default: no filter.
- bandpass_filter¶
Uses a bandpass filter with the default passband before computing moments. If theoretical moments are requested, the spectrum of the model solution is filtered using an ideal bandpass filter. If empirical moments are requested, the Baxter and King (1999) filter is used. Default: no filter.
- bandpass_filter = [HIGHEST_PERIODICITY LOWEST_PERIODICITY]¶
Uses a bandpass filter before computing moments. The passband is set to a periodicity of to LOWEST_PERIODICITY, e.g. \(6\) to \(32\) quarters if the model frequency is quarterly. Default:
[6,32].
- filtered_theoretical_moments_grid = INTEGER¶
When computing filtered theoretical moments (with either option
hp_filteror optionbandpass_filter), this option governs the number of points in the grid for the discrete Inverse Fast Fourier Transform. It may be necessary to increase it for highly autocorrelated processes. Default:512.
- irf = INTEGER¶
Number of periods on which to compute the IRFs. Setting
irf=0suppresses the plotting of IRFs. Default:40.
- irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )¶
The exogenous variables for which to compute IRFs. Default: all.
- relative_irf¶
Requests the computation of normalized IRFs. At first order, the normal shock vector of size one standard deviation is divided by the standard deviation of the current shock and multiplied by 100. The impulse responses are hence the responses to a unit shock of size 1 (as opposed to the regular shock size of one standard deviation), multiplied by 100. Thus, for a loglinearized model where the variables are measured in percent, the IRFs have the interpretation of the percent responses to a 100 percent shock. For example, a response of 400 of output to a TFP shock shows that output increases by 400 percent after a 100 percent TFP shock (you will see that TFP increases by 100 on impact). Given linearity at
order=1, it is straightforward to rescale the IRFs stored inoo_.irfsto any desired size. At higher order, the interpretation is different. Therelative_irfoption then triggers the generation of IRFs as the response to a 0.01 unit shock (corresponding to 1 percent for shocks measured in percent) and no multiplication with 100 is performed. That is, the normal shock vector of size one standard deviation is divided by the standard deviation of the current shock and divided by 100. For example, a response of 0.04 of log output (thus measured in percent of the steady state output level) to a TFP shock also measured in percent then shows that output increases by 4 percent after a 1 percent TFP shock (you will see that TFP increases by 0.01 on impact).
- irf_plot_threshold = DOUBLE¶
Threshold size for plotting IRFs. All IRFs for a particular variable with a maximum absolute deviation from the steady state smaller than this value are not displayed. Default:
1e-10.
- nocorr¶
Don’t print the correlation matrix (printing them is the default).
- nodecomposition¶
Don’t compute (and don’t print) unconditional variance decomposition.
- nofunctions¶
Don’t print the coefficients of the approximated solution (printing them is the default).
- nomoments¶
Don’t print moments of the endogenous variables (printing them is the default).
- nomodelsummary¶
Don’t print the model summary and the covariance of the exogenous shocks (printing them is the default).
- nograph¶
Do not create graphs (which implies that they are not saved to the disk nor displayed). If this option is not used, graphs will be saved to disk (to the format specified by
graph_formatoption, except ifgraph_format=none) and displayed to screen (unlessnodisplayoption is used).
- graph¶
Re-enables the generation of graphs previously shut off with
nograph.
- nodisplay¶
Do not display the graphs, but still save them to disk (unless
nographis used).
- graph_format = FORMAT¶
- graph_format = ( FORMAT, FORMAT... )¶
Specify the file format(s) for graphs saved to disk. Possible values are
eps(the default),pdf,figandnone. Under Octave,figwill use Octave’sofigformat. If the file format is set equal tonone, the graphs are displayed but not saved to the disk.
- noprint
See
noprint.
See
print.
- order = INTEGER¶
Order of Taylor approximation. Note that for third order and above, the
k_order_solveroption is implied and only empirical moments are available (you must provide a value forperiodsoption). Default:2(except after anestimationcommand, in which case the default is the value used for the estimation).
- k_order_solver¶
Use a k-order solver (implemented in C++) instead of the default Dynare solver. This option is not yet compatible with the
bytecodeoption (see Model declaration). Default: disabled for order 1 and 2, enabled for order 3 and above.
- periods = INTEGER
If different from zero, empirical moments will be computed instead of theoretical moments. The value of the option specifies the number of periods to use in the simulations. Values of the initval block, possibly recomputed by
steady, will be used as starting point for the simulation. The simulated endogenous variables are made available to the user in a vector for each variable and in the global matrixoo_.endo_simul(seeoo_.endo_simul). The simulated exogenous variables are made available inoo_.exo_simul(seeoo_.exo_simul). Default:0.
- qz_criterium = DOUBLE¶
Value used to split stable from unstable eigenvalues in reordering the Generalized Schur decomposition used for solving first order problems. Default:
1.000001(except when estimating withlik_initoption equal to1: the default is0.999999in that case; see Estimation based on likelihood).
- qz_zero_threshold = DOUBLE
See
qz_zero_threshold.
- replic = INTEGER¶
Number of simulated series used to compute the IRFs. Default:
1iforder=1, and50otherwise.
- simul_replic = INTEGER¶
Number of series to simulate when empirical moments are requested (i.e.
periods\(>\) 0). Note that if this option is greater than 1, the additional series will not be used for computing the empirical moments but will simply be saved in binary form to the fileFILENAME_simulin theFILENAME/Outputfolder. Default:1.
- solve_algo = INTEGER
See solve_algo, for the possible values and their meaning.
- aim_solver¶
Use the Anderson-Moore Algorithm (AIM) to compute the decision rules, instead of using Dynare’s default method based on a generalized Schur decomposition. This option is only valid for first order approximation. See AIM website for more details on the algorithm.
- conditional_variance_decomposition = INTEGER¶
- conditional_variance_decomposition = [INTEGER1:INTEGER2]¶
- conditional_variance_decomposition = [INTEGER1 INTEGER2 ...]¶
Computes a conditional variance decomposition for the specified period(s). The periods must be strictly positive. Conditional variances are given by \(var(y_{t+k}\vert t)\). For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact.
The results are stored in
oo_.conditional_variance_decomposition(seeoo_.conditional_variance_decomposition). In the presence of measurement error, theoo_.conditional_variance_decompositionfield will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables. The variance decomposition of the measured variables will be stored inoo_.conditional_variance_decomposition_ME(seeoo_.conditional_variance_decomposition_ME). The variance decomposition is only conducted, if theoretical moments are requested, i.e. using theperiods=0option. Only available atorder<3and withoutpruning. In case oforder=2, Dynare provides a second-order accurate approximation to the true second moments based on the linear terms of the second-order solution (see Kim, Kim, Schaumburg and Sims (2008)). Note that the unconditional variance decomposition i.e. at horizon infinity) is automatically conducted if theoretical moments are requested and ifnodecompositionis not set (seeoo_.variance_decomposition).
- pruning¶
Discard higher order terms when iteratively computing simulations of the solution. At second order, Dynare uses the algorithm of Kim, Kim, Schaumburg and Sims (2008), while at third order and higher its generalization by Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018) is used. When specified, theoretical moments are based on the pruned state space, i.e. the computation of second moments uses all terms as in Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018), page 10 as opposed to simply providing a second-order accurate result based on the linear solution as in Kim, Kim, Schaumburg and Sims (2008).
- partial_information¶
Computes the solution of the model under partial information, along the lines of Pearlman, Currie and Levine (1986). Agents are supposed to observe only some variables of the economy. The set of observed variables is declared using the
varobscommand. Note that ifvarobsis not present or contains all endogenous variables, then this is the full information case and this option has no effect. More references can be found here .
- dr = OPTION¶
Determines the method used to compute the decision rule. Possible values for OPTION are:
defaultUses the default method to compute the decision rule based on the generalized Schur decomposition (see Villemot (2011) for more information).
cycle_reductionUses the cycle reduction algorithm of
Bini et al. (2002)to solve the polynomial equation for retrieving the coefficients associated to the endogenous variables in the decision rule. This method is faster than the default one for large scale models.logarithmic_reductionUses the logarithmic reduction algorithm of
Bini et al. (2002)to solve the polynomial equation for retrieving the coefficients associated to the endogenous variables in the decision rule. This method is in general slower than thecycle_reduction.
Default value isdefault.
- dr_cycle_reduction_tol = DOUBLE¶
The convergence criterion used in the cycle reduction algorithm. Its default value is
1e-7.
- dr_cycle_reduction_maxiter = INTEGER¶
The maximum number of iterations used in the cycle reduction algorithm. Its default value is
100.
- dr_logarithmic_reduction_tol = DOUBLE¶
The convergence criterion used in the logarithmic reduction algorithm. Its default value is
1e-12.
- dr_logarithmic_reduction_maxiter = INTEGER¶
The maximum number of iterations used in the logarithmic reduction algorithm. Its default value is
100.
- loglinear¶
See loglinear. Note that ALL variables are log-transformed by using the Jacobian transformation, not only selected ones. Thus, you have to make sure that your variables have strictly positive steady states.
stoch_simulwill display the moments, decision rules, and impulse responses for the log-linearized variables. The decision rules saved inoo_.drand the simulated variables will also be the ones for the log-linear variables.
- tex¶
Requests the printing of results and graphs in TeX tables and graphics that can be later directly included in LaTeX files.
- dr_display_tol = DOUBLE¶
Tolerance for the suppression of small terms in the display of decision rules. Rows where all terms are smaller than
dr_display_tolare not displayed. Default value:1e-6.
- contemporaneous_correlation¶
Saves the contemporaneous correlation between the endogenous variables in
oo_.contemporaneous_correlation. Requires thenocorroption not to be set.
- spectral_density¶
Triggers the computation and display of the theoretical spectral density of the (filtered) model variables. Results are stored in
oo_.SpectralDensity, defined below. Default: do not request spectral density estimates.
- hp_ngrid = INTEGER¶
Deprecated option. It has the same effect as
filtered_theoretical_moments_grid.
Output
This command sets
oo_.dr,oo_.mean,oo_.var,oo_.var_list, andoo_.autocorr, which are described below.If the
periodsoption is present, setsoo_.skewness,oo_.kurtosis, andoo_.endo_simul(seeoo_.endo_simul).If option
irfis different from zero, setsoo_.irfs(see below).If the option
contemporaneous_correlationis different from0, setsoo_.contemporaneous_correlation, which is described below.Example
shocks; var e; stderr 0.0348; end; stoch_simul;
Performs the simulation of the 2nd-order approximation of a model with a single stochastic shock
e, with a standard error of0.0348.Example
stoch_simul(irf=60) y k;
Performs the simulation of a model and displays impulse response functions on 60 periods for variables
yandk.
- MATLAB/Octave variable: oo_.mean¶
After a run ofstoch_simul, contains the mean of the endogenous variables. Contains theoretical mean if theperiodsoption is not present, and simulated mean otherwise. The variables are arranged in declaration order.
- MATLAB/Octave variable: oo_.var¶
After a run ofstoch_simul, contains the variance-covariance of the endogenous variables. Contains theoretical variance if theperiodsoption is not present and simulated variance otherwise. Only available fororder<4. Atorder=2it will be be a second-order accurate approximation (i.e. ignoring terms of order 3 and 4 that would arise when using the full second-order policy function). Atorder=3, theoretical moments are only available withpruning. The variables are arranged in declaration order.
- MATLAB/Octave variable: oo_.var_list¶
The list of variables for which results are displayed.
- MATLAB/Octave variable: oo_.skewness¶
After a run ofstoch_simulcontains the skewness (standardized third moment) of the simulated variables if theperiodsoption is present. The variables are arranged in declaration order.
- MATLAB/Octave variable: oo_.kurtosis¶
After a run ofstoch_simulcontains the excess kurtosis (standardized fourth moment) of the simulated variables if theperiodsoption is present. The variables are arranged in declaration order.
- MATLAB/Octave variable: oo_.autocorr¶
After a run ofstoch_simul, contains a cell array of the autocorrelation matrices of the endogenous variables. The element number of the matrix in the cell array corresponds to the order of autocorrelation. The option ar specifies the number of autocorrelation matrices available. Contains theoretical autocorrelations if theperiodsoption is not present and simulated autocorrelations otherwise. Only available fororder<4. Atorder=2it will be be a second-order accurate approximation. Atorder=3, theoretical moments are only available withpruning. The field is only created if stationary variables are present.The element
oo_.autocorr{i}(k,l)is equal to the correlation between \(y^k_t\) and \(y^l_{t-i}\), where \(y^k\) (resp. \(y^l\)) is the \(k\)-th (resp. \(l\)-th) endogenous variable in the declaration order.Note that if theoretical moments have been requested,
oo_.autocorr{i}is the same thanoo_.gamma_y{i+1}.
- MATLAB/Octave variable: oo_.gamma_y¶
After a run ofstoch_simul, if theoretical moments have been requested (i.e. if theperiodsoption is not present), this variable contains a cell array with the following values (wherearis the value of the option of the same name):oo_.gamma{1}Variance/covariance matrix.
oo_.gamma{i+1}(for i=1:ar)Autocorrelation function. See
oo_.autocorrfor more details. Beware, this is the autocorrelation function, not the autocovariance function.oo_.gamma{ar+2}Unconditional variance decomposition, see
oo_.variance_decomposition.oo_.gamma{ar+3}If a second order approximation has been requested, contains the vector of the mean correction terms.
Only available at
order<4. In caseorder=2, the theoretical second moments are a second order accurate approximation of the true second moments. See conditional_variance_decomposition. Atorder=3, theoretical moments are only available withpruning.
- MATLAB/Octave variable: oo_.variance_decomposition¶
After a run ofstoch_simulwhen requesting theoretical moments (periods=0), contains a matrix with the result of the unconditional variance decomposition (i.e. at horizon infinity). The first dimension corresponds to the endogenous variables (in the order of declaration after the command or inM_.endo_names) and the second dimension corresponds to exogenous variables (in the order of declaration). Numbers are in percent and sum up to 100 across columns. In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables.
- MATLAB/Octave variable: oo_.variance_decomposition_ME¶
Field set after a run ofstoch_simulwhen requesting theoretical moments (periods=0) if measurement error is present. It is similar tooo_.variance_decomposition, but the decomposition will be conducted of the measured variables. The field contains a matrix with the result of the unconditional variance decomposition (i.e. at horizon infinity). The first dimension corresponds to the observed endoogenous variables (in the order of declaration after the command) and the second dimension corresponds to exogenous variables (in the order of declaration), with the last column corresponding to the contribution of measurement error. Numbers are in percent and sum up to 100 across columns.
- MATLAB/Octave variable: oo_.conditional_variance_decomposition¶
After a run ofstoch_simulwith theconditional_variance_decompositionoption, contains a three-dimensional array with the result of the decomposition. The first dimension corresponds to the endogenous variables (in the order of declaration after the command or inM_.endo_namesif not specified), the second dimension corresponds to the forecast horizons (as declared with the option), and the third dimension corresponds to the exogenous variables (in the order of declaration). In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conductedof the actual as opposed to the measured variables.
- MATLAB/Octave variable: oo_.conditional_variance_decomposition_ME¶
Field set after a run ofstoch_simulwith theconditional_variance_decompositionoption if measurement error is present. It is similar tooo_.conditional_variance_decomposition, but the decomposition will be conducted of the measured variables. It contains a three-dimensional array with the result of the decomposition. The first dimension corresponds to the endogenous variables (in the order of declaration after the command or inM_.endo_namesif not specified), the second dimension corresponds to the forecast horizons (as declared with the option), and the third dimension corresponds to the exogenous variables (in the order of declaration), with the last column corresponding to the contribution of the measurement error.
- MATLAB/Octave variable: oo_.contemporaneous_correlation¶
After a run ofstoch_simulwith thecontemporaneous_correlation option, contains theoretical contemporaneous correlations if theperiodsoption is not present, and simulated contemporaneous correlations otherwise. Only available fororder<4. Atorder=2it will be be a second-order accurate approximation. Atorder=3, theoretical moments are only available withpruning. The variables are arranged in declaration order.
- MATLAB/Octave variable: oo_.SpectralDensity¶
After a run ofstoch_simulwith optionspectral_density, contains the spectral density of the model variables. There will be anvarsbynfrequenciessubfieldfreqsstoring the respective frequency grid points ranging from \(0\) to \(2\pi\) and a same sized subfielddensitystoring the corresponding density.
- MATLAB/Octave variable: oo_.irfs¶
After a run ofstoch_simulwith optionirfdifferent from zero, contains the impulse responses, with the following naming convention: VARIABLE_NAME_SHOCK_NAME.For example,
oo_.irfs.gnp_eacontains the effect ongnpof a one-standard deviation shock onea.
- MATLAB/Octave command: IRF_MATRIX=get_irf('EXOGENOUS_NAME' [, 'ENDOGENOUS_NAME']... );¶
Given the name of an exogenous variable, returns the IRFs for the requested endogenous variable(s) (as they are stored inoo_.irfs) in the outputIRF_MATRIX. The periods are stored along the first dimension, with the steady state in the first row. The variables are stored along the second dimension. If no endogenous variables were specified, the matrix contains all variables stored inoo_.irfs.
The approximated solution of a model takes the form of a set of
decision rules or transition equations expressing the current value of
the endogenous variables of the model as function of the previous
state of the model and shocks observed at the beginning of the
period. The decision rules are stored in the structure oo_.dr
which is described below.
- MATLAB/Octave variable: oo_.dr¶
Structure storing the decision rules. The subfields for different orders of approximation are explained below.
- Command: extended_path ;¶
- Command: extended_path(OPTIONS...);
Simulates a stochastic (i.e. rational expectations) model, using the extended path method presented by Fair and Taylor (1983). Time series for the endogenous variables are generated by assuming that the agents believe that there will no more shocks in the following periods.This function first computes a random path for the exogenous variables (stored in
oo_.exo_simul, seeoo_.exo_simul) and then computes the corresponding path for endogenous variables, taking the steady state as starting point. The result of the simulation is stored inoo_.endo_simul(seeoo_.endo_simul). Note that this simulation approach does not solve for the policy and transition equations but for paths for the endogenous variables.Options
- periods = INTEGER
The number of periods for which the simulation is to be computed. No default value, mandatory option.
- solver_periods = INTEGER¶
The number of periods used to compute the solution of the perfect foresight at every iteration of the algorithm. Default:
200.
- order = INTEGER
If order is greater than
0Dynare uses a gaussian quadrature to take into account the effects of future uncertainty; this is called stochastic extended path, see Adjemian and Juillard (2025). Iforder\(=S\) then the time series for the endogenous variables are generated by assuming that the agents believe that there will no more shocks after period \(t+S\). This is an experimental feature and can be quite slow. A non-zero value is not compatible with thebytecodeoption of themodelblock ormodel_optionscommand. Default:0.
- hybrid¶
Use the constant of the second order perturbation reduced form to correct the paths generated by the (stochastic) extended path algorithm.
- lmmcp
Solves the perfect foresight model with a Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra (2004)), which allows to consider inequality constraints on the endogenous variables (such as a ZLB on the nominal interest rate or a model with irreversible investment). For specifying the necessary complementarity conditions, see
lmmcp.
- use_first_order_solution¶
Utilize the model simulation based on a first-order local approximation as the initial guess for the nonlinear solver in each period. If this is not applied, solution in previous period is used.
4.13.2. Typology and ordering of variables¶
Dynare distinguishes four types of endogenous variables:
Purely backward (or purely predetermined) variables
Those that appear only at current and past period in the model, but not at future period (i.e. at \(t\) and \(t-1\) but not \(t+1\)). The number of such variables is equal to
M_.npred.
Purely forward variables
Those that appear only at current and future period in the model, but not at past period (i.e. at \(t\) and \(t+1\) but not \(t-1\)). The number of such variables is stored in
M_.nfwrd.
Mixed variables
Those that appear at current, past and future period in the model (i.e. at \(t\), \(t+1\) and \(t-1\)). The number of such variables is stored in
M_.nboth.
Static variables
Those that appear only at current, not past and future period in the model (i.e. only at \(t\), not at \(t+1\) or \(t-1\)). The number of such variables is stored in
M_.nstatic.
Note that all endogenous variables fall into one of these four categories, since after the creation of auxiliary variables (see Auxiliary variables), all endogenous have at most one lead and one lag. We therefore have the following identity:
M_.npred + M_.both + M_.nfwrd + M_.nstatic = M_.endo_nbr
- MATLAB/Octave variable: M_.state_var¶
Vector of numerical indices identifying the state variables in the vector of declared variables.
M_.endo_names(M_.state_var)therefore yields the name of all variables that are states in the model declaration, i.e. that show up with a lag.
Internally, Dynare uses two orderings of the endogenous variables: the
order of declaration (which is reflected in M_.endo_names), and an
order based on the four types described above, which we will call the
DR-order (“DR” stands for decision rules). Most of the time, the
declaration order is used, but for elements of the decision rules, the
DR-order is used.
The DR-order is the following: static variables appear first, then purely backward variables, then mixed variables, and finally purely forward variables. Inside each category, variables are arranged according to the declaration order.
- MATLAB/Octave variable: oo_.dr.order_var¶
This variables maps DR-order to declaration order.
- MATLAB/Octave variable: oo_.dr.inv_order_var¶
This variable contains the inverse map.
In other words, the k-th variable in the DR-order corresponds to the endogenous
variable numbered oo_.dr.order_var(k) in declaration
order. Conversely, k-th declared variable is numbered
oo_.dr.inv_order_var(k) in DR-order.
Finally, the state variables of the model are the purely backward
variables and the mixed variables. They are ordered in DR-order when
they appear in decision rules elements. There are M_.nspred =
M_.npred + M_.nboth such variables. Similarly, one has M_.nsfwrd =
M_.nfwrd + M_.nboth, and M_.ndynamic = M_.nfwrd + M_.nboth +
M_.npred.
4.13.3. First-order approximation¶
The approximation has the stylized form:
where \(y^s\) is the steady state value of \(y\) and \(y^h_t=y_t-y^s\).
- MATLAB/Octave variable: oo.dr.state_var¶
Vector of numerical indices identifying the state variables in the vector of declared variables, given the current parameter values for which the decision rules have been computed. It may differ from
M_.state_varin case a state variable drops from the model given the current parameterization, because it only gets 0 coefficients in the decision rules. SeeM_.state_var.
The coefficients of the decision rules are stored as follows:
\(y^s\) is stored in
oo_.dr.ys. The vector rows correspond to all endogenous in the declaration order.\(A\) is stored in
oo_.dr.ghx. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to state variables in DR-order, as given byoo_.dr.state_var.\(B\) is stored
oo_.dr.ghu. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to exogenous variables in declaration order.
Of course, the shown form of the approximation is only stylized, because it neglects the required different ordering in \(y^s\) and \(y^h_t\). The precise form of the approximation that shows the way Dynare deals with differences between declaration and DR-order, is
where \(\mathrm{k2}\) selects the state variables, \(y_t\) and \(y^s\) are in declaration order and the coefficient matrices are in DR-order. Effectively, all variables on the right hand side are brought into DR order for computations and then assigned to \(y_t\) in declaration order.
4.13.4. Second-order approximation¶
The approximation has the form:
where \(y^s\) is the steady state value of \(y\), \(y^h_t=y_t-y^s\), and \(\Delta^2\) is the shift effect of the variance of future shocks. For the reordering required due to differences in declaration and DR order, see the first order approximation.
The coefficients of the decision rules are stored in the variables described for first order approximation, plus the following variables:
\(\Delta^2\) is stored in
oo_.dr.ghs2. The vector rows correspond to all endogenous in DR-order.\(C\) is stored in
oo_.dr.ghxx. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of the vector of state variables in DR-order.\(D\) is stored in
oo_.dr.ghuu. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of exogenous variables in declaration order.\(E\) is stored in
oo_.dr.ghxu. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of the vector of state variables (in DR-order) by the vector of exogenous variables (in declaration order).
4.13.5. Third-order approximation¶
The approximation has the form:
where \(y^s\) is the steady state value of \(y\), and
\(z_t\) is a vector consisting of the deviation from the steady
state of the state variables (in DR-order) at date \(t-1\)
followed by the exogenous variables at date \(t\) (in declaration
order). The vector \(z_t\) is therefore of size \(n_z\) =
M_.nspred + M_.exo_nbr.
The coefficients of the decision rules are stored as follows:
\(y^s\) is stored in
oo_.dr.ys. The vector rows correspond to all endogenous in the declaration order.\(G_0\) is stored in
oo_.dr.g_0. The vector rows correspond to all endogenous in DR-order.\(G_1\) is stored in
oo_.dr.g_1. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to state variables in DR-order, followed by exogenous in declaration order.\(G_2\) is stored in
oo_.dr.g_2. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the Kronecker product of state variables (in DR-order), followed by exogenous (in declaration order). Note that the Kronecker product is stored in a folded way, i.e. symmetric elements are stored only once, which implies that the matrix has \(n_z(n_z+1)/2\) columns. More precisely, each column of this matrix corresponds to a pair \((i_1, i_2)\) where each index represents an element of \(z_t\) and is therefore between \(1\) and \(n_z\). Only non-decreasing pairs are stored, i.e. those for which \(i_1 \leq i_2\). The columns are arranged in the lexicographical order of non-decreasing pairs. Also note that for those pairs where \(i_1 \neq i_2\), since the element is stored only once but appears two times in the unfolded \(G_2\) matrix, it must be multiplied by 2 when computing the decision rules.\(G_3\) is stored in
oo_.dr.g_3. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to the third Kronecker power of state variables (in DR-order), followed by exogenous (in declaration order). Note that the third Kronecker power is stored in a folded way, i.e. symmetric elements are stored only once, which implies that the matrix has \(n_z(n_z+1)(n_z+2)/6\) columns. More precisely, each column of this matrix corresponds to a tuple \((i_1, i_2, i_3)\) where each index represents an element of \(z_t\) and is therefore between \(1\) and \(n_z\). Only non-decreasing tuples are stored, i.e. those for which \(i_1 \leq i_2 \leq i_3\). The columns are arranged in the lexicographical order of non-decreasing tuples. Also note that for tuples that have three distinct indices (i.e. \(i_1 \neq i_2\) and \(i_1 \neq i_3\) and \(i_2 \neq i_3\)), since these elements are stored only once but appears six times in the unfolded \(G_3\) matrix, they must be multiplied by 6 when computing the decision rules. Similarly, for those tuples that have two equal indices (i.e. of the form \((a,a,b)\) or \((a,b,a)\) or \((b,a,a)\)), since these elements are stored only once but appears three times in the unfolded \(G_3\) matrix, they must be multiplied by 3 when computing the decision rules.
4.13.6. Higher-order approximation¶
Higher-order approximations are simply a generalization of what is done at order 3.
The steady state is stored in oo_.dr.ys and the constant correction is
stored in oo_.dr.g_0. The coefficient for orders 1, 2, 3, 4… are
respectively stored in oo_.dr.g_0, oo_.dr.g_1, oo_.dr.g_2,
oo_.dr.g_3, oo_.dr.g_4… The columns of those matrices correspond to
multidimensional indices of state variables, in such a way that symmetric
elements are never repeated (for more details, see the description of
oo_.dr.g_3 in the third-order case).
4.14. Occasionally binding constraints (OCCBIN)¶
Dynare allows simulating models with up to two occasionally-binding constraints by relying on a piecewise linear solution as in Guerrieri and Iacoviello (2015). It also allows estimating such models employing either the inversion filter of Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) or the piecewise Kalman filter of Giovannini, Pfeiffer, and Ratto (2021). To trigger computations involving occasionally-binding constraints requires
defining and naming the occasionally-binding constraints using an
occbin_constraintsblockspecifying the model equations for the respective regimes in the
modelblock using appropriate equation tags.potentially specifying a sequence of surprise shocks using a
shocks(surprise)blocksetting up Occbin simulations or estimation with
occbin_setuptriggering a simulation with
occbin_solveror runningestimationorcalib_smoother.
All of these elements are discussed in the following.
- Block: occbin_constraints ;¶
Theoccbin_constraintsblock specifies the occasionally-binding constraints. It contains one or two of the following lines:name ‘STRING’; bind EXPRESSION; [relax EXPRESSION;] [error_bind EXPRESSION;] [error_relax EXPRESSION;]
STRINGis the name of constraint that is used to reference the constraint inrelax/bindequation tags to identify the respective regime (see below). Thebindexpression is mandatory and defines a logical condition that is evaluated in the baseline/steady state regime to check whether the specified constraint becomes binding. In contrast, therelaxexpression is optional and specifies a logical condition that is evaluated in the binding regime to check whether the regime returns to the baseline/steady state regime. If not specified, Dynare will simply check in the binding regime whether thebindexpression evaluates to false. However, there are cases where thebindexpression cannot be evaluated in the binding regime(s), because the variables involved are constant by definition so that e.g. the value of the Lagrange multiplier on the complementary slackness condition needs to be checked. In these cases, it is necessary to provide an explicit condition that can be evaluated in the binding regime that allows to check whether it should be left.Note that the baseline regime denotes the steady state of the model where the economy will settle in the long-run without shocks. For that matter, it may be one where e.g. a borrowing constraint is binding. In that type of setup, the
bindcondition is used to specify the condition when this borrowing constraint becomes non-binding so that the alternative regime is entered.Three things are important to keep in mind when specifying the expressions. First, feasible expressions may only contain contemporaneous endogenous variables. If you want to include leads/lags or exogenous variables, you need to define an auxiliary variable. Second, Dynare will at the current stage not linearly approximate the entered expressions. Because Occbin will work with a linearized model, consistency will often require the user to enter a linearized constraint. Otherwise, the condition employed for checking constraint violations may differ from the one employed within model simulations based on the piecewise-linear model solution. Third, in contrast to the original Occbin replication codes, the variables used in expressions are not automatically demeaned, i.e. they refer to the levels, not deviations from the steady state. To access the steady state level of a variable, the
STEADY_STATE()operator can be used.Finally, it’s worth keeping in mind that for each simulation period, Occbin will check the respective conditions for whether the current regime should be left. Small numerical differences from the cutoff point for a regime can sometimes lead to oscillations between regimes and cause a spurious periodic solution. Such cases may be prevented by introducing a small buffer between the two regimes, e.g.
occbin_constraints; name 'ELB'; bind inom <= iss-1e8; relax inom > iss+1e-8; end;
The
error_bindanderror_relaxoptions are optional and allow specifying numerical criteria for the size of the respective constraint violations employed in numerical routines. By default, Dynare will simply use the absolute value of thebindandrelaxinequalities. But occasionnally, user-specified expressions perform better.Example
occbin_constraints; name 'IRR'; bind log_Invest-log(steady_state(Invest))<log(phi); relax Lambda<0; name 'INEG'; bind log_Invest-log(steady_state(Invest))<0; end;
IRR is a constraint for irreversible investment that becomes binding if investment drops below its steady state by more than 0.025 percent in the non-binding regime. The constraint will be relaxed whenever the associated Lagrange multiplier
Lambdain the binding regime becomes negative. Note that the constraint here takes on a linear form to be consistent with a piecewise linear model solutionThe specification of the model equations belonging to the respective regimes is done in the
modelblock, with equation tags indicating to which regime a particular equation belongs. All equations that differ across regimes must have anametag attached to them that allows uniquely identifying different versions of the same equation. The name of the constraints specified is then used in conjunction with abindorrelaxtag to indicate to which regime a particular equation belongs. In case of more than one occasionally-binding constraint, if an equation belongs to several regimes (e.g. both constraints binding), the constraint name tags must be separated by a comma. If only one name tag is present, the respective equation is assumed to hold for both states of the other constraint.Example
[name='investment',bind='IRR,INEG'] (log_Invest - log(phi*steady_state(Invest))) = 0; [name='investment',relax='IRR'] Lambda=0; [name='investment',bind='IRR',relax='INEG'] (log_Invest - log(phi*steady_state(Invest))) = 0;
The three entered equations for the investment condition define the model equation for all four possible combinations of the two constraints. The first equation defines the model equation in the regime where both the IRR and INEG constraint are binding. The second equation defines the model equation for the regimes where the IRR constraint is non-binding, regardless of whether the INEG constraint is binding or not. Finally, the last equation defines the model equation for the final regime where the IRR constraint is binding, but the INEG one is not.
- Block: shocks(surprise) ;
- Block: shocks(surprise,overwrite);
Theshocks(surprise)block allows specifying a sequence of temporary changes in the value of exogenous variables that in each period come as a surprise to agents, i.e. are not anticipated. Note that to actually use the specified shocks in subsequent commands likeoccbin_solver, the block needs to be followed by a call tooccbin_setup.The block mirrors the perfect foresight syntax in that it should contain one or more occurrences of the following group of three lines:
var VARIABLE_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
Example (with vector values and overwrite option)
shockssequence = randn(100,1)*0.02; shocks(surprise,overwrite); var epsilon; periods 1:100; values (shockssequence); end;
- Command: occbin_setup ;¶
- Command: occbin_setup(OPTIONS...);
Prepares a simulation with occasionally binding constraints. This command will also translate the contents of ashocks(surprise)block for use in subsequent commands.In order to conduct
estimationwith occasionally binding constraints, it needs to be prefaced by a call tooccbin_setupto trigger the use of either the inversion filter or the piecewise Kalman filter (default). An issue that can arise in the context of estimation is a structural shock dropping out of the model in a particular regime. For example, at the zero lower bound on interest rates, the monetary policy shock in the Taylor rule will not appear anymore. This may create a problem if there are then more observables than shocks. The way to handle this issue depends on the type of filter used. The first step is to set the data points for the zero interest rate period to NaN. For the piecewise Kalman filter, the standard deviation of the associated shock needs to be set to 0 for the corresponding periods using theheteroskedastic_shocksblock. This avoids stochastic singularity. However, this approach does not work for the inversion filter as theheteroskedastic_shocksblock does not do anything here. For the inversion filter, as many shocks as observables are required at each point in time. Dynare assumes a one-to-one mapping between the declared shocks invarexoand declared observables invarobs. For example, if the second declared observable is NaN in a given period, Dynare will drop the second declared shock.Warning
If there are missing values, it is imperative for the inversion filter that the declaration order of shocks and observables is conformable. Sticking with our example, if the nominal interest is the second
varobsand is set to NaN, the inversion filter will drop the second declared shock. If that second declared shock is, e.g., a TFP shock, it will be dropped instead of the intended monetary policy shock.Note that models with unit roots will require the user to specify the
diffuse_filteroption as otherwise Blanchard-Kahn errors will be triggered. For the piecewise Kalman filter, the initialization steps in the diffuse filter will always rely on the model solved for the baseline regime, without checking whether this is the actual regime in the first period(s).Example
occbin_setup(likelihood_inversion_filter,smoother_inversion_filter); estimation(smoother,heteroskedastic_filter,...);
The above piece of code sets up an estimation employing the inversion filter for both the likelihood evaluation and the smoother, while also accounting for
heteroskedastic_shocksusing theheteroskedastic_filteroption.Be aware that Occbin has largely command-specific options, i.e. there are separate options to control the behavior of Occbin when called by the smoother or when computing the likelihood. These latter commands will not inherit the options potentially previously set for simulations.
Options
- simul_periods = INTEGER¶
Number of periods of the simulation. Default: 100.
- simul_maxit = INTEGER¶
Maximum number of iterations when trying to find the regimes of the piecewise solution. Default: 30.
- simul_check_ahead_periods = INTEGER¶
Number of periods for which to check ahead for return to the baseline regime. This number should be chosen large enough, because Occbin requires the simulation to return to the baseline regime at the end of time. Default: 200.
- simul_reset_check_ahead_periods¶
Allows to reset
simul_check_ahead_periodsto its specified value at the beginning of each simulation period. Otherwise, the original value may permanently increase endogenously at some point due to regimes that last very long in expectations. This may considerably slow down convergence in subsequent periods. Default: not enabled.
- simul_max_check_ahead_periods = INTEGER¶
If set to a finite number, it enforces the OccBin algorithm to check ahead only for the maximum number of periods (i.e. when we want agents to be myopic beyond some future period) instead of potentially endogenously increasing
simul_check_ahead_periodsever further. Default: Inf.
- simul_curb_retrench¶
Instead of basing the initial regime guess for the current iteration on the last iteration, update the guess only one period at a time. This will slow down the iterations, but may lead to more robust convergence behavior. Default: not enabled.
- simul_periodic_solution¶
Accept a periodic solution where the solution alternates between two sets of results across iterations, i.e. is not found to be unique. This is sometimes caused by spurious numerical errors that lead to oscillations between regiems and may be prevented by allowing for a small buffer in regime transitions. Default: not enabled.
- simul_debug¶
Provide additional debugging information during solving. Default: not enabled.
- smoother_periods = INTEGER¶
Number of periods employed during the simulation when called by the smoother (equivalent of
simul_periods). Default: 100.
- smoother_maxit = INTEGER¶
Maximum number of iterations employed during the simulation when called by the smoother (equivalent of
simul_maxit). Default: 30.
- smoother_check_ahead_periods = INTEGER¶
Number of periods for which to check ahead for return to the baseline regime during the simulation when called by the smoother (equivalent of
simul_check_ahead_periods). Default: 200.
- smoother_max_check_ahead_periods = INTEGER¶
If set to a finite number, it enforces the OccBin algorithm to check ahead only for the maximum number of periods (i.e. when we want agents to be myopic beyond some future period) instead of potentially endogenously increasing
smoother_check_ahead_periodsever further. Equivalent ofsimul_max_check_ahead_periods. Default: Inf.
- smoother_curb_retrench¶
Have the smoother invoke the
simul_curb_retrenchoption during simulations. Default: not enabled.
- smoother_periodic_solution¶
Accept periodic solution where solution alternates between two sets of results (equivalent of
simul_periodic_solution). Default: not enabled.
- likelihood_periods = INTEGER¶
Number of periods employed during the simulation when computing the likelihood (equivalent of
simul_periods). Default: 100.
- likelihood_maxit = INTEGER¶
Maximum number of iterations employed during the simulation when computing the likelihood (equivalent of
simul_maxit). Default: 30.
- likelihood_check_ahead_periods = INTEGER¶
Number of periods for which to check ahead for return to the baseline regime during the simulation when computing the likelihood (equivalent of
simul_check_ahead_periods). Default: 200.
- smoother_max_check_ahead_periods = INTEGER
If set to a finite number, it enforces the OccBin algorithm to check ahead only for the maximum number of periods (i.e. when we want agents to be myopic beyond some future period) instead of potentially endogenously increasing
likelihood_check_ahead_periodsever further. Equivalent ofsimul_max_check_ahead_periods. Default: Inf.
- likelihood_curb_retrench¶
Have the likelihood computation invoke the
simul_curb_retrenchoption during simulations. Default: not enabled.
- likelihood_periodic_solution¶
Accept periodic solution where solution alternates between two sets of results (equivalent of
simul_periodic_solution). Default: not enabled.
- likelihood_inversion_filter¶
Employ the inversion filter of Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) when estimating the model. Default: not enabled.
- likelihood_piecewise_kalman_filter¶
Employ the piecewise Kalman filter of Giovannini, Pfeiffer, and Ratto (2021) when estimating the model. Note that this filter is incompatible with univariate Kalman filters, i.e.
kalman_algo=2,4. Default: enabled.
- likelihood_max_kalman_iterations¶
Maximum number of iterations of the outer loop for the piecewise Kalman filter. Default: 10.
- smoother_inversion_filter¶
Employ the inversion filter of Cuba-Borda, Guerrieri, Iacoviello, and Zhong (2019) when running the smoother. The underlying assumption is that the system starts at the steady state. In this case, the inversion filter will provide the required smoother output. Default: not enabled.
- smoother_piecewise_kalman_filter¶
Employ the piecewise Kalman filter of Giovannini, Pfeiffer, and Ratto (2021) when running the smoother. Default: enabled.
- filter_use_relaxation¶
Triggers relaxation within the guess and verify algorithm used in the update step of the piecewise Kalman filter. When old and new guess regime differ to much, use a new guess closer to the previous guess. In case of multiple solutions, tends to provide an occasionally binding regime with a shorter duration (typically preferable). Specifying this option may slow down convergence. Default: not enabled.
Output
The paths for the exogenous variables are stored into
options_.occbin.simul.SHOCKS.
- Command: occbin_solver ;¶
- Command: occbin_solver(OPTIONS...);
Computes a simulation with occasionally-binding constraints based on a piecewise-linear solution.Note that
occbin_setupmust be called before this command in order for the simulation to take into account previousshocks(surprise)blocks.Options
- simul_periods = INTEGER
See
simul_periods.
- simul_maxit = INTEGER
See
simul_maxit.
- simul_check_ahead_periods = INTEGER
- simul_reset_check_ahead_periods
- simul_max_check_ahead_periods¶
- simul_curb_retrench
See
simul_curb_retrench.
- simul_debug
See
simul_debug.
Output
The command outputs various objects into
oo_.occbin.
- MATLAB/Octave variable: oo_.occbin.simul.piecewise¶
Matrix storing the simulations based on the piecewise-linear solution. The variables are arranged by column, in order of declaration (as inM_.endo_names), while the the rows correspond to thesimul_periods.
- MATLAB/Octave variable: oo_.occbin.simul.linear¶
Matrix storing the simulations based on the linear solution, i.e. ignoring the occasionally binding constraint(s). The variables are arranged column by column, in order of declaration (as inM_.endo_names), while the the rows correspond to thesimul_periods.
- MATLAB/Octave variable: oo_.occbin.simul.shocks_sequence¶
Matrix storing the shock sequence employed during the simulation. The shocks are arranged column by column, with their order inM_.exo_namesstored inoo_.occbin.exo_pos. The the rows correspond to the number of shock periods specified in ashocks(surprise)block, which may be smaller thansimul_periods.
- MATLAB/Octave variable: oo_.occbin.simul.regime_history¶
Structure storing information on the regime history, conditional on the shock that happened in the respective period (stored along the rows).typeis equal to eithersmootherorsimul, depending on whether the output comes from a run of simulations or the smoother. The subfieldregimecontains a vector storing the regime state, while the the subfieldregimestartindicates the expected start of the respective regime state. For example, if row 40 contains[1,0]forregime2and[1,6]forregimestart2, it indicates that - after the shock in period 40 has occurred - the second constraint became binding (1) and is expected to revert to non-binding (0) after six periods including the current one, i.e. period 45.
- MATLAB/Octave variable: oo_.occbin.simul.ys¶
Vector of steady state values
- Command: occbin_graph [VARIABLE_NAME...];¶
- Command: occbin_graph(OPTIONS...) [VARIABLE_NAME...];
Plots a graph comparing the simulation results of the piecewise-linear solution with the occasionally binding contraints to the linear solution ignoring the constraint.Options
- noconstant¶
Omit the steady state in the graphs.
- Command: occbin_write_regimes ;¶
- Command: occbin_write_regimes(OPTIONS...);
Write the information on the regime history stored inoo_.occbin.simul.regime_historyoroo_.occbin.smoother.regime_historyinto an Excel file stored in theFILENAME/Outputfolder.Options
- periods = INTEGER
Number of periods for which to write the expected regime durations. Default: write all available periods.
- filename = FILENAME¶
Name of the Excel file to write. Default:
FILENAME_occbin_regimes.
- simul
Selects the regime history from the last run of simulations. Default: enabled.
- smoother¶
Selects the regime history from the last run of the smoother. Default: use
simul.
4.15. Estimation based on likelihood¶
Provided that you have observations on some endogenous variables, it is possible to use Dynare to estimate some or all parameters. Both maximum likelihood (as in Ireland (2004)) and Bayesian techniques (as in Fernández-Villaverde and Rubio-Ramírez (2004), Rabanal and Rubio-Ramirez (2003), Schorfheide (2000) or Smets and Wouters (2003)) are available. Using Bayesian methods, it is possible to estimate DSGE models, VAR models, or a combination of the two techniques called DSGE-VAR.
Note that in order to avoid stochastic singularity, you must have at least as many shocks or measurement errors in your model as you have observed variables.
- Command: varobs VARIABLE_NAME...;
This command lists the name of observed endogenous variables for the estimation procedure. These variables must be available in the data file (see estimation).Alternatively, this command is also used in conjunction with the
partial_informationoption ofstoch_simul, for declaring the set of observed variables when solving the model under partial information.Only one instance of
varobsis allowed in a model file. If one needs to declare observed variables in a loop, the macro processor can be used as shown in the second example below.Example
varobs C y rr;
Declares endogenous variables
C,yandrras observed variables.Example (with a macro processor loop)
varobs @#for co in countries GDP_@{co} @#endfor ;
- Block: observation_trends ;¶
This block specifies linear trends for observed variables as functions of model parameters. In case theloglinearoption is used, this corresponds to a linear trend in the logged observables, i.e. an exponential trend in the level of the observables.Each line inside of the block should be of the form:
VARIABLE_NAME(EXPRESSION);
In most cases, variables shouldn’t be centered when
observation_trendsis used.Example
observation_trends; Y (eta); P (mu/eta); end;
- Block: estimated_params ;¶
- Block: estimated_params(overwrite) ;
This block lists all parameters to be estimated and specifies bounds and priors as necessary.Each line corresponds to an estimated parameter.
In a maximum likelihood or a method of moments estimation, each line follows this syntax:
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME , INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ];
In a Bayesian MCMC or a penalized method of moments estimation, each line follows this syntax:
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT [, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE, PRIOR_MEAN, PRIOR_STANDARD_ERROR [, PRIOR_3RD_PARAMETER [, PRIOR_4TH_PARAMETER [, SCALE_PARAMETER ] ] ];
The first part of the line consists of one of the four following alternatives:
stderr VARIABLE_NAMEIndicates that the standard error of the exogenous variable VARIABLE_NAME, or of the observation error/measurement errors associated with endogenous observed variable VARIABLE_NAME, is to be estimated.
corr VARIABLE_NAME1, VARIABLE_NAME2Indicates that the correlation between the exogenous variables VARIABLE_NAME1 and VARIABLE_NAME2, or the correlation of the observation errors/measurement errors associated with endogenous observed variables VARIABLE_NAME1 and VARIABLE_NAME2, is to be estimated. Note that correlations set by previous
shocksblocks or estimation commands are kept at their value set prior to estimation if they are not estimated again subsequently. Thus, the treatment is the same as in the case of deep parameters set during model calibration and not estimated.PARAMETER_NAMEThe name of a model parameter to be estimated
DSGE_PRIOR_WEIGHTSpecial name for the weigh of the DSGE model in DSGE-VAR model.
The rest of the line consists of the following fields, some of them being optional:
- INITIAL_VALUE¶
Specifies a starting value for the posterior mode optimizer or the maximum likelihood estimation. If unset, defaults to the prior mean.
- LOWER_BOUND¶
Specifies a lower bound for the parameter value in maximum likelihood estimation. In a Bayesian estimation context, sets a lower bound only effective while maximizing the posterior kernel. This lower bound does not modify the shape of the prior density, and is only aimed at helping the optimizer in identifying the posterior mode (no consequences for the MCMC). For some prior densities (namely inverse gamma, gamma, uniform, beta or Weibull) it is possible to shift the support of the prior distributions to the left or the right using
prior_3rd_parameter. In this case the prior density is effectively modified (note that the truncated Gaussian density is not implemented in Dynare). If unset, defaults to minus infinity (ML) or the natural lower bound of the prior (Bayesian estimation).
- UPPER_BOUND¶
Same as
lower_bound, but specifying an upper bound instead.
- PRIOR_SHAPE¶
A keyword specifying the shape of the prior density. The possible values are:
beta_pdf,gamma_pdf,normal_pdf,uniform_pdf,inv_gamma_pdf,inv_gamma1_pdf,inv_gamma2_pdfandweibull_pdf. Note thatinv_gamma_pdfis equivalent toinv_gamma1_pdf.
- PRIOR_MEAN¶
The mean of the prior distribution.
- PRIOR_STANDARD_ERROR¶
The standard error of the prior distribution.
- PRIOR_3RD_PARAMETER¶
A third parameter of the prior used for generalized beta distribution, generalized gamma, generalized Weibull, the truncated normal, and for the uniform distribution. Default:
-Inffor normal distribution,0otherwise.
- PRIOR_4TH_PARAMETER¶
A fourth parameter of the prior used for generalized beta distribution, the truncated normal, and for the uniform distribution. Default:
Inffor normal distribution,1otherwise.
- SCALE_PARAMETER¶
A parameter specific scale parameter for the jumping distribution’s covariance matrix of the Metropolis-Hasting algorithm.
Note that INITIAL_VALUE, LOWER_BOUND, UPPER_BOUND, PRIOR_MEAN, PRIOR_STANDARD_ERROR, PRIOR_3RD_PARAMETER, PRIOR_4TH_PARAMETER and SCALE_PARAMETER can be any valid EXPRESSION. Some of them can be empty, in which Dynare will select a default value depending on the context and the prior shape.
In case of the uniform distribution, it can be specified either by providing an upper and a lower bound using
PRIOR_3RD_PARAMETERandPRIOR_4TH_PARAMETERor via mean and standard deviation usingPRIOR_MEAN,PRIOR_STANDARD_ERROR. The other two will automatically be filled out. Note that providing both sets of hyperparameters will yield an error message.As one uses options more towards the end of the list, all previous options must be filled: for example, if you want to specify SCALE_PARAMETER, you must specify
PRIOR_3RD_PARAMETERandPRIOR_4TH_PARAMETER. Use empty values, if these parameters don’t apply.Example
corr eps_1, eps_2, 0.5, , , beta_pdf, 0, 0.3, -1, 1;
Sets a generalized beta prior for the correlation between
eps_1andeps_2with mean0and variance0.3. By settingPRIOR_3RD_PARAMETERto-1andPRIOR_4TH_PARAMETERto1the standard beta distribution with support[0,1]is changed to a generalized beta with support[-1,1]. Note that LOWER_BOUND and UPPER_BOUND are left empty and thus default to-1and1, respectively. The initial value is set to0.5.Example
corr eps_1, eps_2, 0.5, -0.5, 1, beta_pdf, 0, 0.3, -1, 1;
Sets the same generalized beta distribution as before, but now truncates this distribution to
[-0.5,1]through the use of LOWER_BOUND and UPPER_BOUND.Parameter transformation
Sometimes, it is desirable to estimate a transformation of a parameter appearing in the model, rather than the parameter itself. It is of course possible to replace the original parameter by a function of the estimated parameter everywhere is the model, but it is often unpractical.
In such a case, it is possible to declare the parameter to be estimated in the parameters statement and to define the transformation, using a pound sign (#) expression (see Model declaration).
Example
parameters bet; model; # sig = 1/bet; c = sig*c(+1)*mpk; end; estimated_params; bet, normal_pdf, 1, 0.05; end;
It is possible to have several
estimated_paramsblocks. By default, subsequent blocks are concatenated with the previous ones; this can be useful when building models in a modular fashion (see alsoestimated_params_removefor that use case). However, if anestimated_paramsblock has theoverwriteoption, its contents becomes the new list of estimated parameters, cancelling previous blocks; this can be useful when doing several estimations in a single.modfile.
- Block: estimated_params_init ;¶
- Block: estimated_params_init(OPTIONS...);
This block declares numerical initial values for the optimizer when these ones are different from the prior mean. It should be specified after theestimated_paramsblock as otherwise the specified starting values are overwritten by the latter.Each line has the following syntax:
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, INITIAL_VALUE;
Options
- use_calibration¶
For not specifically initialized parameters, use the deep parameters and the elements of the covariance matrix specified in the
shocksblock from calibration as starting values for estimation. For components of theshocksblock that were not explicitly specified during calibration or which violate the prior, the prior mean is used.
See
estimated_params, for the meaning and syntax of the various components.
- Block: estimated_params_bounds ;¶
This block declares lower and upper bounds for parameters in maximum likelihood estimation.Each line has the following syntax:
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND;
See
estimated_params, for the meaning and syntax of the various components.
- Block: estimated_params_remove ;¶
This block partially undoes the effect of a previousestimated_paramsblock, by removing some parameters from the estimation.Each line has the following syntax:
stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME;
- Command: estimation [VARIABLE_NAME...];¶
- Command: estimation(OPTIONS...) [VARIABLE_NAME...];
This command runs Bayesian or maximum likelihood estimation.The following information will be displayed by the command:
Results from posterior optimization (also for maximum likelihood)
Marginal log data density
Posterior mean and highest posterior density interval (shortest credible set) from posterior simulation
Convergence diagnostic table when only one MCM chain is used or Metropolis-Hastings convergence graphs documented in Pfeifer (2014) in case of multiple MCM chains
Table with numerical inefficiency factors of the MCMC
Graphs with prior, posterior, and mode
Graphs of smoothed shocks, smoothed observation errors, smoothed and historical variables
Note that the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) will only be computed on the variables listed after the
estimationcommand. Alternatively, one can choose to compute these quantities on all endogenous or on all observed variables (seeconsider_all_endogenous,consider_all_endogenous_and_auxiliary, andconsider_only_observedoptions below). If no variable is listed after the estimation command, then Dynare will interactively ask which variable set to use.Also, during the MCMC (Bayesian estimation with
mh_replic\(>0\)) a (graphical or text) waiting bar is displayed showing the progress of the Monte-Carlo and the current value of the acceptance ratio. Note that if theload_mh_fileoption is used (see below) the reported acceptance ratio does not take into account the draws from the previous MCMC. In the literature there is a general agreement for saying that the acceptance ratio should be close to one third or one quarter. If this not the case, you can stop the MCMC (Ctrl-C) and change the value of optionmh_jscale(see below).Note that by default Dynare generates random numbers using the algorithm
mt199937ar(i.e. Mersenne Twister method) with a seed set equal to0. Consequently the MCMCs in Dynare are deterministic: one will get exactly the same results across different Dynare runs (ceteris paribus). For instance, the posterior moments or posterior densities will be exactly the same. This behaviour allows to easily identify the consequences of a change on the model, the priors or the estimation options. But one may also want to check that across multiple runs, with different sequences of proposals, the returned results are almost identical. This should be true if the number of iterations (i.e. the value ofmh_replic) is important enough to ensure the convergence of the MCMC to its ergodic distribution. In this case the default behaviour of the random number generators in not wanted, and the user should set the seed according to the system clock before the estimation command using the following command:set_dynare_seed('clock');
so that the sequence of proposals will be different across different runs.
Finally, Dynare does not always properly distinguish between maximum likelihood and Bayesian estimation in its field names. While there is an important conceptual distinction between frequentist confidence intervals and Bayesian highest posterior density intervals (HPDI) as well as between posterior density and likelilhood, Dynare sometimes uses the Bayesian terms as a stand-in in its display of maximum likelihood results. An example is the storage of the output of the
forecastoption ofestimationwith ML, which will useHPDinf/HPDsupto denote the confidence interval.Algorithms
The Monte Carlo Markov Chain (MCMC) diagnostics are generated by the estimation command if
mh_replicis larger than 2000 and if optionnodiagnosticis not used. By default, the convergence diagnostics of Geweke (block_iter1992,1999) is computed for each chain. It uses a chi-square test to compare the means of the first and last draws specified bygeweke_intervalafter discarding the burn-in ofmh_drop. The test is computed using variance estimates under the assumption of no serial correlation as well as using tapering windows specified intaper_steps. Ifmh_nblocksis larger than 1, the convergence diagnostics of Brooks and Gelman (1998) are also provided. As described in section 3 of Brooks and Gelman (1998) the univariate convergence diagnostics are based on comparing pooled and within MCMC moments (Dynare displays the second and third order moments, and the length of the Highest Probability Density interval covering 80% of the posterior distribution). Due to computational reasons, the multivariate convergence diagnostic does not follow Brooks and Gelman (1998) strictly, but rather applies their idea for univariate convergence diagnostics to the range of the posterior likelihood function instead of the individual parameters. The posterior kernel is used to aggregate the parameters into a scalar statistic whose convergence is then checked using the Brooks and Gelman (1998) univariate convergence diagnostic.The inefficiency factors are computed as in Giordano et al.(2011) based on Parzen windows as in e.g. Andrews (1991).
Options
- datafile = FILENAME
The datafile: a
.mfile, a.matfile, a.csvfile, or a.xls/.xlsxfile (under Octave, the io package from Octave-Forge is required for the.csvand.xlsxformats and the.xlsfile extension is not supported). Note that the base name (i.e. without extension) of the datafile has to be different from the base name of the model file. If there are several files named FILENAME, but with different file endings, the file name must be included in quoted strings and provide the file ending like:estimation(datafile='../fsdat_simul.mat',...);
- dirname = FILENAME¶
Directory in which to store
estimationoutput. To pass a subdirectory of a directory, you must quote the argument. Default:<mod_file>.
- xls_sheet = QUOTED_STRING¶
The name of the sheet with the data in an Excel file.
- xls_range = RANGE¶
The range with the data in an Excel file. For example,
xls_range=B2:D200.
- nobs = INTEGER
The number of observations following
first_obsto be used. Default: all observations in the file afterfirst_obs.
- nobs = [INTEGER1:INTEGER2]¶
Runs a recursive estimation and forecast for samples of size ranging of
INTEGER1toINTEGER2. Optionforecastmust also be specified. The forecasts are stored in theRecursiveForecastfield of the results structure (seeRecursiveForecast). The respective results structuresoo_are saved inoo_recursive_(seeoo_recursive_) and are indexed with the respective sample length.
- first_obs = INTEGER¶
The number of the first observation to be used. In case of estimating a DSGE-VAR,
first_obsneeds to be larger than the number of lags. Default:1.
- first_obs = [INTEGER1:INTEGER2]¶
Runs a rolling window estimation and forecast for samples of fixed size
nobsstarting with the first observation ranging fromINTEGER1toINTEGER2. Optionforecastmust also be specified. This option is incompatible with requesting recursive forecasts using an expanding window (seenobs). The respective results structuresoo_are saved inoo_recursive_(seeoo_recursive_) and are indexed with the respective first observation of the rolling window.
- prefilter = INTEGER¶
A value of 1 means that the estimation procedure will demean each data series by its empirical mean. If the loglinear option without the
logdataoption is requested, the data will first be logged and then demeaned. Default:0, i.e. no prefiltering.
- presample = INTEGER¶
The number of observations after
first_obsto be skipped before evaluating the likelihood. These presample observations do not enter the likelihood, but are used as a training sample for starting the Kalman filter iterations. This option is incompatible with estimating a DSGE-VAR. Default:0.
- loglinear
Computes a log-linear approximation of the model instead of a linear approximation. As always in the context of estimation, the data must correspond to the definition of the variables used in the model (see Pfeifer (2013) for more details on how to correctly specify observation equations linking model variables and the data). If you specify the loglinear option, Dynare will take the logarithm of both your model variables and of your data as it assumes the data to correspond to the original non-logged model variables. The displayed posterior results like impulse responses, smoothed variables, and moments will be for the logged variables, not the original un-logged ones. Default: computes a linear approximation.
- logdata¶
Dynare applies the \(log\) transformation to the provided data if a log-linearization of the model is requested (
loglinear) unlesslogdataoption is used. This option is necessary if the user provides data already in logs, otherwise the \(log\) transformation will be applied twice (this may result in complex data).
- plot_priors = INTEGER¶
Control the plotting of priors.
0No prior plot.
1Prior density for each estimated parameter is plotted. It is important to check that the actual shape of prior densities matches what you have in mind. Ill-chosen values for the prior standard density can result in absurd prior densities.
Default value is1.
- nograph
See
nograph.
- posterior_nograph¶
Suppresses the generation of graphs associated with Bayesian IRFs (
bayesian_irf), posterior smoothed objects (smoother), and posterior forecasts (forecast).
- posterior_graph¶
Re-enables the generation of graphs previously shut off with
posterior_nograph.
- nodisplay
See
nodisplay.
- graph_format = FORMAT
- graph_format = ( FORMAT, FORMAT... )
See
graph_format.
- no_init_estimation_check_first_obs¶
Do not check for stochastic singularity in first period. If used, ESTIMATION CHECKS does not return an error if the check fails only in first observation. This should only be used when observing stock variables (e.g. capital) in first period, on top of their associated flow (e.g. investment). Using this option may lead to a crash or provide undesired/wrong results for badly specified problems (e.g. the additional variable observed in first period is not predetermined).
For advanced use only.
- lik_init = INTEGER¶
Type of initialization of Kalman filter:
1For stationary models, the initial matrix of variance of the error of forecast is set equal to the unconditional variance of the state variables.
2For nonstationary models: a wide prior is used with an initial matrix of variance of the error of forecast diagonal with 10 on the diagonal (follows the suggestion of Harvey and Phillips(1979)).
3For nonstationary models: use a diffuse filter (use rather the
diffuse_filteroption).4The filter is initialized with the fixed point of the Riccati equation.
5Use i) option 2 for the non-stationary elements by setting their initial variance in the forecast error matrix to 10 on the diagonal and all covariances to 0 and ii) option 1 for the stationary elements.
Default value is 1. For advanced use only.
- conditional_likelihood¶
Do not use the kalman filter to evaluate the likelihood, but instead evaluate the conditional likelihood, based on the first order reduced form of the model, by assuming that the initial state vector is at its steady state. This approach requires that:
The number of structural innovations be equal to the number of observed variables.
The absence of measurement errors (as introduced by the Dynare interface, see documentation about the
estimated_paramsblock).The absence of missing observations.
The evaluation of the conditional likelihood is faster and more stable than the evaluation of the likelihood with the Kalman filter. Also this approach does not require special treatment for models with unit roots. Note however that the conditional likelihood is sensitive to the choice for the initial condition, which can be an issue if the data are initially far from the steady state. This option is not compatible with
analytic_derivation.
- conf_sig = DOUBLE¶
Level of significance of the confidence interval used for classical forecasting after estimation. Default: 0.9.
- mh_conf_sig = DOUBLE¶
Confidence/HPD interval used for the computation of prior and posterior statistics like: parameter distributions, prior/posterior moments, conditional variance decomposition, impulse response functions, Bayesian forecasting. Default:
0.9.
- mh_replic = INTEGER¶
Number of replications for each chain of the Metropolis-Hastings algorithm. The number of draws should be sufficient to achieve convergence of the MCMC and to meaningfully compute posterior objects. Default:
20000.
- sub_draws = INTEGER¶
Number of draws from the MCMC that are used to compute posterior distribution of various objects (smoothed variable, smoothed shocks, forecast, moments, IRF). The draws used to compute these posterior moments are sampled uniformly in the estimated empirical posterior distribution (i.e. draws of the MCMC).
sub_drawsshould be smaller than the total number of MCMC draws available. Default:min(posterior_max_subsample_draws, (Total number of draws)*(number of chains) ).
- posterior_max_subsample_draws = INTEGER¶
Maximum number of draws from the MCMC used to compute posterior distribution of various objects (smoothed variable, smoothed shocks, forecast, moments, IRF), if not overriden by option
sub_draws. Default:1200.
- mh_nblocks = INTEGER¶
Number of parallel chains for Metropolis-Hastings algorithm. Default:
2.
- mh_drop = DOUBLE¶
The fraction of initially generated parameter vectors to be dropped as a burn-in before using posterior simulations. Default:
0.5.
- mh_jscale = DOUBLE¶
The scale parameter of the jumping distribution’s covariance matrix (Metropolis-Hastings or TaRB-algorithm). This option must be tuned to obtain, ideally, an acceptance ratio of 25%-33%. Basically, the idea is to increase the variance of the jumping distribution if the acceptance ratio is too high, and decrease the same variance if the acceptance ratio is too low. In some situations it may help to consider parameter-specific values for this scale parameter. This can be done in the
estimated_paramsblock.Note that
mode_compute=6will tune the scale parameter to achieve an acceptance rate of AcceptanceRateTarget. The resulting scale parameter will be saved into a file namedMODEL_FILENAME_mh_scale.matin theFILENAME/Outputfolder. This file can be loaded in subsequent runs via theposterior_sampler_optionsoption scale_file. Bothmode_compute=6andscale_filewill overwrite any value specified inestimated_paramswith the tuned value. Default:2.38/sqrt(n).Note also that for the Random Walk Metropolis Hastings algorithm, it is possible to use option
mh_tune_jscale, to automatically tune the value ofmh_jscale. In this case, themh_jscaleoption must not be used.
- mh_init_scale = DOUBLE (deprecated)¶
The scale to be used for drawing the initial value of the Metropolis-Hastings chain. Generally, the starting points should be overdispersed for the Brooks and Gelman (1998) convergence diagnostics to be meaningful. Default:
2*mh_jscale.It is important to keep in mind that
mh_init_scaleis set at the beginning of Dynare execution, i.e. the default will not take into account potential changes inmh_jscaleintroduced by eithermode_compute=6or theposterior_sampler_optionsoption scale_file. Ifmh_init_scaleis too wide during initalization of the posterior sampler so that 100 tested draws are inadmissible (e.g. Blanchard-Kahn conditions are always violated), Dynare will request user input of a newmh_init_scalevalue with which the next 100 draws will be drawn and tested. If thenointeractiveoption has been invoked, the program will instead automatically decreasemh_init_scaleby 10 percent after 100 futile draws and try another 100 draws. This iterative procedure will take place at most 10 times, at which point Dynare will abort with an error message.
- mh_init_scale_factor = DOUBLE¶
The multiple of
mh_jscaleused for drawing the initial value of the Metropolis-Hastings chain. Generally, the starting points should be overdispersed for the Brooks and Gelman (1998) convergence diagnostics to be meaningful. Default:2If
mh_init_scale_factoris too wide during initalization of the posterior sampler so that 100 tested draws are inadmissible (e.g. Blanchard-Kahn conditions are always violated), Dynare will request user input of a newmh_init_scale_factorvalue with which the next 100 draws will be drawn and tested. If thenointeractiveoption has been invoked, the program will instead automatically decreasemh_init_scale_factorby 10 percent after 100 futile draws and try another 100 draws. This iterative procedure will take place at most 10 times, at which point Dynare will abort with an error message.
- mh_tune_jscale [= DOUBLE]¶
Automatically tunes the scale parameter of the jumping distribution’s covariance matrix (Metropolis-Hastings), so that the overall acceptance ratio is close to the desired level. Default value is
0.33. It is not possible to match exactly the desired acceptance ratio because of the stochastic nature of the algorithm (the proposals and the initial conditions of the markov chains ifmh_nblocks>1). This option is only available for the Random Walk Metropolis Hastings algorithm. Must not be used in conjunction withmh_jscale = DOUBLE.
- mh_tune_guess = DOUBLE¶
Specifies the initial value for the
mh_tune_jscaleoption. Default:2.38/sqrt(n). Must not be set ifmh_tune_jscaleis not used.
- mh_recover¶
Attempts to recover a Metropolis-Hastings simulation that crashed prematurely, starting with the last available saved
mh-file. Shouldn’t be used together withload_mh_fileor a differentmh_replicthan in the crashed run. Since Dynare 4.5 the proposal density from the previous run will automatically be loaded. In older versions, to assure a neat continuation of the chain with the same proposal density, you should provide themode_fileused in the previous run or the same user-definedmcmc_jumping_covariancewhen using this option. Note that under Octave, a neat continuation of the crashed chain with the respective last random number generator state is currently not supported.
- mh_posterior_mode_estimation¶
Skip optimizer-based mode-finding and instead compute the mode based on a run of a MCMC. The MCMC will start at the prior mode and use the prior variances to compute the inverse Hessian.
- mode_file = FILENAME¶
Name of the file containing previous value for the mode. When computing the mode, Dynare stores the mode (
xparam1) and the hessian (hh, only ifcova_compute=1) in a file calledMODEL_FILENAME_mode.matin theFILENAME/Outputfolder. After a successful run of the estimation command, themode_filewill be disabled to prevent other function calls from implicitly using an updated mode file. Thus, if the.modfile contains subsequentestimationcommands, themode_fileoption, if desired, needs to be specified again.
- mode_compute = INTEGER | FUNCTION_NAME¶
Specifies the optimizer for the mode computation:
0The mode isn’t computed. When the
mode_fileoption is specified, the mode is simply read from that file.When
mode_fileoption is not specified, Dynare reports the value of the log posterior (log likelihood) evaluated at the initial value of the parameters.When
mode_fileis not specified and there is noestimated_paramsblock, but thesmootheroption is used, it is a roundabout way to compute the smoothed value of the variables of a model with calibrated parameters.1Uses
fminconoptimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge, version 1.6 or above, is installed).2Uses the continuous simulated annealing global optimization algorithm described in Corana et al.(1987) and Goffe et al.(1994).
3Uses
fminuncoptimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed).4Uses Chris Sims’s
csminwel.5Uses Marco Ratto’s
newrat. This value is not compatible with non linear filters or DSGE-VAR models. This is a slice optimizer: most iterations are a sequence of univariate optimization step, one for each estimated parameter or shock. Usescsminwelfor line search in each step.6Uses a Monte-Carlo based optimization routine (see https://archives.dynare.org/DynareWiki/MonteCarloOptimization for more details).
7Uses
fminsearch, a simplex-based optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed).8Uses Dynare implementation of the Nelder-Mead simplex-based optimization routine (generally more efficient than the MATLAB or Octave implementation available with
mode_compute=7).9Uses the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) algorithm of Hansen and Kern (2004), an evolutionary algorithm for difficult non-linear non-convex optimization.
10Uses the
simpsaalgorithm, based on the combination of the non-linear simplex and simulated annealing algorithms as proposed by Cardoso, Salcedo and Feyo de Azevedo (1996).11Currently not in use. The Liu and West (2020) filter that used to be available under this option value is now triggered with
posterior_sampling_method='online'.12Uses the
particleswarmoptimization routine (available under MATLAB if the Global Optimization Toolbox is installed; not available under Octave).13Uses the
lsqnonlinnon-linear least squares optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed). Only supported formethod_of_moments.101Uses the SolveOpt algorithm for local nonlinear optimization problems proposed by Kuntsevich and Kappel (1997).
102Uses
simulannealbndoptimization routine (available under MATLAB if the Global Optimization Toolbox is installed; not available under Octave)FUNCTION_NAMEIt is also possible to give a FUNCTION_NAME to this option, instead of an INTEGER. In that case, Dynare takes the return value of that function as the posterior mode.
Default value is5.
- additional_optimizer_steps = [INTEGER]¶
- additional_optimizer_steps = [INTEGER1:INTEGER2]¶
- additional_optimizer_steps = [INTEGER1 INTEGER2 ...]¶
Vector of additional minimization algorithms run after
mode_compute. Default: no additional optimization iterations.
- silent_optimizer¶
Instructs Dynare to run mode computing/optimization silently without displaying results or saving files in between. Useful when running loops.
- mcmc_jumping_covariance = OPTION¶
Tells Dynare which covariance to use for the proposal density of the MCMC sampler. OPTION can be one of the following:
hessianUses the Hessian matrix computed at the mode.
prior_varianceUses the prior variances. No infinite prior variances are allowed in this case.
identity_matrixUses an identity matrix.
FILENAMELoads an arbitrary user-specified covariance matrix from
FILENAME.mat. The covariance matrix must be saved in a variable namedjumping_covariance, must be square, positive definite, and have the same dimension as the number of estimated parameters.Note that the covariance matrices are still scaled with
mh_jscale. Default value ishessian.
- mode_check¶
Tells Dynare to plot the posterior density for values around the computed mode for each estimated parameter in turn. This is helpful to diagnose problems with the optimizer. Note that for
order>1the likelihood function resulting from the particle filter is not differentiable anymore due to the resampling step. For this reason, themode_checkplot may look wiggly.
- mode_check_neighbourhood_size = DOUBLE¶
Used in conjunction with option
mode_check, gives the width of the window around the posterior mode to be displayed on the diagnostic plots. This width is expressed in percentage deviation. TheInfvalue is allowed, and will trigger a plot over the entire domain (see alsomode_check_symmetric_plots). Default:0.5.
- mode_check_symmetric_plots = INTEGER¶
Used in conjunction with option
mode_check, if set to1, tells Dynare to ensure that the check plots are symmetric around the posterior mode. A value of0allows to have asymmetric plots, which can be useful if the posterior mode is close to a domain boundary, or in conjunction withmode_check_neighbourhood_size = Infwhen the domain in not the entire real line. Default:1.
- mode_check_number_of_points = INTEGER¶
Number of points around the posterior mode where the posterior kernel is evaluated (for each parameter). Default is
20.
- prior_trunc = DOUBLE¶
Probability of extreme values of the prior density in each tail that is ignored when computing bounds for the parameters. Default:
1e-10forposterior_sampling_method=sliceand0otherwise .
- huge_number = DOUBLE¶
Value for replacing infinite values in the definition of (prior) bounds when finite values are required for computational reasons. Default:
1e7.
- load_mh_file¶
Tells Dynare to add to previous Metropolis-Hastings simulations instead of starting from scratch. Since Dynare 4.5 the proposal density from the previous run will automatically be loaded. In older versions, to assure a neat continuation of the chain with the same proposal density, you should provide the
mode_fileused in the previous run or the same user-definedmcmc_jumping_covariancewhen using this option. Shouldn’t be used together withmh_recover. Note that under Octave, a neat continuation of the chain with the last random number generator state of the already present draws is currently not supported.
- load_results_after_load_mh¶
This option is available when loading a previous MCMC run without adding additional draws, i.e. when
load_mh_fileis specified withmh_replic=0. It tells Dynare to load the previously computed convergence diagnostics, marginal data density, and posterior statistics from an existing_resultsfile instead of recomputing them.
- mh_initialize_from_previous_mcmc¶
This option allows to pick initial values for new MCMC from a previous one, where the model specification, the number of estimated parameters, (some) prior might have changed (so a situation where
load_mh_filewould not work). If an additional parameter is estimated, it is automatically initialized from prior_draw. Note that, if this option is used to skip the optimization step, you should use a sampling method which does not require a proposal density, like slice. Otherwise, optimization should always be done beforehand or a mode file with an appropriate posterior covariance matrix should be used.
- mh_initialize_from_previous_mcmc_directory = FILENAME¶
If
mh_initialize_from_previous_mcmcis set, users must provide here the path to the standard FNAME folder from where to load prior definitions and last MCMC values to be used to initialize the new MCMC.Example: if previous project directory is
/my_previous_dirand FNAME ismymodel, users should set the option asmh_initialize_from_previous_mcmc_directory = '/my_previous_dir/mymodel'Dynare will then look for the last record file into
/my_previous_dir/mymodel/metropolis/mymodel_mh_history_<LAST>.matand for the prior definition file into
/my_previous_dir/mymodel/prior/definition.mat
- mh_initialize_from_previous_mcmc_record = FILENAME¶
If
mh_initialize_from_previous_mcmcis set, and whenever the standard file or directory tree is not applicable to load initial values, users may directly provide here the path to the record file from which to load values to be used to initialize the new MCMC.
- mh_initialize_from_previous_mcmc_prior = FILENAME¶
If
mh_initialize_from_previous_mcmcis set, and whenever the standard file or directory tree is not applicable to load initial values, users may directly provide here the path to the prior definition file, to get info in the priors used in previous MCMC.
- optim = (NAME, VALUE, ...)¶
A list of NAME and VALUE pairs. Can be used to set options for the optimization routines. The set of available options depends on the selected optimization routine (i.e. on the value of option
mode_compute):1, 3, 7, 12, 13Available options are given in the documentation of the MATLAB Optimization Toolbox or in Octave’s documentation.
2Available options are:
'initial_step_length'Initial step length. Default:
1.'initial_temperature'Initial temperature. Default:
15.'MaxIter'Maximum number of function evaluations. Default:
100000.'neps'Number of final function values used to decide upon termination. Default:
10.'ns'Number of cycles. Default:
10.'nt'Number of iterations before temperature reduction. Default:
10.'step_length_c'Step length adjustment. Default:
0.1.'TolFun'Stopping criteria. Default:
1e-8.'rt'Temperature reduction factor. Default:
0.1.'verbosity'Controls verbosity of display during optimization, ranging from
0(silent) to3(each function evaluation). Default:14Available options are:
'InitialInverseHessian'Initial approximation for the inverse of the Hessian matrix of the posterior kernel (or likelihood). Obviously this approximation has to be a square, positive definite and symmetric matrix. Default:
'1e-4*eye(nx)', where nx is the number of parameters to be estimated.'MaxIter'Maximum number of iterations. Default:
1000.'NumgradAlgorithm'Possible values are
2,3and5, respectively, corresponding to the two, three and five points formula used to compute the gradient of the objective function (see Abramowitz and Stegun (1964)). Values13and15are more experimental. If perturbations on the right and the left increase the value of the objective function (we minimize this function) then we force the corresponding element of the gradient to be zero. The idea is to temporarily reduce the size of the optimization problem. Default:2.'NumgradEpsilon'Size of the perturbation used to compute numerically the gradient of the objective function. Default:
1e-6.'TolFun'Stopping criteria. Default:
1e-7.'verbosity'Controls verbosity of display during optimization. Set to
0to set to silent. Default:1.'SaveFiles'Controls saving of intermediate results during optimization. Set to
0to shut off saving. Default:1.5Available options are:
'Hessian'Triggers three types of Hessian computations.
0: outer product gradient;1: default Dynare Hessian routine;2: ’mixed’ outer product gradient, where diagonal elements are obtained using second order derivation formula and outer product is used for correlation structure. Both {0} and {2} options require univariate filters, to ensure using maximum number of individual densities and a positive definite Hessian. Both {0} and {2} are quicker than default Dynare numeric Hessian, but provide decent starting values for Metropolis for large models (option {2} being more accurate than {0}). Default:1.'MaxIter'Maximum number of iterations. Default:
1000.'TolFun'Stopping criteria. Default:
1e-5for numerical derivatives,1e-7for analytic derivatives.'robust'Trigger more robust but computationally more expensive line search. Default:
false.'TolGstep'Tolerance parameter used for tuning gradient step. Default: same value as
TolFun.'TolGstepRel'Parameter used for tuning gradient step, governing the tolerance relative to the functions value. Default: not triggered.
'verbosity'Controls verbosity of display during optimization. Set to
0to set to silent. Default:1.'SaveFiles'Controls saving of intermediate results during optimization. Set to
0to shut off saving. Default:1.6Available options are:
'AcceptanceRateTarget'A real number between zero and one. The scale parameter of the jumping distribution is adjusted so that the effective acceptance rate matches the value of option
'AcceptanceRateTarget'. Default:1.0/3.0.'InitialCovarianceMatrix'Initial covariance matrix of the jumping distribution. It is also used to initialize the covariance matrix during recursive updating. Default is
'previous'if optionmode_fileis used,'prior'otherwise. The user can also specify'identity', which will use an identity matrix with a diagonal of 0.1.'nclimb-mh'Number of iterations in the last MCMC (climbing mode). Default:
200000.'ncov-mh'Number of iterations used for updating the covariance matrix of the jumping distribution. Default:
20000.'nscale-mh'Maximum number of iterations used for adjusting the scale parameter of the jumping distribution. Default:
200000.'NumberOfMh'Number of MCMC run sequentially. Default:
3.8Available options are:
'InitialSimplexSize'Initial size of the simplex, expressed as percentage deviation from the provided initial guess in each direction. Default:
.05.'MaxIter'Maximum number of iterations. Default:
5000.'MaxFunEvals'Maximum number of objective function evaluations. No default.
'MaxFunvEvalFactor'Set
MaxFunvEvalsequal toMaxFunvEvalFactortimes the number of estimated parameters. Default:500.'TolFun'Tolerance parameter (w.r.t the objective function). Default:
1e-4.'TolX'Tolerance parameter (w.r.t the instruments). Default:
1e-4.'verbosity'Controls verbosity of display during optimization. Set to
0to set to silent. Default:1.9Available options are:
'CMAESResume'Resume previous run. Requires the
variablescmaes.matfrom the last run. Set to1to enable. Default:0.'MaxIter'Maximum number of iterations.
'MaxFunEvals'Maximum number of objective function evaluations. Default:
Inf.'TolFun'Tolerance parameter (w.r.t the objective function). Default:
1e-7.'TolX'Tolerance parameter (w.r.t the instruments). Default:
1e-7.'verbosity'Controls verbosity of display during optimization. Set to
0to set to silent. Default:1.'SaveFiles'Controls saving of intermediate results during optimization. Set to
0to shut off saving. Default:1.10Available options are:
'EndTemperature'Terminal condition w.r.t the temperature. When the temperature reaches
EndTemperature, the temperature is set to zero and the algorithm falls back into a standard simplex algorithm. Default:0.1.'MaxIter'Maximum number of iterations. Default:
5000.'MaxFunvEvals'Maximum number of objective function evaluations. No default.
'TolFun'Tolerance parameter (w.r.t the objective function). Default:
1e-4.'TolX'Tolerance parameter (w.r.t the instruments). Default:
1e-4.'verbosity'Controls verbosity of display during optimization. Set to
0to set to silent. Default:1.101Available options are:
'LBGradientStep'Lower bound for the stepsize used for the difference approximation of gradients. Default:
1e-11.'MaxIter'Maximum number of iterations. Default:
15000'SpaceDilation'Coefficient of space dilation. Default:
2.5.'TolFun'Tolerance parameter (w.r.t the objective function). Default:
1e-6.'TolX'Tolerance parameter (w.r.t the instruments). Default:
1e-6.'verbosity'Controls verbosity of display during optimization. Set to
0to set to silent. Default:1.102Available options are given in the documentation of the MATLAB Global Optimization Toolbox.
Example
To change the defaults of
csminwel(mode_compute=4):estimation(..., mode_compute=4,optim=('NumgradAlgorithm',3,'TolFun',1e-5),...);
- nodiagnostic¶
Does not compute the convergence diagnostics for Metropolis-Hastings. Default: diagnostics are computed and displayed.
- bayesian_irf¶
Triggers the computation of the posterior distribution of IRFs. The length of the IRFs are controlled by the
irfoption. Results are stored inoo_.PosteriorIRF.dsge(see below for a description of this variable). Not compatible with OccBin.
- relative_irf
See
relative_irf.
- dsge_var = DOUBLE¶
Triggers the estimation of a DSGE-VAR model, where the weight of the DSGE prior of the VAR model is calibrated to the value passed (see Del Negro and Schorfheide (2004)). It represents the ratio of dummy over actual observations. To assure that the prior is proper, the value must be bigger than \((k+n)/T\), where \(k\) is the number of estimated parameters, \(n\) is the number of observables, and \(T\) is the number of observations.
NB: The previous method of declaring
dsge_prior_weightas a parameter and then calibrating it is now deprecated and will be removed in a future release of Dynare. Some of objects arising during estimation are stored with their values at the mode inoo_.dsge_var.posterior_mode.
- dsge_var¶
Triggers the estimation of a DSGE-VAR model, where the weight of the DSGE prior of the VAR model will be estimated (as in Adjemian et al.(2008)). The prior on the weight of the DSGE prior,
dsge_prior_weight, must be defined in theestimated_paramssection.NB: The previous method of declaring
dsge_prior_weightas a parameter and then placing it inestimated_paramsis now deprecated and will be removed in a future release of Dynare.
- dsge_varlag = INTEGER¶
The number of lags used to estimate a DSGE-VAR model. Default:
4.
- posterior_sampling_method = NAME¶
Selects the sampler used to sample from the posterior distribution during Bayesian estimation. Default:
’random_walk_metropolis_hastings’.'random_walk_metropolis_hastings'Instructs Dynare to use the Random-Walk Metropolis-Hastings. In this algorithm, the proposal density is recentered to the previous draw in every step.
'dime_mcmc'Instructs Dynare to use the Differential-Independence Mixture Ensemble (“DIME”) MCMC sampler of Boehl (2022) instead of the standard Random-Walk Metropolis-Hastings. DIME is robust for odd shaped, multimodal, black-box distributions and shown to require significantly less likelihood evaluations than alternative samplers. Many chains run simultaneously, thereby further increasing sampling speed. The algorithm is based on a gradient-free global multi-start optimizer and does not require any posterior mode density maximization prior to MCMC sampling. DIME proposals are generated from an endogenous and adaptive proposal distribution, thereby providing close-to-optimal proposal distributions for black box target distributions without manual fine-tuning. Does not yet support
moments_varendo,bayesian_irf, andsmoother.Note that, since DIME is using parameter transformations, setting parameter bounds is often counterproductive. The
prior_truncoption is disabled and set to zero.'tailored_random_block_metropolis_hastings'Instructs Dynare to use the Tailored randomized block (TaRB) Metropolis-Hastings algorithm proposed by Chib and Ramamurthy (2010) instead of the standard Random-Walk Metropolis-Hastings. In this algorithm, at each iteration the estimated parameters are randomly assigned to different blocks. For each of these blocks a mode-finding step is conducted. The inverse Hessian at this mode is then used as the covariance of the proposal density for a Random-Walk Metropolis-Hastings step. If the numerical Hessian is not positive definite, the generalized Cholesky decomposition of Schnabel and Eskow (1990) is used, but without pivoting. The TaRB-MH algorithm massively reduces the autocorrelation in the MH draws and thus reduces the number of draws required to representatively sample from the posterior. However, this comes at a computational cost as the algorithm takes more time to run.
'independent_metropolis_hastings'Use the Independent Metropolis-Hastings algorithm where the proposal distribution - in contrast to the Random Walk Metropolis-Hastings algorithm - does not depend on the state of the chain.
'slice'Instructs Dynare to use the Slice sampler of Planas, Ratto, and Rossi (2015). Note that
'slice'is incompatible withprior_trunc=0.Whereas one Metropolis-Hastings iteration requires one evaluation of the posterior, one slice iteration requires \(neval\) evaluations, where as a rule of thumb \(neval=7\times npar\) with \(npar\) denoting the number of estimated parameters. Spending the same computational budget of \(N\) posterior evaluations in the slice sampler then implies setting
mh_replic=N/neval.Note that the slice sampler will typically return less autocorrelated Monte Carlo Markov Chain draws than the MH-algorithm. Its relative (in)efficiency can be investigated via the reported inefficiency factors.
'hssmc'Instructs Dynare to use the Herbst and Schorfheide (2014) version of the Sequential Monte-Carlo sampler instead of the standard Random-Walk Metropolis-Hastings. Does not yet support
moments_varendo,bayesian_irf, andsmoother.'dsmh'Instructs Dynare to use the Dynamic Striated Metropolis Hastings sampler proposed by Waggoner, Wu and Zha (2016) instead of the standard Random-Walk Metropolis-Hastings.
'online'Instructs Dynare to treat the (estimated) parameters as state variables and estimate them jointly with the original state variables of the model using a nonlinear filter. The algorithm implemented in Dynare is described in Liu and West (2001), and works with
korder local approximations of the model.
- posterior_sampler_options = (NAME, VALUE, ...)¶
A list of NAME and VALUE pairs. Can be used to set options for the posterior sampling methods. The set of available options depends on the selected posterior sampling routine (i.e. on the value of option
posterior_sampling_method):'random_walk_metropolis_hastings'Available options are:
'proposal_distribution'Specifies the statistical distribution used for the proposal density.
'rand_multivariate_normal'Use a multivariate normal distribution. This is the default.
'rand_multivariate_student'Use a multivariate student distribution.
'student_degrees_of_freedom'Specifies the degrees of freedom to be used with the multivariate student distribution. Default:
3.'use_mh_covariance_matrix'Indicates to use the covariance matrix of the draws from a previous MCMC run to define the covariance of the proposal distribution. Requires the
load_mh_fileoption to be specified. Default:0.'scale_file'Provides the name of a
_mh_scale.matfile storing the tuned scale factor from a previous run ofmode_compute=6.'save_tmp_file'Save the MCMC draws into a
_mh_tmp_blckfile at the refresh rate of the status bar instead of just saving the draws when the current_mh*_blckfile is full. Default:0'dime_mcmc'Available options are:
'niter'Number of iterations. Default value is 1500.
'nchain'Number of chains/particles. A range between \(4 ndim\) and \(6 ndim\) is recommended. For very difficult problems, double the number of chains. Default value is: \(5 ndim\).
'tune'Number of ensemble iterations to keep after burnin. Default value is chosen to obtain at least 50,000 samples.
'aimh_prob'Probability to draw a global transition kernel. By default this is set to \(0.1\). It is usually not necessary to change this value.
'df_proposal_dist'Degrees of freedom of the multivariate t distribution used for global kernel proposals. Defaults to \(10\).
'rho'Decay parameter for the mean and covariances of the global transistion kernel. Defaults to \(0.999\).
'gamma'Mean stretch factor for the proposal vector. By default, it is \(2.38 / \sqrt{2\,\mathrm{ndim}}\) as recommended by ter Braak (2006)
'sigma'Standard deviation of the Gaussian used to stretch the proposal vector. This is normally negligible. Defaults to \(1e-5\).
'independent_metropolis_hastings'Takes the same options as in the case of
random_walk_metropolis_hastings.'slice'Available options are:
'rotated'Triggers rotated slice iterations using a covariance matrix from initial burn-in iterations. Requires either
use_mh_covariance_matrixorslice_initialize_with_mode. Default:0.'mode_files'For multimodal posteriors, provide the name of a file containing a
nparambynmodesvariable calledxparamsstoring the different modes. This array must have one column vector per mode and the estimated parameters along the row dimension. With this info, the code will automatically trigger therotatedandmodeoptions. Default:[].'slice_initialize_with_mode'The default for slice is to set
mode_compute=0and start the chain(s) from a random location in the prior space. This option first runs the mode-finder and then starts the chain from the mode. Together withrotated, it will use the inverse Hessian from the mode to perform rotated slice iterations. Default:0.'initial_step_size'Sets the initial size of the interval in the stepping-out procedure as fraction of the prior support, i.e. the size will be
initial_step_size * (UB-LB).initial_step_sizemust be a real number in the interval[0,1]. Default:0.8.'use_mh_covariance_matrix'See use_mh_covariance_matrix. Must be used with
'rotated'. Default:0.'save_tmp_file'See save_tmp_file. Default:
1.'tailored_random_block_metropolis_hastings'Available options are:
'proposal_distribution'Specifies the statistical distribution used for the proposal density. See proposal_distribution.
new_block_probability = DOUBLESpecifies the probability of the next parameter belonging to a new block when the random blocking in the TaRB Metropolis-Hastings algorithm is conducted. The higher this number, the smaller is the average block size and the more random blocks are formed during each parameter sweep. Default:
0.25.mode_compute = INTEGERSpecifies the mode-finder run in every iteration for every block of the TaRB Metropolis-Hastings algorithm. See
mode_compute. Default:4.optim = (NAME, VALUE,...)Specifies the options for the mode-finder used in the TaRB Metropolis-Hastings algorithm. See
optim.'scale_file'See scale_file..
'save_tmp_file'See save_tmp_file. Default:
1.'hssmc'Available options are:
'particles'Number of particles. Default value is: 20000.
'steps'Number of weights \(\phi_i\in[0,1]\) on the likelihood function used to define a sequence of tempered likelihoods. This parameter is denoted \(N_{\phi}\) in Herbst and Schorfheide (2014), and we have \(\phi_1=0\) and \(\phi_{N_\phi}=1\). Default value is: 25.
'lambda'Positive parameter controling the sequence of weights \(\phi_i\), Default value is: 2. Weights are defined by:
\[\phi_i = \left(\frac{i-1}{N_{\phi}-1}\right)^{\lambda}\]for \(i=1,\ldots,N_{\phi}\). Usually we set \(\lambda>1\), so that \(\Delta \phi_i = \phi_i-\phi_{i-1}\) is increasing with \(i\).
'target'Acceptance rate target. Default value is: .25.
'scale'Scale parameter in the mutation step (on the proposal covariance matrix of the MH iteration). Default value is: .5.
- moments_varendo¶
Triggers the computation of the posterior distribution of the theoretical moments of the endogenous variables. Results are stored in
oo_.PosteriorTheoreticalMoments(seeoo_.PosteriorTheoreticalMoments). The number of lags in the autocorrelation function is controlled by thearoption. Not compatible with OccBin.
- contemporaneous_correlation
See
contemporaneous_correlation. Results are stored inoo_.PosteriorTheoreticalMoments. Note that thenocorroption has no effect.
- no_posterior_kernel_density¶
Shuts off the computation of the kernel density estimator for the posterior objects (see density field).
- conditional_variance_decomposition = INTEGER
- conditional_variance_decomposition = [INTEGER1:INTEGER2]
- conditional_variance_decomposition = [INTEGER1 INTEGER2 ...]
Computes the posterior distribution of the conditional variance decomposition for the specified period(s). The periods must be strictly positive. Conditional variances are given by \(var(y_{t+k}\vert t)\). For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact. The results are stored in
oo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecomposition.. Note that this option requires the optionmoments_varendoto be specified. In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables. The variance decomposition of the measured variables will be stored inoo_.PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecompositionME.
- filtered_vars¶
Triggers the computation of the posterior distribution of filtered endogenous variables/one-step ahead forecasts, i.e. \(E_{t}{y_{t+1}}\). Results are stored in
oo_.FilteredVariables(see below for a description of this variable)
- smoother
Triggers the computation of the posterior distribution of smoothed endogenous variables and shocks, i.e. the expected value of variables and shocks given the information available in all observations up to the final date (\(E_{T}{y_t}\)). Results are stored in
oo_.SmoothedVariables,oo_.SmoothedShocksandoo_.SmoothedMeasurementErrors. Also triggers the computation ofoo_.UpdatedVariables, which contains the estimation of the expected value of variables given the information available at the current date (\(E_{t}{y_t}\)). See below for a description of all these variables.
- smoother_redux¶
Triggers a faster computation of the smoothed endogenous variables and shocks for large models. It runs the smoother only for the state variables (i.e. with the same representation used for likelihood computations) and computes the remaining variables ex-post. Static unobserved objects (filtered, smoothed, updated, k-step ahead) are recovered, but there are exceptions to a full recovery, depending on how static unobserved variables depend on the restricted state space adopted. For example, lagged shocks which are ONLY used to recover NON-observed static variables will not be recovered). For such exceptions, only the following output is provided:
FilteredVariablesKStepAhead: will be fully recoveredSmoothedVariables,FilteredVariables,UpdatedVariables: recovered for all periods beyond periodd+1,where
ddenotes the number of diffuse filtering steps.
FilteredVariablesKStepAheadVariances,Variance, andState_uncertaintycannot be recovered, and ZERO is provided as output.If you need variances for those variables, either do not set the option, or declare the variable as observed, using NaNs as data points.
- forecast = INTEGER¶
Computes the posterior distribution of a forecast on INTEGER periods after the end of the sample used in estimation. If no Metropolis-Hastings is computed, the result is stored in variable
oo_.forecastand corresponds to the forecast at the posterior mode. If a Metropolis-Hastings is computed, the distribution of forecasts is stored in variablesoo_.PointForecastandoo_.MeanForecast. See Forecasting, for a description of these variables. Not compatible with OccBin.
- tex
See
tex.
- kalman_algo = INTEGER¶
0Automatically use the Multivariate Kalman Filter for stationary models and the Multivariate Diffuse Kalman Filter for non-stationary models.
1Use the Multivariate Kalman Filter.
2Use the Univariate Kalman Filter.
3Use the Multivariate Diffuse Kalman Filter.
4Use the Univariate Diffuse Kalman Filter.
Default value is
0. In case of missing observations of single or all series, Dynare treats those missing values as unobserved states and uses the Kalman filter to infer their value (see e.g. Durbin and Koopman (2012), Ch. 4.10) This procedure has the advantage of being capable of dealing with observations where the forecast error variance matrix becomes singular for some variable(s). If this happens, the respective observation enters with a weight of zero in the log-likelihood, i.e. this observation for the respective variable(s) is dropped from the likelihood computations (for details see Durbin and Koopman (2012), Ch. 6.4 and 7.2.5 and Koopman and Durbin (2000)). If the use of a multivariate Kalman filter is specified and a singularity is encountered, Dynare by default automatically switches to the univariate Kalman filter for this parameter draw. This behavior can be changed via theuse_univariate_filters_if_singularity_is_detectedoption.
- fast_kalman_filter¶
Select the fast Kalman filter using Chandrasekhar recursions as described by
Herbst (2015). This setting is only used withkalman_algo=1orkalman_algo=3. In case of using the diffuse Kalman filter (kalman_algo=3/lik_init=3), the observables must be stationary. This option is not yet compatible withanalytic_derivation.
- kalman_tol = DOUBLE¶
Numerical tolerance for determining the singularity of the covariance matrix of the prediction errors during the Kalman filter (minimum allowed reciprocal of the matrix condition number). Default value is
1e-10.
- diffuse_kalman_tol = DOUBLE¶
Numerical tolerance for determining the singularity of the covariance matrix of the prediction errors (\(F_{\infty}\)) and the rank of the covariance matrix of the non-stationary state variables (\(P_{\infty}\)) during the Diffuse Kalman filter. Default value is
1e-6.
- filter_covariance¶
Saves the series of one step ahead error of forecast covariance matrices. With Metropolis, they are saved in
oo_.FilterCovariance, otherwise inoo_.Smoother.Variance. Saves also k-step ahead error of forecast covariance matrices iffilter_step_aheadis set.
- filter_step_ahead = [INTEGER1:INTEGER2]¶
- filter_step_ahead = [INTEGER1 INTEGER2 ...]¶
Triggers the computation k-step ahead filtered values, i.e. \(E_{t}{y_{t+k}}\). Stores results in
oo_.FilteredVariablesKStepAhead. Also stores 1-step ahead values inoo_.FilteredVariables.oo_.FilteredVariablesKStepAheadVariancesis stored iffilter_covariance.
- filter_decomposition¶
Triggers the computation of the shock decomposition of the above k-step ahead filtered values. Stores results in
oo_.FilteredVariablesShockDecomposition.
- smoothed_state_uncertainty¶
Triggers the computation of the variance of smoothed estimates, i.e. \(var_T(y_t)\). Stores results in
oo_.Smoother.State_uncertainty.
- diffuse_filter¶
Uses the diffuse Kalman filter (as described in Durbin and Koopman (2012) and Koopman and Durbin (2003) for the multivariate and Koopman and Durbin (2000) for the univariate filter) to estimate models with non-stationary observed variables. This option will also reset the
qz_criteriumto count unit root variables towards the stable variables. Trying to estimate a model with unit roots will otherwise result in a Blanchard-Kahn error.When
diffuse_filteris used thelik_initoption ofestimationhas no effect.When there are nonstationary exogenous variables in a model, there is no unique deterministic steady state. For instance, if productivity is a pure random walk:
\[a_t = a_{t-1} + e_t\]any value of \(\bar a\) of \(a\) is a deterministic steady state for productivity. Consequently, the model admits an infinity of steady states. In this situation, the user must help Dynare in selecting one steady state, except if zero is a trivial model’s steady state, which happens when the
linearoption is used in the model declaration. The user can either provide the steady state to Dynare using asteady_state_modelblock (or writing a steady state file) if a closed form solution is available, seesteady_state_model, or specify some constraints on the steady state, see equation_tag_for_conditional_steady_state, so that Dynare computes the steady state conditionally on some predefined levels for the non stationary variables. In both cases, the idea is to use dummy values for the steady state level of the exogenous non stationary variables.Note that the nonstationary variables in the model must be integrated processes (their first difference or k-difference must be stationary).
- heteroskedastic_filter¶
Runs filter, likelihood, and smoother using heteroskedastic definitions provided in a
heteroskedastic_shocksblock.
- selected_variables_only¶
Only run the classical smoother on the variables listed just after the
estimationcommand. This option is incompatible with requesting classical frequentist forecasts and will be overridden in this case. When using Bayesian estimation, the smoother is by default only run on the declared endogenous variables. Default: run the smoother on all the declared endogenous variables.
- cova_compute = INTEGER¶
When
0, the covariance matrix of estimated parameters is not computed after the computation of posterior mode (or maximum likelihood). This increases speed of computation in large models during development, when this information is not always necessary. Of course, it will break all successive computations that would require this covariance matrix. Otherwise, if this option is equal to1, the covariance matrix is computed and stored in variablehhofMODEL_FILENAME_mode.mat. Default is1.
- solve_algo = INTEGER
See solve_algo.
- order = INTEGER
Order of approximation around the deterministic steady state. When greater than 1, the likelihood is evaluated with a particle or nonlinear filter (see Fernández-Villaverde and Rubio-Ramírez (2005)). Default is
1, i.e. the likelihood of the linearized model is evaluated using a standard Kalman filter.
- irf = INTEGER
See
irf. Only used ifbayesian_irfis passed.
- irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )
See
irf_shocks. Only used ifbayesian_irfis passed.
- irf_plot_threshold = DOUBLE
See
irf_plot_threshold. Only used ifbayesian_irfis passed.
- aim_solver
See
aim_solver.
- dr = OPTION
See
dr. Default:default, i.e. generalized Schur decomposition.
- dr_cycle_reduction_tol = DOUBLE
See
dr_cycle_reduction_tol. Default:1e-7.
- dr_cycle_reduction_maxiter = INTEGER
See
dr_cycle_reduction_maxiter. Default:100.
- dr_logarithmic_reduction_tol = DOUBLE
See
dr_logarithmic_reduction_tol. Default:1e-12.
- dr_logarithmic_reduction_maxiter = INTEGER
See
dr_logarithmic_reduction_maxiter. Default:100.
- lyapunov = OPTION¶
Determines the algorithm used to solve the Lyapunov equation to initialized the variance-covariance matrix of the Kalman filter using the steady-state value of state variables. Possible values for OPTION are:
defaultUses the default solver for Lyapunov equations based on Bartels-Stewart algorithm.
fixed_pointUses a fixed point algorithm to solve the Lyapunov equation. This method is faster than the
defaultone for large scale models, but it could require a large amount of iterations.doublingUses a doubling algorithm to solve the Lyapunov equation (
disclyap_fast). This method is faster than the two previous one for large scale models.square_root_solverUses a square-root solver for Lyapunov equations (
dlyapchol). This method is fast for large scale models (available under MATLAB if the Control System Toolbox is installed; available under Octave if the control package from Octave-Forge is installed)Default value is
default.
- lyapunov_fixed_point_tol = DOUBLE¶
This is the convergence criterion used in the fixed point Lyapunov solver. Its default value is
1e-10.
- lyapunov_doubling_tol = DOUBLE¶
This is the convergence criterion used in the doubling algorithm to solve the Lyapunov equation. Its default value is
1e-16.
- use_penalized_objective_for_hessian¶
Use the penalized objective instead of the objective function to compute numerically the hessian matrix at the mode. The penalties decrease the value of the posterior density (or likelihood) when, for some perturbations, Dynare is not able to solve the model (issues with steady state existence, Blanchard and Kahn conditions, …). In pratice, the penalized and original objectives will only differ if the posterior mode is found to be near a region where the model is ill-behaved. By default the original objective function is used.
- analytic_derivation¶
Triggers estimation with analytic gradient at
order=1. The final hessian at the mode is also computed analytically. Only works for stationary models without missing observations, i.e. forkalman_algo<3. Optimizers that rely on analytic gradients aremode_compute=1,3,4,5,101.
- ar = INTEGER
See
ar. Only useful in conjunction with optionmoments_varendo.
- endogenous_prior¶
Use endogenous priors as in Christiano, Trabandt and Walentin (2011). The procedure is motivated by sequential Bayesian learning. Starting from independent initial priors on the parameters, specified in the
estimated_paramsblock, the standard deviations observed in a “pre-sample”, taken to be the actual sample, are used to update the initial priors. Thus, the product of the initial priors and the pre-sample likelihood of the standard deviations of the observables is used as the new prior (for more information, see the technical appendix of Christiano, Trabandt and Walentin (2011)). This procedure helps in cases where the regular posterior estimates, which minimize in-sample forecast errors, result in a large overprediction of model variable variances (a statistic that is not explicitly targeted, but often of particular interest to researchers).
- use_univariate_filters_if_singularity_is_detected = INTEGER¶
Decide whether Dynare should automatically switch to univariate filter if a singularity is encountered in the likelihood computation (this is the behaviour if the option is equal to
1). Alternatively, if the option is equal to0, Dynare will not automatically change the filter, but rather use a penalty value for the likelihood when such a singularity is encountered. Default:1.
- keep_kalman_algo_if_singularity_is_detected¶
With the default
use_univariate_filters_if_singularity_is_detected=1, Dynare will switch to the univariate Kalman filter when it encounters a singular forecast error variance matrix during Kalman filtering. Upon encountering such a singularity for the first time, all subsequent parameter draws and computations will automatically rely on univariate filter, i.e. Dynare will never try the multivariate filter again. Use thekeep_kalman_algo_if_singularity_is_detectedoption to have theuse_univariate_filters_if_singularity_is_detectedonly affect the behavior for the current draw/computation.
- rescale_prediction_error_covariance¶
Rescales the prediction error covariance in the Kalman filter to avoid badly scaled matrix and reduce the probability of a switch to univariate Kalman filters (which are slower). By default no rescaling is done.
- qz_zero_threshold = DOUBLE
See
qz_zero_threshold.
- taper_steps = [INTEGER1 INTEGER2 ...]¶
Percent tapering used for the spectral window in the Geweke (1992,1999) convergence diagnostics (requires
mh_nblocks=1). The tapering is used to take the serial correlation of the posterior draws into account. Default:[4 8 15].
- brooks_gelman_plotrows = INTEGER¶
Number of parameters to depict along the rows of the figures depicting the Brooks and Gelman (1998) convergence diagnostics. Default: 3.
- geweke_interval = [DOUBLE DOUBLE]¶
Percentage of MCMC draws at the beginning and end of the MCMC chain taken to compute the Geweke (1992,1999) convergence diagnostics (requires
mh_nblocks=1) after discarding the firstmh_drop = DOUBLEpercent of draws as a burnin. Default: [0.2 0.5].
- raftery_lewis_diagnostics¶
Triggers the computation of the Raftery and Lewis (1992) convergence diagnostics. The goal is deliver the number of draws required to estimate a particular quantile of the CDF
qwith precisionrwith a probabilitys. Typically, one wants to estimate theq=0.025percentile (corresponding to a 95 percent HPDI) with a precision of 0.5 percent (r=0.005) with 95 percent certainty (s=0.95). The defaults can be changed viaraftery_lewis_qrs. Based on the theory of first order Markov Chains, the diagnostics will provide a required burn-in (M), the number of draws after the burnin (N) as well as a thinning factor that would deliver a first order chain (k). The last line of the table will also deliver the maximum over all parameters for the respective values.
- raftery_lewis_qrs = [DOUBLE DOUBLE DOUBLE]¶
Sets the quantile of the CDF
qthat is estimated with precisionrwith a probabilitysin the Raftery and Lewis (1992) convergence diagnostics. Default:[0.025 0.005 0.95].
- consider_all_endogenous¶
Compute the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) on all the endogenous variables. This is equivalent to manually listing all the endogenous variables after the
estimationcommand.
- consider_all_endogenous_and_auxiliary¶
Compute the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) on all the endogenous variables and the auxiliary variables introduced by the preprocessor. This option is useful when e.g. running
smoother2histvalon the results of the Kalman smoother.
- consider_only_observed¶
Compute the posterior moments, smoothed variables, k-step ahead filtered variables and forecasts (when requested) on all the observed variables. This is equivalent to manually listing all the observed variables after the
estimationcommand.
- number_of_particles = INTEGER¶
Number of particles used when evaluating the likelihood of a non linear state space model. Default:
1000.
- resampling = OPTION¶
Determines if resampling of the particles is done. Possible values for OPTION are:
noneNo resampling.
systematicResampling at each iteration, this is the default value.
genericResampling if and only if the effective sample size is below a certain level defined by
resampling_threshold*number_of_particles.
- resampling_threshold = DOUBLE¶
A real number between zero and one. The resampling step is triggered as soon as the effective number of particles is less than this number times the total number of particles (as set by
number_of_particles). This option is effective if and only if optionresamplinghas valuegeneric.
- resampling_method = OPTION¶
Sets the resampling method. Possible values for OPTION are:
kitagawa,stratifiedandsmooth.
- filter_algorithm = OPTION¶
Sets the particle filter algorithm. Possible values for OPTION are:
sisSequential importance sampling algorithm, this is the default value.
apfAuxiliary particle filter.
gfGaussian filter.
gmfGaussian mixture filter.
cpfConditional particle filter.
nlkfUse a standard (linear) Kalman filter algorithm with the nonlinear measurement and state equations.
- proposal_approximation = OPTION¶
Sets the method for approximating the proposal distribution. Possible values for OPTION are:
cubature,montecarloandunscented. Default value isunscented.
- distribution_approximation = OPTION¶
Sets the method for approximating the particle distribution. Possible values for OPTION are:
cubature,montecarloandunscented. Default value isunscented.
- cpf_weights = OPTION¶
Controls the method used to update the weights in conditional particle filter, possible values are
amisanotristani(Amisano et al. (2010)) ormurrayjonesparslow(Murray et al. (2013)). Default value isamisanotristani.
- nonlinear_filter_initialization = INTEGER¶
Sets the initial condition of the nonlinear filters. By default the nonlinear filters are initialized with the unconditional covariance matrix of the state variables, computed with the reduced form solution of the first order approximation of the model. If
nonlinear_filter_initialization=2, the nonlinear filter is instead initialized with a covariance matrix estimated with a stochastic simulation of the reduced form solution of the second order approximation of the model. Both these initializations assume that the model is stationary, and cannot be used if the model has unit roots (which can be seen with thecheckcommand prior to estimation). If the model has stochastic trends, user must usenonlinear_filter_initialization=3, the filters are then initialized with an identity matrix for the covariance matrix of the state variables. Default value isnonlinear_filter_initialization=1(initialization based on the first order approximation of the model).
- particle_filter_options = (NAME, VALUE, ...)¶
A list of NAME and VALUE pairs. Can be used to set some fine-grained options for the particle filter routines. The set of available options depends on the selected filter routine.
More information on particle filter options is available at https://git.dynare.org/Dynare/dynare/-/wikis/Particle-filters.
Available options are:
'pruning'Enable pruning for particle filter-related simulations. Default:
false.'liu_west_delta'Set the value for delta for the Liu/West online filter (
posterior_sampling_method='online'). Default:0.99.'unscented_alpha'Set the value for alpha for unscented transforms. Default:
1.'unscented_beta'Set the value for beta for unscented transforms. Default:
2.'unscented_kappa'Set the value for kappa for unscented transforms. Default:
1.'initial_state_prior_std'Value of the diagonal elements for the initial covariance of the state variables when employing
nonlinear_filter_initialization=3. Default:1.'mixture_state_variables'Number of mixture components in the Gaussian-mixture filter (gmf) for the state variables. Default:
5.'mixture_structural_shocks'Number of mixture components in the Gaussian-mixture filter (gmf) for the structural shocks. Default:
1.'mixture_measurement_shocks'Number of mixture components in the Gaussian-mixture filter (gmf) for the measurement errors. Default:
1.
Note
If no
mh_jscaleparameter is used for a parameter inestimated_params, the procedure usesmh_jscalefor all parameters. Ifmh_jscaleoption isn’t set, the procedure uses0.2for all parameters. Note that ifmode_compute=6is used or theposterior_sampler_optioncalledscale_fileis specified, the values set inestimated_paramswill be overwritten.“Endogenous” prior restrictions
It is also possible to impose implicit “endogenous” priors about IRFs and moments on the model during estimation. For example, one can specify that all valid parameter draws for the model must generate fiscal multipliers that are bigger than 1 by specifying how the IRF to a government spending shock must look like. The prior restrictions can be imposed via
irf_calibrationandmoment_calibrationblocks (see IRF/Moment calibration). The way it works internally is that any parameter draw that is inconsistent with the “calibration” provided in these blocks is discarded, i.e. assigned a prior density of 0. When specifying these blocks, it is important to keep in mind that one won’t be able to easily domodel_comparisonin this case, because the prior density will not integrate to 1.Output
After running estimation, the parameters
M_.paramsand the variance matrixM_.Sigma_eof the shocks are set to the mode for maximum likelihood estimation or posterior mode computation without Metropolis iterations. After estimation with Metropolis iterations (optionmh_replic > 0or optionload_mh_fileset) the parametersM_.paramsand the variance matrixM_.Sigma_eof the shocks are set to the posterior mean.Depending on the options,
estimationstores results in various fields of theoo_structure, described below. In the following variables, we will adopt the following shortcuts for specific field names:MOMENT_NAMEThis field can take the following values:
HPDinfLower bound of a 90% HPD interval. [4]
HPDsupUpper bound of a 90% HPD interval.
HPDinf_MELower bound of a 90% HPD interval [5] for observables when taking measurement error into account (see e.g. Christoffel et al. (2010), p.17).
HPDsup_MEUpper bound of a 90% HPD interval for observables when taking measurement error into account.
MeanMean of the posterior distribution.
MedianMedian of the posterior distribution.
StdStandard deviation of the posterior distribution.
VarianceVariance of the posterior distribution.
decilesDeciles of the distribution.
densityNon parametric estimate of the posterior density following the approach outlined in Skoeld and Roberts (2003). First and second columns are respectively abscissa and ordinate coordinates.
ESTIMATED_OBJECTThis field can take the following values:
measurement_errors_corrCorrelation between two measurement errors.
measurement_errors_stdStandard deviation of measurement errors.
parametersParameters.
shocks_corrCorrelation between two structural shocks.
shocks_stdStandard deviation of structural shocks.
- MATLAB/Octave variable: oo_.MarginalDensity.LaplaceApproximation¶
Variable set by the
estimationcommand. Stores the marginal data density based on the Laplace Approximation.
- MATLAB/Octave variable: oo_.MarginalDensity.ModifiedHarmonicMean¶
Variable set by the
estimation command, if it is used withmh_replic > 0orload_mh_fileoption. Stores the marginal data density based on Geweke (1999) Modified Harmonic Mean estimator.
- MATLAB/Octave variable: oo_.posterior.optimization¶
Variable set by the
estimationcommand if mode-finding is used. Stores the results at the mode. Fields are of the form:oo_.posterior.optimization.OBJECT
where OBJECT is one of the following:
modeParameter vector at the mode.
VarianceInverse Hessian matrix at the mode or MCMC jumping covariance matrix when used with the
MCMC_jumping_covarianceoption.log_densityLog likelihood (ML)/log posterior density (Bayesian) at the mode when used with
mode_compute>0.
- MATLAB/Octave variable: oo_.posterior.metropolis¶
Variable set by the
estimationcommand ifmh_replic>0is used. Fields are of the form:oo_.posterior.metropolis.OBJECT
where OBJECT is one of the following:
meanMean parameter vector from the MCMC.
VarianceCovariance matrix of the parameter draws in the MCMC.
- MATLAB/Octave variable: oo_.FilteredVariables¶
Variable set by the
estimationcommand, if it is used with thefiltered_varsoption.After an estimation without Metropolis, fields are of the form:
oo_.FilteredVariables.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form:
oo_.FilteredVariables.MOMENT_NAME.VARIABLE_NAME
- MATLAB/Octave variable: oo_.FilteredVariablesKStepAhead¶
Variable set by the
estimationcommand, if it is used with thefilter_step_aheadoption. The k-steps are stored along the rows while the columns indicate the respective variables. The third dimension of the array provides the observation for which the forecast has been made. For example, iffilter_step_ahead=[1 2 4]andnobs=200, the element (3,5,204) stores the four period ahead filtered value of variable 5 computed at time t=200 for time t=204. The periods at the beginning and end of the sample for which no forecasts can be made, e.g. entries (1,5,1) and (1,5,204) in the example, are set to zero. Note that in case of Bayesian estimation the variables will be ordered in the order of declaration after the estimation command (or in general declaration order if no variables are specified here). In case of running the classical smoother, the variables will always be ordered in general declaration order. If theselected_variables_onlyoption is specified with the classical smoother, non-requested variables will be simply left out in this order.
- MATLAB/Octave variable: oo_.FilteredVariablesKStepAheadVariances¶
Variable set by the
estimationcommand, if it is used with thefilter_step_ahead option. It is a 4 dimensional array where the k-steps are stored along the first dimension, while the fourth dimension of the array provides the observation for which the forecast has been made. The second and third dimension provide the respective variables. For example, iffilter_step_ahead=[1 2 4]andnobs=200, the element (3,4,5,204) stores the four period ahead forecast error covariance between variable 4 and variable 5, computed at time t=200 for time t=204. Padding with zeros and variable ordering is analogous tooo_.FilteredVariablesKStepAhead.
- MATLAB/Octave variable: oo_.Filtered_Variables_X_step_ahead¶
Variable set by the
estimationcommand, if it is used with thefilter_step_ahead optionin the context of Bayesian estimation. Fields are of the form:oo_.Filtered_Variables_X_step_ahead.VARIABLE_NAME
The n-th entry stores the k-step ahead filtered variable computed at time n for time n+k.
- MATLAB/Octave variable: oo_.FilteredVariablesShockDecomposition¶
Variable set by the
estimationcommand, if it is used with thefilter_step_aheadoption. The k-steps are stored along the rows while the columns indicate the respective variables. The third dimension corresponds to the shocks in declaration order. The fourth dimension of the array provides the observation for which the forecast has been made. For example, iffilter_step_ahead=[1 2 4]andnobs=200, the element (3,5,2,204) stores the contribution of the second shock to the four period ahead filtered value of variable 5 (in deviations from the mean) computed at time t=200 for time t=204. The periods at the beginning and end of the sample for which no forecasts can be made, e.g. entries (1,5,1) and (1,5,204) in the example, are set to zero. Padding with zeros and variable ordering is analogous tooo_.FilteredVariablesKStepAhead.
- MATLAB/Octave variable: oo_.PosteriorIRF.dsge¶
Variable set by the
estimationcommand, if it is used with thebayesian_irfoption. Fields are of the form:oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME
- MATLAB/Octave variable: oo_.SmoothedMeasurementErrors¶
Variable set by the
estimationcommand, if it is used with thesmootheroption. Fields are of the form:oo_.SmoothedMeasurementErrors.VARIABLE_NAME
- MATLAB/Octave variable: oo_.SmoothedShocks¶
Variable set by the
estimationcommand (if used with thesmootheroption), or by thecalib_smoothercommand.After an estimation without Metropolis, or if computed by
calib_smoother, fields are of the form:oo_.SmoothedShocks.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form:
oo_.SmoothedShocks.MOMENT_NAME.VARIABLE_NAME
- MATLAB/Octave variable: oo_.SmoothedVariables¶
Variable set by the
estimationcommand (if used with thesmootheroption), or by thecalib_smoothercommand.After an estimation without Metropolis, or if computed by
calib_smoother, fields are of the form:oo_.SmoothedVariables.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form:
oo_.SmoothedVariables.MOMENT_NAME.VARIABLE_NAME
- MATLAB/Octave command: get_smooth('VARIABLE_NAME' [, 'VARIABLE_NAME']...);¶
Returns the smoothed values of the given endogenous or exogenous variable(s), as they are stored in the
oo_.SmoothedVariablesandoo_.SmoothedShocksvariables.
- MATLAB/Octave variable: oo_.UpdatedVariables¶
Variable set by the
estimationcommand (if used with thesmootheroption), or by thecalib_smoothercommand. Contains the estimation of the expected value of variables given the information available at the current date.After an estimation without Metropolis, or if computed by
calib_smoother, fields are of the form:oo_.UpdatedVariables.VARIABLE_NAME
After an estimation with Metropolis, fields are of the form:
oo_.UpdatedVariables.MOMENT_NAME.VARIABLE_NAME
- MATLAB/Octave command: get_update('VARIABLE_NAME' [, 'VARIABLE_NAME']...);¶
Returns the updated values of the given variable(s), as they are stored in the
oo_.UpdatedVariablesvariable.
- MATLAB/Octave variable: oo_.FilterCovariance¶
Three-dimensional array set by the
estimationcommand if used with thesmootherand Metropolis, if thefilter_covarianceoption has been requested. Contains the series of one-step ahead forecast error covariance matrices from the Kalman smoother. TheM_.endo_nbrtimesM_.endo_nbrtimesT+1array contains the variables in declaration order along the first two dimensions. The third dimension of the array provides the observation for which the forecast has been made. Fields are of the form:oo_.FilterCovariance.MOMENT_NAME
Note that density estimation is not supported.
- MATLAB/Octave variable: oo_.Smoother.Variance¶
Three-dimensional array set by the
estimationcommand (if used with thesmoother) without Metropolis, or by thecalib_smoothercommand, if thefilter_covarianceoption has been requested. Contains the series of one-step ahead forecast error covariance matrices from the Kalman smoother. TheM_.endo_nbrtimesM_.endo_nbrtimesT+1array contains the variables in declaration order along the first two dimensions. The third dimension of the array provides the observation for which the forecast has been made.
- MATLAB/Octave variable: oo_.Smoother.State_uncertainty¶
Three-dimensional array set by the
estimationcommand (if used with thesmootheroption) without Metropolis, or by thecalib_smoothercommand, if thesmoothed_state_uncertaintyoption has been requested. Contains the series of covariance matrices for the state estimate given the full data from the Kalman smoother. TheM_.endo_nbrtimesM_.endo_nbrtimesTarray contains the variables in declaration order along the first two dimensions. The third dimension of the array provides the observation for which the smoothed estimate has been made.
- MATLAB/Octave variable: oo_.Smoother.SteadyState¶
Variable set by the
estimationcommand (if used with thesmoother) without Metropolis, or by thecalib_smoothercommand. Contains the steady state component of the endogenous variables used in the smoother in order of variable declaration.
- MATLAB/Octave variable: oo_.Smoother.TrendCoeffs¶
Variable set by the
estimationcommand (if used with thesmoother) without Metropolis, or by thecalib_smoothercommand. Contains the trend coefficients of the observed variables used in the smoother in order of declaration of the observed variables.
- MATLAB/Octave variable: oo_.Smoother.Trend¶
Variable set by the
estimation command(if used with thesmootheroption), or by thecalib_smoothercommand. Contains the trend component of the variables used in the smoother.Fields are of the form:
oo_.Smoother.Trend.VARIABLE_NAME
- MATLAB/Octave variable: oo_.Smoother.Constant¶
Variable set by the
estimationcommand (if used with thesmootheroption), or by thecalib_smoothercommand. Contains the constant part of the endogenous variables used in the smoother, accounting e.g. for the data mean when using the prefilter option.Fields are of the form:
oo_.Smoother.Constant.VARIABLE_NAME
- MATLAB/Octave variable: oo_.Smoother.loglinear¶
Indicator keeping track of whether the smoother was run with the loglinear option and thus whether stored smoothed objects are in logs.
- MATLAB/Octave variable: oo_.PosteriorTheoreticalMoments¶
Variable set by the
estimationcommand, if it is used with themoments_varendooption. Fields are of the form:oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_NAME.VARIABLE_NAME
where THEORETICAL_MOMENT is one of the following:
covarianceVariance-covariance of endogenous variables.
contemporaneous_correlationContemporaneous correlation of endogenous variables when the
contemporaneous_correlationoption is specified.correlationAuto- and cross-correlation of endogenous variables. Fields are vectors with correlations from 1 up to order
options_.ar.VarianceDecompositionDecomposition of variance (unconditional variance, i.e. at horizon infinity). [6]
VarianceDecompositionMESame as VarianceDecomposition, but contains the decomposition of the measured as opposed to the actual variable. The joint contribution of the measurement error will be saved in a field named
ME.ConditionalVarianceDecompositionOnly if the
conditional_variance_decompositionoption has been specified. In the presence of measurement error, the field will contain the variance contribution after measurement error has been taken out, i.e. the decomposition will be conducted of the actual as opposed to the measured variables.ConditionalVarianceDecompositionMEOnly if the
conditional_variance_decompositionoption has been specified. Same as ConditionalVarianceDecomposition, but contains the decomposition of the measured as opposed to the actual variable. The joint contribution of the measurement error will be saved in a field namesME.
- MATLAB/Octave variable: oo_.posterior_density¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_density.PARAMETER_NAME
- MATLAB/Octave variable: oo_.posterior_hpdinf¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_hpdsup¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_mean¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_mode¶
Variable set by the
estimationcommand during mode-finding. Fields are of the form:oo_.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_std_at_mode¶
Variable set by the
estimationcommand during mode-finding. It is based on the inverse Hessian atoo_.posterior_mode. Fields are of the form:oo_.posterior_std_at_mode.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_std¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_var¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_var.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_median¶
Variable set by the
estimationcommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_median.ESTIMATED_OBJECT.VARIABLE_NAME
Example
Here are some examples of generated variables:
oo_.posterior_mode.parameters.alp oo_.posterior_mean.shocks_std.ex oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso
- MATLAB/Octave variable: oo_.dsge_var.posterior_mode¶
Structure set by the
dsge_varoption of theestimationcommand after mode_compute.The following fields are saved:
PHI_tildeStacked posterior DSGE-BVAR autoregressive matrices at the mode (equation (28) of Del Negro and Schorfheide (2004)).
SIGMA_u_tildePosterior covariance matrix of the DSGE-BVAR at the mode (equation (29) of Del Negro and Schorfheide (2004)).
iXXPosterior population moments in the DSGE-BVAR at the mode ( \(inv(\lambda T \Gamma_{XX}^*+ X'X)\)).
priorStructure storing the DSGE-BVAR prior.
PHI_starStacked prior DSGE-BVAR autoregressive matrices at the mode (equation (22) of Del Negro and Schorfheide (2004)).
SIGMA_starPrior covariance matrix of the DSGE-BVAR at the mode (equation (23) of Del Negro and Schorfheide (2004)).
ArtificialSampleSizeSize of the artifical prior sample ( \(inv(\lambda T)\)).
DFPrior degrees of freedom ( \(inv(\lambda T-k-n)\)).
iGXX_starInverse of the theoretical prior “covariance” between X and X (\(\Gamma_{xx}^*\) in Del Negro and Schorfheide (2004)).
- MATLAB/Octave variable: oo_.RecursiveForecast¶
Variable set by the
forecastoption of theestimationcommand when used with the nobs = [INTEGER1:INTEGER2] option (seenobs).Fields are of the form:
oo_.RecursiveForecast.FORECAST_OBJECT.VARIABLE_NAME
where
FORECAST_OBJECTis one of the following [7] :MeanMean of the posterior forecast distribution.
HPDinf/HPDsupUpper/lower bound of the 90% HPD interval taking into account only parameter uncertainty (corresponding to
oo_.MeanForecast).HPDTotalinf/HPDTotalsup.Upper/lower bound of the 90% HPD interval taking into account both parameter and future shock uncertainty (corresponding to
oo_.PointForecast)VARIABLE_NAMEcontains a matrix of the following size: number of time periods for which forecasts are requested using thenobs = [INTEGER1:INTEGER2]option times the number of forecast horizons requested by the forecast option. i.e., the row indicates the period at which the forecast is performed and the column the respective k-step ahead forecast. The starting periods are sorted in ascending order, not in declaration order.
- MATLAB/Octave variable: oo_.convergence.geweke¶
Variable set by the convergence diagnostics of the
estimationcommand. There is a subfield in the struct array for each MCMC chain.Fields are of the form:
oo_.convergence.geweke.VARIABLE_NAME.DIAGNOSTIC_OBJECT
where DIAGNOSTIC_OBJECT is one of the following:
posteriormeanMean of the posterior parameter distribution.
posteriorstdStandard deviation of the posterior parameter distribution.
nse_iidNumerical standard error (NSE) under the assumption of iid draws.
rne_iidRelative numerical efficiency (RNE) under the assumption of iid draws.
nse_taper_xNumerical standard error (NSE) when using an x% taper.
rne_taper_xRelative numerical efficiency (RNE) when using an x% taper.
pooled_meanMean of the parameter when pooling the beginning and end parts of the chain specified in
geweke_intervaland weighting them with their relative precision. It is a vector containing the results under the iid assumption followed by the ones using thetaper_stepsoption (seetaper_steps).pooled_nseNSE of the parameter when pooling the beginning and end parts of the chain and weighting them with their relative precision. See
pooled_mean.prob_chi2_testp-value of a chi-squared test for equality of means in the beginning and the end of the MCMC chain. See
pooled_mean. A value above 0.05 indicates that the null hypothesis of equal means and thus convergence cannot be rejected at the 5 percent level. Differing values along thetaper_stepssignal the presence of significant autocorrelation in draws. In this case, the estimates using a higher tapering are usually more reliable.
- MATLAB/Octave variable: oo_.convergence.raftery_lewis¶
Variable set by the convergence diagnostics of the
estimationcommand when used withraftery_lewis_diagnosticsoption (seeraftery_lewis_diagnostics). There is a subfield in the struct array for each MCMC chain. Contains the results of the test in individual fields.
- Command: unit_root_vars VARIABLE_NAME...;¶
This command is deprecated. Useestimationoptiondiffuse_filterinstead for estimating a model with non-stationary observed variables orsteadyoptionnocheckto preventsteadyto check the steady state returned by your steady state file.
Dynare also has the ability to estimate Bayesian VARs:
- Command: bvar_density ;¶
Computes the marginal density of an estimated BVAR model, using Minnesota priors.See
bvar-a-la-sims.pdf, which comes with Dynare distribution, for more information on this command.
- Command: bvar_irf ;¶
Computes the impulse responses of an estimated BVAR model, using Minnesota priors.See
bvar-a-la-sims.pdf, which comes with Dynare distribution, for more information on this command.
4.16. Estimation based on moments¶
Provided that you have observations on some endogenous variables or their dynamic behavior following structural shocks, Dynare provides a suite of tools for parameter estimation utilizing the method of moments approach. This includes the Simulated Method of Moments (SMM), the Generalized Method of Moments (GMM), and Impulse Response Function Matching (IRF matching). Each of these methods offers a distinct strategy for estimating some or all parameters by minimizing the distances between unconditional model objects (moments or impulse responses) and their empirical counterparts.
GMM and SMM estimation
For SMM Dynare computes model moments via stochastic simulations based on the perturbation approximation up to any order, whereas for GMM model moments are computed in closed-form based on the pruned state-space representation of the perturbation solution up to third order. The implementation of SMM is inspired by Born and Pfeifer (2014) and Ruge-Murcia (2012), whereas the one for GMM is adapted from Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018) and Mutschler (2018). Successful estimation heavily relies on the accuracy and efficiency of the perturbation approximation, so it is advised to tune this as much as possible (see Computing the stochastic solution). The method of moments estimator is consistent and asymptotically normally distributed given certain regularity conditions (see Duffie and Singleton (1993) for SMM and Hansen (1982) for GMM). For instance, it is required to have at least as many moment conditions as estimated parameters (over-identified or just identified). Moreover, the Jacobian of the moments with respect to the estimated parameters needs to have full rank. Performing identification analysis helps to check this regularity condition.
In the over-identified case of declaring more moment conditions than estimated parameters, the
choice of weighting_matrix
matters for the efficiency of the estimation, because the estimated
orthogonality conditions are random variables with unequal variances and
usually non-zero cross-moment covariances. A weighting matrix allows to
re-weight moments to put more emphasis on moment conditions that are
more informative or better measured (in the sense of having a smaller
variance). To achieve asymptotic efficiency, the weighting matrix needs to
be chosen such that, after appropriate scaling, it has a probability limit
proportional to the inverse of the covariance matrix of the limiting
distribution of the vector of orthogonality conditions. Dynare uses a
Newey-West-type estimator with a Bartlett kernel to compute an estimate of this
so-called optimal weighting matrix. Note that in this over-identified case,
it is advised to perform the estimation in at least two stages by setting
e.g. weighting_matrix=['DIAGONAL','DIAGONAL']
so that the computation of the optimal weighting matrix benefits from the
consistent estimation of the previous stages. The optimal weighting matrix
is used to compute standard errors and the J-test of overidentifying
restrictions, which tests whether the model and selection of moment
conditions fits the data sufficiently well. If the null hypothesis of a
“valid” model is rejected, then something is (most likely) wrong with either your model
or selection of orthogonality conditions.
In case the (presumed) global minimum of the moment distance function is
located in a region of the parameter space that
is typically considered unlikely (dilemma of absurd parameters), you may
opt to choose the penalized_estimator option.
Similar to adding priors to the likelihood, this option incorporates prior
knowledge (i.e. the prior mean) as additional moment restrictions and
weights them by their prior precision to guide the minimization algorithm
to more plausible regions of the parameter space. Ideally, these regions are
characterized by only slightly worse values of the objective function. Note that
adding prior information comes at the cost of a loss in efficiency of the estimator.
IRF matching
Dynare employs a user-specified simulation_method to compute the impulse response function (IRF) for observable variables with respect to the structural shocks. Currently, only stochastic simulations based on the perturbation method are supported and it is advised to fine-tune the perturbation approximation as much as possible for optimal results (see Computing the stochastic solution for guidance).
The core idea of IRF matching is then to treat empirical impulse responses (e.g. given from a SVAR or local projection estimation) as data and select model parameters that align the model’s IRFs closely with their empirical counterparts. Dynare supports both Frequentist and Bayesian IRF matching approaches, using the same optimization and sampling techniques as those applied in likelihood-based estimation (sharing many options with the estimation command). The Frequentist approach to this is inspired by the work of Christiano, Eichenbaum, and Evans (2005), while the Bayesian method adapts from Christiano, Trabandt, and Walentin (2010). A crucial element in IRF matching is the choice of the weighting matrix, which influences how the distances between model-generated and empirical IRFs are weighted in the estimation process. It is common practice to employ a diagonal weighting matrix, with the diagonal elements set to the inverse of the estimated variance of the respective empirical impulse response, thereby prioritizing more precisely estimated IRFs. While it’s possible to also specify weights using covariances between different IRF components (possibly with shrinking), this is less common due to the complex interpretation involved (cross effects of different variables or different shocks or both).
Importantly, it is the user’s responsibility to supply (1) the values of the empirical IRFs intended for matching and (2) their importance by choosing an appropriate weighting matrix. Dynare does not perform the SVAR or local projection estimation, it treats the empirical IRFs as given.
4.16.1. Method of moments specific blocks¶
- Command: varobs VARIABLE_NAME...;
Required. All variables used in thematched_moments,matched_irfs, ormatched_irfs_weightsblock need to be observable. See varobs for more details.
- Block: matched_moments ;¶
This block specifies the product moments which are used in estimation. Currently, only linear product moments (e.g. \(E[y_t], E[y_t^2], E[x_t y_t], E[y_t y_{t-1}], E[y_t^3 x^2_{t-4}]\)) are supported. For other functions like \(E[\log(y_t)e^{x_t}]\) you need to declare auxiliary endogenous variables.Each line inside of the block should be of the form:
VARIABLE_NAME(LEAD/LAG)^POWER*VARIABLE_NAME(LEAD/LAG)^POWER*...*VARIABLE_NAME(LEAD/LAG)^POWER;
where VARIABLE_NAME is the name of a declared observable variable, LEAD/LAG is either a negative integer for lags or a positive one for leads, and POWER is a positive integer indicating the exponent on the variable. You can omit LEAD/LAG equal to 0 or POWER equal to 1.
Example
For \(E[c_t], E[y_t], E[c_t^2], E[c_t y_t], E[y_t^2], E[c_t c_{t+3}], E[y_{t+1}^2 c^3_{t-4}], E[c^3_{t-5} y_{t}^2]\) use the following block:
matched_moments; c; y; c*c; c*y; y^2; c*c(3); y(1)^2*c(-4)^3; c(-5)^3*y(0)^2; end;
Limitations
1. For GMM, Dynare can only compute the theoretical mean, covariance, and autocovariances (i.e. first and second moments). Higher-order moments are only supported for SMM.
2. By default, the product moments are not demeaned, unless the
prefilteroption is set to 1. That is, by default, c*c corresponds to \(E[c_t^2]\) and not to \(Var[c_t]=E[c_t^2]-E[c_t]^2\).Output
Dynare translates the
matched_momentsblock into a cell arrayM_.matched_momentswhere:the first column contains a vector of indices for the chosen variables in declaration order
the second column contains the corresponding vector of leads and lags
the third column contains the corresponding vector of powers
During the estimation phase, Dynare will eliminate all redundant or duplicate orthogonality conditions in
M_.matched_momentsand display which conditions were removed. In the example above, this would be the case for the last row, which is the same as the second-to-last one. The original block is saved inM_.matched_moments_orig.
- Block: matched_irfs ;¶
- Block: matched_irfs(overwrite);
This block specifies the values and diagonal weights of the empirical IRFs that are matched in estimation. Theoverwriteoption replaces the currentmatched_irfsblock with the new one.Each line inside of the block should be of the form:
var ENDOGENOUS_NAME; varexo EXOGENOUS_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...; weights DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
ENDOGENOUS_NAME is the name of a declared observable variable, whereas EXOGENOUS_NAME is the name of an exogenous variable. It is possible to specify individual horizons or a range of specified periods as lists with the
periodskeyword. Note that for each entry a corresponding entry invaluesneeds to be provided; that isvaluesis a list of the same length asperiods. If only one value is specified, it is used at all correspondingperiodsin the list.weightsare optional and specify the diagonal element of the corresponding entry in the weighting matrix. Typically, these are set to the inverse of the variance of the empirical IRF. If only one weight is specified, it is used at all correspondingperiodsin the list. If not specified, the weight defaults to 1. Forvaluesandweightsyou can use expressions (e.g. variables or anonymous functions in the workspace) by by putting paranthesis around them. A new statement is started with either thevarorvarexokeyword.Example You can either enter the values directly or load them from variables in the workspace.
- ::
% MATLAB expressions that can be used xx = [23,24,25]; ww = [51,52]; irfs_eR = @(j) IRFFF(2:15,j); % gdp is the 3th column of IRFFF weights_eR = @(j) 1./(IRFFFSE(2:15,j).^2); R_eR = IRFFF(1:15,3); weight_R_eR = 1./(IRFFFSE(1:15,3).^2);
matched_irfs; var gdp; varexo eR; periods 2:15; values (irfs_eR(3)); weights (weights_eR(3)); var R; varexo eR; periods 1:15; values (R_eR); weights (weight_R_eR); var y; varexo eD; periods 5; values 7; weights 25; var r; varexo eD; periods 1,2; values 17,18; weights 37,38; var c; varexo eA; periods 3:5; values (xx); var y; varexo eA; periods 1:2; values 30; weights (ww);
varexo eR; var w; periods 1, 13:15, 2:12; values 2, (xx), 15; weights 3, (xx), 4; end;
Limitations
Output
Dynare translates the
matched_irfsblock into a cell array where the rows correspond to the statements in the blockM_.matched_irfswhere:the first column contains the names of the endogenous variables
the second column contains the names of the exogenous variables
the third column contains a nested cell array that contains the list of horizons, values and weights.
- Block: matched_irfs_weights ;¶
- Block: matched_irfs_weights(overwrite) ;
This optional block specifies elements of the weighting matrix used for IRF matching. Theoverwriteoption replaces the currentmatched_irfs_weightsblock with the new one.The weighting matrix is initialized as a diagonal matrix with ones on the diagonal. Each line inside of the block should be of the form:
ENDOGENOUS_NAME_1(HORIZON_1), EXOGENOUS_NAME_1, ENDOGENOUS_NAME_2(HORIZON_2), EXOGENOUS_NAME_2, WEIGHT;
where ENDOGENOUS_NAME_1 and ENDOGENOUS_NAME_2 are the names of declared observable variables, EXOGENOUS_NAME_1 and EXOGENOUS_NAME_2 are the names of exogenous variables, HORIZON_1 and HORIZON_2 are integers indicating the horizon of the IRFs and WEIGHT is a double value of the weight one wants to assign to the covariance between the two specified IRFs.
Example You can either enter the values directly or load them from variables in the workspace.
matched_irfs_weights; c(1), e_A, c(1), e_A, 20; y(3), e_R, y(2), e_R, (empIRFsCovInv_yR3_yR2); end;
Limitations
Output
Dynare translates the
matched_irfs_weigthsblock into a cell arrayM_.matched_irfs_weightswhere:the first column contains the names of the first endogenous variables
the second column contains the names of the first exogenous variables
the third column contains the horizons of the IRFs for the first endogneous variable
the fourth column contains the names of the second endogenous variables
the fifth column contains the names of the second exogenous variables
the sixth column contains the horizons of IRFs for the second endogenous variable
the seventh column contains the vector of weights
All values that are not specified will be either one (if they are on the diagonal) or zero (if they are not on the diagonal). Symmetry is respected, so one does not need to specify both
c(1), e_A, y(3), e_R, WEIGHTandy(3), e_R, c(1), e_A, WEIGHT. Default: empty cell.
- Block: estimated_params ;
Required. This block lists all parameters to be estimated and specifies bounds and priors as necessary. Seeestimated_paramsfor details and syntax.
- Block: estimated_params_init ;
Optional. This block declares numerical initial values for the optimizer when these ones are different from the prior mean. Seeestimated_params_initfor details and syntax.
- Block: estimated_params_bounds ;
Optional. This block declares lower and upper bounds for parameters in maximum likelihood estimation. Seeestimated_params_boundsfor details and syntax.
4.16.2. method_of_moments command¶
- Command: method_of_moments(OPTIONS...);¶
This command runs the method of moments estimation. The following information will be displayed in the command window:Overview of options chosen by the user
Estimation results for each stage and iteration
Value of minimized moment distance objective function
Result of the J-test (for SMM/GMM)
Comparison plot of model IRFs and empirical IRFs (for IRF matching)
Table of data moments/IRFs and estimated model moments/IRFs
4.16.2.1. Necessary Options¶
- mom_method = SMM|GMM|IRF_MATCHING¶
“Simulated Method of Moments” is triggered by SMM, “Generalized Method of Moments” by GMM and “Impulse Response Function Matching” by IRF_MATCHING.
- datafile = FILENAME
The name of the file containing the data (for GMM and SMM only). See
datafilefor the meaning and syntax. For IRF matching, the data is specified in thematched_irfsblock.
4.16.2.2. Common Options¶
- order = INTEGER
Order of perturbation approximation. For GMM only orders 1|2|3 are supported. For SMM and IRF matching, you can choose an arbitrary order. Note that the order set in other functions will not overwrite the default. Default:
1.
- pruning
Discard higher order terms when iteratively computing simulations of the solution. See
pruningfor more details. Default: not set for SMM and IRF matching, always set for GMM.
- verbose¶
Display and store intermediate estimation results in
oo_.mom. Default: not set.Common options for SMM and GMM
- penalized_estimator¶
This option includes deviations of the estimated parameters from the prior mean as additional moment restrictions and weights them by their prior precision. Default: not set.
- weighting_matrix = ['WM1','WM2',...,'WMn']¶
Determines the weighting matrix used at each estimation stage. The number of elements will define the number of stages, i.e.
weighting_matrix = ['DIAGONAL','DIAGONAL','OPTIMAL']performs a three-stage estimation. Possible values forWMare:
IDENTITY_MATRIXSets the weighting matrix equal to the identity matrix.
OPTIMALUses the optimal weighting matrix computed by a Newey-West-type estimate with a Bartlett kernel. At the first stage, the data-moments are used as initial estimate of the model moments, whereas at subsequent stages the previous estimate of model moments will be used when computing the optimal weighting matrix.
DIAGONALUses the diagonal of the
OPTIMALweighting matrix. This choice puts weights on the specified moments instead of on their linear combinations.
FILENAMEThe name of the MAT-file (extension
.mat) containing a user-specified weighting matrix. The file must include a positive definite square matrix calledweighting_matrixwith both dimensions equal to the number of orthogonality conditions.Default value is
['DIAGONAL','OPTIMAL'].
- weighting_matrix_scaling_factor = DOUBLE¶
Scaling of weighting matrix in objective function. This value should be chosen to obtain values of the objective function in a reasonable numerical range to prevent over- and underflows. Default:
1.
- bartlett_kernel_lag = INTEGER¶
Bandwidth of kernel for computing the optimal weighting matrix. Default:
20.
- se_tolx = DOUBLE¶
Step size for numerical differentiation when computing standard errors with a two-sided finite difference method. Default:
1e-5.
4.16.2.3. SMM specific options¶
- burnin = INTEGER¶
Number of periods dropped at the beginning of simulation. Default:
500.
- bounded_shock_support¶
Trim shocks in simulations to \(\pm 2\) standard deviations. Default: not set.
- seed = INTEGER¶
Common seed used in simulations. Default:
24051986.
- simulation_multiple = INTEGER¶
Multiple of data length used for simulation. Default:
7.
4.16.2.4. GMM specific options¶
4.16.2.5. IRF matching specific options¶
- simulation_method = METHOD¶
Method to compute IRFs. Possible values for
METHODare:
STOCH_SIMULSimulate the model with stochastic simulations and compute IRFs as the difference between the simulated and steady state values. See
stoch_simulfor more details.
- irf_matching_file = FILENAME¶
A MATLAB file containing additional transformations on the model IRFs. This enables more flexibility in matching the model IRFs to the empirical IRFs, e.g. by adding constants to model IRFs, multiplying them with factors, taking the cumulative sum, creating ratios etc. See
NK_irf_matching_file.min the examples directory for an example. Default: empty, i.e. model IRFs exactly match empirical IRFs.
- add_tiny_number_to_cholesky = DOUBLE¶
In case of a non-positive definite covariance matrix, a tiny number is added to the Cholesky factor to avoid numerical problems when computing IRFs. Default: 1e-14.
- drop = INTEGER
Truncation when computing IRFs with perturbation at orders greater than 1. Default:
100.
- relative_irf
Requests the computation of normalized IRFs. See
relative_irffor more details. Default: false.
- replic = INTEGER
Number of simulated series used to compute the IRFs. Default:
1iforder=1, and50otherwise.
4.16.2.6. General options¶
- dirname = FILENAME
Directory in which to store
estimationoutput. Seedirnamefor more details. Default:<mod_file>.
- graph_format = FORMAT
Specify the file format(s) for graphs saved to disk. See
graph_formatfor more details. Default:eps.
- nodisplay
See
nodisplay. Default: not set.
- nograph
See
nograph. Default: not set.
- noprint
See
noprint. Default: not set.
- plot_priors = INTEGER
Control the plotting of priors. See
plot_priorsfor more details. Default:1, i.e. plot priors.
- prior_trunc = DOUBLE
See
prior_truncfor more details. Default:1e-10.
- tex
See
tex. Default: not set.
4.16.2.7. Data options¶
- prefilter = INTEGER
A value of 1 means that the estimation procedure will demean each data series by its empirical mean and each model moment by its theoretical mean. See
prefilterfor more details. Default:0, i.e. no prefiltering.
- first_obs = INTEGER
See
first_obs. Default:1.
- nobs = INTEGER
See
nobs. Default: all observations are considered.
- logdata
See
logdata. Default: not set.
- xls_sheet = QUOTED_STRING
See
xls_sheet. Default:1.
- xls_range = RANGE
See
xls_range. Default: empty.
4.16.2.8. Optimization options¶
- mode_file = FILENAME
Name of the file containing previous value for the mode. See
mode_file. Default: empty.
- mode_compute = INTEGER | FUNCTION_NAME
See
mode_compute. Default:13for GMM and SMM and5for IRF matching.
- additional_optimizer_steps = [INTEGER]
- additional_optimizer_steps = [INTEGER1:INTEGER2]
- additional_optimizer_steps = [INTEGER1 INTEGER2]¶
Vector of additional minimization algorithms run after
mode_compute. Ifverboseoption is set, then the additional estimation results are saved into theoo_.momstructure prefixed with verbose_. Default: empty, i.e. no additional optimization iterations.
- optim = (NAME, VALUE, ...)
See
optim. Default: empty.
- analytic_jacobian¶
Use analytic Jacobian in optimization, only available for GMM and gradient-based optimizers. Default: not set.
- huge_number = DOUBLE
See
huge_number. Default:1e7.
- silent_optimizer
See
silent_optimizer. Default: not set.
- use_penalized_objective_for_hessian
See
use_penalized_objective_for_hessian. Default: not set.
4.16.2.9. Bayesian estimation options¶
General options
- posterior_sampling_method = NAME
See
posterior_sampling_method. Default:random_walk_metropolis_hastings.
- posterior_sampler_options = (NAME, VALUE, ...)
See
posterior_sampler_options. Default: not set.
- mh_posterior_mode_estimation
See
mh_posterior_mode_estimation. Default: not set.
- cova_compute = INTEGER
See
cova_compute. Default:1.
- mcmc_jumping_covariance = OPTION
See
mcmc_jumping_covariance. Default:hessian.
- mh_replic = INTEGER
See
mh_replic. Default:0.
- mh_nblocks = INTEGER
See
mh_nblocks. Default:2.
- mh_jscale = DOUBLE
See
mh_jscale. Default:2.38divided by the square root of the number of estimated parameters.
- mh_tune_jscale [= DOUBLE]
See
mh_tune_jscale. Default:0.33.
- mh_tune_guess = DOUBLE
See
mh_tune_guess. Default:2.38divided by the square root of the number of estimated parameters.
- mh_conf_sig = DOUBLE
See
mh_conf_sig. Default:0.9.
- mh_drop = DOUBLE
See
mh_drop. Default:0.5.
- mh_init_scale_factor = DOUBLE
See
mh_init_scale_factor. Default:2.
- no_posterior_kernel_density
See
no_posterior_kernel_density. Default: not set.
- posterior_max_subsample_draws = INTEGER
See
posterior_max_subsample_draws. Default:1200.
- sub_draws = INTEGER
See
sub_draws. Default:min(posterior_max_subsample_draws, (Total number of draws)*(number of chains) ).
MCMC initialization and recovery
- load_mh_file
See
load_mh_file. Default: not set.
- load_results_after_load_mh
See
load_results_after_load_mh. Default: not set.
- mh_initialize_from_previous_mcmc
See
mh_initialize_from_previous_mcmc. Default: not set.
- mh_initialize_from_previous_mcmc_directory = FILENAME
See
mh_initialize_from_previous_mcmc_directory. Default: empty.
- mh_initialize_from_previous_mcmc_prior = FILENAME
See
mh_initialize_from_previous_mcmc_prior. Default: empty.
- mh_initialize_from_previous_mcmc_record = FILENAME
See
mh_initialize_from_previous_mcmc_record. Default: empty.
- mh_recover
See
mh_recover. Default: not set.
Convergence diagnostics
- nodiagnostic
See
nodiagnostic. Default: not set.
- brooks_gelman_plotrows = INTEGER
See
brooks_gelman_plotrows. Default:3.
- geweke_interval = [DOUBLE DOUBLE]
See
geweke_interval. Default:[0.2 0.5].
- taper_steps = [INTEGER1 INTEGER2 ...]
See
taper_steps. Default:[4 8 15].
- raftery_lewis_diagnostics
See
raftery_lewis_diagnostics. Default: not set.
- raftery_lewis_qrs = [DOUBLE DOUBLE DOUBLE]
See
raftery_lewis_qrs. Default:[0.025 0.005 0.95].
4.16.2.10. Numerical algorithms options¶
- aim_solver
See
aim_solver. Default: not set.
- k_order_solver
See
k_order_solver. Default: disabled for order 1 and 2, enabled for order 3 and above.
- dr = OPTION
See
dr. Default:default, i.e. generalized Schur decomposition.
- dr_cycle_reduction_tol = DOUBLE
See
dr_cycle_reduction_tol. Default:1e-7.
- dr_cycle_reduction_maxiter = INTEGER
See
dr_cycle_reduction_maxiter. Default:100.
- dr_logarithmic_reduction_tol = DOUBLE
See
dr_logarithmic_reduction_tol. Default:1e-12.
- dr_logarithmic_reduction_maxiter = INTEGER
See
dr_logarithmic_reduction_maxiter. Default:100.
- lyapunov = OPTION
See
lyapunov. Default:default, i.e. based on Bartlets-Stewart algorithm.
- lyapunov_complex_threshold = DOUBLE¶
See
lyapunov_complex_threshold. Default:1e-15.
- lyapunov_fixed_point_tol = DOUBLE
See
lyapunov_fixed_point_tol. Default:1e-10.
- lyapunov_doubling_tol = DOUBLE
See
lyapunov_doubling_tol. Default:1e-16.
- qz_criterium = DOUBLE
See
qz_criterium. For unit roots (only possible at order=1) set e.g. to 1.000001. Default:0.999999as it is assumed that the observables are weakly stationary.
- qz_zero_threshold = DOUBLE
See
qz_zero_threshold. Default:1e-6.
- schur_vec_tol = DOUBLE¶
Tolerance level used to find nonstationary variables in Schur decomposition of the transition matrix. Default:
1e-11.
- mode_check
Plots univariate slices through the moments distance objective function around the computed minimum for each estimated parameter. This is helpful to diagnose problems with the optimizer. Default: not set.
- mode_check_neighbourhood_size = DOUBLE
See
mode_check_neighbourhood_size. Default:0.5.
- mode_check_symmetric_plots = INTEGER
See
mode_check_symmetric_plots. Default:1.
- mode_check_number_of_points = INTEGER
See
mode_check_number_of_points. Default:20.
4.16.3. Method of moments specific outputs¶
method_of_momentsstores user options in a structure called options_mom_ in the global workspace. After running the estimation, the parametersM_.paramsand the covariance matrices of the shocksM_.Sigma_eand of the measurement errorsM_.Hare set to the parameters that either minimize the quadratic moments distance objective function or at the posterior mean in case of Bayesian MCMC estimation. The estimation results are stored in a subfolder ofdirnamecalled method_of_moments. Moreover, output is stored in theoo_.momstructure with the following fields:
Common outputs
- MATLAB/Octave variable: oo_.mom.data_moments¶
Variable set by the
method_of_momentscommand. Stores the mean of the selected empirical moments/IRFs of data. NaN values due to leads/lags or missing data are omitted when computing the mean for moments. Vector of dimension equal to the number of orthogonality conditions or IRFs.
- MATLAB/Octave variable: oo_.mom.model_moments¶
Variable set by the
method_of_momentscommand. Stores the implied selected model moments or IRFs given the current parameter guess. Model moments are computed in closed-form from the pruned state-space system for GMM, whereas for SMM these are based on averages of simulated data. Model IRFs are computed from the specified simulation_method. Vector of dimension equal to the number of orthogonality conditions.
- MATLAB/Octave variable: oo_.mom.model_moments_params_derivs¶
Variable set by the
method_of_momentscommand. Stores the analytically computed Jacobian matrix of the derivatives of the model moments with respect to the estimated parameters. Only for GMM withanalytic_standard_errors. Matrix with dimension equal to the number of orthogonality conditions times number of estimated parameters.
- MATLAB/Octave variable: oo_.mom.weighting_info¶
Variable set by the
method_of_momentscommand. Stores the currently used weighting matrix (W), its Cholesky factor (Sw), and an indicator whether the weighting matrix is the optimal one (Woptflag). The inverse (Winv) and its log determinant (Winv_logdet) are also stored.
- MATLAB/Octave variable: oo_.mom.Q¶
Variable set by the
method_of_momentscommand. Stores the scalar value of the quadratic moment’s distance objective function.
- MATLAB/Octave variable: oo_.mom.verbose¶
Structure that contains intermediate estimation results if
verboseis used.
SMM and GMM specific outputs
- MATLAB/Octave variable: oo_.mom.m_data¶
Variable set by the
method_of_momentscommand. Stores the selected empirical moments at each point in time. NaN values due to leads/lags or missing data are replaced by the corresponding mean of the moment. Matrix of dimension time periods times number of orthogonality conditions.
- MATLAB/Octave variable: oo_.mom.gmm_mode¶
- MATLAB/Octave variable: oo_.mom.smm_mode¶
Variables set by the
method_of_momentscommand when estimating with GMM or SMM. Stores the estimated values of the final stage. The structures contain the following fields:
measurement_errors_corr: estimated correlation between two measurement errors
measurement_errors_std: estimated standard deviation of measurement errors
parameters: estimated model parameters
shocks_corr: estimated correlation between two structural shocks.
shocks_std: estimated standard deviation of structural shocks.
- MATLAB/Octave variable: oo_.mom.gmm_std_at_mode¶
- MATLAB/Octave variable: oo_.mom.smm_std_at_mode¶
Variables set by the
method_of_momentscommand when estimating with GMM or SMM. Stores the estimated standard errors of the final stage. The structures contain the following fields:
measurement_errors_corr: standard error of estimated correlation between two measurement errors
measurement_errors_std: standard error of estimated standard deviation of measurement errors
parameters: standard error of estimated model parameters
shocks_corr: standard error of estimated correlation between two structural shocks.
shocks_std: standard error of estimated standard deviation of structural shocks.
- MATLAB/Octave variable: oo_.mom.J_test¶
Variable set by the
method_of_momentscommand. Structure where the value of the test statistic is saved into a field calledj_stat, the degress of freedom into a field calleddegrees_freedomand the p-value of the test statistic into a field calledp_val.
IRF matching specific outputs
- MATLAB/Octave variable: oo_.mom.irf_model_varobs¶
Variable set by the
method_of_momentscommand. Stores all the implied model impulse response functions (not only the matched ones) and is used for the comparison plot. Array of dimension equal to number of observables by number of shocks by maximum horizon.
Bayesian specific outputs
- MATLAB/Octave variable: oo_.mom.prior¶
Variable set by the
method_of_momentscommand if Bayesian estimation is used. Stores information of the joint prior. Fields are of the form:oo_.mom.prior.OBJECTwhere OBJECT is one of the following:
meanPrior mean parameter vector.
modePrior mode parameter vector.
varianceCovariance matrix of joint prior.
hyperparametersVectors of hyperparameters of the prior distributions stored in fields
firstandsecond.
- MATLAB/Octave variable: oo_.mom.posterior.optimization¶
Variable set by the
method_of_momentscommand if mode-finding is used. Stores the results at the mode. Fields are of the form:oo_.mom.posterior.optimization.OBJECTwhere OBJECT is one of the following:
modeParameter vector at the mode.
VarianceInverse Hessian matrix at the mode or MCMC jumping covariance matrix when used with the
MCMC_jumping_covarianceoption.
log_densityLog likelihood (ML)/log posterior density (Bayesian) at the mode when used with
mode_compute>0.
- MATLAB/Octave variable: oo_.mom.posterior.metropolis¶
Variable set by the
method_of_momentscommand ifmh_replic>0is used. Fields are of the form:oo_.mom.posterior.metropolis.OBJECTwhere OBJECT is one of the following:
meanMean parameter vector from the MCMC.
VarianceCovariance matrix of the parameter draws in the MCMC.
- MATLAB/Octave variable: oo_.mom.prior_density¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.prior_density.PARAMETER_NAME
- MATLAB/Octave variable: oo_.mom.posterior_density¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_density.PARAMETER_NAME
- MATLAB/Octave variable: oo_.mom.posterior_hpdinf¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_hpdsup¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_mean¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_mode¶
Variable set by the
method_of_momentscommand during mode-finding. Fields are of the form:oo_.mom.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_std_at_mode¶
Variable set by the
method_of_momentscommand during mode-finding. It is based on the inverse Hessian atoo_.mom.posterior_mode. Fields are of the form:oo_.mom.posterior_std_at_mode.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_std¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_variance¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_variance.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_median¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_median.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.posterior_deciles¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Fields are of the form:oo_.mom.posterior_deciles.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.mom.MarginalDensity.LaplaceApproximation¶
Variable set by the
method_of_momentscommand. Stores the marginal data density based on the Laplace Approximation.
- MATLAB/Octave variable: oo_.mom.MarginalDensity.ModifiedHarmonicMean¶
Variable set by the
method_of_momentscommand, if it is used withmh_replic > 0orload_mh_fileoption. Stores the marginal data density based on Geweke (1999) Modified Harmonic Mean estimator.
4.17. Model Comparison¶
- Command: model_comparison FILENAME[(DOUBLE)]...;¶
- Command: model_comparison(marginal_density = ESTIMATOR) FILENAME[(DOUBLE)]...;
This command computes odds ratios and estimate a posterior density over a collection of models (see e.g. Koop (2003), Ch. 1). The priors over models can be specified as the DOUBLE values, otherwise a uniform prior over all models is assumed. In contrast to frequentist econometrics, the models to be compared do not need to be nested. However, as the computation of posterior odds ratios is a Bayesian technique, the comparison of models estimated with maximum likelihood is not supported.It is important to keep in mind that model comparison of this type is only valid with proper priors. If the prior does not integrate to one for all compared models, the comparison is not valid. This may be the case if part of the prior mass is implicitly truncated because Blanchard and Kahn conditions (instability or indeterminacy of the model) are not fulfilled, or because for some regions of the parameters space the deterministic steady state is undefined (or Dynare is unable to find it). The compared marginal densities should be renormalized by the effective prior mass, but this not done by Dynare: it is the user’s responsibility to make sure that model comparison is based on proper priors. Note that, for obvious reasons, this is not an issue if the compared marginal densities are based on Laplace approximations.
Options
- marginal_density = ESTIMATOR¶
Specifies the estimator for computing the marginal data density. ESTIMATOR can take one of the following two values:
laplacefor the Laplace estimator ormodifiedharmonicmeanfor the Geweke (1999) Modified Harmonic Mean estimator. Default value:laplace
Output
The results are stored in
oo_.Model_Comparison, which is described below.Example
model_comparison my_model(0.7) alt_model(0.3);
This example attributes a 70% prior over
my_modeland 30% prior overalt_model.
- MATLAB/Octave variable: oo_.Model_Comparison¶
Variable set by the
model_comparisoncommand. Fields are of the form:oo_.Model_Comparison.FILENAME.VARIABLE_NAME
where FILENAME is the file name of the model and VARIABLE_NAME is one of the following:
Prior(Normalized) prior density over the model.
Log_Marginal_DensityLogarithm of the marginal data density.
Bayes_RatioRatio of the marginal data density of the model relative to the one of the first declared model
Posterior_Model_ProbabilityPosterior probability of the respective model.
4.18. Shock Decomposition¶
- Command: shock_decomposition [VARIABLE_NAME]...;¶
- Command: shock_decomposition(OPTIONS...) [VARIABLE_NAME]...;
This command computes the historical shock decomposition for a given sample based on the Kalman smoother, i.e. it decomposes the historical deviations of the endogenous variables from their respective steady state values into the contribution coming from the various shocks. Thevariable_namesprovided govern for which variables the decomposition is plotted.Note that this command must come after either
estimation(in case of an estimated model) orstoch_simul(in case of a calibrated model).Options
- parameter_set = OPTION¶
Specify the parameter set to use for running the smoother. Possible values for OPTION are:
calibrationprior_modeprior_meanposterior_modeposterior_meanposterior_medianmle_mode
Note that the parameter set used in subsequent commands like
stoch_simulwill be set to the specifiedparameter_set. Default value:posterior_meanif Metropolis has been run,mle_modeif MLE has been run.
- datafile = FILENAME
See datafile. Useful when computing the shock decomposition on a calibrated model.
- first_obs = INTEGER
See
first_obs.
- nobs = INTEGER
See
nobs.
- prefilter = INTEGER
See
prefilter.
- loglinear
See
loglinear.
- diffuse_kalman_tol = DOUBLE
See
diffuse_kalman_tol.
- diffuse_filter
See
diffuse_filter.
- xls_sheet = QUOTED_STRING
See
xls_sheet.
- xls_range = RANGE
See
xls_range.
- use_shock_groups [= NAME]¶
Uses shock grouping defined by the string instead of individual shocks in the decomposition. The groups of shocks are defined in the
shock_groupsblock. If no group name is given,defaultis assumed.
- colormap = VARIABLE_NAME¶
Controls the
colormapused for the shocks decomposition graphs. VARIABLE_NAME must be the name of a MATLAB/Octave variable that has been declared beforehand and whose value will be passed to the MATLAB/Octavecolormapfunction (see the MATLAB/Octave manual for the list of acceptable values).
- nograph
See
nograph. Suppresses the display and creation only within theshock_decompositioncommand, but does not affect other commands. Seeplot_shock_decompositionfor plotting graphs.
- init_state = BOOLEAN¶
If equal to 0, the shock decomposition is computed conditional on the smoothed state variables in period
0, i.e. the smoothed shocks starting in period 1 are used. If equal to1, the shock decomposition is computed conditional on the smoothed state variables in period 1. Default:0.
- with_epilogue¶
If set, then also compute the decomposition for variables declared in the
epilogueblock (see Epilogue Variables).
Output
- MATLAB/Octave variable: oo_.shock_decomposition¶
The results are stored in the field
oo_.shock_decomposition, which is a three dimensional array. The first dimension contains theM_.endo_nbrendogenous variables. The second dimension stores in the firstM_.exo_nbrcolumns the contribution of the respective shocks. ColumnM_.exo_nbr+1stores the contribution of the initial conditions, while columnM_.exo_nbr+2stores the smoothed value of the respective endogenous variable in deviations from their steady state, i.e. the mean and trends are subtracted. The third dimension stores the time periods. Both the variables and shocks are stored in the order of declaration, i.e.M_.endo_namesandM_.exo_names, respectively.
- Block: shock_groups ;¶
- Block: shock_groups(OPTIONS...);
Shocks can be regrouped for the purpose of shock decomposition. The composition of the shock groups is written in a block delimited byshock_groupsandend.Each line defines a group of shocks as a list of exogenous variables:
SHOCK_GROUP_NAME = VARIABLE_1 [[,] VARIABLE_2 [,]...]; 'SHOCK GROUP NAME' = VARIABLE_1 [[,] VARIABLE_2 [,]...];
Options
- name = NAME
Specifies a name for the following definition of shock groups. It is possible to use several
shock_groupsblocks in a model file, each grouping being identified by a different name. This name must in turn be used in theshock_decompositioncommand. If no name is given,defaultis used.
Example
varexo e_a, e_b, e_c, e_d; ... shock_groups(name=group1); supply = e_a, e_b; 'aggregate demand' = e_c, e_d; end; shock_decomposition(use_shock_groups=group1);
This example defines a shock grouping with the name
group1, containing a set of supply and demand shocks and conducts the shock decomposition for these two groups.
- Command: realtime_shock_decomposition [VARIABLE_NAME]...;¶
- Command: realtime_shock_decomposition(OPTIONS...) [VARIABLE_NAME]...;
This command computes the realtime historical shock decomposition for a given sample based on the Kalman smoother. For each period \(T=[\texttt{presample},\ldots,\texttt{nobs}]\), it recursively computes three objects:Real-time historical shock decomposition \(Y(t\vert T)\) for \(t=[1,\ldots,T]\), i.e. without observing data in \([T+1,\ldots,\texttt{nobs}]\). This results in a standard shock decomposition being computed for each additional datapoint becoming available after
presample.Forecast shock decomposition \(Y(T+k\vert T)\) for \(k=[1,\ldots,forecast]\), i.e. the \(k\)-step ahead forecast made for every \(T\) is decomposed in its shock contributions.
Real-time conditional shock decomposition of the difference between the real-time historical shock decomposition and the forecast shock decomposition. If
vintageis equal to0, it computes the effect of shocks realizing in period \(T\), i.e. decomposes \(Y(T\vert T)-Y(T\vert T-1)\). Put differently, it conducts a \(1\)-period ahead shock decomposition from \(T-1\) to \(T\), by decomposing the update step of the Kalman filter. Ifvintage>0and smaller thannobs, the decomposition is conducted of the forecast revision \(Y(T+k\vert T+k)-Y(T+k\vert T)\).
Like
shock_decompositionit decomposes the historical deviations of the endogenous variables from their respective steady state values into the contribution coming from the various shocks. Thevariable_namesprovided govern for which variables the decomposition is plotted.Note that this command must come after either
estimation(in case of an estimated model) orstoch_simul(in case of a calibrated model).Options
- parameter_set = OPTION
See
parameter_setfor possible values.
- datafile = FILENAME
See datafile.
- first_obs = INTEGER
See
first_obs.
- nobs = INTEGER
See
nobs.
- use_shock_groups [= NAME]
See
use_shock_groups.
- colormap = VARIABLE_NAME
See
colormap.
- nograph
See
nograph. Only shock decompositions are computed and stored inoo_.realtime_shock_decomposition,oo_.conditional_shock_decompositionandoo_.realtime_forecast_shock_decompositionbut no plot is made (Seeplot_shock_decomposition).
- presample = INTEGER
Data point above which recursive realtime shock decompositions are computed, i.e. for \(T=[\texttt{presample+1} \ldots \texttt{nobs}]\).
- forecast = INTEGER
Compute shock decompositions up to \(T+k\) periods, i.e. get shock contributions to k-step ahead forecasts.
- save_realtime = INTEGER_VECTOR¶
Choose for which vintages to save the full realtime shock decomposition. Default:
0.
- fast_realtime = INTEGER¶
- fast_realtime = [INTEGER1:INTEGER2]¶
- fast_realtime = [INTEGER1 INTEGER2 ...]¶
Runs the smoother only for the data vintages provided by the specified integer (vector).
- with_epilogue
See
with_epilogue.
Output
- MATLAB/Octave variable: oo_.realtime_shock_decomposition¶
Structure storing the results of realtime historical decompositions. Fields are three-dimensional arrays with the first two dimension equal to the ones of
oo_.shock_decomposition. The third dimension stores the time periods and is therefore of sizeT+forecast. Fields are of the form:oo_.realtime_shock_decomposition.OBJECT
where OBJECT is one of the following:
poolStores the pooled decomposition, i.e. for every real-time shock decomposition terminal period \(T=[\texttt{presample},\ldots,\texttt{nobs}]\) it collects the last period’s decomposition \(Y(T\vert T)\) (see also
plot_shock_decomposition). The third dimension of the array will have sizenobs+forecast.time_*Stores the vintages of realtime historical shock decompositions if
save_realtimeis used. For example, ifsave_realtime=[5]andforecast=8, the third dimension will be of size13.
- MATLAB/Octave variable: oo_.realtime_conditional_shock_decomposition¶
Structure storing the results of real-time conditional decompositions. Fields are of the form:
oo_.realtime_conditional_shock_decomposition.OBJECT
where OBJECT is one of the following:
poolStores the pooled real-time conditional shock decomposition, i.e. collects the decompositions of \(Y(T\vert T)-Y(T\vert T-1)\) for the terminal periods \(T=[\texttt{presample},\ldots,\texttt{nobs}]\). The third dimension is of size
nobs.time_*Store the vintages of \(k\)-step conditional forecast shock decompositions \(Y(t\vert T+k)\), for \(t=[T \ldots T+k]\). See
vintage. The third dimension is of size1+forecast.
- MATLAB/Octave variable: oo_.realtime_forecast_shock_decomposition¶
Structure storing the results of realtime forecast decompositions. Fields are of the form:
oo_.realtime_forecast_shock_decomposition.OBJECT
where
OBJECTis one of the following:poolStores the pooled real-time forecast decomposition of the \(1\)-step ahead effect of shocks on the \(1\)-step ahead prediction, i.e. \(Y(T\vert T-1)\).
time_*Stores the vintages of \(k\)-step out-of-sample forecast shock decompositions, i.e. \(Y(t\vert T)\), for \(t=[T \ldots T+k]\). See
vintage.
- Command: plot_shock_decomposition [VARIABLE_NAME]...;¶
- Command: plot_shock_decomposition(OPTIONS...) [VARIABLE_NAME]...;
This command plots the historical shock decomposition already computed byshock_decompositionorrealtime_shock_decomposition. For that reason, it must come after one of these commands. Thevariable_namesprovided govern which variables the decomposition is plotted for.Further note that, unlike the majority of Dynare commands, the options specified below are overwritten with their defaults before every call to
plot_shock_decomposition. Hence, if you want to reuse an option in a subsequent call toplot_shock_decomposition, you must pass it to the command again.Options
- use_shock_groups [= NAME]
See
use_shock_groups.
- colormap = VARIABLE_NAME
See
colormap.
- nodisplay
See
nodisplay.
- nograph
See
nograph.
- graph_format = FORMAT
- graph_format = ( FORMAT, FORMAT... )
See
graph_format.
- detail_plot¶
Plots shock contributions using subplots, one per shock (or group of shocks). Default: not activated
- interactive¶
Under MATLAB, add uimenus for detailed group plots. Default: not activated
- screen_shocks¶
For large models (i.e. for models with more than 16 shocks), plots only the shocks that have the largest historical contribution for chosen selected
variable_names. Historical contribution is ranked by the mean absolute value of all historical contributions.
- steadystate¶
If passed, the the \(y\)-axis value of the zero line in the shock decomposition plot is translated to the steady state level. Default: not activated
- type = qoq | yoy | aoa¶
For quarterly data, valid arguments are:
qoqfor quarter-on-quarter plots,yoyfor year-on-year plots of growth rates,aoafor annualized variables, i.e. the value in the last quarter for each year is plotted. Default value: empty, i.e. standard period-on-period plots (qoqfor quarterly data).
- fig_name = STRING¶
Specifies a user-defined keyword to be appended to the default figure name set by
plot_shock_decomposition. This can avoid to overwrite plots in case of sequential calls toplot_shock_decomposition.
- write_xls¶
Saves shock decompositions to Excel file in the main directory, named
FILENAME_shock_decomposition_TYPE_FIG_NAME.xls. This option requires your system to be configured to be able to write Excel files. [8]
- realtime = INTEGER¶
Which kind of shock decomposition to plot. INTEGER can take the following values:
0: standard historical shock decomposition. Seeshock_decomposition.1: realtime historical shock decomposition. Seerealtime_shock_decomposition.2: conditional realtime shock decomposition. Seerealtime_shock_decomposition.3: realtime forecast shock decomposition. Seerealtime_shock_decomposition.
If no vintage is requested, i.e.
vintage=0then the pooled objects fromrealtime_shock_decompositionwill be plotted and the respective vintage otherwise. Default:0.
- vintage = INTEGER¶
Selects a particular data vintage in \([presample,\ldots,nobs]\) for which to plot the results from
realtime_shock_decompositionselected via therealtimeoption. If the standard historical shock decomposition is selected (realtime=0),vintagewill have no effect. Ifvintage=0the pooled objects fromrealtime_shock_decompositionwill be plotted. Ifvintage>0, it plots the shock decompositions for vintage \(T=\texttt{vintage}\) under the following scenarios:realtime=1: the full vintage shock decomposition \(Y(t\vert T)\) for \(t=[1,\ldots,T]\)realtime=2: the conditional forecast shock decomposition from \(T\), i.e. plots \(Y(T+j\vert T+j)\) and the shock contributions needed to get to the data \(Y(T+j)\) conditional on \(T=\) vintage, with \(j=[0,\ldots,\texttt{forecast}]\).realtime=3: plots unconditional forecast shock decomposition from \(T\), i.e. \(Y(T+j\vert T)\), where \(T=\texttt{vintage}\) and \(j=[0,\ldots,\texttt{forecast}]\).
Default:
0.
- plot_init_date = DATE¶
If passed, plots decomposition using
plot_init_dateas initial period. Default: first observation in estimation
- plot_end_date = DATE¶
If passed, plots decomposition using
plot_end_dateas last period. Default: last observation in estimation
- diff¶
If passed, plot the decomposition of the first difference of the list of variables. If used in combination with
flip, thediffoperator is first applied. Default: not activated
- flip¶
If passed, plot the decomposition of the opposite of the list of variables. If used in combination with
diff, thediffoperator is first applied. Default: not activated
- max_nrows¶
Maximum number of rows in the subplot layout of detailed shock decomposition graphs. Note that columns are always 3. Default: 6
- with_epilogue
See
with_epilogue.
- init2shocks¶
- init2shocks = NAME¶
Use the information contained in an
init2shocksblock, in order to attribute initial conditions to shocks. The name of the block can be explicitly given, otherwise it defaults to thedefaultblock.
- Block: init2shocks ;
- Block: init2shocks(OPTIONS...);
This blocks gives the possibility of attributing the initial condition of endogenous variables to the contribution of exogenous variables in the shock decomposition.For example, in an AR(1) process, the contribution of the initial condition on the process variable can naturally be assigned to the innovation of the process.
Each line of the block should have the syntax:
VARIABLE_1 [,] VARIABLE_2;
Where VARIABLE_1 is an endogenous variable whose initial condition will be attributed to the exogenous VARIABLE_2.
The information contained in this block is used by the
plot_shock_decompositioncommand when given theinit2shocksoption.Options
- name = NAME
Specifies a name for the block, that can be referenced from
plot_shock_decomposition, so that several such blocks can coexist in a single model file. If the name is unspecified, it defaults todefault.
Example
var y y_s R pie dq pie_s de A y_obs pie_obs R_obs; varexo e_R e_q e_ys e_pies e_A; ... model; dq = rho_q*dq(-1)+e_q; A = rho_A*A(-1)+e_A; ... end; ... init2shocks; dq e_q; A e_A; end; shock_decomposition(nograph); plot_shock_decomposition(init2shocks) y_obs R_obs pie_obs dq de;
In this example, the initial conditions of
dqandAwill be respectively attributed toe_qande_A.
- Command: initial_condition_decomposition [VARIABLE_NAME]...;¶
- Command: initial_condition_decomposition(OPTIONS...) [VARIABLE_NAME]...;
This command computes and plots the decomposition of the effect of smoothed initial conditions of state variables. Thevariable_namesprovided govern which variables the decomposition is plotted for.Further note that, unlike the majority of Dynare commands, the options specified below are overwritten with their defaults before every call to
initial_condition_decomposition. Hence, if you want to reuse an option in a subsequent call toinitial_condition_decomposition, you must pass it to the command again.Options
- colormap = VARIABLE_NAME
See
colormap.
- nodisplay
See
nodisplay.
- graph_format = FORMAT
- graph_format = ( FORMAT, FORMAT... )
See
graph_format.
- detail_plot
Plots shock contributions using subplots, one per shock (or group of shocks). Default: not activated
- steadystate
If passed, the the \(y\)-axis value of the zero line in the shock decomposition plot is translated to the steady state level. Default: not activated
- type = qoq | yoy | aoa
For quarterly data, valid arguments are:
qoqfor quarter-on-quarter plots,yoyfor year-on-year plots of growth rates,aoafor annualized variables, i.e. the value in the last quarter for each year is plotted. Default value: empty, i.e. standard period-on-period plots (qoqfor quarterly data).
- fig_name = STRING
Specifies a user-defined keyword to be appended to the default figure name set by
plot_shock_decomposition. This can avoid to overwrite plots in case of sequential calls toplot_shock_decomposition.
- write_xls
Saves shock decompositions to Excel file in the main directory, named
FILENAME_shock_decomposition_TYPE_FIG_NAME_initval.xls. This option requires your system to be configured to be able to write Excel files. [8]
- plot_init_date = DATE
If passed, plots decomposition using
plot_init_dateas initial period. Default: first observation in estimation
- plot_end_date = DATE
If passed, plots decomposition using
plot_end_dateas last period. Default: last observation in estimation
- diff
If passed, plot the decomposition of the first difference of the list of variables. If used in combination with
flip, thediffoperator is first applied. Default: not activated
- flip
If passed, plot the decomposition of the opposite of the list of variables. If used in combination with
diff, thediffoperator is first applied. Default: not activated
- Command: squeeze_shock_decomposition [VARIABLE_NAME]...;¶
For large models, the size of the information stored by shock decompositions (especially various settings of realtime decompositions) may become huge. This command allows to squeeze this information in two possible ways:Automatic (default): only the variables for which plotting has been explicitly required with
plot_shock_decompositionwill have their decomposition left inoo_after this command is run;If a list of variables is passed to the command, then only those variables will have their decomposition left in
oo_after this command is run.
4.19. Calibrated Smoother¶
Dynare can also run the smoother on a calibrated model:
- Command: calib_smoother [VARIABLE_NAME]...;¶
- Command: calib_smoother(OPTIONS...) [VARIABLE_NAME]...;
This command computes the smoothed variables (and possible the filtered variables) on a calibrated model.A datafile must be provided, and the observable variables declared with
varobs. The smoother is based on a first-order approximation of the model.By default, the command computes the smoothed variables and shocks and stores the results in
oo_.SmoothedVariablesandoo_.SmoothedShocks. It also fillsoo_.UpdatedVariables.Options
- datafile = FILENAME
See datafile.
- filtered_vars
Triggers the computation of filtered variables. See
filtered_vars, for more details.
- filter_step_ahead = [INTEGER1:INTEGER2]
See
filter_step_ahead.
- prefilter = INTEGER
See
prefilter.
- parameter_set = OPTION
See
parameter_setfor possible values. Default:calibration.
- loglinear
See loglinear.
- first_obs = INTEGER
See
first_obs.
- filter_decomposition
See
filter_decomposition.
- filter_covariance
See
filter_covariance.
- smoother_redux
See
smoother_redux.
- kalman_algo = INTEGER
See
kalman_algo.
- diffuse_filter = INTEGER¶
See
diffuse_filter.
- diffuse_kalman_tol = DOUBLE
See
diffuse_kalman_tol.
- xls_sheet = QUOTED_STRING
See
xls_sheet.
- xls_range = RANGE
See
xls_range.
- heteroskedastic_filter
- nobs = INTEGER
- nobs = [INTEGER1:INTEGER2]
See
nobs.
4.20. Forecasting¶
On a calibrated model, forecasting is done using the forecast
command. On an estimated model, use the forecast option of
estimation command.
It is also possible to compute forecasts on a calibrated or estimated
model for a given constrained path of the future endogenous
variables. This is done, from the reduced form representation of the
DSGE model, by finding the structural shocks that are needed to match
the restricted paths. Use conditional_forecast,
conditional_forecast_paths and plot_conditional_forecast for
that purpose.
Finally, it is possible to do forecasting with a Bayesian VAR using
the bvar_forecast command.
- Command: forecast [VARIABLE_NAME...];¶
- Command: forecast(OPTIONS...) [VARIABLE_NAME...];
This command computes a simulation of a stochastic model from an arbitrary initial point.When the model also contains deterministic exogenous shocks, the simulation is computed conditionally to the agents knowing the future values of the deterministic exogenous variables.
forecastmust be called afterstoch_simul.forecastplots the trajectory of endogenous variables. When a list of variable names follows the command, only those variables are plotted. A 90% confidence interval is plotted around the mean trajectory. Use optionconf_sigto change the level of the confidence interval.Options
- periods = INTEGER
Number of periods of the forecast. Default:
5.
- conf_sig = DOUBLE
Level of significance for confidence interval. Default:
0.90.
- nograph
See
nograph.
- nodisplay
See
nodisplay.
- graph_format = FORMAT
- graph_format = ( FORMAT, FORMAT... )
Initial Values
forecastcomputes the forecast taking as initial values the values specified inhistval(seehistval). When nohistvalblock is present, the initial values are the one stated ininitval. Wheninitvalis followed by commandsteady, the initial values are the steady state (seesteady).Output
The results are stored in
oo_.forecast, which is described below.Example
varexo_det tau; varexo e; ... shocks; var e; stderr 0.01; var tau; periods 1:9; values -0.15; end; stoch_simul(irf=0); forecast;
- MATLAB/Octave variable: oo_.forecast¶
Variable set by the
forecastcommand, or by theestimationcommand if used with theforecastoption and ML or if no Metropolis-Hastings has been computed (in that case, the forecast is computed for the posterior mode). Fields are of the form:oo_.forecast.FORECAST_MOMENT.VARIABLE_NAME
where
FORECAST_MOMENTis one of the following:HPDinfLower bound of a 90% HPD interval [9] of forecast due to parameter uncertainty, but ignoring the effect of measurement error on observed variables. In case of ML, it stores the lower bound of the confidence interval.
HPDsupUpper bound of a 90% HPD forecast interval due to parameter uncertainty, but ignoring the effect of measurement error on observed variables. In case of ML, it stores the upper bound of the confidence interval.
HPDinf_MELower bound of a 90% HPD interval [10] of forecast for observed variables due to parameter uncertainty and measurement error. In case of ML, it stores the lower bound of the confidence interval.
HPDsup_MEUpper bound of a 90% HPD interval of forecast for observed variables due to parameter uncertainty and measurement error. In case of ML, it stores the upper bound of the confidence interval.
MeanMean of the posterior distribution of forecasts.
- MATLAB/Octave variable: oo_.PointForecast¶
Set by the
estimationcommand, if it is used with theforecastoption and if eithermh_replic > 0or theload_mh_fileoption are used.Contains the distribution of forecasts taking into account the uncertainty about both parameters and shocks.
Fields are of the form:
oo_.PointForecast.MOMENT_NAME.VARIABLE_NAME
- MATLAB/Octave variable: oo_.MeanForecast¶
Set by the
estimationcommand, if it is used with theforecastoption and if eithermh_replic > 0orload_mh_fileoption are used.Contains the distribution of forecasts where the uncertainty about shocks is averaged out. The distribution of forecasts therefore only represents the uncertainty about parameters.
Fields are of the form:
oo_.MeanForecast.MOMENT_NAME.VARIABLE_NAME
- Command: conditional_forecast(OPTIONS...);¶
This command computes forecasts on an estimated or calibrated model for a given constrained path of some future endogenous variables. This is done using the reduced form first order state-space representation of the DSGE model by finding the structural shocks that are needed to match the restricted paths. Consider the augmented state space representation that stacks both predetermined and non-predetermined variables into a vector \(y_{t}\):\[y_t=Ty_{t-1}+R\varepsilon_t\]Both \(y_t\) and \(\varepsilon_t\) are split up into controlled and uncontrolled ones, and we assume without loss of generality that the constrained endogenous variables and the controlled shocks come first :
\[\begin{split}\begin{pmatrix} y_{c,t}\\ y_{u,t} \end{pmatrix} = \begin{pmatrix} T_{c,c} & T_{c,u}\\ T_{u,c} & T_{u,u} \end{pmatrix} \begin{pmatrix} y_{c,t-1}\\ y_{u,t-1} \end{pmatrix} + \begin{pmatrix} R_{c,c} & R_{c,u}\\ R_{u,c} & R_{u,u} \end{pmatrix} \begin{pmatrix} \varepsilon_{c,t}\\ \varepsilon_{u,t} \end{pmatrix}\end{split}\]where matrices \(T\) and \(R\) are partitioned consistently with the vectors of endogenous variables and innovations. Provided that matrix \(R_{c,c}\) is square and full rank (a necessary condition is that the number of free endogenous variables matches the number of free innovations), given \(y_{c,t}\), \(\varepsilon_{u,t}\) and \(y_{t-1}\) the first block of equations can be solved for \(\varepsilon_{c,t}\):
\[\varepsilon_{c,t} = R_{c,c}^{-1}\bigl( y_{c,t} - T_{c,c}y_{c,t-1} - T_{c,u}y_{u,t-1} - R_{c,u}\varepsilon_{u,t}\bigr)\]and \(y_{u,t}\) can be updated by evaluating the second block of equations:
\[y_{u,t} = T_{u,c}y_{c,t-1} + T_{u,u}y_{u,t-1} + R_{u,c}\varepsilon_{c,t} + R_{u,u}\varepsilon_{u,t}\]By iterating over these two blocks of equations, we can build a forecast for all the endogenous variables in the system conditional on paths for a subset of the endogenous variables. If the distribution of the free innovations \(\varepsilon_{u,t}\) is provided (i.e. some of them have positive variances) this exercise is replicated (the number of replication is controlled by the option
replicdescribed below) by drawing different sequences of free innovations. The result is a predictive distribution for the uncontrolled endogenous variables, \(y_{u,t}\), that Dynare will use to report confidence bands around the point conditional forecast.A few things need to be noted. First, the controlled exogenous variables are set to zero for the uncontrolled periods. This implies that there is no forecast uncertainty arising from these exogenous variables in uncontrolled periods. Second, by making use of the first order state space solution, even if a higher-order approximation was performed, the conditional forecasts will be based on a first order approximation. Since the controlled exogenous variables are identified on the basis of the reduced form model (i.e. after solving for the expectations), they are unforeseen shocks from the perspective of the agents in the model. That is, agents expect the endogenous variables to return to their respective steady state levels but are surprised in each period by the realisation of shocks keeping the endogenous variables along a predefined (unexpected) path. Fourth, if the structural innovations are correlated, because the calibrated or estimated covariance matrix has non zero off diagonal elements, the results of the conditional forecasts will depend on the ordering of the innovations (as declared after
varexo). As in VAR models, a Cholesky decomposition is used to factorise the covariance matrix and identify orthogonal impulses. It is preferable to declare the correlations in the model block (explicitly imposing the identification restrictions), unless you are satisfied with the implicit identification restrictions implied by the Cholesky decomposition.This command has to be called after
estimationorstoch_simul.Use
conditional_forecast_pathsblock to give the list of constrained endogenous, and their constrained future path. Optioncontrolled_varexois used to specify the structural shocks which will be matched to generate the constrained path.Use
plot_conditional_forecastto graph the results.Options
- parameter_set = OPTION
See
parameter_setfor possible values. No default value, mandatory option.
- controlled_varexo = (VARIABLE_NAME...)¶
Specify the exogenous variables to use as control variables. No default value, mandatory option.
- periods = INTEGER
Number of periods of the forecast. Default:
40.periodscannot be smaller than the number of constrained periods.
- replic = INTEGER
Number of simulations used to compute the conditional forecast uncertainty. Default:
5000.
- conf_sig = DOUBLE
Level of significance for confidence interval. Default:
0.80.
Output
The results are stored in
oo_.conditional_forecast, which is described below.Example
var y a; varexo e u; ... estimation(...); conditional_forecast_paths; var y; periods 1:3, 4:5; values 2, 5; var a; periods 1:5; values 3; end; conditional_forecast(parameter_set = calibration, controlled_varexo = (e, u), replic = 3000); plot_conditional_forecast(periods = 10) a y;
- MATLAB/Octave variable: oo_.conditional_forecast.cond¶
Variable set by the
conditional_forecastcommand. It stores the conditional forecasts. Fields areperiods+1by1vectors storing the steady state (time 0) and the subsequentperiodsforecasts periods. Fields are of the form:oo_.conditional_forecast.cond.FORECAST_MOMENT.VARIABLE_NAME
where FORECAST_MOMENT is one of the following:
MeanMean of the conditional forecast distribution.
ciConfidence interval of the conditional forecast distribution. The size corresponds to
conf_sig.
- MATLAB/Octave variable: oo_.conditional_forecast.uncond¶
Variable set by the
conditional_forecastcommand. It stores the unconditional forecasts. Fields are of the form:oo_.conditional_forecast.uncond.FORECAST_MOMENT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.conditional_forecast.instruments¶
Variable set by the
conditional_forecastcommand. Stores the names of the exogenous instruments.
- MATLAB/Octave variable: oo_.conditional_forecast.controlled_variables¶
Variable set by the
conditional_forecastcommand. Stores the position of the constrained endogenous variables in declaration order.
- MATLAB/Octave variable: oo_.conditional_forecast.controlled_exo_variables¶
Variable set by the
conditional_forecastcommand. Stores the values of the controlled exogenous variables underlying the conditional forecasts to achieve the constrained endogenous variables. Fields are[number of constrained periods]by1vectors and are of the form:oo_.conditional_forecast.controlled_exo_variables.FORECAST_MOMENT.SHOCK_NAME
- MATLAB/Octave variable: oo_.conditional_forecast.graphs¶
Variable set by the
conditional_forecastcommand. Stores the information for generating the conditional forecast plots.
- Block: conditional_forecast_paths ;¶
Describes the path of constrained endogenous, before callingconditional_forecast. The syntax is similar to deterministic shocks inshocks, seeconditional_forecastfor an example and Shocks on exogenous variables for the detailed syntax reference.Note that you need to specify the full path for all constrained endogenous variables between the first and last specified period. If an intermediate period is not specified, a value of 0 is assumed. That is, if you specify only values for periods 1 and 3, the values for period 2 will be 0. Currently, it is not possible to have uncontrolled intermediate periods.
It is however possible to have different number of controlled periods for different variables. In that case, the order of declaration of endogenous controlled variables and of
controlled_varexomatters: if the second endogenous variable is controlled for less periods than the first one, the secondcontrolled_varexoisn’t set for the last periods.In case of the presence of
observation_trends, the specified controlled path for these variables needs to include the trend component. When using the loglinear option, it is necessary to specify the logarithm of the controlled variables.
- Block: filter_initial_state ;¶
This block specifies the initial values of the endogenous states at the beginning of the Kalman filter recursions. That is, if the Kalman filter recursion starts with time t=1 being the first observation, this block provides the state estimate at time 0 given information at time 0, \(E_0(x_0)\). If nothing is specified, the initial condition is assumed to be at the steady state (which is the unconditional mean for a stationary model).This block is terminated by
end;.Each line inside of the block should be of the form:
VARIABLE_NAME(INTEGER)=EXPRESSION;
EXPRESSIONis any valid expression returning a numerical value and can contain parameter values. This allows specifying relationships that will be honored during estimation.INTEGERrefers to the lag with which a variable appears. By convention in Dynare, period 1 is the first period. Going backwards in time, the first period before the start of the simulation is period 0, then period -1, and so on. Note that thefilter_initial_stateblock does not take non-state variables.Example
filter_initial_state; k(0)= ((1/bet-(1-del))/alp)^(1/(alp-1))*l_ss; P(0)=2.5258; m(0)= mst; end;
- Command: plot_conditional_forecast [VARIABLE_NAME...];¶
- Command: plot_conditional_forecast(periods = INTEGER) [VARIABLE_NAME...];
Plots the conditional (plain lines) and unconditional (dashed lines) forecasts.To be used after
conditional_forecast.Options
- periods = INTEGER
Number of periods to be plotted. Default: equal to periods in
conditional_forecast. The number of periods declared inplot_conditional_forecastcannot be greater than the one declared inconditional_forecast.
- Command: bvar_forecast ;¶
This command computes (out-of-sample) forecasts for an estimated BVAR model, using Minnesota priors.See
bvar-a-la-sims.pdf, which comes with Dynare distribution, for more information on this command.
If the model contains strong non-linearities or if some perfectly
expected shocks are considered, the forecasts and the conditional
forecasts can be computed using an extended path method. The forecast
scenario describing the shocks and/or the constrained paths on some
endogenous variables should be build. The first step is the forecast
scenario initialization using the function init_plan:
- MATLAB/Octave command: HANDLE = init_plan(DATES);¶
Creates a new forecast scenario for a forecast period (indicated as a dates class, see dates class members). This function return a handle on the new forecast scenario.
The forecast scenario can contain some simple shocks on the exogenous
variables. This shocks are described using the function
basic_plan:
- MATLAB/Octave command: HANDLE = basic_plan(HANDLE, 'VAR_NAME', 'SHOCK_TYPE', DATES, MATLAB VECTOR OF DOUBLE);¶
Adds to the forecast scenario a shock on the exogenous variable indicated between quotes in the second argument. The shock type has to be specified in the third argument between quotes:
'surprise'in case of an unexpected shock or'perfect_foresight'for a perfectly anticipated shock. The fourth argument indicates the period of the shock using a dates class (see dates class members). The last argument is the shock path indicated as a MATLAB vector of double. This function return the handle of the updated forecast scenario.
The forecast scenario can also contain a constrained path on an
endogenous variable. The values of the related exogenous variable
compatible with the constrained path are in this case computed. In
other words, a conditional forecast is performed. This kind of shock
is described with the function flip_plan:
- MATLAB/Octave command: HANDLE = flip_plan(HANDLE, 'VAR_NAME', 'VAR_NAME', 'SHOCK_TYPE', DATES, MATLAB VECTOR OF DOUBLE);¶
Adds to the forecast scenario a constrained path on the endogenous variable specified between quotes in the second argument. The associated exogenous variable provided in the third argument between quotes, is considered as an endogenous variable and its values compatible with the constrained path on the endogenous variable will be computed. The nature of the expectation on the constrained path has to be specified in the fourth argument between quotes:
'surprise'in case of an unexpected path or'perfect_foresight'for a perfectly anticipated path. The fifth argument indicates the period where the path of the endogenous variable is constrained using a dates class (see dates class members). The last argument contains the constrained path as a MATLAB vector of double. This function return the handle of the updated forecast scenario.
Once the forecast scenario if fully described, the forecast is
computed with the command det_cond_forecast:
- MATLAB/Octave command: DSERIES = det_cond_forecast(HANDLE, DSERIES, DATES);¶
Computes the forecast or the conditional forecast using an extended path method for the given forecast scenario (first argument). The first argument is a handle to the forecast scenario as created by init_plan and associated commands. The second argument contains the past values of the endogenous and the path of exogenous variables, in a dseries object (see dseries class members). The third argument is the date range of the forecast, typically the range used when creating the scenario handle. This function returns a dataset containing the historical and forecast values for the endogenous and exogenous variables.
Example
% conditional forecast using extended path method % with perfect foresight on r path var y r; varexo e u; ... smoothed = dseries('smoothed_variables.csv'); frng = 2013Q4:2029Q4; fplan = init_plan(frng); fplan = flip_plan(fplan, 'y', 'u', 'surprise', frng(1:4), [1 1.1 1.2 1.1]); fplan = flip_plan(fplan, 'r', 'e', 'perfect_foresight', frng(1:4), [2 1.9 1.9 1.9]); dset_forecast = det_cond_forecast(fplan, smoothed, frng); plot(dset_forecast.{'y','u'}); plot(dset_forecast.{'r','e'});
- Command: smoother2histval ;¶
- Command: smoother2histval(OPTIONS...);
The purpose of this command is to construct initial conditions (for a subsequent simulation) that are the smoothed values of a previous estimation.
More precisely, after an estimation run with the
smootheroption,smoother2histvalwill extract the smoothed values (fromoo_.SmoothedVariables, and possibly fromoo_.SmoothedShocksif there are lagged exogenous), and will use these values to construct initial conditions (as if they had been manually entered throughhistval).Options
- period = INTEGER¶
Period number to use as the starting point for the subsequent simulation. It should be between 1 and the number of observations that were used to produce the smoothed values. Default: the last observation.
- infile = FILENAME¶
Load the smoothed values from a
_results.matfile created by a previous Dynare run. Default: use the smoothed values currently in the global workspace.
- invars = ( VARIABLE_NAME [VARIABLE_NAME ...] )¶
A list of variables to read from the smoothed values. It can contain state endogenous variables, and also exogenous variables having a lag. Default: all the state endogenous variables, and all the exogenous variables with a lag.
- outfile = FILENAME¶
Write the initial conditions to a file. Default: write the initial conditions in the current workspace, so that a simulation can be performed.
- outvars = ( VARIABLE_NAME [VARIABLE_NAME ...] )¶
A list of variables which will be given the initial conditions. This list must have the same length than the list given to
invars, and there will be a one-to-one mapping between the two list. Default: same value as optioninvars.
Use cases
There are three possible ways of using this command:
Everything in a single file: run an estimation with a smoother, then run
smoother2histval(without theinfileandoutfileoptions), then run a stochastic simulation.In two files: in the first file, run the smoother and then run
smoother2histvalwith theoutfileoption; in the second file, runhistval_fileto load the initial conditions, and run a (deterministic or stochastic) simulation.In two files: in the first file, run the smoother; in the second file, run
smoother2histvalwith theinfileoption equal to the_results.matfile created by the first file, and then run a (deterministic or stochastic) simulation.
4.21. Optimal policy¶
Dynare has tools to compute optimal policies for various types of
objectives. You can either solve for optimal policy under
commitment with ramsey_model, for optimal policy under discretion
with discretionary_policy or for optimal simple rules with osr
(also implying commitment).
- Command: planner_objective MODEL_EXPRESSION ;¶
This command declares the policy maker objective, for use withramsey_modelordiscretionary_policy.You need to give the one-period objective, not the discounted lifetime objective. The discount factor is given by the
planner_discountoption oframsey_modelanddiscretionary_policy. The objective function can only contain current endogenous variables and no exogenous ones. This limitation is easily circumvented by defining an appropriate auxiliary variable in the model.With
ramsey_model, you are not limited to quadratic objectives: you can give any arbitrary nonlinear expression.With
discretionary_policy, the objective function must be quadratic.
- Command: evaluate_planner_objective ;¶
- Command: evaluate_planner_objective(OPTIONS...);
This command computes, displays, and stores the value of the planner objective function under Ramsey policy or discretion in
oo_.planner_objective_value. It will provide both unconditional welfare and welfare conditional on the initial (i.e. period 0) values of the endogenous and exogenous state variables inherited by the planner. In a deterministic context, the respective initial values are set usinginitvalorhistval(depending on the exact context).In a stochastic context, if no initial state values have been specified with
histval, their values are taken to be the steady state values. Because conditional welfare is computed conditional on optimal policy by the planner in the first endogenous period (period 1), it is conditional on the information set in the period 1. This information set includes both the predetermined states inherited from period 0 (specified viahistvalfor both endogenous and lagged exogenous states) as well as the period 1 values of the exogenous shocks. The latter are specified using the perfect foresight syntax of theshocksblock.At the current stage, the stochastic context does not support the
pruningoption. Atorder>3, only the computation of conditional welfare with steady state Lagrange multipliers is supported. Note that atorder=2, the output is based on the second-order accurate approximation of the variance stored inoo_.var.Options
- periods = INTEGER
The value of the option specifies the number of periods to use in the simulations in the computation of unconditional welfare at higher order.
Default:
10000.
- drop = INTEGER
The number of burn-in draws out of
periodsdiscarded before computing the unconditional welfare at higher order. Default:1000.
Example (stochastic context)
var a ...; varexo u; model; a = rho*a(-1)+u+u(-1); ... end; histval; u(0)=1; a(0)=-1; end; shocks; var u; stderr 0.008; var u; periods 1; values 1; end; evaluate_planner_objective;
- MATLAB/Octave variable: oo_.planner_objective_value.unconditional¶
Scalar storing the value of unconditional welfare. In a perfect foresight context, it corresponds to welfare in the long-run, approximated as welfare in the terminal simulation period.
- MATLAB/Octave variable: oo_.planner_objective_value.conditional¶
In a perfect foresight context, this field will be a scalar storing the value of welfare conditional on the specified initial condition and zero initial Lagrange multipliers.
In a stochastic context, it will have two subfields:
- MATLAB/Octave variable: oo_.planner_objective_value.conditional.steady_initial_multiplier¶
Stores the value of the planner objective when the initial Lagrange multipliers associated with the planner’s problem are set to their steady state values (see
ramsey_policy).- MATLAB/Octave variable: oo_.planner_objective_value.conditional.zero_initial_multiplier¶
Stores the value of the planner objective when the initial Lagrange multipliers associated with the planner’s problem are set to 0, i.e. it is assumed that the planner exploits its ability to surprise private agents in the first period of implementing Ramsey policy. This value corresponds to the planner implementing optimal policy for the first time and committing not to re-optimize in the future.
4.21.1. Optimal policy under commitment (Ramsey)¶
Dynare allows to automatically compute optimal policy choices of a Ramsey planner
who takes the specified private sector equilibrium conditions into account and commits
to future policy choices. Doing so requires specifying the private sector equilibrium
conditions in the model block and a planner_objective as well as potentially some
instruments to facilitate computations.
Warning
Be careful when employing forward-looking auxiliary variables in the context
of timeless perspective Ramsey computations. They may alter the problem the Ramsey
planner will solve for the first period, although they seemingly leave the private
sector equilibrium unaffected. The reason is the planner optimizes with respect to variables
dated t and takes the value of time 0 variables as given, because they are predetermined.
This set of initially predetermined variables will change with forward-looking definitions.
Thus, users are strongly advised to use model-local variables instead.
Example
Consider a perfect foresight example where the Euler equation for the return to capital is given by
1/C=beta*1/C(+1)*(R(+1)+(1-delta))The job of the Ramsey planner in period
1is to choose \(C_1\) and \(R_1\), taking as given \(C_0\). The above equation may seemingly equivalently be written as1/C=beta*1/C(+1)*(R_cap); R_cap=R(+1)+(1-delta);due to perfect foresight. However, this changes the problem of the Ramsey planner in the first period to choosing \(C_1\) and \(R_1\), taking as given both \(C_0\) and \(R^{cap}_0\). Thus, the relevant return to capital in the Euler equation of the first period is not a choice of the planner anymore due to the forward-looking nature of the definition in the second line!
A correct specification would be to instead define
R_capas a model-local variable:1/C=beta*1/C(+1)*(R_cap); #R_cap=R(+1)+(1-delta);
- Command: ramsey_model(OPTIONS...);¶
This command computes the First Order Conditions for maximizing the policy maker objective function subject to the constraints provided by the equilibrium path of the private economy.The planner objective must be declared with the
planner_objectivecommand.This command only creates the expanded model, it doesn’t perform any computations. It needs to be followed by other instructions to actually perform desired computations. Examples are calls to
steadyto compute the steady state of the Ramsey economy, tostoch_simulwith various approximation orders to conduct stochastic simulations based on perturbation solutions, toestimationin order to estimate models under optimal policy with commitment, and to perfect foresight simulation routines.See Auxiliary variables, for an explanation of how Lagrange multipliers are automatically created.
Options
This command accepts the following options:
- planner_discount = EXPRESSION¶
Declares or reassigns the discount factor of the central planner
optimal_policy_discount_factor. Default:1.0.
- planner_discount_latex_name = LATEX_NAME¶
Sets the LaTeX name of the
optimal_policy_discount_factorparameter.
- instruments = (VARIABLE_NAME,...)¶
Declares instrument variables for the computation of the steady state under optimal policy. Requires a
steady_state_modelblock or a_steadystate.mfile. See below.
Steady state
Dynare takes advantage of the fact that the Lagrange multipliers appear linearly in the equations of the steady state of the model under optimal policy. Nevertheless, it is in general very difficult to compute the steady state with simply a numerical guess in
initvalfor the endogenous variables.It greatly facilitates the computation, if the user provides an analytical solution for the steady state (in
steady_state_modelblock or in a_steadystate.mfile). In this case, it is necessary to provide a steady state solution CONDITIONAL on the value of the instruments in the optimal policy problem and declared with the optioninstruments. The initial value of the instrument for steady state finding in this case is set withinitval. Note that computing and displaying steady state values using thesteadycommand or calls toresidmust come after theramsey_modelstatement and theinitvalblock.Note that choosing the instruments is partly a matter of interpretation and you can choose instruments that are handy from a mathematical point of view but different from the instruments you would refer to in the analysis of the paper. A typical example is choosing inflation or nominal interest rate as an instrument.
- Block: ramsey_constraints ;¶
This block lets you define constraints on the variables in the Ramsey problem. The constraints take the form of a variable, an inequality operator (> or <) and a constant.Example
ramsey_constraints; i > 0; end;
- Command: ramsey_policy [VARIABLE_NAME...];¶
- Command: ramsey_policy(OPTIONS...) [VARIABLE_NAME...];
This command is deprecated and formally equivalent to the calling sequenceramsey_model; stoch_simul; evaluate_planner_objective;
It computes an approximation of the policy that maximizes the policy maker’s objective function subject to the constraints provided by the equilibrium path of the private economy and under commitment to this optimal policy. The Ramsey policy is computed by approximating the equilibrium system around the perturbation point where the Lagrange multipliers are at their steady state, i.e. where the Ramsey planner acts as if the initial multipliers had been set to 0 in the distant past, giving them time to converge to their steady state value. Consequently, the optimal decision rules are computed around this steady state of the endogenous variables and the Lagrange multipliers.
Note that the variables in the list after the
ramsey_policyorstoch_simulcommand can also contain multiplier names, but in a case-sensititve way (e.g.MULT_1). In that case, Dynare will for example display the IRFs of the respective multipliers whenirf>0.The planner objective must be declared with the
planner_objectivecommand.Options
This command accepts all options of
stoch_simul, plus:- planner_discount = EXPRESSION
See
planner_discount.
- instruments = (VARIABLE_NAME,...)
Declares instrument variables for the computation of the steady state under optimal policy. Requires a
steady_state_modelblock or a_steadystate.mfile. See below.
Output
This command generates all the output variables of
stoch_simul. For specifying the initial values for the endogenous state variables (except for the Lagrange multipliers), see above.Steady state
See
Ramsey steady state.
4.21.2. Optimal policy under discretion¶
- Command: discretionary_policy [VARIABLE_NAME...];¶
- Command: discretionary_policy(OPTIONS...) [VARIABLE_NAME...];
This command computes an approximation of the optimal policy under discretion. The algorithm implemented is essentially an LQ solver, and is described by Dennis (2007).You must ensure that your objective is quadratic. Regarding the model, it must either be linear or solved at first order with an analytical steady state provided. In the first case, you should set the
linearoption of themodelblock ormodel_optionscommand.It is possible to use the
estimationcommand after thediscretionary_policycommand, in order to estimate the model with optimal policy under discretion andevaluate_planner_objectiveto compute welfare.Options
This command accepts the same options as
ramsey_policy, plus:- discretionary_tol = NON-NEGATIVE DOUBLE¶
Sets the tolerance level used to assess convergence of the solution algorithm. Default:
1e-7.
- maxit = INTEGER
Maximum number of iterations. Default:
3000.
4.21.3. Optimal Simple Rules (OSR)¶
- Command: osr [VARIABLE_NAME...];¶
- Command: osr(OPTIONS...) [VARIABLE_NAME...];
This command computes optimal simple policy rules for linear-quadratic problems of the form:\[\min_\gamma E(y'_tWy_t)\]such that:
\[A_1 E_ty_{t+1}+A_2 y_t+ A_3 y_{t-1}+C e_t=0\]where:
\(E\) denotes the unconditional expectations operator;
\(\gamma\) are parameters to be optimized. They must be elements of the matrices \(A_1\), \(A_2\), \(A_3\), i.e. be specified as parameters in the
paramscommand and be entered in themodelblock;\(y\) are the endogenous variables, specified in the
varcommand, whose (co)-variance enters the loss function;\(e\) are the exogenous stochastic shocks, specified in the
varexo- ommand;\(W\) is the weighting matrix;
The linear quadratic problem consists of choosing a subset of model parameters to minimize the weighted (co)-variance of a specified subset of endogenous variables, subject to a linear law of motion implied by the first order conditions of the model. A few things are worth mentioning. First, \(y\) denotes the selected endogenous variables’ deviations from their steady state, i.e. in case they are not already mean 0 the variables entering the loss function are automatically demeaned so that the centered second moments are minimized. Second,
osronly solves linear quadratic problems of the type resulting from combining the specified quadratic loss function with a first order approximation to the model’s equilibrium conditions. The reason is that the first order state-space representation is used to compute the unconditional (co)-variances. Hence,osrwill automatically selectorder=1. Third, because the objective involves minimizing a weighted sum of unconditional second moments, those second moments must be finite. In particular, unit roots in \(y\) are not allowed.The subset of the model parameters over which the optimal simple rule is to be optimized, \(\gamma\), must be listed with
osr_params.The weighting matrix \(W\) used for the quadratic objective function is specified in the
optim_weightsblock. By attaching weights to endogenous variables, the subset of endogenous variables entering the objective function, \(y\), is implicitly specified.The linear quadratic problem is solved using the numerical optimizer specified with
opt_algo.Options
The
osrcommand will subsequently runstoch_simuland accepts the same options, including restricting the endogenous variables by listing them after the command, asstoch_simul(see Stochastic solution and simulation) plus- opt_algo = INTEGER¶
Specifies the optimizer for minimizing the objective function. The same solvers as for
mode_compute(seemode_compute) are available, except for5,6, and10.
- optim = (NAME, VALUE, ...)
A list of NAME`` and VALUE pairs. Can be used to set options for the optimization routines. The set of available options depends on the selected optimization routine (i.e. on the value of option
opt_algo). Seeoptim.
- maxit = INTEGER
Determines the maximum number of iterations used in
opt_algo=4. This option is now deprecated and will be removed in a future release of Dynare. Useoptiminstead to set optimizer-specific values. Default:1000.
- tolf = DOUBLE
Convergence criterion for termination based on the function value used in
opt_algo=4. Iteration will cease when it proves impossible to improve the function value by more than tolf. This option is now deprecated and will be removed in a future release of Dynare. Useoptiminstead to set optimizer-specific values. Default:1e-7.
- analytic_derivation
Triggers estimation with analytic gradient of the objective function.
- analytic_derivation_mode = INTEGER¶
See :opt:analytic_derivation_mode.
- silent_optimizer
See
silent_optimizer.
- huge_number = DOUBLE
Value for replacing the infinite bounds on parameters by finite numbers. Used by some optimizers for numerical reasons (see
huge_number). Users need to make sure that the optimal parameters are not larger than this value. Default:1e7.
The value of the objective is stored in the variable
oo_.osr.objective_functionand the value of parameters at the optimum is stored inoo_.osr.optim_params. See below for more details.After running
osrthe parameters entering the simple rule will be set to their optimal value so that subsequent runs ofstoch_simulwill be conducted at these values.
- Command: osr_params PARAMETER_NAME...;¶
This command declares parameters to be optimized byosr.
- Block: optim_weights ;¶
This block specifies quadratic objectives for optimal policy problems.More precisely, this block specifies the nonzero elements of the weight matrix \(W\) used in the quadratic form of the objective function in
osr.An element of the diagonal of the weight matrix is given by a line of the form:
VARIABLE_NAME EXPRESSION;
An off-the-diagonal element of the weight matrix is given by a line of the form:
VARIABLE_NAME, VARIABLE_NAME EXPRESSION;
Example
var y inflation r; varexo y_ inf_; parameters delta sigma alpha kappa gammarr gammax0 gammac0 gamma_y_ gamma_inf_; delta = 0.44; kappa = 0.18; alpha = 0.48; sigma = -0.06; gammarr = 0; gammax0 = 0.2; gammac0 = 1.5; gamma_y_ = 8; gamma_inf_ = 3; model(linear); y = delta * y(-1) + (1-delta)*y(+1)+sigma *(r - inflation(+1)) + y_; inflation = alpha * inflation(-1) + (1-alpha) * inflation(+1) + kappa*y + inf_; r = gammax0*y(-1)+gammac0*inflation(-1)+gamma_y_*y_+gamma_inf_*inf_; end; shocks; var y_; stderr 0.63; var inf_; stderr 0.4; end; optim_weights; inflation 1; y 1; y, inflation 0.5; end; osr_params gammax0 gammac0 gamma_y_ gamma_inf_; osr y;
- Block: osr_params_bounds ;¶
This block declares lower and upper bounds for parameters in the optimal simple rule. If not specified the optimization is unconstrained.Each line has the following syntax:
PARAMETER_NAME, LOWER_BOUND, UPPER_BOUND;
Note that the use of this block requires the use of a constrained optimizer, i.e. setting
opt_algoto1,2,5or9.Example
osr_params_bounds; gamma_inf_, 0, 2.5; end; osr(opt_algo=9) y;
- MATLAB/Octave variable: oo_.osr.objective_function¶
After an execution of the
osrcommand, this variable contains the value of the objective under optimal policy.
- MATLAB/Octave variable: oo_.osr.optim_params¶
After an execution of the
osrcommand, this variable contains the value of parameters at the optimum, stored in fields of the formoo_.osr.optim_params.PARAMETER_NAME.
- MATLAB/Octave variable: M_.osr.param_names¶
After an execution of the
osrcommand, this cell contains the names of the parameters.
- MATLAB/Octave variable: M_.osr.param_indices¶
After an execution of the
osrcommand, this vector contains the indices of the OSR parameters inM_.params.
- MATLAB/Octave variable: M_.osr.param_bounds¶
After an execution of the
osrcommand, this two by number of OSR parameters matrix contains the lower and upper bounds of the parameters in the first and second column, respectively.
- MATLAB/Octave variable: M_.osr.variable_weights¶
After an execution of the
osrcommand, this sparse matrix contains the weighting matrix associated with the variables in the objective function.
- MATLAB/Octave variable: M_.osr.variable_indices¶
After an execution of the
osrcommand, this vector contains the indices of the variables entering the objective function inM_.endo_names.
4.22. Sensitivity and identification analysis¶
Dynare provides an interface to the global sensitivity analysis (GSA) toolbox (developed by the Joint Research Center (JRC) of the European Commission), which is now part of the official Dynare distribution. The GSA toolbox can be used to answer the following questions:
What is the domain of structural coefficients assuring the stability and determinacy of a DSGE model?
Which parameters mostly drive the fit of, e.g., GDP and which the fit of inflation? Is there any conflict between the optimal fit of one observed series versus another?
How to represent in a direct, albeit approximated, form the relationship between structural parameters and the reduced form of a rational expectations model?
The discussion of the methodologies and their application is described in Ratto (2008).
With respect to the previous version of the toolbox, in order to work properly, the GSA toolbox no longer requires that the Dynare estimation environment is set up.
4.22.1. Performing sensitivity analysis¶
- Command: sensitivity ;¶
- Command: sensitivity(OPTIONS...);
This command triggers sensitivity analysis on a DSGE model.Sampling Options
- Nsam = INTEGER¶
Size of the Monte-Carlo sample. Default:
2048.
- ilptau = INTEGER¶
If equal to
1, use \(LP_\tau\) quasi-Monte-Carlo. If equal to0, use LHS Monte-Carlo. Default:1.
- pprior = INTEGER¶
If equqal to
1, sample from the prior distributions. If equal to0, sample from the multivariate normal \(N(\bar{\theta},\Sigma)\), where \(\bar{\theta}\) is the posterior mode and \(\Sigma=H^{-1}\), \(H\) is the Hessian at the mode. Default:1.
- prior_range = INTEGER¶
If equal to
1, sample uniformly from prior ranges. If equal to0, sample from prior distributions. Default:1.
- morris = INTEGER¶
If equal to
0, ANOVA mapping (Type I error) If equal to1, Screening analysis (Type II error). If equal to2, Analytic derivatives (similar to Type II error, only valid when identification=1). The ANOVA mapping requires the SS-ANOVA-R MATLAB Toolbox available at https://joint-research-centre.ec.europa.eu/system/files/2025-01/ss_anova_recurs.zip Default:1whenidentification=1,0otherwise.
- morris_nliv = INTEGER¶
Number of levels in Morris design. Default:
6.
- morris_ntra = INTEGER¶
Number trajectories in Morris design. Default:
20.
- ppost = INTEGER¶
If equal to
1, use Metropolis posterior sample. If equal to0, do not use Metropolis posterior sample. Default:0.NB: This overrides any other sampling option.
- neighborhood_width = DOUBLE¶
When
pprior=0andppost=0, allows for the sampling of parameters around the value specified in themode_file, in the range \(\texttt{xparam1} \pm \left \vert \texttt{xparam1} \times \texttt{neighborhood\_width} \right \vert\). Default:0.
Stability Mapping Options
- stab = INTEGER¶
If equal to
1, perform stability mapping. If equal to0, do not perform stability mapping. Default:1.
- load_stab = INTEGER¶
If equal to
1, load a previously created sample. If equal to0, generate a new sample. Default:0.
- alpha2_stab = DOUBLE¶
Critical value for correlations \(\rho\) in filtered samples: plot couples of parmaters with \(\left\vert\rho\right\vert>\)
alpha2_stab. Default:0.
- pvalue_ks = DOUBLE¶
The threshold \(pvalue\) for significant Kolmogorov-Smirnov test (i.e. plot parameters with \(pvalue<\)
pvalue_ks). Default:0.001.
- pvalue_corr = DOUBLE¶
The threshold \(pvalue\) for significant correlation in filtered samples (i.e. plot bivariate samples when \(pvalue<\)
pvalue_corr). Default:1e-5.
Reduced Form Mapping Options
- redform = INTEGER¶
If equal to
1, prepare Monte-Carlo sample of reduced form matrices. If equal to0, do not prepare Monte-Carlo sample of reduced form matrices. Default:0.
- load_redform = INTEGER¶
If equal to
1, load previously estimated mapping. If equal to0, estimate the mapping of the reduced form model. Default:0.
- logtrans_redform = INTEGER¶
If equal to
1, use log-transformed entries. If equal to0, use raw entries. Default:0.
- threshold_redform = [DOUBLE DOUBLE]¶
The range over which the filtered Monte-Carlo entries of the reduced form coefficients should be analyzed. The first number is the lower bound and the second is the upper bound. An empty vector indicates that these entries will not be filtered. Default: empty.
- ksstat_redform = DOUBLE¶
Critical value for Smirnov statistics \(d\) when reduced form entries are filtered. Default:
0.001.
- alpha2_redform = DOUBLE¶
Critical value for correlations \(\rho\) when reduced form entries are filtered. Default:
1e-5.
- namendo = (VARIABLE_NAME...)¶
List of endogenous variables. ‘:’ indicates all endogenous variables. Default: empty.
- namlagendo = (VARIABLE_NAME...)¶
List of lagged endogenous variables. ‘:’ indicates all lagged endogenous variables. Analyze entries [namendo \(\times\) namlagendo] Default: empty.
- namexo = (VARIABLE_NAME...)¶
List of exogenous variables. ‘:’ indicates all exogenous variables. Analyze entries [namendo \(\times\) namexo]. Default: empty.
RMSE Options
- rmse = INTEGER¶
If equal to
1, perform RMSE analysis. If equal to0, do not perform RMSE analysis. Default:0.
- load_rmse = INTEGER¶
If equal to
1, load previous RMSE analysis. If equal to0, make a new RMSE analysis. Default:0.
- lik_only = INTEGER¶
If equal to
1, compute only likelihood and posterior. If equal to0, compute RMSE’s for all observed series. Default:0.
- var_rmse = (VARIABLE_NAME...)¶
List of observed series to be considered. ‘:’ indicates all observed variables. Default:
varobs.
- pfilt_rmse = DOUBLE¶
Filtering threshold for RMSE’s. Default:
0.1.
- istart_rmse = INTEGER¶
Value at which to start computing RMSE’s (use
2to avoid big intitial error). Default:presample+1.
- alpha_rmse = DOUBLE¶
Critical value for Smirnov statistics \(d\): plot parameters with \(d>\)
alpha_rmse. Default:0.001.
- alpha2_rmse = DOUBLE¶
Critical value for correlation \(\rho\): plot couples of parmaters with \(\left\vert\rho\right\vert=\)
alpha2_rmse. Default:1e-5.
- datafile = FILENAME
See datafile.
- nobs = INTEGER
- nobs = [INTEGER1:INTEGER2]
See
nobs.
- first_obs = INTEGER
See
first_obs.
- prefilter = INTEGER
See
prefilter.
- presample = INTEGER
See
presample.
- nograph
See
nograph.
- nodisplay
See
nodisplay.
- graph_format = FORMAT
- graph_format = ( FORMAT, FORMAT... )
See
graph_format.
- conf_sig = DOUBLE
See conf_sig.
- loglinear
See loglinear.
- mode_file = FILENAME
See
mode_file.
- kalman_algo = INTEGER
See
kalman_algo.
Identification Analysis Options
- identification = INTEGER¶
If equal to
1, performs identification analysis (forcingredform=0andmorris=1) If equal to0, no identification analysis. Default:0.
- morris = INTEGER
See
morris.
- morris_nliv = INTEGER
See
morris_nliv.
- morris_ntra = INTEGER
See
morris_ntra.
- load_ident_files = INTEGER¶
Loads previously performed identification analysis. Default:
0.
- useautocorr = INTEGER¶
Use autocorrelation matrices in place of autocovariance matrices in moments for identification analysis. Default:
0.
- ar = INTEGER
Maximum number of lags for moments in identification analysis. Default:
1.
- diffuse_filter = INTEGER
See
diffuse_filter.
- Command: dynare_sensitivity ;¶
- Command: dynare_sensitivity(OPTIONS...);
This is a deprecated alias for thesensitivitycommand.
4.22.2. IRF/Moment calibration¶
The irf_calibration and moment_calibration blocks allow
imposing implicit “endogenous” priors about IRFs and moments on the
model. The way it works internally is that any parameter draw that is
inconsistent with the “calibration” provided in these blocks is
discarded, i.e. assigned a prior density of 0. In the context of
dynare_sensitivity, these restrictions allow tracing out which
parameters are driving the model to satisfy or violate the given
restrictions.
IRF and moment calibration can be defined in irf_calibration and
moment_calibration blocks:
- Block: irf_calibration ;¶
- Block: irf_calibration(OPTIONS...);
This block allows defining IRF calibration criteria and is terminated byend;. To set IRF sign restrictions, the following syntax is used:VARIABLE_NAME(INTEGER), EXOGENOUS_NAME, -; VARIABLE_NAME(INTEGER:INTEGER), EXOGENOUS_NAME, +;
To set IRF restrictions with specific intervals, the following syntax is used:
VARIABLE_NAME(INTEGER), EXOGENOUS_NAME, [EXPRESSION, EXPRESSION]; VARIABLE_NAME(INTEGER:INTEGER), EXOGENOUS_NAME, [EXPRESSION, EXPRESSION];
When
(INTEGER:INTEGER)is used, the restriction is considered to be fulfilled by a logical OR. A list of restrictions must always be fulfilled with logical AND.Options
- relative_irf
See
relative_irf.
Example
irf_calibration; y(1:4), e_ys, [-50, 50]; //[first year response with logical OR] @#for ilag in 21:40 R_obs(@{ilag}), e_ys, [0, 6]; //[response from 5th to 10th years with logical AND] @#endfor end;
- Block: moment_calibration ;¶
- Block: moment_calibration(OPTIONS...);
This block allows defining moment calibration criteria. This block is terminated byend;, and contains lines of the form:VARIABLE_NAME1, VARIABLE_NAME2(+/-INTEGER), [EXPRESSION, EXPRESSION]; VARIABLE_NAME1, VARIABLE_NAME2(+/-INTEGER), +/-; VARIABLE_NAME1, VARIABLE_NAME2(+/-(INTEGER:INTEGER)), [EXPRESSION, EXPRESSION]; VARIABLE_NAME1, VARIABLE_NAME2((-INTEGER:+INTEGER)), [EXPRESSION, EXPRESSION];
When
(INTEGER:INTEGER)is used, the restriction is considered to be fulfilled by a logical OR. A list of restrictions must always be fulfilled with logical AND. The moment restrictions generally apply to auto- and cross-correlations between variables. The only exception is a restriction on the unconditional variance of an endogenous variable, specified as shown in the example below.Example
moment_calibration; y_obs,y_obs, [0.5, 1.5]; //[unconditional variance] y_obs,y_obs(-(1:4)), +; //[sign restriction for first year autocorrelation with logical OR] @#for ilag in -2:2 y_obs,R_obs(@{ilag}), -; //[-2:2 cross correlation with logical AND] @#endfor @#for ilag in -4:4 y_obs,pie_obs(@{ilag}), -; //[-4_4 cross correlation with logical AND] @#endfor end;
4.22.3. Performing identification analysis¶
- Command: identification ;¶
- Command: identification(OPTIONS...);
This command triggers:Theoretical identification analysis based on
moments as in Iskrev (2010)
spectral density as in Qu and Tkachenko (2012)
minimal system as in Komunjer and Ng (2011)
reduced-form solution and linear rational expectation model as in Ratto and Iskrev (2011)
Note that for orders 2 and 3, all identification checks are based on the pruned state space system as in Mutschler (2015). That is, theoretical moments and spectrum are computed from the pruned ABCD-system, whereas the minimal system criteria is based on the first-order system, but augmented by the theoretical (pruned) mean at order 2 or 3.
Identification strength analysis based on (theoretical or simulated) curvature of moment information matrix as in Ratto and Iskrev (2011)
Parameter checks based on nullspace and multicorrelation coefficients to determine which (combinations of) parameters are involved
General Options
- order = 1|2|3¶
Order of approximation. At orders 2 and 3 identification is based on the pruned state space system. Note that the order set in other functions does not overwrite the default. Default:
1.
- parameter_set = OPTION
See
parameter_setfor possible values. Default:prior_mean.
- prior_mc = INTEGER¶
Size of Monte-Carlo sample. Default:
1.
- prior_range = INTEGER
Triggers uniform sample within the range implied by the prior specifications (when
prior_mc>1). Default:0.
- advanced = INTEGER¶
If set to
1, shows a more detailed analysis, comprised of an analysis for the linearized rational expectation model as well as the associated reduced form solution. Further performs a bruteforce search of the groups of parameters best reproducing the behavior of each single parameter. The maximum dimension of the group searched is triggered bymax_dim_cova_group. Default:0.
- max_dim_cova_group = INTEGER¶
In the brute force search (performed when
advanced=1) this option sets the maximum dimension of groups of parameters that best reproduce the behavior of each single model parameter. Default:2.
- gsa_sample_file = INTEGER|FILENAME¶
If equal to
0, do not use sample file. If equal to1, triggers gsa prior sample. If equal to2, triggers gsa Monte-Carlo sample (i.e. loads a sample corresponding topprior=0andppost=0in thedynare_sensitivityoptions). If equal toFILENAMEuses the provided path to a specific user defined sample file. Default:0.
- diffuse_filter
Deals with non-stationary cases. See
diffuse_filter.
Numerical Options
- analytic_derivation_mode = INTEGER
Different ways to compute derivatives either analytically or numerically. Possible values are:
0: efficient sylvester equation method to compute analytical derivatives
1: kronecker products method to compute analytical derivatives (only at order=1)
-1: numerical two-sided finite difference method to compute all identification Jacobians (numerical tolerance level is equal tooptions_.dynatol.x)
-2: numerical two-sided finite difference method to compute derivatives of steady state and dynamic model numerically, the identification Jacobians are then computed analytically (numerical tolerance level is equal tooptions_.dynatol.x)Default:
0.
- normalize_jacobians = INTEGER¶
If set to
1: Normalize Jacobian matrices by rescaling each row by its largest element in absolute value. Normalize Gram (or Hessian-type) matrices by transforming into correlation-type matrices. Default:1
- tol_rank = DOUBLE¶
Tolerance level used for rank computations. Default:
1.e-10.
- tol_deriv = DOUBLE¶
Tolerance level for selecting non-zero columns in Jacobians. Default:
1.e-8.
- tol_sv = DOUBLE¶
Tolerance level for selecting non-zero singular values. Default:
1.e-3.
- schur_vec_tol = DOUBLE
See
schur_vec_tol.
Identification Strength Options
- no_identification_strength¶
Disables computations of identification strength analysis based on sample information matrix.
- periods = INTEGER
When the analytic Hessian is not available (i.e. with missing values or diffuse Kalman filter or univariate Kalman filter), this triggers the length of stochastic simulation to compute Simulated Moments Uncertainty. Default:
300.
- replic = INTEGER
When the analytic Hessian is not available, this triggers the number of replicas to compute Simulated Moments Uncertainty. Default:
100.
Moments Options
- no_identification_moments¶
Disables computations of identification check based on Iskrev (2010)’s J, i.e. derivative of first two moments.
- ar = INTEGER
Number of lags of computed autocovariances/autocorrelations (theoretical moments) in Iskrev (2010)’s J criteria. Default:
1.
- useautocorr = INTEGER
If equal to
1, compute derivatives of autocorrelation. If equal to0, compute derivatives of autocovariances. Default:0.
Spectrum Options
- no_identification_spectrum¶
Disables computations of identification check based on Qu and Tkachenko (2012)’s G, i.e. Gram matrix of derivatives of first moment plus outer product of derivatives of spectral density.
- grid_nbr = INTEGER¶
Number of grid points in [-pi;pi] to approximate the integral to compute Qu and Tkachenko (2012)’s G criteria. Default:
5000.
Minimal State Space System Options
- no_identification_minimal¶
Disables computations of identification check based on Komunjer and Ng (2011)’s D, i.e. minimal state space system and observational equivalent spectral density transformations.
Misc Options
- nograph
See
nograph.
- nodisplay
See
nodisplay.
- graph_format = FORMAT
- graph_format = ( FORMAT, FORMAT... )
See
graph_format.
- tex
See
tex.
Debug Options
- load_ident_files = INTEGER
If equal to
1, allow Dynare to load previously computed analyzes. Default:0.
- lik_init = INTEGER
See
lik_init.
- kalman_algo = INTEGER
See
kalman_algo.
- no_identification_reducedform¶
Disables computations of identification check based on steady state and reduced-form solution.
- checks_via_subsets = INTEGER¶
If equal to
1: finds problematic parameters in a bruteforce fashion: It computes the rank of the Jacobians for all possible parameter combinations. If the rank condition is not fullfilled, these parameter sets are flagged as non-identifiable. The maximum dimension of the group searched is triggered bymax_dim_subsets_groups. Default:0.
- max_dim_subsets_groups = INTEGER¶
Sets the maximum dimension of groups of parameters for which the above bruteforce search is performed. Default:
4.
4.22.4. Types of analysis and output files¶
The sensitivity analysis toolbox includes several types of
analyses. Sensitivity analysis results are saved locally in
<mod_file>/gsa, where <mod_file>.mod is the name of the Dynare
model file.
4.22.4.1. Sampling¶
The following binary files are produced:
<mod_file>_prior.mat: this file stores information about the analyses performed sampling from the prior, i.e.pprior=1andppost=0;
<mod_file>_mc.mat: this file stores information about the analyses performed sampling from multivariate normal, i.e.pprior=0andppost=0;
<mod_file>_post.mat: this file stores information about analyses performed using the Metropolis posterior sample, i.e.ppost=1.
4.22.4.2. Stability Mapping¶
Figure files produced are of the form <mod_file>_prior_*.fig and
store results for stability mapping from prior Monte-Carlo samples:
<mod_file>_prior_stable.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample fulfilling Blanchard-Kahn conditions (blue color) with the cdf of the rest of the sample (red color), i.e. either instability or indeterminacy or the solution could not be found (e.g. the steady state solution could not be found by the solver);
<mod_file>_prior_indeterm.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample producing indeterminacy (red color) with the cdf of the rest of the sample (blue color);
<mod_file>_prior_unstable.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample producing explosive roots (red color) with the cdf of the rest of the sample (blue color);
<mod_file>_prior_wrong.fig: plots of the Smirnov test and the correlation analyses confronting the cdf of the sample where the solution could not be found (e.g. the steady state solution could not be found by the solver - red color) with the cdf of the rest of the sample (blue color);
<mod_file>_prior_calib.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where IRF/moment restrictions are matched (blue color) with the cdf where IRF/moment restrictions are NOT matched (red color);
Similar conventions apply for <mod_file>_mc_*.fig files, obtained
when samples from multivariate normal are used.
4.22.4.3. IRF/Moment restrictions¶
The following binary files are produced:
<mod_file>_prior_restrictions.mat: this file stores information about the IRF/moment restriction analysis performed sampling from the prior ranges, i.e.pprior=1andppost=0;
<mod_file>_mc_restrictions.mat: this file stores information about the IRF/moment restriction analysis performed sampling from multivariate normal, i.e.pprior=0andppost=0;
<mod_file>_post_restrictions.mat: this file stores information about IRF/moment restriction analysis performed using the Metropolis posterior sample, i.e.ppost=1.
Figure files produced are of the form
<mod_file>_prior_irf_calib_*.fig and
<mod_file>_prior_moment_calib_*.fig and store results for mapping
restrictions from prior Monte-Carlo samples:
<mod_file>_prior_irf_calib_<ENDO_NAME>_vs_<EXO_NAME>_<PERIOD>.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where the individual IRF restriction<ENDO_NAME>vs.<EXO_NAME>at period(s)<PERIOD>is matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)
<mod_file>_prior_irf_calib_<ENDO_NAME>_vs_<EXO_NAME>_ALL.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where ALL the individual IRF restrictions for the same couple<ENDO_NAME>vs.<EXO_NAME>are matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)
<mod_file>_prior_irf_restrictions.fig: plots visual information on the IRF restrictions compared to the actual Monte Carlo realization from prior sample.
<mod_file>_prior_moment_calib_<ENDO_NAME1>_vs_<ENDO_NAME2>_<LAG>.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where the individual acf/ccf moment restriction<ENDO_NAME1>vs.<ENDO_NAME2>at lag(s)<LAG>is matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)
<mod_file>_prior_moment_calib_<ENDO_NAME>_vs_<EXO_NAME>_ALL.fig: plots of the Smirnov test and the correlation analyses splitting the sample fulfilling Blanchard-Kahn conditions, by confronting the cdf of the sample where ALL the individual acf/ccf moment restrictions for the same couple<ENDO_NAME1>vs.<ENDO_NAME2>are matched (blue color) with the cdf where the IRF restriction is NOT matched (red color)
<mod_file>_prior_moment_restrictions.fig: plots visual information on the moment restrictions compared to the actual Monte Carlo realization from prior sample.
Similar conventions apply for <mod_file>_mc_*.fig and
<mod_file>_post_*.fig files, obtained when samples from
multivariate normal or from posterior are used.
4.22.4.4. Reduced Form Mapping¶
When the option threshold_redform is not set, or it is empty (the
default), this analysis estimates a multivariate smoothing spline
ANOVA model (the ’mapping’) for the selected entries in the transition
matrix of the shock matrix of the reduce form first order solution of
the model. This mapping is done either with prior samples or with MC
samples with neighborhood_width. Unless neighborhood_width is
set with MC samples, the mapping of the reduced form solution forces
the use of samples from prior ranges or prior distributions, i.e.:
pprior=1 and ppost=0. It uses 250 samples to optimize
smoothing parameters and 1000 samples to compute the fit. The rest of
the sample is used for out-of-sample validation. One can also load a
previously estimated mapping with a new Monte-Carlo sample, to look at
the forecast for the new Monte-Carlo sample.
The following synthetic figures are produced:
<mod_file>_redform_<endo name>_vs_lags_*.fig: shows bar charts of the sensitivity indices for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus lagged endogenous variables (namlagendo); suffixlogindicates the results for log-transformed entries;
<mod_file>_redform_<endo name>_vs_shocks_*.fig: shows bar charts of the sensitivity indices for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus exogenous variables (namexo); suffixlogindicates the results for log-transformed entries;
<mod_file>_redform_gsa(_log).fig: shows bar chart of all sensitivity indices for each parameter: this allows one to notice parameters that have a minor effect for any of the reduced form coefficients.
Detailed results of the analyses are shown in the subfolder
<mod_file>/gsa/redform_prior for prior samples and in
<mod_file>/gsa/redform_mc for MC samples with option
neighborhood_width, where the detailed results of the estimation
of the single functional relationships between parameters
\(\theta\) and reduced form coefficient (denoted as \(y\)
hereafter) are stored in separate directories named as:
<namendo>_vs_<namlagendo>, for the entries of the transition matrix;
<namendo>_vs_<namexo>, for entries of the matrix of the shocks.
The following files are stored in each directory (we stick with prior sample but similar conventions are used for MC samples):
<mod_file>_prior_<namendo>_vs_<namexo>.fig: histogram and CDF plot of the MC sample of the individual entry of the shock matrix, in sample and out of sample fit of the ANOVA model;
<mod_file>_prior_<namendo>_vs_<namexo>_map_SE.fig: for entries of the shock matrix it shows graphs of the estimated first order ANOVA terms \(y = f(\theta_i)\) for each deep parameter \(\theta_i\);
<mod_file>_prior_<namendo>_vs_<namlagendo>.fig: histogram and CDF plot of the MC sample of the individual entry of the transition matrix, in sample and out of sample fit of the ANOVA model;
<mod_file>_prior_<namendo>_vs_<namlagendo>_map_SE.fig: for entries of the transition matrix it shows graphs of the estimated first order ANOVA terms \(y = f(\theta_i)\) for each deep parameter \(\theta_i\);
<mod_file>_prior_<namendo>_vs_<namexo>_map.mat,<mod_file>_<namendo>_vs_<namlagendo>_map.mat: these files store info in the estimation;
When option logtrans_redform is set, the ANOVA estimation is
performed using a log-transformation of each y. The ANOVA mapping is
then transformed back onto the original scale, to allow comparability
with the baseline estimation. Graphs for this log-transformed case,
are stored in the same folder in files denoted with the _log
suffix.
When the option threshold_redform is set, the analysis is
performed via Monte Carlo filtering, by displaying parameters that
drive the individual entry y inside the range specified in
threshold_redform. If no entry is found (or all entries are in the
range), the MCF algorithm ignores the range specified in
threshold_redform and performs the analysis splitting the MC
sample of y into deciles. Setting threshold_redform=[-inf inf]
triggers this approach for all y’s.
Results are stored in subdirectories of <mod_file>/gsa/redform_prior named
<mod_file>_prior_<namendo>_vs_<namlagendo>_threshold, for the entries of the transition matrix;
<mod_file>_prior_<namendo>_vs_<namexo>_threshold, for entries of the matrix of the shocks.
The files saved are named:
<mod_file>_prior_<namendo>_vs_<namexo>_threshold.fig,<mod_file>_<namendo>_vs_<namlagendo>_threshold.fig: graphical outputs;
<mod_file>_prior_<namendo>_vs_<namexo>_threshold.mat,<mod_file>_<namendo>_vs_<namlagendo>_threshold.mat: info on the analysis;
4.22.4.5. RMSE¶
The RMSE analysis can be performed with different types of sampling options:
When
pprior=1andppost=0, the toolbox analyzes the RMSEs for the Monte-Carlo sample obtained by sampling parameters from their prior distributions (or prior ranges): this analysis provides some hints about what parameter drives the fit of which observed series, prior to the full estimation;When
pprior=0andppost=0, the toolbox analyzes the RMSEs for a multivariate normal Monte-Carlo sample, with covariance matrix based on the inverse Hessian at the optimum: this analysis is useful when maximum likelihood estimation is done (i.e. no Bayesian estimation);When
ppost=1the toolbox analyzes the RMSEs for the posterior sample obtained by Dynare’s Metropolis procedure.
The use of cases 2 and 3 requires an estimation step beforehand. To
facilitate the sensitivity analysis after estimation, the
dynare_sensitivity command also allows you to indicate some
options of the estimation command. These are:
datafile
nobs
first_obs
prefilter
presample
nograph
nodisplay
graph_format
conf_sig
loglinear
mode_file
Binary files produced my RMSE analysis are:
<mod_file>_prior_*.mat: these files store the filtered and smoothed variables for the prior Monte-Carlo sample, generated when doing RMSE analysis (pprior=1andppost=0);
<mode_file>_mc_*.mat: these files store the filtered and smoothed variables for the multivariate normal Monte-Carlo sample, generated when doing RMSE analysis (pprior=0andppost=0).
Figure files <mod_file>_rmse_*.fig store results for the RMSE analysis.
<mod_file>_rmse_prior*.fig: save results for the analysis using prior Monte-Carlo samples;
<mod_file>_rmse_mc*.fig: save results for the analysis using multivariate normal Monte-Carlo samples;
<mod_file>_rmse_post*.fig: save results for the analysis using Metropolis posterior samples.
The following types of figures are saved (we show prior sample to fix ideas, but the same conventions are used for multivariate normal and posterior):
<mod_file>_rmse_prior_params_*.fig: for each parameter, plots the cdfs corresponding to the best 10% RMSEs of each observed series (only those cdfs below the significance thresholdalpha_rmse);
<mod_file>_rmse_prior_<var_obs>_*.fig: if a parameter significantly affects the fit ofvar_obs, all possible trade-off’s with other observables for same parameter are plotted;
<mod_file>_rmse_prior_<var_obs>_map.fig: plots the MCF analysis of parameters significantly driving the fit the observed seriesvar_obs;
<mod_file>_rmse_prior_lnlik*.fig: for each observed series, plots in BLUE the cdf of the log-likelihood corresponding to the best 10% RMSEs, in RED the cdf of the rest of the sample and in BLACK the cdf of the full sample; this allows one to see the presence of some idiosyncratic behavior;
<mod_file>_rmse_prior_lnpost*.fig: for each observed series, plots in BLUE the cdf of the log-posterior corresponding to the best 10% RMSEs, in RED the cdf of the rest of the sample and in BLACK the cdf of the full sample; this allows one to see idiosyncratic behavior;
<mod_file>_rmse_prior_lnprior*.fig: for each observed series, plots in BLUE the cdf of the log-prior corresponding to the best 10% RMSEs, in RED the cdf of the rest of the sample and in BLACK the cdf of the full sample; this allows one to see idiosyncratic behavior;
<mod_file>_rmse_prior_lik.fig: whenlik_only=1, this shows the MCF tests for the filtering of the best 10% log-likelihood values;
<mod_file>_rmse_prior_post.fig: whenlik_only=1, this shows the MCF tests for the filtering of the best 10% log-posterior values.
4.22.4.6. Screening Analysis¶
Screening analysis does not require any additional options with respect to those listed in Sampling Options. The toolbox performs all the analyses required and displays results.
The results of the screening analysis with Morris sampling design are
stored in the subfolder <mod_file>/gsa/screen. The data file
<mod_file>_prior stores all the information of the analysis
(Morris sample, reduced form coefficients, etc.).
Screening analysis merely concerns reduced form coefficients. Similar synthetic bar charts as for the reduced form analysis with Monte-Carlo samples are saved:
<mod_file>_redform_<endo name>_vs_lags_*.fig: shows bar charts of the elementary effect tests for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus lagged endogenous variables (namlagendo);
<mod_file>_redform_<endo name>_vs_shocks_*.fig: shows bar charts of the elementary effect tests for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus exogenous variables (namexo);
<mod_file>_redform_screen.fig: shows bar chart of all elementary effect tests for each parameter: this allows one to identify parameters that have a minor effect for any of the reduced form coefficients.
4.22.4.7. Identification Analysis¶
Setting the option identification=1, an identification analysis
based on theoretical moments is performed. Sensitivity plots are
provided that allow to infer which parameters are most likely to be
less identifiable.
Prerequisite for properly running all the identification routines, is
the keyword identification; in the Dynare model file. This keyword
triggers the computation of analytic derivatives of the model with
respect to estimated parameters and shocks. This is required for
option morris=2, which implements Iskrev (2010) identification
analysis.
For example, the placing:
identification;
dynare_sensitivity(identification=1, morris=2);
in the Dynare model file triggers identification analysis using analytic derivatives as in Iskrev (2010), jointly with the mapping of the acceptable region.
The identification analysis with derivatives can also be triggered by the single command:
identification;
This does not do the mapping of acceptable regions for the model and
uses the standard random sampler of Dynare. Additionally, using only
identification; adds two additional identification checks: namely,
of Qu and Tkachenko (2012) based on the spectral density and of
Komunjer and Ng (2011) based on the minimal state space system.
It completely offsets any use of the sensitivity analysis toolbox.
4.23. Markov-switching SBVAR¶
Given a list of variables, observed variables and a data file, Dynare can be used to solve a Markov-switching SBVAR model according to Sims, Waggoner and Zha (2008). [11] Having done this, you can create forecasts and compute the marginal data density, regime probabilities, IRFs, and variance decomposition of the model.
The commands have been modularized, allowing for multiple calls to the
same command within a <mod_file>.mod file. The default is to use
<mod_file> to tag the input (output) files used (produced) by the
program. Thus, to call any command more than once within a
<mod_file>.mod file, you must use the *_tag options described
below.
- Command: markov_switching(OPTIONS...);¶
Declares the Markov state variable information of a Markov-switching SBVAR model.Options
- chain = INTEGER¶
The Markov chain considered. Default:
none.
- number_of_regimes = INTEGER¶
Specifies the total number of regimes in the Markov Chain. This is a required option.
- duration = DOUBLE | [ROW VECTOR OF DOUBLES]¶
The duration of the regimes or regimes. This is a required option. When passed a scalar real number, it specifies the average duration for all regimes in this chain. When passed a vector of size equal
number_of_regimes, it specifies the average duration of the associated regimes (1:number_of_regimes) in this chain. An absorbing state can be specified through therestrictionsoption.
- restrictions = [[ROW VECTOR OF 3 DOUBLES],[ROW VECTOR OF 3 DOUBLES],...]¶
Provides restrictions on this chain’s regime transition matrix. Its vector argument takes three inputs of the form:
[current_period_regime, next_period_regime, transition_probability].The first two entries are positive integers, and the third is a non-negative real in the set [0,1]. If restrictions are specified for every transition for a regime, the sum of the probabilities must be 1. Otherwise, if restrictions are not provided for every transition for a given regime the sum of the provided transition probabilities msut be <1. Regardless of the number of lags, the restrictions are specified for parameters at time
tsince the transition probability for a parameter at t is equal to that of the parameter att-1.
In case of estimating a MS-DSGE model, [12] in addition the following options are allowed:
- parameters = [LIST OF PARAMETERS]¶
This option specifies which parameters are controlled by this Markov Chain.
- number_of_lags = DOUBLE¶
Provides the number of lags that each parameter can take within each regime in this chain.
Example
markov_switching(chain=1, duration=2.5, restrictions=[[1,3,0],[3,1,0]]);
Specifies a Markov-switching BVAR with a first chain with 3 regimes that all have a duration of 2.5 periods. The probability of directly going from regime 1 to regime 3 and vice versa is 0.
Example
markov_switching(chain=2, number_of_regimes=3, duration=[0.5, 2.5, 2.5], parameter=[alpha, rho], number_of_lags=2, restrictions=[[1,3,0],[3,3,1]]);
Specifies a Markov-switching DSGE model with a second chain with 3 regimes that have durations of 0.5, 2.5, and 2.5 periods, respectively. The switching parameters are
alphaandrho. The probability of directly going from regime 1 to regime 3 is 0, while regime 3 is an absorbing state.
- Command: svar(OPTIONS...);¶
Each Markov chain can control the switching of a set of parameters. We allow the parameters to be divided equation by equation and by variance or slope and intercept.Options
- coefficients¶
Specifies that only the slope and intercept in the given equations are controlled by the given chain. One, but not both, of
coefficientsorvariancesmust appear. Default:none.
- variances¶
Specifies that only variances in the given equations are controlled by the given chain. One, but not both, of
coefficientsorvariancesmust appear. Default:none.
- equations¶
Defines the equation controlled by the given chain. If not specified, then all equations are controlled by
chain. Default:none.
- chain = INTEGER
Specifies a Markov chain defined by
markov_switching. Default:none.
- Command: sbvar(OPTIONS...);¶
To be documented. For now, see the wiki: https://archives.dynare.org/DynareWiki/SbvarOptionsOptions
datafile,freq,initial_year,initial_subperiod,final_year,final_subperiod,data,vlist,vlistlog,vlistper,restriction_fname,nlags,cross_restrictions,contemp_reduced_form,real_pseudo_forecast,no_bayesian_prior,dummy_obs,nstates,indxscalesstates,alpha,beta,gsig2_lmdm,q_diag,flat_prior,ncsk,nstd,ninv,indxparr,indxovr,aband,indxap,apband,indximf,indxfore,foreband,indxgforhat,indxgimfhat,indxestima,indxgdls,eq_ms,cms,ncms,eq_cms,tlindx,tlnumber,cnum,forecast,coefficients_prior_hyperparameters
- Block: svar_identification ;¶
This block is terminated byend;and contains lines of the form:UPPER_CHOLESKY; LOWER_CHOLESKY; EXCLUSION CONSTANTS; EXCLUSION LAG INTEGER; EQUATION INTEGER, VARIABLE_NAME [[,] VARIABLE_NAME...]; RESTRICTION EQUATION INTEGER, EXPRESSION = EXPRESSION;
To be documented. For now, see the wiki: https://archives.dynare.org/DynareWiki/MarkovSwitchingInterface
- Command: ms_estimation(OPTIONS...);¶
Triggers the creation of an initialization file for, and the estimation of, a Markov-switching SBVAR model. At the end of the run, the \(A^0\), \(A^+\), \(Q\) and \(\zeta\) matrices are contained in theoo_.msstructure.General Options
- file_tag = FILENAME¶
The portion of the filename associated with this run. This will create the model initialization file,
init_<file_tag>.dat. Default:<mod_file>.
- output_file_tag = FILENAME¶
The portion of the output filename that will be assigned to this run. This will create, among other files,
est_final_<output_file_tag>.out,est_intermediate_<output_file_tag>.out. Default:<file_tag>.
- no_create_init¶
Do not create an initialization file for the model. Passing this option will cause the Initialization Options to be ignored. Further, the model will be generated from the output files associated with the previous estimation run (i.e.
est_final_<file_tag>.out,est_intermediate_<file_tag>.outorinit_<file_tag>.dat, searched for in sequential order). This functionality can be useful for continuing a previous estimation run to ensure convergence was reached or for reusing an initialization file. NB: If this option is not passed, the files from the previous estimation run will be overwritten. Default: off (i.e. create initialization file)
Initialization Options
- coefficients_prior_hyperparameters = [DOUBLE1 DOUBLE2 ... DOUBLE6]¶
Sets the hyper parameters for the model. The six elements of the argument vector have the following interpretations:
1Overall tightness for \(A^0\) and \(A^+\).
2Relative tightness for \(A^+\).
3Relative tightness for the constant term.
4Tightness on lag decay (range: 1.2 - 1.5); a faster decay produces better inflation process.
5Weight on nvar sums of coeffs dummy observations (unit roots).
6Weight on single dummy initial observation including constant.
Default:
[1.0 1.0 0.1 1.2 1.0 1.0]
- freq = INTEGER | monthly | quarterly | yearly¶
Frequency of the data (e.g.
monthly, 12). Default:4.
- initial_year = INTEGER¶
The first year of data. Default:
none.
- initial_subperiod = INTEGER¶
The first period of data (i.e. for quarterly data, an integer in
[1,4]). Default:1.
- final_year = INTEGER¶
The last year of data. Default: Set to encompass entire dataset.
- final_subperiod = INTEGER¶
The final period of data (i.e. for monthly data, an integer in
[1,12]. Default: When final_year is also missing, set to encompass entire dataset; whenfinal_yearis indicated, set to the maximum number of subperiods given the frequency (i.e. 4 for quarterly data, 12 for monthly,…).
- datafile = FILENAME
See datafile.
- xls_sheet = QUOTED_STRING
See
xls_sheet.
- xls_range = RANGE
See
xls_range.
- nlags = INTEGER¶
The number of lags in the model. Default:
1.
- cross_restrictions¶
Use cross \(A^0\) and \(A^+\) restrictions. Default:
off.
- contemp_reduced_form¶
Use contemporaneous recursive reduced form. Default:
off.
- no_bayesian_prior¶
Do not use Bayesian prior. Default:
off(i.e. use Bayesian prior).
- alpha = INTEGER¶
Alpha value for squared time-varying structural shock lambda. Default:
1.
- beta = INTEGER¶
Beta value for squared time-varying structural shock lambda. Default:
1.
- gsig2_lmdm = INTEGER¶
The variance for each independent \(\lambda\) parameter under
SimsZharestrictions. Default:50^2.
- specification = sims_zha | none¶
This controls how restrictions are imposed to reduce the number of parameters. Default:
Random Walk.
Estimation Options
- convergence_starting_value = DOUBLE¶
This is the tolerance criterion for convergence and refers to changes in the objective function value. It should be rather loose since it will gradually be tightened during estimation. Default:
1e-3.
- convergence_ending_value = DOUBLE¶
The convergence criterion ending value. Values much smaller than square root machine epsilon are probably overkill. Default:
1e-6.
- convergence_increment_value = DOUBLE¶
Determines how quickly the convergence criterion moves from the starting value to the ending value. Default:
0.1.
- max_iterations_starting_value = INTEGER¶
This is the maximum number of iterations allowed in the hill-climbing optimization routine and should be rather small since it will gradually be increased during estimation. Default:
50.
- max_iterations_increment_value = DOUBLE¶
Determines how quickly the maximum number of iterations is increased. Default:
2.
- max_block_iterations = INTEGER¶
The parameters are divided into blocks and optimization proceeds over each block. After a set of blockwise optimizations are performed, the convergence criterion is checked and the blockwise optimizations are repeated if the criterion is violated. This controls the maximum number of times the blockwise optimization can be performed. Note that after the blockwise optimizations have converged, a single optimization over all the parameters is performed before updating the convergence value and maximum number of iterations. Default:
100.
- max_repeated_optimization_runs = INTEGER¶
The entire process described by
max_block_iterationsis repeated until improvement has stopped. This is the maximum number of times the process is allowed to repeat. Set this to0to not allow repetitions. Default:10.
- function_convergence_criterion = DOUBLE¶
The convergence criterion for the objective function when
max_repeated_optimizations_runsis positive. Default:0.1.
- parameter_convergence_criterion = DOUBLE¶
The convergence criterion for parameter values when
max_repeated_optimizations_runsis positive. Default:0.1.
- number_of_large_perturbations = INTEGER¶
The entire process described by
max_block_iterationsis repeated with random starting values drawn from the posterior. This specifies the number of random starting values used. Set this to0to not use random starting values. A larger number should be specified to ensure that the entire parameter space has been covered. Default:5.
- number_of_small_perturbations = INTEGER¶
The number of small perturbations to make after the large perturbations have stopped improving. Setting this number much above
10is probably overkill. Default:5.
- number_of_posterior_draws_after_perturbation = INTEGER¶
The number of consecutive posterior draws to make when producing a small perturbation. Because the posterior draws are serially correlated, a small number will result in a small perturbation. Default:
1.
- max_number_of_stages = INTEGER¶
The small and large perturbation are repeated until improvement has stopped. This specifies the maximum number of stages allowed. Default:
20.
- random_function_convergence_criterion = DOUBLE¶
The convergence criterion for the objective function when
number_of_large_perturbationsis positive. Default:0.1.
- random_parameter_convergence_criterion = DOUBLE¶
The convergence criterion for parameter values when
number_of_large_perturbationsis positive. Default:0.1.
Example
ms_estimation(datafile=data, initial_year=1959, final_year=2005, nlags=4, max_repeated_optimization_runs=1, max_number_of_stages=0); ms_estimation(file_tag=second_run, datafile=data, initial_year=1959, final_year=2005, nlags=4, max_repeated_optimization_runs=1, max_number_of_stages=0); ms_estimation(file_tag=second_run, output_file_tag=third_run, no_create_init, max_repeated_optimization_runs=5, number_of_large_perturbations=10);
- Command: ms_simulation ;¶
- Command: ms_simulation(OPTIONS...);
Simulates a Markov-switching SBVAR model.Options
- file_tag = FILENAME
The portion of the filename associated with the
ms_estimationrun. Default:<mod_file>.
- output_file_tag = FILENAME
The portion of the output filename that will be assigned to this run. Default:
<file_tag>.
- mh_replic = INTEGER
The number of draws to save. Default:
10,000.
- drop = INTEGER
The number of burn-in draws. Default:
0.1*mh_replic*thinning_factor.
- thinning_factor = INTEGER¶
The total number of draws is equal to
thinning_factor*mh_replic+drop. Default:1.
- adaptive_mh_draws = INTEGER¶
Tuning period for Metropolis-Hastings draws. Default:
30,000.
- save_draws¶
Save all elements of \(A^0\), \(A^+\), \(Q\), and \(\zeta\), to a file named
draws_<<file_tag>>.outwith each draw on a separate line. A file that describes how these matrices are laid out is contained indraws_header_<<file_tag>>.out. A file calledload_flat_file.mis provided to simplify loading the saved files into the corresponding variablesA0,Aplus,Q, andZetain your MATLAB/Octave workspace. Default:off.
Example
ms_simulation(file_tag=second_run); ms_simulation(file_tag=third_run, mh_replic=5000, thinning_factor=3);
- Command: ms_compute_mdd ;¶
- Command: ms_compute_mdd(OPTIONS...);
Computes the marginal data density of a Markov-switching SBVAR model from the posterior draws. At the end of the run, the Muller and Bridged log marginal densities are contained in theoo_.msstructure.Options
- file_tag = FILENAME
See
file_tag.
- output_file_tag = FILENAME
See
output_file_tag.
- simulation_file_tag = FILENAME¶
The portion of the filename associated with the simulation run. Default:
<file_tag>.
- proposal_type = INTEGER¶
The proposal type:
1Gaussian.
2Power.
3Truncated Power.
4Step.
5Truncated Gaussian.
Default:
3
- proposal_lower_bound = DOUBLE¶
The lower cutoff in terms of probability. Not used for
proposal_typein[1,2]. Required for all other proposal types. Default:0.1.
- proposal_upper_bound = DOUBLE¶
The upper cutoff in terms of probability. Not used for
proposal_typeequal to1. Required for all other proposal types. Default:0.9.
- mdd_proposal_draws = INTEGER¶
The number of proposal draws. Default:
100,000.
- mdd_use_mean_center¶
Use the posterior mean as center. Default:
off.
- Command: ms_compute_probabilities ;¶
- Command: ms_compute_probabilities(OPTIONS...);
Computes smoothed regime probabilities of a Markov-switching SBVAR model. Output.epsfiles are contained in<output_file_tag/Output/Probabilities>.Options
- file_tag = FILENAME
See
file_tag.
- output_file_tag = FILENAME
See
output_file_tag.
- filtered_probabilities¶
Filtered probabilities are computed instead of smoothed. Default:
off.
- real_time_smoothed¶
Smoothed probabilities are computed based on time
tinformation for \(0\le t\le nobs\). Default:off
- Command: ms_irf ;¶
- Command: ms_irf(OPTIONS...);
Computes impulse response functions for a Markov-switching SBVAR model. Output.epsfiles are contained in<output_file_tag/Output/IRF>, while data files are contained in<output_file_tag/IRF>.Options
- file_tag = FILENAME
See
file_tag.
- output_file_tag = FILENAME
See
output_file_tag.
- simulation_file_tag = FILENAME
See
simulation_file_tag.
- horizon = INTEGER¶
The forecast horizon. Default:
12.
- filtered_probabilities
Uses filtered probabilities at the end of the sample as initial conditions for regime probabilities. Only one of
filtered_probabilities,regimeandregimesmay be passed. Default:off.
- error_band_percentiles = [DOUBLE1 ...]¶
The percentiles to compute. Default:
[0.16 0.50 0.84]. Ifmedianis passed, the default is[0.5].
- shock_draws = INTEGER¶
The number of regime paths to draw. Default:
10,000.
- shocks_per_parameter = INTEGER¶
The number of regime paths to draw under parameter uncertainty. Default:
10.
- thinning_factor = INTEGER
Only \(1/ \texttt{thinning\_factor}\) of the draws in posterior draws file are used. Default:
1.
- free_parameters = NUMERICAL_VECTOR¶
A vector of free parameters to initialize theta of the model. Default: use estimated parameters
- parameter_uncertainty¶
Calculate IRFs under parameter uncertainty. Requires that
ms_simulationhas been run. Default:off.
- regime = INTEGER¶
Given the data and model parameters, what is the ergodic probability of being in the specified regime. Only one of
filtered_probabilities,regimeandregimesmay be passed. Default:off.
- regimes¶
Describes the evolution of regimes. Only one of
filtered_probabilities,regimeandregimesmay be passed. Default:off.
- median¶
A shortcut to setting
error_band_percentiles=[0.5]. Default:off.
- Command: ms_forecast ;¶
- Command: ms_forecast(OPTIONS...);
Generates forecasts for a Markov-switching SBVAR model. Output.epsfiles are contained in<output_file_tag/Output/Forecast>, while data files are contained in<output_file_tag/Forecast>.Options
- file_tag = FILENAME
See
file_tag.
- output_file_tag = FILENAME
See
output_file_tag.
- simulation_file_tag = FILENAME
See
simulation_file_tag.
- data_obs_nbr = INTEGER¶
The number of data points included in the output. Default:
0.
- error_band_percentiles = [DOUBLE1 ...]
See
error_band_percentiles.
- shock_draws = INTEGER
See
shock_draws.
- shocks_per_parameter = INTEGER
See
shocks_per_parameter.
- thinning_factor = INTEGER
See
thinning_factor.
- free_parameters = NUMERICAL_VECTOR
See
free_parameters.
- parameter_uncertainty
- regime = INTEGER
See
regime.
- regimes
See
regimes.
- median
See
median.
- horizon = INTEGER
See
horizon.
- Command: ms_variance_decomposition ;¶
- Command: ms_variance_decomposition(OPTIONS...);
Computes the variance decomposition for a Markov-switching SBVAR model. Output.epsfiles are contained in<output_file_tag/Output/Variance_Decomposition>, while data files are contained in<output_file_tag/Variance_Decomposition>.Options
- file_tag = FILENAME
See
file_tag.
- output_file_tag = FILENAME
See
output_file_tag.
- simulation_file_tag = FILENAME
See
simulation_file_tag.
- horizon = INTEGER
See
horizon.
- filtered_probabilities
- no_error_bands¶
Do not output percentile error bands (i.e. compute mean). Default:
off(i.e. output error bands)
- error_band_percentiles = [DOUBLE1 ...]
See
error_band_percentiles.
- shock_draws = INTEGER
See
shock_draws.
- shocks_per_parameter = INTEGER
See
shocks_per_parameter.
- thinning_factor = INTEGER
See
thinning_factor.
- free_parameters = NUMERICAL_VECTOR
See
free_parameters.
- parameter_uncertainty
- regime = INTEGER
See
regime.
- regimes
See
regimes.
4.24. Epilogue Variables¶
- Block: epilogue ;
The epilogue block is useful for computing output variables of interest that may not be necessarily defined in the model (e.g. various kinds of real/nominal shares or relative prices, or annualized variables out of a quarterly model).
It can also provide several advantages in terms of computational efficiency and flexibility:
You can calculate variables in the epilogue block after smoothers/simulations have already been run without adding the new definitions and equations and rerunning smoothers/simulations. Even posterior smoother subdraws can be recycled for computing epilogue variables without rerunning subdraws with the new definitions and equations.
You can also reduce the state space dimension in data filtering/smoothing. Assume, for example, you want annualized variables as outputs. If you define an annual growth rate in a quarterly model, you need lags up to order 7 of the associated quarterly variable; in a medium/large scale model this would just blow up the state dimension and increase by a huge amount the computing time of a smoother.
The epilogue block is terminated by end; and contains lines of the
form:
NAME = EXPRESSION;
- Example
epilogue; // annualized level of y ya = exp(y)+exp(y(-1))+exp(y(-2))+exp(y(-3)); // annualized growth rate of y gya = ya/ya(-4)-1; end;
4.25. Semi-structural models¶
Dynare provides tools for semi-structural models, in the vain of the FRB/US
model (see Brayton and Tinsley (1996)), where expectations are not necessarily
model consistent but based on a VAR auxiliary model. In the following, it is
assumed that each equation is written as VARIABLE = EXPRESSION or
T(VARIABLE) = EXPRESSION where T(VARIABLE) stands for a transformation
of an endogenous variable (log or diff). This representation, where each
equation determines the endogenous variable on the LHS, can be exploited when
simulating the model (see algorithms 12 and 14 in solve_algo)
and is mandatory to define auxiliary models used for computing expectations (see
below).
4.25.1. Auxiliary models¶
The two auxiliary models defined in this section are linear backward-looking models
used to form expectations. Both models can be recast as VAR(1)-processes and
therefore offer isomorphic ways of specifying the expectations process, but differ
in their convenience of specifying features like cointegration and error correction.
var_model directly specifies a VAR, while trend_component_model allows to define
a trend target to which the endogenous variables may be attracted in the long-run
(i.e. an error correction model).
- Command: var_model(OPTIONS...);¶
Picks equations in themodelblock to form a VAR model. This model can be used as an auxiliary model invar_expectation_modelorpac_model. It must be of the following form:\[Y_t = \mathbf{c} + \sum_{i=1}^p A_i Y_{t-i} + \varepsilon_t\]or
\[A_0 Y_t = \mathbf{c} + \sum_{i=1}^p A_i Y_{t-i} + \varepsilon_t\]if the VAR is structural (see below), where \(Y_t\) and \(\varepsilon_t\) are \(n\times 1\) vectors, \(\mathbf{c}\) is a \(n\times 1\) vector of parameters, \(A_i\) (\(i=0,\ldots,p\)) are \(n\times n\) matrices of parameters, and \(A_0\) is non singular square matrix. Vector \(\mathbf{c}\) and matrices \(A_i\) (\(i=0,\ldots,p\)) are set by parsing the equations in the
modelblock. Then, Dynare builds a VAR(1)-companion form model for \(\mathcal{Y}_t = (1, Y_t, \ldots, Y_{t-p+1})'\) as:\[\begin{split}\begin{pmatrix} 1\\ Y_t\\ Y_{t-1}\\ \vdots\\ \vdots\\ Y_{t-p+1} \end{pmatrix} = \underbrace{ \begin{pmatrix} 1 & 0_n' & \ldots & \ldots & \ldots & 0_n'\\ \mathbf{c} & A_1 & A_2 & \ldots & \ldots & A_p\\ 0_n & I_n & O_n & \ldots & \ldots & O_n\\ 0_n & O_n & I_n & O_n & \ldots & O_n\\ \vdots & O_n & \ddots & \ddots & \ddots & \vdots \\ 0_n & O_n & \ldots & O_n & I_n & O_n \end{pmatrix}}_{\mathcal{C}} \begin{pmatrix} 1\\ Y_{t-1}\\ Y_{t-2}\\ \vdots\\ \vdots\\ Y_{t-p} \end{pmatrix} + \underbrace{ \begin{pmatrix} 0\\ \varepsilon_t\\ 0_n\\ \vdots\\ \vdots\\ 0_n \end{pmatrix}}_{\mathcal{\epsilon}_t}\end{split}\]assuming that we are dealing with a reduced form VAR (otherwise, the right-hand side would additionally be premultiplied by \(A_0^{-1}.\) to obtain the reduced for representation). If the VAR does not have a constant, we remove the first line of the system and the first column of the companion matrix \(\mathcal{C}.\) Dynare only saves the companion matrix, since that is the only information required to compute the expectations.
- MATLAB/Octave variable: oo_.var.MODEL_NAME.CompanionMatrix¶
Reduced form companion matrix of the
var_model.
Options
- model_name = STRING¶
Name of the VAR model, which will be referenced in
var_expectation_modelorpac_modelas anauxiliary_model. Needs to be a valid MATLAB field name.- eqtags = [QUOTED_STRING[, QUOTED_STRING[, ...]]]¶
List of equations in the
modelblock (referenced using the equation tagname) used to build the VAR model.- structural¶
By default the VAR model is not structural, i.e. each equation must contain exactly one contemporaneous variable (on the LHS). If the
structuraloption is provided then any variable defined in the system can appear at time \(t\) in each equation. Internally Dynare will rewrite this model as a reduced form VAR (by inverting the implied matrix \(A_0\)).Example
var_model(model_name = toto, eqtags = [ 'X', 'Y', 'Z' ]); model; [ name = 'X' ] x = a*x(-1) + b*x(-2) + c*z(-2) + e_x; [ name = 'Z' ] z = f*z(-1) + e_z; [ name = 'Y' ] y = d*y(-2) + e*z(-1) + e_y; end;
- Command: trend_component_model(OPTIONS...);¶
Picks equations in the model block to form a trend component model. This model can be used as an auxiliary model invar_expectation_modelorpac_model. It must be of the following form:\[\begin{split}\begin{cases} \Delta X_t &= A_0 (X_{t-1}-C_0 Z_{t-1}) + \sum_{i=1}^p A_i \Delta X_{t-i} + \varepsilon_t\\ Z_t &= Z_{t-1} + \eta_t \end{cases}\end{split}\]where \(X_t\) and \(Z_t\) are \(n\times 1\) and \(m\times 1\) vectors of endogenous variables. \(Z_t\) defines the trend target to whose linear combination \(C_0 Z_t\) the endogenous variables \(X_t\) will be attracted, provided the implied error correction matrix \(A_0\) is negative definite. \(\varepsilon_t\) and \(\eta_t\) are \(n\times 1\) and \(m\times 1\) vectors of exogenous variables, \(A_i\) (\(i=0,\ldots,p\)) are \(n\times n\) matrices of parameters, and \(C_0\) is a \(n\times m\) matrix. This model can also be cast into a VAR(1) model by first rewriting it in levels. Let \(Y_t = (X_t',Z_t')'\) and \(\zeta_t = (\varepsilon_t',\eta_t')'\). Then we have:
\[Y_t = \sum_{i=1}^{p+1} B_i Y_{t-i} + \zeta_t\]with
\[\begin{split}B_1 = \begin{pmatrix} I_n+A_0+A_1 & -\Lambda\\ O_{m,n} & I_m \end{pmatrix}\end{split}\]where \(\Lambda = A_0C_0\),
\[\begin{split}B_i = \begin{pmatrix} A_i-A_{i-1} & O_{n,m}\\ O_{m,n} & O_m \end{pmatrix}\end{split}\]for \(i=2,\ldots,p\), and
\[\begin{split}B_{p+1} = \begin{pmatrix} -A_p & O_{n,m}\\ O_{m,n} & O_m \end{pmatrix}\end{split}\]This VAR(p+1) in levels can again be rewritten as a VAR(1)-companion model form.
- MATLAB/Octave variable: oo_.trend_component.MODEL_NAME.CompanionMatrix¶
Reduced form companion matrix of the
trend_component_model.
Options
- model_name = STRING
Name of the trend component model, will be referenced in
var_expectation_modelorpac_modelas anauxiliary_model. Needs to be a valid MATLAB field name.- eqtags = [QUOTED_STRING[, QUOTED_STRING[, ...]]]
List of equations in the
modelblock (referenced using the equation tagname) used to build the trend component model.- targets = [QUOTED_STRING[, QUOTED_STRING[, ...]]]¶
List of targets, corresponding to the variables in vector \(Z_t\), referenced using the equation tag
name) of the associated equation in themodelblock.targetmust be a subset ofeqtags.Example
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']); model; [name='eq:x1'] diff(x1) = a_x1_0*(x1(-1)-x1bar(-1))+a_x1_0_*(x2(-1)-x2bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1; [name='eq:x2'] diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2; [name='eq:x1bar'] x1bar = x1bar(-1) + ex1bar; [name='eq:x2bar'] x2bar = x2bar(-1) + ex2bar; end;
4.25.2. VAR expectations¶
Suppose we wish to forecast a variable \(y_t\) and that \(y_t\) is an element of vector of variables \(\mathcal{Y}_t\) whose law of motion is described by a VAR(1) model \(\mathcal{Y}_t = \mathcal{C}\mathcal{Y}_{t-1}+\epsilon_t\). More generally, \(y_t\) may be a linear combination of the scalar variables in \(\mathcal{Y}_t\). Let the vector \(\alpha\) be such that \(y_t = \alpha'\mathcal{Y}_t\) (\(\alpha\) is a selection vector if \(y_t\) is a variable in \(\mathcal{Y}_t\), i.e. a column of an identity matrix, or an arbitrary vector defining the weights of a linear combination). Then the best prediction, in the sense of the minimisation of the RMSE, for \(y_{t+h}\) given the information set at \(t-\tau\) (which we assume to include all observables up to time \(t-\tau\), \(\mathcal{Y}_{\underline{t-\tau}}\)) is:
\[y_{t+h|t-\tau} = \mathbb E[y_{t+h}|\mathcal{Y}_{\underline{t-\tau}}] = \alpha\mathcal{C}^{h+\tau} \mathcal{Y}_{t-\tau}\]
In a semi-structural model, variables appearing in \(t+h\) (e.g. the expected output gap in a dynamic IS curve or expected inflation in a New Keynesian Phillips curve) will be replaced by the expectation implied by an auxiliary VAR model. Another use case is for the computation of permanent incomes. Typically, consumption will depend on something like:
\[\sum_{h=0}^{\infty} \beta^h y_{t+h|t-\tau}\]
Assuming that \(0<\beta<1\) and knowing the limit of geometric series, the conditional expectation of this variable can be evaluated based on the same auxiliary model:
\[\mathbb E \left[\sum_{h=0}^{\infty} \beta^h y_{t+h}\Biggl| \mathcal{Y}_{\underline{t-\tau}}\right] = \alpha \mathcal{C}^\tau(I-\beta\mathcal{C})^{-1}\mathcal{Y}_{t-\tau}\]
Finite discounted sums can also be considered.
- Command: var_expectation_model(OPTIONS...);¶
Declares a model used to forecast an endogenous variable or linear combination of variables in \(t+h\). More generally, the same model can be used to forecast the discounted flow of a variable or a linear expression of variables:\[\sum_{h=a}^b \beta^{h-\tau}\mathbb E[y_{t+h}|\mathcal{Y}_{\underline{t-\tau}}]\]where \((a,b)\in\mathbb N^2\) with \(a<b\) and \(a<\infty\), \(\beta\in(0,1]\) is a discount factor, and \(\tau\) is a finite positive integer.
Options
- model_name = STRING
Name of the VAR based expectation model, which will be referenced in the
modelblock.- auxiliary_model = STRING¶
Name of the associated auxiliary model, defined with
var_modelortrend_component_model.- expression = VARIABLE_NAME | EXPRESSION¶
Name of the variable or expression (linear combination of variables) to be expected.
- discount = PARAMETER_NAME | DOUBLE¶
Discount factor (\(\beta\)).
- horizon = INTEGER | [INTEGER:INTEGER]¶
If the value of
horizonis a finite integer scalar, the following expectation is computed:\[\beta^{h-\tau}\mathbb E[y_{t+h}|\mathcal{Y}_{\underline{t-\tau}}]\]otherwise the value is a range of periods \(a:b\) over which the expected discounted sum is computed (the upper bound can be
Inf).- time_shift = INTEGER¶
Shift of the information set (\(\tau\)), default value is 0.
- Operator: var_expectation (NAME_OF_VAR_EXPECTATION_MODEL);¶
This operator is used instead of a leaded variable, e.g.X(1), in themodelblock to substitute a model-consistent forecast with a forecast based on a VAR model.Example
var_model(model_name=toto, eqtags=['X', 'Y', 'Z']); var_expectation_model(model_name=varexp, expression=x, auxiliary_model_name=toto, horizon=1, discount=beta); model; [name='X'] x = a*x(-1) + b*x(-2) + c*z(-2) + e_x; [name='Z'] z = f*z(-1) + e_z; [name='Y'] y = d*y(-2) + e*z(-1) + e_y; foo = .5*foo(-1) + var_expectation(varexp); end;
In this example
var_expectation(varexp)stands for the one step ahead expectation ofx, as a replacement forx(1).
- MATLAB/Octave command: var_expectation.initialize(NAME_OF_VAR_EXPECTATION_MODEL);¶
Initialise thevar_expectation_modelby building the companion matrix of the associated auxiliaryvar_model. Needs to be executed before attempts to simulate or estimate the model.
- MATLAB/Octave command: var_expectation.update(NAME_OF_VAR_EXPECTATION_MODEL);¶
Update/compute the reduced form parameters ofvar_expectation_model. Needs to be executed before attempts to simulate or estimate the model and requires the auxiliaryvar_modelto have previously been initialized.Example (continued)
var_expectation.initialize('varexp'); var_expectation.update('varexp');
Warning
Changes to the parameters of the underlying auxiliary var_model require calls to
var_expectation.initialize and var_expectation.update to become effective. Changes to the
var_expectation_model or its associated parameters require a call to var_expectation.update.
4.25.3. PAC equation¶
In its simplest form, a PAC equation breaks down changes in a variable of interest \(y\) into three contributions: (i) the lagged deviation from a target \(y^{\star}\), (ii) the lagged changes in the variable \(y\), and (iii) the expected changes in the target \(y^{\star}\):
\[\Delta y_t = a_0(y_{t-1}^{\star}-y_{t-1}) + \sum_{i=1}^{m-1} a_i \Delta y_{t-i} + \sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i} +\varepsilon_t\]
Brayton et alii (2000) shows how such an equation can be derived from the minimisation of a quadratic cost function penalising expected deviations from the target and non-smoothness of \(y\), where future costs are discounted (with discount factor \(\beta\)). They also show that the parameters \((d_i)_{i\in\mathbb N}\) are non-linear functions of the \(m\) parameters \(a_i\) and the discount factor \(\beta\). To simulate or estimate this equation we need to figure out how to determine the expected changes of the target. This can be done as in the previous section using VAR based expectations, or considering model consistent expectations (MCE).
To ensure that the endogenous variable \(y\) is equal to its target \(y^{\star}\) in the (deterministic) long run, i.e. that the error correction term is zero in the long run, we can optionally add a growth neutrality correction to this equation. Suppose that \(g\) is the long run growth rate, for \(y\) and \(y^{\star}\), then in the long run (assuming that the data are in logs) we must have:
\[ \begin{align}\begin{aligned}g = a_0(y^{\star}_{\infty}-y_{\infty}) + g\sum_{i=1}^{m-1} a_i + g\sum_{i=0}^{\infty} d_i\\\Leftrightarrow a_0(y^{\star}_{\infty}-y_{\infty}) = \left(1-\sum_{i=1}^{m-1} a_i-\sum_{i=0}^{\infty} d_i\right) g\end{aligned}\end{align} \]
Unless additional restrictions are placed on the coefficients \((a_i)_{i=0}^{m-1}\), i.e. on the form of the minimised cost function, there is no reason for the right-hand side to be zero. Instead, we can optionally add the right hand side to the PAC equation, to ensure that the error correction term is asymptotically zero.
The PAC equations can be generalised by adding exogenous variables. This can be done in two, non exclusive, manners. We can replace the PAC equation by a convex combination of the original PAC equation (derived from an optimisation program) and a linear expression involving exogenous variables (referred as the rule of thumb part as opposed to the part derived from the minimisation of a cost function; not to be confused with exogenous shocks):
\[\Delta y_t = \lambda \left(a_0(y_{t-1}^{\star}-y_{t-1}) + \sum_{i=1}^{m-1} a_i \Delta y_{t-i} + \sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i}\right) + (1-\lambda)\gamma'X_t +\varepsilon_t\]
where \(\lambda\in[0,1]\) is the weight of the pure PAC equation, \(\gamma\) is a \(k\times 1\) vector of parameters, and \(X_t\) a \(k\times 1\) vector of variables in the rule of thumb part. Or we can simply add the exogenous variables to the PAC equation (without the weight \(\lambda\)):
\[\Delta y_t = a_0(y_{t-1}^{\star}-y_{t-1}) + \sum_{i=1}^{m-1} a_i \Delta y_{t-i} + \sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i} + \gamma'X_t +\varepsilon_t\]
- Command: pac_model(OPTIONS...);¶
Declares a PAC model. A.modfile can have more than one PAC model or PAC equation, but each PAC equation must be associated to a different PAC model.Options
- model_name = STRING
Name of the PAC model, will be referenced in the
modelblock.- auxiliary_model = STRING
Name of the associated auxiliary model, defined with
var_modelortrend_component_model, to compute the VAR based expectations for the expected changes in the target, i.e. to evaluate \(\sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i}\). The infinite sum will then be replaced by a linear combination, defined by a vector \(h\), of the variables involved in the companion representation of the auxiliary model. The weights defining the linear combination are nonlinear functions of the \((a_i)_{i=0}^{m-1}\) coefficients in the PAC equation. This option is not mandatory, if absent Dynare understands that the expected changes of the target have to be computed under the MCE assumption. This is done by rewriting recursively the infinite sum as shown in equation 10 of Brayton et alii (2000).- discount = PARAMETER_NAME | DOUBLE
Discount factor (\(\beta\)) for future expected costs appearing in the definition of the cost function.
- growth = PARAMETER_NAME | VARIABLE_NAME | EXPRESSION | DOUBLE¶
If present a growth neutrality correction is added to the PAC equation. The user must ensure that the provided value (or long term level if a variable or expression is given) is consistent with the asymptotic growth rate of the endogenous variable.
- kind = dd | dl¶
Instructs Dynare how to compute the vector \(h\), the weights defining the linear combination of the companion VAR variables. The default value
ddmust be used if the target appears in first difference in the auxiliary model, see equation (A.79) in Brayton et alii (2000), while valuedlmust be used if the target shows up in level in the auxiliary model, equation (A.74) in Brayton et alii (2000).- auxname = STRING¶
Name the auxiliary variable, created by the preprocessor, that will define the expectation term in the PAC equation.
- Operator: pac_expectation (NAME_OF_PAC_MODEL);¶
This operator is used instead of the infinite sum, \(\sum_{i=0}^{\infty} d_i \Delta y^{\star}_{t+i}\), in a PAC equation defined in themodelblock. Depending on the assumption regarding the formation of expectations, it will be replaced by a linear combination of the variables involved in the companion representation of the auxiliary model or by a recursive forward equation.
The PAC equation target can be composite and defined as a weighted sum
of stationary and non stationary components. Such a target requires an
additional equation in the model block, with the target variable on
the left hand-side and the components in the right hand-side. Each
component must be an endogenous variable in the auxiliary model. The
characteristics of each component must be described in the
pac_target_info block, see below, and the
pac_target_nonstationary operator must be used in the error
correction term of the PAC equation to link the target to the provided
description. Note that composite targets make only sense if the
auxiliary model is not a trend component model (where all the
variables are non stationary).
- Block: pac_target_info(NAME_OF_PAC_MODEL);¶
This block enables the user to provide the properties of each component of a target in PAC models with a composite target. TheNAME_OF_PAC_MODELargument refers to a PAC model (must match the value of optionmodel_namein the declaration of a PAC model).On the first line of the block, the name of the composite target variable must be provided using the following syntax:
target VARIABLE_NAME ;
where
VARIABLE_NAMEis a declared endogenous variable, its associated equation is not part of the auxiliary model but all the components (the variables on the right hand-side) must be defined in the auxiliary model. Next, the following line declares the name of the auxilary variable that will appear in the error correction term, this variable contains only the non stationary components of the target:auxname_target_nonstationary NAME ;
The block should contain the following group of lines for each stationary component:
component STATIONARY_VARIABLE_NAME ; kind ll ; auxname AUX_VAR_NAME ;
where
STATIONARY_VARIABLE_NAMEis the name of a stationary variable appearing in the right hand-side of the equation defining the targetVARIABLE_NAME. The second line instructs Dynare that the component appears in levels in the auxiliary model and in the PAC expectations. The third line specifies the name of the auxiliary variable created by Dynare for the component of the PAC expectation related toSTATIONARY_VARIABLE_NAME.The block should contain the following group of lines for each nonstationary component:
component NONSTATIONARY_VARIABLE_NAME ; kind dd | dl ; auxname AUX_VAR_NAME ; growth PARAMETER_NAME | VARIABLE_NAME | EXPRESSION | DOUBLE ;
where
NONSTATIONARY_VARIABLE_NAMEis the name of a nonstationary variable appearing in the right hand-side of the equation defining the targetVARIABLE_NAME. The second line instructs Dynare on how to calculate the weights that define the linear combination of the companion VAR variables. Use valueddif the target appears in first difference in the auxiliary model, ordlif the target shows up in level in the auxiliary model. The third line sets the name of the auxiliary variable created by Dynare for the component of the PAC expectation related toNONSTATIONARY_VARIABLE_NAME. The fourth line is mandatory if a growth neutrality correction is required. The provided value for this option must be consistent with the asymptotic growth rate of the PAC endogenous variable.
- Operator: pac_target_nonstationary (NAME_OF_PAC_MODEL);¶
This operator is only required in presence of a composite target in the PAC equation. The operator, used in the error correction term of the PAC equation, selects the non stationary components of the target.
- MATLAB/Octave command: pac.initialize(NAME_OF_PAC_MODEL);¶
- MATLAB/Octave command: pac.update(NAME_OF_PAC_MODEL);¶
Same as in the previous section for the VAR expectations, initialise the PAC model, by building the companion matrix of the auxiliary model, and computes the reduced form parameters of the PAC equation (the weights in the linear combination of the variables involved in the companion representation of the auxiliary model, or the parameters of the recursive representation of the infinite sum in the MCE case).
Example (trend component auxiliary model)
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']); pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman); model; [name='eq:y'] y = (1-rho_1-rho_2)*diff(x2(-1)) + rho_1*y(-1) + rho_2*y(-2) + ey; [name='eq:x1'] diff(x1) = a_x1_0*(x1(-1)-x1bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1; [name='eq:x2'] diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2; [name='eq:x1bar'] x1bar = x1bar(-1) + ex1bar; [name='eq:x2bar'] x2bar = x2bar(-1) + ex2bar; [name='zpac'] diff(z) = e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman) + ez; end; pac.initialize('pacman'); pac.update.expectation('pacman');
Example (VAR auxiliary model and composite target)
var_model(model_name=toto, eqtags=['eq:x', 'eq:y']); pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman); pac_target_info(pacman); target v; auxname_target_nonstationary vns; component y; auxname pv_y_; kind ll; component x; growth diff(x(-1)); auxname pv_dx_; kind dd; end; model; [name='eq:y'] y = a_y_1*y(-1) + a_y_2*diff(x(-1)) + b_y_1*y(-2) + b_y_2*diff(x(-2)) + ey ; [name='eq:x'] diff(x) = b_x_1*y(-2) + b_x_2*diff(x(-1)) + ex ; [name='eq:v'] v = x + d_y*y ; // Composite PAC target, no residuals here only variables defined in the auxiliary model. [name='zpac'] diff(z) = e_c_m*(pac_target_nonstationary(pacman)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman) + ez; end; pac.initialize('pacman'); pac.update.expectation('pacman');
4.25.4. Estimation of a PAC equation¶
The PAC equation, introduced in the previous section, can be estimated. This equation is nonlinear with respect to the estimated parameters \((a_i)_{i=0}^{m-1}\), since the reduced form parameters (in the computation of the infinite sum) are nonlinear functions of the autoregressive parameters and the error correction parameter. Brayton et alii (2000) shows how to estimate the PAC equation by iterative OLS. Although this approach is implemented in Dynare, mainly for comparison purposes, we also propose NLS estimation, which is much preferable (asymptotic properties of NLS being more solidly grounded).
Note that it is currently not feasible to estimate the PAC equation jointly with the remaining parameters of the model using e.g. Bayesian techniques. Thus, estimation of the PAC equation can only be conducted conditional on the values of the parameters of the auxiliary model.
Warning
The estimation routines described below require the option
json=compute be passed to the preprocessor (via the command line
or at the top of the .mod file, see Dynare invocation).
- MATLAB/Octave command: pac.estimate.nls(EQNAME, GUESS, DATA, RANGE[, ALGO]);¶
- MATLAB/Octave command: pac.estimate.iterative_ols(EQNAME, GUESS, DATA, RANGE);¶
Trigger the NLS or iterative OLS estimation of a PAC equation.EQNAMEis a row char array designating the PAC equation to be estimated (the PAC equation must have a name specified with an equation tag).DATAis adseriesobject containing the data required for the estimation (i.e. data for all the endogenous and exogenous variables in the equation). The residual values of the PAC equation (which correspond to a defined varexo) must also be a member ofDATA, but filled withNaNvalues.RANGEis adatesobject defining the time span of the sample.ALGOis a row char array used to select the method (or minimisation algorithm) for NLS. Possible values are :'fmincon','fminunc','fminsearch','lsqnonlin','particleswarm','csminwel','simplex','annealing', and'GaussNewton'. The first four algorithms require the Mathworks Optimisation toolbox. The fifth algorithm requires the Mathworks Global Optimisation toolbox. When the optimisation algorithm allows it, we impose constraints on the error correction parameter, which must be positive and smaller than 1 (it the case for'fmincon','lsqnonlin','particleswarm', and'annealing'). The default optimisation algorithm is'csminwel'.GUESSis a structure containing the initial guess values for the estimated parameters. Each field is the name of a parameter in the PAC equation and holds the initial guess for this parameter. If some parameters are calibrated, then they should not be members of theGUESSstructure (and values have to be provided in the.modfile before the call to the estimation routine).For the NLS routine the estimation results are displayed in a table after the estimation. For both the NLS and iterative OLS routines, the results are saved in
oo_(under the fieldsnlsoriterative_ols). Also, the values of the parameters are updated inM_.params.
Example (continued)
// Set the initial guess for the estimated parameters eparams.e_c_m = .9; eparams.c_z_1 = .5; eparams.c_z_2 = .2; // Define the dataset used for estimation edata = TrueData; edata.ez = dseries(NaN); // Set to NaN the residual of the equation. pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+200, 'annealing');
Warning
The specification of GUESS and DATA involves the use of structures.
As such, their subfields will not be cleared across Dynare runs as the structures
stay in the workspace. Be careful to clear these structures from the memory
(e.g. within the .mod file) when e.g. changing which parameters are calibrated.
4.26. Displaying and saving results¶
Dynare has comments to plot the results of a simulation and to save the results.
- Command: rplot VARIABLE_NAME...;¶
Plots the simulated path of one or several variables, as stored inoo_.endo_simulby eitherperfect_foresight_solver,simul(see Deterministic simulation) orstoch_simulwith optionperiods(see Stochastic solution and simulation). The variables are plotted in levels.
- Command: dynatype(FILENAME) [VARIABLE_NAME...];¶
This command prints the listed endogenous or exogenous variables in a text file named FILENAME. If no VARIABLE_NAME is listed, all endogenous variables are printed.
- Command: dynasave(FILENAME) [VARIABLE_NAME...];¶
This command saves the listed endogenous or exogenous variables in a binary file named FILENAME. If no VARIABLE_NAME is listed, all endogenous variables are saved.In MATLAB or Octave, variables saved with the
dynasavecommand can be retrieved by the command:load(FILENAME,'-mat')
4.27. Macro processing language¶
It is possible to use “macro” commands in the .mod file for performing
tasks such as: including modular source files, replicating blocks of equations
through loops, conditionally executing some code, writing indexed sums or
products inside equations…
The Dynare macro-language provides a new set of macro-commands which can be
used in .mod files. It features:
File inclusion
Loops (
forstructure)Conditional inclusion (
if/then/elsestructures)Expression substitution
This macro-language is totally independent of the basic
Dynare language, and is processed by a separate component of the
Dynare pre-processor. The macro processor transforms a .mod file
with macros into a .mod file without macros (doing
expansions/inclusions), and then feeds it to the Dynare parser. The
key point to understand is that the macro processor only does text
substitution (like the C preprocessor or the PHP language). Note that
it is possible to see the output of the macro processor by using the
savemacro option of the dynare command (see Dynare invocation).
The macro processor is invoked by placing macro directives in the .mod
file. Directives begin with an at-sign followed by a pound sign (@#). They
produce no output, but give instructions to the macro processor. In most cases,
directives occupy exactly one line of text. If needed, two backslashes (\\)
at the end of the line indicate that the directive is continued on the next
line. Macro directives following // are not interpreted by the macro
processor. For historical reasons, directives in commented blocks, ie
surrounded by /* and */, are interpreted by the macro processor. The
user should not rely on this behavior. The main directives are:
@#includepath, paths to search for files that are to be included,
@#include, for file inclusion,
@#define, for defining a macro processor variable,
@#if, @#ifdef, @#ifndef, @#elseif, @#else, @#endiffor conditional statements,
@#for, @#endforfor constructing loops.
The macro processor maintains its own list of variables (distinct from model
variables and MATLAB/Octave variables). These macro-variables are assigned
using the @#define directive and can be of the following basic types:
boolean, real, string, tuple, function, and array (of any of the previous
types).
4.27.1. Macro expressions¶
Macro-expressions can be used in two places:
Inside macro directives, directly;
In the body of the
.modfile, between an @-sign and curly braces (like@{expr}): the macro processor will substitute the expression with its value
It is possible to construct macro-expressions that can be assigned to macro-variables or used within a macro-directive. The expressions are constructed using literals (i.e.fixed values) of the basic types (boolean, real, string, tuple, array), comprehensions, macro-variables, macro-functions, and standard operators.
Note
Elsewhere in the manual, MACRO_EXPRESSION designates an expression constructed as explained in this section.
Boolean
The following operators can be used on booleans:
Comparison operators:
==,!=Logical operators:
&&,||,!
Real
The following operators can be used on reals:
Arithmetic operators:
+,-,*,/,^Comparison operators:
<,>,<=,>=,==,!=Logical operators:
&&,||,!Ranges with an increment of
1:REAL1:REAL2(for example,1:4is equivalent to real array[1, 2, 3, 4]).Changed in version 4.6: Previously, putting brackets around the arguments to the colon operator (e.g.
[1:4]) had no effect. Now,[1:4]will create an array containing an array (i.e.[ [1, 2, 3, 4] ]).Ranges with user-defined increment:
REAL1:REAL2:REAL3(for example,6:-2.1:-1is equivalent to real array[6, 3.9, 1.8, -0.3]).Functions:
max,min,mod,exp,log,log10,sin,cos,tan,asin,acos,atan,sqrt,cbrt,sign,floor,ceil,trunc,erf,erfc,gamma,lgamma,round,normpdf,normcdf. NBlncan be used instead oflog
String
String literals have to be enclosed by double quotes (like "name").
The following operators can be used on strings:
Comparison operators:
<,>,<=,>=,==,!=Concatenation of two strings:
+Extraction of substrings: if
sis a string, thens[3]is a string containing only the third character ofs, ands[4:6]contains the characters from 4th to 6thFunction:
length
Tuple
Tuples are enclosed by parentheses and elements are separated by commas (like
(a,b,c) or (1,2,3)).
The following operators can be used on tuples:
Comparison operators:
==,!=Functions:
empty,length
Array
Arrays are enclosed by brackets, and their elements are separated
by commas (like [1,[2,3],4] or ["US", "FR"]).
The following operators can be used on arrays:
Comparison operators:
==,!=Dereferencing: if
vis an array, thenv[2]is its 2nd elementConcatenation of two arrays:
+Set union of two arrays:
|Set intersection of two arrays:
&Difference
-: returns the first operand from which the elements of the second operand have been removed.Cartesian product of two arrays:
*Cartesian product of one array N times:
^NExtraction of sub-arrays: e.g.
v[4:6]Testing membership of an array:
inoperator (for example:"b"in["a", "b", "c"]returns1)Functions:
empty,sum,length
Comprehension
Comprehension syntax is a shorthand way to make arrays from other arrays. There are three different ways the comprehension syntax can be employed: filtering, mapping, and filtering and mapping.
Filtering
Filtering allows one to choose those elements from an array for which a certain condition hold.
Example
Create a new array, choosing the even numbers from the array
1:5:[ i in 1:5 when mod(i,2) == 0 ]would result in:
[2, 4]
Mapping
Mapping allows you to apply a transformation to every element of an array.
Example
Create a new array, squaring all elements of the array
1:5:[ i^2 for i in 1:5 ]would result in:
[1, 4, 9, 16, 25]
Filtering and Mapping
Combining the two preceding ideas would allow one to apply a transformation to every selected element of an array.
Example
Create a new array, squaring all even elements of the array
1:5:[ i^2 for i in 1:5 when mod(i,2) == 0]would result in:
[4, 16]Further Examples
[ (j, i+1) for (i,j) in (1:2)^2 ] [ (j, i+1) for (i,j) in (1:2)*(1:2) when i < j ]would result in:
[(1, 2), (2, 2), (1, 3), (2, 3)] [(2, 2)]
Function
Functions can be defined in the macro processor using the @#define
directive (see below). A function is evaluated at the time it is invoked during
the macroprocessing stage, not
at define time. Functions can be included in expressions and the operators that
can be combined with them depend on their return type.
Checking variable type
Given a variable name or literal, you can check the type it evaluates to using
the following functions: isboolean, isreal, isstring, istuple,
and isarray.
Examples
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Casting between types
Variables and literals of one type can be cast into another type. Some type changes are straightforward (e.g. changing a real to a string) whereas others have certain requirements (e.g. to cast an array to a real it must be a one element array containing a type that can be cast to real).
Examples
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Casts can be used in expressions:
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4.27.2. Macro directives¶
- Macro directive: @#includepath "PATH"¶
- Macro directive: @#includepath MACRO_EXPRESSION
This directive adds the path contained in PATH to the list of those to search when looking for a.modfile specified by@#include. If provided with a MACRO_EXPRESSION argument, the argument must evaluate to a string. Note that these paths are added after any paths passed using-I.Example
@#includepath "/path/to/folder/containing/modfiles" @#includepath folders_containing_mod_files
- Macro directive: @#include "FILENAME"¶
- Macro directive: @#include MACRO_EXPRESSION
This directive simply includes the content of another file in its place; it is exactly equivalent to a copy/paste of the content of the included file. If provided with a MACRO_EXPRESSION argument, the argument must evaluate to a string. Note that it is possible to nest includes (i.e. to include a file from an included file). The file will be searched for in the current directory. If it is not found, the file will be searched for in the folders provided by-Iand@#includepath.Example
@#include "modelcomponent.mod" @#include location_of_modfile
- Macro directive: @#define MACRO_VARIABLE¶
- Macro directive: @#define MACRO_VARIABLE = MACRO_EXPRESSION
- Macro directive: @#define MACRO_FUNCTION = MACRO_EXPRESSION
Defines a macro-variable or macro function.Example
@#define var // Equals true @#define x = 5 // Real @#define y = "US" // String @#define v = [ 1, 2, 4 ] // Real array @#define w = [ "US", "EA" ] // String array @#define u = [ 1, ["EA"] ] // Mixed array @#define z = 3 + v[2] // Equals 5 @#define t = ("US" in w) // Equals true @#define f(x) = " " + x + y // Function `f` with argument `x` // returns the string ' ' + x + 'US'
Example
@#define x = 1 @#define y = [ "B", "C" ] @#define i = 2 @#define f(x) = x + " + " + y[i] @#define i = 1 model; A = @{y[i] + f("D")}; end;
The latter is strictly equivalent to:
model; A = BD + B; end;
- Macro directive: @#if MACRO_EXPRESSION¶
- Macro directive: @#ifdef MACRO_VARIABLE¶
- Macro directive: @#ifndef MACRO_VARIABLE¶
- Macro directive: @#elseif MACRO_EXPRESSION¶
- Macro directive: @#else¶
- Macro directive: @#endif¶
Conditional inclusion of some part of the.modfile. The lines between@#if,@#ifdef, or@#ifndefand the next@#elseif,@#elseor@#endifis executed only if the condition evaluates totrue. Following the@#ifbody, zero or more@#elseifbranches are allowed. An@#elseifcondition is only evaluated if the preceding@#ifor@#elseifcondition(s) evaluated tofalse. The@#elsebranch is optional and only evaluated if all@#ifand@#elseifstatements evaluate to false.Note that when using
@#ifdef, the condition will evaluate totrueif the MACRO_VARIABLE has been previously defined, regardless of its value. Conversely,@#ifndefwill evaluate to true if the MACRO_VARIABLE has not yet been defined.Note that when using
@#elseifyou can check whether or not a variable has been defined by using thedefinedoperator. Hence, to enter the body of an@#elseifbranch if the variableXhas not been defined, you would write:@#elseif !defined(X).Note that if a real appears as the result of the MACRO_EXPRESSION, it will be interpreted as a boolean; a value of
0is interpreted asfalse, otherwise it is interpreted astrue. Further note that because of the imprecision of reals, extra care must be taken when testing them in the MACRO_EXPRESSION. For example,exp(log(5)) == 5will evaluate tofalse. Hence, when comparing real values, you should generally use a non-zero tolerance around the value desired, e.g.exp(log(5)) > 5-1e-14 && exp(log(5)) < 5+1e-14Example
Choose between two alternative monetary policy rules using a macro-variable:
@#define linear_mon_pol = false // 0 would be treated the same ... model; @#if linear_mon_pol i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); @#else i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; @#endif ... end;
This would result in:
... model; i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; ... end;
Example
Choose between two alternative monetary policy rules using a macro-variable. The only difference between this example and the previous one is the use of
@#ifdefinstead of@#if:@#define linear_mon_pol = false // 0 would be treated the same ... model; @#ifdef linear_mon_pol i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); @#else i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; @#endif ... end;
Although
linear_mon_polcontains the valuefalsebecause@#ifdefonly checks that the variable has been defined, the linear monetary policy is output::This would result in:... model; i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); ... end;
- Macro directive: @#for MACRO_VARIABLE in MACRO_EXPRESSION¶
- Macro directive: @#for MACRO_VARIABLE in MACRO_EXPRESSION when MACRO_EXPRESSION
- Macro directive: @#for MACRO_TUPLE in MACRO_EXPRESSION
- Macro directive: @#for MACRO_TUPLE in MACRO_EXPRESSION when MACRO_EXPRESSION
- Macro directive: @#endfor¶
Loop construction for replicating portions of the.modfile. Note that this construct can enclose variable/parameter declarations, computational tasks, but not a model declaration.Example
model; @#for country in [ "home", "foreign" ] GDP_@{country} = A * K_@{country}^a * L_@{country}^(1-a); @#endfor end;
The latter is equivalent to:
model; GDP_home = A * K_home^a * L_home^(1-a); GDP_foreign = A * K_foreign^a * L_foreign^(1-a); end;
Example
model; @#for (i, j) in ["GDP"] * ["home", "foreign"] @{i}_@{j} = A * K_@{j}^a * L_@{j}^(1-a); @#endfor end;
The latter is equivalent to:
model; GDP_home = A * K_home^a * L_home^(1-a); GDP_foreign = A * K_foreign^a * L_foreign^(1-a); end;
Example
@#define countries = ["US", "FR", "JA"] @#define nth_co = "US" model; @#for co in countries when co != nth_co (1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; @#endfor E_@{nth_co} = 1; end;
The latter is equivalent to:
model; (1+i_FR) = (1+i_US) * E_FR(+1) / E_FR; (1+i_JA) = (1+i_US) * E_JA(+1) / E_JA; E_US = 1; end;
- Macro directive: @#echo MACRO_EXPRESSION¶
Asks the preprocessor to display some message on standard output. The argument must evaluate to a string.
- Macro directive: @#error MACRO_EXPRESSION¶
Asks the preprocessor to display some error message on standard output and to abort. The argument must evaluate to a string.
- Macro directive: @#echomacrovars¶
- Macro directive: @#echomacrovars MACRO_VARIABLE_LIST
- Macro directive: @#echomacrovars(save) MACRO_VARIABLE_LIST
Asks the preprocessor to display the value of all macro variables up until this point. If thesaveoption is passed, then values of the macro variables are saved tooptions_.macrovars_line_<<line_numbers>>. IfNAME_LISTis passed, only display/save variables and functions with that name.Example
@#define A = 1 @#define B = 2 @#define C(x) = x*2 @#echomacrovars A C D
The output of the command above is:
Macro Variables: A = 1 Macro Functions: C(x) = (x * 2)
4.27.3. Typical usages¶
4.27.3.1. Modularization¶
The @#include directive can be used to split .mod files into
several modular components.
Example setup:
modeldesc.mod
Contains variable declarations, model equations, and shocks declarations.
simul.mod
Includes
modeldesc.mod, calibrates parameter,s and runs stochastic simulations.
estim.mod
Includes
modeldesc.mod, declares priors on parameters, and runs Bayesian estimation.
Dynare can be called on simul.mod and estim.mod, but it makes
no sense to run it on modeldesc.mod.
The main advantage is that you don’t have to copy/paste the whole model (during initial development) or changes to the model (during development).
4.27.3.2. Indexed sums of products¶
The following example shows how to construct a moving average:
@#define window = 2
var x MA_x;
...
model;
...
MA_x = @{1/(2*window+1)}*(
@#for i in -window:window
+x(@{i})
@#endfor
);
...
end;
After macro processing, this is equivalent to:
var x MA_x;
...
model;
...
MA_x = 0.2*(
+x(-2)
+x(-1)
+x(0)
+x(1)
+x(2)
);
...
end;
4.27.3.3. Multi-country models¶
Here is a bare bones example for a multi-country model:
@#define countries = [ "US", "EA", "AS", "JP", "RC" ]
@#define nth_co = "US"
@#for co in countries
var Y_@{co} K_@{co} L_@{co} i_@{co} E_@{co} ...;
parameters a_@{co} ...;
varexo ...;
@#endfor
model;
@#for co in countries
Y_@{co} = K_@{co}^a_@{co} * L_@{co}^(1-a_@{co});
...
@#if co != nth_co
(1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; // UIP relation
@#else
E_@{co} = 1;
@#endif
@#endfor
end;
4.27.3.4. Endogeneizing parameters¶
When calibrating the model, it may be useful to pin down parameters by targeting endogenous objects.
For example, suppose production is defined by a CES function:
\[y_t = \left(\alpha^{1/\xi} \ell_t^{1-1/\xi}+(1-\alpha)^{1/\xi}k_t^{1-1/\xi}\right)^{\xi/(\xi-1)}\]
and the labor share in GDP is defined as:
\[\textrm{lab\_rat}_t = (w_t \ell_t)/(p_t y_t)\]
In the model, \(\alpha\) is a (share) parameter and \(lab\_rat_t\) is an endogenous variable.
It is clear that setting a value for \(\alpha\) is not straightforward. But we have real world data for \(lab\_rat_t\) and it is clear that these two objects are economically linked.
The solution is to use a method called variable flipping, which consists in changing the way of computing the steady state. During this computation, \(\alpha\) will be made an endogenous variable and the steady state value \(lab\_rat\) of the dynamic variable \(lab\_rat_t\) will be made a parameter. An economically sensible value will be calibrated for \(lab\_rat\), and the solution algorithm will deduce the implied value for \(\alpha\).
An implementation could consist of the following files:
modeqs.mod
This file contains variable declarations and model equations. The code for the declaration of \(\alpha\) and
lab_ratwould look like:@#if steady var alpha; parameter lab_rat; @#else parameter alpha; var lab_rat; @#endif
steadystate.mod
This file computes the steady state. It begins with:
@#define steady = 1 @#include "modeqs.mod"Then it initializes parameters (including
lab_rat, excluding \(\alpha\)), computes the steady state (using guess values for endogenous, including \(\alpha\)), then saves values of parameters and variables at steady state in a file, using thesave_params_and_steady_statecommand.
simulate.mod
This file computes the simulation. It begins with:
@#define steady = 0 @#include "modeqs.mod"Then it loads values of parameters and variables at steady state from file, using the
load_params_and_steady_statecommand, and computes the simulations.
4.27.4. MATLAB/Octave loops versus macro processor loops¶
Suppose you have a model with a parameter \(\rho\) and you want to run simulations for three values: \(\rho = 0.8, 0.9, 1\). There are several ways of doing this:
With a MATLAB/Octave loop
rhos = [ 0.8, 0.9, 1]; for i = 1:length(rhos) set_param_value('rho',rhos(i)); stoch_simul(order=1); if info(1)~=0 error('Simulation failed for parameter draw') end endHere the loop is not unrolled, MATLAB/Octave manages the iterations. This is interesting when there are a lot of iterations. It is strongly advised to always check whether the error flag
info(1)==0to prevent erroneously relying on stale results from previous iterations.
With a macro processor loop (case 1)
rhos = [ 0.8, 0.9, 1]; @#for i in 1:3 set_param_value('rho',rhos(@{i})); stoch_simul(order=1); if info(1)~=0 error('Simulation failed for parameter draw') end @#endforThis is very similar to the previous example, except that the loop is unrolled. The macro processor manages the loop index but not the data array (
rhos).
With a macro processor loop (case 2)
@#for rho_val in [ 0.8, 0.9, 1] rho = @{rho_val}; stoch_simul(order=1); if info(1)~=0 error('Simulation failed for parameter draw') end @#endforThe advantage of this method is that it uses a shorter syntax, since the list of values is directly given in the loop construct. The inconvenience is that you can not reuse the macro array in MATLAB/Octave.
4.28. Verbatim inclusion¶
Pass everything contained within the verbatim block to the
<mod_file>.m file.
- Block: verbatim ;
By default, whenever Dynare encounters code that is not understood by the parser, it is directly passed to the preprocessor output.In order to force this behavior you can use the
verbatimblock. This is useful when the code you want passed to the driver file contains tokens recognized by the Dynare preprocessor.Example
verbatim; % Anything contained in this block will be passed % directly to the driver file, including comments var = 1; end;
4.29. Misc commands¶
- Command: set_dynare_seed(INTEGER)¶
- Command: set_dynare_seed('default')
- Command: set_dynare_seed('clock')
- Command: set_dynare_seed('reset')
- Command: set_dynare_seed('ALGORITHM', INTEGER)
Sets the seed used for random number generation. It is possible to set a given integer value, to use a default value, or to use the clock (by using the latter, one will therefore get different results across different Dynare runs). Theresetoption serves to reset the seed to the value set by the lastset_dynare_seedcommand. On MATLAB 7.8 or above, it is also possible to choose a specific algorithm for random number generation; accepted values aremcg16807,mlfg6331_64,mrg32k3a,mt19937ar(the default),shr3congandswb2712.
- Command: save_params_and_steady_state(FILENAME);¶
For all parameters, endogenous and exogenous variables, stores their value in a text file, using a simple name/value associative table.for parameters, the value is taken from the last parameter initialization.
for exogenous, the value is taken from the last
initvalblock.for endogenous, the value is taken from the last steady state computation (or, if no steady state has been computed, from the last
initvalblock).
Note that no variable type is stored in the file, so that the values can be reloaded with
load_params_and_steady_statein a setup where the variable types are different.The typical usage of this function is to compute the steady-state of a model by calibrating the steady-state value of some endogenous variables (which implies that some parameters must be endogeneized during the steady-state computation).
You would then write a first
.modfile which computes the steady state and saves the result of the computation at the end of the file, usingsave_params_and_steady_state.In a second file designed to perform the actual simulations, you would use
load_params_and_steady_statejust after your variable declarations, in order to load the steady state previously computed (including the parameters which had been endogeneized during the steady state computation).The need for two separate
.modfiles arises from the fact that the variable declarations differ between the files for steady state calibration and for simulation (the set of endogenous and parameters differ between the two); this leads to differentvarandparametersstatements.Also note that you can take advantage of the
@#includedirective to share the model equations between the two files (see Macro processing language).
- Command: load_params_and_steady_state(FILENAME);¶
For all parameters, endogenous and exogenous variables, loads their value from a file created withsave_params_and_steady_state.for parameters, their value will be initialized as if they had been calibrated in the
.modfile.for endogenous and exogenous variables, their value will be initialized as they would have been from an
initvalblock .
This function is used in conjunction with
save_params_and_steady_state; see the documentation of that function for more information.
- Command: compilation_setup(OPTIONS);¶
When the
use_dlloption is present, Dynare uses the GCC compiler that was distributed with it to compile the static and dynamic C files produced by the preprocessor. You can use this option to change the compiler, flags, and libraries used.Options
- compiler = FILENAME¶
The path to the compiler.
- substitute_flags = QUOTED_STRING¶
The flags to use instead of the default flags.
- add_flags = QUOTED_STRING¶
The flags to use in addition to the default flags. If
substitute_flagsis passed, these flags are added to the flags specified there.
- substitute_libs = QUOTED_STRING¶
The libraries to link against instead of the default libraries.
- add_libs = QUOTED_STRING¶
The libraries to link against in addition to the default libraries. If
substitute_libsis passed, these libraries are added to the libraries specified there.
- MATLAB/Octave command: dynare_version ;¶
Output the version of Dynare that is currently being used (i.e. the one that is highest on the MATLAB/Octave path).
- MATLAB/Octave command: write_latex_definitions ;¶
Writes the names, LaTeX names and long names of model variables to tables in a file named<<M_.fname>>_latex_definitions.tex. Requires the following LaTeX packages:longtable.
- MATLAB/Octave command: write_latex_parameter_table ;¶
Writes the LaTeX names, parameter names, and long names of model parameters to a table in a file named<<M_.fname>>_latex_parameters.tex.The command writes the values of the parameters currently stored. Thus, if parameters are set or changed in the steady state computation, the command should be called after asteadycommand to make sure the parameters were correctly updated. The long names can be used to add parameter descriptions. Requires the following LaTeX packages:longtable, booktabs.
- MATLAB/Octave command: write_latex_prior_table ;¶
Writes descriptive statistics about the prior distribution to a LaTeX table in a file named<<M_.fname>>_latex_priors_table.tex. The command writes the prior definitions currently stored. Thus, this command must be invoked after theestimated_paramsblock. If priors are defined over the measurement errors, the command must also be preceeded by the declaration of the observed variables (withvarobs). The command displays a warning if no prior densities are defined (ML estimation) or if the declaration of the observed variables is missing. Requires the following LaTeX packages:longtable, booktabs.
- MATLAB/Octave command: collect_latex_files ;¶
Writes a LaTeX file named<<M_.fname>>_TeX_binder.texthat collects all TeX output generated by Dynare into a file. This file can be compiled usingpdflatexand automatically tries to load all required packages. Requires the following LaTeX packages:breqn,psfrag,graphicx,epstopdf,longtable,booktabs,caption,float,amsmath,amsfonts,amssymb, andmorefloats.
Footnotes