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 comment
var 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 ValuesInf
and-Inf
are 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.
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.var
commands 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
model
block, 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) y
statement, two endogenous will be created (y
andLOG_y
), and an auxiliary equation linking the two will also be added (equal toLOG_y = log(y)
). Moreover, every occurence ofy
in 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, var
andparameters
commands. The deflator is assumed to be multiplicative; for an additive deflator, uselog_deflator
. This option can be used together with thelog
option (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 thelog
option, 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_name
options 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
.m
file 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.varexo
commands 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_det
commands 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_simul
will 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.
parameters
commands 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
.mod
file: the type change is effective after, but also before, thechange_type
command. 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,
alpha
andbeta
will be endogenous andy
andw
will 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_variables
is 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_variables
command. 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 isk
and 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_var
instead.Trend variables are required if the user wants to be able to write a nonstationary model in the
model
block. Thetrend_var
command must appear before the var command that references the trend variable.trend_var
commands 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
model
block, 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_factor
keyword. 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_state
as 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 (ifendval
is followed bysteady
) or simply the terminal state provided by the user (ifendval
is 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_function
commands 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_provided
option is used in the sameexternal_function
command. 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
parameters
command, 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 bymodel
andend
keywords.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_policy
ordiscretionary_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;
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. 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 aname
tag, using a syntax like:model; [name = 'Budget constraint']; c + k = k^theta*A; end;
Here,
name
is the keyword indicating that the tag names the equation. If an equation of the model is tagged with a name, theresid
command 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',mcp = 'r > -1.94478'] 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
model
blocks, 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_options
command.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
mex
command (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 thedynare
command (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:0
All the endogenous variables are considered as feedback variables.
1
The 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.
2
In addition of variables with
mfs = 1
the 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.3
In addition of variables with
mfs = 2
the 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 themfs
option 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_VAR
will 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
model
block.The purpose of this statement is to specify the options that apply to the whole model, when there are several
model
blocks, so as to restore the symmetry between those blocks (since otherwise onemodel
block 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
model
block.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
name
tag; 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
endogenous
tag, 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,
dummy1
will be turned into an exogenous, anddummy2
anddummy3
will 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
.mod
file 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
.mod
file 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
EXPECTATION
operators 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
.mod
file 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_model
for details).Also note that this command will not output the contents of the optional
steady_state_model
block (seesteady_state_model
); it will rather output a static version (i.e. without leads and lags) of the dynamicmodel
declared in the model block. To write the LaTeX contents of thesteady_state_model
seewrite_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
.mod
file 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
.tex
file will differ from the model declared by the user in some dimensions (seewrite_latex_dynamic_model
for 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...);
Theinitval
block 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 ofhistval
andendval
blocks 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 aninitval
block with the presence ofhistval
andendval
blocks in perfect foresight simulations, it is strongly recommended to check that the constructedoo_.endo_simul
andoo_.exo_simul
variables contain the desired values after runningperfect_foresight_setup
and 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
initval
block is terminated byend;
and contains lines of the form:VARIABLE_NAME = EXPRESSION;
In a deterministic (i.e. perfect foresight) modelFirst, both the
oo_.endo_simul
andoo_.exo_simul
variables 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 ahistval
block (and therefore absence of anendval
block), thishistval
block will provide/overwrite the historical values for the state variables (lags) by setting the first column/row ofoo_.endo_simul
andoo_.exo_simul
. This implies that theinitval
block in the presence ofhistval
only sets the terminal values for the variables with leads and provides initial values for the perfect foresight solver.Because of these various functions of
initval
it is often necessary to provide values for all the endogenous variables in aninitval
block. 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 theinitval
block, a zero value is assumed. It is particularly important to keep this in mind when specifying exogenous variables usingvarexo
that are not allowed to take on the value of zero, like e.g. TFP.Note that if the
initval
block is immediately followed by asteady
command, its semantics are slightly changed. Thesteady
command 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 theinitval
block. These steady state values conditional on the declared exogenous variables are then written intooo_.endo_simul
and take up the potential roles as historical and terminal conditions as well as starting values for the solver. Aninitval
block followed bysteady
is therefore formally equivalent to aninitval
block 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
initval
is to provide initial guess values for the non-linear solver in the steady state computation. Note that if theinitval
block is not followed bysteady
, the steady state computation will still be triggered by subsequent commands (stoch_simul
,estimation
…).As such,
initval
allows 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 theperiods
option is specified.forecast
as the initial point at which the forecasts are computed.conditional_forecast
as 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;
Theendval
block makes only sense in a deterministic model and cannot be used together withhistval
. Similar to theinitval
command, it will fill both theoo_.endo_simul
andoo_.exo_simul
variables storing the endogenous and exogenous variables with the values provided by this block. If noinitval
block 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
initval
block is present,initval
will provide the historical values for the variables (if there are states/lags), whileendval
will 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
endval
block, the value assumed is that of the lastinitval
block orsteady
command (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_simul
andoo_.exo_simul
variables after runningperfect_foresight_setup
and before runningperfect_foresight_solver
to see whether the desired outcome has been achieved.Like
initval
, if theendval
block is immediately followed by asteady
command, its semantics are slightly changed. Thesteady
command 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 theendval
block. These steady state values conditional on the declared exogenous variables are then written intooo_.endo_simul
and therefore take up the potential roles as historical and terminal conditions as well as starting values for the solver. Anendval
block followed bysteady
is therefore formally equivalent to anendval
block 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
.c
is a forward-looking variable and the exogenous variablex
appears with a lead in the expected return of physical capital, whilek
is a purely backward-looking (state) variable.The initial equilibrium is computed by
steady
conditional onx=1
, and the terminal one conditional onx=2
. Theinitval
block sets the initial condition fork
(since it is the only backward-looking variable), while theendval
block 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 theendval
block. 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
c
andx
, since both appear with a lead, and an initial condition fork
, since it appears with a lag.Setting
x=1.1
in theendval
block without ashocks
block implies that technology is at \(1.1\) in \(t=1\) and stays there forever, becauseendval
is filling all entries ofoo_.endo_simul
andoo_.exo_simul
except for the very first one, which stores the initial conditions and was set to \(0\) by theinitval
block when not explicitly specifying a value for it.Because the law of motion for capital is backward-looking, we need an initial condition for
k
at time \(0\). Due to the presence ofendval
, this cannot be done via ahistval
block, but rather must be specified in theinitval
block. Similarly, because the Euler equation is forward-looking, we need a terminal condition forc
at \(t=201\), which is specified in theendval
block.As can be seen, it is not necessary to specify
c
andx
in theinitval
block andk
in theendval
block, because they have no impact on the results. Due to the optimization problem in the first period being to choosec,k
at \(t=1\) given the predetermined capital stockk
inherited from \(t=0\) as well as the current and future values for technologyx
, the values forc
andx
at time \(t=0\) play no role. The same applies to the choice ofc,k
at time \(t=200\), which does not depend onk
at \(t=201\). As the Euler equation shows, that choice only depends on current capital as well as future consumptionc
and 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
x
andc
are 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
steady
afterinitval
andendval
. 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 theinitval
andendval
blocks. The endogenous variables at \(t=0\) and \(t=201\) are then set to the corresponding steady state equilibrium values.The fact that
c
at \(t=0\) andk
at \(t=201\) specified ininitval
andendval
are 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 usingrplot
or manually plottingoo_.endo_simul
.
- Block: histval ;¶
- Block: histval(OPTIONS...);
In a deterministic perfect foresight contextIn models with lags on more than one period, the
histval
block permits to specify different historical initial values for different periods of the state variables. In this case, theinitval
block takes over the role of specifying terminal conditions and starting values for the solver. Note that thehistval
block 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
histval
block are assumed to have a value of zero at period 0 and before. Note thathistval
cannot 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,
histval
is used to set the historical conditions for the two lags of the endogenous variablex
, stored in the first column ofoo_.endo_simul
. Theinitval
block is used to set the terminal condition for the forward looking variablec
, stored in the last column ofoo_.endo_simul
. Moreover, theinitval
block defines the starting values for the perfect foresight solver for both endogenous variablesc
andx
.In a stochastic simulation context
In the context of stochastic simulations,
histval
allows 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 theperiods
option 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, thehistval
block nevertheless takes the unlogged starting values.In
forecast
as the initial point at which the forecasts are computed. When using the loglinear option, thehistval
block nevertheless takes the unlogged starting values.In
conditional_forecast
for a calibrated model as the initial point at which the conditional forecasts are computed. When using the loglinear option, thehistval
block 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 lastinitval
orendval
block (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,
steady
uses the first observation loaded byinitval_file
as guess value to solve for the steady state of the model. This first observation is determined by thefirst_obs
option when it is used.Don’t mix
initval_file
withinitval
statements. However, afterinitval_file
, you can modify the historical initial values withhistval
orhistval_file
statement.There can be several
initval_file
statements 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
.xls
or.xlsx
): for each endogenous and exogenous variable, the file must contain a column of the same name. NB: Octave only supports the.xlsx
file 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 observaton 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_obs
observation).
- 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
,x
ande
. 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.csv
starting 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.csv
are 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
.xls
or.xlsx
): for each endogenous and exogenous variable, the file must contain a column of the same name. NB: Octave only supports the.xlsx
file 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_obs
observation).
- 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.csv
starting 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 theoverwrite
option.In deterministic context
For deterministic simulations, the
shocks
block specifies temporary changes in the value of exogenous variables. For permanent shocks, use anendval
block.The block 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) ]...;
It is possible to specify shocks which last several periods and which can vary over time. The
periods
keyword accepts a list of several dates or date ranges, which must be matched by as many shock values in thevalues
keyword. Note that a range in theperiods
keyword can be matched by only one value in thevalues
keyword. Ifvalues
represents a scalar, the same value applies to the whole range. Ifvalues
represents 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
is from 0 to the number ofperiods
specified inperfect_foresight_setup
.Warning
Note that the first endogenous simulation period is period 1. 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
initval
orendval
(depending on the exact context). Users should always verify the correct setting ofoo_.exo_simul
afterperfect_foresight_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;
In stochastic contextFor stochastic simulations, the
shocks
block 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
varobs
command to be specified before theshocks
block.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_model
ordiscretionary_policy
context, 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_simul
will compute the rational expectation solution adding future information to the state space (nothing is shown in the output ofstoch_simul
) andforecast
will 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 theshocks
block for deterministic shocks, except that the numeric values given will be interpreted in a multiplicative way. For example, if a value of1.05
is given as shock value for some exogenous at some date, it means 5% above its steady state value.If no
endval
block is present, the steady state as specified in theinitval
block is used as the basis for the multiplication. If anendval
block is present, the terminal steady state as specified in theendval
block will be used as the basis for the multiplication (unless therelative_to_initval
option is passed).The syntax is the same as
shocks
in 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
endval
block is present, the initial steady state as specified in theinitval
block 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. Ifstd0
is the usual standard error forshock1
, then:using a scale factor in period
t
implies:std(shock1|t)=std0(shock1)*scale(t)
using a provided value in period
t
implies:std(shock1|t)=value(t)
.
The block has a similar syntax as the
shocks
block 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:
scales
andvalues
cannot be simultaneously set for the same shock in the same period, but it is possible to setvalues
for some periods andscales
for other periods for the same shock. There can be only onescales
andvalues
directive 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 previousshocks
block.
- 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 ashocks
block, 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 usedsteady
displays 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
initval
orendval
block.steady
uses an iterative procedure and takes as initial guess the value of the endogenous variables set in the previousinitval
orendval
block.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
maxit
is 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)
- solve_algo = INTEGER¶
Determines the non-linear solver to use. Possible values for the option are:
0
Use
fsolve
(under MATLAB, only available if you have the Optimization Toolbox; always available under Octave).1
Use a Newton-like algorithm with line-search.
2
Splits the model into recursive blocks and solves each block in turn using the same solver as value
1
.3
Use Chris Sims’ solver.
4
Splits the model into recursive blocks and solves each block in turn using a trust-region solver with autoscaling.
5
Newton algorithm with a sparse Gaussian elimination (SPE) solver at each iteration (requires
bytecode
option, see Model declaration).6
Newton algorithm with a sparse LU solver at each iteration (requires
bytecode
and/orblock
option, see Model declaration).7
Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires
bytecode
and/orblock
option, see Model declaration).8
Newton algorithm with a Stabilized Bi-Conjugate Gradient (BiCGStab) solver at each iteration (requires bytecode and/or block option, see Model declaration).
9
Trust-region algorithm with autoscaling (same as value
4
, but applied to the entire model, without splitting).10
Levenberg-Marquardt mixed complementarity problem (LMMCP) solver (Kanzow and Petra (2004)). The complementarity conditions are specified with an
mcp
equation tag, seelmmcp
.11
PATH mixed complementarity problem solver of Ferris and Munson (1999). The complementarity conditions are specified with an
mcp
equation tag, seelmmcp
. 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.12
Computes 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_solver
command with purely backward, forward or static models, or with routines for semi-structural models, and it must not be combined with optionblock
of themodel
block. Also note that for those models, the block decomposition is performed as ifmfs=3
had been passed to themodel
block, 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).14
Same 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_setup
block. This option can take three possible values:0
Do not use homotopy.
1
In 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_steps
option); the problem is solved as many times as there are steps.2
Same 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.3
Dynare 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_steps
contains 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_mode
option for more details. Default is10
.
- homotopy_force_continue = INTEGER¶
This option controls what happens when homotopy fails.
0
steady
fails with an error message1
steady
keeps 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_model
block. 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 (lowmarkowitz
values) and the one that preserves maximum sparsity (highmarkowitz
values). Default: 0.5.
- fsolve_options = (NAME, VALUE, ...)¶
A list of NAME and VALUE pairs. Can be used to set options for the
fsolve
routine, 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 offsolve
in the MATLAB or Octave manual. Note that Dynare already uses the values of themaxit
,tolf
andtolx
options of thesteady
command 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 theJacobian
orSpecifyObjectiveGradient
option.
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
var
command (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
initval
orendval
block. Exogenous variables are ordered in the order of declaration used in thevarexo
command (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_mode
of 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_setup
block 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
initval
block (or from the preceedingendval
block if there is one before thehomotopy_setup
block).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
initval
orendval
block).The
from_initval_to_endval
option 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 theendval
block that describes the terminal steady state. In that case, in the subsequentsteady
command, Dynare will perform a homotopy from the initial to the terminal steady state (technically, using this option is equivalent to writing ahomotopy_setup
block where all exogenous variables are asked to transition from their values in theinitval
to their values in theendval
block). When this option is used, thehomotopy_setup
block 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_endval
option).If the homotopy fails, a possible solution is to increase the number of steps (given in
homotopy_steps
option ofsteady
).Example
In the following example, Dynare will first compute the steady state for the initial values (
gam=0.5
andx=1
), and then subdivide the problem into 50 smaller problems to find the steady state for the final values (gam=2
andx=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_model
block, which is described below in more details. See alsofs2000.mod
in theexamples
directory 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
.mod
file is calledFILENAME.mod
, the steady state file must be calledFILENAME_steadystate.m
. SeeNK_baseline_steadystate.m
in 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_model
block also works with deterministic models. Aninitval
block and, when necessary, anendval
block, is used to set the value of the exogenous variables. Eachinitval
orendval
block must be followed bysteady
to execute the function created bysteady_state_model
and 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
,endval
orsteady
statement. 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
check
returns the eigenvalues in the global variableoo_.dr.eigval
.
- MATLAB/Octave variable: oo_.dr.eigval¶
Contains the eigenvalues of the model, as computed by the
check
command.
- 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_info
indicates 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 FORWARD
In 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 BACKWARD
The 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 x
The 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,
x
denotes the subtype of the block.x
is equal toSIMPLE
if the block has only one equation. If several equations appear in the block,x
is equal toCOMPLETE
.SOLVE FORWARD x
The 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,
x
denotes the subtype of the block.x
is equal toSIMPLE
if the block has only one equation. If several equations appear in the block,x
is equal toCOMPLETE
.SOLVE TWO BOUNDARIES x
The 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,
x
denotes the subtype of the block.x
is equal toSIMPLE
if the block has only one equation. If several equations appear in the block,x
is 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_dynamic
orblock_static
options.
- 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 thebytecode
option of themodel
block.
- 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 thebytecode
option of themodel
block.
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
t
is zero, causingperfect_foresight_solver
to generally fail. This is due to the use of the Lagrange multiplierLambda
as 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
,endval
andshocks
blocks 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.
- datafile = FILENAME
Used to specify path for all endogenous and exogenous variables. Strictly equivalent to
initval_file
.
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_setup
must 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_homotopy
option 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
maxit
is50
.
- 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:
0
Use a Newton algorithm with a direct sparse LU solver at each iteration, applied to the stacked system of all equations in all periods (Default).
1
Use 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 theblock
option 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 BACKWARD
orSOLVE FORWARD
, seemodel_info
), since LBJ only makes sense on blocks with both leads and lags (of typeSOLVE TWO BOUNDARIES
).2
Use a Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration, applied on the stacked system of all equations in all periods (requires
bytecode
and/orblock
option, see Model declaration)3
Use 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 (requires
bytecode
and/orblock
option, see Model declaration).4
Use 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
bytecode
and/orblock
option, see Model declaration).5
Use 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 (requires
bytecode
option, see Model declaration).6
Synonymous for
stack_solve_algo=1
. Kept for backward compatibility.7
Allows 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
,7
and8
, which requirebytecode
and/orblock
options, 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.
- 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 theblock
nor thebytecode
option of themodel
block is specified); for those models, the values12
and14
are 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
0
and1
. 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 \(\frac{y(s^*)-y(0)}{s^*}\).
- 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
0.01
by default, but can be modified by passing some other value to the option.
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_fallback
or thehomotopy_marginal_linearization_fallback
option. 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.
- markowitz = DOUBLE
Value of the Markowitz criterion, used to select the pivot. 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 ZLB on the nominal interest rate or a model with irreversible investment). This option is equivalent to
stack_solve_algo=7
andsolve_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:\[ \begin{align}\begin{aligned}\begin{split}LB = X &\Rightarrow F(X)>0\\\end{split}\\\begin{split}LB\leq X \leq UB &\Rightarrow F(X)=0\\\end{split}\\X =UB &\Rightarrow F(X)<0.\end{aligned}\end{align} \]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 attaching an equation tag (see Model declaration) with the
mcp
keyword to the affected equations. This tag states that the equation to which the tag is attached has to hold unless the inequality constraint within the tag is binding.For instance, a ZLB on the nominal interest rate would be specified as follows in the model block:
model; ... [mcp = 'r > -1.94478'] r = rho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e; ... end;
where
1.94478
is the steady state level of the nominal interest rate andr
is 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 ther
is fixed at-1.94478
(thereby being equivalent to a complementary slackness condition). By restricting the value ofr
coming out of this equation, themcp
tag also avoids usingmax(r,-1.94478)
for other occurrences ofr
in the rest of the model. Two things are important to keep in mind. First, because themcp
tag effectively replaces a complementary slackness condition, it cannot be simply attached to any equation. Rather, 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=RHS
asresidual=LHS-RHS
, while an implicit equationLHS
is 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) + e
is belowr=-1.94478
. In contrast, specifying the equation asrho*r(-1) + (1-rho)*(gpi*Infl+gy*YGap) + e = r;`
would be wrong.
Note that in the current implementation, the content of the
mcp
equation tag is not parsed by the preprocessor. The inequalities must therefore be as simple as possible: an endogenous variable, followed by a relational operator, followed by a number (not a variable, parameter or expression).
- 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 option
stack_solve_algo==0
.
- 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
,endval
or a subsequentsteady
. Only available with optionstack_solve_algo==0
orstack_solve_algo==7
.
- endval_steady¶
In scenarios with a permanent shock, specifies that the terminal condition is a steady state, even if the
steady
command has not been called after theendval
block. As a consequence, theperfect_foresight_solver
command will compute the terminal steady state itself (given the value of the exogenous variables given in theendval
block). In practice, this option is useful when the permanent shock is very large, in which case the homotopy procedure insideperfect_foresight_solver
will 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).
- steady_solve_algo = INTEGER¶
See solve_algo. Used when computing the terminal steady state when option
endval_steady
has been specified to theperfect_foresight_setup
command.
- steady_tolf = DOUBLE¶
See tolf. Used when computing the terminal steady state when option
endval_steady
has been specified to theperfect_foresight_setup
command.
- steady_tolx = DOUBLE¶
See tolx. Used when computing the terminal steady state when option
endval_steady
has been specified to theperfect_foresight_setup
command.
- steady_maxit = INTEGER¶
See maxit. Used when computing the terminal steady state when option
endval_steady
has been specified to theperfect_foresight_setup
command.
- steady_markowitz = DOUBLE¶
See markowitz. Used when computing the terminal steady state when option
endval_steady
has been specified to theperfect_foresight_setup
command.
Output
The simulated endogenous variables are available in global matrix
oo_.endo_simul
.
- Command: simul ;¶
- Command: simul(OPTIONS...);
This command is deprecated. It is strictly equivalent to a call toperfect_foresight_setup
followed by a call toperfect_foresight_solver
.Options
Accepts all the options of
perfect_foresight_setup
andperfect_foresight_solver
.
- MATLAB/Octave variable: oo_.endo_simul¶
This variable stores the result of a deterministic simulation (computed byperfect_foresight_solver
orsimul
) or of a stochastic simulation (computed bystoch_simul
with the periods option 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 theperiods
option: 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
,simul
,stoch_simul
orextended_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 theperiods
option: 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: oo_.initial_steady_state¶
If a permanent shock is simulated through the use of both
initval
andendval
blocks, this variable contains the initial steady state, as determined by theinitval
block (when followed by asteady
command). This variable has the same structure asoo_.steady_state
(and this latter variable contains the terminal steady state, if theendval
block is followed by asteady
command).
- MATLAB/Octave variable: oo_.initial_exo_steady_state¶
If a permanent shock is simulated through the use of both
initval
andendval
blocks, this variable contains the initial steady state of the exogenous variables, as specified in theinitval
block. 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).
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) ;
- Block: shocks(learnt_in=INTEGER,overwrite) ;
Theshocks(learnt_in=INTEGER)
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 regularshocks
block:var VARIABLE_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; 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
values
keyword can be replaced by theadd
keyword; similarly, if the new information consists of a multiplication of the previously-anticipated value, thevalues
keyword can be replaced by themultiply
keyword.The
overwrite
option says that this block cancels and replaces previousshocks
andmshocks
blocks that have the samelearnt_in
option.Note that a
shocks(learnt_in=1)
block is equivalent to a regularshocks
block.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,
x
is 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),
x
is 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),
x
is expected to be equal to 1.3 in period 4, and to 2.8 in period 5.
- Block: endval(learnt_in=INTEGER) ;
Theendval(learnt_in=INTEGER)
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 regularendval
block.Also note that, similarly to the regular
endval
block, 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
endval
blocks without thelearnt_in
option). 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
x
will be 1.1;the terminal condition for
y
will be 0.1 above the terminal condition fory
that was expected from the perspective of period 2;the terminal condition for
z
will be 2 times the terminal condition forz
that 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
,y
orz
specified through a regularshocks
block 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.
- Block: mshocks(learnt_in=INTEGER) ;
- Block: mshocks(learnt_in=INTEGER,OPTIONS...) ;
Themshocks(learnt_in=INTEGER)
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 regularmshocks
block:var VARIABLE_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...;
As in the regular
mshocks
block (without thelearnt_in
option), the values are interpreted as a multiplicative factor over the steady state value of the exogenous variable (the latter being taken either from theinitval
orendval
, seemshocks
for the details).If the terminal steady state as specified in the
endval
block is used as a basis for the multiplication, its value as anticipated from the period given in thelearnt_in
option will be used.Note that a
mshocks(learnt_in=1)
block is equivalent to a regularmshocks
block.Options
- overwrite
This block cancels and replaces previous
shocks
andmshocks
blocks that have the samelearnt_in
option.
- relative_to_initval
Same meaning as in the regular
mshocks
block.
Example
mshocks(learnt_in=2); var x; periods 3:4; values 1.1; end;
This syntax means that from the perspective of period 2,
x
in periods 3 and 4 is expected to be equal to 1.1 times its steady state. If there is noendval
block, the initial steady state as given byinitval
is used; if there is anendval
block, the terminal steady state as anticipated from the perspective of period 2 is used (as specified in the relevantendval(learnt_in=…
block)).
- 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
,endval
andshocks
blocks (the latter two types of blocks typically used with thelearnt_in
option); alternatively, the information about future shocks can be given in a CSV file using thedatafile
option.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
endval
block is ignored. Said otherwise, the equivalent of optionendval_steady
of theperfect_foresight_setup
command is always implicitly enabled.Options
- periods = INTEGER
Number of periods of the simulation.
- datafile = FILENAME
Used to specify the information about future shocks and their anticipation through a CSV file, as an alternative to
shocks
andendval
blocks.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
p
is the value of theperiods
option andk
is the number of exogenous variables, then the CSV file hasp+3
lines andk×p+1
columns.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 ands
on the second line. Of course, values in cells corresponding tot<s
are ignored.
Output
oo_.exo_simul
andoo_.endo_simul
are 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
datafile
option (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_setup
must 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 periods` option of
perfect_foresight_with_expectation_errors_setup
; as a consequence, the simulated paths as stored inoo_.endo_simul
will 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).
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_state
andoo_.exo_steady_state
.
- 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_setup
has 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 indexedk
in 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_setup
has 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)
.
4.13. 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_simul
computes 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 thevarexo
command.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
1
and 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
shocks
command (see Shocks on exogenous variables).When a list of
VARIABLE_NAME
is 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_simul
and the workspace. Here, no periods are dropped. Default:100
.
- hp_filter = DOUBLE¶
Uses HP filter with \(\lambda =\)
DOUBLE
before 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 \(\lambda =\)
DOUBLE
described 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_filter
or 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=0
suppresses 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_.irfs
to any desired size. At higher order, the interpretation is different. Therelative_irf
option 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_format
option, except ifgraph_format=none
) and displayed to screen (unlessnodisplay
option 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
nograph
is 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
,fig
andnone
. Under Octave,fig
will use Octave’sofig
format. 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_solver
option is implied and only empirical moments are available (you must provide a value forperiods
option). Default:2
(except after anestimation
command, 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
bytecode
option (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_init
option equal to1
: the default is0.999999
in 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:
1
iforder=1
, and50
otherwise.
- 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_simul
in theFILENAME/Output
folder. 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_decomposition
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_.conditional_variance_decomposition_ME
(seeoo_.conditional_variance_decomposition_ME
). The variance decomposition is only conducted, if theoretical moments are requested, i.e. using theperiods=0
option. Only available atorder<3
and 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 ifnodecomposition
is 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
varobs
command. Note that ifvarobs
is 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:
default
Uses the default method to compute the decision rule based on the generalized Schur decomposition (see Villemot (2011) for more information).
cycle_reduction
Uses 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_reduction
Uses 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_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_simul
will display the moments, decision rules, and impulse responses for the log-linearized variables. The decision rules saved inoo_.dr
and 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_tol
are not displayed. Default value:1e-6
.
- contemporaneous_correlation¶
Saves the contemporaneous correlation between the endogenous variables in
oo_.contemporaneous_correlation
. Requires thenocorr
option 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
periods
option is present, setsoo_.skewness
,oo_.kurtosis
, andoo_.endo_simul
(seeoo_.endo_simul
).If option
irf
is different from zero, setsoo_.irfs
(see below).If the option
contemporaneous_correlation
is 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
y
andk
.
- MATLAB/Octave variable: oo_.mean¶
After a run ofstoch_simul
, contains the mean of the endogenous variables. Contains theoretical mean if theperiods
option 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 theperiods
option is not present and simulated variance otherwise. Only available fororder<4
. Atorder=2
it 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_simul
contains the skewness (standardized third moment) of the simulated variables if theperiods
option is present. The variables are arranged in declaration order.
- MATLAB/Octave variable: oo_.kurtosis¶
After a run ofstoch_simul
contains the excess kurtosis (standardized fourth moment) of the simulated variables if theperiods
option 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 theperiods
option is not present and simulated autocorrelations otherwise. Only available fororder<4
. Atorder=2
it 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 theperiods
option is not present), this variable contains a cell array with the following values (wherear
is 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_.autocorr
for 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_simul
when 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_simul
when 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_simul
with theconditional_variance_decomposition
option, 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_names
if 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_simul
with theconditional_variance_decomposition
option 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_names
if 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_simul
with thecontemporaneous_correlation option
, contains theoretical contemporaneous correlations if theperiods
option is not present, and simulated contemporaneous correlations otherwise. Only available fororder<4
. Atorder=2
it 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_simul
with optionspectral_density
, contains the spectral density of the model variables. There will be anvars
bynfrequencies
subfieldfreqs
storing the respective frequency grid points ranging from \(0\) to \(2\pi\) and a same sized subfielddensity
storing the corresponding density.
- MATLAB/Octave variable: oo_.irfs¶
After a run ofstoch_simul
with optionirf
different from zero, contains the impulse responses, with the following naming convention: VARIABLE_NAME_SHOCK_NAME.For example,
oo_.irfs.gnp_ea
contains the effect ongnp
of 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
0
Dynare uses a gaussian quadrature to take into account the effects of future uncertainty. 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 thebytecode
option of themodel
block. 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
mcp
tag, seelmmcp
.
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_var
in 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_constraints
blockspecifying the model equations for the respective regimes in the
model
block using appropriate equation tags.potentially specifying a sequence of surprise shocks using a
shocks(surprise)
blocksetting up Occbin simulations or estimation with
occbin_setup
triggering a simulation with
occbin_solver
or runningestimation
orcalib_smoother
.
All of these elements are discussed in the following.
- Block: occbin_constraints ;¶
Theoccbin_constraints
block 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;]
STRING
is the name of constraint that is used to reference the constraint inrelax
/bind
equation tags to identify the respective regime (see below). Thebind
expression 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, therelax
expression 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 thebind
expression evaluates to false. However, there are cases where thebind
expression 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
bind
condition 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_bind
anderror_relax
options 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 thebind
andrelax
inequalities. 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
Lambda
in 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
model
block, with equation tags indicating to which regime a particular equation belongs. All equations that differ across regimes must have aname
tag 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 abind
orrelax
tag 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
estimation
with occasionally binding constraints, it needs to be prefaced by a call tooccbin_setup
to 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 of stochastic singularity if there are then more observables than shocks. To avoid this issue, the data points for the zero interest rate should be set to NaN and the standard deviation of the associated shock set to 0 for the corresponding periods using theheteroskedastic_shocks
block.Note that models with unit roots will require the user to specify the
diffuse_filter
option 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_shocks
using theheteroskedastic_filter
option.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_periods
to 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_periods
ever 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_periods
ever further. Equivalent ofsimul_max_check_ahead_periods
. Default: Inf.
- smoother_curb_retrench¶
Have the smoother invoke the
simul_curb_retrench
option 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_periods
ever further. Equivalent ofsimul_max_check_ahead_periods
. Default: Inf.
- likelihood_curb_retrench¶
Have the likelihood computation invoke the
simul_curb_retrench
option 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_setup
must 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_names
stored 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).type
is equal to eithersmoother
orsimul
, depending on whether the output comes from a run of simulations or the smoother. The subfieldregime
contains a vector storing the regime state, while the the subfieldregimestart
indicates the expected start of the respective regime state. For example, if row 40 contains[1,0]
forregime2
and[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_history
oroo_.occbin.smoother.regime_history
into an Excel file stored in theFILENAME/Output
folder.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_information
option ofstoch_simul
, for declaring the set of observed variables when solving the model under partial information.Only one instance of
varobs
is 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
,y
andrr
as 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 theloglinear
option 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_trends
is 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_NAME
Indicates 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_NAME2
Indicates 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
shocks
blocks 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_NAME
The name of a model parameter to be estimated
DSGE_PRIOR_WEIGHT
Special 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_pdf
andweibull_pdf
. Note thatinv_gamma_pdf
is 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:
-Inf
for normal distribution,0
otherwise.
- PRIOR_4TH_PARAMETER¶
A fourth parameter of the prior used for generalized beta distribution, the truncated normal, and for the uniform distribution. Default:
Inf
for normal distribution,1
otherwise.
- 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_PARAMETER
andPRIOR_4TH_PARAMETER
or 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_PARAMETER
andPRIOR_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_1
andeps_2
with mean0
and variance0.3
. By settingPRIOR_3RD_PARAMETER
to-1
andPRIOR_4TH_PARAMETER
to1
the 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-1
and1
, 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_params
blocks. By default, subsequent blocks are concatenated with the previous ones; this can be useful when building models in a modular fashion (see alsoestimated_params_remove
for that use case). However, if anestimated_params
block has theoverwrite
option, its contents becomes the new list of estimated parameters, cancelling previous blocks; this can be useful when doing several estimations in a single.mod
file.
- 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_params
block 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
shocks
block from calibration as starting values for estimation. For components of theshocks
block 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_params
block, 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
estimation
command. 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_observed
options 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_file
option 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
forecast
option ofestimation
with ML, which will useHPDinf/HPDsup
to denote the confidence interval.Algorithms
The Monte Carlo Markov Chain (MCMC) diagnostics are generated by the estimation command if
mh_replic
is larger than 2000 and if optionnodiagnostic
is 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_interval
after 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_nblocks
is 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
.m
file, a.mat
file, a.csv
file, or a.xls/.xlsx
file (under Octave, the io package from Octave-Forge is required for the.csv
and.xlsx
formats and the.xls
file 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
estimation
output. 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_obs
to 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
INTEGER1
toINTEGER2
. Optionforecast
must also be specified. The forecasts are stored in theRecursiveForecast
field 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_obs
needs 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
nobs
starting with the first observation ranging fromINTEGER1
toINTEGER2
. Optionforecast
must 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
logdata
option 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_obs
to 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
) unlesslogdata
option 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.
0
No prior plot.
1
Prior 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:
1
For stationary models, the initial matrix of variance of the error of forecast is set equal to the unconditional variance of the state variables.
2
For 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)).
3
For nonstationary models: use a diffuse filter (use rather the
diffuse_filter
option).4
The filter is initialized with the fixed point of the Riccati equation.
5
Use 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_params
block).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_draws
should 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_params
block.Note that
mode_compute=6
will 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.mat
in theFILENAME/Output
folder. This file can be loaded in subsequent runs via theposterior_sampler_options
option scale_file. Bothmode_compute=6
andscale_file
will overwrite any value specified inestimated_params
with 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_jscale
option 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_scale
is set at the beginning of Dynare execution, i.e. the default will not take into account potential changes inmh_jscale
introduced by eithermode_compute=6
or theposterior_sampler_options
option scale_file. Ifmh_init_scale
is 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
value with which the next 100 draws will be drawn and tested. If thenointeractive
option has been invoked, the program will instead automatically decreasemh_init_scale
by 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_jscale
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
If
mh_init_scale_factor
is 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_factor
value with which the next 100 draws will be drawn and tested. If thenointeractive
option has been invoked, the program will instead automatically decreasemh_init_scale_factor
by 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_jscale
option. Default:2.38/sqrt(n)
. Must not be set ifmh_tune_jscale
is 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_file
or a differentmh_replic
than 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_file
used in the previous run or the same user-definedmcmc_jumping_covariance
when 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.mat
in theFILENAME/Output
folder. After a successful run of the estimation command, themode_file
will be disabled to prevent other function calls from implicitly using an updated mode file. Thus, if the.mod
file contains subsequentestimation
commands, themode_file
option, if desired, needs to be specified again.
- mode_compute = INTEGER | FUNCTION_NAME¶
Specifies the optimizer for the mode computation:
0
The mode isn’t computed. When the
mode_file
option is specified, the mode is simply read from that file.When
mode_file
option is not specified, Dynare reports the value of the log posterior (log likelihood) evaluated at the initial value of the parameters.When
mode_file
is not specified and there is noestimated_params
block, but thesmoother
option is used, it is a roundabout way to compute the smoothed value of the variables of a model with calibrated parameters.1
Uses
fmincon
optimization 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).2
Uses the continuous simulated annealing global optimization algorithm described in Corana et al.(1987) and Goffe et al.(1994).
3
Uses
fminunc
optimization routine (available under MATLAB if the Optimization Toolbox is installed; available under Octave if the optim package from Octave-Forge is installed).4
Uses Chris Sims’s
csminwel
.5
Uses 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. Usescsminwel
for line search in each step.6
Uses a Monte-Carlo based optimization routine (see https://archives.dynare.org/DynareWiki/MonteCarloOptimization for more details).
7
Uses
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).8
Uses Dynare implementation of the Nelder-Mead simplex-based optimization routine (generally more efficient than the MATLAB or Octave implementation available with
mode_compute=7
).9
Uses the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) algorithm of Hansen and Kern (2004), an evolutionary algorithm for difficult non-linear non-convex optimization.
10
Uses the
simpsa
algorithm, based on the combination of the non-linear simplex and simulated annealing algorithms as proposed by Cardoso, Salcedo and Feyo de Azevedo (1996).11
This is not strictly speaking an optimization algorithm. The (estimated) parameters are treated as state variables and estimated 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
k
order local approximations of the model.12
Uses the
particleswarm
optimization routine (available under MATLAB if the Global Optimization Toolbox is installed; not available under Octave).13
Uses the
lsqnonlin
non-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
.101
Uses the SolveOpt algorithm for local nonlinear optimization problems proposed by Kuntsevich and Kappel (1997).
102
Uses
simulannealbnd
optimization routine (available under MATLAB if the Global Optimization Toolbox is installed; not available under Octave)FUNCTION_NAME
It 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:
hessian
Uses the Hessian matrix computed at the mode.
prior_variance
Uses the prior variances. No infinite prior variances are allowed in this case.
identity_matrix
Uses an identity matrix.
FILENAME
Loads 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>1
the likelihood function resulting from the particle filter is not differentiable anymore due to the resampling step. For this reason, themode_check
plot 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. TheInf
value 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 of0
allows 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 = Inf
when 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 that is ignored when computing bounds for the parameters. Default:
1e-32
.
- 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_file
used in the previous run or the same user-definedmcmc_jumping_covariance
when 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_file
is specified withmh_replic=0
. It tells Dynare to load the previously computed convergence diagnostics, marginal data density, and posterior statistics from an existing_results
file 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_file
would 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_mcmc
is 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_dir
and 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>.mat
and 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_mcmc
is 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_mcmc
is 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, 13
Available options are given in the documentation of the MATLAB Optimization Toolbox or in Octave’s documentation.
2
Available 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:1
4
Available 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
,3
and5
, respectively, corresponding to the two, three and five points formula used to compute the gradient of the objective function (see Abramowitz and Stegun (1964)). Values13
and15
are 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
0
to set to silent. Default:1
.'SaveFiles'
Controls saving of intermediate results during optimization. Set to
0
to shut off saving. Default:1
.5
Available 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-5
for numerical derivatives,1e-7
for 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
0
to set to silent. Default:1
.'SaveFiles'
Controls saving of intermediate results during optimization. Set to
0
to shut off saving. Default:1
.6
Available 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_file
is 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
.8
Available 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
MaxFunvEvals
equal toMaxFunvEvalFactor
times 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
0
to set to silent. Default:1
.9
Available options are:
'CMAESResume'
Resume previous run. Requires the
variablescmaes.mat
from the last run. Set to1
to 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
0
to set to silent. Default:1
.'SaveFiles'
Controls saving of intermediate results during optimization. Set to
0
to shut off saving. Default:1
.10
Available 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
0
to set to silent. Default:1
.101
Available 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
0
to set to silent. Default:1
.102
Available 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
irf
option. Results are stored inoo_.PosteriorIRF.dsge
(see below for a description of this variable).
- 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_weight
as 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_params
section.NB: The previous method of declaring
dsge_prior_weight
as a parameter and then placing it inestimated_params
is 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.
'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.
- 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_file
option to be specified. Default:0
.'scale_file'
Provides the name of a
_mh_scale.mat
file storing the tuned scale factor from a previous run ofmode_compute=6
.'save_tmp_file'
Save the MCMC draws into a
_mh_tmp_blck
file at the refresh rate of the status bar instead of just saving the draws when the current_mh*_blck
file is full. Default:0
'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_matrix
orslice_initialize_with_mode
. Default:0
.'mode_files'
For multimodal posteriors, provide the name of a file containing a
nparam
bynmodes
variable calledxparams
storing 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 therotated
andmode
options. Default:[]
.'slice_initialize_with_mode'
The default for slice is to set
mode_compute=0
and 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_size
must 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 = DOUBLE
Specifies 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 = INTEGER
Specifies 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 thear
option.
- contemporaneous_correlation
See
contemporaneous_correlation
. Results are stored inoo_.PosteriorTheoreticalMoments
. Note that thenocorr
option 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_varendo
to 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_.SmoothedShocks
andoo_.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
d
denotes the number of diffuse filtering steps.
FilteredVariablesKStepAheadVariances
,Variance
, andState_uncertainty
cannot 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_.forecast
and corresponds to the forecast at the posterior mode. If a Metropolis-Hastings is computed, the distribution of forecasts is stored in variablesoo_.PointForecast
andoo_.MeanForecast
. See Forecasting, for a description of these variables.
- tex
See
tex
.
- kalman_algo = INTEGER¶
0
Automatically use the Multivariate Kalman Filter for stationary models and the Multivariate Diffuse Kalman Filter for non-stationary models.
1
Use the Multivariate Kalman Filter.
2
Use the Univariate Kalman Filter.
3
Use the Multivariate Diffuse Kalman Filter.
4
Use 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_detected
option.
- fast_kalman_filter¶
Select the fast Kalman filter using Chandrasekhar recursions as described by
Herbst (2015)
. This setting is only used withkalman_algo=1
orkalman_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_ahead
is 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_.FilteredVariablesKStepAheadVariances
is 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_criterium
to 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_filter
is used thelik_init
option ofestimation
has 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
linear
option is used in the model declaration. The user can either provide the steady state to Dynare using asteady_state_model
block (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_shocks
block.
- selected_variables_only¶
Only run the classical smoother on the variables listed just after the
estimation
command. 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 variablehh
ofMODEL_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_irf
is passed.
- irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] )
See
irf_shocks
. Only used ifbayesian_irf
is passed.
- irf_plot_threshold = DOUBLE
See
irf_plot_threshold
. Only used ifbayesian_irf
is passed.
- aim_solver
See
aim_solver
.
- 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:
default
Uses the default solver for Lyapunov equations based on Bartels-Stewart algorithm.
fixed_point
Uses a fixed point algorithm to solve the Lyapunov equation. This method is faster than the
default
one for large scale models, but it could require a large amount of iterations.doubling
Uses 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_solver
Uses 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_params
block, 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_detected
option to have theuse_univariate_filters_if_singularity_is_detected
only 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 = DOUBLE
percent 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
q
with precisionr
with a probabilitys
. Typically, one wants to estimate theq=0.025
percentile (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
q
that is estimated with precisionr
with a probabilitys
in 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
estimation
command.
- 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
smoother2histval
on 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
estimation
command.
- 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:
none
No resampling.
systematic
Resampling at each iteration, this is the default value.
generic
Resampling 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 optionresampling
has valuegeneric
.
- resampling_method = OPTION¶
Sets the resampling method. Possible values for OPTION are:
kitagawa
,stratified
andsmooth
.
- filter_algorithm = OPTION¶
Sets the particle filter algorithm. Possible values for OPTION are:
sis
Sequential importance sampling algorithm, this is the default value.
apf
Auxiliary particle filter.
gf
Gaussian filter.
gmf
Gaussian mixture filter.
cpf
Conditional particle filter.
nlkf
Use 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
,montecarlo
andunscented
. Default value isunscented
.
- distribution_approximation = OPTION¶
Sets the method for approximating the particle distribution. Possible values for OPTION are:
cubature
,montecarlo
andunscented
. 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 thecheck
command 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. 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_jscale
parameter is used for a parameter inestimated_params
, the procedure usesmh_jscale
for all parameters. Ifmh_jscale
option isn’t set, the procedure uses0.2
for all parameters. Note that ifmode_compute=6
is used or theposterior_sampler_option
calledscale_file
is specified, the values set inestimated_params
will 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_calibration
andmoment_calibration
blocks (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_comparison
in this case, because the prior density will not integrate to 1.Output
After running estimation, the parameters
M_.params
and the variance matrixM_.Sigma_e
of 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 > 0
or optionload_mh_file
set) the parametersM_.params
and the variance matrixM_.Sigma_e
of the shocks are set to the posterior mean.Depending on the options,
estimation
stores results in various fields of theoo_
structure, described below. In the following variables, we will adopt the following shortcuts for specific field names:MOMENT_NAME
This field can take the following values:
HPDinf
Lower bound of a 90% HPD interval. [4]
HPDsup
Upper bound of a 90% HPD interval.
HPDinf_ME
Lower 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_ME
Upper bound of a 90% HPD interval for observables when taking measurement error into account.
Mean
Mean of the posterior distribution.
Median
Median of the posterior distribution.
Std
Standard deviation of the posterior distribution.
Variance
Variance of the posterior distribution.
deciles
Deciles of the distribution.
density
Non 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_OBJECT
This field can take the following values:
measurement_errors_corr
Correlation between two measurement errors.
measurement_errors_std
Standard deviation of measurement errors.
parameters
Parameters.
shocks_corr
Correlation between two structural shocks.
shocks_std
Standard deviation of structural shocks.
- MATLAB/Octave variable: oo_.MarginalDensity.LaplaceApproximation¶
Variable set by the
estimation
command. 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 > 0
orload_mh_file
option. Stores the marginal data density based on Geweke (1999) Modified Harmonic Mean estimator.
- MATLAB/Octave variable: oo_.posterior.optimization¶
Variable set by the
estimation
command 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:
mode
Parameter vector at the mode.
Variance
Inverse Hessian matrix at the mode or MCMC jumping covariance matrix when used with the
MCMC_jumping_covariance
option.log_density
Log 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
estimation
command ifmh_replic>0
is used. Fields are of the form:oo_.posterior.metropolis.OBJECT
where OBJECT is one of the following:
mean
Mean parameter vector from the MCMC.
Variance
Covariance matrix of the parameter draws in the MCMC.
- MATLAB/Octave variable: oo_.FilteredVariables¶
Variable set by the
estimation
command, if it is used with thefiltered_vars
option.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
estimation
command, if it is used with thefilter_step_ahead
option. 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_only
option 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
estimation
command, 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
estimation
command, if it is used with thefilter_step_ahead option
in 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
estimation
command, if it is used with thefilter_step_ahead
option. 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
estimation
command, if it is used with thebayesian_irf
option. Fields are of the form:oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME
- MATLAB/Octave variable: oo_.SmoothedMeasurementErrors¶
Variable set by the
estimation
command, if it is used with thesmoother
option. Fields are of the form:oo_.SmoothedMeasurementErrors.VARIABLE_NAME
- MATLAB/Octave variable: oo_.SmoothedShocks¶
Variable set by the
estimation
command (if used with thesmoother
option), or by thecalib_smoother
command.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
estimation
command (if used with thesmoother
option), or by thecalib_smoother
command.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_.SmoothedVariables
andoo_.SmoothedShocks
variables.
- MATLAB/Octave variable: oo_.UpdatedVariables¶
Variable set by the
estimation
command (if used with thesmoother
option), or by thecalib_smoother
command. 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_.UpdatedVariables
variable.
- MATLAB/Octave variable: oo_.FilterCovariance¶
Three-dimensional array set by the
estimation
command if used with thesmoother
and Metropolis, if thefilter_covariance
option has been requested. Contains the series of one-step ahead forecast error covariance matrices from the Kalman smoother. TheM_.endo_nbr
timesM_.endo_nbr
timesT+1
array 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
estimation
command (if used with thesmoother
) without Metropolis, or by thecalib_smoother
command, if thefilter_covariance
option has been requested. Contains the series of one-step ahead forecast error covariance matrices from the Kalman smoother. TheM_.endo_nbr
timesM_.endo_nbr
timesT+1
array 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
estimation
command (if used with thesmoother
option) without Metropolis, or by thecalib_smoother
command, if thesmoothed_state_uncertainty
option has been requested. Contains the series of covariance matrices for the state estimate given the full data from the Kalman smoother. TheM_.endo_nbr
timesM_.endo_nbr
timesT
array 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
estimation
command (if used with thesmoother
) without Metropolis, or by thecalib_smoother
command. 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
estimation
command (if used with thesmoother
) without Metropolis, or by thecalib_smoother
command. 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 thesmoother
option), or by thecalib_smoother
command. 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
estimation
command (if used with thesmoother
option), or by thecalib_smoother
command. 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
estimation
command, if it is used with themoments_varendo
option. Fields are of the form:oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_NAME.VARIABLE_NAME
where THEORETICAL_MOMENT is one of the following:
covariance
Variance-covariance of endogenous variables.
contemporaneous_correlation
Contemporaneous correlation of endogenous variables when the
contemporaneous_correlation
option is specified.correlation
Auto- and cross-correlation of endogenous variables. Fields are vectors with correlations from 1 up to order
options_.ar
.VarianceDecomposition
Decomposition of variance (unconditional variance, i.e. at horizon infinity). [6]
VarianceDecompositionME
Same 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
.ConditionalVarianceDecomposition
Only if the
conditional_variance_decomposition
option 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.ConditionalVarianceDecompositionME
Only if the
conditional_variance_decomposition
option 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
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.posterior_density.PARAMETER_NAME
- MATLAB/Octave variable: oo_.posterior_hpdinf¶
Variable set by the
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_hpdsup¶
Variable set by the
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_mean¶
Variable set by the
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_mode¶
Variable set by the
estimation
command 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
estimation
command 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
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_var¶
Variable set by the
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.posterior_var.ESTIMATED_OBJECT.VARIABLE_NAME
- MATLAB/Octave variable: oo_.posterior_median¶
Variable set by the
estimation
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_var
option of theestimation
command after mode_compute.The following fields are saved:
PHI_tilde
Stacked posterior DSGE-BVAR autoregressive matrices at the mode (equation (28) of Del Negro and Schorfheide (2004)).
SIGMA_u_tilde
Posterior covariance matrix of the DSGE-BVAR at the mode (equation (29) of Del Negro and Schorfheide (2004)).
iXX
Posterior population moments in the DSGE-BVAR at the mode ( \(inv(\lambda T \Gamma_{XX}^*+ X'X)\)).
prior
Structure storing the DSGE-BVAR prior.
PHI_star
Stacked prior DSGE-BVAR autoregressive matrices at the mode (equation (22) of Del Negro and Schorfheide (2004)).
SIGMA_star
Prior covariance matrix of the DSGE-BVAR at the mode (equation (23) of Del Negro and Schorfheide (2004)).
ArtificialSampleSize
Size of the artifical prior sample ( \(inv(\lambda T)\)).
DF
Prior degrees of freedom ( \(inv(\lambda T-k-n)\)).
iGXX_star
Inverse 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
forecast
option of theestimation
command when used with the nobs = [INTEGER1:INTEGER2] option (seenobs
).Fields are of the form:
oo_.RecursiveForecast.FORECAST_OBJECT.VARIABLE_NAME
where
FORECAST_OBJECT
is one of the following [7] :Mean
Mean of the posterior forecast distribution.
HPDinf/HPDsup
Upper/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_NAME
contains 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
estimation
command. 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:
posteriormean
Mean of the posterior parameter distribution.
posteriorstd
Standard deviation of the posterior parameter distribution.
nse_iid
Numerical standard error (NSE) under the assumption of iid draws.
rne_iid
Relative numerical efficiency (RNE) under the assumption of iid draws.
nse_taper_x
Numerical standard error (NSE) when using an x% taper.
rne_taper_x
Relative numerical efficiency (RNE) when using an x% taper.
pooled_mean
Mean of the parameter when pooling the beginning and end parts of the chain specified in
geweke_interval
and weighting them with their relative precision. It is a vector containing the results under the iid assumption followed by the ones using thetaper_steps
option (seetaper_steps
).pooled_nse
NSE 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_test
p-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_steps
signal 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
estimation
command when used withraftery_lewis_diagnostics
option (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. Useestimation
optiondiffuse_filter
instead for estimating a model with non-stationary observed variables orsteady
optionnocheck
to preventsteady
to 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_weights
block 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
prefilter
option 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_moments
block into a cell arrayM_.matched_moments
where: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_moments
and 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. Theoverwrite
option replaces the currentmatched_irfs
block 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
periods
keyword. Note that for each entry a corresponding entry invalues
needs to be provided; that isvalues
is a list of the same length asperiods
. If only one value is specified, it is used at all correspondingperiods
in the list.weights
are 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 correspondingperiods
in the list. If not specified, the weight defaults to 1. Forvalues
andweights
you 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 thevar
orvarexo
keyword.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_irfs
block into a cell array where the rows correspond to the statements in the blockM_.matched_irfs
where: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. Theoverwrite
option replaces the currentmatched_irfs_weights
block 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_weigths
block into a cell arrayM_.matched_irfs_weights
where: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, WEIGHT
andy(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_params
for 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_init
for details and syntax.
- Block: estimated_params_bounds ;
Optional. This block declares lower and upper bounds for parameters in maximum likelihood estimation. Seeestimated_params_bounds
for 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
datafile
for the meaning and syntax. For IRF matching, the data is specified in thematched_irfs
block.
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
pruning
for 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 forWM
are:
IDENTITY_MATRIX
Sets the weighting matrix equal to the identity matrix.
OPTIMAL
Uses 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.
DIAGONAL
Uses the diagonal of the
OPTIMAL
weighting matrix. This choice puts weights on the specified moments instead of on their linear combinations.
FILENAME
The name of the MAT-file (extension
.mat
) containing a user-specified weighting matrix. The file must include a positive definite square matrix calledweighting_matrix
with 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
METHOD
are:
STOCH_SIMUL
Simulate the model with stochastic simulations and compute IRFs as the difference between the simulated and steady state values. See
stoch_simul
for 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.m
in 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_irf
for more details. Default: false.
- replic = INTEGER
Number of simulated series used to compute the IRFs. Default:
1
iforder=1
, and50
otherwise.
4.16.2.6. General options¶
- dirname = FILENAME
Directory in which to store
estimation
output. Seedirname
for more details. Default:<mod_file>
.
- graph_format = FORMAT
Specify the file format(s) for graphs saved to disk. See
graph_format
for 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_priors
for more details. Default:1
, i.e. plot priors.
- prior_trunc = DOUBLE
See
prior_trunc
for 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
prefilter
for 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:13
for GMM and SMM and5
for 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
. Ifverbose
option is set, then the additional estimation results are saved into theoo_.mom
structure 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:2.38
divided 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:2.38
divided 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_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:0.999999
as 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_moments
stores user options in a structure called options_mom_ in the global workspace. After running the estimation, the parametersM_.params
and the covariance matrices of the shocksM_.Sigma_e
and of the measurement errorsM_.H
are 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 ofdirname
called method_of_moments. Moreover, output is stored in theoo_.mom
structure with the following fields:
Common outputs
- MATLAB/Octave variable: oo_.mom.data_moments¶
Variable set by the
method_of_moments
command. 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_moments
command. 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_moments
command. 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_moments
command. 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_moments
command. 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
verbose
is used.
SMM and GMM specific outputs
- MATLAB/Octave variable: oo_.mom.m_data¶
Variable set by the
method_of_moments
command. 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_moments
command 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_moments
command 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_moments
command. Structure where the value of the test statistic is saved into a field calledj_stat
, the degress of freedom into a field calleddegrees_freedom
and 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_moments
command. 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_moments
command 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:
mean
Prior mean parameter vector.
mode
Prior mode parameter vector.
variance
Covariance matrix of joint prior.
hyperparameters
Vectors of hyperparameters of the prior distributions stored in fields
first
andsecond
.
- MATLAB/Octave variable: oo_.mom.posterior.optimization¶
Variable set by the
method_of_moments
command 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:
mode
Parameter vector at the mode.
Variance
Inverse Hessian matrix at the mode or MCMC jumping covariance matrix when used with the
MCMC_jumping_covariance
option.
log_density
Log 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_moments
command ifmh_replic>0
is used. Fields are of the form:oo_.mom.posterior.metropolis.OBJECTwhere OBJECT is one of the following:
mean
Mean parameter vector from the MCMC.
Variance
Covariance matrix of the parameter draws in the MCMC.
- MATLAB/Octave variable: oo_.mom.prior_density¶
Variable set by the
method_of_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.mom.prior_density.PARAMETER_NAME
- MATLAB/Octave variable: oo_.mom.posterior_density¶
Variable set by the
method_of_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. Fields are of the form:oo_.mom.posterior_density.PARAMETER_NAME
- MATLAB/Octave variable: oo_.mom.posterior_hpdinf¶
Variable set by the
method_of_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command 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_moments
command 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_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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_moments
command. Stores the marginal data density based on the Laplace Approximation.
- MATLAB/Octave variable: oo_.mom.MarginalDensity.ModifiedHarmonicMean¶
Variable set by the
method_of_moments
command, if it is used withmh_replic > 0
orload_mh_file
option. 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:
laplace
for the Laplace estimator ormodifiedharmonicmean
for 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_model
and 30% prior overalt_model
.
- MATLAB/Octave variable: oo_.Model_Comparison¶
Variable set by the
model_comparison
command. 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_Density
Logarithm of the marginal data density.
Bayes_Ratio
Ratio of the marginal data density of the model relative to the one of the first declared model
Posterior_Model_Probability
Posterior 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_names
provided 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:
calibration
prior_mode
prior_mean
posterior_mode
posterior_mean
posterior_median
mle_mode
Note that the parameter set used in subsequent commands like
stoch_simul
will be set to the specifiedparameter_set
. Default value:posterior_mean
if Metropolis has been run,mle_mode
if 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_groups
block. If no group name is given,default
is assumed.
- colormap = VARIABLE_NAME¶
Controls the
colormap
used 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/Octavecolormap
function (see the MATLAB/Octave manual for the list of acceptable values).
- nograph
See
nograph
. Suppresses the display and creation only within theshock_decomposition
command, but does not affect other commands. Seeplot_shock_decomposition
for 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
epilogue
block (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_nbr
endogenous variables. The second dimension stores in the firstM_.exo_nbr
columns the contribution of the respective shocks. ColumnM_.exo_nbr+1
stores the contribution of the initial conditions, while columnM_.exo_nbr+2
stores 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_names
andM_.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_groups
andend
.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_groups
blocks in a model file, each grouping being identified by a different name. This name must in turn be used in theshock_decomposition
command. If no name is given,default
is 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
vintage
is 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>0
and smaller thannobs
, the decomposition is conducted of the forecast revision \(Y(T+k\vert T+k)-Y(T+k\vert T)\).
Like
shock_decomposition
it decomposes the historical deviations of the endogenous variables from their respective steady state values into the contribution coming from the various shocks. Thevariable_names
provided 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_set
for 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_decomposition
andoo_.realtime_forecast_shock_decomposition
but 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:
pool
Stores 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_realtime
is 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:
pool
Stores 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
OBJECT
is one of the following:pool
Stores 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_decomposition
orrealtime_shock_decomposition
. For that reason, it must come after one of these commands. Thevariable_names
provided 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:
qoq
for quarter-on-quarter plots,yoy
for year-on-year plots of growth rates,aoa
for 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 (qoq
for 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=0
then the pooled objects fromrealtime_shock_decomposition
will 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_decomposition
selected via therealtime
option. If the standard historical shock decomposition is selected (realtime=0
),vintage
will have no effect. Ifvintage=0
the pooled objects fromrealtime_shock_decomposition
will 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_date
as initial period. Default: first observation in estimation
- plot_end_date = DATE¶
If passed, plots decomposition using
plot_end_date
as 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
, thediff
operator 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
, thediff
operator 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
init2shocks
block, in order to attribute initial conditions to shocks. The name of the block can be explicitly given, otherwise it defaults to thedefault
block.
- 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_decomposition
command when given theinit2shocks
option.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
dq
andA
will be respectively attributed toe_q
ande_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_names
provided 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:
qoq
for quarter-on-quarter plots,yoy
for year-on-year plots of growth rates,aoa
for 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 (qoq
for 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_date
as initial period. Default: first observation in estimation
- plot_end_date = DATE
If passed, plots decomposition using
plot_end_date
as 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
, thediff
operator 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
, thediff
operator 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_decomposition
will 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_.SmoothedVariables
andoo_.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_set
for 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.
forecast
must be called afterstoch_simul
.forecast
plots 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_sig
to 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
forecast
computes the forecast taking as initial values the values specified inhistval
(seehistval
). When nohistval
block is present, the initial values are the one stated ininitval
. Wheninitval
is 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
forecast
command, or by theestimation
command if used with theforecast
option 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_MOMENT
is one of the following:HPDinf
Lower 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.
HPDsup
Upper 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_ME
Lower 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_ME
Upper 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.
Mean
Mean of the posterior distribution of forecasts.
- MATLAB/Octave variable: oo_.PointForecast¶
Set by the
estimation
command, if it is used with theforecast
option and if eithermh_replic > 0
or theload_mh_file
option 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
estimation
command, if it is used with theforecast
option and if eithermh_replic > 0
orload_mh_file
option 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 \(\varepsilon_{c,t}\):
\[\varepsilon_{c,t} = R_{c,c}^{-1}\bigl( y_{c,t} - T_{c,c}y_{c,t} - T_{c,u}y_{u,t} - R_{c,u}\varepsilon_{u,t}\bigr)\]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
replic
described below) by drawing different sequences of free innovations. The result is a predictive distribution for the uncontrolled e