Next: Estimation, Previous: Deterministic simulation, Up: The Model file [Contents][Index]
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), Collard
and Juillard (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.
• Computing the stochastic solution: | ||
• Typology and ordering of variables: | ||
• First order approximation: | ||
• Second order approximation: | ||
• Third order approximation: |
Description
stoch_simul
solves a stochastic (i.e. rational
expectations) model, using perturbation techniques.
More precisely, stoch_simul
computes a Taylor approximation of
the decision and transition functions for the 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 the varexo
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 on the DynareWiki.
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 .
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.
The stoch_simul
command with a first order approximation can benefit from the block decomposition of the model (see block).
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 = 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 = DOUBLE described in Stock and Watson (1999) before computing moments. This option is only available with simulated moments. Default: no filter.
hp_ngrid = INTEGER
Number of points in the grid for the discrete Inverse Fast Fourier
Transform used in the HP filter computation. It may be necessary to
increase it for highly autocorrelated processes. Default: 512
.
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 HIGHEST_PERIODICITY
to LOWEST_PERIODICITY
, e.g. to quarters if the model frequency is quarterly.
Default: [6,32]
.
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 in oo_.irfs
to any desired size.
At higher order, the interpretation is different. The relative_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).
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 if graph_format=none
) and displayed to screen
(unless nodisplay
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
and none
(under Octave,
only eps
and none
are available). If the file format is set equal to
none
, the graphs are displayed but not saved to the disk.
noprint
Don’t print anything. Useful for loops.
print
Print results (opposite of noprint
).
order = INTEGER
Order of Taylor approximation. Acceptable values are 1
,
2
and 3
. Note that for third order,
k_order_solver
option is implied and only empirical moments are
available (you must provide a value for periods
option). Default: 2
(except after an estimation
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 otherwise
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
matrix oo_.endo_simul
(see oo_.endo_simul). The simulated
exogenous variables are made available in oo_.exo_simul
(see oo_.exo_simul). Default: 0
.
qz_criterium = DOUBLE
Value used to split stable from unstable eigenvalues in reordering the
Generalized Schur decomposition used for solving 1^st order
problems. Default: 1.000001
(except when estimating with
lik_init
option equal to 1
: the default is
0.999999
in that case; see Estimation).
qz_zero_threshold = DOUBLE
See qz_zero_threshold.
replic = INTEGER
Number of simulated series used to compute the IRFs. Default: 1
if order
=1
, and 50
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
file FILENAME_simul. 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
See below.
conditional_variance_decomposition = [INTEGER1:INTEGER2]
See below.
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
. 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
(see oo_.conditional_variance_decomposition). The variance decomposition is only conducted, if theoretical moments are requested, i.e. using the periods=0
-option. In case of order=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 if nodecomposition
is not set (see oo_.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 its generalization by Andreasen, Fernández-Villaverde and Rubio-Ramírez (2013) is used.
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 if varobs
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 at
http://www.dynare.org/DynareWiki/PartialInformation.
sylvester = OPTION
Determines the algorithm used to solve the Sylvester equation for block decomposed model. Possible values for OPTION
are:
default
Uses the default solver for Sylvester equations (gensylv
) based
on Ondra Kamenik’s algorithm (see
the
Dynare Website for more information).
fixed_point
Uses a fixed point algorithm to solve the Sylvester equation (gensylv_fp
). This method is faster than the default
one for large scale models.
Default value is default
sylvester_fixed_point_tol = DOUBLE
It is the convergence criterion used in the fixed point Sylvester solver. Its default value is 1e-12.
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 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 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 the cycle_reduction
.
Default value is default
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
in oo_.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 the nocorr
-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
Output
This command sets oo_.dr
, oo_.mean
, oo_.var
and
oo_.autocorr
, which are described below.
If option periods
is present, sets oo_.skewness
,
oo_.kurtosis
, and oo_.endo_simul
(see oo_.endo_simul), and also saves the simulated variables in
MATLAB/Octave vectors of the global workspace with the same name as
the endogenous variables.
If option irf
is different from zero, sets oo_.irfs
(see below) and also saves the IRFs in MATLAB/Octave vectors of
the global workspace (this latter way of accessing the IRFs is
deprecated and will disappear in a future version).
If the option contemporaneous_correlation
is different from 0, sets
oo_.contemporaneous_correlation
, which is described below.
Example 1
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 of
0.0348.
Example 2
stoch_simul(irf=60) y k;
Performs the simulation of a model and displays impulse
response functions on 60 periods for variables y
and k
.
After a run of stoch_simul
, contains the mean of the endogenous
variables. Contains theoretical mean if the periods
option is
not present, and simulated mean otherwise. The variables are arranged
in declaration order.
After a run of stoch_simul
, contains the variance-covariance of
the endogenous variables. Contains theoretical variance if the
periods
option is not present (or an approximation thereof for order=2
),
and simulated variance
otherwise. The variables are arranged in declaration order.
After a run of stoch_simul
contains the skewness (standardized third moment)
of the simulated variables if the periods
option is present.
The variables are arranged in declaration order.
After a run of stoch_simul
contains the excess kurtosis (standardized fourth moment)
of the simulated variables if the periods
option is present.
The variables are arranged in declaration order.
After a run of stoch_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 the periods
option is not present (or an approximation thereof for order=2
), and
simulated autocorrelations otherwise. The field is only created if stationary variables are present.
The element oo_.autocorr{i}(k,l)
is equal to the correlation
between and , where
(resp. ) is the -th (resp. -th) endogenous
variable in the declaration order.
Note that if theoretical moments have been requested,
oo_.autocorr{i}
is the same than oo_.gamma_y{i+1}
.
After a run of stoch_simul
, if theoretical moments have been
requested (i.e. if the periods
option is not present), this
variable contains a cell array with the following values (where
ar
is the value of the option of the same name):
oo_.gamma{1}
Variance/co-variance 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{nar+2}
Unconditional variance decomposition see oo_.variance_decomposition
oo_.gamma{nar+3}
If a second order approximation has been requested, contains the vector of the mean correction terms.
In case of order=2
, the theoretical second moments are a second order
accurate approximation of the true second moments, see conditional_variance_decomposition
.
After a run of stoch_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) and
the second dimension corresponds to exogenous variables (in the order of declaration).
Numbers are in percent and sum up to 100 across columns.
After a run of stoch_simul
with the
conditional_variance_decomposition
option, contains a
three-dimensional array with the result of the decomposition. The
first dimension corresponds to forecast horizons (as declared with the
option), the second dimension corresponds to endogenous variables (in
the order of declaration), the third dimension corresponds to
exogenous variables (in the order of declaration).
After a run of stoch_simul
with the
contemporaneous_correlation
option, contains theoretical contemporaneous correlations if the
periods
option is not present (or an approximation thereof for order=2
),
and simulated contemporaneous correlations otherwise. The variables are arranged in declaration order.
After a run of stoch_simul
with option spectral_density
contains the spectral density
of the model variables. There will be a nvars
by nfrequencies
subfield
freqs
storing the respective frequency grid points ranging from 0 to 2*pi and a
same sized subfield density
storing the corresponding density.
After a run of stoch_simul
with option irf
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 on gnp
of a one standard deviation shock on ea
.
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.
Description
extended_path
solves 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
, see oo_.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
in oo_.endo_simul
(see oo_.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. If order
=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. Default: 0
.
hybrid
Use the constant of the second order perturbation reduced form to correct the paths generated by the (stochastic) extended path algorithm.
Next: First order approximation, Previous: Computing the stochastic solution, Up: Stochastic solution and simulation [Contents][Index]
Dynare distinguishes four types of endogenous variables:
Those that appear only at current and past period in the model, but
not at future period (i.e. at and but not
). The number of such variables is equal to
M_.npred
.
Those that appear only at current and future period in the model, but
not at past period (i.e. at and but not
). The number of such variables is stored in
M_.nfwrd
.
Those that appear at current, past and future period in the model
(i.e. at , and ). The number of such
variables is stored in M_.nboth
.
Those that appear only at current, not past and future period in the
model (i.e. only at , not at or
). 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
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.
Variable oo_.dr.order_var
maps DR-order to declaration
order, and variable oo_.dr.inv_order_var
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
.
Next: Second order approximation, Previous: Typology and ordering of variables, Up: Stochastic solution and simulation [Contents][Index]
The approximation has the stylized form:
where is the steady state value of and .
The coefficients of the decision rules are stored as follows:
oo_.dr.ys
. The vector rows
correspond to all endogenous in the declaration order.
oo_.dr.ghx
. The matrix rows correspond to all
endogenous in DR-order. The matrix columns correspond to state
variables in DR-order.
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 and . The precise form of the approximation that shows the way Dynare deals with differences between declaration and DR-order, is
where selects the state variables, and 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 in declaration order.
Next: Third order approximation, Previous: First order approximation, Up: Stochastic solution and simulation [Contents][Index]
The approximation has the form:
where is the steady state value of , , and 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:
oo_.dr.ghs2
. The vector rows
correspond to all endogenous in DR-order.
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.
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.
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).
Previous: Second order approximation, Up: Stochastic solution and simulation [Contents][Index]
The approximation has the form:
where is the steady state value of , and is a
vector consisting of the deviation from the steady state of the state
variables (in DR-order) at date followed by the exogenous variables at
date (in declaration order). The vector is
therefore of size = M_.nspred +
M_.exo_nbr
.
The coefficients of the decision rules are stored as follows:
oo_.dr.ys
. The vector rows
correspond to all endogenous in the declaration order.
oo_.dr.g_0
. The
vector rows correspond to all endogenous in DR-order.
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.
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 columns. More
precisely, each column of this matrix corresponds to a pair
where each index represents an element of and is therefore between
and . Only non-decreasing pairs are stored, i.e. those for
which . The columns are arranged in the lexicographical order
of non-decreasing pairs. Also note that for those pairs where , since the element is stored only once but appears two times in
the unfolded matrix, it must be multiplied by 2 when computing the
decision rules.
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
columns. More precisely, each column of this matrix corresponds to a
tuple
where each index represents an element of
and is therefore between and . Only
non-decreasing tuples are stored, i.e. those for which
. The columns are arranged in the lexicographical
order of non-decreasing tuples. Also note that for tuples that have
three distinct indices (i.e. and and ), since these elements are stored
only once but appears six times in the unfolded 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 or or
), since these elements are stored only once but appears
three times in the unfolded matrix, they must be multiplied
by 3 when computing the decision rules.
Previous: Second order approximation, Up: Stochastic solution and simulation [Contents][Index]