stoch_simul

stoch_simul — computes the solution and simulates the model

Synopsis

stoch_simul [(OPTION [, OPTION...])] [VARIABLE_NAME...] ;

Options

ar = INTEGER

Order of autocorrelation coefficients to compute and to print. Default: 5

drop = INTEGER

Number of points dropped at the beginning of simulation before computing the summary statistics. Default: 100

hp_filter = INTEGER

Uses HP filter with λ = INTEGER before computing moments. Note that this option is currently not available when computing empirical moments (see periods option). 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

irf = INTEGER

Number of periods on which to compute the IRFs. Setting irf=0, suppresses the plotting of IRF's. Default: 40

relative_irf

Requests the computation of normalized IRFs in percentage of the standard error of each shock

linear

Indicates that the original model is linear (put it rather in the model command)

nocorr

Don't print the correlation matrix (printing them is the default)

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

Doesn't do the graphs. Useful for loops

noprint

Don't print anything. Useful for loops

print

Print results (opposite of the above)

order = INTEGER

Order of Taylor approximation. Acceptable values are 1, 2 and 3. Note that for third order, k_order_solver option is implied (in particular, you must specify the use_dll option on the model block and you need a working compilation environment, see Section 1, “Software requirements” for more details), and only empirical moments are available (you must provide a value for periods option). Default: 2

k_order_solver

Use a k-order solver, implemented in C++, instead of the default Dynare solver. When using this option, you must specify the use_dll option on the model block, and you need a working compilation environment, i.e. a working mex command (see Section 1, “Software requirements” for more details). 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. The variables in the oo_.endo_simul matrix, in their order of declaration (as in M_.endo_names) Default: 0

qz_criterium = DOUBLE

Value used to split stable from unstable eigenvalues in reordering the Generalized Schur decomposition used for solving 1st order problems. Default: 1.000001

replic = INTEGER

Number of simulated series used to compute the IRFs. Default: 1 if order=1, and 50 otherwise

simul_seed = INTEGER

Specifies a seed for the random generator so as to obtain the same random sample at each run of the program. Otherwise a different sample is used for each run. Default: seed not specified

simul_algo = INTEGER

Obsolete. Use only the default = 0

solve_algo = INTEGER

See there 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). Conditional variances are given by var(yt+k|t). For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact.

Description

stoch_simul computes a Taylor approximation of the decision and transition functions for the model, 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. 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 the nonlinear solver. The function should be called with the name of the .mod file followed by _steadystate. See fs2000a_steadystate.m in examples/fs2000 directory.

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.

Variance decomposition, correlation, autocorrelation are only displayed for variables with 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 %.

Currently, the IRFs are only plotted for 12 variables. Select the ones you want to see, if your model contains more than 12 endogenous variables.

Currently, the HP filter is only available when computing theoretical moments, not for for moments of simulated variables.

The covariance matrix of the shocks is specified either with the shocks command or with the Sigma_e command.

When a list of VARIABLE_NAME is specified, results are displayed only for these variables.

Auxiliary variables for leads and lags

For a stochastic model, Dynare will perform a transformation of the model so that there is only one lead and one lag on endogenous, and no lead/lag on exogenous.

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_LEAG or AUX_EXO_LAG respectively).

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.

Decision rules

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 oberved at the beginning of the period. The decision rules are stored in the structure oo_.dr which is described below.

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 oo_.dr.npred - oo_.dr.nboth.

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 oo_.dr.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 oo_.dr.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 oo_.dr.nstatic.

Note that all endogenous variables fall into one of these four categories, since after the creation of auxiliary variables, all endogenous have at most one lead and one lag. We therefore have the following identity: oo_.dr.npred + oo_.dr.nfwrd + oo_.dr.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 orderer in DR-order when they appear in decision rules elements. There are oo_.dr.npred such variables.

First order approximation

The approximation has the form:

yt = ys + A yht-1 + B ut

where ys is the steady state value of y and yht=yt-ys.

The coefficients of the decision rules are stored as follows:

  • ys 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.

  • 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.

Second order approximation

The approximation has the form:

yt = ys + 0.5 Δ2 + A yht-1 + B ut + 0.5 C (yht-1⊗yht-1) + 0.5 D (ut⊗ut) + E (yht-1⊗ut)

where ys is the steady state value of y, yht=yt-ys, and Δ2 is the shift effect of the variance of future shocks.

The coefficients of the decision rules are stored in the variables described for first order approximation, plus the following variables:

  • Δ2 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).

Third order approximation

The approximation has the form:

yt = ys + G0 + G1 zt + G2 (zt⊗ zt) + G3 (zt⊗ zt ⊗ zt)

where ys is the steady state value of y, and zt 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 zt is therefore of size nz = oo_.dr.npred + M_.exo_nbr.

The coefficients of the decision rules are stored as follows:

  • ys is stored in oo_.dr.ys. The vector rows correspond to all endogenous in the declaration order.

  • G0 is stored in oo_.dr.g_0. The vector rows correspond to all endogenous in DR-order.

  • G1 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.

  • G2 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 nz(nz+1)/2 columns. More precisely, each column of this matrix corresponds to a pair (i1, i2) where each index represents an element of zt and is therefore between 1 and nz. Only non-decreasing pairs are stored, i.e. those for which i1 ≤ i2. The columns are arranged in the lexicographical order of non-decreasing pairs. Also note that for those pairs where i1 ≠ i2, since the element is stored only once but appears two times in the unfolded G2 matrix, it must be multiplied by 2 when computing the decision rules.

  • G3 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 nz(nz+1)(nz+2)/6 columns. More precisely, each column of this matrix corresponds to a tuple (i1, i2, i3) where each index represents an element of zt and is therefore between 1 and nz. Only non-decreasing tuples are stored, i.e. those for which i1 ≤ i2 ≤ i3. The columns are arranged in the lexicographical order of non-decreasing tuples. Also note that for tuples that have three distinct indices (i.e. i1 ≠ i2 and i1 ≠ i3 and i2 ≠ i3), since these elements are stored only once but appears six times in the unfolded G3 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 G3 matrix, they must be multiplied by 3 when computing the decision rules.

Other output variables

stoch_simul sets other fields in global variable oo_. The descriptive statistics are theoretical moments when no simulation is requested and otherwise represent the moments of the simulated variables.

  • The mean of the endogenous variables is available in the vector oo_.mean. The variables are arranged in declaration order.

  • The matrix of variance-covariance of the endogenous variables in the matrix oo_.var. The variables are arranged in declaration order.

  • The matrix of autocorrelation of the endogenous variables are made available in cell array oo_.autocorr. 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.

  • Simulated variables, when they have been computed, are available in MATLAB®/Octave vectors in the global workspace with the same name as the endogenous variables. They are also available in the oo_.endo_simul matrix. The series are arranged by row, in declaration order of the variable names

  • Impulse responses, when they have been computed, are available in oo_.irfs, with the following naming convention: VARIABLE_NAME_SHOCK_NAME.

    (DEPRECATED) They are currently also available in MATLAB®/Octave vectors in the global workspace, however they will disappear there in a future version.

    Example: oo_.irfs.gnp_ea contains the effect on gnp of a one standard deviation shock on ea.

Examples

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(linear,irf=60) y k;
  

Performs the simulation of a linear model and displays impulse response functions on 60 periods for variables y and k.