DYNARE MANUAL Version 3.0 DYNARE MichelJuillard CEPREMAP and University Paris 8 michel.juillard@cepremap.cnrs.fr
142 rue du Chevaleret 75013ParisFrance
1996, 2005 Michel Juillard Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions. Dynare var varexo varexo_det parameters model initval endval histval shocks periods simul check stoch_simul estimated_params estimated_params_init estimated_params_bounds varobs observation_trends estimation rplot dynasave dynatype unit_root_vars olr olr_inst optim_weights osr osr_params @define forecast @if...@elseif...@else...@end Preface Dynare is a pre-processor and a collection of Matlab, Scilab or Gauss routines which solve, simulate and estimate non-linear models with forward looking variables. It is the result of research carried at CEPREMAP by several people (see Laffargue, 1990, Boucekkine, 1995, and Juillard, 1996, Collard and Juillard 2001a and 2001b). 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, Dynare uses a Newton-type algorithm, first proposed by Laffargue (1990), instead of a first order technique like the one proposed by Fair and Taylor (1983), and used in earlier generation simulation programs. We believe this approach to be in general both faster and more robust. The details of the algorithm used in Dynare can be found in Juillard (1996). In a stochastic context, Dynare computes one or several simulations corresponding to a random draw of the shocks. Starting with version 2.3 (not available for Gauss), Dynare uses a second order Taylor approximation of the expectation functions (see Judd, 1996, Collard and Juillard, 2001a, 2001b, and Schmitt-Grohe and Uribe, 2002). Starting with version 3.0, it is possible to use Dynare to estimate model parameters either by maximum likelihood as in Ireland (2004) or using a Bayesian approach as in Rabanal and Rubio-Ramirez (2002), Schorfheide (2000) or Smets and Wouters (2002). Currently the development team of Dynare is composed of S. Adjemian, M. Juillard and O. Kamenik. Several parts of Dynare use or have strongly benefited from publicly available programs by F. Collard, L. Ingber, P. Klein, S. Sakata, F. Schorfheide, C. Sims, P. Soederlind and R. Wouters.
Changes December 30, 2005 added details about parameter transformation in and in added conditional compilation commands and enhanced output section of command added exogenous deterministic shocks in stochastic models. See , , , added a forecast command for calibrated models. See . October 14, 2005 added syntax for computing optimal policy. See , , , , . added syntax for estimating correlation between two shocks or two measurment errors in , and July 20, 2005 Expanded description of statement changed the default for nonlinear solver in added a mention of the possibility to write explicitly a steady state function in , , and added a brief Ouput section in corrected misleading description of option prefilter in added variance decomposition among the statistics computed with option moments_varendo in tex option in isn't yet implemented May 3, 2005 added option noprint in modified option irf in modified option simul_seed in March 6, 2005 corrected typos in equations for 1st and 2nd order approximation formulas in . temporarily removed description of output variables in as old content was outdated and the new one isn't ready yet. added cross-references
Introduction In order to give instructions to Dynare, the user has to write a model file whose file name must terminate by ".mod". This file contains the description of the model and the computing tasks required by the user. In practice, the handling of your model file is done in two steps: in the first one, the model and the processing instructions written by the user in a model file are interpreted and the proper Gauss, Matlab or Scilab instructions are generated; in the second step, the program actually runs the computations. Both steps are triggered by a single keyword: Dynare. Software requirements This version of Dynare works only under Windows 98/NT/2000/XP. For a Unix version, please, write me. The Matlab version has been written with Matlab 6.5.1. The Scilab version has been tested with Scilab 3.0. The Gauss version of Dynare has been written with Gauss version 3.2. It most likely doesn't work with previous versions. Installation In case of update from a previous version, it is a good idea to copy the old version in a backup directory so as to be able to revert to it in case of problems. None of the previous files are usefull anymore, so you are strongly encouraged to remove them from directory c:\dynare. The Matlab version of Dynare lets you now easily have different versions of Dynare on your computer. After installation, Dynare can be used in any directory on your computer. It is best practive to keep your model files in directories different from the one containing the Dynare toolbox. That way you can upgrade Dynare and discard the previous version without having to worry about your own files. Installing the Matlab version Starting with version 3.0, by default, Dynare is installed in a directory whose name contains the version number. For example Dynare_v3.0 This directory contains several sub-directories, among which matlab, doc and examples. After unpacking the archive, start the Matlab program and use the menu File/Set path to add the path to Dynare matlab subdirectory. For example c:\dynare_v3.0\matlab Installing the Scilab version Unpack the zip file in the directory c:\ (If you want to use another directory, see below). The Scilab version in automatically installed in c:\dynare\scilab. Then, find the scilab.star file, in the top directory of your Scilab distribution. Edit this file and add the following line after similar statements: load('c:/dynare/scilab/lib'); If you installed Dynare for Scilab in a directory different from c:\dynare\scilab, change the above instructions accordingly and edit the following line in Dynare.sci command = 'c:\dynare\scilab\dynare_s '+fname; Then, restart Scilab and run the command uplib(). Installing the Gauss version Unpack the zip file in the directory c:\ (If you want to use another directory, see below). The Gauss version in automatically installed in c:\dynare\gauss. If you had any previous version of Dynare, use the Gauss editor or any text editor to remove all references to it from the library file user.lcg. After unpacking the archive, start the Gauss program and type the following: library pgraph lib user c:\dynare\gauss\dynare.src lib user c:\dynare\gauss\dynare1.src lib user c:\dynare\gauss\dynare2.src lib user c:\dynare\gauss\dynare3.src If you installed Dynare for Gauss in a directory different from c:\dynare\gauss, change the above instructions accordingly and edit the following line in Dynare.src declare string PARSER = "c:\\dynare\\gauss\\dynare_g "; Commands Dynare commands are either single instructions or a block of instructions. Each single instructions or block elements are terminated by ;. Block of instructions are terminated by end;. Most Dynare commands have arguments and several accept options, indicated in parentheses after the command keyword. In the description of Dynare commands, the following conventions are observed: optional arguments or options are indicated between square brackets [] repreated arguments are indicated by ellipses ... INTEGER indicates an integer number DOUBLE indicates a double precision number. The following syntaxes are valid: 1.1e3, 1.1E3, 1.1d3, 1.1D3. EXPRESSION indicates a mathematical expression valid in the underlying language (Matlab, Scilab or Gauss) VARIABLE_NAME indicates a variable name starting with an alphabetical character and can't contain ()+-*/^=!;:@#. or accentuated characters PARAMETER_NAME indicates a parameter name starting with an alphabetical charcater and can't contain ()+-*/^=!;:@#. or accentuated characters FILENAME indicates a file name valid under your operating system (Windows, Linux or Unix) Executing Dynare dynare dynare executes Dynare dynare FILENAME[.mod] Description dynare executes instruction included in filename.mod. filename.mod is the name of the model file containing the model and the processing instructions. Details In Matlab, dynare creates three intermediary files: filename.m with the instructions for the simulations filename_ff.m with the dynamic model equations filename_fff.m with the long run static model equations In Scilab, dynare creates three intermediary files: filename.sci with the instructions for the simulations filename_ff.sci with the dynamic model equations filename_fff.sci with the long run static model equations In Gauss, dynare creates an intermediary file filename.gau with the instructions for the simulations. The Gauss version still accepts the former .mdl extension, but it is now deprecated. These files may be looked at to understand errors reported at the simulation stage. Output Depending on the computing tasks requested in the *.mod file, executing command dynare will leave in the workspace variables containing results available for further processing. More details are given under the relevant computing tasks. Under Matlab, some results are also saved in a file called FILENAME_results.mat. Currently, this file contains when available the structures dr_ and oo_. Examples dynare ramst or dynare ramst.mod General declarations General declarations of variables and parameters are made with the following commands: (deprecated) periods periods specifies the number of simulation periods periods INTEGER; Description This command is now deprecated (but will still work for older model files). It is not necessary when no simulation is performed and is replaced by an option PERIODS in SIMUL and STOCH_SIMUL. Set the number of periods in the simulation. The periods are numbered from 1 to INTEGER. In perfect foresight simulations, it is assumed that all future events are perfectly known at the beginning of period 1. Example periods 100; var var declares endogenous variables var VARIABLE_NAME , VARIABLE_NAME ; Description This required command declares the endogenous variables in the model. The variable names must start with a letter and can't contain the following characters : ()+-*/^=!;:@#. or accentuated characters. In Gauss, setting _longname = 1 allows the use of more than 8 characters in the variable names and makes a distinction between lower and upper case letters. Example var c gnp q1 q2; varexo varexo declares exogenous variables varexo VARIABLE_NAME , VARIABLE_NAME ; Description This optional command declares the exogenous variables in the model. See command for the syntax of VARIABLE_NAME. Exogenous variables are required if the user wants to be able to apply shocks to her model. Example varexo m gov; varexo_det varexo_det declares exogenous deterministic variables in a stochastic model varexo_det VARIABLE_NAME , VARIABLE_NAME ; Description This optional command declares exogenous deterministic variables in a stochastic model. See command for the syntax of VARIABLE_NAME. 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 will compute the rational expectation solution adding future information to the state space (nothing is shown in the output of ) and will compute a simulation conditional on initial conditions and future information. Example varexo m gov; varexo_det tau; parameters parameters declares parameters parameters PARAMETER_NAME , PARAMETER_NAME ; Description This optional command declares parameters used in the model, in variable initialization or in shock declarations. The parameters must then be assigned values using standard syntax of underlying matrix programming language. Be carefull not to use names reserved by Dynare or the underlying language (Matlab, Scilab or Gauss). Example parameters alpha, bet; alpha = 0.3; bet = sqrt(2); Model declaration The model is declared inside a block. model model declares the model equations model (linear) ; MATLAB EXPRESSION; MATLAB EXPRESSION; end; EQUATION; EQUATION; end; #EXPRESSION = #EXPRESSION ; EXPRESSION = EXPRESSION ; Description The equations of the model are written in a block delimited by model; and end;. There must be as many equations as there are endogenous variables in the model, except when used to compute the unconstrained optimal policy with olr. The lead and lag of the variables are written in parenthesis immediately after the variable name. Leads or lags of more than one period are allowed. All the functions available in Matlab, Scilab or Gauss, respectively, are recognized. Each equation must be terminated by a semicolon (;). 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. It is possible to include arbitrary Matlab expressions in a model. It must be preceeded by a pound sign (#) as the first character of the line. This is particularily usefull to declare tansformation of parameters for estimation purpose (see ). The option linear declares the model as being linear. It avoids to have to declare initial values for computing the steady state and it sets automatically order=1 in stoch_simul. Example 1 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 2 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 3 model(linear); # b = 1/c; x = a*x(-1)+b*y(+1)+e_x; y = d*y(-1)+e_y; end; Initial and terminal conditions In many contexts, it is necessary to compute the steady state of a non-linear model specifies then numerical initial values for the non-linear solver. Used in perfect foresight mode, the types of forward-loking 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, trigger a shock in first period, and study the trajectory of return at the initial equilbrium. To do that, one needs and (see next section). Another one is to study, how an economy, starting from arbitrary initial conditions converges toward equilibrium. To do that, one needs and ; For models with lags on more than one period, the command permits to specify different historical initial values in different periods. initval initval specifies numerical starting values for finding the steady state and/or initial values for simulations initval; VARIABLE_NAME = EXPRESSION; VARIABLE_NAME = EXPRESSION; end; Description EXPRESSION is any valid expression returning a numerical value and can contain already initialized variable names. The initval; ... end; block serves two purposes. It set the initial and, possibly, terminal conditions for the simulation and provides numerical initialization for various computation tasks (, , ). Theoreticaly, initial conditions are only necessary for lagged variables. However, as initval provides also numerical initialization, it is necessary to provide values for all variables in the model, except if the model is declared as linear. For stochastic models, it isn't necessary to delcare 0 as initial values for exogneous stochastic variables as it is the only possible value. When the initval block is followed by the command , it is not necessary to provide exact initialization values for the endogenous variables. will use the values provided in the initval block as initial guess in the non-linear equation solver and computes exact values for the endogenous variables at the steady state. The steady state is defined by keeping constant the value of the exogenous variables. Example initval; c = 1.2; k = 12; x = 1; end; steady; endval endval specifies terminal values for deterministic simulations endval; VARIABLE_NAME = EXPRESSION; VARIABLE_NAME = EXPRESSION; end; Description EXPRESSION is any valid expression returning a numerical value and can contain already initialized variable names. The optional endval; ... end; block serves two purposes. It set the terminal conditions for the simulation with the LBJ alogrithm, when those differ from the initial conditions. When it is the case, the endval block also provides the numerical initialization for various computation tasks (, ), starting in period 1. Theoreticaly, terminal conditions are required in the LBJ algorithm only for forward variables. However, as endval provides also numerical initialization, it is necessary to provide values for all variables in the model. When the endval block is followed by the command , it is not necessary to provide exact values for the endogenous variables. will use the values provided in the endval block as initial guess in the non-linear equation solver and computes exact values for the endogenous variables at the steady state. The steady state is defined by keeping constant the value of the exogenous variables. Example var c k; varexo x; ... initval; c = 1.2; k = 12; x = 1; end; steady; endval; c = 2; k = 20; x = 2; end; steady; The initial equilibrium is comptuted by for x=1, and the terminal one, for x=2. histval histval specifies historical values before the start of a simulation histval; VARIABLE_NAME (INTEGER) = EXPRESSION; VARIABLE_NAME (INTEGER) = EXPRESSION; end; Description EXPRESSION is any valid expression returning a numerical value and can contain already initialized variable names. In models with lags on more than one period, the optional histval; ... end; block permits to specify different historical initial values for different periods. 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. If your lagged variables are linked by identities, be careful to satisfy these identities when you set historical initial values. 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; 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 and . 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 . 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 command. Or, the entire matrix can be direclty entered with . Note that, starting with version 2.5.2, the direct specification of the internal matrix Sigma_e_, prone to errors, is discouraged. 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. shocks shocks specifies shocks on deterministic or stochastic exogenous variables shocks (OPTION,); DETERMINISTIC SHOCK STATEMENT STOCHASTIC SHOCK STATEMENT DETERMINISTIC SHOCK STATEMENT STOCHASTIC SHOCK STATEMENT end; var VARIABLE_NAME; periods PERIOD STATEMENT; values EXPRESSION; INTEGER : INTEGER INTEGER : INTEGER ; VARIANCE STATEMENT COVARIANCE STATEMENT STANDARD ERROR STATEMENT var VARIABLE_NAME = EXPRESSION; var VARIABLE_NAME , VARIABLE_NAME = EXPRESSION; var VARIABLE_NAME; stderr EXPRESSION; Options shocks_file = FILENAME: reads sequence of deterministic shocks from FILENAME. It can be either a *.m or a *.mat file. The file must create vectors with the same names as the deterministic exogenous variables. Description In deterministic context For deterministic simulations, the shocks block specifies temporary changes in the value of an exogenous variables. For permanent shocks, use an block. When specifying shocks on several periods, the values EXPRESSION must return either a scalar value common to all periods with a shock or a column vector with as many elements as there are periods in the periods statement just before it. Example shocks; var e; periods 1; values 0.5; var u; periods 4:5; values 0; var v; periods 4 5 6; values 0; var u; periods 4 5 6; values 1 1.1 0.9; end; In stochastic context For stochastic simulations (available only in the Matlab or Scilab versions), the shocks block specifies the non zero elements of the covariance matrix of the shocks. Example shocks; var e = 0.000081; var e,u = phi*0.009*0.009; var u = 0.000081; var v; stderr 0.009; end; See also Sigma_e Mixing determininistic 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 will compute the rational expectation solution adding future information to the state space (nothing is shown in the output of ) and 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; Sigma_e Sigma_e specifies directly the covariance matrix of the stochastic shocks Sigma_e = [MATRIX ELEMENT ,MATRIX ELEMENT ;MATRIX ELEMENT]; INTEGER DOUBLE (EXPRESSION) WARNING: the matrix elements are actually written beween square brackets ([]). Here, the initial [ and final ] don't have the meaning of "optional element" as elsewhere. Description The matrix of variance-covariance of the shocks can be directly specified as a upper (or lower) triangular matrix. Dynare builds the corresponding symmetrix matrix. Each row of the triangular matrix, except the last one, must be terminated by a semi-colon ';'. For a given element, an EXPRESSION using predefined parameters is allowed but must be placed between parentheses. THE ORDER OF THE COVARIANCES IN THE MATRIX IS THE SAME AS THE ONE USED IN THE VAREXO DECLARATION. In previous versions, it was possible to directly set Dynare's internal covariance matrix Sigma_e_. This is still possible for compatibility with older .mod files, but STRONGLY DISCOURAGED as too prone to error. When setting Sigma_e_ directly, the order of the exogenous shocks is the ALPHABETICAL order of their names. Example varexo u, e; ... Sigma_e = [ 0.81 (phi*0.9*0.009); 0.000081]; where the variance of u is 0.81, the variance of e, 0.000081, and the correlation between e and u is phi. Solving and simulating Dynare has special commands for the computation of the static equilibrium of the model (, of the eigenvalues of the linearized model () for dynamics local analysis, of a deterministic simulation () and for solving and/or simulating a stochastic model (). steady steady copmutes the steady state of a model steady (solve_algo = 0 1 2 ) ; Options solve_algo = 0: uses Matlab Optimization Toolbox FSOLVE solve_algo = 1: uses Dynare's own nonlinear equation solver solve_algo = 2: splits the model into recursive blocks and solves each block in turn. (Thanks to Manfred Gilli for showing me Matlab's function DMPERM) (this is the default since Dynare version 3.046). Description Computes the equilibrium value of the endogenous variables for the value of the exogenous variables specified in the previous or block. steady uses an iterative procedure and takes as initial guess the value of the endogenous variables set in the previous or 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. If you know how to compute the steady state for your model, you can provide a Matlab function doing the computation instead of using steady. The function should be called with the name of the .mod file followed by _steadystate. See fs2000a_steadystate.m in examples/fs2000 directory. Output variables The steeady state is available in ys_. Endogenous variables are ordered alphabeticaly as in lgy_. Examples See and . check check computes the eigenvalues of the (linearized) model check ; Description Computes the eigenvalues of the model linearized around the values specified by the last , or statement. Generally, the eigenvalues are only meaningfull 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. Output variables check returns the eigenvalues in the global variable eigenvalues_. forecast forecast computes a simulation of a stochastic model from a given state forecast (OPTION,) VARIABLE_NAME VARIABLE_NAME ; Options periods = INTEGER: number of periods of the forecast (default = 40) conf_sig = DOUBLE: level of significance for confidence interval (default = 0.90) Description forecast computes a simulation of a stochastic model from an arbitrary initial point. When the model also contains deterministic exogenous shocks, the simulation is computed conditionaly to the agents knowing the future values of the deterministic exogenous variables. forecast must be called after . forecast plots the trajectory of endogenous variables. When a list of variable names follows the command, only those variables are ploted. A 90% confidence interval is ploted around the mean trajectory. Use option conf_sig to change the level of the confidence interval. Output variables The following variables are set in structure oo_: oo_.forecast.Mean.VARIABLE NAME: mean forecast of endogenous variables oo_.forecast.HPDinf.VARIABLE NAME: lower bound of a confidence interval around the forecast oo_.forecast.HPDsup.VARIABLE NAME: upper bound of a confidence interval around the forecast oo_.forecast.Exogenous.VARIABLE NAME: trajectory of the deterministic exogenous variables 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; simul simul simulates a deterministic model simul (periods=INTEGER) ; Description Triggers the computation of a deterministic simulation of the model for the number of periods set in the option periods=. simul uses a Newton method to solve simultaneously all the equations for every period (see Juillard, 1996). Output variables the simulated variables are available in global matrix y_. The variables are arranged row by row, in alphabetical order. stoch_simul stoch_simul computes the solution and simulates the model stoch_simul (OPTION,) VARIABLE_NAME VARIABLE_NAME ; Options ar = INTEGER: Order of autocorrelation coefficients to compute and to print (default = 5) n dr_algo = 0 | 1: specifies the algorithm used for computing the quadratic approximation of the decision rules: 0: uses a pure perturbation approach as in Schmitt-Grohe and Uribe (2002) (default) 1: moves the point around which the Taylor expansion is computed toward the means of the distribution as in Collard and Juillard (2001) 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 lambda = INTEGER before computing moments (default: no filter) hp_ngrid = INTEGER: number of points in the grid for the discreet 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 (default = 40). Setting IRF=0, suppresses the plotting of IRF's. 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: doesn't print the correlation matrix (printing them is the default) nofunctions: doesn't print the coefficients of the approximated solution (printing them is the default) nomoments: doesn't print moments of the endogenous variables (printing them is the default) noprint: cancel any printing. Usefull for loops. order = 1 | 2 : order of Taylor approximation (default = 2) periods = INTEGER: specifies the number of periods to use in simulations. At order=1, no simulation is necessary to compute theoretical moments and IRFs. A number of periods larger than one triggers automatically option simul (default = 0). qz_criterium = INTEGER | 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: computes a stochastic simulation of the model for the number of periods specified in the periods statement. Uses values, possibly recomputed by , as initial values for the simulation. The simulated endogenous variables are made available to the user in a vector for each variable and in the global matrix y_. The variables are ordered alphabeticaly in the y_ matrix (default: no simulation) simul_seed = INTEGER|DOUBLE|(EXPRESSION): 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). Note that if you use an EXPRESSION rather than an INTEGER or a DOUBLE, the EXPRESSION must be in parenthesis. all steady options (see ) When a list of VARIABLE_NAMEs is specified, results are displayed only for these variables. 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 command. The Taylor approximation is computed around the steady state (except whith option dr_algo=1). If you know how to compute the steady state for your model, you can provide a Matlab 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. Variance decomposition, correlation, autocorrelation are only displayed for variables with positive variance. Impulse response functions are only ploted for variables with response larger than 1e-10. Currently, the IRF's are only ploted 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 command or with the command. 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. First order approximation
yt = ys + A yht-1 + B ut where ys is the steady state value of y and yht=yt-ys.
Second order approximation
yt = ys + 0.5Δ2 + A yht-1 + B ut + 0.5C(yht-1⊗yht-1) + 0.5D(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.
Output variables stoch_simul sets several 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 coefficients of the decision rules are stored in global structuredr_. Here is the correspondance with the symbols used in the above description of the decision rules: Decision rule coefficients ys: dr_.ys. The vector rows correspond to variables in alphabetical order of the variable names. Δ2: dr_.ghs2. The vector rows correspond to re-ordered variables (see below). A: dr_.ghx. The matrix rows correspond to re-ordered variables. The matrix columns correspond to state variables (see below). B: dr_.ghu. The matrix rows correspond to re-ordered variables (see below). The matrix columns correspond to exogenous variables in alphabetical order. C: dr_.ghxx. The matrix rows correspond to re-ordered variables. The matrix columns correspond to the Kronecker product of the vector of state variables (see below). D: dr_.ghuu. The matrix rows correspond to re-ordered variables (see below). The matrix columns correspond to the Kronecker product of exogenous variables in alphabetical order. E: dr_.ghxu. The matrix rows correspond to re-ordered variables. The matrix columns correspond to the Kronecker product of the vector of state variables (see below) by the vector of exogenous variables in alphabetical order. When reordered, the variables are stored in the following order: static variables, purely predetermined variables (variables that appear only at the current and lagged periods in the model), variables that are both predetermined and forward-looking (variables that appear at the current, future and lagged periods in the model), purely forward-looking variables (variables that appear only at the current and future periods in the model). In each category, the variables are arranged alphabetically. The state variables of the model are purely predetermined variables and variables that are both predetermined and forward-looking. They are ordered in that order. When there are lags on more than one period, the state variables are ordered first according to their lag: first variables from the previous period, then variables from two periods before and so on. Note also that when a variable appears in the model at a lag larger than one period, it is automatically included at all inferior lags. The mean of the endogenous variables is available in the vector oo_.mean. The variables are arranged in alphabetical order. The matrix of variance-covariance of the endogenous variables in the matrix oo_.var. The variables are arranged in alphabetical 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 (default ar=5) specifies the number of autocorrelation matrices available. Simulated variables, when they have been computed, are available in Matlab vectors with the same name as the endogenous variables. Impulse responses, when they have been computed, are available in Matlab vectors witht the following naming convention VARIABLE_NAME_shock name. gnp_ea contains the effect on gnp of a one standard deviation shock on ea. 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. Estimation Provided that you have observations on some endogenous variables, it is possible to use Dynare to estimate some or all parameters. Both maximum likelihood and Bayesian techniques are available. 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. varobs varobs lists the observed variables varobs VARIABLE_NAME VARIABLE_NAME ; Description varobs lists the name of observed endogenous variables for the estimation procedure. These variables must be available in the data file (see ). Example varobs C y rr; observation_trends observation_trends specifies linear trends for observed variables observation_trends; VARIABLE_NAME (EXPRESSION); end; Description observation_trends specifies trends for observed variables as functions of model parameters. In most cases, variables shouldn't be centered when observation_trends is used. Example observation_trends; Y (eta); P (mu/eta); end; estimated_params estimated_params specifies the estimated parameters and their prior Syntax I (maximum likelihood estimation) estimated_params; stderr VARIABLE_NAME corr VARIABLE_NAME_1, VARIABLE_NAME_2 PARAMETER_NAME , INITIAL_VALUE , LOWER_BOUND , UPPER_BOUND ; ... end; Syntax II (Bayesian estimation) estimated_params; stderr VARIABLE_NAME corr VARIABLE_NAME_1, VARIABLE_NAME_2 PARAMETER_NAME , PRIOR_SHAPE , PRIOR_MEAN , PRIOR_STANDARD_ERROR , PRIOR_3RD_PARAMETER , PRIOR_4TH_PARAMETER , SCALE_PARAMETER ; ... end; Description The estimated_params;....end; block lists all parameters to be estimated and specifies bounds and priors as necessary. Estimated parameter specification Each line corresponds to an estimated parameter and follows this syntax: stderr is a keyword indicating that the standard error of the exogenous variable, VARIABLE_NAME, or of the observation error associated with endogenous observed variable, VARIABLE_NAME, is to be estimated corr is a keyword indicating that the correlation between the exogenous variables, VARIABLE_NAME_1 and VARIABLE_NAME_2, or the correlation of the observation errors associated with endogenous observed variables, VARIABLE_NAME_1 and VARIABLE_NAME_2, is to be estimated PARAMETER_NAME is the name of a model parameter to be estimated INITIAL_VALUE specifies a starting value for maximum likelihood estimation LOWER_BOUND specifies a lower bound for the parameter value in maximum likelihood estimation UPPER_BOUND specifies an upper bound for the parameter value in maximum likelihood estimation PRIOR_SHAPE is prior density among beta_pdf, gamma_pdf, normal_pdf, inv_gamma_pdf, inv_gamma1_pdf, inv_gamma2_pdf, uniform_pdf PRIOR_MEAN is the mean of the prior distribution PRIOR_STANDARD_ERROR is the standard error of the prior distribution PRIOR_3RD_PARAMETER is a third parameter of the prior used for generalized beta distribution, generalized gamma and for the uniform distribution (default 0) PRIOR_4TH_PARAMETER is a fourth parameter of the prior used for generalized beta distribution, generalized gamma and for the uniform distribution (default 1) SCALE_PARAMETER is the scale parameter to be used for the jump distribution of the Metropolis-Hasting algorithm At minimum, one must specify the name of the parameter and an initial guess. That will trigger unconstrained maximum likelihood estimation. As one uses options more towards the end of the list, all previous options must be filled: if you want to specify jscale, you must specify prior_p3 and prior_p4. Use default values, if these parameters don't apply. 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 statement and to define the transformation at the top of the section, as a Matlab expression. The first character of the line must be a pound sign (#). Example parameters bet; model; # sig = 1/bet; c = sig*c(+1)*mpk; end; estimated_params; bet,normal_pdf,1,0.05; end; estimated_params_init estimated_params_init specifies initial values for optimization estimated_params_init; stderr VARIABLE_NAME corr VARIABLE_NAME_1, VARIABLE_NAME_2 PARAMETER_NAME , INITIAL_VALUE ; ... end; Description The estimated_params_init;....end; block declares numerical initial values for the optimizer when these ones are different from the prior mean Estimated parameter initial value specification Each line corresponds to an estimated parameter and follows this syntax: stderr is a keyword indicating that the standard error of the exogenous variable, VARIABLE_NAME, or of the observation error associated with endogenous observed variable, VARIABLE_NAME, is to be estimated corr is a keyword indicating that the correlation between the exogenous variables, VARIABLE_NAME_1 and VARIABLE_NAME_2, or the correlation of the observation errors associated with endogenous observed variables, VARIABLE_NAME_1 and VARIABLE_NAME_2, is to be estimated PARAMETER_NAME is the name of a model parameter to be estimated INITIAL_VALUE specifies a starting value for maximum likelihood estimation estimated_params_bounds estimated_params_bounds specifies lower and upper bounds for the estimated parameters estimated_params_bounds; stderr VARIABLE_NAME corr VARIABLE_NAME_1, VARIABLE_NAME_2 PARAMETER_NAME , LOWER_BOUND , UPPER_BOUND ; ... end; Description The estimated_params;....end; block lists all parameter to be estimated and specifies bounds and priors when required. Estimated parameter specification Each line corresponds to an estimated parameter and follows this syntax: stderr is a keyword indicating that the standard error of the exogenous variable, VARIABLE_NAME, or of the observation error associated with endogenous observed variable, VARIABLE_NAME, is to be estimated corr is a keyword indicating that the correlation between the exogenous variables, VARIABLE_NAME_1 and VARIABLE_NAME_2, or the correlation of the observation errors associated with endogenous observed variables, VARIABLE_NAME_1 and VARIABLE_NAME_2, is to be estimated PARAMETER_NAME is the name of a model parameter to be estimated LOWER_BOUND specifies a lower bound for the parameter value in maximum likelihood estimation UPPER_BOUND specifies an upper bound for the parameter value in maximum likelihood estimation estimation estimation computes estimation. estimation (OPTIONS) ; OPTIONS datafile = FILENAME: the datafile (a .m file or a .mat file) nobs = INTEGER: the number of observations to be used (default: all observations in the file) nobs = ([INTEGER_1:INTEGER_2]): runs a recursive estimation and forecast for samples of size ranging of INTEGER_1 to INTEGER_2. Option FORECAST must also be specified. first_obs = INTEGER: the number of the first observation to be used (default = 1) prefilter = 1: the estimation procedure demeans the data (default=0, no prefiltering) presample = INTEGER: the number of observations to be skipped before evaluating the likelihood (default = 0) loglinear: computes a log--linear approximation of the model instead of a linear (default) approximation. The data must correspond to the definition of the variables used in the modelx. nograph: no graphs should be plotted lik_init: INTEGER: type of initialization of Kalman filter. 1 (default): 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. conf_sig = {INTEGER | DOUBLE}: the level for the confidence intervals reported in the results (default = 0.90) mh_replic = INTEGER: number of replication for Metropolis Hasting algorithm. For the time being, mh_replic should be larger than 1200 (default = 20000.) mh_nblocks = INTEGER: number of paralletl chains for Metropolis Hasting algorithm (default = 2). mh_drop = DOUBLE: the fraction of initially generated parameter vectors to be dropped before using posterior simulations (default = 0.5) mh_jscale = DOUBLE: the scale to be used for the jumping distribution in MH algorithm. The default value is rarely satisfactory. This option must be tune to obtain, ideally, an accpetation rate of 25% in the Metropolis-Hastings algorithm (default = 0.2). mh_init_scale=DOUBLE: the scale to be used for drawing the initial value of the Metropolis-Hastings chain (default=2*mh_scale). 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) in a file called MODEL NAME_mode. mode_compute=INTEGER: specifies the optimizer for the mode computation. 0: the mode isn't computed. mode_file must be specified 1: uses Matlab fmincon. 2: uses Lester Ingber's Adaptive Simulated Annealing. 3: uses Matlab fminunc. 4 (default): uses Chris Sim's csminwel. mode_check: when mode_check is set, Dynare plots the posterior density for values around the computed mode for each estimated parameter in turn. This is helpful to diagnose problems with the optimizer. prior_trunc=DOUBLE: probability of extreme values of the prior density that is ignored when computing bounds for the parameters (default=1e-32). load_mh_file: when load_mh_file is declared, Dynare adds to previous Metropolis-Hastings simulations instead of starting from scratch. optim=(fmincon options): can be used to set options for fmincon, the optimizing function of Matlab Optimizaiton toolbox. Use Matlab syntax for these options (default: ('display','iter','LargeScale','off','MaxFunEvals',100000,'TolFun',1e-8,'TolX',1e-6)) nodiagnostic: doesn't compute the convergence diagnostics for Metropolis (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 moments_varendo triggers the computation of the posterior distribution of the theoretical moments of the endogenous variables filtered_vars triggers the computation of the posterior distribution of filtered endogenous variables and shocks smoother triggers the computation of the posterior distribution of smoothered endogenous variables and shocks forecast = INTEGER computes the posterior distribution of a forecast on INTEGER periods after the end of the sample used in estimation tex requests the printing of results and graphs in TeX tables and graphics that can be later directly included in Latex files (not yet implemented) All options for If no mh_jscale parameter is used in estimated_params, the procedure uses mh_jscale for all parameters. If mh_jscale option isn't set, the procedure uses 0.2 for all parameters. Results results from posterior optimization (also for maximum likelihood) marginal log density mean and shortest confidence interval from posterior simulation Metropolis-Hastings convergence graphs that still need to be documented graphs with prior, posterior and mode graphs of smoothed shocks, smoothed observation errors, smoothed and historical variables Output After running estimation, the parameters and the variance matrix of the shocks are set to the mode for maximum likelihood estimation or posterior mode computation without Metropolis iterations. After estimation with Metropolis iterations (option mh_replic > 0 or option load_mh_file set) the parameters and the variance matrix of the shocks are set to the posterior mean. Depending on the options, estimation stores results in the following fields of structure oo_: Content of <varname>oo_</varname> Field 1Field 2Field 3Field 4Field 5Required optionsForecastSee Variable nameMarginalDensityLaplaceApproximationAlways providedModifiedHarmonicMean> 0 or PosteriorFilteredVariablesSee Variable namePosteriorIRFDsgeSee IRF name: name of endogenous variable '_' name of shockPosteriorSmoothedObservationErrorsSee Variable namePosteriorSmoothedShocksSee Variable namePosteriorSmoothedVariablesSee Variable namePosteriorTheoreticalMomentsSee See See Variable nameposterior_densityParameter name> 0 or posterior_hpdinfSee Variable name> 0 or posterior_hpdsupSee Variable name> 0 or posterior_meanSee Variable name> 0 or posterior_modeSee Variable name> 0 or posterior_stdSee Variable name> 0 or
Moments of forecasts Field nameDescriptionHPDinfLower bound of a 90% HPD intervalSee option to change the size of the HPD interval of forecast due to parameter uncertaintyHPDsupLower bound of a 90% HPD interval due to parameter uncertaintyHPDTotalinfLower bound of a 90% HPD interval of forecast due to parameter uncertainty and future shocksHPDTotalsupLower bound of a 90% HPD interval due to parameter uncertainty and future shocksMeanMean of the posterior distribution of forecastsMedianMedian of the posterior distribution of forecastsStd Standard deviation of the posterior distribution of forecasts
 Moments Names Field nameDescriptionHPDinfLower bound of a 90% HPD intervalSee option to change the size of the HPD intervalHPDsupUpper bound of a 90% HPD interval MeanMean of the posterior distributionMedianMedian of the posterior distributionStd Standard deviation of the posterior distribution
 Theoretical Moments Field nameDescriptionAutocorrelationAutocorrelation of endogenous variablesThe autocorrlation coefficients are computed for the number of periods specified in option .CorrelationCorrelation between two endogenous variablesDecompDecomposition of varianceWhen the shocks are correlated, it is the decomposition of orthogonalized shocks via Cholesky decompostion according to the order of declaration of shocks (see ). ExpectationExpectation of endogenous variablesVariance(co-)variance of endogenous variables
 Estimated objects Field nameDescriptionmeasurement_errors_corrCorrelation between two measurement errorsmeasurement_errors_stdStandard deviation of measurement errorsparametersParametersshocks_corrCorrelation between two structural shocksshocks_stdStandard deviation of structural shocks
 Examples oo_.posterior_mode.parameters.alp oo_.posterior_mean.shocks_std.ex oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso Note on steady state computation If you know how to compute the steady state for your model, you can provide a Matlab function doing the computation instead of using steady. The function should be called with the name of the .mod file followed by _steadystate. See fs2000a_steadystate.m in examples/fs2000 directory. unit_root_vars unit_root_vars declares unit-root variables for estimation unit_root_vars VARIABLE_NAME VARIABLE_NAME ; Description unit_root_vars is used to declare unit-root variables of a model so that a diffuse prior can be used in the initialization of the Kalman filter for these variables only. For stationary variables, the unconditional covariance matrix of these variables is used for initialization. The algorithm to compute a true diffuse prior is taken from Durbin and Koopman (2001, 2003). When unit_root_vars is used the lik_init option of has no effect. When there are nonstationary variables in a model, there is no unique deterministic steady state. The user must supply a Matlab function that computes the steady state values of the stationary variables in the model and returns dummy values for the nonstationary ones. The function should be called with the name of the .mod file followed by _steadystate. See fs2000a_steadystate.m in examples/fs2000 directory. Note that the nonstationary variables in the model must be integrated processes(their first difference or k-difference must be stationary). Optimal policy Dynare has tools to compute optimal policies for quadratic objectives. You can either solve for optimal policy under commitment with or for optimal simple rule with . olr olr computes optimal policy under commitment olr (OPTION,) VARIABLE_NAME VARIABLE_NAME ; OPTIONS olr_beta=VALUE sets the value of the discount factor for the intertemporal optimization problem All options for Description olr computes optimal policies under commitment (Ramsey plans) for linear--quadratic problems of the form
maxu E0Σt=0∞βt(yt′W11yt+2yt′W12ut+ut′W22ut) s.t. A1Et(yt+1)+A2yt+A3yt-1+But+Cet=0
with y: endogenous variables u: policiy instrument e: exogenous stochastic shocks β: discount factor The policy instruments must be listed with . The quadratic objectives must be listed with . Multipliers are automatically added to the model. Note, however, that the representation isn't minimal and that, in the solution, some multipliers could be sustituted off. Forward-looking endogenous variables don't need to be present in the dynamics of the economy. Dynare automatically builds the corresponding linear rational expectation model and solves it as with . olr_inst olr_inst declares instruments for optimal policy under commitment olr_inst VARIABLE_NAME VARIABLE_NAME ; Description olr_inst declares instruments for optimal policy computed by . optim_weights optim_weights specifies quadratic objectives for optimal policy problems optim_weights; VARIANCE STATEMENT COVARIANCE STATEMENT STANDARD ERROR STATEMENT end; var VARIABLE_NAME = EXPRESSION; var VARIABLE_NAME , VARIABLE_NAME = EXPRESSION; var VARIABLE_NAME; stderr EXPRESSION; Description optim_weights secifies the nonzero elements of the quadratic weight matrices for the objectives in and osr osr computes optimal simple policy rules osr (OPTION,) VARIABLE_NAME VARIABLE_NAME ; OPTIONS All options for Description osr computes optimal simple policy rules for linear--quadratic problems of the form
maxγ E(yt′Wyt) s.t. A1Et(yt+1)+A2yt+A3yt-1+Cet=0
with γ: parameters to be optimized. They must be elements of matrices A1, A2, A3. y: endogenous variables e: exogenous stochastic shocks The parameters to be optimized must be listed with . The quadratic objectives must be listed with . This problem is solved using a numerical optimizer. osr_params osr_params declares the parameters to be optimized for optimal simple rules osr_params PARAMETER_NAME PARAMETER_NAME ; Description osr_params declares parameters to be optimized by . Displaying and saving results Dynare has comments to plot the results of a simulation and to save the results. rplot rplot plot variables rplot VARIABLE_NAME VARIABLE_NAME ; Description Plots one or several variables dynatype dynatype print simulated variables dynatype (FILENAME) VARIABLE_NAME VARIABLE_NAME ; Description dynatype prints the listed variables in a text file named FILENAME. If no VARIABLE_NAME are listed, all endogenous variables are printed. dynasave dynasave save simulated variables in a binary file dynasave (FILENAME) VARIABLE_NAME VARIABLE_NAME ; Description dynasave saves the listed variables in a binary file named FILENAME. If no VARIABLE_NAME are listed, all endogenous variables are saved. In Matlab, variables saved with the dynasave command can be retrieved by the Matlab command load -mat FILENAME. Conditional compilation Dynare has the following commands to choose which part of the *.mod file is executed. This is useful to maintain several versions of a model in the same *.mod file. @define @define defines a macro @define VARIABLE_NAME INTEGER ; Description @define defines a macro with name VARIABLE_NAME and value INTEGER. This macro can be used later in the *.mod file only in @if or @elseif statements. The macros can't be used to replace arbitrary part of codes like in C, for example. Example @define version 1; @if ... @elseif ... @else ... @end @if ... @elseif ... @else ... @end defines conditional compilation of the *.mod file @if VARIABLE_NAME LOGICAL_OPERATOR INTEGER ; ... Description LOGICAL_OPERATOR are == equal != not equal < lesser than > greater than <= lesser or equal than >= greater or equal than These commands let the user define which part of the *.mod file should be handled by Dynare Example @define version 1; parameters alph bet; alph = 0.3; @if version == 1; bet = 0.9; @elseif version == 2; bet = 0.95; @else; bet = 0.98; @end; Examples Fabrice Collard (GREMAQ, University of Toulouse) has written a guide to stochastic simulations with Dynare entitled "Dynare in Practice" which is in guide.pdf. Boucekkine Raouf 1995 An alternative methodology for solving nonlinear forward-looking models Journal of Economic Dynamics and Control 19 711-734 Collard Fabrice Juillard Michel 2001 Accuracy of stochastic perturbation methods: The case of asset pricing models Journal of Economic Dynamics and Control 25 979-999 Collard Fabrice Juillard Michel 2001 A Higher-Order Taylor Expansion Approach to Simulation of Stochastic Forward-Looking Models with an Application to a Non-Linear Phillips Curve Computational Economics 17 125-139 Durbin J. Koopman S.J. 2001 Time Series Analysis by State Space Methods Oxford University Press Fair Ray Taylor John 1983 Solution and Maximum Likelihood Estimation of Dynamic Nonlinear Rational Expectation Models Econometrica 51 1169-1185 Fernandez-Villaverde Jesus Rubio-Ramirez Juan 2004 Comparing Dynamic Equilibrium Economies to Data: A Bayesian Approach Journal of Econometrics 123 153-187 Ireland Peter 2004 A Method for Taking Models to the Data Journal of Economic Dynamics and Control 28 1205-26 Judd Kenneth 1996 Approximation, Perturbation, and Projection Methods in Economic Analysis Amman Hans Kendrick David Rust John Handbook of Computational Economics 1996 North Holland Press 511-585 Juillard Michel 1996 Dynare: A program for the resolution and simulation of dynamic models with forward variables through the use of a relaxation algorithm CEPREMAP Couverture Orange 9602 Koopman S.J. Durbin J. 2003 Filtering and Smoothing of State Vector for Diffuse State Space Models Journal of Time Series Analysis 24 85-98 Laffargue Jean-Pierre Résolution d'un modèle macroéconomique avec anticipations rationnelles 1990 Annales d'Economie et Statistique 17 97-119 Lubik Thomas Schorfheide Frank 2003 Do Central Banks Target Exchange Rates? A Structural Investigation University of Pennsylvania Rabanal Pau Rubio-Ramirez Juan 2003 Comparing New Keynesian Models of the Business Cycle: A Bayesian Approach Atlanta Fed Working Paper 2001-22a, rev 2003 Schorfheide Frank 2000 Loss Function-based evaluation of DSGE models Journal of Applied Econometrics 15 645-70 Schmitt-Grohe Stephanie Uribe Martin 2002 Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function Rutgers University Smets Frank Wouters Rafael 2002 An Estimated Stochastic Dynamic General Equilibrium Model of the Euro Area European Central Bank ECB Working Paper 171