Dynare Working Papers Series http://www.dynare.org/wp/ Dynare: Reference Manual Version 4 Stéphane Adjemian Houtan Bastani Fréderic Karamé Michel Juillard Junior Maih Ferhat Mihoubi George Perendia Marco Ratto Sébastien Villemot Working Paper no. 1 Initial revision: April 2011 This revision: June 2012 142, rue du Chevaleret — 75013 Paris — France http://www.cepremap.ens.fr Dynare Reference Manual, version 4.3.0 Stéphane Adjemian Houtan Bastani Michel Juillard Junior Maih Ferhat Mihoubi George Perendia Marco Ratto Sébastien Villemot c 1996-2012, Dynare Team. Copyright Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license can be found at http://www.gnu.org/licenses/fdl.txt. i Table of Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 2 What is Dynare ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Documentation sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Citing Dynare in your research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Installation and configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 2.2 Software requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Installation of Dynare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 On Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 On Debian GNU/Linux and Ubuntu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 On Mac OS X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 For other systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 For MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 For GNU Octave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Some words of warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 3 4 4 4 4 4 5 3 Dynare invocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 The Model file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.1 4.2 4.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Variable declarations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3.1 Parameters and variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.1.1 Inside the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.1.2 Outside the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3.3.1 Built-in Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3.3.2 External Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 Parameter initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.5 Model declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.6 Auxiliary variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.7 Initial and terminal conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.8 Shocks on exogenous variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.9 Other general declarations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.10 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.10.1 Finding the steady state with Dynare nonlinear solver . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.10.2 Using a steady state file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.11 Getting information about the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.12 Deterministic simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.13 Stochastic solution and simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.13.1 Computing the stochastic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.13.2 Typology and ordering of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.13.3 First order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.13.4 Second order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.13.5 Third order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.14 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ii 4.15 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.16 Optimal policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Sensitivity and identification analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.2 Stability Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.3 Reduced Form Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.4 RMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.5 Screening Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.6 Identification Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17.7 Performing Sensitivity and Identification Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Markov-switching SBVAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Displaying and saving results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Macro-processing language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.1 Macro expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.2 Macro directives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.3 Typical usages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.3.1 Modularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.3.2 Indexed sums or products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.3.3 Multi-country models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.3.4 Endogeneizing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20.4 MATLAB/Octave loops versus macro-processor loops . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 Misc commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 54 58 60 60 60 61 61 63 63 63 68 76 77 77 78 80 80 80 81 81 82 83 The Configuration File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1 5.2 Dynare Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Parallel Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7 Dynare internal documentation and unitary tests . . . . . . . . . . . 89 8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Command and Function Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Variable Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Chapter 1: Introduction 1 1 Introduction 1.1 What is Dynare ? Dynare is a software platform for handling a wide class of economic models, in particular dynamic stochastic general equilibrium (DSGE) and overlapping generations (OLG) models. The models solved by Dynare include those relying on the rational expectations hypothesis, wherein agents form their expectations about the future in a way consistent with the model. But Dynare is also able to handle models where expectations are formed differently: on one extreme, models where agents perfectly anticipate the future; on the other extreme, models where agents have limited rationality or imperfect knowledge of the state of the economy and, hence, form their expectations through a learning process. In terms of types of agents, models solved by Dynare can incorporate consumers, productive firms, governments, monetary authorities, investors and financial intermediaries. Some degree of heterogeneity can be achieved by including several distinct classes of agents in each of the aforementioned agent categories. Dynare offers a user-friendly and intuitive way of describing these models. It is able to perform simulations of the model given a calibration of the model parameters and is also able to estimate these parameters given a dataset. In practice, the user will write a text file containing the list of model variables, the dynamic equations linking these variables together, the computing tasks to be performed and the desired graphical or numerical outputs. A large panel of applied mathematics and computer science techniques are internally employed by Dynare: multivariate nonlinear solving and optimization, matrix factorizations, local functional approximation, Kalman filters and smoothers, MCMC techniques for Bayesian estimation, graph algorithms, optimal control, . . . Various public bodies (central banks, ministries of economy and finance, international organisations) and some private financial institutions use Dynare for performing policy analysis exercises and as a support tool for forecasting exercises. In the academic world, Dynare is used for research and teaching purposes in postgraduate macroeconomics courses. Dynare is a free software, which means that it can be downloaded free of charge, that its source code is freely available, and that it can be used for both non-profit and for-profit purposes. Most of the source files are covered by the GNU General Public Licence (GPL) version 3 or later (there are some exceptions to this, see the file ‘license.txt’ in Dynare distribution). It is available for the Windows, Mac and Linux platforms and is fully documented through a user guide and a reference manual. Part of Dynare is programmed in C++, while the rest is written using the MATLAB programming language. The latter implies that commercially-available MATLAB software is required in order to run Dynare. However, as an alternative to MATLAB, Dynare is also able to run on top of GNU Octave (basically a free clone of MATLAB): this possibility is particularly interesting for students or institutions who cannot afford, or do not want to pay for, MATLAB and are willing to bear the concomitant performance loss. The development of Dynare is mainly done at Cepremap by a core team of researchers who devote part of their time to software development. Currently the development team of Dynare is composed of Stéphane Adjemian (Université du Maine, Gains and Cepremap), Houtan Bastani (Cepremap), Michel Juillard (Banque de France), Frédéric Karamé (Université d’Évry, Epee and Cepremap), Junior Maih (Norges Bank), Ferhat Mihoubi (Université d’Évry, Epee and Cepremap), George Perendia, Johannes Pfeifer, Marco Ratto (JRC) and Sébastien Villemot (Cepremap and Paris School of Economics). Increasingly, the developer base is expanding, as tools developed by researchers outside of Cepremap are integrated into Dynare. Financial support is provided by Cepremap, Banque de France and DSGE-net (an international research network for DSGE modeling). The Dynare project also received funding through the Seventh Framework Programme for Research (FP7) of the European Commission’s Socio-economic Sciences and Humanities (SSH) Program from October 2008 to September 2011 under grant agreement SSH-CT-2009-225149. Chapter 1: Introduction 2 Interaction between developers and users of Dynare is central to the project. A web forum is available for users who have questions about the usage of Dynare or who want to report bugs. Training sessions are given through the Dynare Summer School, which is organized every year and is attended by about 40 people. Finally, priorities in terms of future developments and features to be added are decided in cooperation with the institutions providing financial support. 1.2 Documentation sources The present document is the reference manual for Dynare. It documents all commands and features in a systematic fashion. New users should rather begin with Dynare User Guide (Mancini (2007)), distributed with Dynare and also available from the official Dynare web site. Other useful sources of information include the Dynare wiki and the Dynare forums. 1.3 Citing Dynare in your research If you would like to refer to Dynare in a research article, the recommended way is to cite the present manual, as follows: Stéphane Adjemian, Houtan Bastani, Michel Juillard, Ferhat Mihoubi, George Perendia, Marco Ratto and Sébastien Villemot (2011), “Dynare: Reference Manual, Version 4,” Dynare Working Papers, 1, CEPREMAP Note that citing the Dynare Reference Manual in your research is a good way to help the Dynare project. If you want to give a URL, use the address of the Dynare website: http://www.dynare.org. Chapter 2: Installation and configuration 3 2 Installation and configuration 2.1 Software requirements Packaged versions of Dynare are available for Windows XP/Vista/Seven, Debian GNU/Linux, Ubuntu and Mac OS X Leopard/Snow Leopard. Dynare should work on other systems, but some compilation steps are necessary in that case. In order to run Dynare, you need one of the following: • MATLAB version 7.0 (R14) or above; • GNU Octave version 3.2.2 or above. Some installation instructions for GNU Octave can be found on the Dynare Wiki. The following optional extensions are also useful to benefit from extra features, but are in no way required: • If under MATLAB: the optimization toolbox, the statistics toolbox, the control system toolbox; • If under GNU Octave, the following Octave-Forge packages: optim, io, java, statistics, control. If you plan to use the use_dll option of the model command, you will need to install the necessary requirements for compiling MEX files on your machine. If you are using MATLAB under Windows, install a C++ compiler on your machine and configure it with MATLAB: see instructions on the Dynare wiki. Users of Octave under Linux should install the package for MEX file compilation (under Debian or Ubuntu, it is called ‘liboctave-dev’ or ‘octave3.2-headers’). If you are using Octave or MATLAB under Mac OS X, you should install the latest version of XCode: see instructions on the Dynare wiki. Mac OS X Octave users will also need to install gnuplot if they want graphing capabilities. Users of MATLAB under Linux and Mac OS X, and users of Octave under Windows, normally need to do nothing, since a working compilation environment is available by default. 2.2 Installation of Dynare After installation, Dynare can be used in any directory on your computer. It is best practice 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. 2.2.1 On Windows Execute the automated installer called ‘dynare-4.x.y-win.exe’ (where 4.x.y is the version number), and follow the instructions. The default installation directory is ‘c:\dynare\4.x.y’. After installation, this directory will contain several sub-directories, among which are ‘matlab’, ‘mex’ and ‘doc’. The installer will also add an entry in your Start Menu with a shortcut to the documentation files and uninstaller. Note that you can have several versions of Dynare coexisting (for example in ‘c:\dynare’), as long as you correctly adjust your path settings (see Section 2.3.3 [Some words of warning], page 5). 2.2.2 On Debian GNU/Linux and Ubuntu Please refer to the Dynare Wiki for detailed instructions. Dynare will be installed under ‘/usr/share/dynare’ and ‘/usr/lib/dynare’. Documentation will be under ‘/usr/share/doc/dynare’. Chapter 2: Installation and configuration 4 2.2.3 On Mac OS X Execute the automated installer called ‘dynare-4.x.y-macosx-10.5+10.6.pkg’ (where 4.x.y is the version number), and follow the instructions. The default installation directory is ‘/Applications/Dynare/4.x.y’. Please refer to the Dynare Wiki for detailed instructions. After installation, this directory will contain several sub-directories, among which are ‘matlab’, ‘mex’ and ‘doc’. Note that you can have several versions of Dynare coexisting (for example in ‘/Applications/Dynare’), as long as you correctly adjust your path settings (see Section 2.3.3 [Some words of warning], page 5). 2.2.4 For other systems You need to download Dynare source code from the Dynare website and unpack it somewhere. Then you will need to recompile the pre-processor and the dynamic loadable libraries. Please refer to Dynare Wiki. 2.3 Configuration 2.3.1 For MATLAB You need to add the ‘matlab’ subdirectory of your Dynare installation to MATLAB path. You have two options for doing that: • Using the addpath command in the MATLAB command window: Under Windows, assuming that you have installed Dynare in the standard location, and replacing 4.x.y with the correct version number, type: addpath c:\dynare\4.x.y\matlab Under Debian GNU/Linux or Ubuntu, type: addpath /usr/share/dynare/matlab Under Mac OS X, assuming that you have installed Dynare in the standard location, and replacing 4.x.y with the correct version number, type: addpath /Applications/Dynare/4.x.y/matlab MATLAB will not remember this setting next time you run it, and you will have to do it again. • Via the menu entries: Select the “Set Path” entry in the “File” menu, then click on “Add Folder. . . ”, and select the ‘matlab’ subdirectory of your Dynare installation. Note that you should not use “Add with Subfolders. . . ”. Apply the settings by clicking on “Save”. Note that MATLAB will remember this setting next time you run it. 2.3.2 For GNU Octave You need to add the ‘matlab’ subdirectory of your Dynare installation to Octave path, using the addpath at the Octave command prompt. Under Windows, assuming that you have installed Dynare in the standard location, and replacing “4.x.y” with the correct version number, type: addpath c:\dynare\4.x.y\matlab Under Debian GNU/Linux or Ubuntu, there is no need to use the addpath command; the packaging does it for you. Under Mac OS X, assuming that you have installed Dynare in the standard location, and replacing “4.x.y” with the correct version number, type: Chapter 2: Installation and configuration 5 addpath /Applications/Dynare/4.x.y/matlab If you are using an Octave version strictly older than 3.2.0, you will also want to tell to Octave to accept the short syntax (without parentheses and quotes) for the dynare command, by typing: mark_as_command dynare If you don’t want to type this command every time you run Octave, you can put it in a file called ‘.octaverc’ in your home directory (under Windows this will generally by ‘c:\Documents and Settings\USERNAME\’). This file is run by Octave at every startup. 2.3.3 Some words of warning You should be very careful about the content of your MATLAB or Octave path. You can display its content by simply typing path in the command window. The path should normally contain system directories of MATLAB or Octave, and some subdirectories of your Dynare installation. You have to manually add the ‘matlab’ subdirectory, and Dynare will automatically add a few other subdirectories at runtime (depending on your configuration). You must verify that there is no directory coming from another version of Dynare than the one you are planning to use. You have to be aware that adding other directories to your path can potentially create problems, if some of your M-files have the same names than Dynare files. Your files would then override Dynare files, and make Dynare unusable. Chapter 3: Dynare invocation 6 3 Dynare invocation In order to give instructions to Dynare, the user has to write a model file whose filename extension must be ‘.mod’. This file contains the description of the model and the computing tasks required by the user. Its contents is described in Chapter 4 [The Model file], page 9. Once the model file is written, Dynare is invoked using the dynare command at the MATLAB or Octave prompt (with the filename of the ‘.mod’ given as argument). In practice, the handling of the 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 MATLAB or GNU Octave instructions are generated; in the second step, the program actually runs the computations. Boths steps are triggered automatically by the dynare command. dynare FILENAME[.mod] [OPTIONS . . . ] [MATLAB/Octave command] Description This command launches Dynare and executes the instructions included in ‘FILENAME.mod’. This user-supplied file contains the model and the processing instructions, as described in Chapter 4 [The Model file], page 9. dynare begins by launching the preprocessor on the ‘.mod’ file. By default (unless use_dll option has been given to model), the preprocessor creates three intermediary files: ‘FILENAME.m’ Contains variable declarations, and computing tasks ‘FILENAME_dynamic.m’ Contains the dynamic model equations ‘FILENAME_static.m’ Contains the long run static model equations These files may be looked at to understand errors reported at the simulation stage. dynare will then run the computing tasks by executing ‘FILENAME.m’. Options noclearall By default, dynare will issue a clear all command to MATLAB or Octave, thereby deleting all workspace variables; this options instructs dynare not to clear the workspace debug Instructs the preprocessor to write some debugging information about the scanning and parsing of the ‘.mod’ file notmpterms Instructs the preprocessor to omit temporary terms in the static and dynamic files; this generally decreases performance, but is used for debugging purposes since it makes the static and dynamic files more readable savemacro[=FILENAME] Instructs dynare to save the intermediary file which is obtained after macroprocessing (see Section 4.20 [Macro-processing language], page 77); the saved output will go in the file specified, or if no file is specified in ‘FILENAME-macroexp.mod’ onlymacro Instructs the preprocessor to only perform the macro-processing step, and stop just after. Mainly useful for debugging purposes or for using the macro-processor independently of the rest of Dynare toolbox. Chapter 3: Dynare invocation 7 nolinemacro Instructs the macro-preprocessor to omit line numbering information in the intermediary ‘.mod’ file created after the maco-processing step. Useful in conjunction with savemacro when one wants that to reuse the intermediary ‘.mod’ file, without having it cluttered by line numbering directives. nolog Instructs Dynare to no create a logfile of this run in ‘FILENAME.log’. The default is to create the logfile. warn_uninit Display a warning for each variable or parameter which is not initialized. See Section 4.4 [Parameter initialization], page 15, or [load params and steady state], page 83 for initialization of parameters. See Section 4.7 [Initial and terminal conditions], page 19, or [load params and steady state], page 83 for initialization of endogenous and exogenous variables. console Activate console mode: Dynare will not use graphical waitbars for long computations. Note that this option is only useful under MATLAB, since Octave does not provide graphical waitbar capabilities. cygwin Tells Dynare that your MATLAB is configured for compiling MEX files with Cygwin (see Section 2.1 [Software requirements], page 3). This option is only available under Windows, and is used in conjunction with use_dll. msvc Tells Dynare that your MATLAB is configured for compiling MEX files with Microsoft Visual C++ (see Section 2.1 [Software requirements], page 3). This option is only available under Windows, and is used in conjunction with use_dll. parallel[=CLUSTER_NAME] Tells Dynare to perform computations in parallel. If CLUSTER NAME is passed, Dynare will use the specified cluster to perform parallel computations. Otherwise, Dynare will use the first cluster specified in the configuration file. See Chapter 5 [The Configuration File], page 84, for more information about the configuration file. conffile=FILENAME Specifies the location of the configuration file if it differs from the default. See Chapter 5 [The Configuration File], page 84, for more information about the configuration file and its default location. parallel_slave_open_mode Instructs Dynare to leave the connection to the slave node open after computation is complete, closing this connection only when Dynare finishes processing. parallel_test Tests the parallel setup specified in the configuration file without executing the ‘.mod’ file. See Chapter 5 [The Configuration File], page 84, for more information about the configuration file. -DMACRO_VARIABLE=MACRO_EXPRESSION Defines a macro-variable from the command line (the same effect as using the Macro directive @#define in a model file, see Section 4.20 [Macro-processing language], page 77). 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. The M_, oo_ and options_ structures are also saved in a file called ‘FILENAME_results.mat’. Chapter 3: Dynare invocation 8 Example dynare ramst dynare ramst.mod savemacro The output of Dynare is left into three main variables in the MATLAB/Octave workspace: [MATLAB/Octave variable] M_ Structure containing various informations about the model. [MATLAB/Octave variable] Structure contains the values of the various options used by Dynare during the computation. options_ oo_ Structure containing the various results of the computations. [MATLAB/Octave variable] Chapter 4: The Model file 9 4 The Model file 4.1 Conventions A model file contains a list of commands and of blocks. Each command and each element of a block is terminated by a semicolon (;). Blocks are terminated by end;. Most Dynare commands have arguments and several accept options, indicated in parentheses after the command keyword. Several options are separated by commas. In the description of Dynare commands, the following conventions are observed: • optional arguments or options are indicated between square brackets: ‘[]’; • repreated arguments are indicated by ellipses: “. . . ”; • mutually exclusive arguments are separated by vertical bars: ‘|’; • INTEGER indicates an integer number; • DOUBLE indicates a double precision number. The following syntaxes are valid: 1.1e3, 1.1E3, 1.1d3, 1.1D3; • NUMERICAL VECTOR indicates a vector of numbers separated by spaces, enclosed by square brackets; • EXPRESSION indicates a mathematical expression valid outside the model description (see Section 4.3 [Expressions], page 12); • MODEL EXPRESSION indicates a mathematical expression valid in the model description (see Section 4.3 [Expressions], page 12 and Section 4.5 [Model declaration], page 16); • MACRO EXPRESSION designates an expression of the macro-processor (see Section 4.20.1 [Macro expressions], page 77); • 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 character and can’t contain: ‘()+-*/^=!;:@#.’ or accentuated characters; • LATEX NAME indicates a valid LaTeX expression in math mode (not including the dollar signs); • FUNCTION NAME indicates a valid MATLAB function name; • FILENAME indicates a filename valid in the underlying operating system; it is necessary to put it between quotes when specifying the extension or if the filename contains a non-alphanumeric character; 4.2 Variable declarations Declarations of variables and parameters are made with the following commands: var VARIABLE_NAME [$LATEX_NAME$] . . . ; var (deflator = MODEL_EXPRESSION) VARIABLE_NAME [$LATEX_NAME$] . . . ; [Command] [Command] Description This required command declares the endogenous variables in the model. See Section 4.1 [Conventions], page 9, for the syntax of VARIABLE NAME and MODEL EXPRESSION. Optionally it is possible to give a LaTeX name to the variable or, if it is nonstationary, provide information regarding its deflator. var commands can appear several times in the file and Dynare will concatenate them. Options Chapter 4: The Model file 10 If the model is nonstationary and is to be written as such in the model block, Dynare will need the trend deflator for the appropriate endogenous variables in order to stationarize the model. The trend deflator must be provided alongside the variables that follow this trend. deflator = MODEL_EXPRESSION The expression used to detrend an endogenous variable. All trend variables, endogenous variables and parameters referenced in MODEL EXPRESSION must already have been declared by the trend_var, var and parameters commands. Example var c gnp q1 q2; var(deflator=A) i b; varexo VARIABLE_NAME [$LATEX_NAME$] . . . ; [Command] Description This optional command declares the exogenous variables in the model. See Section 4.1 [Conventions], page 9, for the syntax of VARIABLE NAME. Optionally it is possible to give a LaTeX name to the variable. Exogenous variables are required if the user wants to be able to apply shocks to her model. varexo commands can appear several times in the file and Dynare will concatenate them. Example varexo m gov; varexo_det VARIABLE_NAME [$LATEX_NAME$] . . . ; [Command] Description This optional command declares exogenous deterministic variables in a stochastic model. See Section 4.1 [Conventions], page 9, for the syntax of VARIABLE NAME. Optionally it is possible to give a LaTeX name to the variable. It is possible to mix deterministic and stochastic shocks to build models where agents know from the start of the simulation about future exogenous changes. In that case stoch_simul will compute the rational expectation solution adding future information to the state space (nothing is shown in the output of stoch_simul) and forecast will compute a simulation conditional on initial conditions and future information. varexo_det commands can appear several times in the file and Dynare will concatenate them. Example varexo m gov; varexo_det tau; parameters PARAMETER_NAME [$LATEX_NAME$] . . . ; [Command] Description This command declares parameters used in the model, in variable initialization or in shocks declarations. See Section 4.1 [Conventions], page 9, for the syntax of PARAMETER NAME. Optionally it is possible to give a LaTeX name to the parameter. Chapter 4: The Model file 11 The parameters must subsequently be assigned values (see Section 4.4 [Parameter initialization], page 15). parameters commands can appear several times in the file and Dynare will concatenate them. Example parameters alpha, bet; change_type (var | varexo | varexo det | parameters) VARIABLE_NAME | PARAMETER_NAME . . . ; [Command] Description Changes the types of the specified variables/parameters to another type: endogenous, exogenous, exogenous deterministic or parameter. It is important to understand that this command has a global effect on the ‘.mod’ file: the type change is effective after, but also before, the change_type command. This command is typically used when flipping some variables for steady state calibration: typically a separate model file is used for calibration, which includes the list of variable declarations with the macro-processor, and flips some variable. Example var y, w; parameters alpha, bet; ... change_type(var) alpha, bet; change_type(parameters) y, w; Here, in the whole model file, alpha and beta will be endogenous and y and w will be parameters. predetermined_variables VARIABLE_NAME . . . ; [Command] Description In Dynare, the default convention is that the timing of a variable reflects when this variable is decided. The typical example is for capital stock: since the capital stock used at current period is actually decided at the previous period, then the capital stock entering the production function is k(-1), and the law of motion of capital must be written: k = i + (1-delta)*k(-1) Put another way, for stock variables, the default in Dynare is to use a “stock at the end of the period” concept, instead of a “stock at the beginning of the period” convention. The predetermined_variables is used to change that convention. The endogenous variables declared as predetermined variables are supposed to be decided one period ahead of all other endogenous variables. For stock variables, they are supposed to follow a “stock at the beginning of the period” convention. Example The following two program snippets are strictly equivalent. Using default Dynare timing convention: var y, k, i; ... model; y = k(-1)^alpha; k = i + (1-delta)*k(-1); Chapter 4: The Model file 12 ... end; Using the alternative timing convention: var y, k, i; predetermined_variables k; ... model; y = k^alpha; k(+1) = i + (1-delta)*k; ... end; trend_var (growth factor = MODEL_EXPRESSION) VARIABLE_NAME [$LATEX_NAME$] . . . ; [Command] Description This optional command declares the trend variables in the model. See Section 4.1 [Conventions], page 9, for the syntax of MODEL EXPRESSION and VARIABLE NAME. Optionally it is possible to give a LaTeX name to the variable. Trend variables are required if the user wants to be able to write a nonstationary model in the model block. The trend_var command must appear before the var command that references the trend variable. trend_var commands can appear several times in the file and Dynare will concatenate them. If the model is nonstationary and is to be written as such in the model block, Dynare will need the growth factor of every trend variable in order to stationarize the model. The growth factor must be provided within the declaration of the trend variable, using the growth_factor keyword. All endogenous variables and parameters referenced in MODEL EXPRESSION must already have been declared by the var and parameters commands. Example trend_var (growth_factor=gA) A; 4.3 Expressions Dynare distinguishes between two types of mathematical expressions: those that are used to describe the model, and those that are used outside the model block (e.g. for initializing parameters or variables, or as command options). In this manual, those two types of expressions are respectively denoted by MODEL EXPRESSION and EXPRESSION. Unlike MATLAB or Octave expressions, Dynare expressions are necessarily scalar ones: they cannot contain matrices or evaluate to matrices1 . Expressions can be constructed using integers (INTEGER), floating point numbers (DOUBLE), parameter names (PARAMETER NAME), variable names (VARIABLE NAME), operators and functions. The following special constants are also accepted in some contexts: inf [Constant] Represents infinity. 1 Note that arbitrary MATLAB or Octave expressions can be put in a ‘.mod’ file, but those expressions have to be on separate lines, generally at the end of the file for post-processing purposes. They are not interpreted by Dynare, and are simply passed on unmodified to MATLAB or Octave. Those constructions are not addresses in this section. Chapter 4: The Model file nan 13 [Constant] “Not a number”: represents an undefined or unrepresentable value. 4.3.1 Parameters and variables Parameters and variables can be introduced in expressions by simply typing their names. The semantics of parameters and variables is quite different whether they are used inside or outside the model block. 4.3.1.1 Inside the model Parameters used inside the model refer to the value given through parameter initialization (see Section 4.4 [Parameter initialization], page 15) or homotopy_setup when doing a simulation, or are the estimated variables when doing an estimation. Variables used in a MODEL EXPRESSION denote current period values when neither a lead or a lag is given. A lead or a lag can be given by enclosing an integer between parenthesis just after the variable name: a positive integer means a lead, a negative one means a lag. Leads or lags of more than one period are allowed. For example, if c is an endogenous variable, then c(+1) is the variable one period ahead, and c(-2) is the variable two periods before. When specifying the leads and lags of endogenous variables, it is important to respect the following convention: in Dynare, the timing of a variable reflects when that variable is decided. A control variable — which by definition is decided in the current period — must have no lead. A predetermined variable — which by definition has been decided in a previous period — must have a lag. A consequence of this is that all stock variables must use the “stock at the end of the period” convention. Please refer to Mancini-Griffoli (2007) for more details and concrete examples. Leads and lags are primarily used for endogenous variables, but can be used for exogenous variables. They have no effect on parameters and are forbidden for local model variables (see Section 4.5 [Model declaration], page 16). 4.3.1.2 Outside the model When used in an expression outside the model block, a parameter or a variable simply refers to the last value given to that variable. More precisely, for a parameter it refers to the value given in the corresponding parameter initialization (see Section 4.4 [Parameter initialization], page 15); for an endogenous or exogenous variable, it refers to the value given in the most recent initval or endval block. 4.3.2 Operators The following operators are allowed in both MODEL EXPRESSION and EXPRESSION : • binary arithmetic operators: +, -, *, /, ^ • unary arithmetic operators: +, • binary comparison operators (which evaluate to either 0 or 1): <, >, <=, >=, ==, != The following special operators are accepted in MODEL EXPRESSION (but not in EXPRESSION ): STEADY_STATE (MODEL_EXPRESSION) [Operator] This operator is used to take the value of the enclosed expression at the steady state. A typical usage is in the Taylor rule, where you may want to use the value of GDP at steady state to compute the output gap. EXPECTATION (INTEGER) (MODEL_EXPRESSION) [Operator] This operator is used to take the expectation of some expression using a different information set than the information available at current period. For example, EXPECTATION(-1)(x(+1)) is equal to the expected value of variable x at next period, using the information set available Chapter 4: The Model file 14 at the previous period. See Section 4.6 [Auxiliary variables], page 19, for an explanation of how this operator is handled internally and how this affects the output. 4.3.3 Functions 4.3.3.1 Built-in Functions The following standard functions are supported internally for both MODEL EXPRESSION and EXPRESSION : exp (x) [Function] Natural exponential. log (x) ln (x) [Function] [Function] Natural logarithm. log10 (x) [Function] Base 10 logarithm. sqrt (x) [Function] Square root. abs (x) [Function] Absolute value. sign (x) [Function] Signum function. sin (x) cos (x) tan (x) asin (x) acos (x) atan (x) [Function] [Function] [Function] [Function] [Function] [Function] Trigonometric functions. max (a, b) min (a, b) [Function] [Function] Maximum and minimum of two reals. normcdf (x) normcdf (x, mu, sigma) [Function] [Function] Gaussian cumulative density function, with mean mu and standard deviation sigma. Note that normcdf(x) is equivalent to normcdf(x,0,1). normpdf (x) normpdf (x, mu, sigma) [Function] [Function] Gaussian probability density function, with mean mu and standard deviation sigma. Note that normpdf(x) is equivalent to normpdf(x,0,1). erf (x) Gauss error function. [Function] Chapter 4: The Model file 15 4.3.3.2 External Functions Any other user-defined (or built-in) MATLAB or Octave function may be used in both a MODEL EXPRESSION and an EXPRESSION, provided that this function has a scalar argument as a return value. To use an external function in a MODEL EXPRESSION, one must declare the function using the external_function statement. This is not necessary for external functions used in an EXPRESSION. external_function (OPTIONS . . . ); [Command] Description This command declares the external functions used in the model block. It is required for every unique function used in the model block. external_function commands can appear several times in the file and must come before the model block. Options name = NAME The name of the function, which must also be the name of the M-/MEX file implementing it. This option is mandatory. nargs = INTEGER The number of arguments of the function. If this option is not provided, Dynare assumes nargs = 1. first_deriv_provided [= NAME] If NAME is provided, this tells Dynare that the Jacobian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to name returns the Jacobian as its second output argument. second_deriv_provided [= NAME] If NAME is provided, this tells Dynare that the Hessian is provided as the only output of the M-/MEX file given as the option argument. If NAME is not provided, this tells Dynare that the M-/MEX file specified by the argument passed to name returns the Hessian as its third output argument. NB: This option can only be used if the first_deriv_provided option is used in the same external_function command. Example external_function(name = funcname); external_function(name = otherfuncname, nargs = 2, first_deriv_provided, second_deriv_provided); external_function(name = yetotherfuncname, nargs = 3, first_deriv_provided = funcname_deriv); 4.4 Parameter initialization When using Dynare for computing simulations, it is necessary to calibrate the parameters of the model. This is done through parameter initialization. The syntax is the following: PARAMETER_NAME = EXPRESSION; Here is an example of calibration: Chapter 4: The Model file 16 parameters alpha, bet; beta = 0.99; alpha = 0.36; A = 1-alpha*beta; Internally, the parameter values are stored in M_.params: [MATLAB/Octave variable] Contains the values of model parameters. The parameters are in the order that was used in the parameters command. M_.params 4.5 Model declaration The model is declared inside a model block: model ; model (OPTIONS . . . ); [Block] [Block] Description The equations of the model are written in a block delimited by model and end keywords. There must be as many equations as there are endogenous variables in the model, except when computing the unconstrained optimal policy with ramsey_policy or discretionary_policy. The syntax of equations must follow the conventions for MODEL EXPRESSION as described in Section 4.3 [Expressions], page 12. Each equation must be terminated by a semicolon (‘;’). A normal equation looks like: MODEL_EXPRESSION = MODEL_EXPRESSION; When the equations are written in homogenous form, it is possible to omit the ‘=0’ part and write only the left hand side of the equation. A homogenous equation looks like: MODEL_EXPRESSION; Inside the model block, Dynare allows the creation of model-local variables, which constitute a simple way to share a common expression between several equations. The syntax consists of a pound sign (#) followed by the name of the new model local variable (which must not be declared as in Section 4.2 [Variable declarations], page 9), an equal sign, and the expression for which this new variable will stand. Later on, every time this variable appears in the model, Dynare will substitute it by the expression assigned to the variable. Note that the scope of this variable is restricted to the model block; it cannot be used outside. A model local variable declaration looks like: # VARIABLE_NAME = MODEL_EXPRESSION; Options 2 linear Declares the model as being linear. It spares oneself from having to declare initial values for computing the steady state, and it sets automatically order=1 in stoch_ simul. use_dll Instructs the preprocessor to create dynamic loadable libraries (DLL) containing the model equations and derivatives, instead of writing those in M-files. You need a working compilation environment, i.e. a working mex command (see Section 2.1 [Software requirements], page 3 for more details). Using this option can result in faster simulations or estimations, at the expense of some initial compilation time.2 In particular, for big models, the compilation step can be very time-consuming, and use of this option may be counter-productive in those cases. Chapter 4: The Model file 17 block Perform the block decomposition of the model, and exploit it in computations (steady-state, deterministic simulation, stochastic simulation with first order approximation and estimation). See Dynare wiki for details on the algorithms used in deterministic simulation and steady-state computation. bytecode Instead of M-files, use a bytecode representation of the model, i.e. a binary file containing a compact representation of all the equations. cutoff = DOUBLE Threshold under which a jacobian element is considered as null during the model normalization. Only available with option block. Default: 1e-15 mfs = INTEGER Controls the handling of minimum feedback set of endogenous variables. Only available with option block. Possible values: 0 All the endogenous variables are considered as feedback variables (Default). 1 The endogenous variables assigned to equation naturally normalized (i.e. of the form x = f (Y ) where x does not appear in Y ) are potentially recursive variables. All the other variables are forced to belong to the set of feedback variables. 2 In addition of variables with mfs = 1 the endogenous variables related to linear equations which could be normalized are potential recursive variables. All the other variables are forced to belong to the set of feedback variables. 3 In addition of variables with mfs = 2 the endogenous variables related to non-linear equations which could be normalized are potential recursive variables. All the other variables are forced to belong to the set of feedback variables. no_static Don’t create the static model file. This can be useful for models which don’t have a steady state. Example 1: elementary RBC model var c k; varexo x; parameters aa alph bet delt gam; model; c = - k + aa*x*k(-1)^alph + (1-delt)*k(-1); c^(-gam) = (aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam)/(1+bet); end; Example 2: use of model local variables The following program: model; # gamma = 1 - 1/sigma; u1 = c1^gamma/gamma; u2 = c2^gamma/gamma; end; . . . is formally equivalent to: Chapter 4: The Model file 18 model; u1 = c1^(1-1/sigma)/(1-1/sigma); u2 = c2^(1-1/sigma)/(1-1/sigma); end; Example 3: a linear model model(linear); x = a*x(-1)+b*y(+1)+e_x; y = d*y(-1)+e_y; end; Dynare has the ability to output the list of model equations to a LaTeX file, using the write_ latex_dynamic_model command. The static model can also be written with the write_latex_ static_model command. write_latex_dynamic_model ; [Command] Description This command creates a LaTeX file containing the (dynamic) model. If your ‘.mod’ file is ‘FILENAME.mod’, then Dynare will create a file called ‘FILENAME_dynamic.tex’, containing the list of all the dynamic model equations. If LaTeX names were given for variables and parameters (see Section 4.2 [Variable declarations], page 9), then those will be used; otherwise, the plain text names will be used. Time subscripts (t, t+1, t-1, . . . ) will be appended to the variable names, as LaTeX subscripts. Note that the model written in the TeX file will differ from the model declared by the user in the following dimensions: • the timing convention of predetermined variables (see [predetermined variables], page 11) will have been changed to the default Dynare timing convention; in other words, variables declared as predetermined will be lagged on period back, • the expectation operators (see [expectation], page 13) will have been removed, replaced by auxiliary variables and new equations as explained in the documentation of the operator, • endogenous variables with leads or lags greater or equal than two will have been removed, replaced by new auxiliary variables and equations, • for a stochastic model, exogenous variables with leads or lags will also have been replaced by new auxiliary variables and equations. Compiling the TeX file requires the following Latex packages: geometry, fullpage, breqn. write_latex_static_model ; [Command] Description This command creates a LaTeX file containing the static model. If your ‘.mod’ file is ‘FILENAME.mod’, then Dynare will create a file called ‘FILENAME_static.tex’, containing the list of all the equations of the steady state model. If LaTeX names were given for variables and parameters (see Section 4.2 [Variable declarations], page 9), then those will be used; otherwise, the plain text names will be used. Note that the model written in the TeX file will differ from the model declared by the user in the some dimensions (see [write latex dynamic model], page 18 for details). Also note that this command will not output the contents of the optional steady_state_model block (see [steady state model], page 29); it will rather output a static version (i.e. without leads and lags) of the dynamic model declared in the model block. Chapter 4: The Model file 19 Compiling the TeX file requires the following Latex packages: geometry, fullpage, breqn. 4.6 Auxiliary variables The model which is solved internally by Dynare is not exactly the model declared by the user. In some cases, Dynare will introduce auxiliary endogenous variables—along with corresponding auxiliary equations—which will appear in the final output. The main transformation concerns leads and lags. Dynare will perform a transformation of the model so that there is only one lead and one lag on endogenous variables and, in the case of a stochastic model, no leads/lags on exogenous variables. This transformation is achieved by the creation of auxiliary variables and corresponding equations. For example, if x(+2) exists in the model, Dynare will create one auxiliary variable AUX_ ENDO_LEAD = x(+1), and replace x(+2) by AUX_ENDO_LEAD(+1). A similar transformation is done for lags greater than 2 on endogenous (auxiliary variables will have a name beginning with AUX_ENDO_LAG), and for exogenous with leads and lags (auxiliary variables will have a name beginning with AUX_EXO_LEAD or AUX_EXO_LAG respectively). Another transformation is done for the EXPECTATION operator. For each occurence of this operator, Dynare creates an auxiliary variable defined by a new equation, and replaces the expectation operator by a reference to the new auxiliary variable. For example, the expression EXPECTATION(1)(x(+1)) is replaced by AUX_EXPECT_LAG_1(-1), and the new auxiliary variable is declared as AUX_EXPECT_LAG_1 = x(+2). Auxiliary variables are also introduced by the preprocessor for the ramsey_policy command. In this case, they are used to represent the Lagrange multipliers when first order conditions of the Ramsey problem are computed. The new variables take the form MULT_i, where i represents the constraint with which the multiplier is associated (counted from the order of declaration in the model block). Once created, all auxiliary variables are included in the set of endogenous variables. The output of decision rules (see below) is such that auxiliary variable names are replaced by the original variables they refer to. The number of endogenous variables before the creation of auxiliary variables is stored in M_ .orig_endo_nbr, and the number of endogenous variables after the creation of auxiliary variables is stored in M_.endo_nbr. See Dynare Wiki for more technical details on auxiliary variables. 4.7 Initial and terminal conditions For most simulation exercises, it is necessary to provide initial (and possibly terminal) conditions. It is also necessary to provide initial guess values for non-linear solvers. This section describes the statements used for those purposes. In many contexts (determistic or stochastic), it is necessary to compute the steady state of a non-linear model: initval then specifies numerical initial values for the non-linear solver. The command resid can be used to compute the equation residuals for the given initial values. Used in perfect foresight mode, the types of forward-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 initval and shocks (see Section 4.8 [Shocks on exogenous variables], page 23. Another one is to study, how an economy, starting from arbitrary initial conditions converges toward equilibrium. To do that, one needs initval and endval. For models with lags on more than one period, the command histval permits to specify different historical initial values for periods before the beginning of the simulation. Chapter 4: The Model file initval ; 20 [Block] Description The initval block serves two purposes: declaring the initial (and possibly terminal) conditions in a simulation exercise, and providing guess values for non-linear solvers. This block is terminated by end;, and contains lines of the form: VARIABLE_NAME = EXPRESSION; In a deterministic (i.e. perfect foresight) model First, it provides the initial conditions for all the endogenous and exogenous variables at all the periods preceeding the first simulation period (unless some of these initial values are modified by histval). Second, in the absence of an endval block, it sets the terminal conditions for all the periods succeeding the last simulation period. Third, in the absence of an endval block, it provides initial guess values at all simulation dates for the non-linear solver implemented in simul. For this last reason, it necessary to provide values for all the endogenous variables in an initval block (even though, theoretically, initial conditions are only necessary for lagged variables). If some variables, endogenous or exogenous, are not mentionned in the initval block, a zero value is assumed. Note that if the initval block is immediately followed by a steady command, its semantics is changed. The steady command will compute the steady state of the model for all the endogenous variables, assuming that exogenous variables are kept constant to the value declared in the initval block, and using the values declared for the endogenous as initial guess values for the non-linear solver. An initval block followed by steady is formally equivalent to an initval block with the same values for the exogenous, and with the associated steady state values for the endogenous. In a stochastic model The main purpose of initval is to provide initial guess values for the non-linear solver in the steady state computation. Note that if the initval block is not followed by steady, the steady state computation will still be triggered by subsequent commands (stoch_simul, estimation. . . ). It is not necessary to declare 0 as initial value for exogenous stochastic variables, since it is the only possible value. This steady state will be used as the initial condition at all the periods preceeding the first simulation period for the two possible types of simulations in stochastic mode: • in stoch_simul, if the periods options is specified • in forecast (in this case, note that it is still possible to modify some of these initial values with histval) Example initval; c = 1.2; k = 12; x = 1; end; steady; Chapter 4: The Model file endval ; 21 [Block] Description This block is terminated by end;, and contains lines of the form: VARIABLE_NAME = EXPRESSION; The endval block makes only sense in a determistic model, and serves two purposes. First, it sets the terminal conditions for all the periods succeeding the last simulation period. Second, it provides initial guess values at all the simulation dates for the non-linear solver implemented in simul. For this last reason, it necessary to provide values for all the endogenous variables in an endval block (even though, theoretically, initial conditions are only necessary for forward variables). If some variables, endogenous or exogenous, are not mentionned in the endval block, a zero value is assumed. Note that if the endval block is immediately followed by a steady command, its semantics is changed. The steady command will compute the steady state of the model for all the endogenous variables, assuming that exogenous variables are kept constant to the value declared in the endval block, and using the values declared for the endogenous as initial guess values for the non-linear solver. An endval block followed by steady is formally equivalent to an endval block with the same values for the exogenous, and with the associated steady state values for the endogenous. 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 computed by steady for x=1, and the terminal one, for x=2. histval ; [Block] Description In models with lags on more than one period, the histval block permits to specify different historical initial values for different periods. This block is terminated by end;, and contains lines of the form: VARIABLE_NAME(INTEGER) = EXPRESSION; Chapter 4: The Model file 22 EXPRESSION is any valid expression returning a numerical value and can contain already initialized variable names. By convention in Dynare, period 1 is the first period of the simulation. Going backward in time, the first period before the start of the simulation is period 0, then period -1, and so on. If your lagged variables are linked by identities, be careful to satisfy these identities when you set historical initial values. Variables not initialized in the histval block are assumed to have a value of zero at period 0 and before. Note that this behavior differs from the case where there is no histval block, where all variables are initialized at their steady state value at period 0 and before (except when a steady command doesn’t follow an initval block). 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; resid ; [Command] This command will display the residuals of the static equations of the model, using the values given for the endogenous in the last initval or endval block (or the steady state file if you provided one, see Section 4.10 [Steady state], page 26). initval_file (filename = FILENAME); [Command] Description In a deterministic setup, this command is used to specify a path for all endogenous and exogenous variables. The length of these paths must be equal to the number of simulation periods, plus the number of leads and the number of lags of the model (for example, with 50 simulation periods, in a model with 2 lags and 1 lead, the paths must have a length of 53). Note that these paths cover two different things: • the constraints of the problem, which are given by the path for exogenous and the initial and terminal values for endogenous • the initial guess for the non-linear solver, which is given by the path for endogenous variables for the simulation periods (excluding initial and terminal conditions) The command accepts three file formats: Chapter 4: The Model file 23 • M-file (extension ‘.m’): for each endogenous and exogenous variable, the file must contain a row vector of the same name. • MAT-file (extension ‘.mat’): same as for M-files. • Excel file (extension ‘.xls’): for each endogenous and exogenous, the file must contain a column of the same name (supported under Octave if the io and java packages from Octave-Forge are installed, along with a Java Runtime Environment). Warning The extension must be omitted in the command argument. Dynare will automatically figure out the extension and select the appropriate file type. 4.8 Shocks on exogenous variables In a deterministic context, when one wants to study the transition of one equilibrium position to another, it is equivalent to analyze the consequences of a permanent shock and this in done in Dynare through the proper use of initval and endval. Another typical experiment is to study the effects of a temporary shock after which the system goes back to the original equilibrium (if the model is stable. . . ). A temporary shock is a temporary change of value of one or several exogenous variables in the model. Temporary shocks are specified with the command shocks. In a stochastic framework, the exogenous variables take random values in each period. In Dynare, these random values follow a normal distribution with zero mean, but it belongs to the user to specify the variability of these shocks. The non-zero elements of the matrix of variancecovariance of the shocks can be entered with the shocks command. Or, the entire matrix can be direclty entered with Sigma_e (this use is however deprecated). 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 ; [Block] In deterministic context For deterministic simulations, the shocks block specifies temporary changes in the value of exogenous variables. For permanent shocks, use an endval block. The block should contain one or more occurrences of the following group of three lines: var VARIABLE_NAME; periods INTEGER[:INTEGER] [[,] INTEGER[:INTEGER]]...; values DOUBLE | (EXPRESSION) [[,] DOUBLE | (EXPRESSION) ]...; It is possible to specify shocks which last several periods and which can vary over time. The periods keyword accepts a list of several dates or date ranges, which must be matched by as many shock values in the values keyword. Note that a range in the periods keyword can be matched by only one value in the values keyword. If values represents a scalar, the same value applies to the whole range. If values represents a vector, it must have as many elements as there are periods in the range. Note that shock values are not restricted to numerical constants: arbitrary expressions are also allowed, but you have to enclose them inside parentheses. Here is an example: shocks; var e; periods 1; Chapter 4: The Model file 24 values 0.5; var u; periods 4:5; values 0; var v; periods 4:5 6 7:9; values 1 1.1 0.9; var w; periods 1 2; values (1+p) (exp(z)); end; A second example with a vector of values: xx = [1.2; 1.3; 1]; shocks; var e; periods 1:3; values (xx); end; In stochastic context For stochastic simulations, the shocks block specifies the non zero elements of the covariance matrix of the shocks of exogenous variables. You can use the following types of entries in the block: var VARIABLE_NAME; stderr EXPRESSION; Specifies the standard error of a variable. var VARIABLE_NAME = EXPRESSION; Specifies the variance error of a variable. var VARIABLE_NAME, VARIABLE_NAME = EXPRESSION; Specifies the covariance of two variables. corr VARIABLE_NAME, VARIABLE_NAME = EXPRESSION; Specifies the correlation of two variables. In an estimation context, it is also possible to specify variances and covariances on endogenous variables: in that case, these values are interpreted as the calibration of the measurement errors on these variables. Here is an example: shocks; var e = 0.000081; var u; stderr 0.009; corr e, u = 0.8; var v, w = 2; end; 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 stoch_simul will compute the rational expectation solution adding future information to the state space (nothing is shown in the output of stoch_simul) and forecast will compute a simulation conditional on initial conditions and future information. Chapter 4: The Model file 25 Here is an 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; mshocks ; [Block] The purpose of this block is similar to that of the shocks block for deterministic shocks, except that the numeric values given will be interpreted in a multiplicative way. For example, if a value of 1.05 is given as shock value for some exogenous at some date, it means 5% above its steady state value (as given by the last initval or endval block). The syntax is the same than shocks in a deterministic context. This command is only meaningful in two situations: • on exogenous variables with a non-zero steady state, in a deterministic setup, • on deterministic exogenous variables with a non-zero steady state, in a stochastic setup. Sigma_e [Special variable] Warning The use of this special variable is deprecated and is strongly discouraged. You should use a shocks block instead. Description This special variable specifies directly the covariance matrix of the stochastic shocks, as an 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 arbitrary EXPRESSION is allowed (instead of a simple constant), but in that case you need to enclose the expression in parentheses. The order of the covariances in the matrix is the same as the one used in the varexo declaration. Example varexo u, e; ... Sigma_e = [ 0.81 (phi*0.9*0.009); 0.000081]; This sets the variance of u to 0.81, the variance of e to 0.000081, and the correlation between e and u to phi. Chapter 4: The Model file 26 4.9 Other general declarations dsample INTEGER [INTEGER]; [Command] Reduces the number of periods considered in subsequent output commands. periods INTEGER; [Command] 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. This command sets 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; 4.10 Steady state There are two ways of computing the steady state (i.e. the static equilibrium) of a model. The first way is to let Dynare compute the steady state using a nonlinear Newton-type solver; this should work for most models, and is relatively simple to use. The second way is to give more guidance to Dynare, using your knowledge of the model, by providing it with a “steady state file”. 4.10.1 Finding the steady state with Dynare nonlinear solver steady ; steady (OPTIONS . . . ); [Command] [Command] Description This command computes the steady state of a model using a nonlinear Newton-type solver and displays it. When a steady state file is used steady displays the steady state and checks that it is a solution of the static model. More precisely, it computes the equilibrium value of the endogenous variables for the value of the exogenous variables specified in the previous initval or endval block. steady uses an iterative procedure and takes as initial guess the value of the endogenous variables set in the previous initval or endval block. For complicated models, finding good numerical initial values for the endogenous variables is the trickiest part of finding the equilibrium of that model. Often, it is better to start with a smaller model and add new variables one by one. Options maxit = INTEGER Determines the maximum number of iterations used in the non-linear solver. The default value of maxit is 10. The maxit option is shared with the simul command. So a change in maxit in a steady command will also be considered in the following simul commands. solve_algo = INTEGER Determines the non-linear solver to use. Possible values for the option are: Chapter 4: The Model file 27 0 Use fsolve (under MATLAB, only available if you have the Optimization Toolbox; always available under Octave) 1 Use Dynare’s own nonlinear equation solver 2 Splits the model into recursive blocks and solves each block in turn 3 Use Chris Sims’ solver 4 Similar to value 2, except that it deals differently with nearly singular Jacobian 5 Newton algorithm with a sparse Gaussian elimination (SPE) (requires bytecode option, see Section 4.5 [Model declaration], page 16) 6 Newton algorithm with a sparse LU solver at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16) 7 Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16; not available under Octave) 8 Newton algorithm with a Stabilized Bi-Conjugate Gradient (BICGSTAB) solver at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16) Default value is 2. homotopy_mode = INTEGER Use a homotopy (or divide-and-conquer) technique to solve for the steady state. If you use this option, you must specify a homotopy_setup block. This option can take three possible values: 1 In this mode, all the parameters are changed simultaneously, and the distance between the boudaries for each parameter is divided in as many intervals as there are steps (as defined by homotopy_steps option); the problem is solves as many times as there are steps. 2 Same as mode 1, except that only one parameter is changed at a time; the problem is solved as many times as steps times number of parameters. 3 Dynare tries first the most extreme values. If it fails to compute the steady state, the interval between initial and desired values is divided by two for all parameters. Every time that it is impossible to find a steady state, the previous interval is divided by two. When it succeeds to find a steady state, the previous interval is multiplied by two. In that last case homotopy_steps contains the maximum number of computations attempted before giving up. homotopy_steps = INTEGER Defines the number of steps when performing a homotopy. See homotopy_mode option for more details. homotopy_force_continue = INTEGER This option controls what happens when homotopy fails. 0 steady fails with an error message 1 steady keeps the values of the last homotopy step that was successful and continues. BE CAREFUL: parameters and/or exogenous variables are NOT at the value expected by the user Chapter 4: The Model file 28 Default is 0. nocheck Don’t check the steady state values when they are provided explicitely either by a steady state file or a steady_state_model block. This is useful for models with unit roots as, in this case, the steady state is not unique or doesn’t exist. Example See Section 4.7 [Initial and terminal conditions], page 19. After computation, the steady state is available in the following variable: oo_.steady_state [MATLAB/Octave variable] Contains the computed steady state. Endogenous variables are ordered in order of declaration used in var command (which is also the order used in M_.endo_names). homotopy_setup ; [Block] Description This block is used to declare initial and final values when using a homotopy method. It is used in conjunction with the option homotopy_mode of the steady command. The idea of homotopy (also called divide-and-conquer by some authors) is to subdivide the problem of finding the steady state into smaller problems. It assumes that you know how to compute the steady state for a given set of parameters, and it helps you finding the steady state for another set of parameters, by incrementally moving from one to another set of parameters. The purpose of the homotopy_setup block is to declare the final (and possibly also the initial) values for the parameters or exogenous that will be changed during the homotopy. It should contain lines of the form: VARIABLE_NAME, EXPRESSION, EXPRESSION; This syntax specifies the initial and final values of a given parameter/exogenous. There is an alternative syntax: VARIABLE_NAME, EXPRESSION; Here only the final value is specified for a given parameter/exogenous; the initial value is taken from the preceeding initval block. A necessary condition for a successful homotopy is that Dynare must be able to solve the steady state for the initial parameters/exogenous without additional help (using the guess values given in the initval block). If the homotopy fails, a possible solution is to increase the number of steps (given in homotopy_ steps option of steady). Example In the following example, Dynare will first compute the steady state for the initial values (gam=0.5 and x=1), and then subdivide the problem into 50 smaller problems to find the steady state for the final values (gam=2 and x=2). var c k; varexo x; parameters alph gam delt bet aa; alph=0.5; delt=0.02; Chapter 4: The Model file 29 aa=0.5; bet=0.05; model; c + k - aa*x*k(-1)^alph - (1-delt)*k(-1); c^(-gam) - (1+bet)^(-1)*(aa*alph*x(+1)*k^(alph-1) + 1 - delt)*c(+1)^(-gam); end; initval; x = 1; k = ((delt+bet)/(aa*x*alph))^(1/(alph-1)); c = aa*x*k^alph-delt*k; end; homotopy_setup; gam, 0.5, 2; x, 2; end; steady(homotopy_mode = 1, homotopy_steps = 50); 4.10.2 Using a steady state file If you know how to compute the steady state for your model, you can provide a MATLAB/Octave function doing the computation instead of using steady. If your MOD-file is called ‘FILENAME.mod’, the steady state file should be called ‘FILENAME_steadystate.m’. Again, there are two options for creating this file: • The easiest way is to write a steady_state_model block. • You can write the corresponding Matlab function by hand. See ‘fs2000_steadystate.m’ in the ‘examples’ directory for an example. This option gives a bit more flexibility, at the expense of a heavier programming burden and a lesser efficiency. steady_state_model ; [Block] Description When the analytical solution of the model is known, this command can be used to help Dynare find the steady state in a more efficient and reliable way, especially during estimation where the steady state has to be recomputed for every point in the parameter space. Each line of this block consists of a variable (either an endogenous, a temporary variable or a parameter) which is assigned an expression (which can contain parameters, exogenous at the steady state, or any endogenous or temporary variable already declared above). Each line therefore looks like: VARIABLE_NAME = EXPRESSION; Note that it is also possible to assign several variables at the same time, if the main function in the right hand side is a MATLAB/Octave function returning several arguments: [ VARIABLE_NAME, VARIABLE_NAME... ] = EXPRESSION; Dynare will automatically generate a steady state file using the information provided in this block. Steady state file for deterministic models steady_state_model block works also with deterministic models. An initval block and, when necessary, an endval block, is used to set the value of the exogenous variables. Each initval Chapter 4: The Model file 30 or endval block must be followed by steady to execute the function created by steady_state_ model and set the initial, respectively terminal, steady state. Example var m P c e W R k d n l gy_obs gp_obs y dA; varexo e_a e_m; parameters alp bet gam mst rho psi del; ... // parameter calibration, (dynamic) model declaration, shock calibration... ... steady_state_model; dA = exp(gam); gst = 1/dA; // A temporary variable m = mst; // Three other temporary variables khst = ( (1-gst*bet*(1-del)) / (alp*gst^alp*bet) )^(1/(alp-1)); xist = ( ((khst*gst)^alp - (1-gst*(1-del))*khst)/mst )^(-1); nust = psi*mst^2/( (1-alp)*(1-psi)*bet*gst^alp*khst^alp ); n P k = xist/(nust+xist); = xist + nust; = khst*n; l c d y R = = = = = psi*mst*n/( (1-psi)*(1-n) ); mst/P; l - mst + 1; k^alp*n^(1-alp)*gst^alp; mst/bet; // You can use MATLAB functions which return several arguments [W, e] = my_function(l, n); gp_obs = m/dA; gy_obs = dA; end; steady; 4.11 Getting information about the model check ; check (solve algo = INTEGER) ; [Command] [Command] Description Computes the eigenvalues of the model linearized around the values specified by the last initval, endval or steady statement. Generally, the eigenvalues are only meaningful if the linearization is done around a steady state of the model. It is a device for local analysis in the neighborhood of this steady state. Chapter 4: The Model file 31 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. Options solve_algo = INTEGER See [solve algo], page 26, for the possible values and their meaning. Output check returns the eigenvalues in the global variable oo_.dr.eigval. [MATLAB/Octave variable] Contains the eigenvalues of the model, as computed by the check command. oo_.dr.eigval model_info ; model_info (OPTIONS . . . ); [Command] [Command] Description This command provides information about: • the normalization of the model: an endogenous variable is attributed to each equation of the model; • the block structure of the model: for each block model info indicates its type, the equations number and endogenous variables belonging to this block. This command can only be used in conjunction with the block option of the model block. There are five different types of blocks depending on the simulation method used: ‘EVALUATE FORWARD’ In this case the block contains only equations where endogenous variable attributed to the equation appears currently on the left hand side and where no forward looking endogenous variables appear. The block has the form: yj,t = fj (yt , yt−1 , . . . , yt−k ). ‘EVALUATE BACKWARD’ The block contains only equations where endogenous variable attributed to the equation appears currently on the left hand side and where no backward looking endogenous variables appear. The block has the form: yj,t = fj (yt , yt+1 , . . . , yt+k ). ‘SOLVE FORWARD x’ The block contains only equations where endogenous variable attributed to the equation does not appear currently on the left hand side and where no forward looking endogenous variables appear. The block has the form: gj (yj,t , yt , yt−1 , . . . , yt−k ) = 0. x is equal to ‘SIMPLE’ if the block has only one equation. If several equation appears in the block, x is equal to ‘COMPLETE’. ‘SOLVE FORWARD x’ The block contains only equations where endogenous variable attributed to the equation does not appear currently on the left hand side and where no backward looking endogenous variables appear. The block has the form: gj (yj,t , yt , yt+1 , . . . , yt+k ) = 0. x is equal to ‘SIMPLE’ if the block has only one equation. If several equation appears in the block, x is equal to ‘COMPLETE’. Chapter 4: The Model file 32 ‘SOLVE TWO BOUNDARIES x’ The block contains equations depending on both forward and backward variables. The block looks like: gj (yj,t , yt , yt−1 , . . . , yt−k , yt , yt+1 , . . . , yt+k ) = 0. x is equal to ‘SIMPLE’ if the block has only one equation. If several equation appears in the block, x is equal to ‘COMPLETE’. Options ’static’ Prints out the block decomposition of the static model. Without ’static’ option model info displays the block decomposition of the dynamic model. ’incidence’ Displays the gross incidence matrix and the reordered incidence matrix of the block decomposed model. print_bytecode_dynamic_model ; [Command] Prints the equations and the Jacobian matrix of the dynamic model stored in the bytecode binary format file. Can only be used in conjunction with the bytecode option of the model block. print_bytecode_static_model ; [Command] Prints the equations and the Jacobian matrix of the static model stored in the bytecode binary format file. Can only be used in conjunction with the bytecode option of the model block. 4.12 Deterministic simulation 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) and Boucekkine (1995), 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 can be found in Juillard (1996). simul ; simul (OPTIONS . . . ); [Command] [Command] Description Triggers the computation of a deterministic simulation of the model for the number of periods set in the option periods. Options periods = INTEGER Number of periods of the simulation maxit = INTEGER Determines the maximum number of iterations used in the non-linear solver. The default value of maxit is 10. The maxit option is shared with the steady command. So a change in maxit in a simul command will also be considered in the following steady commands. Chapter 4: The Model file 33 stack_solve_algo = INTEGER Algorithm used for computing the solution. Possible values are: 0 Newton method to solve simultaneously all the equations for every period, using sparse matrices (Default). 1 Use a Newton algorithm with a sparse LU solver at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16). 2 Use a Newton algorithm with a Generalized Minimal Residual (GMRES) solver at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16; not available under Octave) 3 Use a Newton algorithm with a Stabilized Bi-Conjugate Gradient (BICGSTAB) solver at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16). 4 Use a Newton algorithm with a optimal path length at each iteration (requires bytecode and/or block option, see Section 4.5 [Model declaration], page 16). 5 Use a Newton algorithm with a sparse Gaussian elimination (SPE) solver at each iteration (requires bytecode option, see Section 4.5 [Model declaration], page 16). 6 Use the historical algorithm proposed in Juillard (1996): it is slower than stack_solve_algo=0, but may be less memory consuming on big models (not available with bytecode and/or block options). markowitz = DOUBLE Value of the Markowitz criterion, used to select the pivot. Only used when stack_ solve_algo = 5. Default: 0.5. minimal_solving_periods = INTEGER Specify the minimal number of periods where the model has to be solved, before using a constant set of operations for the remaining periods. Only used when stack_ solve_algo = 5. Default: 1. datafile = FILENAME If the variables of the model are not constant over time, their initial values, stored in a text file, could be loaded, using that option, as initial values before a deteministic simulation. Output The simulated endogenous variables are available in global matrix oo_.endo_simul. [MATLAB/Octave variable] This variable stores the result of a deterministic simulation (computed by simul) or of a stochastic simulation (computed by stoch_simul with the periods option or by extended_path). The variables are arranged row by row, in order of declaration (as in M_.endo_names). Note that this variable also contains initial and terminal conditions, so it has more columns than the value of periods option. oo_.endo_simul [MATLAB/Octave variable] This variable stores the path of exogenous variables during a simulation (computed by simul, stoch_simul or extended_path). The variables are arranged in columns, in order of declaration (as in M_.endo_names). Periods are in rows. Note that this convention regarding columns and rows is the opposite of the convention for oo_.endo_simul! oo_.exo_simul Chapter 4: The Model file 34 4.13 Stochastic solution and simulation In a stochastic context, Dynare computes one or several simulations corresponding to a random draw of the shocks. The main algorithm for solving stochastic models relies on a Taylor approximation, up to third order, of the expectation functions (see Judd (1996), Collard and Juillard (2001a), Collard and Juillard (2001b), and Schmitt-Grohé and Urı́be (2004)). The details of the Dynare implementation of the first order solution are given in Villemot (2011). Such a solution is computed using the stoch_simul command. As an alternative, it is possible to compute a simulation to a stochastic model using the extended path method presented by Fair and Taylor (1983). This method is especially useful when there are strong nonlinearities or binding constraints. Such a solution is computed using the extended_path command. 4.13.1 Computing the stochastic solution stoch_simul [VARIABLE_NAME . . . ]; stoch_simul (OPTIONS . . . ) [VARIABLE_NAME . . . ]; [Command] [Command] 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 Section 4.10 [Steady state], page 26). 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 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. The covariance matrix of the shocks is specified with the shocks command (see Section 4.8 [Shocks on exogenous variables], page 23). 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], page 17). Options Chapter 4: The Model file 35 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 = DOUBLE Uses HP filter with λ = DOUBLE before computing 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. irf = INTEGER Number of periods on which to compute the IRFs. Setting irf=0, suppresses the plotting of IRF’s. 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 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 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) and displayed to screen (unless nodisplay option is used). nodisplay Do not display the graphs, but still save them to disk (unless nograph is used). graph_format = FORMAT Specify the file format for graphs saved to disk. Possible values are eps (the default), pdf and fig (the latter is not available under Octave). 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 Section 4.5 [Model declaration], page 16. Default: disabled for order 1 and 2, enabled otherwise Chapter 4: The Model file 36 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], page 33). The simulated exogenous variables are made available in oo_.exo_simul (see [oo .exo simul], page 33). 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 Section 4.14 [Estimation], page 41). 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], page 26, 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 var(yt+k |t). For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact. The results are stored in oo_.conditional_ variance_decomposition (see [oo .conditional variance decomposition], page 41). pruning Discard higher order terms when iteratively computing simulations of the solution, as in Kim, Kim, Schaumburg and Sims (2008). partial_information Computes the solution of the model under partial information, along the lines of Pearlman, Currie and Levine (1986). Agents are supposed to observe only some variables of the economy. The set of observed variables is declared using the varobs command. Note that 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. Chapter 4: The Model file 37 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 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. Output This command sets oo_.dr, oo_.mean, oo_.var and oo_.autocorr, which are described below. If option periods is present, sets oo_.endo_simul (see [oo .endo simul], page 33), and also saves the simulated variables in MATLAB/Octave vectors of the global workspace with the same name as the endogenous variables. If options 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). 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. [MATLAB/Octave variable] After a run of stoch_simul, contains the mean of the endogenous variables. Contains theoretical mean if the periods option is not present, and empirical mean otherwise. The variables are arranged in declaration order. oo_.mean [MATLAB/Octave variable] After a run of stoch_simul, contains the variance-covariance of the endogenous variables. Contains theoretical variance if the periods option is not present, and empirical variance otherwise. The variables are arranged in declaration order. oo_.var [MATLAB/Octave variable] 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 oo_.autocorr Chapter 4: The Model file 38 order of autocorrelation. The option ar specifies the number of autocorrelation matrices available. Contains theoretical autocorrelations if the periods option is not present, and empirical autocorrelations otherwise. l The element oo_.autocorr{i}(k,l) is equal to the correlation between ytk and yt−i , where y k l (resp. y ) is the k-th (resp. l-th) endogenous variable in the declaration order. Note that if theoretical moments have been requested, oo_.autocorr{i} is the same than oo_ .gamma_y{i+1}. [MATLAB/Octave variable] 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_y oo_.gamma{1} Variance/co-variance matrix. oo_.gamma{i+1} (for i=1:ar) Autocorrelation function. see [oo .autocorr], page 37 for more details. Beware, this is the autocorrelation function, not the autocovariance function. oo_.gamma{nar+2} Variance decomposition. oo_.gamma{nar+3} If a second order approximation has been requested, contains the vector of the mean correction terms. [MATLAB/Octave variable] 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. oo_.irfs The approximated solution of a model takes the form of a set of decision rules or transition equations expressing the current value of the endogenous variables of the model as function of the previous state of the model and shocks oberved at the beginning of the period. The decision rules are stored in the structure oo_.dr which is described below. extended_path ; extended_path (OPTIONS . . . ) ; [Command] [Command] Description extended_path solves a stochastic (i.e. rational expectations) model, using the extended path method presented by Fair and Taylor (1983). This function first computes a random path for the exogenous variables (stored in oo_.exo_ simul, see [oo .exo simul], page 33) 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], page 33). 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 approximate solution at every iteration of the algorithm. Default: 200. Chapter 4: The Model file 39 4.13.2 Typology and ordering of variables Dynare distinguishes four types of endogenous variables: Purely backward (or purely predetermined) variables Those that appear only at current and past period in the model, but not at future period (i.e. at t and t − 1 but not t + 1). The number of such variables is equal to 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 (see Section 4.6 [Auxiliary variables], page 19), 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. 4.13.3 First order approximation The approximation has the form: h yt = y s + Ayt−1 + But where y s is the steady state value of y and yth = yt − y s . The coefficients of the decision rules are stored as follows: • y s is stored in oo_.dr.ys. The vector rows correspond to all endogenous in the declaration order. • A is stored in oo_.dr.ghx. The matrix rows correspond to all endogenous in DR-order. The matrix columns correspond to state variables in DR-order. • 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. Chapter 4: The Model file 40 4.13.4 Second order approximation The approximation has the form: h h h h yt = y s + 0.5Δ2 + Ayt−1 + But + 0.5C(yt−1 ⊗ yt−1 ) + 0.5D(ut ⊗ ut ) + E(yt−1 ⊗ ut ) s h s 2 where y is the steady state value of y, yt = yt − y , and Δ 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). 4.13.5 Third order approximation The approximation has the form: yt = y s + G0 + G1 zt + G2 (zt ⊗ zt ) + G3 (zt ⊗ zt ⊗ zt ) where y s 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: • y s 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 6= 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 Chapter 4: The Model file 41 that have three distinct indices (i.e. i1 6= i2 and i1 6= i3 and i2 6= 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. [MATLAB/Octave variable] 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). oo_.conditional_variance_decomposition 4.14 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 (as in Ireland (2004)) and Bayesian techniques (as in Rabanal and Rubio-Ramirez (2003), Schorfheide (2000) or Smets and Wouters (2003)) are available. Using Bayesian methods, it is possible to estimate DSGE models, VAR models, or a combination of the two techniques called DSGE-VAR. Note that in order to avoid stochastic singularity, you must have at least as many shocks or measurement errors in your model as you have observed variables. The estimation using a first order approximation can benefit from the block decomposition of the model (see [block], page 17). varobs VARIABLE_NAME . . . ; [Command] Description This command lists the name of observed endogenous variables for the estimation procedure. These variables must be available in the data file (see [estimation cmd], page 44). Alternatively, this command is also used in conjunction with the partial_information option of stoch_simul, for declaring the set of observed variables when solving the model under partial information. Only one instance of varobs is allowed in a model file. If one needs to declare observed variables in a loop, the macroprocessor can be used as shown in the second example below. Simple example varobs C y rr; Example with a loop varobs @#for co in countries GDP_@{co} @#endfor ; observation_trends ; [Block] Description This block specifies linear trends for observed variables as functions of model parameters. Each line inside of the block should be of the form: Chapter 4: The Model file 42 VARIABLE_NAME(EXPRESSION); In most cases, variables shouldn’t be centered when observation_trends is used. Example observation_trends; Y (eta); P (mu/eta); end; estimated_params ; [Block] Description This block lists all parameters to be estimated and specifies bounds and priors as necessary. Each line corresponds to an estimated parameter. In a maximum likelihood estimation, each line follows this syntax: stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME , INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND ]; In a Bayesian estimation, each line follows this syntax: stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME | DSGE_PRIOR_WEIGHT [, INITIAL_VALUE [, LOWER_BOUND, UPPER_BOUND]], PRIOR_SHAPE, PRIOR_MEAN, PRIOR_STANDARD_ERROR [, PRIOR_3RD_PARAMETER [, PRIOR_4TH_PARAMETER [, SCALE_PARAMETER ] ] ]; The first part of the line consists of one of the three following alternatives: stderr VARIABLE_NAME Indicates 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 VARIABLE_NAME1, VARIABLE_NAME2 Indicates that the correlation between the exogenous variables VARIABLE NAME1 and VARIABLE NAME2, or the correlation of the observation errors associated with endogenous observed variables VARIABLE NAME1 and VARIABLE NAME2, is to be estimated PARAMETER_NAME The name of a model parameter to be estimated DSGE_PRIOR_WEIGHT ... The rest of the line consists of the following fields, some of them being optional: 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 A keyword specifying the shape of the prior density. The possible values are: beta_pdf, gamma_pdf, normal_pdf, uniform_pdf, inv_gamma_pdf, inv_gamma1_ pdf, inv_gamma2_pdf. Note that inv_gamma_pdf is equivalent to inv_gamma1_pdf Chapter 4: The Model file 43 PRIOR_MEAN The mean of the prior distribution PRIOR_STANDARD_ERROR The standard error of the prior distribution PRIOR_3RD_PARAMETER A third parameter of the prior used for generalized beta distribution, generalized gamma and for the uniform distribution. Default: 0 PRIOR_4TH_PARAMETER A fourth parameter of the prior used for generalized beta distribution and for the uniform distribution. Default: 1 SCALE_PARAMETER The scale parameter to be used for the jump distribution of the Metropolis-Hasting algorithm Note that INITIAL VALUE, LOWER BOUND, UPPER BOUND, PRIOR MEAN, PRIOR STANDARD ERROR, PRIOR 3RD PARAMETER, PRIOR 4TH PARAMETER and SCALE PARAMETER can be any valid EXPRESSION. Some of them can be empty, in which Dynare will select a default value depending on the context and the prior shape. As one uses options more towards the end of the list, all previous options must be filled: for example, if you want to specify SCALE PARAMETER, you must specify PRIOR 3RD PARAMETER and PRIOR 4TH PARAMETER. Use empty values, if these parameters don’t apply. Example The following line: corr eps_1, eps_2, 0.5, , , beta_pdf, 0, 0.3, -1, 1; sets a generalized beta prior for the correlation between eps_1 and eps_2 with mean 0 and variance 0.3. By setting PRIOR 3RD PARAMETER to -1 and PRIOR 4TH PARAMETER to 1 the standard beta distribution with support [0,1] is changed to a generalized beta with support [-1,1]. Note that LOWER BOUND and UPPER BOUND are left empty and thus default to -1 and 1, respectively. The initial value is set to 0.5. Similarly, the following line: corr eps_1, eps_2, 0.5, -0.5, 1, beta_pdf, 0, 0.3, -1, 1; sets the same generalized beta distribution as before, but now truncates this distribution to [-0.5,1] through the use of LOWER BOUND and UPPER BOUND. Hence, the prior does not integrate to 1 anymore. Parameter transformation Sometimes, it is desirable to estimate a transformation of a parameter appearing in the model, rather than the parameter itself. It is of course possible to replace the original parameter by a function of the estimated parameter everywhere is the model, but it is often unpractical. In such a case, it is possible to declare the parameter to be estimated in the parameters statement and to define the transformation, using a pound sign (#) expression (see Section 4.5 [Model declaration], page 16). Example parameters bet; model; Chapter 4: The Model file 44 # sig = 1/bet; c = sig*c(+1)*mpk; end; estimated_params; bet, normal_pdf, 1, 0.05; end; estimated_params_init ; [Block] This block declares numerical initial values for the optimizer when these ones are different from the prior mean. Each line has the following syntax: stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME , INITIAL_VALUE; See [estimated params], page 42, for the meaning and syntax of the various components. estimated_params_bounds ; [Block] This block declares lower and upper bounds for parameters in maximum likelihood estimation. Each line has the following syntax: stderr VARIABLE_NAME | corr VARIABLE_NAME_1, VARIABLE_NAME_2 | PARAMETER_NAME , LOWER_BOUND, UPPER_BOUND; See [estimated params], page 42, for the meaning and syntax of the various components. estimation [VARIABLE_NAME . . . ]; estimation (OPTIONS . . . ) [VARIABLE_NAME . . . ]; [Command] [Command] Description This command runs Bayesian or maximum likelihood estimation. The following information will be displayed by the command: • results from posterior optimization (also for maximum likelihood) • marginal log 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 Options datafile = FILENAME The datafile: a ‘.m’ file, a ‘.mat’ file or, a ‘.xls’ file (the latter format is supported under Octave if the io and java packages from Octave-Forge are installed, along with a Java Runtime Environment) xls_sheet = NAME The name of the sheet with the data in an Excel file xls_range = RANGE The range with the data in an Excel file nobs = INTEGER The number of observations to be used. Default: all observations in the file Chapter 4: The Model file 45 nobs = [INTEGER1:INTEGER2] Runs a recursive estimation and forecast for samples of size ranging of INTEGER1 to INTEGER2. Option forecast must also be specified first_obs = INTEGER The number of the first observation to be used. Default: 1 prefilter = INTEGER A value of 1 means that the estimation procedure will demean the data. Default: 0, i.e. 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 approximation. The data must correspond to the definition of the variables used in the model. Default: computes a linear approximation plot_priors = INTEGER Control the plotting of priors: 0 No prior plot 1 Prior density for each estimated parameter is plotted. It is important to check that the actual shape of prior densities matches what you have in mind. Ill choosen values for the prior standard density can result in absurd prior densities. Default value is 1. nograph See [nograph], page 35. nodisplay See [nodisplay], page 35. graph_format = FORMAT See [graph format], page 35. lik_init = INTEGER Type of initialization of Kalman filter: 1 For stationary models, the initial matrix of variance of the error of forecast is set equal to the unconditional variance of the state variables 2 For nonstationary models: a wide prior is used with an initial matrix of variance of the error of forecast diagonal with 10 on the diagonal 3 For nonstationary models: use a diffuse filter (use rather the diffuse_ filter option) 4 The filter is initialized with the fixed point of the Riccati equation Default value is 1. For advanced use only. lik_algo = INTEGER For internal use and testing only. conf_sig = DOUBLE See [conf sig], page 55. mh_replic = INTEGER Number of replications for Metropolis-Hastings algorithm. For the time being, mh_ replic should be larger than 1200. Default: 20000 Chapter 4: The Model file 46 sub_draws = INTEGER number of draws from the Metropolis iterations that are used to compute posterior distribution of various objects (smoothed variable, smoothed shocks, forecast, moments, IRF). sub_draws should be smaller than the total number of Metropolis draws available. Default: min(1200,0.25*Total number of draws) mh_nblocks = INTEGER Number of parallel chains for Metropolis-Hastings algorithm. Default: 2 mh_drop = DOUBLE The fraction of initially generated parameter vectors to be dropped before using posterior simulations. Default: 0.5 mh_jscale = DOUBLE The scale to be used for the jumping distribution in Metropolis-Hastings algorithm. The default value is rarely satisfactory. This option must be tuned to obtain, ideally, an acceptation 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 mh_recover Attempts to recover a Metropolis-Hastings simulation that crashed prematurely. Shouldn’t be used together with load_mh_file mh_mode = INTEGER ... 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_FILENAME_mode.mat’ mode_compute = INTEGER | FUNCTION_NAME Specifies the optimizer for the mode computation: 0 The mode isn’t computed. When mode_file option is specified, the mode is simply read from that file. When mode_file option is not specified, Dynare reports the value of the log posterior (log likelihood) evaluated at the initial value of the parameters. When mode_file option is not specified and there is no estimated_ params block, but the smoother option is used, it is a roundabout way to compute the smoothed value of the variables of a model with calibrated parameters. 1 Uses fmincon optimization routine (not available under Octave) 2 Value no longer used 3 Uses fminunc optimization routine 4 Uses Chris Sims’s csminwel 5 Uses Marco Ratto’s newrat. This value is not compatible with non linear filters or DSGE-VAR models 6 Uses a Monte-Carlo based optimization routine (see Dynare wiki for more details) Chapter 4: The Model file 47 7 Uses fminsearch, a simplex based optimization routine (available under MATLAB if the optimization toolbox is installed; available under Octave if the optim package from Octave-Forge is installed) 8 Uses Dynare implementation of the Nelder-Mead simplex based optimization routine (generally more efficient than the MATLAB or Octave implementation available with mode_compute=7) 9 Uses the CMA-ES (Covariance Matrix Adaptation Evolution Strategy) algorithm, an evolutionary algorithm for difficult non-linear non-convex optimization FUNCTION_NAME It is also possible to give a FUNCTION NAME to this option, instead of an INTEGER. In that case, Dynare takes the return value of that function as the posterior mode. Default value is 4. mode_check Tells Dynare to plot the posterior density for values around the computed mode for each estimated parameter in turn. This is helpful to diagnose problems with the optimizer 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 Tells Dynare to add to previous Metropolis-Hastings simulations instead of starting from scratch. Shouldn’t be used together with mh_recover optim = (fmincon options) Can be used to set options for fmincon, the optimizing function of MATLAB Optimization toolbox. Use MATLAB’s syntax for these options. Default: (’display’,’iter’,’LargeScale’,’off’,’MaxFunEvals’,100000,’TolFun’,1e8,’TolX’,1e-6) nodiagnostic Doesn’t compute the convergence diagnostics for Metropolis-Hastings. Default: diagnostics are computed and displayed bayesian_irf Triggers the computation of the posterior distribution of IRFs. The length of the IRFs are controlled by the irf option. Results are stored in oo_.PosteriorIRF.dsge (see below for a description of this variable) dsge_var Triggers the estimation of a DSGE-VAR model, where the weight of the DSGE prior of the VAR model will be estimated. The prior on the weight of the DSGE prior, dsge_prior_weight, must be defined in the estimated_params section. NB: The previous method of declaring dsge_prior_weight as a parameter and then placing it in estimated_params is now deprecated and will be removed in a future release of Dynare. dsge_var = DOUBLE Triggers the estimation of a DSGE-VAR model, where the weight of the DSGE prior of the VAR model is calibrated to the value passed. NB: The previous method of declaring dsge_prior_weight as a parameter and then calibrating it is now deprecated and will be removed in a future release of Dynare. Chapter 4: The Model file 48 dsge_varlag = INTEGER The number of lags used to estimate a DSGE-VAR model. Default: 4. moments_varendo Triggers the computation of the posterior distribution of the theoretical moments of the endogenous variables. Results are stored in oo_.PosteriorTheoreticalMoments (see below for a description of this variable) conditional_variance_decomposition = INTEGER See below. conditional_variance_decomposition = [INTEGER1:INTEGER2] See below. conditional_variance_decomposition = [INTEGER1 INTEGER2 ...] Computes the posterior distribution of the conditional variance decomposition for the specified period(s). The periods must be strictly positive. Conditional variances are given by var(yt+k |t). For period 1, the conditional variance decomposition provides the decomposition of the effects of shocks upon impact. The results are stored in oo_ .PosteriorTheoreticalMoments.dsge.ConditionalVarianceDecomposition, but currently there is not output. Note that this option requires the option moments_varendo to be specified. filtered_vars Triggers the computation of the posterior distribution of filtered endogenous variables and shocks. Results are stored in oo_.FilteredVariables (see below for a description of this variable) smoother Triggers the computation of the posterior distribution of smoothered endogenous variables and shocks. Results are stored in oo_.SmoothedVariables, oo_ .SmoothedShocks and oo_.SmoothedMeasurementErrors. Also triggers the computation of oo_.UpdatedVariables, which contains the estimation of the expected value of variables given the information available at the current date. See below for a description of all these variables. forecast = INTEGER Computes the posterior distribution of a forecast on INTEGER periods after the end of the sample used in estimation. If no Metropolis-Hastings is computed, the result is stored in variable oo_.forecast and corresponds to the forecast at the posterior mode. If a Metropolis-Hastings is computed, the distribution of forecasts is stored in variables oo_.PointForecast and oo_.MeanForecast. See Section 4.15 [Forecasting], page 54) for a description of these variables. 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) kalman_algo = INTEGER ... kalman_tol = DOUBLE ... filter_covariance Saves the series of one step ahead error of forecast covariance matrices. filter_step_ahead = [INTEGER1:INTEGER2] Triggers the computation k-step ahead filtered values. results in oo_.FilteredVariablesKStepAhead and .FilteredVariablesKStepAheadVariances. Stores oo_ Chapter 4: The Model file 49 filter_decomposition Triggers the computation of the shock decomposition of the above k-step ahead filtered values. constant ... noconstant ... diffuse_filter Uses the diffuse Kalman filter (as described in Durbin and Koopman (2001) and Koopman and Durbin (2003)) to estimate models with non-stationary observed variables. When diffused_filter is used the lik_init option of estimation has no effect. When there are nonstationary variables in a model, there is no unique deterministic steady state. The user must supply a MATLAB/Octave 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 ‘fs2000_steadystate.m’ in ‘examples’ directory for an example. Note that the nonstationary variables in the model must be integrated processes (their first difference or k-difference must be stationary). selected_variables_only Only run the smoother on the variables listed just after the estimation command. Default: run the smoother on all the declared endogenous variables. cova_compute = INTEGER When 0, the covariance matrix of estimated parameters is not computed after the computation of posterior mode (or maximum likelihood). This increases speed of computation in large models during development, when this information is not always necessary. Of course, it will break all successive computations that would require this covariance matrix. Default is 1. solve_algo = INTEGER See [solve algo], page 26. order = INTEGER Order of approximation, either 1 or 2. When equal to 2, the likelihood is evaluated with a particle filter based on a second order approximation of the model (see Fernandez-Villaverde and Rubio-Ramirez (2005)). Default is 1, ie the lilkelihood of the linearized model is evaluated using a standard Kalman filter. irf = INTEGER See [irf], page 35. Only used if [bayesian irf], page 47 is passed. irf_shocks = ( VARIABLE_NAME [[,] VARIABLE_NAME ...] ) See [irf shocks], page 35. Only used if [bayesian irf], page 47 is passed. Cannot be used with [dsge var], page 47. aim_solver See [aim solver], page 36. sylvester = OPTION See [sylvester], page 37. sylvester_fixed_point_tol = DOUBLE See [sylvester fixed point tol], page 37. Chapter 4: The Model file 50 lyapunov = OPTION Determines the algorithm used to solve the Laypunov equation to initialized the variance-covariance matrix of the Kalman filter using the steady-state value of state variables. Possible values for OPTION are: default Uses the default solver for Lyapunov equations based on Bartels-Stewart algorithm. fixed_point Uses a fixed point algorithm to solve the Lyapunov equation. This method is faster than the default one for large scale models, but it could require a large amount of iterations. doubling Uses a doubling algorithm to solve the Lyapunov equation (disclyap_ fast). This method is faster than the two previous one for large scale models. square_root_solver Uses a square-root solver for Lyapunov equations (dlyapchol). This method is fast for large scale models (available under MATLAB if the control system toolbox is installed; available under Octave if the control package from Octave-Forge is installed) Default value is default lyapunov_fixed_point_tol = DOUBLE This is the convergence criterion used in the fixed point lyapunov solver. Its default value is 1e-10. lyapunov_doubling_tol = DOUBLE This is the convergence criterion used in the doubling algorithm to solve the lyapunov equation. Its default value is 1e-16. analytic_derivation Triggers estimation with analytic gradient. The final hessian is also computed analytically. Only works for stationary models without missing observations. Note 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. Output After running estimation, the parameters M_.params and the variance matrix M_.Sigma_e of the shocks are set to the mode for maximum likelihood estimation or posterior mode computation without Metropolis iterations. After estimation with Metropolis iterations (option mh_replic > 0 or option load_mh_file set) the parameters M_.params and the variance matrix M_.Sigma_e of the shocks are set to the posterior mean. Depending on the options, estimation stores results in various fields of the oo_ structure, described below. Running the smoother with calibrated parameters It is possible to compute smoothed value of the endogenous variables and the shocks with calibrated parameters, without estimation proper. For this usage, there should be no estimated_ params block. Observed variables must be declared. A dataset must be specified in Chapter 4: The Model file 51 the estimation instruction. In addition, use the following options: mode_compute=0,mh_ replic=0,smoother. Currently, there is no specific output for this usage of the estimation command. The results are made available in fields of oo_ structure. An example is available in ‘./tests/smoother/calibrated_model.mod’. In the following variables, we will adopt the following shortcuts for specific field names: MOMENT NAME This field can take the following values: HPDinf Lower bound of a 90% HPD interval3 HPDsup Upper bound of a 90% HPD interval Mean Mean of the posterior distribution Median Median of the posterior distribution Std Standard deviation of the posterior distribution deciles Deciles of the distribution. density Non parametric estimate of the posterior density. First and second columns are respectively abscissa and ordinate coordinates. ESTIMATED OBJECT This field can take the following values: measurement_errors_corr Correlation between two measurement errors measurement_errors_std Standard deviation of measurement errors parameters Parameters shocks_corr Correlation between two structural shocks shocks_std Standard deviation of structural shocks oo_.MarginalDensity.LaplaceApproximation [MATLAB/Octave variable] Variable set by the estimation command. [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. oo_.MarginalDensity.ModifiedHarmonicMean [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the filtered_vars option. Fields are of the form: oo_.FilteredVariables.VARIABLE_NAME oo_.FilteredVariables [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the filter_step_ahead option. oo_.FilteredVariablesKStepAhead [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the filter_step_ahead option. oo_.FilteredVariablesKStepAheadVariances 3 See option [conf sig], page 55 to change the size of the HPD interval Chapter 4: The Model file 52 [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the bayesian_irf option. Fields are of the form: oo_.PosteriorIRF.dsge.MOMENT_NAME.VARIABLE_NAME_SHOCK_NAME oo_.PosteriorIRF.dsge [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the smoother option. Fields are of the form: oo_.SmoothedMeasurementErrors.VARIABLE_NAME oo_.SmoothedMeasurementErrors [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the smoother option. Fields are of the form: oo_.SmoothedShocks.VARIABLE_NAME oo_.SmoothedShocks [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the smoother option. Fields are of the form: oo_.SmoothedVariables.VARIABLE_NAME oo_.SmoothedVariables [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the smoother option. Contains the estimation of the expected value of variables given the information available at the current date. Fields are of the form: oo_.UpdatedVariables.VARIABLE_NAME oo_.UpdatedVariables [MATLAB/Octave variable] Variable set by the estimation command, if it is used with the moments_varendo option. Fields are of the form: oo_.PosteriorTheoreticalMoments.dsge.THEORETICAL_MOMENT.ESTIMATED_OBJECT.MOMENT_ NAME.VARIABLE_NAME where THEORETICAL MOMENT is one of the following: oo_.PosteriorTheoreticalMoments covariance Variance-covariance of endogenous variables correlation Correlation between endogenous variables VarianceDecomposition Decomposition of variance4 ConditionalVarianceDecomposition Only if the conditional_variance_decomposition option has been specified [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form: oo_.posterior_density.PARAMETER_NAME oo_.posterior_density [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form: oo_.posterior_hpdinf.ESTIMATED_OBJECT.VARIABLE_NAME oo_.posterior_hpdinf 4 When the shocks are correlated, it is the decomposition of orthogonalized shocks via Cholesky decompostion according to the order of declaration of shocks (see Section 4.2 [Variable declarations], page 9) Chapter 4: The Model file 53 [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form: oo_.posterior_hpdsup.ESTIMATED_OBJECT.VARIABLE_NAME oo_.posterior_hpdsup [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form: oo_.posterior_mean.ESTIMATED_OBJECT.VARIABLE_NAME oo_.posterior_mean [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form: oo_.posterior_mode.ESTIMATED_OBJECT.VARIABLE_NAME oo_.posterior_mode [MATLAB/Octave variable] Variable set by the estimation command, if it is used with mh_replic > 0 or load_mh_file option. Fields are of the form: oo_.posterior_std.ESTIMATED_OBJECT.VARIABLE_NAME oo_.posterior_std Here are some examples of generated variables: oo_.posterior_mode.parameters.alp oo_.posterior_mean.shocks_std.ex oo_.posterior_hpdsup.measurement_errors_corr.gdp_conso model_comparison FILENAME[(DOUBLE)] . . . ; model_comparison (marginal density = laplace | modifiedharmonicmean) FILENAME[(DOUBLE)] . . . ; [Command] [Command] Description This command computes odds ratios and estimate a posterior density over a collection of models. The priors over models can be specified as the DOUBLE values, otherwise a uniform prior is assumed. Example model_comparison my_model(0.7) alt_model(0.3); This example attributes a 70% prior over my_model and 30% prior over alt_model. shock_decomposition [VARIABLE_NAME] . . . ; shock_decomposition (OPTIONS . . . ) [VARIABLE_NAME] . . . ; [Command] [Command] Description This command computes and displays shock decomposition according to the model for a given sample. Note that this command must come after either estimation (in case of an estimated model) or stoch_simul (in case of a calibrated model). Options parameter_set = PARAMETER_SET Specify the parameter set to use for running the smoother. The PARAMETER SET can take one of the following five values: calibration, prior_ mode, prior_mean, posterior_mode, posterior_mean, posterior_median. Default value: posterior_mean if Metropolis has been run, else posterior_mode. Chapter 4: The Model file 54 shocks = (VARIABLE_NAME [VARIABLE_NAME ...] [ ; VARIABLE_NAME [VARIABLE_NAME ...] ...] ) ... labels = ( VARIABLE_NAME [VARIABLE_NAME ...] ) ... datafile = FILENAME See [datafile], page 44. Useful when computing the shock decomposition on a calibrated model. unit_root_vars VARIABLE_NAME . . . ; [Command] This command is deprecated. Use estimation option diffuse_filter instead for estimating a model with non-stationary observed variables or steady option nocheck to prevent steady to check the steady state returned by your steady state file. Dynare also has the ability to estimate Bayesian VARs: bvar_density ; [Command] Computes the marginal density of an estimated BVAR model, using Minnesota priors. See ‘bvar-a-la-sims.pdf’, which comes with Dynare distribution, for more information on this command. calib_smoother [VARIABLE_NAME] . . . ; calib_smoother (OPTIONS . . . ) [VARIABLE_NAME] . . . ; [Command] [Command] Description This command computes the smoothed variables (and possible the filtered variables) on a calibrated model. A datafile must be provided, and the observable variables declared with varobs. The smoother is based on a first-order approximation of the model. By default, the command computes the smoothed variables and shocks and stores the results in oo_.SmoothedVariables and oo_.SmoothedShocks. It also fills oo_.UpdatedVariables. Options datafile = FILENAME See [datafile], page 44. filtered_vars Triggers the computation of filtered variables. See [filtered vars], page 48 for more details. filter_step_ahead = [INTEGER1:INTEGER2] See [filter step ahead], page 48. 4.15 Forecasting On a calibrated model, forecasting is done using the forecast command. On an estimated model, use the forecast option of estimation command. It is also possible to compute forecasts on a calibrated or estimated model for a given constrained path of the future endogenous variables. This is done, from the reduced form representation of the DSGE model, by finding the structural shocks that are needed to match the restricted paths. Use conditional_forecast, conditional_forecast_paths and plot_conditional_forecast for that purpose. Finally, it is possible to do forecasting with a Bayesian VAR using the bvar_forecast command. Chapter 4: The Model file forecast [VARIABLE_NAME . . . ]; forecast (OPTIONS . . . ) [VARIABLE_NAME . . . ]; 55 [Command] [Command] Description This command computes a simulation of a stochastic model from an arbitrary initial point. When the model also contains deterministic exogenous shocks, the simulation is computed conditionaly to the agents knowing the future values of the deterministic exogenous variables. forecast must be called after stoch_simul. forecast plots the trajectory of endogenous variables. When a list of variable names follows the command, only those variables are plotted. A 90% confidence interval is plotted around the mean trajectory. Use option conf_sig to change the level of the confidence interval. Options periods = INTEGER Number of periods of the forecast. Default: 40 conf_sig = DOUBLE Level of significance for confidence interval. Default: 0.90 nograph See [nograph], page 35. nodisplay See [nodisplay], page 35. graph_format = FORMAT See [graph format], page 35. Initial Values forecast computes the forecast taking as initial values the values specified in histval (see Section 4.7 [Initial and terminal conditions], page 19). When no histval block is present, the initial values are the one stated in initval. When initval is followed by command steady, the initial values are the steady state (see Section 4.10 [Steady state], page 26). Output The results are stored in oo_.forecast, which is described below. Example varexo_det tau; varexo e; ... shocks; var e; stderr 0.01; var tau; periods 1:9; values -0.15; end; stoch_simul(irf=0); forecast; Chapter 4: The Model file 56 [MATLAB/Octave variable] Variable set by the forecast command, or by the estimation command if used with the forecast option and if no Metropolis-Hastings has been computed (in that case, the forecast is computed for the posterior mode). Fields are of the form: oo_.forecast.FORECAST_MOMENT.VARIABLE_NAME where FORECAST MOMENT is one of the following: oo_.forecast HPDinf Lower bound of a 90% HPD interval5 of forecast due to parameter uncertainty HPDsup Lower bound of a 90% HPD interval due to parameter uncertainty HPDTotalinf Lower bound of a 90% HPD interval of forecast due to parameter uncertainty and future shocks (only with the estimation command) HPDTotalsup Lower bound of a 90% HPD interval due to parameter uncertainty and future shocks (only with the estimation command) Mean Mean of the posterior distribution of forecasts Median Median of the posterior distribution of forecasts Std Standard deviation of the posterior distribution of forecasts [MATLAB/Octave variable] Set by the estimation command, if it is used with the forecast option and if either mh_replic > 0 or load_mh_file option is used. Contains the distribution of forecasts taking into account the uncertainty about both parameters and shocks. Fields are of the form: oo_.PointForecast.MOMENT_NAME.VARIABLE_NAME oo_.PointForecast [MATLAB/Octave variable] Set by the estimation command, if it is used with the forecast option and if either mh_replic > 0 or load_mh_file option is used. Contains the distribution of forecasts where the uncertainty about shocks is averaged out. The distribution of forecasts therefore only represents the uncertainty about parameters. Fields are of the form: oo_.MeanForecast.MOMENT_NAME.VARIABLE_NAME oo_.MeanForecast conditional_forecast (OPTIONS . . . ) [VARIABLE_NAME . . . ]; [Command] Description This command computes forecasts on an estimated model for a given constrained path of some future endogenous variables. This is done, from the reduced form representation of the DSGE model, by finding the structural shocks that are needed to match the restricted paths. This command has to be called after estimation. Use conditional_forecast_paths block to give the list of constrained endogenous, and their constrained future path. Option controlled_varexo is used to specify the structural shocks which will be matched to generate the constrained path. Use plot_conditional_forecast to graph the results. Options 5 See option [conf sig], page 55 to change the size of the HPD interval Chapter 4: The Model file 57 parameter_set = calibration | prior_mode | prior_mean | posterior_mode | posterior_mean | posterior_median Specify the parameter set to use for the forecasting. No default value, mandatory option. controlled_varexo = (VARIABLE_NAME...) Specify the exogenous variables to use as control variables. No default value, mandatory option. periods = INTEGER Number of periods of the forecast. Default: 40. periods cannot be less than the number of constrained periods. replic = INTEGER Number of simulations. Default: 5000. conf_sig = DOUBLE Level of significance for confidence interval. Default: 0.80 Example var y a varexo e u; ... estimation(...); conditional_forecast_paths; var y; periods 1:3, 4:5; values 2, 5; var a; periods 1:5; values 3; end; conditional_forecast(parameter_set = calibration, controlled_varexo = (e, u), replic = plot_conditional_forecast(periods = 10) a y; conditional_forecast_paths ; [Block] Describes the path of constrained endogenous, before calling conditional_forecast. The syntax is similar to deterministic shocks in shocks, see conditional_forecast for an example. The syntax of the block is the same than the deterministic shocks in the shocks blocks (see Section 4.8 [Shocks on exogenous variables], page 23). plot_conditional_forecast [VARIABLE_NAME . . . ]; plot_conditional_forecast (periods = INTEGER) [VARIABLE_NAME . . . ]; Description Plots the conditional (plain lines) and unconditional (dashed lines) forecasts. To be used after conditional_forecast. Options [Command] [Command] Chapter 4: The Model file 58 periods = INTEGER Number of periods to be plotted. Default: equal to periods in conditional_ forecast. The number of periods declared in plot_conditional_forecast cannot be greater than the one declared in conditional_forecast. bvar_forecast ; [Command] This command computes in-sample or out-sample forecasts for an estimated BVAR model, using Minnesota priors. See ‘bvar-a-la-sims.pdf’, which comes with Dynare distribution, for more information on this command. 4.16 Optimal policy Dynare has tools to compute optimal policies for various types of objectives. You can either solve for optimal policy under commitment with ramsey_policy, for optimal policy under discretion with discretionary_policy or for optimal simple rule with osr. osr [VARIABLE_NAME . . . ]; osr (OPTIONS . . . ) [VARIABLE_NAME . . . ]; [Command] [Command] Description This command computes optimal simple policy rules for linear-quadratic problems of the form: maxγ E(yt0 W yt ) such that: A1 Et yt+1 + A2 yt + A3 yt−1 + Cet = 0 where: • γ are parameters to be optimized. They must be elements of matrices A1 , A2 , A3 ; • y are the endogenous variables; • e are the exogenous stochastic shocks; The parameters to be optimized must be listed with osr_params. The quadratic objectives must be listed with optim_weights. This problem is solved using a numerical optimizer. Options This command accept the same options than stoch_simul (see Section 4.13.1 [Computing the stochastic solution], page 34). The value of the objective is stored in the variable oo_.osr.objective_function, which is described below. osr_params PARAMETER_NAME . . . ; [Command] This command declares parameters to be optimized by osr. optim_weights ; [Block] This block specifies quadratic objectives for optimal policy problems More precisely, this block specifies the nonzero elements of the quadratic weight matrices for the objectives in osr. A element of the diagonal of the weight matrix is given by a line of the form: VARIABLE_NAME EXPRESSION; An off-the-diagonal element of the weight matrix is given by a line of the form: VARIABLE_NAME, VARIABLE_NAME EXPRESSION; Chapter 4: The Model file 59 [MATLAB/Octave variable] After an execution of the osr command, this variable contains the value of the objective under optimal policy. oo_.osr.objective_function ramsey_policy [VARIABLE_NAME . . . ]; ramsey_policy (OPTIONS . . . ) [VARIABLE_NAME . . . ]; [Command] [Command] Description This command computes the first order approximation of the policy that maximizes the policy maker objective function submitted to the constraints provided by the equilibrium path of the economy. The planner objective must be declared with the planner_objective command. See Section 4.6 [Auxiliary variables], page 19, for an explanation of how this operator is handled internally and how this affects the output. Options This command accepts all options of stoch_simul, plus: planner_discount = EXPRESSION Declares the discount factor of the central planner. Default: 1.0 instruments = (VARIABLE_NAME,...) Declares instrument variables for the computation of the steady state under optimal policy. Requires a steady_state_model block or a ..._steadystate.m file. See below. Note that only first order approximation is available (i.e. order=1 must be specified). Output This command generates all the output variables of stoch_simul. In addition, it stores the value of planner objective function under Ramsey policy in oo_ .planner_objective_value. Steay state Dynare takes advantage of the fact that the Lagrange multipliers appear linearly in the equations of the steady state of the model under optimal policy. Nevertheless, it is in general very difficult to compute the steady state with simply a numerical guess in initval for the endogenous variables. It greatly facilitates the computation, if the user provides an analytical solution for the steady state (in steady_state_model block or in a ..._steadystate.m file). In this case, it is necessary to provide a steady state solution CONDITIONAL on the value of the instruments in the optimal policy problem and declared with option instruments. Note that choosing the instruments is partly a matter of interpretation and you can choose instruments that are handy from a mathematical point of view but different from the instruments you would refer to in the analysis of the paper. Typical example is choosing inflation or nominal interest rate as an instrument. discretionary_policy [VARIABLE_NAME . . . ]; discretionary_policy (OPTIONS . . . ) [VARIABLE_NAME . . . ]; [Command] [Command] Description This command computes an approximation of the optimal policy under discretion. The algorithm implemented is essentially an LQ solver, and is described by Dennis (2007). Chapter 4: The Model file 60 Options This command accepts the same options than ramsey_policy, plus: discretionary_tol = NON-NEGATIVE DOUBLE Sets the tolerance level used to assess convergence of the solution algorithm. Default: 1e-7. planner_objective MODEL_EXPRESSION; [Command] This command declares the policy maker objective, for use with ramsey_policy or discretionary_policy. You need to give the one-period objective, not the discounted lifetime objective. The discount factor is given by the planner_discount option of ramsey_policy and discretionary_policy. Note that with this command you are not limited to quadratic objectives: you can give any arbitrary nonlinear expression. 4.17 Sensitivity and identification analysis Dynare provides an interface to the global sensitivity analysis (GSA) toolbox (developed by the Joint Research Center (JRC) of the European Commission), which is now part of the official Dynare distribution. The GSA toolbox can be used to answer the following questions: 1. What is the domain of structural coefficients assuring the stability and determinacy of a DSGE model? 2. Which parameters mostly drive the fit of, e.g., GDP and which the fit of inflation? Is there any conflict between the optimal fit of one observed series versus another? 3. How to represent in a direct, albeit approximated, form the relationship between structural parameters and the reduced form of a rational expectations model? The discussion of the methodologies and their application is described in Ratto (2008). With respect to the previous version of the toolbox, in order to work properly, the GSA toolbox no longer requires that the Dynare estimation environment is setup. Sensitivity analysis results are saved locally in <mod_file>/GSA, where <mod_file>.mod is the name of the DYNARE model file. 4.17.1 Sampling The following binary files are produced: • <mod_file>_prior.mat: this file stores information about the analyses performed sampling from the prior ranges, i.e. pprior=1 and ppost=0; • <mod_file>_mc.mat: this file stores information about the analyses performed sampling from multivariate normal, i.e. pprior=0 and ppost=0; • <mod_file>_post.mat: this file stores information about analyses performed using the Metropolis posterior sample, i.e. ppost=1. 4.17.2 Stability Mapping Figure files produced are of the form <mod_file>_prior_*.fig and store results for stability mapping from prior Monte-Carlo samples: • <mod_file>_prior_stab_SA_*.fig: plots of the Smirnov test analyses confronting the cdf of the sample fulfilling Blanchard-Kahn conditions with the cdf of the rest of the sample; • <mod_file>_prior_stab_indet_SA_*.fig: plots of the Smirnov test analyses confronting the cdf of the sample producing indeterminacy with the cdf of the original prior sample; • <mod_file>_prior_stab_unst_SA_*.fig: plots of the Smirnov test analyses confronting the cdf of the sample producing unstable (explosive roots) behavior with the cdf of the original prior sample; Chapter 4: The Model file 61 • <mod_file>_prior_stable_corr_*.fig: plots of bivariate projections of the sample fulfilling Blanchard-Kahn conditions; • <mod_file>_prior_indeterm_corr_*.fig: plots of bivariate projections of the sample producing indeterminacy; • <mod_file>_prior_unstable_corr_*.fig: plots of bivariate projections of the sample producing instability; • <mod_file>_prior_unacceptable_corr_*.fig: plots of bivariate projections of the sample producing unacceptable solutions, i.e. either instability or indeterminacy or the solution could not be found (e.g. the steady state solution could not be found by the solver). Similar conventions apply for <mod_file>_mc_*.fig files, obtained when samples from multivariate normal are used. 4.17.3 Reduced Form Mapping The mapping of the reduced form solution forces the use of samples from prior ranges or prior distributions, i.e.: pprior=1 and ppost=0. It uses 250 samples to optimize smoothing parameters and 1000 samples to compute the fit. The rest of the sample is used for out-of-sample validation. One can also load a previously estimated mapping with a new Monte-Carlo sample, to look at the forecast for the new Monte-Carlo sample. The following synthetic figures are produced: • <mod_file>_redform_<endo name>_vs_lags_*.fig: shows bar charts of the sensitivity indices for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus lagged endogenous variables (namlagendo); suffix log indicates the results for log-transformed entries; • <mod_file>_redform_<endo name>_vs_shocks_*.fig: shows bar charts of the sensitivity indices for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus exogenous variables (namexo); suffix log indicates the results for log-transformed entries; • <mod_file>_redform_GSA(_log).fig: shows bar chart of all sensitivity indices for each parameter: this allows one to notice parameters that have a minor effect for any of the reduced form coefficients. Detailed results of the analyses are shown in the subfolder <mod_file>/GSA/redform_stab, where the detailed results of the estimation of the single functional relationships between parameters θ and reduced form coefficient are stored in separate directories named as: • <namendo>_vs_<namlagendo>: for the entries of the transition matrix; • <namendo>_vs_<namexo>: for entries of the matrix of the shocks. Moreover, analyses for log-transformed entries are denoted with the following suffixes (y denotes the generic reduced form coefficient): • log: y ∗ = log(y); • minuslog: y ∗ = log(−y); • logsquared: y ∗ = log(y 2 ) for symmetric fat tails; • logskew: y ∗ = log(|y + λ|) for asymmetric fat tails. The optimal type of transformation is automatically selected without the need of user intervention. 4.17.4 RMSE The RMSE analysis can be performed with different types of sampling options: Chapter 4: The Model file 62 1. When pprior=1 and ppost=0, the toolbox analyzes the RMSEs for the Monte-Carlo sample obtained by sampling parameters from their prior distributions (or prior ranges): this analysis provides some hints about what parameter drives the fit of which observed series, prior to the full estimation; 2. When pprior=0 and ppost=0, the toolbox analyzes the RMSEs for a multivariate normal Monte-Carlo sample, with covariance matrix based on the inverse Hessian at the optimum: this analysis is useful when maximum likelihood estimation is done (i.e. no Bayesian estimation); 3. When ppost=1 the toolbox analyzes the RMSEs for the posterior sample obtained by Dynare’s Metropolis procedure. The use of cases 2 and 3 requires an estimation step beforehand. To facilitate the sensitivity analysis after estimation, the dynare_sensitivity command also allows you to indicate some options of the estimation command. These are: • datafile • nobs • first_obs • prefilter • presample • nograph • nodisplay • graph_format • conf_sig • loglinear • mode_file Binary files produced my RMSE analysis are: • <mod_file>_prior_*.mat: these files store the filtered and smoothed variables for the prior Monte-Carlo sample, generated when doing RMSE analysis (pprior=1 and ppost=0); • <mode_file>_mc_*.mat: these files store the filtered and smoothed variables for the multivariate normal Monte-Carlo sample, generated when doing RMSE analysis (pprior=0 and ppost=0). Figure files <mod_file>_rmse_*.fig store results for the RMSE analysis. • <mod_file>_rmse_prior*.fig: save results for the analysis using prior Monte-Carlo samples; • <mod_file>_rmse_mc*.fig: save results for the analysis using multivariate normal MonteCarlo samples; • <mod_file>_rmse_post*.fig: save results for the analysis using Metropolis posterior samples. The following types of figures are saved (we show prior sample to fix ideas, but the same conventions are used for multivariate normal and posterior): • <mod_file>_rmse_prior_*.fig: for each parameter, plots the cdfs corresponding to the best 10% RMSEs of each observed series; • <mod_file>_rmse_prior_dens_*.fig: for each parameter, plots the pdfs corresponding to the best 10% RMESs of each observed series; • <mod_file>_rmse_prior_<name of observedseries>_corr_*.fig: for each observed series plots the bi-dimensional projections of samples with the best 10% RMSEs, when the correlation is significant; • <mod_file>_rmse_prior_lnlik*.fig: for each observed series, plots in red the cdf of the log-likelihood corresponding to the best 10% RMSEs, in green the cdf of the rest of the sample and in blue the cdf of the full sample; this allows one to see the presence of some idiosyncratic behavior; Chapter 4: The Model file 63 • <mod_file>_rmse_prior_lnpost*.fig: for each observed series, plots in red the cdf of the log-posterior corresponding to the best 10% RMSEs, in green the cdf of the rest of the sample and in blue the cdf of the full sample; this allows one to see idiosyncratic behavior; • <mod_file>_rmse_prior_lnprior*.fig: for each observed series, plots in red the cdf of the log-prior corresponding to the best 10% RMSEs, in green the cdf of the rest of the sample and in blue the cdf of the full sample; this allows one to see idiosyncratic behavior; • <mod_file>_rmse_prior_lik_SA_*.fig: when lik_only=1, this shows the Smirnov tests for the filtering of the best 10% log-likelihood values; • <mod_file>_rmse_prior_post_SA_*.fig: when lik_only=1, this shows the Smirnov test for the filtering of the best 10% log-posterior values. 4.17.5 Screening Analysis Screening analysis does not require any additional options with respect to those listed in [Sampling Options], page 64. The toolbox performs all the analyses required and displays results. The results of the screening analysis with Morris sampling design are stored in the subfolder <mod_file>/GSA/SCREEN. The data file <mod_file>_prior stores all the information of the analysis (Morris sample, reduced form coefficients, etc.). Screening analysis merely concerns reduced form coefficients. Similar synthetic bar charts as for the reduced form analysis with Monte-Carlo samples are saved: • <mod_file>_redform_<endo name>_vs_lags_*.fig: shows bar charts of the elementary effect tests for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus lagged endogenous variables (namlagendo); • <mod_file>_redform_<endo name>_vs_shocks_*.fig: shows bar charts of the elementary effect tests for the ten most important parameters driving the reduced form coefficients of the selected endogenous variables (namendo) versus exogenous variables (namexo); • <mod_file>_redform_screen.fig: shows bar chart of all elementary effect tests for each parameter: this allows one to identify parameters that have a minor effect for any of the reduced form coefficients. 4.17.6 Identification Analysis Setting the option identification=1, an identification analysis based on theoretical moments is performed. Sensitivity plots are provided that allow to infer which parameters are most likely to be less identifiable. Prerequisite for properly running all the identification routines, is the keyword identification; in the Dynare model file. This keyword triggers the computation of analytic derivatives of the model with respect to estimated parameters and shocks. This is required for option morris=2, which implements Iskrev (2010) identification analysis. For example, the placing identification; dynare_sensitivity(identification=1, morris=2); in the Dynare model file trigger identification analysis using analytic derivatives Iskrev (2010), jointly with the mapping of the acceptable region. The identification analysis with derivatives can also be triggered by the commands identification; This does not do the mapping of acceptable regions for the model and uses the standard random sampler of Dynare. It completely offsets any use of the sensitivity analysis toolbox. 4.17.7 Performing Sensitivity and Identification Analysis dynare_sensitivity ; dynare_sensitivity (OPTIONS . . . ); Description [Command] [Command] Chapter 4: The Model file 64 This command triggers sensitivity analysis on a DSGE model. Options Sampling Options nsam = INTEGER Size of the Monte-Carlo sample. Default: 2048 ilptau = INTEGER If equal to 1, use LPτ quasi-Monte-Carlo. If equal to 0, use LHS Monte-Carlo. Default: 1 pprior = INTEGER If equal to 1, sample from the prior distributions. If equal to 0, sample from the multivariate normal N (θ̄, Σ), where θ̄ is the posterior mode and Σ = H −1 , H is the Hessian at the mode. Default: 1 prior_range = INTEGER If equal to 1, sample uniformly from prior ranges. If equal to 0, sample from prior distributions. Default: 1 morris = INTEGER If equal to 0, ANOVA mapping (Type I error) If equal to 1, Screening analysis (Type II error) If equal to 2, Analytic derivatives (similar to Type II error, only valid when identification=1).Default: 1 when identification=1, 0 otherwise morris_nliv = INTEGER Number of levels in Morris design. Default: 6 morris_ntra = INTEGER Number trajectories in Morris design. Default: 20 ppost = INTEGER If equal to 1, use Metropolis posterior sample. If equal to 0, do not use Metropolis posterior sample. NB: This overrides any other sampling option. Default: 0 neighborhood_width = DOUBLE When pprior=0 and ppost=0, allows for the sampling eters around the value specified in the mode_file, in xparam1± |xparam1 × neighborhood_width|. Default: 0 of paramthe range Stability Mapping Options stab = INTEGER If equal to 1, perform stability mapping. If equal to 0, do not perform stability mapping. Default: 1 load_stab = INTEGER If equal to 1, load a previously created sample. If equal to 0, generate a new sample. Default: 0 alpha2_stab = DOUBLE Critical value for correlations ρ in filtered samples: plot couples of parmaters with |ρ| > alpha2_stab. Default: 0.3 ksstat = DOUBLE Critical value for Smirnov statistics d: plot parameters with d > ksstat. Default: 0.1 Chapter 4: The Model file 65 pvalue_ks = DOUBLE The threshold pvalue for significant Kolmogorov-Smirnov test (i.e. plot parameters with pvalue < pvalue_ks). Default: 0.001 pvalue_corr = DOUBLE The threshold pvalue for significant correlation in filtered samples (i.e. plot bivariate samples when pvalue < pvalue_corr). Default: 0.001 Reduced Form Mapping Options redform = INTEGER If equal to 1, prepare Monte-Carlo sample of reduced form matrices. If equal to 0, do not prepare Monte-Carlo sample of reduced form matrices. Default: 0 load_redform = INTEGER If equal to 1, load previously estimated mapping. If equal to 0, estimate the mapping of the reduced form model. Default: 0 logtrans_redform = INTEGER If equal to 1, use log-transformed entries. If equal to 0, use raw entries. Default: 0 threshold_redform = [DOUBLE DOUBLE] The range over which the filtered Monte-Carlo entries of the reduced form coefficients should be analyzed. The first number is the lower bound and the second is the upper bound. An empty vector indicates that these entries will not be filtered. Default: empty ksstat_redform = DOUBLE Critical value for Smirnov statistics d when reduced form entries are filtered. Default: 0.1 alpha2_redform = DOUBLE Critical value for correlations ρ when reduced form entries are filtered. Default: 0.3 namendo = (VARIABLE_NAME...) List of endogenous variables. ‘:’ indicates all endogenous variables. Default: empty namlagendo = (VARIABLE_NAME...) List of lagged endogenous variables. ‘:’ indicates all lagged endogenous variables. Analyze entries [namendo×namlagendo] Default: empty namexo = (VARIABLE_NAME...) List of exogenous variables. ‘:’ indicates all exogenous variables. Analyze entries [namendo×namexo]. Default: empty RMSE Options rmse = INTEGER If equal to 1, perform RMSE analysis. If equal to 0, do not perform RMSE analysis. Default: 0 load_rmse = INTEGER If equal to 1, load previous RMSE analysis. If equal to 0, make a new RMSE analysis. Default: 0 lik_only = INTEGER If equal to 1, compute only likelihood and posterior. If equal to 0, compute RMSE’s for all observed series. Default: 0 Chapter 4: The Model file 66 var_rmse = (VARIABLE_NAME...) List of observed series to be considered. ‘:’ indicates all observed variables. Default: varobs pfilt_rmse = DOUBLE Filtering threshold for RMSE’s. Default: 0.1 istart_rmse = INTEGER Value at which to start computing RMSE’s (use 2 to avoid big intitial error). Default: presample+1 alpha_rmse = DOUBLE Critical value for Smirnov statistics d: plot parameters with d > alpha_rmse. Default: 0.002 alpha2_rmse = DOUBLE Critical value for correlation ρ: plot couples of parmaters with |ρ| = alpha2_rmse. Default: 1.0 datafile = FILENAME See [datafile], page 44. nobs = INTEGER nobs = [INTEGER1:INTEGER2] See [nobs], page 44. first_obs = INTEGER See [first obs], page 45. prefilter = INTEGER See [prefilter], page 45. presample = INTEGER See [presample], page 45. nograph See [nograph], page 35. nodisplay See [nodisplay], page 35. graph_format = FORMAT See [graph format], page 35. conf_sig = DOUBLE See [conf sig], page 55. loglinear See [loglinear], page 45. mode_file = FILENAME See [mode file], page 46. kalman_algo = INTEGER See [kalman algo], page 48. Identification Analysis Options identification = INTEGER If equal to 1, performs identification anlysis (forcing redform=0 and morris=1) If equal to 0, no identification analysis. Default: 0 morris = INTEGER See [morris], page 64. Chapter 4: The Model file 67 morris_nliv = INTEGER See [morris nliv], page 64. morris_ntra = INTEGER See [morris ntra], page 64. load_ident_files = INTEGER Loads previously performed identification analysis. Default: 0 useautocorr = INTEGER Use autocorrelation matrices in place of autocovariance matrices in moments for identification analysis. Default: 0 ar = INTEGER Maximum number of lags for moments in identification analysis. Default: 1 lik_init = INTEGER See [lik init], page 45. identification ; identification (OPTIONS . . . ); [Command] [Command] Description This command triggers identification analysis. Options ar = INTEGER Number of lags of computed autocorrelations (theoretical moments). Default: 1 useautocorr = INTEGER If equal to 1, compute derivatives of autocorrelation. If equal to 0, compute derivatives of autocovariances. Default: 0 load_ident_files = INTEGER If equal to 1, allow Dynare to load previously computed analyzes. Default: 0 prior_mc = INTEGER Size of Monte-Carlo sample. Default: 1 prior_range = INTEGER Triggers uniform sample within the range implied by the prior specifications (when prior_mc>1). Default: 0 advanced = INTEGER Shows a more detailed analysis, comprised of an analysis for the linearized rational expectation model as well as the associated reduced form solution. Further performs a brute force search of the groups of parameters best reproducing the behavior of each single parameter. The maximum dimension of the group searched is triggered by max_dim_cova_group. Default: 0 max_dim_cova_group = INTEGER In the brute force search (performed when advanced=1) this option sets the maximum dimension of groups of parameters that best reproduce the behavior of each single model parameter. Default: 2 periods = INTEGER When the analytic Hessian is not available (i.e. with missing values or diffuse Kalman filter or univariate Kalman filter), this triggers the length of stochastic simulation to compute Simulated Moments Uncertainty. Default: 300 Chapter 4: The Model file 68 replic = INTEGER When the analytic Hessian is not available, this triggers the number of replicas to compute Simulated Moments Uncertainty. Default: 100 gsa_sample_file = INTEGER If equal to 0, do not use sample file. If equal to 1, triggers gsa prior sample. If equal to 2, triggers gsa Monte-Carlo sample (i.e. loads a sample corresponding to pprior=0 and ppost=0 in the dynare_sensitivity options). Default: 0 gsa_sample_file = FILENAME Uses the provided path to a specific user defined sample file. Default: 0 parameter_set = calibration | prior_mode | prior_mean | posterior_mode | posterior_mean | posterior_median Specify the parameter set to use. Default: prior_mean lik_init = INTEGER See [lik init], page 45. kalman_algo = INTEGER See [kalman algo], page 48. nograph See [nograph], page 35. nodisplay See [nodisplay], page 35. graph_format = FORMAT See [graph format], page 35. 4.18 Markov-switching SBVAR Given a list of variables, observed variables and a data file, Dynare can be used to solve a Markovswitching SBVAR model according to Sims, Waggoner and Zha (2008). Having done this, you can create forecasts and compute the marginal data density, regime probabilities, IRFs, and variance decomposition of the model. The commands have been modularized, allowing for multiple calls to the same command within a <mod_file>.mod file. The default is to use <mod_file> to tag the input (output) files used (produced) by the program. Thus, to call any command more than once within a <mod_file>.mod file, you must use the *_tag options described below. markov_switching (OPTIONS . . . ); [Command] Description Declares the Markov state variable information of a Markov-switching SBVAR model. Options chain = INTEGER The Markov chain. Default: none state = INTEGER This state has duration equal to duration. Exactly one of state and number_of_ states must be passed. Default: none number_of_states = INTEGER Total number of states. Implies that all states have the same duration. Exactly one of state and number_of_states must be passed. Default: none duration = DOUBLE | inf The duration of the state or states. Default: none Chapter 4: The Model file svar (OPTIONS . . . ); 69 [Command] Description Each Makov chain can control the switching of a set of parameters. We allow the parameters to be divided equation by equation and by variance or slope and intercept. Options coefficients Specifies that only the slope and intercept in the given equations are controlled by the given chain. One, but not both, of coefficients or variances must appear. Default: none variances Specifies that only variances in the given equations are controlled by the given chain. One, but not both, of coefficients or variances must appear. Default: none equations Defines the equation controlled by the given chain. If not specificed, then all equations are controlled by chain. Default: none chain = INTEGER Specifies a Markov chain defined by [markov switching], page 68. Default: none ms_estimation (OPTIONS . . . ); [Command] Description Triggers the creation of an initialization file for, and the estimation of, a Markov-switching SBVAR model. At the end of the run, the A0 , A+ , Q and ζ matrices are contained in the oo_.ms structure. Options General Options file_tag = FILENAME The portion of the filename associated with this run. This will create the model initialization file, init_<file_tag>.dat. Default: <mod_file> output_file_tag = FILENAME The portion of the output filename that will be assigned to this run. This will create, among other files, est_final_<output_file_tag>.out, est_intermediate_ <output_file_tag>.out. Default: <file_tag> no_create_init Do not create an initialization file for the model. Passing this option will cause the Initialization Options to be ignored. Further, the model will be generated from the output files associated with the previous estimation run (i.e. est_final_ <file_tag>.out, est_intermediate_<file_tag>.out or init_<file_tag>.dat, searched for in sequential order). This functionality can be useful for continuing a previous estimation run to ensure convergence was reached or for reusing an initialization file. NB: If this option is not passed, the files from the previous estimation run will be overwritten. Default: off (i.e. create initialization file) Initialization Options Chapter 4: The Model file 70 coefficients_prior_hyperparameters = [DOUBLE1 DOUBLE2 DOUBLE3 DOUBLE4 DOUBLE5 DOUBLE6] Sets the hyper parameters for the model. The six elements of the argument vector have the following interpretations: Position Interpretation 1 Overall tightness for A0 and A+ 2 Relative tightness for A+ 3 Relative tightness for the constant term 4 Tightness on lag decay (range: 1.2 - 1.5); a faster decay produces better inflation process 5 Weight on nvar sums of coeffs dummy observations (unit roots) 6 Weight on single dummy initial observation including constant Default: [1.0 1.0 0.1 1.2 1.0 1.0] freq = INTEGER | monthly | quarterly | yearly Frequency of the data (e.g. monthly, 12). Default: 4 initial_year = INTEGER The first year of data. Default: none initial_subperiod = INTEGER The first period of data (i.e. for quarterly data, an integer in [1,4]). Default: 1 final_year = INTEGER The last year of data. Default: none final_subperiod = INTEGER The final period of data (i.e. for monthly data, an integer in [1,12]. Default: 4 datafile = FILENAME See [datafile], page 44. xls_sheet = NAME See [xls sheet], page 44. xls_range = RANGE See [xls range], page 44. nlags = INTEGER The number of lags in the model. Default: 1 cross_restrictions Use cross A0 and A+ restrictions. Default: off contemp_reduced_form Use contemporaneous recursive reduced form. Default: off no_bayesian_prior Do not use bayesian prior. Default: off (i.e. use bayesian prior) alpha = INTEGER Alpha value for squared time-varying structural shock lambda. Default: 1 beta = INTEGER Beta value for squared time-varying structural shock lambda. Default: 1 gsig2_lmdm = INTEGER The variance for each independent λ parameter under SimsZha restrictions. Default: 50^2 Chapter 4: The Model file 71 specification = sims_zha | none This controls how restrictions are imposed to reduce the number of parameters. Default: Random Walk Estimation Options convergence_starting_value = DOUBLE This is the tolerance criterion for convergence and refers to changes in the objective function value. It should be rather loose since it will gradually be tighened during estimation. Default: 1e-3 convergence_ending_value = DOUBLE The convergence criterion ending value. Values much smaller than square root machine epsilon are probably overkill. Default: 1e-6 convergence_increment_value = DOUBLE Determines how quickly the convergence criterion moves from the starting value to the ending value. Default: 0.1 max_iterations_starting_value = INTEGER This is the maximum number of iterations allowed in the hill-climbing optimization routine and should be rather small since it will gradually be increased during estimation. Default: 50 max_iterations_increment_value = DOUBLE Determines how quickly the maximum number of iterations is increased. Default: 2 max_block_iterations = INTEGER The parameters are divided into blocks and optimization proceeds over each block. After a set of blockwise optimizations are performed, the convergence criterion is checked and the blockwise optimizations are repeated if the criterion is violated. This controls the maximum number of times the blockwise optimization can be performed. Note that after the blockwise optimizations have converged, a single optimization over all the parameters is performed before updating the convergence value and maximum number of iterations. Default: 100 max_repeated_optimization_runs = INTEGER The entire process described by [max block iterations], page 71 is repeated until improvement has stopped. This is the maximum number of times the process is allowed to repeat. Set this to 0 to not allow repetitions. Default: 10 function_convergence_criterion = DOUBLE The convergence criterion for the objective function when max_repeated_ optimizations_runs is positive. Default: 0.1 parameter_convergence_criterion = DOUBLE The convergence criterion for parameter optimizations_runs is positive. Default: 0.1 values when max_repeated_ number_of_large_perturbations = INTEGER The entire process described by [max block iterations], page 71 is repeated with random starting values drawn from the posterior. This specifies the number of random starting values used. Set this to 0 to not use random starting values. A larger number should be specified to ensure that the entire parameter space has been covererd. Default: 5 number_of_small_perturbations = INTEGER The number of small perturbations to make after the large perturbations have stopped improving. Setting this number much above 10 is probably overkill. Default: 5 Chapter 4: The Model file 72 number_of_posterior_draws_after_perturbation = INTEGER The number of consecutive posterior draws to make when producing a small perturbation. Because the posterior draws are serially correlated, a small number will result in a small perturbation. Default: 1 max_number_of_stages = INTEGER The small and large perturbation are repeated until improvement has stopped. This specifices the maximum number of stages allowed. Default: 20 random_function_convergence_criterion = DOUBLE The convergence criterion for the objective function when number_of_large_ perturbations is positive. Default: 0.1 random_parameter_convergence_criterion = DOUBLE The convergence criterion for parameter values when number_of_large_ perturbations is positive. Default: 0.1 Example ms_estimation(datafile=data, initial_year=1959, final_year=2005, nlags=4, max_repeated_optimization_runs=1, max_number_of_stages=0); ms_estimation(file_tag=second_run, datafile=data, initial_year=1959, final_year=2005, nlags=4, max_repeated_optimization_runs=1, max_number_of_stages=0); ms_estimation(file_tag=second_run, output_file_tag=third_run, no_create_init, max_repeated_optimization_runs=5, number_of_large_perturbations=10); ms_simulation ; ms_simulation (OPTIONS . . . ); [Command] [Command] Description Simulates a Markov-switching SBVAR model. Options file_tag = FILENAME The portion of the filename associated with the ms_estimation run. Default: <mod_ file> output_file_tag = FILENAME The portion of the output filename that will be assigned to this run. Default: <file_tag> mh_replic = INTEGER The number of draws to save. Default: 10,000 drop = INTEGER The number of burn-in draws. Default: 0.1*mh_replic*thinning_factor thinning_factor = INTEGER The total number of draws is equal to thinning_factor*mh_replic+drop. Default: 1 adaptive_mh_draws = INTEGER Tuning period for Metropolis-Hasting draws. Default: 30,000 Chapter 4: The Model file 73 Example ms_simulation(file_tag=second_run); ms_simulation(file_tag=third_run, mh_replic=5000, thinning_factor=3); ms_compute_mdd ; ms_compute_mdd (OPTIONS . . . ); [Command] [Command] Description Computes the marginal data density of a Markov-switching SBVAR model from the posterior draws. At the end of the run, the Muller and Bridged log marginal densities are contained in the oo_.ms structure. Options file_tag = FILENAME See [file tag], page 72. output_file_tag = FILENAME See [output file tag], page 72. simulation_file_tag = FILENAME The portion of the filename associated with the simulation run. Defualt: <file_ tag> proposal_type = INTEGER The proposal type: 1 Gaussian 2 Power 3 Truncated Power 4 Step 5 Truncated Gaussian Default: 3 proposal_lower_bound = DOUBLE The lower cutoff in terms of probability. Not used for proposal_type in [1,2]. Required for all other proposal types. Default: 0.1 proposal_upper_bound = DOUBLE The upper cutoff in terms of probability. Not used for proposal_type equal to 1. Required for all other proposal types. Default: 0.9 mdd_proposal_draws = INTEGER The number of proposal draws. Default: 100,000 mdd_use_mean_center Use the posterior mean as center. Default: off ms_compute_probabilities ; ms_compute_probabilities (OPTIONS . . . ); [Command] [Command] Description Computes smoothed regime probabilities of a Markov-switching SBVAR model. Output .eps files are contained in <output_file_tag/Output/Probabilities>. Options Chapter 4: The Model file 74 file_tag = FILENAME See [file tag], page 72. output_file_tag = FILENAME See [output file tag], page 72. filtered_probabilities Filtered probabilities are computed instead of smoothed. Default: off real_time_smoothed Smoothed probabilities are computed based on time t information for 0 ≤ t ≤ nobs. Default: off ms_irf ; ms_irf (OPTIONS . . . ); [Command] [Command] Description Computes impulse response functions for a Markov-switching SBVAR model. Output .eps files are contained in <output_file_tag/Output/IRF>, while data files are contained in <output_ file_tag/IRF>. Options file_tag = FILENAME See [file tag], page 72. output_file_tag = FILENAME See [output file tag], page 72. simulation_file_tag = FILENAME See [simulation file tag], page 73. horizon = INTEGER The forecast horizon. Default: 12 filtered_probabilities Uses filtered probabilities at the end of the sample as initial conditions for regime probabilities. Only one of filtered_probabilities, regime and regimes may be passed. Default: off error_band_percentiles = [DOUBLE1 ...] The percentiles to compute. Default: [0.16 0.50 0.84]. If median is passed, the default is [0.5] shock_draws = INTEGER The number of regime paths to draw. Default: 10,000 shocks_per_parameter = INTEGER The number of regime paths to draw under parameter uncertainty. Default: 10 thinning_factor = INTEGER Only 1/thinning_factor of the draws in posterior draws file are used. Default: 1 free_parameters = NUMERICAL_VECTOR A vector of free parameters to initialize theta of the model. Default: use estimated parameters parameter_uncertainty Calculate IRFs under parameter uncertainty. Requires that ms_simulation has been run. Default: off Chapter 4: The Model file 75 regime = INTEGER Given the data and model parameters, what is the ergodic probability of being in the specified regime. Only one of filtered_probabilities, regime and regimes may be passed. Default: off regimes Describes the evolution of regimes. Only one of filtered_probabilities, regime and regimes may be passed. Default: off median A shortcut to setting error_band_percentiles=[0.5]. Default: off ms_forecast ; ms_forecast (OPTIONS . . . ); [Command] [Command] Description Generates forecasts for a Markov-switching SBVAR model. Output .eps files are contained in <output_file_tag/Output/Forecast>, while data files are contained in <output_file_ tag/Forecast>. Options file_tag = FILENAME See [file tag], page 72. output_file_tag = FILENAME See [output file tag], page 72. simulation_file_tag = FILENAME See [simulation file tag], page 73. data_obs_nbr = INTEGER The number of data points included in the output. Default: 0 error_band_percentiles = [DOUBLE1 ...] See [error band percentiles], page 74. shock_draws = INTEGER See [shock draws], page 74. shocks_per_parameter = INTEGER See [shocks per parameter], page 74. thinning_factor = INTEGER See [thinning factor], page 74. free_parameters = NUMERICAL_VECTOR See [free parameters], page 74. parameter_uncertainty See [parameter uncertainty], page 74. regime = INTEGER See [regime], page 75. regimes See [regimes], page 75. median See [median], page 75. Chapter 4: The Model file ms_variance_decomposition ; ms_variance_decomposition (OPTIONS . . . ); 76 [Command] [Command] Description Computes the variance decomposition for a Markov-switching SBVAR model. Output .eps files are contained in <output_file_tag/Output/Variance_Decomposition>, while data files are contained in <output_file_tag/Variance_Decomposition>. Options file_tag = FILENAME See [file tag], page 72. output_file_tag = FILENAME See [output file tag], page 72. simulation_file_tag = FILENAME See [simulation file tag], page 73. horizon = INTEGER See [horizon], page 74. filtered_probabilities See [filtered probabilities], page 74. no_error_bands Do not output percentile error bands (i.e. compute mean). Default: off (i.e. output error bands) error_band_percentiles = [DOUBLE1 ...] See [error band percentiles], page 74. shock_draws = INTEGER See [shock draws], page 74. shocks_per_parameter = INTEGER See [shocks per parameter], page 74. thinning_factor = INTEGER See [thinning factor], page 74. free_parameters = NUMERICAL_VECTOR See [free parameters], page 74. parameter_uncertainty See [parameter uncertainty], page 74. regime = INTEGER See [regime], page 75. regimes See [regimes], page 75. 4.19 Displaying and saving results Dynare has comments to plot the results of a simulation and to save the results. rplot VARIABLE_NAME . . . ; [Command] Plots the simulated path of one or several variables, as stored in oo .endo simul by either simul (see Section 4.12 [Deterministic simulation], page 32) or stoch simul with option periods (see Section 4.13.1 [Computing the stochastic solution], page 34). The variables are plotted in levels. Chapter 4: The Model file dynatype (FILENAME) [VARIABLE_NAME . . . ]; This command prints the listed variables in a text file named FILENAME. ABLE NAME is listed, all endogenous variables are printed. 77 [Command] If no VARI- dynasave (FILENAME) [VARIABLE_NAME . . . ]; [Command] This command saves the listed variables in a binary file named FILENAME. If no VARIABLE NAME are listed, all endogenous variables are saved. In MATLAB or Octave, variables saved with the dynasave command can be retrieved by the command: load -mat FILENAME 4.20 Macro-processing language It is possible to use “macro” commands in the ‘.mod’ file for doing the following tasks: including modular source files, replicating blocks of equations through loops, conditionally executing some code, writing indexed sums or products inside equations. . . The Dynare macro-language provides a new set of macro-commands which can be inserted inside ‘.mod’ files. It features: • file inclusion • loops (for structure) • conditional inclusion (if/then/else structures) • expression substitution Technically, this macro language is totally independent of the basic Dynare language, and is processed by a separate component of the Dynare pre-processor. The macro processor transforms a ‘.mod’ file with macros into a ‘.mod’ file without macros (doing expansions/inclusions), and then feeds it to the Dynare parser. The key point to understand is that the macro-processor only does text substitution (like the C preprocessor or the PHP language). Note that it is possible to see the output of the macro-processor by using the savemacro option of the dynare command (see Chapter 3 [Dynare invocation], page 6). The macro-processor is invoked by placing macro directives in the ‘.mod’ file. Directives begin with an at-sign followed by a pound sign (@#). They produce no output, but give instructions to the macro-processor. In most cases, directives occupy exactly one line of text. In case of need, two anti-slashes (\\) at the end of the line indicates that the directive is continued on the next line. The main directives are: • @#include, for file inclusion, • @#define, for defining a macro-processor variable, • @#if, @#ifdef, @#else, @#endif for conditional statements, • @#for, @#endfor for constructing loops. The macro-processor maintains its own list of variables (distinct of model variables and of MATLAB/Octave variables). These macro-variables are assigned using the @#define directive, and can be of four types: integer, character string, array of integers, array of strings. 4.20.1 Macro expressions It is possible to construct macro-expressions which can be assigned to macro-variables or used within a macro-directive. The expressions are constructed using literals of the four basic types (integers, strings, arrays of strings, arrays of integers), macro-variables names and standard operators. String literals have to be enclosed between double quotes (like "name"). Arrays are enclosed within brackets, and their elements are separated by commas (like [1,2,3] or ["US", "EA"]). Note that there is no boolean type: false is represented by integer zero and true is any non-null integer. Chapter 4: The Model file 78 The following operators can be used on integers: • arithmetic operators: +, -, *, / • comparison operators: <, >, <=, >=, ==, != • logical operators: &&, ||, ! • integer ranges, using the following syntax: INTEGER1:INTEGER2 (for example, 1:4 is equivalent to integer array [1,2,3,4]) The following operators can be used on strings: • comparison operators: ==, != • concatenation of two strings: + • extraction of substrings: if s is a string, then s[3] is a string containing only the third character of s, and s[4:6] contains the characters from 4th to 6th The following operators can be used on arrays: • dereferencing: if v is an array, then v[2] is its 2nd element • concatenation of two arrays: + • difference -: returns the first operand from which the elements of the second operand have been removed • extraction of sub-arrays: e.g. v[4:6] • testing membership of an array: in operator (for example: "b" in ["a", "b", "c"] returns 1) Macro-expressions can be used at two places: • inside macro directives, directly; • in the body of the .mod file, between an at-sign and curly braces (like @{expr}): the macro processor will substitute the expression with its value. In the following, MACRO EXPRESSION designates an expression constructed as explained above. 4.20.2 Macro directives [Macro directive] This directive simply includes the content of another file at the place where it is inserted. It is exactly equivalent to a copy/paste of the content of the included file. Note that it is possible to nest includes (i.e. to include a file from an included file). @#include "FILENAME" Example @#include "modelcomponent.mod" @#define MACRO_VARIABLE = MACRO_EXPRESSION Defines a macro-variable. Example 1 @#define @#define @#define @#define @#define @#define Example 2 x y v w z t = = = = = = 5 // Integer "US" // String [ 1, 2, 4 ] // Integer array [ "US", "EA" ] // String array 3 + v[2] // Equals 5 ("US" in w) // Equals 1 (true) [Macro directive] Chapter 4: The Model file 79 @#define x = [ "B", "C" ] @#define i = 2 model; A = @{x[i]}; end; is strictly equivalent to: model; A = C; end; [Macro directive] [Macro directive] [Macro directive] [Macro directive] Conditional inclusion of some part of the ‘.mod’ file. The lines between @#if or @#ifdef and the next @#else or @#endif is executed only if the condition evaluates to a non-null integer. The @#else branch is optional and, if present, is only evaluated if the condition evaluates to 0. @#if MACRO_EXPRESSION @#ifdef MACRO_VARIABLE @#else @#endif Example Choose between two alternative monetary policy rules using a macro-variable: @#define linear_mon_pol = 0 // or 1 ... model; @#if linear_mon_pol i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); @#else i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; @#endif ... end; Example Choose between two alternative monetary policy rules using a macro-variable. As linear_mon_ pol was not previously defined in this example, the second equation will be chosen: model; @#ifdef linear_mon_pol i = w*i(-1) + (1-w)*i_ss + w2*(pie-piestar); @#else i = i(-1)^w * i_ss^(1-w) * (pie/piestar)^w2; @#endif ... end; @#for MACRO_VARIABLE in MACRO_EXPRESSION @#endfor [Macro directive] [Macro directive] Loop construction for replicating portions of the ‘.mod’ file. Note that this construct can enclose variable/parameters declaration, computational tasks, but not a model declaration. Example Chapter 4: The Model file 80 model; @#for country in [ "home", "foreign" ] GDP_@{country} = A * K_@{country}^a * L_@{country}^(1-a); @#endfor end; is equivalent to: model; GDP_home = A * K_home^a * L_home^(1-a); GDP_foreign = A * K_foreign^a * L_foreign^(1-a); end; [Macro directive] Asks the preprocessor to display some message on standard output. The argument must evaluate to a string. @#echo MACRO_EXPRESSION [Macro directive] Asks the preprocessor to display some error message on standard output and to abort. The argument must evaluate to a string. @#error MACRO_EXPRESSION 4.20.3 Typical usages 4.20.3.1 Modularization The @#include directive can be used to split ‘.mod’ files into several modular components. Example setup: ‘modeldesc.mod’ Contains variable declarations, model equations and shocks declarations ‘simul.mod’ Includes ‘modeldesc.mod’, calibrates parameters and runs stochastic simulations ‘estim.mod’ Includes ‘modeldesc.mod’, declares priors on parameters and runs bayesian estimation Dynare can be called on ‘simul.mod’ and ‘estim.mod’, but it makes no sense to run it on ‘modeldesc.mod’. The main advantage is that it is no longer needed to manually copy/paste the whole model (at the beginning) or changes to the model (during development). 4.20.3.2 Indexed sums or products The following example shows how to construct a moving average: @#define window = 2 var x MA_x; ... model; ... MA_x = 1/@{2*window+1}*( @#for i in -window:window +x(@{i}) @#endfor ); ... end; After macro-processing, this is equivalent to: Chapter 4: The Model file 81 var x MA_x; ... model; ... MA_x = 1/5*( +x(-2) +x(-1) +x(0) +x(1) +x(2) ); ... end; 4.20.3.3 Multi-country models Here is a skeleton example for a multi-country model: @#define countries = [ "US", "EA", "AS", "JP", "RC" ] @#define nth_co = "US" @#for co in countries var Y_@{co} K_@{co} L_@{co} i_@{co} E_@{co} ...; parameters a_@{co} ...; varexo ...; @#endfor model; @#for co in countries Y_@{co} = K_@{co}^a_@{co} * L_@{co}^(1-a_@{co}); ... @# if co != nth_co (1+i_@{co}) = (1+i_@{nth_co}) * E_@{co}(+1) / E_@{co}; // UIP relation @# else E_@{co} = 1; @# endif @#endfor end; 4.20.3.4 Endogeneizing parameters When doing the steady state calibration of the model, it may be useful to consider a parameter as an endogenous (and vice-versa). For example, suppose production is defined by a CES function: ξ/(ξ−1) y = α1/ξ `1−1/ξ + (1 − α)1/ξ k 1−1/ξ The labor share in GDP is defined as: lab_rat = (w`)/(py) In the model, α is a (share) parameter, and lab_rat is an endogenous variable. It is clear that calibrating α is not straigthforward; but on the contrary, we have real world data for lab_rat, and it is clear that these two variables are economically linked. The solution is to use a method called variable flipping, which consist in changing the way of computing the steady state. During this computation, α will be made an endogenous variable and lab_rat will be made a parameter. An economically relevant value will be calibrated for lab_rat, and the solution algorithm will deduce the implied value for α. Chapter 4: The Model file 82 An implementation could consist of the following files: ‘modeqs.mod’ This file contains variable declarations and model equations. The code for the declaration of α and lab_rat would look like: @#if steady var alpha; parameter lab_rat; @#else parameter alpha; var lab_rat; @#endif ‘steady.mod’ This file computes the steady state. It begins with: @#define steady = 1 @#include "modeqs.mod" Then it initializes parameters (including lab_rat, excluding α, computes the steady state (using guess values for endogenous, including α, then saves values of parameters and endogenous at steady state in a file, using the save_params_and_steady_state command. ‘simul.mod’ This file computes the simulation. It begins with: @#define steady = 0 @#include "modeqs.mod" Then it loads values of parameters and endogenous at steady state from file, using the load_params_and_steady_state command, and computes the simulations. 4.20.4 MATLAB/Octave loops versus macro-processor loops Suppose you have a model with a parameter ρ, and you want to make simulations for three values: ρ = 0.8, 0.9, 1. There are several ways of doing this: With a MATLAB/Octave loop rhos = [ 0.8, 0.9, 1]; for i = 1:length(rhos) rho = rhos(i); stoch_simul(order=1); end Here the loop is not unrolled, MATLAB/Octave manages the iterations. This is interesting when there are a lot of iterations. With a macro-processor loop (case 1) rhos = [ 0.8, 0.9, 1]; @#for i in 1:3 rho = rhos(@{i}); stoch_simul(order=1); @#endfor This is very similar to previous example, except that the loop is unrolled. The macroprocessor manages the loop index but not the data array (rhos). With a macro-processor loop (case 2) @#for rho_val in [ "0.8", "0.9", "1"] rho = @{rho_val}; Chapter 4: The Model file 83 stoch_simul(order=1); @#endfor The advantage of this method is that it uses a shorter syntax, since list of values directly given in the loop construct. Note that values are given as character strings (the macro-processor does not know floating point values. The inconvenient is that you can not reuse an array stored in a MATLAB/Octave variable. 4.21 Misc commands (INTEGER) (’default’) (’reset’) (’ALGORITHM’, INTEGER) Sets the seed used for random number generation. set_dynare_seed set_dynare_seed set_dynare_seed set_dynare_seed [Command] [Command] [Command] [Command] save_params_and_steady_state (FILENAME); [Command] For all parameters, endogenous and exogenous variables, stores their value in a text file, using a simple name/value associative table. • for parameters, the value is taken from the last parameter initialization • for exogenous, the value is taken from the last initval block • for endogenous, the value is taken from the last steady state computation (or, if no steady state has been computed, from the last initval block) Note that no variable type is stored in the file, so that the values can be reloaded with load_ params_and_steady_state in a setup where the variable types are different. The typical usage of this function is to compute the steady-state of a model by calibrating the steady-state value of some endogenous variables (which implies that some parameters must be endogeneized during the steady-state computation). You would then write a first ‘.mod’ file which computes the steady state and saves the result of the computation at the end of the file, using save_params_and_steady_state. In a second file designed to perform the actual simulations, you would use load_params_and_ steady_state just after your variable declarations, in order to load the steady state previously computed (including the parameters which had been endogeneized during the steady state computation). The need for two separate ‘.mod’ files arises from the fact that the variable declarations differ between the files for steady state calibration and for simulation (the set of endogenous and parameters differ between the two); this leads to different var and parameters statements. Also note that you can take advantage of the @#include directive to share the model equations between the two files (see Section 4.20 [Macro-processing language], page 77). load_params_and_steady_state (FILENAME); [Command] For all parameters, endogenous and exogenous variables, loads their value from a file created with save_params_and_steady_state. • for parameters, their value will be initialized as if they had been calibrated in the ‘.mod’ file • for endogenous and exogenous, their value will be initialized as they would have been from an initval block This function is used in conjunction with save_params_and_steady_state; see the documentation of that function for more information. Chapter 5: The Configuration File 84 5 The Configuration File The configuration file is used to provide Dynare with information not related to the model (and hence not placed in the model file). At the moment, it is only used when using Dynare to run parallel computations. On Linux and Mac OS X, the default location of the configuration file is ‘$HOME/.dynare’, while on Windows it is ‘%APPDATA%\dynare.ini’ (typically ‘C:\Documents and Settings\USERNAME\Application Data\dynare.ini’ under Windows XP, or ‘C:\Users\USERNAME\AppData\dynare.ini’ under Windows Vista or Windows 7). You can specify a non standard location using the conffile option of the dynare command (see Chapter 3 [Dynare invocation], page 6). The parsing of the configuration file is case-sensitive and it should take the following form, with each option/choice pair placed on a newline: [command0] option0 = choice0 option1 = choice1 [command1] option0 = choice0 option1 = choice1 The configuration file follows a few conventions (self-explanatory conventions such as USER NAME have been excluded for concision): COMPUTER NAME Indicates the valid name of a server (e.g. localhost, server.cepremap.org) or an IP address. DRIVE NAME Indicates a valid drive name in Windows, without the trailing colon (e.g. C). PATH Indicates a valid path in /home/user/dynare/matlab/). the underlying operating system (e.g. PATH AND FILE Indicates a valid path to a file in the underlying operating system (e.g. /usr/local/MATLAB/R2010b/bin/matlab). BOOLEAN Is true or false. 5.1 Dynare Configuration This section explains how to configure Dynare for general processing. Currently, there is only one option available. [hooks] [Configuration block] Description The [hooks] block can be used to specify configuration options that will be used when running dynare. Options GlobalInitFile = PATH_AND_FILE The location of the global initialization file to be run at the end of global_ initialization.m Chapter 5: The Configuration File 85 Example [hooks] GlobalInitFile = /home/usern/dynare/myInitFile.m 5.2 Parallel Configuration This section explains how to configure Dynare for parallelizing some tasks which require very little inter-process communication. The parallelization is done by running several MATLAB or Octave processes, either on local or on remote machines. Communication between master and slave processes are done through SMB on Windows and SSH on UNIX. Input and output data, and also some short status messages, are exchanged through network filesystems. Currently the system works only with homogenous grids: only Windows or only Unix machines. The following routines are currently parallelized: • the Metropolis-Hastings algorithm; • the Metropolis-Hastings diagnostics; • the posterior IRFs; • the prior and posterior statistics; • some plotting routines. Note that creating the configuration file is not enough in order to trigger parallelization of the computations: you also need to specify the parallel option to the dynare command. For more details, and for other options related to the parallelization engine, see see Chapter 3 [Dynare invocation], page 6. You also need to verify that the following requirements are met by your cluster (which is composed of a master and of one or more slaves): For a Windows grid • a standard Windows network (SMB) must be in place; • PsTools must be installed in the path of the master Windows machine; • the Windows user on the master machine has to be user of any other slave machine in the cluster, and that user will be used for the remote computations. For a UNIX grid • SSH must be installed on the master and on the slave machines; • SSH keys must be installed so that the SSH connection from the master to the slaves can be done without passwords, or using an SSH agent We now turn to the description of the configuration directives: [cluster] [Configuration block] Description When working in parallel, [cluster] is required to specify the group of computers that will be used. It is required even if you are only invoking multiple processes on one computer. Options Name = CLUSTER_NAME The reference name of this cluster. Chapter 5: The Configuration File 86 Members = NODE_NAME[(WEIGHT)] NODE_NAME[(WEIGHT)] ... A list of nodes that comprise the cluster with an optional computing weight specified for that node. The computing weight indicates how much more powerful one node is with respect to the others (e.g. n1(2) n2(1) n3(3), means that n1 is two times more powerful than n2 whereas n3 is three times more powerful than n2). Each node is separated by at least one space and the weights are in parenthesis with no spaces separating them from their node. Example [cluster] Name = c1 Members = n1 n2 n3 [cluster] Name = c2 Members = n1(4) n2 n3 [node] [Configuration block] Description When working in parallel, [node] is required for every computer that will be used. The options that are required differ, depending on the underlying operating system and whether you are working locally or remotely. Options Name = NODE_NAME The reference name of this node. CPUnbr = INTEGER | [INTEGER:INTEGER] If just one integer is passed, the number of processors to use. If a range of integers is passed, the specific processors to use (processor counting is defined to begin at one as opposed to zero). Note that using specific processors is only possible under Windows; under Linux and Mac OS X, if a range is passed the same number of processors will be used but the range will be adjusted to begin at one. ComputerName = COMPUTER_NAME The name or IP address of the node. If you want to run locally, use localhost (case-sensitive). Port = INTEGER The port number to connect to on the node. The default is empty, meaning that the connection will be made to the default SSH port (22). UserName = USER_NAME The username used to log into a remote system. Required for remote runs on all platforms. Password = PASSWORD The password used to log into the remote system. Required for remote runs originating from Windows. RemoteDrive = DRIVE_NAME The drive to be used for remote computation. Required for remote runs originating from Windows. Chapter 5: The Configuration File 87 RemoteDirectory = PATH The directory to be used for remote computation. Required for remote runs on all platforms. DynarePath = PATH The path to the ‘matlab’ subdirectory within the Dynare installation directory. The default is the empty string. MatlabOctavePath = PATH_AND_FILE The path to the MATLAB or Octave executable. The default value is matlab. SingleCompThread = BOOLEAN Whether or not to disable MATLAB’s native multithreading. The default value is true. Option meaningless under Octave. OperatingSystem = OPERATING_SYSTEM The operating system associated with a node. Only necessary when creating a cluster with nodes from different operating systems. Possible values are unix or windows. There is no default value. Example [node] Name = n1 ComputerName = localhost CPUnbr = 1 [node] Name = n2 ComputerName = dynserv.cepremap.org CPUnbr = 5 UserName = usern RemoteDirectory = /home/usern/Remote DynarePath = /home/usern/dynare/matlab MatlabOctavePath = matlab [node] Name = n3 ComputerName = dynserv.dynare.org Port = 3333 CPUnbr = [2:4] UserName = usern RemoteDirectory = /home/usern/Remote DynarePath = /home/usern/dynare/matlab MatlabOctavePath = matlab Chapter 6: Examples 88 6 Examples Dynare comes with a database of example ‘.mod’ files, which are designed to show a broad range of Dynare features, and are taken from academic papers for most of them. You should have these files in the ‘examples’ subdirectory of your distribution. Here is a short list of the examples included. For a more complete description, please refer to the comments inside the files themselves. ‘ramst.mod’ An elementary real business cycle (RBC) model, simulated in a deterministic setup. ‘example1.mod’ ‘example2.mod’ Two examples of a small RBC model in a stochastic setup, presented in Collard (2001) (see the file ‘guide.pdf’ which comes with Dynare). ‘fs2000.mod’ A cash in advance model, estimated by Schorfheide (2000). ‘fs2000_nonstationary.mod’ The same model than ‘fs2000.mod’, but written in non-stationary form. Detrending of the equations is done by Dynare. ‘bkk.mod’ Multi-country RBC model with time to build, presented in Backus, Kehoe and Kydland (1992). ‘agtrend.mod’ Small open economy RBC model with shocks to the growth trend, presented in Aguiar and Gopinath (2004). Chapter 7: Dynare internal documentation and unitary tests 89 7 Dynare internal documentation and unitary tests One can obtain internal documentation of matlab/octave’s routines or perform unitary tests using the internals command. This is a new feature, and, at this time, will work properly for a small number of routines. At the top of the (available) matlab/octave routines a commented block for the internal documentation is written in the GNU texinfo documentation format. This block is processed by calling texinfo from matlab. Consequently, texinfo has to be installed on your machine. internals FLAG ROUTINENAME[.m] [MATLAB/Octave command] Description Depending on the FLAG this command prints internal documentation of a matlab/octave routine or triggers unitary tests associated to this routine. Flags --info Prints on screen the internal documentation of ROUTINENAME (if this routine exists and if this routine has texinfo internal documentation header). If the command is executed in the matlab directory of Dynare, then the path to ROUTINENAME has to be provided. --test Performs the unitary test associated to ROUTINENAME (if this routine exists and if the matalab/octave m file has unitary test sections). Example internals --info particle/local_state_iteration internals --test particle/local_state_iteration Chapter 8: Bibliography 90 8 Bibliography • Aguiar, Mark and Gopinath, Gita (2004): “Emerging Market Business Cycles: The Cycle is the Trend,” NBER Working Paper, 10734 • Backus, David K., Patrick J. Kehoe, and Finn E. Kydland (1992): “International Real Business Cycles,” Journal of Political Economy, 100(4), 745–775 • Boucekkine, Raouf (1995): “An alternative methodology for solving nonlinear forward-looking models,” Journal of Economic Dynamics and Control, 19, 711–734 • Collard, Fabrice (2001): “Stochastic simulations with Dynare: A practical guide” • Collard, Fabrice and Michel Juillard (2001a): “Accuracy of stochastic perturbation methods: The case of asset pricing models,” Journal of Economic Dynamics and Control, 25, 979–999 • Collard, Fabrice and Michel Juillard (2001b): “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 • Dennis, Richard (2007): “Optimal Policy In Rational Expectations Models: New Solution Algorithms,” Macroeconomic Dynamics, 11(1), 31–55 • Durbin, J. and S. J. Koopman (2001), Time Series Analysis by State Space Methods, Oxford University Press • Fair, Ray and John Taylor (1983): “Solution and Maximum Likelihood Estimation of Dynamic Nonlinear Rational Expectation Models,” Econometrica, 51, 1169–1185 • Fernández-Villaverde, Jesús and Juan Rubio-Ramı́rez (2004): “Comparing Dynamic Equilibrium Economies to Data: A Bayesian Approach,” Journal of Econometrics, 123, 153–187 • Fernández-Villaverde, Jesús and Juan Rubio-Ramı́rez (2005): “Estimating Dynamic Equilibrium Economies: Linear versus Nonlinear Likelihood,” Journal of Applied Econometrics, 20, 891–910 • Ireland, Peter (2004): “A Method for Taking Models to the Data,” Journal of Economic Dynamics and Control, 28, 1205–26 • Iskrev, Nikolay (2010): “Local identification in DSGE models,” Journal of Monetary Economics, 57(2), 189–202 • Judd, Kenneth (1996): “Approximation, Perturbation, and Projection Methods in Economic Analysis”, in Handbook of Computational Economics, ed. by Hans Amman, David Kendrick, and John Rust, 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 • Kim, Jinill, Sunghyun Kim, Ernst Schaumburg, and Christopher A. Sims (2008): “Calculating and using second-order accurate solutions of discrete time dynamic equilibrium models,” Journal of Economic Dynamics and Control, 32(11), 3397–3414 • Koopman, S. J. and J. Durbin (2003): “Filtering and Smoothing of State Vector for Diffuse State Space Models,” Journal of Time Series Analysis, 24(1), 85–98 • Laffargue, Jean-Pierre (1990): “Résolution d’un modèle macroéconomique avec anticipations rationnelles”, Annales d’Économie et Statistique, 17, 97–119 • Lubik, Thomas and Frank Schorfheide (2007): “Do Central Banks Respond to Exchange Rate Movements? A Structural Investigation,” Journal of Monetary Economics, 54(4), 1069–1087 • Mancini-Griffoli, Tommaso (2007): “Dynare User Guide: An introduction to the solution and estimation of DSGE models” • Pearlman, Joseph, David Currie, and Paul Levine (1986): “Rational expectations models with partial information,” Economic Modelling, 3(2), 90–105 Chapter 8: Bibliography 91 • Rabanal, Pau and Juan Rubio-Ramirez (2003): “Comparing New Keynesian Models of the Business Cycle: A Bayesian Approach,” Federal Reserve of Atlanta, Working Paper Series, 2003-30. • Ratto, Marco (2008): “Analysing DSGE models with global sensitivity analysis”, Computational Economics, 31, 115–139 • Schorfheide, Frank (2000): “Loss Function-based evaluation of DSGE models,” Journal of Applied Econometrics, 15(6), 645–670 • Schmitt-Grohé, Stephanie and Martin Urı́be (2004): “Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function,” Journal of Economic Dynamics and Control, 28(4), 755–775 • Sims, Christopher A., Daniel F. Waggoner and Tao Zha (2008): “Methods for inference in large multiple-equation Markov-switching models,” Journal of Econometrics, 146, 255–274 • Smets, Frank and Rafael Wouters (2003): “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1(5), 1123–1175 • Villemot, Sébastien (2011): “Solving rational expectations models at first order: what Dynare does,” Dynare Working Papers, 2, CEPREMAP Command and Function Index 92 Command and Function Index F @ @#define . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#else . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#endfor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#endif . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#ifdef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . @#include . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 80 79 79 79 80 79 79 79 78 [ [cluster] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 [hooks] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 [node] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 A abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . acos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . asin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . atan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 14 14 14 B bvar_density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 bvar_forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 C calib_smoother . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . change_type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . conditional_forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . conditional_forecast_paths . . . . . . . . . . . . . . . . . . . . . cos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 11 30 56 57 14 D discretionary_policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 dsample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 dynare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 dynare_sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 dynasave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 dynatype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 E endval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . erf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . estimated_params . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . estimated_params_bounds . . . . . . . . . . . . . . . . . . . . . . . . . estimated_params_init . . . . . . . . . . . . . . . . . . . . . . . . . . . estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EXPECTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . extended_path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . external_function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 14 42 44 44 44 14 13 38 15 forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 H histval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 homotopy_setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 I identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . initval_file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . internals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 12 20 22 89 L ln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . load_params_and_steady_state . . . . . . . . . . . . . . . . . . . log . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . log10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 83 14 14 M markov_switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . model_comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . model_info . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ms_compute_mdd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ms_compute_probabilities . . . . . . . . . . . . . . . . . . . . . . . . ms_estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ms_forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ms_irf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ms_simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ms_variance_decomposition . . . . . . . . . . . . . . . . . . . . . . mshocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 14 14 16 53 31 73 73 69 75 74 72 76 25 N nan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 normcdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 normpdf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 O observation_trends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . optim_weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . osr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . osr_params . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 58 58 58 P parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . planner_objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plot_conditional_forecast . . . . . . . . . . . . . . . . . . . . . . predetermined_variables . . . . . . . . . . . . . . . . . . . . . . . . . print_bytecode_dynamic_model . . . . . . . . . . . . . . . . . . . 10 26 60 57 11 32 Command and Function Index 93 print_bytecode_static_model . . . . . . . . . . . . . . . . . . . . 32 svar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 R T ramsey_policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 resid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 rplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 tan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 trend_var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 U S save_params_and_steady_state . . . . . . . . . . . . . . . . . . . set_dynare_seed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shock_decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . simul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sqrt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . steady . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . STEADY_STATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . steady_state_model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stoch_simul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit_root_vars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 83 83 53 23 14 32 14 14 26 13 29 34 V var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 varexo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 varexo_det . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 varobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 W write_latex_dynamic_model . . . . . . . . . . . . . . . . . . . . . . 18 write_latex_static_model . . . . . . . . . . . . . . . . . . . . . . . . 18 Variable Index 94 Variable Index M M_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 M_.endo_nbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 M_.orig_endo_nbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 M_.params . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16, 50 M_.Sigma_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 O oo_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 oo_.autocorr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 oo_.conditional_variance_decomposition. . . . . . . . 41 oo_.dr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 oo_.dr.eigval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 oo_.dr.g_0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.g_1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.g_2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.g_3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.ghs2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.ghu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.ghuu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.ghx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.ghxu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.ghxx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 oo_.dr.inv_order_var . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.nboth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.nfwrd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.npred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.nstatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.order_var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 oo_.dr.ys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39, 40 oo_.endo_simul. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 36 oo_.exo_simul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 oo_.FilteredVariables . . . . . . . . . . . . . . . . . . . . . . . 48, 51 oo_.FilteredVariablesKStepAhead . . . . . . . . . . . . 48, 51 oo_.FilteredVariablesKStepAheadVariances . . 48, 51 oo_.forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48, 56 oo_.gamma_y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 oo_.irfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 oo_.MarginalDensity.LaplaceApproximation . . . . . 51 oo_.MarginalDensity.ModifiedHarmonicMean . . . . . 51 oo_.mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 oo_.MeanForecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 oo_.osr.objective_function . . . . . . . . . . . . . . . . . . . . . 59 oo_.planner_objective_value . . . . . . . . . . . . . . . . . . . . 59 oo_.PointForecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 oo_.posterior_density . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 oo_.posterior_hpdinf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 oo_.posterior_hpdsup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 oo_.posterior_mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 oo_.posterior_mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 oo_.posterior_std . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 oo_.PosteriorIRF.dsge . . . . . . . . . . . . . . . . . . . . . . . 47, 52 oo_.PosteriorTheoreticalMoments . . . . . . . . . . . . 48, 52 oo_.SmoothedMeasurementErrors . . . . . . . . . . . . . . 48, 52 oo_.SmoothedShocks . . . . . . . . . . . . . . . . . . . . . . . 48, 52, 54 oo_.SmoothedVariables . . . . . . . . . . . . . . . . . . . . 48, 52, 54 oo_.steady_state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 oo_.UpdatedVariables . . . . . . . . . . . . . . . . . . . . . 48, 52, 54 oo_.var . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 options_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 S Sigma_e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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