-
-| Revision |
- Version |
- Date |
- Description of changes |
-
-
-
-|
- | 1.3.7
- | 2008/01/15
- |
-
- |
| | | Corrected a serious bug in centralizing a
-decision rule. This bug implies that all results based on simulations
-of the decision rule were wrong. However results based on stochastic
-fix points were correct. Thanks to Wouter J. den Haan and Joris de Wind!
-
- |
| | | Added options --centralize and --no-centralize.
-
- |
| | | Corrected an error of a wrong
-variance-covariance matrix in real-time simulations (thanks to Pawel
-Zabzcyk).
-
- |
| | | Corrected a bug of integer overflow in refined
-faa Di Bruno formula if one of refinements is empty. This bug appeared
-when solving models without forward looking variables.
-
- |
| | | Corrected a bug in the Sylvester equation
-formerly working only for models with forward looking variables.
-
- |
| | | Corrected a bug in global check printout.
-
- |
| | | Added generating a dump file.
-
- |
| | | Fixed a bug of forgetting repeated assignments
-(for example in parameter settings and initval).
-
- |
| | | Added a diff operator to the parser.
-
- |
-| 1539
- | 1.3.6
- | 2008/01/03
- |
-
- |
| | | Corrected a bug of segmentation faults for long
-names and path names.
-
- |
| | | Changed a way how random numbers are
-generated. Dynare++ uses a separate instance of Mersenne twister for
-each simulation, this corrects a flaw of additional randomness caused
-by operating system scheduler. This also corrects a strange behaviour
-of random generator on Windows, where each simulation was getting the
-same sequence of random numbers.
-
- |
| | | Added calculation of conditional distributions
-controlled by --condper and --condsim.
-
- |
| | | Dropped creating unfoled version of decision
-rule at the end. This might consume a lot of memory. However,
-simulations might be slower for some models.
-
- |
-| 1368
- | 1.3.5
- | 2007/07/11
- |
-
- |
| | | Corrected a bug of useless storing all derivative
-indices in a parser. This consumed a lot of memory for large models.
-
- |
| | | Added an option --ss-tol controlling a
-tolerance used for convergence of a non-linear solver.
-
- |
| | | Corrected buggy interaction of optimal policy
-and forward looking variables with more than one period.
-
- |
| | | Variance matrices can be positive
-semidefinite. This corrects a bug of throwing an error if estimating
-approximation errors on ellipse of the state space with a
-deterministic variable.
-
- |
| | | Implemented simulations with statistics
-calculated in real-time. Options --rtsim and --rtper.
-
- |
-| 1282
- | 1.3.4
- | 2007/05/15
- |
-
- |
| | | Corrected a bug of wrong representation of NaN in generated M-files.
-
- |
| | | Corrected a bug of occassionaly wrong evaluation of higher order derivatives of integer powers.
-
- |
| | | Implemented automatic handling of terms involving multiple leads.
-
- |
| | | Corrected a bug in the numerical integration, i.e. checking of the precision of the solution.
-
- |
-| 1090
- | 1.3.3
- | 2006/11/20
- |
-
- |
| | | Corrected a bug of non-registering an auxiliary variable in initval assignments.
-
- |
-| 988
- | 1.3.2
- | 2006/10/11
- |
-
- |
| | | Corrected a few not-serious bugs: segfault on
-some exception, error in parsing large files, error in parsing
-matrices with comments, a bug in dynare_simul.m
-
- |
| | | Added posibility to specify a list of shocks for
-which IRFs are calculated
-
- |
| | | Added --order command line switch
-
- |
| | | Added writing two MATLAB files for steady state
-calcs
-
- |
| | | Implemented optimal policy using keyword
-planner_objective and planner_discount
-
- |
| | | Implemented an R interface to Dynare++ algorithms
-(Tamas Papp)
-
- |
| | | Highlevel code reengineered to allow for
-different model inputs
-
- |
-| 799
- | 1.3.1
- | 2006/06/13
- |
-
- |
| | | Corrected few bugs: in error functions, in linear algebra module.
-
- |
| | | Updated dynare_simul.
-
- |
| | | Updated the tutorial.
-
- |
| | | Corrected an error in summing up tensors where
-setting up the decision rule derivatives. Thanks to Michel
-Juillard. The previous version was making deterministic effects of
-future volatility smaller than they should be.
-
- |
-| 766
- | 1.3.0
- | 2006/05/22
- |
-
- |
| | | The non-linear solver replaced with a new one.
-
- |
| | | The parser and derivator replaced with a new
-code. Now it is possible to put expressions in parameters and initval
-sections.
-
- |
-| 752
- | 1.2.2
- | 2006/05/22
- |
-
- |
| | | Added an option triggering/suppressing IRF calcs..
-
- |
| | | Newton algortihm is now used for fix-point calculations.
-
- |
| | | Vertical narrowing of tensors in Faa Di Bruno
-formula to avoid multiplication with zeros..
-
- |
-| 436
- | 1.2.1
- | 2005/08/17
- |
-
- |
| | | Faa Di Bruno for sparse matrices optimized. The
-implementation now accommodates vertical refinement of function stack
-in order to fit a corresponding slice to available memory. In
-addition, zero slices are identified. For some problems, this implies
-significant speedup.
-
- |
| | | Analytic derivator speedup.
-
- |
| | | Corrected a bug in the threading code. The bug
-stayed concealed in Linux 2.4.* kernels, and exhibited in Linux 2.6.*,
-which has a different scheduling. This correction also allows using
-detached threads on Windows.
-
- |
-| 410
- | 1.2
- | 2005/07/29
- |
-
- |
| | | Added Dynare++ tutorial.
-
- |
| | | Changed and enriched contents of MAT-4 output
-file.
-
- |
| | | Corrected a bug of wrong variable indexation
-resulting in an exception. The error occurred if a variable appeared
-at time t-1 or t+1 and not at t.
-
- |
| | | Added MATLAB interface, which allows simulation
-of a decision rule in MATLAB.
-
- |
| | | Got rid of Matrix Template Library.
-
- |
| | | Added checking of model residuals by the
-numerical integration. Three methods: checking along simulation path,
-checking along shocks, and on ellipse of states.
-
- |
| | | Corrected a bug in calculation of higher moments
-of Normal dist.
-
- |
| | | Corrected a bug of wrong drawing from Normal dist
-with non-zero covariances.
-
- |
| |
- | Added numerical integration module. Product and Smolyak
-quadratures over Gauss-Hermite and Gauss-Legendre, and quasi Monte
-Carlo.
-
- |
-| 152
- | 1.1
- | 2005/04/22
- |
-
- |
| |
- | Added a calculation of approximation at a stochastic steady state
-(still experimental).
-
- |
| |
- | Corrected a bug in Cholesky decomposition of variance-covariance
-matrix with off-diagonal elements.
-
- |
-| 89
- | 1.01
- | 2005/02/23
- |
-
- |
| |
- | Added version printout.
-
- |
| |
- | Corrected the bug of multithreading support for P4 HT processors running on Win32.
-
- |
| |
- | Enhanced Kronecker product code resulting in approx. 20% speedup.
-
- |
| |
- | Implemented vertical stack container refinement, and another
-method for sparse folded Faa Di Bruno (both not used yet).
-
- |
-| 5
- | 1.0
- | 2005/02/23
- | The first released version.
-
- |
-
-
diff --git a/dynare++/doc/changelog-sylv-old.html b/dynare++/doc/changelog-sylv-old.html
deleted file mode 100644
index 39571f73a..000000000
--- a/dynare++/doc/changelog-sylv-old.html
+++ /dev/null
@@ -1,140 +0,0 @@
-
-
-
-
-
-| Tag |
- Date |
- Description/Changes |
-
-
- |
-2003/09/10 |
-Initial version solving triangular system put to repository |
-
-
- |
- |
-Implemented solution of general case. |
-
-
- |
- |
-Implemented a memory pool (Paris). |
-
-
- |
- |
-Implemented MEX interface to the routine (Paris). |
-
-
- |
- |
-Implemented QuasiTriangularZero (Paris) (not fully used yet). |
-
-
-| rel-1 |
-2003/10-02 |
-Version sent to Michel. |
-
-
- |
- |
-Inheritance streamlined, QuasiTriangular inherits from GeneralMatrix. |
-
-
- |
- |
-Implemented block diagonalization algorithm. |
-
-
- |
- |
-Solution routines rewritten so that the output rewrites input,
-considerable memory improvement. |
-
-
- |
- |
-MEX interface now links with LAPACK library from MATLAB. |
-
-
- |
- |
-Added a hack to MEX library loading in order to avoid MATLAB crash in Wins. |
-
-
-| rel-2 |
-2003/10/15 |
-Version sent to Michel. |
-
-
- |
- |
-KronUtils now rewrite input by output using less memory. |
-
-
- |
- |
-Added iterative solution algorithm (doubling). |
-
-
- |
- |
-Introduced abstraction for set of parameters (SylvParams). |
-
-
- |
- |
-Algorithm enabled to solve problems with singular C. |
-
-
- |
- |
-Implemented a class chooser chossing between QuasiTriangularZero,
-and QuasiTriangular (padded with zero) depending on size of the
-problem. Full use of QuasiTriangularZero. |
-
-
- |
- |
-Reimplemented QuasiTriangular::solve, offdiagonal elements are
-eleiminated by gauss with partial pivoting, not by transformation of
-complex eigenvalues. More stable for ill conditioned eigenvalues. |
-
-
- |
- |
-Reimplemented calculation of eliminating vectors, much more
-numerically stable now. |
-
-
- |
- |
-Implemented algorithm for ordering of eigenvalues (not used now,
-no numerical improvements). |
-
-
-| rel-3 |
-2003/12/4 |
-Version sent to Michel. |
-
-
- |
- |
-GeneralMatrix separated for use outside, in sylv module we use
-its subclass SylvMatrix. Implemented ConstGeneralMatrix (useful outside).
- |
-
-
-| rel-4 |
-2004/6/4 |
-Version, which was moved to pythie.cepremap.cnrs.fr repository. |
-
-
-
-
diff --git a/dynare++/doc/dynare++-ramsey.tex b/dynare++/doc/dynare++-ramsey.tex
deleted file mode 100644
index 00b73ce8f..000000000
--- a/dynare++/doc/dynare++-ramsey.tex
+++ /dev/null
@@ -1,157 +0,0 @@
-\documentclass[10pt]{article}
-\usepackage{array,natbib,times}
-\usepackage{amsmath, amsthm, amssymb}
-
-%\usepackage[pdftex,colorlinks]{hyperref}
-
-\begin{document}
-
-\title{Implementation of Ramsey Optimal Policy in Dynare++, Timeless Perspective}
-
-\author{Ondra Kamen\'\i k}
-
-\date{June 2006}
-\maketitle
-
-\textbf{Abstract:} This document provides a derivation of Ramsey
-optimal policy from timeless perspective and describes its
-implementation in Dynare++.
-
-\section{Derivation of the First Order Conditions}
-
-Let us start with an economy populated by agents who take a number of
-variables exogenously, or given. These may include taxes or interest
-rates for example. These variables can be understood as decision (or control)
-variables of the timeless Ramsey policy (or social planner). The agent's
-information set at time $t$ includes mass-point distributions of these
-variables for all times after $t$. If $i_t$ denotes an interest rate
-for example, then the information set $I_t$ includes
-$i_{t|t},i_{t+1|t},\ldots,i_{t+k|t},\ldots$ as numbers. In addition
-the information set includes all realizations of past exogenous
-innovations $u_\tau$ for $\tau=t,t-1,\ldots$ and distibutions
-$u_\tau\sim N(0,\Sigma)$ for $\tau=t+1,\ldots$. These information sets will be denoted $I_t$.
-
-An information set including only the information on past realizations
-of $u_\tau$ and future distributions of $u_\tau\sim N(0\sigma)$ will
-be denoted $J_t$. We will use the following notation for expectations
-through these sets:
-\begin{eqnarray*}
-E^I_t[X] &=& E(X|I_t)\\
-E^J_t[X] &=& E(X|J_t)
-\end{eqnarray*}
-
-The agents optimize taking the decision variables of the social
-planner at $t$ and future as given. This means that all expectations
-they form are conditioned on the set $I_t$. Let $y_t$ denote a vector
-of all endogenous variables including the planer's decision
-variables. Let the number of endogenous variables be $n$. The economy
-can be described by $m$ equations including the first order conditions
-and transition equations:
-\begin{equation}\label{constr}
-E_t^I\left[f(y_{t-1},y_t,y_{t+1},u_t)\right] = 0.
-\end{equation}
-This lefts $n-m$
-the planner's control variables. The solution of this problem is a
-decision rule of the form:
-\begin{equation}\label{agent_dr}
-y_t=g(y_{t-1},u_t,c_{t|t},c_{t+1|t},\ldots,c_{t+k|t},\ldots),
-\end{equation}
-where $c$ is a vector of planner's control variables.
-
-Each period the social planner chooses the vector $c_t$ to maximize
-his objective such that \eqref{agent_dr} holds for all times following
-$t$. This would lead to $n-m$ first order conditions with respect to
-$c_t$. These first order conditions would contain unknown derivatives
-of endogenous variables with respect to $c$, which would have to be
-retrieved from the implicit constraints \eqref{constr} since the
-explicit form \eqref{agent_dr} is not known.
-
-The other way to proceed is to assume that the planner is so dumb that
-he is not sure what are his control variables. So he optimizes with
-respect to all $y_t$ given the constraints \eqref{constr}. If the
-planner's objective is $b(y_{t-1},y_t,y_{t+1},u_t)$ with a discount rate
-$\beta$, then the optimization problem looks as follows:
-\begin{align}
-\max_{\left\{y_\tau\right\}^\infty_t}&E_t^J
-\left[\sum_{\tau=t}^\infty\beta^{\tau-t}b(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]\notag\\
-&\rm{s.t.}\label{planner_optim}\\
-&\hskip1cm E^I_\tau\left[f(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]=0\quad\rm{for\ }
-\tau=\ldots,t-1,t,t+1,\ldots\notag
-\end{align}
-Note two things: First, each constraint \eqref{constr} in
-\eqref{planner_optim} is conditioned on $I_\tau$ not $I_t$. This is
-very important, since the behaviour of agents at period $\tau=t+k$ is
-governed by the constraint using expectations conditioned on $t+k$,
-not $t$. The social planner knows that at $t+k$ the agents will use
-all information available at $t+k$. Second, the constraints for the
-planner's decision made at $t$ include also constraints for agent's
-behaviour prior to $t$. This is because the agent's decision rules are
-given in the implicit form \eqref{constr} and not in the explicit form
-\eqref{agent_dr}.
-
-Using Lagrange multipliers, this can be rewritten as
-\begin{align}
-\max_{y_t}E_t^J&\left[\sum_{\tau=t}^\infty\beta^{\tau-t}b(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right.\notag\\
-&\left.+\sum_{\tau=-\infty}^{\infty}\beta^{\tau-t}\lambda^T_\tau E_\tau^I\left[f(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]\right],
-\label{planner_optim_l}
-\end{align}
-where $\lambda_t$ is a vector of Lagrange multipliers corresponding to
-constraints \eqref{constr}. Note that the multipliers are multiplied
-by powers of $\beta$ in order to make them stationary. Taking a
-derivative wrt $y_t$ and putting it to zero yields the first order
-conditions of the planner's problem:
-\begin{align}
-E^J_t\left[\vphantom{\frac{\int^(_)}{\int^(\_)}}\right.&\frac{\partial}{\partial y_t}b(y_{t-1},y_t,y_{t+1},u_t)+
-\beta L^{+1}\frac{\partial}{\partial y_{t-1}}b(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\beta^{-1}\lambda_{t-1}^TE^I_{t-1}\left[L^{-1}\frac{\partial}{\partial y_{t+1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]\notag\\
-&+\lambda_t^TE^I_t\left[\frac{\partial}{\partial y_{t}}f(y_{t-1},y_t,y_{t+1},u_t)\right]\notag\\
-&+\beta\lambda_{t+1}^TE^I_{t+1}\left[L^{+1}\frac{\partial}{\partial y_{t-1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]
-\left.\vphantom{\frac{\int^(_)}{\int^(\_)}}\right]
- = 0,\label{planner_optim_foc}
-\end{align}
-where $L^{+1}$ and $L^{-1}$ are one period lead and lag operators respectively.
-
-Now we have to make a few assertions concerning expectations
-conditioned on the different information sets to simplify
-\eqref{planner_optim_foc}. Recall the formula for integration through
-information on which another expectation is conditioned, this is:
-$$E\left[E\left[u|v\right]\right] = E[u],$$
-where the outer expectation integrates through $v$. Since $J_t\subset
-I_t$, by easy application of the above formula we obtain
-\begin{eqnarray}
-E^J_t\left[E^I_t\left[X\right]\right] &=& E^J_t\left[X\right]\quad\rm{and}\notag\\
-E^J_t\left[E^I_{t-1}\left[X\right]\right] &=& E^J_t\left[X\right]\label{e_iden}\\
-E^J_t\left[E^I_{t+1}\left[X\right]\right] &=& E^J_{t+1}\left[X\right]\notag
-\end{eqnarray}
-Now, the last term of \eqref{planner_optim_foc} needs a special
-attention. It is equal to
-$E^J_t\left[\beta\lambda^T_{t+1}E^I_{t+1}[X]\right]$. If we assume
-that the problem \eqref{planner_optim} has a solution, then there is a
-deterministic function from $J_{t+1}$ to $\lambda_{t+1}$ and so
-$\lambda_{t+1}\in J_{t+1}\subset I_{t+1}$. And the last term is equal
-to $E^J_{t}\left[E^I_{t+1}[\beta\lambda^T_{t+1}X]\right]$, which is
-$E^J_{t+1}\left[\beta\lambda^T_{t+1}X\right]$. This term can be
-equivalently written as
-$E^J_{t}\left[\beta\lambda^T_{t+1}E^J_{t+1}[X]\right]$. The reason why
-we write the term in this way will be clear later. All in all, we have
-\begin{align}
-E^J_t\left[\vphantom{\frac{\int^(_)}{\int^(\_)}}\right.&\frac{\partial}{\partial y_t}b(y_{t-1},y_t,y_{t+1},u_t)+
-\beta L^{+1}\frac{\partial}{\partial y_{t-1}}b(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\beta^{-1}\lambda_{t-1}^TL^{-1}\frac{\partial}{\partial y_{t+1}}f(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\lambda_t^T\frac{\partial}{\partial y_{t}}f(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\beta\lambda_{t+1}^TE^J_{t+1}\left[L^{+1}\frac{\partial}{\partial y_{t-1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]
-\left.\vphantom{\frac{\int^(_)}{\int^(\_)}}\right]
- = 0.\label{planner_optim_foc2}
-\end{align}
-Note that we have not proved that \eqref{planner_optim_foc} and
-\eqref{planner_optim_foc2} are equivalent. We proved only that if
-\eqref{planner_optim_foc} has a solution, then
-\eqref{planner_optim_foc2} is equivalent (and has the same solution).
-
-%%- \section{Implementation}
-%%-
-%%- The user inputs $b(y_{t-1},y_t,y_{t+1},u_t)$, $\beta$, and agent's
-%%- first order conditions \eqref{constr}. The algorithm has to produce
-%%- \eqref{planner_optim_foc2}.
-%%-
-\end{document}
diff --git a/dynare++/doc/dynare++-tutorial.tex b/dynare++/doc/dynare++-tutorial.tex
deleted file mode 100644
index d94475a2a..000000000
--- a/dynare++/doc/dynare++-tutorial.tex
+++ /dev/null
@@ -1,1512 +0,0 @@
-\documentclass[10pt]{article}
-\usepackage{array,natbib}
-\usepackage{amsmath, amsthm, amssymb}
-
-\usepackage[pdftex,colorlinks]{hyperref}
-
-\begin{document}
-
-\title{DSGE Models with Dynare++. A Tutorial.}
-
-\author{Ondra Kamen\'\i k}
-
-\date{First version: February 2011 \\ This version: September 2020}
-\maketitle
-
-\tableofcontents
-
-\section{Setup}
-
-The Dynare++ setup procedure is pretty straightforward as Dynare++ is included in the Dynare installation
-packages which can be downloaded from \url{https://www.dynare.org}. Take the following steps:
-\begin{enumerate}
-\item Add the {\tt dynare++} subdirectory of the root Dynare installation directory to the your
-operating system path. This ensures that your OS will find the {\tt dynare++} executable.
-\item If you have MATLAB and want to run custom simulations (see \ref{custom}),
- then you need to add to your MATLAB path the {\tt dynare++} subdirectory of
- the root Dynare installation directory, and also directory containing the
- \texttt{dynare\_simul\_} MEX file (note the trailing underscore). The easiest
- way to add the latter is to run Dynare once in your MATLAB session (even
- without giving it any MOD file).
-\end{enumerate}
-
-\section{Sample Session}
-
-As an example, let us take a simple DSGE model with time to build, whose dynamic
-equilibrium is described by the following first order conditions:
-
-\begin{align*}
-&c_t\theta h_t^{1+\psi} = (1-\alpha)y_t\cr
-&\beta E_t\left[\frac{\exp(b_t)c_t}{\exp(b_{t+1})c_{t+1}}
-\left(\exp(b_{t+1})\alpha\frac{y_{t+1}}{k_{t+1}}+1-\delta\right)\right]=1\cr
-&y_t=\exp(a_t)k_t^\alpha h_t^{1-\alpha}\cr
-&k_{t}=\exp(b_{t-1})(y_{t-1}-c_{t-1})+(1-\delta)k_{t-1}\cr
-&a_t=\rho a_{t-1}+\tau b_{t-1}+\epsilon_t\cr
-&b_t=\tau a_{t-1}+\rho b_{t-1}+\nu_t
-\end{align*}
-
-\label{timing}
-The convention is that the timing of a variable reflects when this variable
-is decided. Dynare++ therefore uses a ``stock at the end of the
-period'' notation for predetermined state variables (see the Dynare manual for details).
-
-The timing of this model is that the exogenous shocks $\epsilon_t$,
-and $\nu_t$ are observed by agents in the beginning of period $t$ and
-before the end of period $t$ all endogenous variables with index $t$
-are decided. The expectation operator $E_t$ works over the information
-accumulated just before the end of the period $t$ (this includes
-$\epsilon_t$, $\nu_t$ and all endogenous variables with index $t$).
-
-The exogenous shocks $\epsilon_t$ and $\nu_t$ are supposed to be
-serially uncorrelated with zero means and time-invariant
-variance-covariance matrix. In Dynare++, these variables are called
-exogenous; all other variables are endogenous. Now we are prepared to
-start writing a model file for Dynare++, which is an ordinary text
-file and could be created with any text editor.
-
-The model file starts with a preamble declaring endogenous and
-exogenous variables, parameters, and setting values of the
-parameters. Note that one can put expression on right hand sides. The
-preamble follows:
-
-{\small
-\begin{verbatim}
-var Y, C, K, A, H, B;
-varexo EPS, NU;
-
-parameters beta, rho, alpha, delta, theta, psi, tau;
-alpha = 0.36;
-rho = 0.95;
-tau = 0.025;
-beta = 1/(1.03^0.25);
-delta = 0.025;
-psi = 0;
-theta = 2.95;
-\end{verbatim}
-}
-
-The section setting values of the parameters is terminated by a
-beginning of the {\tt model} section, which states all the dynamic
-equations. A timing convention of a Dynare++ model is the same as the
-timing of our example model, so we may proceed with writing the model
-equations. The time indexes of $c_{t-1}$, $c_t$, and $c_{t+1}$ are
-written as {\tt C(-1)}, {\tt C}, and {\tt C(1)} resp. The {\tt model}
-section looks as follows:
-
-{\small
-\begin{verbatim}
-model;
-C*theta*H^(1+psi) = (1-alpha)*Y;
-beta*exp(B)*C/exp(B(1))/C(1)*
- (exp(B(1))*alpha*Y(1)/K(1)+1-delta) = 1;
-Y = exp(A)*K^alpha*H^(1-alpha);
-K = exp(B(-1))*(Y(-1)-C(-1)) + (1-delta)*K(-1);
-A = rho*A(-1) + tau*B(-1) + EPS;
-B = tau*A(-1) + rho*B(-1) + NU;
-end;
-\end{verbatim}
-}
-
-At this point, almost all information that Dynare++ needs has been
-provided. Only three things remain to be specified: initial values of
-endogenous variables for non-linear solver, variance-covariance matrix
-of the exogenous shocks and order of the Taylor approximation. Since
-the model is very simple, there is a closed form solution for the
-deterministic steady state. We use it as initial values for the
-non-linear solver. Note that the expressions on the right hand-sides in
-{\tt initval} section can reference values previously calculated. The
-remaining portion of the model file looks as follows:
-
-{\small
-\begin{verbatim}
-initval;
-A = 0;
-B = 0;
-H = ((1-alpha)/(theta*(1-(delta*alpha)
- /(1/beta-1+delta))))^(1/(1+psi));
-Y = (alpha/(1/beta-1+delta))^(alpha/(1-alpha))*H;
-K = alpha/(1/beta-1+delta)*Y;
-C = Y - delta*K;
-end;
-
-vcov = [
- 0.0002 0.00005;
- 0.00005 0.0001
-];
-
-order = 7;
-\end{verbatim}
-}
-
-Note that the order of rows/columns of the variance-covariance matrix
-corresponds to the ordering of exogenous variables in the {\tt varexo}
-declaration. Since the {\tt EPS} was declared first, its variance is
-$0.0002$, and the variance of {\tt NU} is $0.0001$.
-
-Let the model file be saved as {\tt example1.mod}. Now we are prepared
-to solve the model. At the operating system command
-prompt\footnote{Under Windows it is a {\tt cmd} program, under Unix it
-is any shell} we issue a command:
-
-{\small
-\begin{verbatim}
-dynare++ example1.mod
-\end{verbatim}
-}
-
-When the program is finished, it produces two output files: a journal
-file {\tt example1.jnl} and a MATLAB MAT-4 {\tt example1.mat}. The
-journal file contains information about time, memory and processor
-resources needed for all steps of solution. The output file is more
-interesting. It contains various simulation results. It can be loaded
-into MATLAB or Octave and examined.
-
-Let us first examine the contents of the MAT file:
-{\small
-\begin{verbatim}
->> load example1.mat
->> who
-
-Your variables are:
-
-dyn_g_1 dyn_i_Y dyn_npred
-dyn_g_2 dyn_irfm_EPS_mean dyn_nstat
-dyn_g_3 dyn_irfm_EPS_var dyn_shocks
-dyn_g_4 dyn_irfm_NU_mean dyn_ss
-dyn_g_5 dyn_irfm_NU_var dyn_state_vars
-dyn_i_A dyn_irfp_EPS_mean dyn_steady_states
-dyn_i_B dyn_irfp_EPS_var dyn_vars
-dyn_i_C dyn_irfp_NU_mean dyn_vcov
-dyn_i_EPS dyn_irfp_NU_var dyn_vcov_exo
-dyn_i_H dyn_mean
-dyn_i_K dyn_nboth
-dyn_i_NU dyn_nforw
-\end{verbatim}
-}
-
-All the variables coming from one MAT file have a common prefix. In
-this case it is {\tt dyn}, which is Dynare++ default. The prefix can
-be changed, so that the multiple results could be loaded into one MATLAB
-session.
-
-In the default setup, Dynare++ solves the Taylor approximation to the
-decision rule and calculates unconditional mean and covariance of the
-endogenous variables, and generates impulse response functions. The
-mean and covariance are stored in {\tt dyn\_mean} and {\tt
-dyn\_vcov}. The ordering of the endogenous variables is given by {\tt
-dyn\_vars}.
-
-In our example, the ordering is
-
-{\small
-\begin{verbatim}
->> dyn_vars
-dyn_vars =
-H
-A
-Y
-C
-K
-B
-\end{verbatim}
-}
-
-and unconditional mean and covariance are
-
-{\small
-\begin{verbatim}
->> dyn_mean
-dyn_mean =
- 0.2924
- 0.0019
- 1.0930
- 0.8095
- 11.2549
- 0.0011
->> dyn_vcov
-dyn_vcov =
- 0.0003 0.0006 0.0016 0.0004 0.0060 0.0004
- 0.0006 0.0024 0.0059 0.0026 0.0504 0.0012
- 0.0016 0.0059 0.0155 0.0069 0.1438 0.0037
- 0.0004 0.0026 0.0069 0.0040 0.0896 0.0016
- 0.0060 0.0504 0.1438 0.0896 2.1209 0.0405
- 0.0004 0.0012 0.0037 0.0016 0.0405 0.0014
-\end{verbatim}
-}
-
-The ordering of the variables is also given by indexes starting with
-{\tt dyn\_i\_}. Thus the mean of capital can be retrieved as
-
-{\small
-\begin{verbatim}
->> dyn_mean(dyn_i_K)
-ans =
- 11.2549
-\end{verbatim}
-}
-
-\noindent and covariance of labor and capital by
-
-{\small
-\begin{verbatim}
->> dyn_vcov(dyn_i_K,dyn_i_H)
-ans =
- 0.0060
-\end{verbatim}
-}
-
-The impulse response functions are stored in matrices as follows
-\begin{center}
-\begin{tabular}{|l|l|}
-\hline
-matrix& response to\\
-\hline
-{\tt dyn\_irfp\_EPS\_mean}& positive impulse to {\tt EPS}\\
-{\tt dyn\_irfm\_EPS\_mean}& negative impulse to {\tt EPS}\\
-{\tt dyn\_irfp\_NU\_mean}& positive impulse to {\tt NU}\\
-{\tt dyn\_irfm\_NU\_mean}& negative impulse to {\tt NU}\\
-\hline
-\end{tabular}
-\end{center}
-All shocks sizes are one standard error. Rows of the matrices
-correspond to endogenous variables, columns correspond to
-periods. Thus capital response to a positive shock to {\tt EPS} can be
-plotted as
-
-{\small
-\begin{verbatim}
-plot(dyn_irfp_EPS_mean(dyn_i_K,:));
-\end{verbatim}
-}
-
-The data is in units of the respective variables, so in order to plot
-the capital response in percentage changes from the decision rule's
-fix point (which is a vector {\tt dyn\_ss}), one has to issue the
-commands:
-
-{\small
-\begin{verbatim}
-Kss=dyn_ss(dyn_i_K);
-plot(100*dyn_irfp_EPS_mean(dyn_i_K,:)/Kss);
-\end{verbatim}
-}
-
-The plotted impulse response shows that the model is pretty persistent
-and that the Dynare++ default for a number of simulated periods is not
-sufficient. In addition, the model persistence puts in doubt also a
-number of simulations. The Dynare++ defaults can be changed when
-calling Dynare++, in operating system's command prompt, we issue a
-command:
-
-{\small
-\begin{verbatim}
-dynare++ --per 300 --sim 150 example1.mod
-\end{verbatim}
-}
-
-\noindent This sets the number of simulations to $150$ and the number
-of periods to $300$ for each simulation giving $45000$ total simulated
-periods.
-
-\section{Sample Optimal Policy Session}
-\label{optim_tut}
-
-Suppose that one wants to solve the following optimal policy problem
-with timeless perspective.\footnote{See \ref{ramsey} on how to solve
-Ramsey optimality problem within this framework} The following
-optimization problem is how to choose capital taxes financing public
-good to maximize agent's utility from consumption good and public
-good. The problem takes the form:
-\begin{align*}
-\max_{\{\tau_t\}_{t_0}^\infty}
-E_{t_0}\sum_{t=t_0}^\infty &\beta^{t-t_0}\left(u(c_t)+av(g_t)\right)\\
-\hbox{subject\ to}&\\
-u'(c_t) &=
-\beta E_t\left[u'(c_{t+1})\left(1-\delta+f'(k_{t+1})(1-\alpha\tau_{t+1})\right)\right]\\
-K_t &= (1-\delta)K_{t-1} + (f(K_{t-1}) - c_{t-1} - g_{t-1})\\
-g_t &= \tau_t\alpha f(K_t),\\
-\hbox{where\ } t & = \ldots,t_0-1,t_0,t_0+1,\ldots
-\end{align*}
-$u(c_t)$ is utility from consuming the consumption good, $v(g_t)$ is
-utility from consuming the public good, $f(K_t)$ is a production
-function $f(K_t) = Z_tK_t^\alpha$. $Z_t$ is a technology shock modeled
-as AR(1) process. The three constraints come from the first order
-conditions of a representative agent. We suppose that it pursues a
-different objective, namely lifetime utility involving only
-consumption $c_t$. The representative agents chooses between
-consumption and investment. It rents the capital to firms and supplies
-constant amount of labour. All output is paid back to consumer in form
-of wage and capital rent. Only the latter is taxed. We suppose that
-the optimal choice has been taking place from infinite past and will
-be taking place for ever. Further we suppose the same about the
-constraints.
-
-Let us choose the following functional forms:
-\begin{eqnarray*}
-u(c_t) &=& \frac{c_t^{1-\eta}}{1-\eta}\\
-v(g_t) &=& \frac{g_t^{1-\phi}}{1-\phi}\\
-f(K_t) &=& K_t^\alpha
-\end{eqnarray*}
-
-Then the problem can be coded into Dynare++ as follows. We start with
-a preamble which states all the variables, shocks and parameters:
-{\small
-\begin{verbatim}
-var C G K TAU Z;
-
-varexo EPS;
-
-parameters eta beta alpha delta phi a rho;
-
-eta = 2;
-beta = 0.99;
-alpha = 0.3;
-delta = 0.10;
-phi = 2.5;
-a = 0.1;
-rho = 0.7;
-\end{verbatim}
-}
-
-Then we specify the planner's objective and the discount factor in the
-objective. The objective is an expression (possibly including also
-variable leads and lags), and the discount factor must be one single
-declared parameter:
-{\small
-\begin{verbatim}
-planner_objective C^(1-eta)/(1-eta) + a*G^(1-phi)/(1-phi);
-
-planner_discount beta;
-\end{verbatim}
-}
-
-The model section will contain only the constraints of the social
-planner. These are capital accumulation, identity for the public
-product, AR(1) process for $Z_t$ and the first order condition of the
-representative agent (with different objective).
-{\small
-\begin{verbatim}
-model;
-K = (1-delta)*K(-1) + (exp(Z(-1))*K(-1)^alpha - C(-1) - G(-1));
-G = TAU*alpha*K^alpha;
-Z = rho*Z(-1) + EPS;
-C^(-eta) = beta*C(+1)^(-eta)*(1-delta +
- exp(Z(+1))*alpha*K(+1)^(alpha-1)*(1-alpha*TAU(+1)));
-end;
-\end{verbatim}
-}
-
-Now we have to provide a good guess for non-linear solver calculating
-the deterministic steady state. The model's steady state has a closed
-form solution if the taxes are known. So we provide a guess for
-taxation {\tt TAU} and then use the closed form solution for capital,
-public good and consumption:\footnote{Initial guess for Lagrange
-multipliers and some auxiliary variables is calculated automatically. See
-\ref{opt_init} for more details.}
-{\small
-\begin{verbatim}
-initval;
-TAU = 0.70;
-K = ((delta+1/beta-1)/(alpha*(1-alpha*TAU)))^(1/(alpha-1));
-G = TAU*alpha*K^alpha;
-C = K^alpha - delta*K - G;
-Z = 0;
-\end{verbatim}
-}
-
-Finally, we have to provide the order of approximation, and the
-variance-covariance matrix of the shocks (in our case we have only one
-shock):
-{\small
-\begin{verbatim}
-order = 4;
-
-vcov = [
- 0.01
-];
-\end{verbatim}
-}
-
-After this model file has been run, we can load the resulting MAT-file
-into the MATLAB and examine its contents:
-{\small
-\begin{verbatim}
->> load kp1980_2.mat
->> who
-
-Your variables are:
-
-dyn_g_1 dyn_i_MULT1 dyn_nforw
-dyn_g_2 dyn_i_MULT2 dyn_npred
-dyn_g_3 dyn_i_MULT3 dyn_nstat
-dyn_g_4 dyn_i_TAU dyn_shocks
-dyn_i_AUX_3_0_1 dyn_i_Z dyn_ss
-dyn_i_AUX_4_0_1 dyn_irfm_EPS_mean dyn_state_vars
-dyn_i_C dyn_irfm_EPS_var dyn_steady_states
-dyn_i_EPS dyn_irfp_EPS_mean dyn_vars
-dyn_i_G dyn_irfp_EPS_var dyn_vcov
-dyn_i_K dyn_mean dyn_vcov_exo
-dyn_i_MULT0 dyn_nboth
-\end{verbatim}
-}
-
-The data dumped into the MAT-file have the same structure as in the
-previous example of this tutorial. The only difference is that
-Dynare++ added a few more variables. Indeed:
-{\small
-\begin{verbatim}
->> dyn_vars
-dyn_vars =
-MULT1
-G
-MULT3
-C
-K
-Z
-TAU
-AUX_3_0_1
-AUX_4_0_1
-MULT0
-MULT2
-\end{verbatim}
-}
-Besides the five variables declared in the model ({\tt C}, {\tt G},
-{\tt K}, {\tt TAU}, and {\tt Z}), Dy\-na\-re++ added 6 more, four as Lagrange
-multipliers of the four constraints, two as auxiliary variables for
-shifting in time. See \ref{aux_var} for more details.
-
-The structure and the logic of the MAT-file is the same as these new 6
-variables were declared in the model file and the file is examined in
-the same way.
-
-For instance, let us examine the Lagrange multiplier of the optimal
-policy associated with the consumption first order condition. Recall
-that the consumers' objective is different from the policy
-objective. Therefore, the constraint will be binding and the
-multiplier will be non-zero. Indeed, its deterministic steady state,
-fix point and mean are as follows:
-{\small
-\begin{verbatim}
->> dyn_steady_states(dyn_i_MULT3,1)
-ans =
- -1.3400
->> dyn_ss(dyn_i_MULT3)
-ans =
- -1.3035
->> dyn_mean(dyn_i_MULT3)
-ans =
- -1.3422
-\end{verbatim}
-}
-
-\section{What Dynare++ Calculates}
-\label{dynpp_calc}
-
-Dynare++ solves first order conditions of a DSGE model in the recursive form:
-\begin{equation}\label{focs}
-E_t[f(y^{**}_{t+1},y_t,y^*_{t-1},u_t)]=0,
-\end{equation}
-where $y$ is a vector of endogenous variables, and $u$ a vector of
-exogenous variables. Some of elements of $y$ can occur at time $t+1$,
-these are $y^{**}$. Elements of $y$ occurring at time $t-1$ are denoted
-$y^*$. The exogenous shocks are supposed to be serially independent
-and normally distributed $u_t\sim N(0,\Sigma)$.
-
-The solution of this dynamic system is a decision rule
-\[
-y_t=g(y^*_{t-1},u_t)
-\]
-Dynare++ calculates a Taylor approximation of this decision rule of a
-given order. The approximation takes into account deterministic
-effects of future volatility, so a point about which the Taylor
-approximation is done will be different from the fix point $y$ of the rule
-yielding $y=g(y^*,0)$.
-
-The fix point of a rule corresponding to a model with $\Sigma=0$ is
-called {\it deterministic steady state} denoted as $\bar y$. In
-contrast to deterministic steady state, there is no consensus in
-literature how to call a fix point of the rule corresponding to a
-model with non-zero $\Sigma$. I am tempted to call it {\it stochastic
- steady state}, however, it might be confused with unconditional mean
-or with steady distribution. So I will use a term {\it fix point} to
-avoid a confusion.
-
-By default, Dynare++ solves the Taylor approximation about the
-deterministic steady state. Alternatively, Dynare++ can split the
-uncertainty to a few steps and take smaller steps when calculating the
-fix points. This is controlled by an option {\tt --steps}. For the
-brief description of the second method, see \ref{multistep_alg}.
-
-\subsection{Decision Rule Form}
-\label{dr_form}
-
-In case of default solution algorithm (approximation about the
-deterministic steady state $\bar y$), Dynare++ calculates the higher
-order derivatives of the equilibrium rule to get a decision rule of
-the following form. In Einstein notation, it is:
-\[
-y_t-\bar y = \sum_{i=0}^k\frac{1}{i!}\left[g_{(y^*u)^i}\right]
-_{\alpha_1\ldots\alpha_i}
-\prod_{j=1}^i\left[\begin{array}{c} y^*_{t-1}-\bar y^*\\ u_t \end{array}\right]
-^{\alpha_j}
-\]
-
-Note that the ergodic mean will be different from the deterministic
-steady state $\bar y$ and thus deviations $y^*_{t-1}-\bar y^*$ will
-not be zero in average. This implies that in average we will commit
-larger round off errors than if we used the decision rule expressed in
-deviations from a point closer to the ergodic mean. Therefore, by
-default, Dynare++ recalculates this rule and expresses it in
-deviations from the stochastic fix point $y$.
-\[
-y_t-y = \sum_{i=1}^k\frac{1}{i!}\left[\tilde g_{(y^*u)^i}\right]
-_{\alpha_1\ldots\alpha_i}
-\prod_{j=1}^i\left[\begin{array}{c} y^*_{t-1}-y^*\\ u_t \end{array}\right]
-^{\alpha_j}
-\]
-Note that since the rule is centralized around its fix point, the
-first term (for $i=0$) drops out.
-
-Also note, that this rule mathematically equivalent to the rule
-expressed in deviations from the deterministic steady state, and still
-it is an approximation about the deterministic steady state. The fact
-that it is expressed in deviations from a different point should not
-be confused with the algorithm in \ref{multistep_alg}.
-
-This centralization can be avoided by invoking {\tt --no-centralize}
-command line option.
-
-\subsection{Taking Steps in Volatility Dimension}
-\label{multistep_alg}
-
-For models, where volatility of the exogenous shocks plays a big
-role, the approximation about deterministic steady state can be poor,
-since the equilibrium dynamics can be very different from the dynamics
-in the vicinity of the perfect foresight (deterministic steady state).
-
-Therefore, Dynare++ has on option {\tt --steps} triggering a multistep
-algorithm. The algorithm splits the volatility to a given number of
-steps. Dynare++ attempts to calculate approximations about fix points
-corresponding to these levels of volatility. The problem is that if we
-want to calculate higher order approximations about fix points
-corresponding to volatilities different from zero (as in the case of
-deterministic steady state), then the derivatives of lower orders
-depend on derivatives of higher orders with respect to forward looking
-variables. The multistep algorithm in each step approximates the
-missing higher order derivatives with extrapolations based on the
-previous step.
-
-In this way, the approximation of the stochastic fix point and the
-derivatives about this fix point are obtained. It is difficult to a
-priori decide whether this algorithm yields a better decision
-rule. Nothing is guaranteed, and the resulted decision rule should be
-checked with a numerical integration. See \ref{checks}.
-
-\subsection{Simulating the Decision Rule}
-
-After some form of a decision rule is calculated, it is simulated to
-obtain draws from ergodic (unconditional) distribution of endogenous
-variables. The mean and the covariance are reported. There are two
-ways how to calculate the mean and the covariance. The first one is to
-store all simulated samples and calculate the sample mean and
-covariance. The second one is to calculate mean and the covariance in
-the real-time not storing the simulated sample. The latter case is
-described below (see \ref{rt_simul}).
-
-The stored simulated samples are then used for impulse response
-function calculations. For each shock, the realized shocks in these
-simulated samples (control simulations) are taken and an impulse is
-added and the new realization of shocks is simulated. Then the control
-simulation is subtracted from the simulation with the impulse. This is
-done for all control simulations and the results are averaged. As the
-result, we get an expectation of difference between paths with impulse
-and without impulse. In addition, the sample variances are
-reported. They might be useful for confidence interval calculations.
-
-For each shock, Dynare++ calculates IRF for two impulses, positive and
-negative. Size of an impulse is one standard error of a respective
-shock.\footnote{Note that if the exogenous shocks are correlated, Dynare++ will
- ignore the correlation when computing the IRFs, and simulate the impulse on
- each shock independently of the others. Note that Dynare behaves differently
- in this case, and computes IRFs after performing an orthogonalization of the
- shocks (via a Cholesky decomposition of the variance-covariance matrix).}
-
-The rest of this subsection is divided to three parts giving account
-on real-time simulations, conditional simulations, and on the way how
-random numbers are generated resp.
-
-\subsubsection{Simulations With Real-Time Statistics}
-\label{rt_simul}
-
-When one needs to simulate large samples to get a good estimate of
-unconditional mean, simulating the decision rule with statistics
-calculated in real-time comes handy. The main reason is that the
-storing of all simulated samples may not fit into the available
-memory.
-
-The real-time statistics proceed as follows: We model the ergodic
-distribution as having normal distribution $y\sim N(\mu,\Sigma)$. Further,
-the parameters $\mu$ and $\Sigma$ are modelled as:
-\begin{eqnarray*}
- \Sigma &\sim& {\rm InvWishart}_\nu(\Lambda)\\
- \mu|\Sigma &\sim& N(\bar\mu,\Sigma/\kappa) \\
-\end{eqnarray*}
-This model of $p(\mu,\Sigma)$ has an advantage of conjugacy, i.e. a
-prior distribution has the same form as posterior. This property is
-used in the calculation of real-time estimates of $\mu$ and $\Sigma$,
-since it suffices to maintain only the parameters of $p(\mu,\Sigma)$
-conditional observed draws so far. The parameters are: $\nu$,
-$\Lambda$, $\kappa$, and $\bar\mu$.
-
-The mean of $\mu,\Sigma|Y$, where $Y$ are all the draws (simulated
-periods) is reported.
-
-\subsubsection{Conditional Distributions}
-\label{cond_dist}
-
-Starting with version 1.3.6, Dynare++ calculates variable
-distributions $y_t$ conditional on $y_0=\bar y$, where $\bar y$ is the
-deterministic steady state. If triggered, Dynare++ simulates a given
-number of samples with a given number of periods all starting at
-the deterministic steady state. Then for each time $t$, mean
-$E[y_t|y_0=\bar y]$ and variances $E[(y_t-E[y_t|y_0=\bar
-y])(y_t-E[y_t|y_0=\bar y])^T|y_0=\bar y]$ are reported.
-
-\subsubsection{Random Numbers}
-\label{random_numbers}
-
-For generating of the pseudo random numbers, Dynare++ uses Mersenne
-twister by Makoto Matsumoto and Takuji Nishimura. Because of the
-parallel nature of Dynare++ simulations, each simulated sample gets
-its own instance of the twister. Each such instance is seeded before
-the simulations are started. This is to prevent additional randomness
-implied by the operating system's thread scheduler to interfere with
-the pseudo random numbers.
-
-For seeding the individual instances of the Mersenne twister assigned
-to each simulated sample the system (C library) random generator is
-used. These random generators do not have usually very good
-properties, but we use them only to seed the Mersenne twister
-instances. The user can set the initial seed of the system random
-generator and in this way deterministically choose the seeds of all
-instances of the Mersenne twister.
-
-In this way, it is guaranteed that two runs of Dynare++
-with the same seed will yield the same results regardless the
-operating system's scheduler. The only difference may be caused by a
-different round-off errors committed when the same set of samples are
-summed in the different order (due to the operating system's scheduler).
-
-\subsection{Numerical Approximation Checks}
-\label{checks}
-
-Optionally, Dynare++ can run three kinds of checks for Taylor
-approximation errors. All three methods numerically calculate
-the residual of the DSGE equations
-\[
-E[f(g^{**}(g^*(y^*,u),u'),g(y^*,u),y^*,u)|y^*,u]
-\]
-which must be ideally zero for all $y^*$ and $u$. This integral is
-evaluated by either product or Smolyak rule applied to one dimensional
-Gauss--Hermite quadrature. The user does not need to care about the
-decision. An algorithm yielding higher quadrature level and less
-number of evaluations less than a user given maximum is selected.
-
-The three methods differ only by a set of $y^*$ and $u$ where the
-residuals are evaluated. These are:
-\begin{itemize}
-\item The first method calculates the residuals along the shocks for
-fixed $y^*$ equal to the fix point. We let all elements of $u$ be
-fixed at $0$ but one element, which varies from $-\mu\sigma$ to
-$\mu\sigma$, where $\sigma$ is a standard error of the element and
-$\mu$ is the user given multiplier. In this way we can see how the
-approximation error grows if the fix point is disturbed by a shock of
-varying size.
-\item The second method calculates the residuals along a simulation
-path. A random simulation is run, and at each point the residuals are
-reported.
-\item The third method calculates the errors on an ellipse of the
-state variables $y^*$. The shocks $u$ are always zero. The ellipse is
-defined as
-\[\{Ax|\; \Vert x\Vert_2=\mu\},\]
-where $\mu$ is a user given multiplier, and $AA^T=V$ for $V$ being a
-covariance of endogenous variables based on the first order
-approximation. The method calculates the residuals at low discrepancy
-sequence of points on the ellipse. Both the residuals and the points
-are reported.
-\end{itemize}
-
-\section{Optimal Policy with Dynare++}
-\label{optim}
-
-Starting with version 1.3.2, Dynare++ is able to automatically
-generate and then solve the first order conditions for a given
-objective and (possibly) forward looking constraints. Since the
-constraints can be forward looking, the use of this feature will
-mainly be in optimal policy or control.
-
-The only extra thing which needs to be added to the model file is a
-specification of the policy's objective. This is done by two keywords,
-placed not before parameter settings. If the objective is to maximize
-$$E_{t_0}\sum_{t=t_0}^\infty\beta^{t-t_0}\left[\frac{c_t^{1-\eta}}{1-\eta}+
-a\frac{g_t^{1-\phi}}{1-\phi}\right],$$
-then the keywords will be:
-{\small
-\begin{verbatim}
-planner_objective C^(1-eta)/(1-eta) + a*G^(1-phi)/(1-phi);
-
-planner_discount beta;
-\end{verbatim}
-}
-
-Dynare++ parses the file and if the two keywords are present, it
-automatically derives the first order conditions for the problem. The
-first order conditions are put to the form \eqref{focs} and solved. In
-this case, the equations in the {\tt model} section are understood as
-the constraints (they might come as the first order conditions from
-optimizations of other agents) and their number must be less than the
-number of endogenous variables.
-
-This section further describes how the optimal policy first order
-conditions look like, then discusses some issues with the initial
-guess for deterministic steady state, and finally describes how to
-simulate Ramsey policy within this framework.
-
-\subsection{First Order Conditions}
-
-Mathematically, the optimization problem looks as follows:
-\begin{align}
-\max_{\left\{y_\tau\right\}^\infty_t}&E_t
-\left[\sum_{\tau=t}^\infty\beta^{\tau-t}b(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]\notag\\
-&\rm{s.t.}\label{planner_optim}\\
-&\hskip1cm E^I_\tau\left[f(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]=0\quad\rm{for\ }
-\tau=\ldots,t-1,t,t+1,\ldots\notag
-\end{align}
-where $E^I$ is an expectation operator over an information set including,
-besides all the past, all future realizations of policy's control
-variables and distributions of future shocks $u_t\sim
-N(0,\Sigma)$. The expectation operator $E$ integrates over an
-information including only distributions of $u_t$ (besides the past).
-
-Note that the constraints $f$ take place at all times, and they are
-conditioned at the running $\tau$ since the policy knows that the
-agents at time $\tau$ will use all the information available at
-$\tau$.
-
-The maximization problem can be rewritten using Lagrange multipliers as:
-\begin{align}
-\max_{y_t}E_t&\left[\sum_{\tau=t}^\infty\beta^{\tau-t}b(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right.\notag\\
-&\left.+\sum_{\tau=-\infty}^{\infty}\beta^{\tau-t}\lambda^T_\tau E_\tau^I\left[f(y_{\tau-1},y_\tau,y_{\tau+1},u_\tau)\right]\right],
-\label{planner_optim_l}
-\end{align}
-where $\lambda_t$ is a column vector of Lagrange multipliers.
-
-After some manipulations with compounded expectations over different
-information sets, one gets the following first order conditions:
-\begin{align}
-E_t\left[\vphantom{\frac{\int^(_)}{\int^(\_)}}\right.&\frac{\partial}{\partial y_t}b(y_{t-1},y_t,y_{t+1},u_t)+
-\beta L^{+1}\frac{\partial}{\partial y_{t-1}}b(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\beta^{-1}\lambda_{t-1}^TL^{-1}\frac{\partial}{\partial y_{t+1}}f(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\lambda_t^T\frac{\partial}{\partial y_{t}}f(y_{t-1},y_t,y_{t+1},u_t)\notag\\
-&+\beta\lambda_{t+1}^TE_{t+1}\left[L^{+1}\frac{\partial}{\partial y_{t-1}}f(y_{t-1},y_t,y_{t+1},u_t)\right]
-\left.\vphantom{\frac{\int^(_)}{\int^(\_)}}\right]
- = 0,\label{planner_optim_foc2}
-\end{align}
-where $L^{+1}$ is one period lead operator, and $L^{-1}$ is one period lag operator.
-
-Dynare++ takes input corresponding to \eqref{planner_optim},
-introduces the Lagrange multipliers according to
-\eqref{planner_optim_l}, and using its symbolic derivator it compiles
-\eqref{planner_optim_foc2}. The system \eqref{planner_optim_foc2} with
-the constraints from \eqref{planner_optim_l} is then solved in the
-same way as the normal input \eqref{focs}.
-
-\subsection{Initial Guess for Deterministic Steady State}
-\label{opt_init}
-
-Solving deterministic steady state of non-linear dynamic systems is
-not trivial and the first order conditions for optimal policy add
-significant complexity. The {\tt initval} section allows to input the
-initial guess of the non-linear solver. It requires that all user
-declared endogenous variables be initialized. However, in most cases,
-we have no idea what are good initial guesses for the Lagrange
-multipliers.
-
-For this reason, Dynare++ calculates an initial guess of Lagrange
-multipliers using user provided initial guesses of all other
-endogenous variables. It uses the linearity of the Lagrange
-multipliers in the \eqref{planner_optim_foc2}. In its static form,
-\eqref{planner_optim_foc2} looks as follows:
-\begin{align}
-&\frac{\partial}{\partial y_t}b(y,y,y,0)+
-\beta\frac{\partial}{\partial y_{t-1}}b(y,y,y,0)\notag\\
-&+\lambda^T\left[\beta^{-1}\frac{\partial}{\partial y_{t+1}}f(y,y,y,0)
- +\frac{\partial}{\partial y_{t}}f(y,y,y,0)
- +\beta\frac{\partial}{\partial y_{t-1}}f(y,y,y,0)\right]
- = 0\label{planner_optim_static}
-\end{align}
-
-The user is required to provide an initial guess of all declared
-variables (all $y$). Then \eqref{planner_optim_static} becomes an
-overdetermined linear system in $\lambda$, which is solved by means of
-the least squares. The closer the initial guess of $y$ is to the exact
-solution, the closer are the Lagrange multipliers $\lambda$.
-
-The calculated Lagrange multipliers by the least squares are not used,
-if they are set in the {\tt initval} section. In other words, if a
-multiplier has been given a value in the {\tt initval} section, then
-the value is used, otherwise the calculated value is taken.
-
-For even more difficult problems, Dynare++ generates two MATLAB files
-calculating a residual of the static system and its derivative. These
-can be used in MATLAB's {\tt fsolve} or other algorithm to get an
-exact solution of the deterministic steady state. See
-\ref{output_matlab_scripts} for more details.
-
-Finally, Dynare++ might generate a few auxiliary variables. These are
-simple transformations of other variables. They are initialized
-automatically and the user usually does not need to care about it.
-
-\subsection{Optimal Ramsey Policy}
-\label{ramsey}
-
-Dynare++ solves the optimal policy problem with timeless
-perspective. This means that it assumes that the constraints in
-\eqref{planner_optim} are valid from the infinite past to infinite
-future. Dynare++ calculation of ergodic distribution then assumes that
-the policy has been taking place from infinite past.
-
-If some constraints in \eqref{planner_optim} are forward looking, this
-will result in some backward looking Lagrange multipliers. Such
-multipliers imply possibly time inconsistent policy in the states of
-the ``original'' economy, since these backward looking multipliers add
-new states to the ``optimized'' economy. In this respect, the timeless
-perspective means that there is no fixed initial distribution of such
-multipliers, instead, their ergodic distribution is taken.
-
-In contrast, Ramsey optimal policy is started at $t=0$. This means
-that the first order conditions at $t=0$ are different than the first
-order conditions at $t\geq 1$, which are
-\eqref{planner_optim_foc2}. However, it is not difficult to assert
-that the first order conditions at $t=0$ are in the form of
-\eqref{planner_optim_foc2} if all the backward looking Lagrange
-multipliers are set to zeros at period $-1$, i.e. $\lambda_{-1}=0$.
-
-All in all, the solution of \eqref{planner_optim_foc2} calculated by
-Dynare++ can be used as a Ramsey optimal policy solution provided that
-all the backward looking Lagrange multipliers were set to zeros prior
-to the first simulation period. This can be done by setting the
-initial state of a simulation path in {\tt dynare\_simul.m}. If this
-is applied on the example from \ref{optim_tut}, then we may do the
-following in the command prompt:
-{\small
-\begin{verbatim}
->> load kp1980_2.mat
->> shocks = zeros(1,100);
->> ystart = dyn_ss;
->> ystart(dyn_i_MULT3) = 0;
->> r=dynare_simul('kp1980_2.mat',shocks,ystart);
-\end{verbatim}
-}
-This will simulate the economy if the policy was introduced in the
-beginning and no shocks happened.
-
-More information on custom simulations can be obtained by typing:
-{\small
-\begin{verbatim}
-help dynare_simul
-\end{verbatim}
-}
-
-
-\section{Running Dynare++}
-
-This section deals with Dynare++ input. The first subsection
-\ref{dynpp_opts} provides a list of command line options, next
-subsection \ref{dynpp_mod} deals with a format of Dynare++ model file,
-and the last subsection discusses incompatibilities between Dynare
-and Dynare++.
-
-\subsection{Command Line Options}
-\label{dynpp_opts}
-
-The calling syntax of the Dynare++ is
-
-{\small
-\begin{verbatim}
-dynare++ [--help] [--version] [options]