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 1 2 3 4 5 6 \documentclass[11pt,a4paper]{article} \usepackage{amssymb} \usepackage{amsmath} \usepackage{hyperref} \hypersetup{breaklinks=true,pagecolor=white,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue}  Sébastien Villemot committed Apr 12, 2011 7 8 9 \usepackage{natbib} \usepackage{fullpage}  10 11 12  \begin{document}  Sébastien Villemot committed Apr 12, 2011 13 14 \author{S\'ebastien Villemot\thanks{Paris School of Economics and CEPREMAP. E-mail:  Sébastien Villemot committed Oct 01, 2012 15  \href{mailto:sebastien@dynare.org}{\texttt{sebastien@dynare.org}}.}}  Sébastien Villemot committed Apr 12, 2011 16 17 18 19 20 21 22 23  \title{Solving rational expectations models at first order: \\ what Dynare does\thanks{Copyright \copyright~2009,~2011 S\'ebastien Villemot. 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  Sébastien Villemot committed Nov 18, 2011 24 25 26 27 28  at: \url{http://www.gnu.org/licenses/fdl.txt} \newline The author acknowledges funding through the Seventh Framework Programme for Research (FP7) of the European Commission's Socio-economic Sciences and Humanities (SSH) Program under grant agreement SSH-CT-2009-225149.}  Sébastien Villemot committed Apr 12, 2011 29 30 31 } \date{First version: December 2009 \hspace{1cm} This version: April 2011}  32 33 \maketitle  Sébastien Villemot committed Apr 12, 2011 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 \begin{abstract} This paper describes in detail the algorithm implemented in Dynare for computing the first order approximated solution of a nonlinear rational expectations model. The core of the algorithm is a generalized Schur decomposition (also known as the QZ decomposition), as advocated by several authors in the litterature. The contribution of the present paper is to focus on implementation details that make the algorithm more generic and more efficient, especially for large models. \medskip \noindent \textit{JEL classification:} C63; C68 \medskip \noindent \textit{Keywords:} Dynare; Numerical methods; Perturbation; Rational expectations \end{abstract} \section{Introduction} Perturbation techniques are widely used for solving and estimating rational expectation models and Dynare\footnote{Dynare is a free software platform for handling a wide class of economic models. See \url{http://www.dynare.org} and \citet{adjemian/al:2011} for more details.} offers a popular, user-friendly access to these techniques. The purpose of the present paper is to describe in detail the algorithm implemented in Dynare for computing the first order approximated solution of nonlinear rational expectations models.\footnote{This algorithm is available using the \texttt{stoch\_simul} command of Dynare. The original implementation of this algorithm was done by Michel Juillard, using MATLAB, and is available in the \texttt{matlab/dr1.m} file which is distributed with Dynare. Another implementation was done by the author, in C++, in the \texttt{DecisionRules} class, in the \texttt{mex/sources/estimation} directory of the source tree. The notations used in the present paper are closer to the C++ implementation than to the MATLAB implementation.} This algorithm is based on a generalized Schur decomposition---also known as the QZ decomposition---and is therefore essentially a variation on the methods presented by \citet{klein:2000}, \citet{sims:2001} and \citet{uhlig:1999}. The contribution of this paper is to present some implementation details that make the algorithm more generic and more efficient for large models. In particular I describe the algorithm for removing the leads and lags of more than one in a nonlinear model. I also describe a way of reducing the size of the Schur decomposition problem by dealing separately with endogenous variables that appear only at the current date (called \emph{static} endogenous variables in the following). It should be noted that Dynare is able to go further than first order and can deliver second and third order approximation of the solution of rational expectations models. These higher order solutions can be computed recursively using the first order solution as a starting point. For algorithmic details on this issue, the interested reader can refer to \citet{collard/juillard:2001:compecon} or \citet{schmitt-grohe/uribe:2004}. The rest of this paper is organized as follows. Section \ref{sec:model} presents the class of models to be solved and defines a typology of the endogenous variables. Section \ref{sec:first-order} defines the solution to the model and characterizes its first order approximation. Sections \ref{sec:g-y} and \ref{sec:g-u} describe the algorithm used to recover this first order approximation. \section{The problem to be solved} \label{sec:model} \subsection{The model} In the following, we consider a dynamic nonlinear rational expectations model composed of several variables (endogenous and exogenous) and several equations. In compact form, the model is written as: \label{eq:model} \mathbb{E}_t f(y^+_{t+1}, y_t, y^-_{t-1}, u_t) = 0 where $y_t$ is the vector of endogenous variables, $y^+_{t+1}$ (resp. $y^-_{t-1}$) is the subset of variables of $y_t$ that appear with a lead (resp. a lag), and $u_t$ is the vector of exogenous variables. For obvious identification reasons, the model must contain as many equations as there are endogenous variables; let $n$ be this number. For the timing of variables, the convention used here is the same as in Dynare: a variable decided at date $t$ should have a time subscript of $t$. For example, in a typical RBC model, the capital stock used to produce date $t$ output is actually decided at date $t-1$, so it will be written as $k_{t-1}$ using this convention. Thus accordingly, the law of motion of capital will be $k_t = (1-\delta)k_{t-1} + i_t$. Another way of expressing this timing convention is that stock variables should use the end-of-period'' convention. The vector of exogenous variables $u_t$ (of dimension $p$) follows a Markov process: \begin{equation*} u_t = P(u_{t-1}, \varepsilon_t) \end{equation*} where the $\varepsilon_t$ are independent and identically distributed innovations with zero mean and variance-covariance matrix $\Sigma$.  132   Sébastien Villemot committed Apr 12, 2011 133 134 135 Note that the stochastic process adopted here for exogenous variables is more generic than the one allowed in Dynare (which only accepts a white noise for stochastic variables, \textit{i.e.} $u_t = \varepsilon_t$).  136   Sébastien Villemot committed Apr 12, 2011 137 138 139 140 141 142 143 144 \subsection{Typology of variables} All endogenous variables are required to appear at least at one period. However it is not required that all endogenous variables appear at the current period (a weaker condition is actually sufficient, see assumption (\ref{eq:identification}) below). We define four types of endogenous variables:  145 \begin{description}  Sébastien Villemot committed Apr 12, 2011 146 147 148 149 150 151 152 153 154 155 156 157 158 159 \item[Static endogenous variables:] those that appear only at the current period. Their number is $n^s \leq n$, and their indices in the $y_t$ vector are $\zeta^s_j, j=1\ldots n^s$ \item[Purely forward endogenous variables:] those that appear only at the future period, possibly at the current period, but not at the previous period. Their number is $n^{++} \leq n$, and their indices $\zeta^{++}_j, j=1\ldots n^{++}$ \item[Purely backward endogenous variables:] those that appear only at the previous period, possibly at the current period, but not at the future period. Their number is $n^{--} \leq n$, and their indices $\zeta^{--}_j, j=1\ldots n^{--}$ \item[Mixed endogenous variables:] those that appear both at the future and the previous period, and possibly at the current period. Their number is $n^m \leq n$, and their indices $\zeta^m_j, j=1\ldots n^m$  160 161 \end{description}  Sébastien Villemot committed Apr 12, 2011 162 163 These four types of variables form a partition of the endogenous variables, and we therefore have:  164 165 166 167 168 169 \begin{equation*} n^m + n^{++} + n^{--} + n^s = n \end{equation*} We also define: \begin{description}  Sébastien Villemot committed Apr 12, 2011 170 171 172 173 174 175 176 177 178 \item[Forward endogenous variables:] the union of mixed and purely forward endogenous variables. Their number is $n^+ = n^{++} + n^m$, and their indices $\zeta^+_j, j=1\ldots n^+$. \item[Backward endogenous variables:] the union of mixed and purely backward endogenous variables. Their number is $n^- = n^{--} + n^m$, and their indices $\zeta^-_j, j=1\ldots n^-$ \item[Dynamic endogenous variables:] all the variables except static endogenous variables. Their number is $n^d = n - n^s$, and their indices $\zeta^d_j, j=1\ldots n^d$  179 180 \end{description}  Sébastien Villemot committed Apr 12, 2011 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 The seven indices are such that $1 \leq \zeta^k_1 < \zeta^k_2 < \ldots < \zeta^k_{n^k} \leq n$, where $k \in \{ s, +, ++, -, --, m, d \}$. We denote by $y_t = (y_{1,t}, \ldots, y_{n,t})'$ the vector of endogenous variables at date $t$. We denote by $y^k_t = (y_{\zeta^k_1, t}, \ldots, y_{\zeta^k_{n^k}, t})'$ a subvector of endogenous variables, where $k \in \{ s, +, ++, -, --, m, d \}$. We denote by $\beta^+_j, j=1\ldots n^m$ the indices of mixed endogenous variables inside the $\zeta^+_j$ sequence, \textit{i.e.} $\beta^+_j$ is such that $\zeta^+_{\beta^+_j}$ is a mixed endogenous variable. We similarly define $\beta^-_j$ for mixed endogenous variables inside the $\zeta^-_j$ sequence. We similarly define $\pi^+_j$ (resp. $\pi^-_j$) for purely forward (resp. purely backward) endogenous variables inside $\zeta^+_j$ (resp $\zeta^-_j$). Finally, the vector of \emph{state variables} is formed by the union of backward endogenous variables at the previous period and of exogenous variables at the current period, and is therefore of size $n^-+p$. \subsection{Removing extra leads and lags} The form given in equation (\ref{eq:model}) makes the assumption that endogenous variables appear with at most one lead and one lag, and that exogenous variables appear only at the current period. This assumption does not imply any loss of generality, since it is easy to transform a nonlinear model with many leads and lags into an equivalent model of the form given in (\ref{eq:model}), as is detailed below.\footnote{The algorithm described in the present section is implemented in the Dynare preprocessor, since version 4.1. Auxiliary variables are created automatically and will show up at several places in Dynare output; see \citet{adjemian/al:2011} for the names of these variables.} For every variable $x_t$ in the original model whose maximum lag is $x_{t-d-k}$ with $k>0$ (and $d=1$ if $x$ is an endogenous variable or $d=0$ if it is an exogenous variable), the transformation is the following: \begin{itemize} \item introduce $k$ new endogenous variables $z^j_t$, for $j\in\{1,\ldots,k\}$; \item add $k$ new equations to the model: \begin{equation*} \left\{\begin{array}{ll} z^1_t = x_{t-d} & \\ z^j_t = z^{j-1}_{t-1} & \text{for } j\in\{2, \ldots,k\} \end{array}\right.; \end{equation*} \item replace all occurrences of $x_{t-d-j}$ (with $j>0$) in the original model by $z^j_{t-1}$ in the transformed model. \end{itemize}  230   Sébastien Villemot committed Apr 12, 2011 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 The transformation for variables with a lead is a bit more elaborate because one has to handle the fact that there is an expectation operator in front of all equations. The algorithm is as follows: \begin{itemize} \item decompose every equation of the original model in the following form: \begin{equation*} A + \sum_{i\in I} B_i \, \mathbb{E}_t C_i = 0 \end{equation*} where $A$ and the $B_i$ are (possibly nonlinear) expressions containing only current or lagged variables, and the $C_i$ are (possibly nonlinear) expressions which may contain leads; this decomposition is not unique, but one should aim at making the $C_i$ terms as simple as possible; \item for every $C_i$ where there is a lead of 2 or more on an endogenous variable, or a lead on an exogenous variable: \begin{itemize} \item let $k$ be the minimal number of periods so that $C_i^{(-k)}$ has at most one lead on endogenous variables and no lead on exogenous variables (where $C_i^{(-k)}$ stands for the transformation of $C_i$ where all variables have been lagged by $k$ periods); \item introduce $k$ new endogenous variables $z^j_t$, for $j\in\{1,\ldots,k\}$; \item add $k$ new equations to the model: \begin{equation*} \left\{\begin{array}{ll} z^1_t = \mathbb{E}_tC_i^{(-k)} & \\ z^j_t = \mathbb{E}_tz^{j-1}_{t+1} & \text{for } j\in\{2, \ldots,k\} \end{array}\right.; \end{equation*} \item replace all occurrences of $\mathbb{E}_t C_i$ in the original model by $\mathbb{E}_tz^k_{t+1}$ in the transformed model. \end{itemize} \end{itemize} It is straightforward to see that this transformed model is under the form given in (\ref{eq:model}). And by the law of iterated expectations, it is equivalent to the original one.  266   Sébastien Villemot committed Apr 12, 2011 267 268 \section{The solution and its first order approximation} \label{sec:first-order}  269   Sébastien Villemot committed Apr 12, 2011 270 271 We first define the deterministic steady state of the model as the vector $(\bar{y}, \bar{u}, \bar{\varepsilon})$ satisfying:  272 \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 273  \bar{\varepsilon} = 0  274 275 \end{equation*} \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 276  \bar{u} = P(\bar{u}, \bar{\varepsilon})  277 278 279 280 \end{equation*} \begin{equation*} f(\bar{y}^+, \bar{y}, \bar{y}^-, \bar{u}) = 0 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 281 282 283 284 285 286 Finding the deterministic steady state involves the resolution of a multivariate nonlinear system.\footnote{Dynare offers efficient ways of performing this task, but this is out of the scope of the present paper.} Then, finding the rational expectation solution of the model means finding the policy functions (also known as decision rules), which give current endogenous variables as a function of state variables:  287 \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 288  y_t = g(y^-_{t-1}, u_t)  289 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 290 291 Note that, by definition of the deterministic steady state, we have $\bar{y} = g(\bar{y}^-, \bar{u})$.  292   Sébastien Villemot committed Apr 12, 2011 293 The function $g$ is characterized by the following functional equation:  294 295  \label{eq:g-definition}  Sébastien Villemot committed Apr 12, 2011 296  \mathbb{E}_t f\left[g^+(g^-(y^-_{t-1}, u_t), u_{t+1}), g(y^-_{t-1}, u_t), y^-_{t-1}, u_t\right] = 0  297   Sébastien Villemot committed Apr 12, 2011 298 299 where $g^+$ (resp. $g^-$) is the restriction of $g$ to forward (resp. backward) endogenous variables.  300   Sébastien Villemot committed Apr 12, 2011 301 302 303 304 In the general case, this functional equation cannot be solved exactly, and one has to resort to numerical techniques to get an approximated solution. The remainder of this paper describes the first order perturbation technique implemented in Dynare. Let:  305 306 307 308 309 310 \begin{equation*} f_{y^+} = \frac{\partial f}{\partial y^+_{t+1}}, \; f_{y^0} = \frac{\partial f}{\partial y_t}, \; f_{y^-} = \frac{\partial f}{\partial y^-_{t-1}}, \; f_u = \frac{\partial f}{\partial u_t} \end{equation*} \begin{equation*} g_y = \frac{\partial g}{\partial y^-_{t-1}},\;  Sébastien Villemot committed Apr 12, 2011 311 312 313 314 315  g_u = \frac{\partial g}{\partial u_t} \end{equation*} \begin{equation*} P_u = \frac{\partial P}{\partial u_{t-1}}, \: P_\varepsilon = \frac{\partial P}{\partial \varepsilon_t}  316 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 317 318 where the derivatives are taken at $\bar{y}$, $\bar{u}$ and $\bar{\varepsilon}$.  319   Sébastien Villemot committed Apr 12, 2011 320 321 322 323 324 325 326 327 328 The first order approximation of the policy function is therefore: \begin{equation*} \hat{g}(y^-_{t-1}, u_t) = \bar{y} + g_y \hat{y}^-_{t-1} + g_u \hat{u}_t \end{equation*} where $\hat{y}^-_{t-1} = y^-_{t-1} - \bar{y}^-$, $\hat{u}_t = u_t - \bar{u}$, and $g_y$ and $g_u$ are unknowns at this stage. A first order approximation of (\ref{eq:g-definition}) around $\bar{y}$ and $\bar{u}$ gives:  329 \begin{multline*}  Sébastien Villemot committed Apr 12, 2011 330 331 332  f(\bar{y}^+, \bar{y}, \bar{y}^-, \bar{u}) + f_{y^+} [g^+_y(g^-_y \hat{y}^-_{t-1} + g^-_u \hat{u}_t) + g^+_u \mathbb{E}_t [P_u \hat{u}_t + P_\varepsilon \varepsilon_{t+1}] ] \\ + f_{y^0} (g_y \hat{y}^-_{t-1} + g_u \hat{u}_t) + f_{y^-}\hat{y}^-_{t-1} + f_u \hat{u}_t = 0  333 \end{multline*}  Sébastien Villemot committed Apr 12, 2011 334 335 336 337 where $g^+_y$, $g^-_y$, $g^-_u$, $g^+_u$ are the derivatives of the restrictions of $g$ with obvious notation. Computing the expectancy term, taking into account the property of the deterministic steady state, and reorganizing the terms, we obtain:  338 339 340 341 342  \label{eq:first-order} (f_{y^+} g^+_y g^-_y + f_{y^0} g_y + f_{y^-}) \hat{y}^-_{t-1} + (f_{y^+} g^+_yg^-_u+ f_{y^+}g^+_u P_u + f_y g_u + f_u) \hat{u}_t = 0  Sébastien Villemot committed Apr 12, 2011 343 344 345 In the next sections, we exploit this equation in order to recover the unknown coefficients $g_u$ and $g_y$.  346 \section{Recovering $g_y$}  Sébastien Villemot committed Apr 12, 2011 347 \label{sec:g-y}  348   Sébastien Villemot committed Apr 12, 2011 349 350 Taking into account the term multiplying $\hat{y}^-_{t-1}$, equation (\ref{eq:first-order}) imposes:  351 352 353 354 355 356 357 358 359 \begin{equation*} f_{y^+} g^+_y g^-_y + f_{y^0} g_y + f_{y^-} = 0 \end{equation*} This amounts to: \label{eq:gy} f_{y^+} \hat{y}^+_{t+1} + f_{y^0} \hat{y}_t + f_{y^-} \hat{y}^-_{t-1} = 0  Sébastien Villemot committed Apr 12, 2011 360 361 362 363 364 Let $S$ be the $n\times n^s$ submatrix of $f_{y^0}$ where only the columns for static endogenous variables are kept, \textit{i.e.} $S_{i,j} = f_{y^0, i, \zeta^s_j}$. A QR decomposition\footnote{See \citet[section 5.2]{golub/van-loan:1996}.} gives $S = QR$ where $Q$ is an $n\times n$ orthogonal matrix and $R$ an $n\times n^s$ upper triangular matrix.  365 366 367 368  For the model to be identified, we assume that: \label{eq:identification}  Sébastien Villemot committed Apr 12, 2011 369  \mathop{rank}(R) = n^s.  370   Sébastien Villemot committed Apr 12, 2011 371 Thus, equation (\ref{eq:gy}) can be rewritten as:  372 373 374 375  \label{eq:gy-qr} A^+ \hat{y}^+_{t+1} + A^0 \hat{y}_t + A^- \hat{y}^-_{t-1} = 0  Sébastien Villemot committed Apr 12, 2011 376 377 378 where $A^+ = Q'f_{y^+}$, $A^0 = Q'f_{y^0}$ and $A^- = Q'f_{y^-}$. By construction, columns of static endogenous variables in $A^0$ are zero in their lower part: $\forall i > n^s,\forall j\leq n^s, \: A^0_{i,\zeta^s_j} = 0$.  379   Sébastien Villemot committed Apr 12, 2011 380 \subsection{Non-static endogenous variables}  381 382 383 384 385 386  Taking only the $n^d$ lower rows of system (\ref{eq:gy-qr}), we get: \label{eq:gy-no-static} \tilde{A}^+ \hat{y}^+_{t+1} + \tilde{A}^{0+} \hat{y}^+_t + \tilde{A}^{0-} \hat{y}^-_t + \tilde{A}^- \hat{y}^-_{t-1} = 0  Sébastien Villemot committed Apr 12, 2011 387 388 389 where $\tilde{A}^+$ (resp. $\tilde{A}^-$) contains the last $n^d$ rows of $A^+$ (resp. $A^-$). Matrices $\tilde{A}^{0+}$ and $\tilde{A}^{0-}$ can be defined in two ways, depending on where we deal with mixed endogenous variables:  390 391  \begin{itemize}  Sébastien Villemot committed Apr 12, 2011 392 393 394 395 396 397 398 399 400 401 402 403 \item $\tilde{A}^{0+}$ is a submatrix of $A^0$ where only the last $n^d$ rows and the columns for forward endogenous variables are kept ($\tilde{A}^{0+}_{i,j} = A^0_{n^s+i, \zeta^+_j}$), and $\tilde{A}^{0-}$ is such that purely backward columns are taken from $A^0$ ($\tilde{A}^{0-}_{i,\pi^-_j} = A^0_{n^s+i,\zeta^{--}_j}$), and the rest is zero; \item $\tilde{A}^{0-}$ is a submatrix of $A^0$ where only the last $n^d$ rows and the columns for backward endogenous variables are kept ($\tilde{A}^{0-}_{i,j} = A^0_{n^s+i, \zeta^-_j}$), and $\tilde{A}^{0+}$ is such that purely forward columns are taken from $A^0$ ($\tilde{A}^{0+}_{i,\pi^+_j} = A^0_{n^s+i,\zeta^{++}_j}$), and the rest is zero.  404 \end{itemize}  Sébastien Villemot committed Apr 12, 2011 405 406 Note that in equation (\ref{eq:gy-no-static}), static endogenous variables no longer appear.  407 408 409 410  The structural state space representation of (\ref{eq:gy-no-static}) is: \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 411 412 413 414 415 416 417 418  \underbrace{ \left( \begin{matrix} \tilde{A}^{0-} & \tilde{A}^+ \\ I^- & 0 \end{matrix} \right) }_D  419 420  \left( \begin{matrix}  Sébastien Villemot committed Apr 12, 2011 421 422  \hat{y}^-_t \\ \hat{y}^+_{t+1}  423 424  \end{matrix} \right)  Sébastien Villemot committed Apr 12, 2011 425 426 427 428 429 430 431 432 433  = \underbrace{ \left( \begin{matrix} -\tilde{A}^- & -\tilde{A}^{0+} \\ 0 & I^+ \end{matrix} \right) }_E  434 435  \left( \begin{matrix}  Sébastien Villemot committed Apr 12, 2011 436 437  \hat{y}^-_{t-1} \\ \hat{y}^+_t  438 439 440  \end{matrix} \right) \end{equation*}  Sébastien Villemot committed Apr 12, 2011 441 442 443 444 445 446 447 448 where $I^-$ is an $n^m \times n^-$ selection matrix for mixed endogenous variables, such that $I^-_{i,\beta^-_i}=1$, and zero otherwise. Similarly, $I^+$ is an $n^m \times n^+$ matrix, such that $I^+_{i,\beta^+_i}=1$, and zero otherwise. Therefore, $D$ and $E$ are square matrices of size $n^{++}+n^{--}+2n^m$. Using the fact that $\hat{y}^+_{t+1} = g^+_y \hat{y}^-_t$, this can be rewritten as:  449   Sébastien Villemot committed Apr 12, 2011 450  \label{eq:state-space}  451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470  D \left( \begin{matrix} I_{n^-} \\ g^+_y \end{matrix} \right) \hat{y}^-_t = E \left( \begin{matrix} I_{n^-} \\ g^+_y \end{matrix} \right) \hat{y}^-_{t-1} where $I_{n^-}$ is the identity matrix of size $n^-$.  Sébastien Villemot committed Apr 12, 2011 471 472 473 474 A generalized Schur decomposition (also known as the QZ decomposition) of the pencil $(D,E)$ is performed:\footnote{See \citet[section 7.7]{golub/van-loan:1996} for theoretical and practical details on this decomposition.}  475 \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 476 477 478 479 480  \left\{\begin{array}{rcl} D & = & QTZ \\ E & = & QSZ \end{array} \right.  481 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 482 483 484 485 where $T$ is upper triangular, $S$ quasi upper triangular, and $Q$ and $Z$ are orthogonal matrices. The decomposition is done is such a way that stable generalized eigenvalues (modulus less than 1) are in the upper left corner of $T$ and $S$.  486   Sébastien Villemot committed Apr 12, 2011 487 488 489 490 Matrices $T$ and $S$ are block decomposed so that the upper left block of both matrices is square and contains generalized eigenvalues of modulus less than 1, and the lower right block is square and contains generalized eigenvalues of modulus strictly greater than 1.  491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534  Equation (\ref{eq:state-space}) can be rewritten as: \label{eq:state-space-qz} \left( \begin{matrix} T_{11} & T_{12} \\ 0 & T_{22} \end{matrix} \right) \left( \begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix} \right) \left( \begin{matrix} I_{n^-} \\ g^+_y \end{matrix} \right) \hat{y}^-_t = \left( \begin{matrix} S_{11} & S_{12} \\ 0 & S_{22} \end{matrix} \right) \left( \begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix} \right) \left( \begin{matrix} I_{n^-} \\ g^+_y \end{matrix} \right) \hat{y}^-_{t-1}  Sébastien Villemot committed Apr 12, 2011 535 536 537 where $T_{11}$ and $S_{11}$ are square and contain stable generalized eigenvalues, while $T_{22}$ and $S_{22}$ are square and contain explosive generalized eigenvalues.  538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566  To exclude explosive trajectories, we impose: \label{eq:non-explosive} \left( \begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix} \right) \left( \begin{matrix} I_{n^-} \\ g^+_y \end{matrix} \right) = \left( \begin{matrix} X \\ 0 \end{matrix} \right) which implies: \begin{equation*} g^+_y = -(Z_{22})^{-1} Z_{21} \end{equation*}  Sébastien Villemot committed Apr 12, 2011 567 568 569 570 Note that the squareness of $Z_{22}$ is the \citet{blanchard/kahn:1980} \emph{order} condition (\textit{i.e.} the requirement to have as many explosive eigenvalues as forward or mixed endogenous variables), and the non-singularity of $Z_{22}$ is the \citet{blanchard/kahn:1980} \emph{rank} condition.  571   Sébastien Villemot committed Apr 12, 2011 572 573 Using equation (\ref{eq:non-explosive}) and the fact that $\hat{y}^-_t = g^-_y \hat{y}^-_{t-1}$, equation (\ref{eq:state-space-qz}) implies:  574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 \begin{equation*} \left( \begin{matrix} T_{11} & T_{12} \\ 0 & T_{22} \end{matrix} \right) \left( \begin{matrix} X \\ 0 \end{matrix} \right) g^-_y = \left( \begin{matrix} S_{11} & S_{12} \\ 0 & S_{22} \end{matrix} \right) \left( \begin{matrix} X \\ 0 \end{matrix} \right) \end{equation*}  Sébastien Villemot committed Apr 12, 2011 602 603 604 Then, using the fact that solving equation (\ref{eq:non-explosive}) for $X$ gives $X = (Z'_{11})^{-1}$, the upper part of this system gives the solution for $g^-_y$:  605 606 607 608 \begin{equation*} g^-_y = X^{-1} T_{11}^{-1}S_{11}X = Z'_{11}T_{11}^{-1}S_{11}(Z'_{11})^{-1} \end{equation*}  Sébastien Villemot committed Apr 12, 2011 609 610 611 612 Note that mixed variables appear in both $g^+$ and $g^-$: the corresponding lines will be equal across the two matrices by construction. \subsection{Static endogenous variables}  613 614  The $n^s$ upper lines of equation (\ref{eq:gy-qr}) can be written as:  Sébastien Villemot committed Apr 12, 2011 615 616  \label{eq:static-part}  617  \breve{A}^+ \hat{y}^+_{t+1} + \breve{A}^{0d} \hat{y}^d_t + \breve{A}^{0s} \hat{y}^{s}_t + \breve{A}^- \hat{y}^-_{t-1} = 0  Sébastien Villemot committed Apr 12, 2011 618 619 620 621 622 623 624 625 626  where $\breve{A}^+$ (resp. $\breve{A}^-$) contains the first $n^s$ rows of $A^+$ (resp. $A^-$). Matrix $\breve{A}^{0s}$ (resp. $\breve{A}^{0d}$) contains the first $n^s$ rows and only the static (resp. non-static) columns of $A^0$. Recall that $\breve{A}^{0s}$ is a square upper triangular matrix by construction, and it is invertible because of assumption (\ref{eq:identification}). Equation (\ref{eq:static-part}) can be rewritten as:  627 628 629 \begin{equation*} \breve{A}^+ g^+_y g^-_y \hat{y}^-_{t-1} + \breve{A}^{0d} g^d_y \hat{y}^-_{t-1} + \breve{A}^{0s} \hat{y}^{s}_t + \breve{A}^- \hat{y}^-_{t-1} = 0 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 630 631 where $g^d_y$, the restriction of $g_y$ to non-static endogenous variables, is obtained by combining $g^+_y$ and $g^-_y$. We therefore have:  632 633 634 635 636 \begin{equation*} g^s_y = -\left[\breve{A}^{0s}\right]^{-1} \left(\breve{A}^+ g^+_y g^-_y + \breve{A}^{0d} g^d_y + \breve{A}^-\right) \end{equation*} \section{Recovering $g_u$}  Sébastien Villemot committed Apr 12, 2011 637 \label{sec:g-u}  638 639 640  Equation (\ref{eq:first-order}) restricted to $\hat{u}_t$ imposes: \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 641  f_{y^+} g^+_yg^-_u+ f_{y^+}g^+_u P_u + f_y g_u + f_u = 0,  642 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 643 644 645 646 647 648 649 650 and be rewritten as: \begin{equation*} (f_{y^+} g^+_y J^- + f_y) g_u + f_{y^+}J^+ g_u P_u + f_u = 0 \end{equation*} where $J^-$ (resp $J^+$) is an $n^-\times n$ matrix (resp. $n^+\times n$ matrix) selecting only the backward (resp. forward) endogenous variables. In the particular case solved by Dynare, where $P_u = 0$, the solution to this equation is:  651 \begin{equation*}  Sébastien Villemot committed Apr 12, 2011 652  g_u = -(f_{y^+} g^+_y J^- + f_y)^{-1} f_u  653 \end{equation*}  Sébastien Villemot committed Apr 12, 2011 654 655 656 657 In the general case, this equation is a specialized Sylvester equation, which can be solved using the algorithm proposed by \citet{kamenik:2003}\footnote{This paper is distributed with Dynare, in the \texttt{sylvester.pdf} file under the documentation directory.}.  658   Sébastien Villemot committed Apr 12, 2011 659 660 \bibliographystyle{ecta} \bibliography{dr}  661 662  \end{document}