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Cosmetic changes to the documentation of BVAR "à la Sims"

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Sébastien Villemot 2011-09-08 18:09:37 +02:00
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doc/bvar-a-la-sims.tex
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@ -1,21 +1,42 @@
\documentclass[10pt,a4paper]{article}
\documentclass[11pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{url}
\usepackage{hyperref}
\hypersetup{breaklinks=true,pagecolor=white,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue}
\usepackage{fullpage}
\usepackage{textcomp}
\newcommand{\df}{\text{df}}
\begin{document}
\title{BVAR models ``\`a la Sims'' in Dynare}
\author{S\'ebastien Villemot\thanks{CEPREMAP. E-mail: \texttt{sebastien.villemot@ens.fr}}}
\date{September 2007}
\title{BVAR models ``\`a la Sims'' in Dynare\thanks{Copyright \copyright~2007--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
at: \url{http://www.gnu.org/licenses/fdl.txt}
\newline
\indent Many thanks to Christopher Sims for providing his BVAR
MATLAB\textregistered~routines, to St\'ephane Adjemian and Michel Juillard
for their helpful support, and to Marek Jaroci\'nski for reporting a bug.
}}
\author{S\'ebastien Villemot\thanks{Paris School of Economics and
CEPREMAP. E-mail:
\href{mailto:sebastien.villemot@ens.fr}{\texttt{sebastien.villemot@ens.fr}}.}}
\date{First version: September 2007 \hspace{1cm} This version: September 2011}
\maketitle
Dynare incorporates routines for BVAR models estimation, that can be used alone or in parallel with a DSGE estimation.
This document describes their implementation and usage.
\begin{abstract}
Dynare incorporates routines for Bayesian VAR models estimation, using a
flavor of the so-called ``Minnesota priors,''. These routines can be used
alone or in parallel with a DSGE estimation. This document describes their
implementation and usage.
\end{abstract}
If you are impatient to try the software and wish to skip mathematical details, jump to section \ref{dynare-commands}.
@ -94,7 +115,7 @@ The second component of the prior is constructed from the likelihood of $T^*$ du
$$p_2(\Phi, \Sigma) \propto |\Sigma|^{-T^*/2} \exp\left\{-\frac{1}{2}Tr(\Sigma^{-1}(Y^*-X^*\Phi)'(Y^*-X^*\Phi))\right\}$$
The dummy observations are constructed according to Sims' version of the Minnesota prior\footnote{See Doan, Litterman and Sims (1984).}.
The dummy observations are constructed according to Sims' version of the Minnesota prior.\footnote{See Doan, Litterman and Sims (1984).}
Before constructing the dummy observations, one needs to choose values for the following parameters:
\begin{itemize}
@ -394,7 +415,7 @@ f(\Phi,\Sigma | \df,S,\hat{\Phi},\Omega) & = & |\Sigma|^{-(\df + ny + 1)/2} \exp
We also note:
$$F(\df,S,\hat{\Phi},\Omega) = \int f(\Phi,\Sigma | \df,S,\hat{\Phi},\Omega)d\Phi d\Sigma$$
The function $F$ has an analytical form, which is given by the normalization constants of matrix-normal and inverse-Wishart densities\footnote{Function \texttt{matricint} of file \texttt{bvar\_density.m} implements the calculation of the log of $F$.}:
The function $F$ has an analytical form, which is given by the normalization constants of matrix-normal and inverse-Wishart densities:\footnote{Function \texttt{matricint} of file \texttt{bvar\_density.m} implements the calculation of the log of $F$.}
$$F(\df,S,\hat{\Phi},\Omega) = (2\pi)^{\frac{ny\cdot k}{2}} |\Omega|^{\frac{ny}{2}} \cdot 2^{\frac{ny\cdot \df}{2}} \pi^{\frac{ny(ny-1)}{4}} |S|^{-\frac{\df}{2}} \prod_{i=1}^{ny} \Gamma\left(\frac{\df + 1 - i}{2}\right) $$
@ -464,7 +485,7 @@ Note that option \texttt{prefilter} implies option \texttt{noconstant}.
Please also note that if option \texttt{loglinear} had been specified in a previous \texttt{estimation} statement, without option \texttt{logdata}, then the BVAR model will be estimated on the log of the provided dataset, for maintaining coherence with the DSGE estimation procedure.
\emph{Restrictions related to the initialization of lags:} in DSGE estimation routines, the likelihood (and therefore the marginal density) are evaluated starting from the observation numbered \texttt{first\_obs + presample} in the datafile\footnote{\texttt{first\_obs} points to the first observation to be used in the datafile (defaults to 1), and \texttt{presample} indicates how many observations after \texttt{first\_obs} will be used to initialize the Kalman filter (defaults to 0).}. The BVAR estimation routines use the same convention (i.e. the first observation of $Y^+$ will be \texttt{first\_obs + presample}). Since we need $p$ observations to initialize the lags, and since we may also use a training sample, the user must ensure that the following condition holds (estimation will fail otherwise):
\emph{Restrictions related to the initialization of lags:} in DSGE estimation routines, the likelihood (and therefore the marginal density) are evaluated starting from the observation numbered \texttt{first\_obs + presample} in the datafile.\footnote{\texttt{first\_obs} points to the first observation to be used in the datafile (defaults to 1), and \texttt{presample} indicates how many observations after \texttt{first\_obs} will be used to initialize the Kalman filter (defaults to 0).} The BVAR estimation routines use the same convention (i.e. the first observation of $Y^+$ will be \texttt{first\_obs + presample}). Since we need $p$ observations to initialize the lags, and since we may also use a training sample, the user must ensure that the following condition holds (estimation will fail otherwise):
$$\texttt{first\_obs} + \texttt{presample} > \texttt{bvar\_prior\_train} + \text{number\_of\_lags}$$
@ -592,16 +613,5 @@ Schorfheide, Frank (2004), ``\textit{Notes on Model Evaluation}'', Department of
Sims, Christopher (2003), ``\textit{Matlab Procedures to Compute Marginal Data Densities for VARs with Minnesota and Training Sample Priors}'', Department of Economics, Princeton University
\section*{Acknowledgements}
Many thanks to Christopher Sims for his BVAR Matlab routines, and to St\'ephane Adjemian and Michel Juillard for their helpful support.
\section*{License}
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:
\url{http://www.gnu.org/licenses/fdl.txt}
\end{document}

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license.txt
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@ -139,6 +139,16 @@ License: GFDL-1.3+
.
A copy of the license can be found at <http://www.gnu.org/licenses/fdl.txt>
Files: doc/bvar_a_la_sims.tex
Copyright: 2007-2011, Sébastien Villemot
License: GFDL-1.3+
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>
Files: dynare++/*.cweb, dynare++/*.hweb, dynare++/*.cpp, dynare++/*.h,
dynare++/*.tex, dynare++/*.mod, dynare++/*.m, dynare++/*.web, dynare++/*.lex,
dynare++/*.y