487 lines
23 KiB
TeX
487 lines
23 KiB
TeX
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\begin{document}
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\title{Sensitivity Analysis Toolbox for DYNARE\thanks{Copyright \copyright~2012 Dynare
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Team. Permission is granted to copy, distribute and/or modify
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this document under the terms of the GNU Free Documentation
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License, Version 1.3 or any later version published by the Free
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Software Foundation; with no Invariant Sections, no Front-Cover
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Texts, and no Back-Cover Texts. A copy of the license can be found
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at: \url{https://www.gnu.org/licenses/fdl.txt}}}
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\author{Marco Ratto\\
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European Commission, Joint Research Centre \\
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TP361, IPSC, \\21027 Ispra
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(VA) Italy\\
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\texttt{marco.ratto@jrc.ec.europa.eu}
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\thanks{The author gratefully thanks Christophe Planas, Kenneth Judd, Michel Juillard,
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Alessandro Rossi, Frank Schorfheide and the participants to the
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Courses on Global Sensitivity Analysis for Macroeconomic
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Models (Ispra, 2006-2007-2008-2010) for interesting discussions and
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helpful suggestions.}}
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%\date{\today \thanks{Authors gratefully acknowledge the
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%contribution by ... for ...}}
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\maketitle %\tableofcontents
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%-----------------------------------------------------------------------
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\begin{abstract}
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\noindent The Sensitivity Analysis Toolbox for DYNARE is a set of
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MATLAB routines for the analysis of DSGE models with global
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sensitivity analysis. The routines are thought to be used within
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the DYNARE v4 environment.
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\begin{description}
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\item \textbf{Keywords}: Stability Mapping , Reduced form solution, DSGE models,
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Monte Carlo filtering, Global Sensitivity Analysis.
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\end{description}
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\end{abstract}
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\newpage
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% ----------------------------------------------------------------
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\section{Introduction} \label{s:intro}
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The Sensitivity Analysis Toolbox for DYNARE is a collection of
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MATLAB routines implemented to answer the following questions: (i)
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Which is the domain of structural coefficients assuring the
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stability and determinacy of a DSGE model? (ii) Which parameters
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mostly drive the fit of, e.g., GDP and which the fit of inflation?
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Is there any conflict between the optimal fit of one observed
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series versus another one? (iii) How to represent in a direct,
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albeit approximated, form the relationship between structural
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parameters and the reduced form of a rational expectations model?
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The discussion of the methodologies and their application is
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described in \cite{Ratto_CompEcon_2008}.
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\section{Use of the Toolbox}
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The DYNARE parser now recognizes sensitivity analysis commands.
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The syntax is based on a single command:
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\vspace{0.5cm}
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\verb"dynare_sensitivity(option1=<opt1_val>,option2=<opt2_val>,...)"
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\vspace{0.5cm} \noindent with a list of options described in the
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next section.
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With respect to the previous version of the toolbox, in order to
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work properly, the sensitivity analysis Toolbox \emph{no longer}
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needs that the DYNARE estimation environment is set-up.
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Therefore, \verb"dynare_sensitivity" is the only command to run to
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make a sensitivity analysis on a DSGE model\footnote{Of course,
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when the user needs to perform the mapping of the fit with a
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posterior sample, a Bayesian estimation has to be performed
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beforehand}.
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\section{List of options}
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\subsection{Sampling options}
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\begin{tabular}{r|l|l}
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% after \\ : \hline or \cline{col1-col2} \cline{col3-col4} ...
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option name & default & description \\ \hline
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\verb"Nsam"& 2048& Size of MC sample \\
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\verb"ilptau"& 1& 1 = use $LP_\tau$ quasi-Monte Carlo \\
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& & 0 = use LHS Monte Carlo \\
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\verb"pprior"& 1& 1 = sample from prior distributions\\
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& & 0 = sample from multivariate
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normal \\
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& & \hspace{0.5 cm} $N(\hat{\theta},\Sigma)$, $\hat{\theta}$ is posterior mode \\
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& & \hspace{0.5 cm} $\Sigma = H^{-1}$, $H$ is Hessian at the mode\\
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\verb"prior_range"& 1& 1 = sample \textit{uniformly} from prior ranges\\
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& & 0 = sample from prior distributions: \\
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\verb"morris"& 0& 0 = no Morris sampling for screening \\
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& & 1 = Morris sampling for screening \\
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\verb"morris_nliv"& 6& number of levels in Morris design\\
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\verb"morris_ntra"& 20& number of trajectories in Morris design\\
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\verb"ppost"& 0& 0 = don't use Metropolis posterior sample\\
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& & 1 = use Metropolis posterior sample: this \\
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& & \hspace{0.5 cm} overrides any other sampling option! \\
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\verb"neighborhood_width"& []& $\delta$ (real number$>0$) uniform sample in the\\
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& & neighborhood of the posterior mode $\hat{\theta}$ \\
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& & interval width: $\hat{\theta}(1\pm\delta)$ \\\hline
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\end{tabular}
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\subsection{Stability mapping}
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\begin{tabular}{r|l|l}
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option name & default & description \\ \hline
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\verb"stab"& 1& 1 = perform stability mapping \\
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& & 0 = no stability mapping is performed\\
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\verb"load_stab"& 0& 0 = generate a new sample\\
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& & 1 = load a previously created sample \\
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\verb"pvalue_corr"& 0.001& critical p-value for correlations $\rho$ in filtered samples:\\
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& & plot couples of parameters with \\
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& & p-value$<$\verb"pvalue_corr"\\
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\verb"pvalue_ks" & 0.001& critical p-value for Smirnov statistics $d$: \\
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& & plot parameters with p-value$<$\verb"pvalue_ks"\\
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\verb"lik_init" & 1& 1 = the model is stationary (unit roots are `explosive')\\
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& & 3 = the model has unit roots (unit roots are `stable')\\ \hline
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\end{tabular}
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\newpage
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\subsection{Reduced form mapping}% and identification}
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The mapping of the reduced form solution forces the use of samples
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from prior ranges or prior distributions, i.e.:
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\\
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\verb"options_.opt_gsa.pprior=1;"\\
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\verb"options_.opt_gsa.ppost=0;"\\
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It uses 250 samples to optimize smoothing parameters and 1000
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samples to compute the fit. The rest of the sample is used for
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out-of-sample validation. \vspace{0.5cm}
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\begin{tabular}{r|l|l}
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option name & default & description \\ \hline
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\verb"redform"& 0& 0 = don't prepare MC sample of \\
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& & reduced form matrices \\
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& & 1 = prepare MC sample of \\
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& & reduced form matrices \\
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\verb"load_redform"& 0& 0 = estimate the mapping of \\
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& & reduced form model\\
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& & 1 = load previously estimated mapping\\
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\verb"logtrans_redform"& 0& 0 = use raw entries\\
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& & 1 = use log-transformed entries \\
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\verb"threshold_redform"& []& [] = don't filter MC entries \\
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& & of reduced form coefficients\\
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& & [\verb"max" \verb"max"] = analyse filtered \\
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& & entries within the range [\verb"max" \verb"max"] \\
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\verb"ksstat_redform"& 0.001& critical p-value for Smirnov statistics $d$ \\
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& & when \verb"threshold_redform" is active\\
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& & plot parameters with p-value$<$\verb"ksstat_redform"\\
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\verb"alpha2_redform"& 0& critical p-value for correlation $\rho$ \\
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& & when \verb"threshold_redform" is active\\
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& & plot couples of parameters with \\
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& & p-value$<$\verb"alpha2_redform"\\
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\verb"namendo"& () & list of endogenous variables \\
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& : & jolly character to indicate ALL endogenous \\
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\verb"namlagendo"& () & list of lagged endogenous variables:\\
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& & analyse entries [\verb"namendo"$\times$\verb"namlagendo"]\\
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& : & jolly character to indicate ALL endogenous \\
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\verb"namexo"& ()& list of exogenous variables:\\
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& & analyse entries
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[\verb"namendo"$\times$\verb"namexo"]\\
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& : & jolly character to indicate ALL exogenous \\\hline
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\end{tabular}
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\vspace{0.5cm} \\
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One can also load a previously estimated mapping
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with a new MC sample, to look at the forecast for the new MC sample.\\
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\subsection{Mapping the fit}
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The RMSE analysis can be performed with different types of
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sampling options:
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\begin{enumerate}
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\item when \verb"pprior=1" and \verb"ppost=0", the Toolbox analyses the RMSE's for
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the MC sample obtained by sampling parameters from their prior
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distributions (or prior ranges): this analysis provides some hints
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about what parameter drives the fit of which observed series,
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prior to the full estimation;
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\item when \verb"pprior=0" and \verb"ppost=0", the Toolbox analyses the RMSE's for
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a multivariate normal MC sample, with covariance matrix based on
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the inverse Hessian at the optimum: this analysis is useful when
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ML estimation is done (i.e. no Bayesian estimation);
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\item when \verb"ppost=1" the Toolbox analyses
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the RMSE's for the posterior sample obtained by DYNARE's
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Metropolis procedure.
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\end{enumerate}
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The use of cases 2. and 3. requires an estimation step beforehand!
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To facilitate the sensitivity analysis after estimation, the
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\verb"dynare_sensitivity" command also allows to indicate some
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options of \verb"dynare_estimation". These are:
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\begin{itemize}
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\item \verb"datafile"
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\item \verb"mode_file"
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\item \verb"first_obs"
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\item \verb"lik_init"
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\item \verb"nobs"
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\item \verb"prefilter"
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\item \verb"presample"
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\item \verb"loglinear"
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\end{itemize}
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\vspace{1cm}
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\begin{tabular}{r|l|l}
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option name & default & description \\ \hline
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\verb"rmse"& 0& 0 = no RMSE analysis\\
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& & 1 = do RMSE analysis \\
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\verb"load_rmse"& 0& 0 = make a new RMSE analysis\\
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& & 1 = load previous RMSE analysis \\
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\verb"lik_only"& 0& 0 = compute RMSE's for all observed series\\
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& & 1 = compute only likelihood and posterior \\
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\verb"var_rmse"& varobs& list of observed series to be considered\\
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\verb"pfilt_rmse"& 0.1& filtering threshold for RMSE's: default it to\\
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& & filter the best 10\% for each observed series\\
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\verb"istart_rmse"& 1& start computing RMSE's from \verb"istart_rmse":\\
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& & use 2 to avoid big initial error \\
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\verb"alpha_rmse"& 0.001& p-value for Smirnov statistics $d$:\\
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& & plot parameters with p-value$<$\verb"alpha_rmse"\\
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\verb"alpha2_rmse"& 0& p-value for correlation $\rho$\\
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& & plot couples of parameters with
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p-value$<$\verb"alpha2_rmse"\\
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\end{tabular}
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\subsection{Screening analysis}
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The screening analysis does not require any additional options
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with respect to those listed in the `Sampling options':
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\verb"morris", \verb"morris_nliv", \verb"morris_ntra". The Toolbox
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performs all the analyses required and displays results.
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\subsection{Identification analysis}
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Setting the option \verb"identification=1", an identification
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analysis based on theoretical moments is performed. Sensitivity plots are provided that
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allow to infer which parameters are most likely to be less
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identifiable.
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\vspace{1cm}
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\begin{tabular}{r|l|l}
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option name & default & description \\ \hline
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\verb"identification"& 0 & 0 = no identification analysis \\
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& & 1 = performs identification analysis:\\
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& & this forces \verb"redform"=0 and default \verb"morris"=1\\
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\verb"morris"& 1 & 1 = Screening analysis (Type II error)\\
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& & 2 = Analytic derivatives \citep{Iskrev2010,Iskrev2011}\\
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\verb"morris_nliv"& 6 & number of levels in Morris design\\
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\verb"morris_ntra"& 20& number of trajectories in Morris design\\
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\end{tabular}
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\vspace{1cm}
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\noindent For example, the following commands in the DYNARE model file
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\vspace{1cm}
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\noindent\verb"dynare_sensitivity(identification=1, morris=2);"
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\vspace{1cm}
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\noindent trigger the identification analysis using \cite{Iskrev2010,Iskrev2011}, jointly with the mapping of the acceptable region.
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\newpage
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\section{Directory structure}
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Sensitivity analysis results are saved on the hard-disk of the
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computer. The Toolbox uses a dedicated folder called \verb"GSA",
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located in \\
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\\
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\verb"<DYNARE_file>\GSA", \\
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\\
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where \verb"<DYNARE_file>.mod" is the name of the DYNARE model
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file.
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\subsection{Binary data files}
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A set of binary data files is saved in the \verb"GSA" folder:
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\begin{description}
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\item[]\verb"<DYNARE_file>_prior.mat": this file stores
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information about the analyses performed sampling from the prior
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ranges, i.e. \verb"pprior=1" and \verb"ppost=0";
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\item[]\verb"<DYNARE_file>_mc.mat": this file stores
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information about the analyses performed sampling from
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multivariate normal, i.e. \verb"pprior=0" and \verb"ppost=0";
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\item[]\verb"<DYNARE_file>_post.mat": this file stores information
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about analyses performed using the Metropolis posterior sample,
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i.e. \verb"ppost=1".
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\end{description}
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\begin{description}
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\item[]\verb"<DYNARE_file>_prior_*.mat": these files store
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the filtered and smoothed variables for the prior MC sample,
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generated when doing RMSE analysis (\verb"pprior=1" and
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\verb"ppost=0");
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\item[]\verb"<DYNARE_file>_mc_*.mat": these files store
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the filtered and smoothed variables for the multivariate normal MC
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sample, generated when doing RMSE analysis (\verb"pprior=0" and
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\verb"ppost=0").
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\end{description}
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\subsection{Stability analysis}
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Figure files \verb"<DYNARE_file>_prior_*.fig" store results for
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the stability mapping from prior MC samples:
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\begin{description}
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\item[]\verb"<DYNARE_file>_prior_stab_SA_*.fig": plots of the Smirnov
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test analyses confronting the cdf of the sample fulfilling
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Blanchard-Kahn conditions with the cdf of the rest of the sample;
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\item[]\verb"<DYNARE_file>_prior_stab_indet_SA_*.fig": plots of the Smirnov
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test analyses confronting the cdf of the sample producing
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indeterminacy with the cdf of the original prior sample;
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\item[]\verb"<DYNARE_file>_prior_stab_unst_SA_*.fig": plots of the Smirnov
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test analyses confronting the cdf of the sample producing unstable
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(explosive roots) behaviour with the cdf of the original prior
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sample;
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\item[]\verb"<DYNARE_file>_prior_stable_corr_*.fig": plots of
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bivariate projections of the sample fulfilling Blanchard-Kahn
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conditions;
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\item[]\verb"<DYNARE_file>_prior_indeterm_corr_*.fig": plots of
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bivariate projections of the sample producing indeterminacy;
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\item[]\verb"<DYNARE_file>_prior_unstable_corr_*.fig": plots of
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bivariate projections of the sample producing instability;
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\item[]\verb"<DYNARE_file>_prior_unacceptable_corr_*.fig": plots of
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bivariate projections of the sample producing unacceptable
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solutions, i.e. either instability or indeterminacy or the
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solution could not be found (e.g. the steady state solution could
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not be found by the solver).
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\end{description}
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Similar conventions apply for \verb"<DYNARE_file>_mc_*.fig" files,
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obtained when samples from multivariate normal are used.
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\subsection{RMSE analysis}
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Figure files \verb"<DYNARE_file>_rmse_*.fig" store results for the
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RMSE analysis.
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\begin{description}
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\item[]\verb"<DYNARE_file>_rmse_prior*.fig": save results for
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the analysis using prior MC samples;
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\item[]\verb"<DYNARE_file>_rmse_mc*.fig": save results for
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the analysis using multivariate normal MC samples;
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\item[]\verb"<DYNARE_file>_rmse_post*.fig": save results for
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the analysis using Metropolis posterior samples.
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\end{description}
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The following types of figures are saved (we show prior sample to
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fix ideas, but the same conventions are used for multivariate
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normal and posterior):
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\begin{description}
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\item[]\verb"<DYNARE_file>_rmse_prior_*.fig": for each parameter, plots the cdf's
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corresponding to the best 10\% RMES's of each observed series;
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\item[]\verb"<DYNARE_file>_rmse_prior_dens_*.fig": for each parameter, plots the pdf's
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corresponding to the best 10\% RMES's of each observed series;
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\item[]\verb"<DYNARE_file>_rmse_prior_<name of observedseries>_corr_*.fig": for each observed series plots the
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bi-dimensional projections of samples with the best 10\% RMSE's,
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when the correlation is significant;
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\item[]\verb"<DYNARE_file>_rmse_prior_lnlik*.fig": for each observed
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series, plots \emph{in red} the cdf of the log-likelihood
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corresponding to the best 10\% RMSE's, \emph{in green} the cdf of
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the rest of the sample and \emph{in blue }the cdf of the full
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sample; this allows to see the presence of some idiosyncratic
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behaviour;
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\item[]\verb"<DYNARE_file>_rmse_prior_lnpost*.fig": for each observed
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series, plots \emph{in red} the cdf of the log-posterior
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corresponding to the best 10\% RMSE's, \emph{in green} the cdf of
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the rest of the sample and \emph{in blue }the cdf of the full
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sample; this allows to see idiosyncratic behaviour;
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\item[]\verb"<DYNARE_file>_rmse_prior_lnprior*.fig": for each observed
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series, plots \emph{in red} the cdf of the log-prior corresponding
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to the best 10\% RMSE's, \emph{in green} the cdf of the rest of
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the sample and \emph{in blue }the cdf of the full sample; this
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allows to see idiosyncratic behaviour;
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\item[]\verb"<DYNARE_file>_rmse_prior_lik_SA_*.fig": when
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\verb"lik_only=1", this shows the Smirnov tests for the filtering
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of the best 10\% log-likelihood values;
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\item[]\verb"<DYNARE_file>_rmse_prior_post_SA_*.fig": when
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\verb"lik_only=1", this shows the Smirnov test for the filtering
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of the best 10\% log-posterior values.
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\end{description}
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\subsection{Reduced form mapping}
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In the case of the mapping of the reduced form solution, synthetic
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figures are saved in the \verb"\GSA" folder:
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\begin{description}
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\item[]\verb"<DYNARE_file>_redform_<endo name>_vs_lags_*.fig":
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shows bar charts of the sensitivity indices for the \emph{ten most
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important} parameters driving the reduced form coefficients of the
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selected endogenous variables (\verb"namendo") versus lagged
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endogenous variables (\verb"namlagendo"); suffix \verb"log"
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indicates the results for log-transformed entries;
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\item[]\verb"<DYNARE_file>_redform_<endo name>_vs_shocks_*.fig":
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shows bar charts of the sensitivity indices for the \emph{ten most
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important} parameters driving the reduced form coefficients of the
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selected endogenous variables (\verb"namendo") versus exogenous
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variables (\verb"namexo"); suffix \verb"log" indicates the results
|
|
for log-transformed entries;
|
|
\item[]\verb"<DYNARE_file>_redform_GSA(_log).fig": shows bar chart of
|
|
all sensitivity indices for each parameter: this allows to notice
|
|
parameters that have a minor effect for \emph{any} of the reduced
|
|
form coefficients,
|
|
\end{description}
|
|
|
|
Detailed results of the analyses are shown in the subfolder
|
|
\verb"\GSA\redform_stab", where the detailed results of the
|
|
estimation of the single functional relationships between
|
|
parameters $\theta$ and reduced form coefficient are stored in
|
|
separate directories named as:
|
|
\begin{description}
|
|
\item[]\verb"<namendo>_vs_<namlagendo>": for the entries of the
|
|
transition matrix;
|
|
\item[]\verb"<namendo>_vs_<namexo>": for entries of the matrix of
|
|
the shocks.
|
|
\end{description}
|
|
Moreover, analyses for log-transformed entries are denoted with
|
|
the following suffixes ($y$ denotes the generic reduced form
|
|
coefficient):
|
|
\begin{description}
|
|
\item[]\verb"log": $y^*=\log(y)$;
|
|
\item[]\verb"minuslog": $y^*=\log(-y)$;
|
|
\item[]\verb"logsquared": $y^*=\log(y^2)$ for symmetric fat tails;
|
|
\item[]\verb"logskew": $y^*=\log(|y+\lambda|)$ for asymmetric fat tails.
|
|
\end{description}
|
|
The optimal type of transformation is automatically selected
|
|
without the need of any user's intervention.
|
|
|
|
\subsection{Screening analysis}
|
|
The results of the screening analysis with Morris sampling design
|
|
are stored in the subfolder \verb"\GSA\SCREEN". The data file
|
|
\verb"<DYNARE_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
|
|
MC samples are saved:
|
|
\begin{description}
|
|
\item[]\verb"<DYNARE_file>_redform_<endo name>_vs_lags_*.fig":
|
|
shows bar charts of the elementary effect tests for the \emph{ten
|
|
most important} parameters driving the reduced form coefficients
|
|
of the selected endogenous variables (\verb"namendo") versus
|
|
lagged endogenous variables (\verb"namlagendo");
|
|
\item[]\verb"<DYNARE_file>_redform_<endo name>_vs_shocks_*.fig":
|
|
shows bar charts of the elementary effect tests for the \emph{ten
|
|
most important} parameters driving the reduced form coefficients
|
|
of the selected endogenous variables (\verb"namendo") versus
|
|
exogenous variables (\verb"namexo");
|
|
\item[]\verb"<DYNARE_file>_redform_screen.fig": shows bar chart of
|
|
all elementary effect tests for each parameter: this allows to
|
|
identify parameters that have a minor effect for \emph{any} of the
|
|
reduced form coefficients.
|
|
\end{description}
|
|
|
|
|
|
% ----------------------------------------------------------------
|
|
\bibliographystyle{plainnat}
|
|
%\bibliographystyle{amsplain}
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|
%\bibliographystyle{alpha}
|
|
\bibliography{marco}
|
|
|
|
\newpage
|
|
|
|
|
|
\end{document}
|
|
% ----------------------------------------------------------------
|