Add report for JRC contract.
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% Created 2023-11-10 Fri 14:22
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% Intended LaTeX compiler: pdflatex
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\documentclass{amsart}
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\usepackage[utf8]{inputenc}
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\usepackage{graphicx}
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\usepackage{longtable}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{capt-of}
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\usepackage{hyperref}
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\author{Stéphane Adjemian}
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\date{\today}
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\title{}
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\hypersetup{
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pdfauthor={Stéphane Adjemian},
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pdftitle={},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 29.1 (Org mode 9.6.6)},
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pdflang={English}}
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\usepgfplotslibrary{patchplots}
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% \usepackage{grffile}
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% \usepackage{amsmath}
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scaled y ticks = false,
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scaled x ticks = true,
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colorlinks=true,
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filecolor=magenta,
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urlcolor=cyan,
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pdfpagemode=FullScreen,
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pdfauthor={Stéphane Adjemian},
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pdftitle={},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Stéphane Adjemian},
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pdflang={English}
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}
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\urlstyle{same}
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\VerbatimFootnotes
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\setlength{\parindent}{0pt}
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\begin{document}
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\section{Improve deterministic simulation}
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\label{sec:orga51ffcf}
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The purpose of the present item is to improve the performance of the
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Laffargue-Boucekkine-Juillard algorithm (henceceforth LBJ), which can used for
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solving perfect foresight models in Dynare (LBJ is available under the option
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\texttt{stack\_solve\_algo=1} of the \texttt{perfect\_foresight\_solver} command).\newline
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More precisely, the goal is to improve the performance of LBJ when used in
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conjunction with the \texttt{bytecode} and \texttt{block} options of the \texttt{model} block. The
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\texttt{bytecode} option instructs Dynare to use a custom bytecode representation of
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the model when evaluating the model residuals and Jacobian, supposedly faster
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than a pure \texttt{.m} representation. The \texttt{block} option instructs Dynare to
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decompose the model into blocks in order to accelerate the computation via a
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divide-and-conquer approach.\newline
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It turns out that, for historical reasons, the implementation of LBJ in Dynare
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in combination with the \texttt{bytecode} and/or \texttt{block} options was inefficient, and
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the purpose of the present item is therefore to improve the situation and
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deliver better performance for these cases.\newline
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The improvements have been implemented in the unstable version of Dynare. The
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table below reports a performance comparison between before and after those
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changes (respectively corresponding to commits
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\href{https://git.dynare.org/Dynare/dynare/-/commit/af205124766be59fe8bfbb053a91167696a89fb4}{af20512} and
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\href{https://git.dynare.org/Dynare/dynare/-/commit/666aa46dfb8519db326a0671f75bc300f48fb71f}{666aa46}). The comparison was performed using
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a model provided by DG ECFIN (filename \texttt{SMALLMODEL10.dyn}, which has 1364
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equations), applying a permanent shock that is simulated with LBJ over 50
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periods. The test environment consists of an AMD Ryzen 9 7950X CPU, using
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MATLAB R2023b under Debian GNU/Linux. Timings correspond to the full execution
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time of the \texttt{.mod} file, and are reported in seconds.\newline
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\begin{center}
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\begin{tabular}{lrr}
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& Before & Now\\[0pt]
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\hline
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Neither \texttt{bytecode} nor \texttt{block} & 19 & 19\\[0pt]
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\texttt{block} (without \texttt{bytecode}) & 72 & 11\\[0pt]
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\texttt{bytecode} (without \texttt{block}) & 123 & 24\\[0pt]
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\texttt{bytecode} + \texttt{block} & 64 & 18\\[0pt]
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\end{tabular}
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\end{center}
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\bigskip
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As one can see, before the changes implemented under the present item, the LBJ
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implementation with either \texttt{bytecode} or \texttt{block} was much less efficient than
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the one with neither of those options.\newline
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The changes that have been implemented lead to a drastic performance
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improvement: by a factor of more than 3 for \texttt{bytecode} + \texttt{block}, of more than
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5 for \texttt{bytecode} alone, and of more than 6 for \texttt{block} alone.\newline
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Before the change, there were 3 implementations of LBJ in Dynare:
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\begin{enumerate}
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\item a pure \texttt{.m} implementation (\texttt{sim1\_lbj.m}) for the case with neither \texttt{block} nor \texttt{bytecode}
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\item a pure \texttt{.m} implementation (\texttt{solve\_two\_boundaries.m}) for the case with \texttt{block} without \texttt{bytecode}
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\item a C++ implementation in the \texttt{bytecode} MEX for the case with \texttt{bytecode}
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(with or without \texttt{block})
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\end{enumerate}
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Implementation 1 was unchanged, hence the first line of the table shows no
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change in timings.\newline
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Implementation 2 and 3 were both inefficient. We decided to drop implementation
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3, and to improve implementation 2. As a consequence, implementation 1 is now
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used when \texttt{block} is not present (with or without \texttt{bytecode}), and the improved
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implementation 2 is now used when \texttt{block} is present (with or without
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\texttt{bytecode}).\newline
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Under the present situation, one can observe that adding the \texttt{block} option
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always improve performance, which is consistend with expectations (without
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\texttt{bytecode}, it leads to a drop from 19 to 11 seconds; with \texttt{bytecode}, from 24
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to 18 seconds).\newline
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More surprisingly, adding the \texttt{bytecode} option now always degrades
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performance. As a consequence, the current best option is \texttt{block} without
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\texttt{bytecode}.\newline
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It remains to be seen why \texttt{bytecode} degrades performance. First, it should be
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noted that we tested only one specific simulation on one specific model, with
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one specific algorithm (LBJ); results may be different under another scenario.
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The most plausible explanation for these results is that the just in-time (JIT)
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compiler in MATLAB has been improved to the point of making our bytecode
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representation useless. Another possible explanation is that the way the
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Jacobian is returned to the \texttt{.m} routines is less efficient in the \texttt{bytecode}
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case, because the latter does not use the same sparse matrix format as without
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the \texttt{bytecode} option; but it is unlikely that this explains the whole
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difference (actually, this explanation probably only plays a role in the
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\texttt{bytecode} + \texttt{block} option, because in \texttt{bytecode} without \texttt{block} the code
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currently exploits some sparsity).\newline
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As another follow-up to the present project, if further improvements of LBJ
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were deemed necessary, it could be possible to explore the potential for a
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dedicated MEX (probably written in Fortran).
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\section{A MEX file for the Kalman filter}
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\label{sec:org10b97d6}
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In Dynare, the vanilla Kalman filter is currently implemented using MATLAB code
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\texttt{kalman\_filter.m} under \texttt{matlab/kalman/likelihood/}. We coded the equivalent routine
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in Fortran and compiled it into a MEX to assess the associated computational
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gains. The source code is located at
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\texttt{mex/sources/kalman\_filter/kalman\_filter.f08}. The efficiency improvements
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stem from the resort to a unique call to a LU decomposition routine and the
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tailored use of its output components to perform the update step of the
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Kalman filter and compute each period’s contribution to the marginal
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likelihood. In the MATLAB routine, using similar input arguments, an
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operation as costly as the LU decomposition is called at least twice to
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perform each Kalman filter update step (matrix inversion and reciprocal
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condition number calculation) and a full determinant calculation is
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necessary to compute each period’s contribution to the marginal likelihood.\newline
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As commented at the end of the source code, exploiting the symmetry of some
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of the involved matrices using a Cholesky decomposition instead of a LU
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decomposition may bring induce further computational gains. We wrote a first
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test in \texttt{tests/kalman/likelihood}. The main file is \texttt{test\_kalman\_mex.m}, while
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an auxiliary routine lies in \texttt{compare\_kalman\_mex.m}. The test sets up a
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state-space framework with simulated data and allows for different
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specifications of the observation equation through inputs \texttt{Z}, \texttt{Zflag} and \texttt{H}.
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One may provide an index signaling observables among state variables, in
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which case \texttt{Zflag} is 0 and \texttt{Z} is a vector of integer indices. Otherwise, one
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may declare observable variables as a linear combination of state variables,
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in which case \texttt{Z} is a real matrix and \texttt{Zflag} is 1. We test both formulations.
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The structure of observation errors can also vary, being either constantly
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equal to zero or following a multidimensional normal law with specified
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variance-covariance matrix \texttt{H}. In that latter regard, we test 3 setups : (i)
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no observation errors; (ii) observation errors with a diagonal
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variance-covariance matrix; (iii) observation errors with an unrestricted
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variance-covariance matrix.\newline
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Using a setup embedding an 11th Gen Intel(R) Core(TM) i9-11900K @ 3.50GHz with
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32 GB memory (under Debian GNU/Linux with MATLAB R2023b), we compare the outputs
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and the execution speeds of the MATLAB and MEX routines. The results are similar
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across routines and the Fortran implementation provides a small gain, as
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the test log bears witness of:
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\begin{table}[H]
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\begin{tabular}{lccc}
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& No measurement erros & Measurement errors & Correlated measurement errors \\ \hline
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\texttt{Zflag=0} & 1.07 & 1.07 & 1.06 \\
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\texttt{Zflag=1} & 1.03 & 1.01 & 1.01 \\
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\end{tabular}
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\end{table}
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In this test the model has 100 state variables, 12 observed variables and 10
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structural innovations. The sample contains 300 observations. In this example
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the gain is below 10\%, but the measure of the relative advantage of the MEX
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heavily depends on the model specification and the sample size (the larger is
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the proportion of periods before reaching the Kalman steady state, the larger is
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the gain)\footnote{One can easily change the size of the state-space model in the main test routine.}. With the \texttt{fs2000.mod} model provided in the \texttt{examples}
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subfolder, the gain is more substantial (about 20\%). For the Quest-III model,
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also used as a benchmark in section \ref{sec:orgacd17cc}, the evaluation of the
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posterior kernel are 10\% faster with the MEX. Obviously, we ensured the
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consistency of values returned by the MEX for all the considered models. The
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last line of the previous table shows that the advantage of the MEX vanishes
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with \texttt{Zflag=1}. The performance of the MEX routine seems to degrade
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further when utilising a selection matrix in place of a vector of indices to
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choose the observed variables, as compared to the MATLAB code.\newline
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Note that these execution speed wedges are wider when the convergence of the
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gain matrix demands more iterations to reach. Indeed, when the gain matrix no
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longer significantly evolves, both the MATLAB and MEX routines call the
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\texttt{kalman\_filter\_ss.m} routine.\newline
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\section{Initialisation of the Kalman filter}
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\label{sec:orgacd17cc}
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Computing the likelihood of a linear DSGE model's heavily depends on the
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utilisation of the Kalman filter. When the DSGE model's steady state is already
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available in a closed form, the estimation process allocates a significant
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portion of its time to the Kalman filter. In this document, we aim to quantify
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the potential benefits that may arise from considering an alternative
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initialisation of the Kalman filter.\newline
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The reduced form DSGE model is given by:
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\[
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y_t = \bar y + A \left(y_{t-1}-\bar y\right) + B \varepsilon_t
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\]
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where $y_t$ is a $n\times 1$ vector of the endogenous variables defined by the
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DSGE model, $\bar y$ the steady state of these endogenous variables,
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$\varepsilon_t$ a $q\times 1$ vector of Gaussian innovations with zero mean and
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covariance matrix $\Sigma_{\varepsilon}$. Matrices $A$ and $B$, respectively of
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dimensions $n\times n$ and $n\times q$, as the steady state $\bar y$ are
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nonlinear functions of the deep parameters we aim to estimate (along with the
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covariance matrix $\Sigma_{\varepsilon}$). We only observe $p\geq q$ endogenous
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variables, the vector of observed endogenous variables is:
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\[
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y^{\star}_t = Z y_t
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\]
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where $Z$ is a $p\times n$ selection matrix\footnote{Here, without loss of
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generality, we implicitly exclude measurement errors.}. The likelihood, the
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density of the sample, can be expressed as a product of the (Gaussian) conditional
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densities of the prediction errors $v_t$ which are determine recursively with
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the Kalman filter:
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\begin{equation}
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\label{eq:kalman:1}
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v_t = y_t^{\star} - \bar y^{\star}- Z \hat y_t
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\end{equation}
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\begin{equation}
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\label{eq:kalman:2}
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F_t = Z P_t Z'
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\end{equation}
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\begin{equation}
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\label{eq:kalman:3}
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K_t = A P_t A' Z' F_t^{-1}
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\end{equation}
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\begin{equation}
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\label{eq:kalman:4}
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\hat y_{t+1} = A \hat y_t + K_t v_t
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\end{equation}
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\begin{equation}
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\label{eq:kalman:5}
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P_{t+1} = A P_t \left( A - K_t Z\right)' + B \Sigma_{\varepsilon}B'
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\end{equation}
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where $F_t$ is the conditional covariance matrix of the prediction errors, what
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we learn from observation $t$ is given by the Kalman gain matrix $K_t$,
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$\hat y_t$ the vector of endogenous variables in deviations to the steady state,
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and $P_t$ the conditional covariance matrix of $\hat y_t$. Substituting (\ref{eq:kalman:2})
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and (\ref{eq:kalman:3}) into (\ref{eq:kalman:5}) we obtain the following recurrence
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for the conditional covariance matrix of the endogenous variables:
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\begin{equation}
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\label{eq:ricatti}
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P_{t+1} = A P_t A' - A P_t Z' \left(Z P_t Z \right)^{-1}Z A P_t A' + B \Sigma_{\varepsilon}B'
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\end{equation}
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This is the Riccati equation, which consumes a significant portion of the
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computation time during the evaluation of the Kalman filter. Additionally, it is
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a critical point where round-off errors can accumulate significantly. As we
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iterate through these equations, the Riccati equation will ultimately, given a
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sufficiently large sample size, converge to a fixed point denoted as $\bar P$,
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which satisfies the following:
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\begin{equation}
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\label{eq:ricatti:fp}
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A \bar P A' - \bar P - A \bar P Z' \left(Z \bar P Z \right)^{-1}Z A \bar P A' + B \Sigma_{\varepsilon}B' = 0
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\end{equation}
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and the Kalman filter equations fall down into the following system:
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\[
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\begin{cases}
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v_t &= y_t^{\star} - \bar y^{\star}- Z \hat y_t\\
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\hat y_{t+1} &= A \hat y_t + \bar K v_t
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\end{cases}
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\]
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referred to as the steady state Kalman filter, with $\bar F = Z \bar P Z'$ and a
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constant Kalman gain matrix $\bar K = A \bar P A' Z' \bar F^{-1}$. Iterations are
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then significantly faster compared to the system described by
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(\ref{eq:kalman:1})--(\ref{eq:kalman:5}). The number of iterations required to
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(approximately) converge to the steady state does not depend on the data but only
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on the persistence in the model (through matrix $A$).\newline
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Due to its recursive nature, the Kalman filter requires an initial condition
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$\hat y_0$ and $P_0$. Assuming that the model is stationary, a usual choice for
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the initial condition is the (stochastic) fixed point of the reduced form model:
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$\hat y_0 = 0$ and $P_0=P^{\star}$ solving:
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\[
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P^{\star} - A P^{\star} A' - B \Sigma_{\varepsilon} B' = 0
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\]
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which is a Sylvester equation that can be easily solved.\newline
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In this note we consider instead the fixed point of the Ricatti equation
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$\bar P$ as an initial condition for the covariance matrix of the endogenous
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variables. This approach offers a clear computational advantage: we directly jump to the
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steady state Kalman filter without the need to go through
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(\ref{eq:kalman:1})--(\ref{eq:kalman:5}). Clearly this will reduce the time
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required to compute the likelihood, but this will also change the evaluation of
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the likelihood. To assess the impact of this alternative initial condition, we
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conduct experiments using a medium-sized model, quantifying the time savings and
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evaluating their implications on parameter estimation. Dynare
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provides a MEX file, linked to the subroutine \verb+sb02od+ from the
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\href{http://www.slicot.org/}{Slicot library} (version 5.0), to solve
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(\ref{eq:ricatti:fp}) for $\bar{P}$. The use of $\bar P$ as an initial condition
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for the Kalman filter is triggered by option \verb+lik_init=4+ in the
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\verb+estimation+ command.\newline
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We consider\footnote{Under the same environment as in section \ref{sec:org10b97d6}.} the estimation of the Quest-III model (113 endogenous variables, 19
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innovations, 17 observed variables) as an example. All the codes to reproduce
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the results are available in a
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\href{https://git.ithaca.fr/stepan/quest-iii-estimation}{Git repository}. The
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model is estimated under both settings (initialization with $\bar P$, or
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with $P^{\star}$)\footnote{To reproduce the results run \verb+Q3.mod+, provided
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in the root of the Git repository, with macro options
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\verb+-DUSE_RICATTI_FIXED_POINT=true+ and
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\verb+-DUSE_RICATTI_FIXED_POINT=false+.} with \verb+mode_compute=5+. We find
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that the estimation of the model with the fixed point of the reduced form model
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($P^{\star}$) is 33\% slower than the estimation of the same model with the
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fixed point of the Ricatti equation ($\bar P$). Because the objective functions
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are different, the optimiser probably do not behave identically in both cases.
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If we just evaluate the likelihood on the same point (the prior mean), instead
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of running \verb+mode_compute=5+, we find that the evaluation with $P^{\star}$
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is just 21\% slower than the evaluation with $\bar P$, indicating that initialising
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with the fixed point of the Riccati equation renders the objective more amenable to
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optimisation. This is expected since it is widely recognised that
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iterations on (\Ref{eq:ricatti}) exhibit erratic behaviour. The values of the logged
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posterior kernel at the estimated posterior mode are close (7235.92 with
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$\bar P$ and 7237.68 with $P^{\star}$). But much larger differences can be
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observed elsewhere. For instance, at the prior mean the values are -275.58 with
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$\bar P$ and 5.13 with $P^{\star}$. Do these differences in the likelihood
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functions matter? Table \ref{tab:1} shows the estimated values for the
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parameters. Both likelihood functions deliver similar estimates in terms of
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point estimate or variance\footnote{Note however that the estimates for the
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posterior variance should be taken with care since they rely on an asymptotic
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Gaussian approximation. For some parameters, the estimated posterior variance
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is larger than the prior variance... That should not be the case.}. Figures
|
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(\ref{fig:1})--(\ref{fig:7}) display cross sections of the likelihood (black
|
||||
plain curves for the initialisation with $P^{\star}$, red dashed curves for the
|
||||
initialisation with $\bar P$). The conclusion seems to be that even if the
|
||||
likelihood functions are different, inference on the parameters is only
|
||||
marginally affected. At least for a first attempt, we can safely consider the
|
||||
alternative initialisation of the Kalman filter. We should also try to compare
|
||||
posterior densities, running samplers under both settings, this would bring a
|
||||
more definitive answer (at least in the case of Quest-III). Ideally we should
|
||||
also build a Monte-Carlo exercise, on a smaller model. All this is left for
|
||||
future research. Note that the gain in terms of computing time is model
|
||||
\textit{and parameter} dependent. If the standard Kalman filter converges
|
||||
quickly to the steady state the gain is small (this depends on the persistence
|
||||
in the model).
|
||||
|
||||
\newpage
|
||||
|
||||
\begin{longtable}[l]{ l l d{4} d{4} d{4} d{4} d{4} d{4} }
|
||||
\toprule
|
||||
\multicolumn{1}{c}{} & \multicolumn{3}{c}{\textbf{Prior}} & \multicolumn{2}{c}{\textbf{Posterior with $P^{\star}$}} & \multicolumn{2}{c}{\textbf{Posterior with $\bar P$}}\\
|
||||
\cmidrule(rl){2-4}
|
||||
\cmidrule(rl){5-6}
|
||||
\cmidrule(rl){7-8}
|
||||
\textbf{Parameters} & Shape & \multicolumn{1}{c}{Mode} & \multicolumn{1}{c}{SD} & \multicolumn{1}{c}{Mode} & \multicolumn{1}{c}{SD} & \multicolumn{1}{c}{Mode} & \multicolumn{1}{c}{SD} \\
|
||||
\midrule
|
||||
\endhead
|
||||
|
||||
\\
|
||||
\midrule
|
||||
\multicolumn{8}{c}{\tiny{\textbf{Continued on next page}}}\\
|
||||
\endfoot
|
||||
|
||||
\bottomrule
|
||||
\\
|
||||
\caption{\textbf{Estimation results of Quest-III.} Without and with the fixed point of the Ricatti equation as an initial condition of the Kalman filter. Standard deviation estimates rely on the inverse of the Hessian matrix at the estimated posterior mode.}
|
||||
\endlastfoot
|
||||
|
||||
E\_EPS\_C & Gamma & 0.0320 & 0.0300 & 0.0528 & 0.0109 & 0.0521 & 0.0111\\
|
||||
E\_EPS\_ETA & Gamma & 0.0640 & 0.0600 & 0.1163 & 0.0340 & 0.1225 & 0.0347\\
|
||||
E\_EPS\_ETAM & Gamma & 0.0088 & 0.0150 & 0.0183 & 0.0022 & 0.0183 & 0.0023\\
|
||||
E\_EPS\_ETAX & Gamma & 0.0640 & 0.0600 & 0.0274 & 0.0152 & 0.0279 & 0.0159\\
|
||||
E\_EPS\_EX & Gamma & 0.0032 & 0.0030 & 0.0044 & 0.0004 & 0.0043 & 0.0004\\
|
||||
E\_EPS\_G & Gamma & 0.0320 & 0.0300 & 0.0047 & 0.0003 & 0.0047 & 0.0003\\
|
||||
E\_EPS\_IG & Gamma & 0.0320 & 0.0300 & 0.0054 & 0.0004 & 0.0054 & 0.0004\\
|
||||
E\_EPS\_L & Gamma & 0.0320 & 0.0300 & 0.0220 & 0.0052 & 0.0213 & 0.0049\\
|
||||
E\_EPS\_LOL & Gamma & 0.0032 & 0.0030 & 0.0040 & 0.0009 & 0.0045 & 0.0008\\
|
||||
E\_EPS\_M & Gamma & 0.0016 & 0.0015 & 0.0012 & 0.0001 & 0.0012 & 0.0001\\
|
||||
E\_EPS\_RPREME & Gamma & 0.0032 & 0.0030 & 0.0015 & 0.0002 & 0.0014 & 0.0002\\
|
||||
E\_EPS\_RPREMK & Gamma & 0.0032 & 0.0030 & 0.0050 & 0.0016 & 0.0054 & 0.0018\\
|
||||
E\_EPS\_TR & Gamma & 0.0320 & 0.0300 & 0.0022 & 0.0001 & 0.0022 & 0.0001\\
|
||||
E\_EPS\_W & Gamma & 0.0320 & 0.0300 & 0.0396 & 0.0129 & 0.0410 & 0.0136\\
|
||||
E\_EPS\_Y & Gamma & 0.0320 & 0.0300 & 0.0118 & 0.0012 & 0.0120 & 0.0012\\
|
||||
A2E & Beta & 0.0500 & 0.0240 & 0.0378 & 0.0132 & 0.0370 & 0.0130\\
|
||||
G1E & Beta & 0.0000 & 0.6000 & -0.0456 & 0.0977 & -0.0455 & 0.0963\\
|
||||
GAMIE & Gamma & 16.6667 & 20.0000 & 62.7825 & 18.9969 & 66.3819 & 19.6627\\
|
||||
GAMI2E & Gamma & 8.3333 & 10.0000 & 0.6044 & 0.3537 & 0.6161 & 0.3627\\
|
||||
GAMLE & Gamma & 16.6667 & 20.0000 & 54.0067 & 10.8176 & 55.4743 & 11.2277\\
|
||||
GAMPE & Gamma & 16.6667 & 20.0000 & 45.8370 & 13.5341 & 48.2540 & 13.8116\\
|
||||
GAMPME & Gamma & 16.6667 & 20.0000 & 1.3257 & 0.4455 & 1.3501 & 0.4538\\
|
||||
GAMPXE & Gamma & 16.6667 & 20.0000 & 8.9856 & 7.5122 & 9.3195 & 7.9101\\
|
||||
GAMWE & Gamma & 16.6667 & 20.0000 & 0.5913 & 0.3666 & 0.5815 & 0.3625\\
|
||||
GSLAG & Beta & 0.0000 & 0.4000 & -0.4336 & 0.1045 & -0.4321 & 0.1046\\
|
||||
GVECM & Beta & -0.5000 & 0.2000 & -0.1552 & 0.0432 & -0.1591 & 0.0439\\
|
||||
HABE & Beta & 0.7222 & 0.1000 & 0.5381 & 0.0400 & 0.5369 & 0.0398\\
|
||||
HABLE & Beta & 0.7222 & 0.1000 & 0.8731 & 0.0560 & 0.8637 & 0.0591\\
|
||||
IG1E & Beta & 0.0000 & 0.6000 & 0.1802 & 0.0943 & 0.1824 & 0.0924\\
|
||||
IGSLAG & Beta & 0.5000 & 0.2000 & 0.4384 & 0.0819 & 0.4401 & 0.0819\\
|
||||
IGVECM & Beta & -0.5000 & 0.2000 & -0.1330 & 0.0586 & -0.1354 & 0.0582\\
|
||||
ILAGE & Beta & 0.8856 & 0.0750 & 0.8887 & 0.0156 & 0.8856 & 0.0160\\
|
||||
KAPPAE & Gamma & 1.0500 & 0.5000 & 1.6617 & 0.3660 & 1.6443 & 0.3596\\
|
||||
RHOCE & Beta & 0.8856 & 0.0750 & 0.9305 & 0.0242 & 0.9382 & 0.0221\\
|
||||
RHOETA & Beta & 0.5000 & 0.2000 & 0.0771 & 0.0596 & 0.0733 & 0.0563\\
|
||||
RHOETAM & Beta & 0.8856 & 0.0750 & 0.9599 & 0.0138 & 0.9602 & 0.0138\\
|
||||
RHOETAX & Beta & 0.8856 & 0.0750 & 0.8841 & 0.0592 & 0.8824 & 0.0610\\
|
||||
RHOGE & Beta & 0.5000 & 0.2000 & 0.2930 & 0.1116 & 0.2978 & 0.1130\\
|
||||
RHOIG & Beta & 0.8856 & 0.0750 & 0.8789 & 0.0660 & 0.8806 & 0.0645\\
|
||||
RHOLE & Beta & 0.8856 & 0.0750 & 0.9844 & 0.0081 & 0.9865 & 0.0076\\
|
||||
RHOL0 & Beta & 0.9578 & 0.0200 & 0.9360 & 0.0233 & 0.9320 & 0.0214\\
|
||||
RHOPCPM & Beta & 0.5000 & 0.2000 & 0.6987 & 0.1411 & 0.7006 & 0.1403\\
|
||||
RHOPWPX & Beta & 0.5000 & 0.2000 & 0.1889 & 0.0667 & 0.1895 & 0.0671\\
|
||||
RHORPE & Beta & 0.8856 & 0.0750 & 0.9908 & 0.0065 & 0.9943 & 0.0037\\
|
||||
RHORPK & Beta & 0.8856 & 0.0750 & 0.9254 & 0.0270 & 0.9175 & 0.0271\\
|
||||
RHOUCAP0 & Beta & 0.9578 & 0.0200 & 0.9602 & 0.0160 & 0.9560 & 0.0160\\
|
||||
RPREME & Beta & 0.0200 & 0.0080 & 0.0205 & 0.0071 & 0.0204 & 0.0068\\
|
||||
RPREMK & Beta & 0.0200 & 0.0080 & 0.0243 & 0.0030 & 0.0256 & 0.0025\\
|
||||
SE & Beta & 0.8000 & 0.0800 & 0.8540 & 0.0208 & 0.8561 & 0.0206\\
|
||||
SFPE & Beta & 0.5000 & 0.2000 & 0.8831 & 0.0672 & 0.8878 & 0.0656\\
|
||||
SFPME & Beta & 0.5000 & 0.2000 & 0.7211 & 0.1288 & 0.7239 & 0.1277\\
|
||||
SFPXE & Beta & 0.5000 & 0.2000 & 0.9293 & 0.0508 & 0.9299 & 0.0503\\
|
||||
SFWE & Beta & 0.5000 & 0.2000 & 0.7997 & 0.1439 & 0.8045 & 0.1413\\
|
||||
SIGC & Gamma & 1.5000 & 1.0000 & 4.3516 & 1.2195 & 4.3758 & 1.2276\\
|
||||
SIGEXE & Gamma & 1.0500 & 0.5000 & 2.4811 & 0.3204 & 2.4645 & 0.3212\\
|
||||
SIGIME & Gamma & 1.0500 & 0.5000 & 1.1520 & 0.1930 & 1.1564 & 0.1931\\
|
||||
SLC & Beta & 0.5000 & 0.1000 & 0.3293 & 0.0668 & 0.3225 & 0.0668\\
|
||||
TINFE & Beta & 2.0000 & 0.4000 & 1.9073 & 0.1996 & 1.7677 & 0.1378\\
|
||||
TR1E & Beta & 0.0000 & 0.6000 & 0.9235 & 0.0913 & 0.9246 & 0.0908\\
|
||||
RHOTR & Beta & 0.8856 & 0.0750 & 0.8588 & 0.0488 & 0.8576 & 0.0486\\
|
||||
TYE1 & Beta & 0.1500 & 0.2000 & 0.3427 & 0.0920 & 0.3198 & 0.0893\\
|
||||
TYE2 & Beta & 0.1500 & 0.2000 & 0.0720 & 0.0258 & 0.0680 & 0.0253\\
|
||||
WRLAG & Beta & 0.5000 & 0.2000 & 0.2274 & 0.1364 & 0.2258 & 0.1352\\
|
||||
\label{tab:1}
|
||||
\end{longtable}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-1.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:1}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-2.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:2}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-3.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:3}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-4.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:4}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-5.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:5}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-6.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:6}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\scalebox{.75}{
|
||||
\input{cross-posterior-kernel-7.tex}}
|
||||
\caption{Cross sections of the posterior kernel.}
|
||||
\label{fig:7}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\end{document}
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
%%% TeX-master: t
|
||||
%%% End:
|
||||
Loading…
Reference in New Issue