521 lines
13 KiB
TeX
521 lines
13 KiB
TeX
% © Stéphane Adjemian, 2014-2017
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\documentclass[10pt,slidestop]{beamer}
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\synctex=1
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\usepackage{etex}
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\usepackage{fourier-orns}
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\usepackage{ccicons}
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\usepackage{amssymb}
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\usepackage{amstext}
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\usepackage{amsbsy}
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\usepackage{amsopn}
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\usepackage{amscd}
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\usepackage{amsxtra}
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\usepackage{amsthm}
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\usepackage{float}
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\usepackage{color}
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\usepackage{mathrsfs}
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\usepackage{bm}
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\usepackage{lastpage}
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\usepackage[nice]{nicefrac}
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\usepackage{setspace}
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\usepackage{hyperref}
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\usepackage{ragged2e}
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\usepackage{listings}
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\usepackage{fontawesome}
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\usepackage[utf8x]{inputenc}
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\usepackage{array}
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\usepackage[super]{nth}
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\usepackage{algorithm}
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\usepackage{algpseudocode}
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\newcolumntype{x}[1]{>{\centering\arraybackslash\hspace{0pt}}m{#1}}
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\setlength{\parindent}{0pt}
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\pdfcompresslevel=9
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\newcommand{\trace}{\mathrm{tr}}
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\newcommand{\vect}{\mathrm{vec}}
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\newcommand{\tracarg}[1]{\mathrm{tr}\left\{#1\right\}}
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\newcommand{\vectarg}[1]{\mathrm{vec}\left(#1\right)}
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\newcommand{\vecth}[1]{\mathrm{vech}\left(#1\right)}
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\newcommand{\iid}[2]{\mathrm{iid}\left(#1,#2\right)}
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\newcommand{\normal}[2]{\mathcal N\left(#1,#2\right)}
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\newcommand{\dynare}{\href{http://www.dynare.org}{\color{blue}Dynare}}
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\newcommand{\AllSample}{ \mathcal Y_T }
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\newcommand{\sample}{\mathcal Y_T}
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\newcommand{\samplet}[1]{\mathcal Y_{#1}}
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\newcommand{\slidetitle}[1]{\fancyhead[L]{\textsc{#1}}}
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\definecolor{gray}{gray}{0.4}
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\setbeamertemplate{footline}{
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{\hfill\vspace*{1pt}\href{https://creativecommons.org/publicdomain/zero/1.0/}{\cczero}
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\href{https://git.ithaca.fr/stepan/ispra-2023}{\faCodeFork}
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}\hspace{1cm}}
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\setbeamertemplate{navigation symbols}{}
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\setbeamertemplate{blocks}[rounded][shadow=true]
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\newenvironment{notes}
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{\bgroup \justifying\bgroup\tiny\begin{spacing}{1.0}}
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{\end{spacing}\egroup\egroup}
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\begin{document}
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\title{Estimation of nonlinear models with Dynare}
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\author[]{}
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\institute{\texttt{stephane.adjemian@univ-lemans.fr}}
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\date{September, 2023}
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\begin{frame}
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\titlepage{}
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\end{frame}
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\begin{frame}
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\frametitle{Introduction}
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\bigskip
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\begin{itemize}
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\item Full information estimation of models approximated at higher order...\newline
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\item Or, in principle, full information estimation of nonlinear models.\newline
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\item Cannot use the (linear) Kalman filter anymore.\newline
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\item Dynare provides routines to evaluate the likelihood of models approximated at order $k\geq 1$.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{The reduced form model}
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\bigskip
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\[
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\begin{aligned}
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& s_t = f( s_{t-1}, \varepsilon_t; \bm\theta ) \\
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& y_t = g( s_t; \bm\theta) + e_t \\
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\end{aligned}
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\]
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\bigskip
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\begin{itemize}
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\item $f(.)$ and $g(.)$ are the state and measurement equations.\newline
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\item $\bm\theta\in\Theta\subseteq \mathbb R^m$ a vector of $m$ parameters.\newline
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\item Cannot use the (linear) Kalman filter anymore.\newline
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\item $s_t$ and $y_t$ are the vectors of state variables and observed variables.\newline
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\item Innovations $\varepsilon_t$ and $e_t$ are the structural shocks and measurement errors.\newline
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\item $\#(y_t) = \#(e_t)$
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Reduced form with second order approximation}
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\bigskip
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\begin{itemize}
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\item The ``state'' equations:
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\[
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\begin{split}
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s_t = \bar{s}(\bm\theta) & + g_u(\bm\theta) \hat{s}_{t-1} + g_u(\bm\theta) \varepsilon_t \\
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& + 0.5 g_{\sigma\sigma}(\bm\theta) \\
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& + 0.5 g_{yy}(\bm\theta)\left(\hat{s}_{t-1} \otimes \hat{s}_{t-1}\right) \\
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& + 0.5 g_{uu}(\bm\theta) \left(\varepsilon_t \otimes \varepsilon_t\right) \\
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& + 0.5 g_{uy}(\bm\theta) \left(\hat{s}_{t-1} \otimes \varepsilon_t\right)
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\end{split}
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\]
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\medskip
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\item The measurement equations:
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\[
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y_t = Z s_t + e_t
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\]
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\bigskip
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where $Z$ is a selection matrix.
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Properties of the state space model}
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\begin{itemize}
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\item $s_t \sim $ first order Markov process:\newline
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\[
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p\left(s_t | s_{0:t-1} \right) = p(s_t\left| s_{t-1} \right)
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\]
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\bigskip\bigskip
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\item Observations are conditionally independent:\newline
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\[
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p\left( y_t | y_{1:t-1}, s_{0:t} \right) = p\left(y_t | s_t \right)
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\]
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\bigskip\bigskip
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\item[$\Rightarrow$] We cannot evaluate the likelihood, product of $p(y_t|y_{1:t-1})$ densities, without tracking the state variables.
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Nonlinear filter (I)}
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\begin{itemize}
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\item Suppose $p(y_{t-1}|s_{t-1})$ is known $\longrightarrow$ How to compute $p(y_{t}|s_{t})$?\newline
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\bigskip
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\item First we can predict the states in $t$ given information in $t-1$:
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\begin{equation}\label{prediction}
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p\left(s_t|y_{1:t-1}\right) = \int p(s_t|s_{t-1})p\left(s_{t-1}|y_{1:t-1}\right)\mathrm d s_{t-1}
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\end{equation}
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where $p(s_t|s_{t-1})$ is defined by the state equations:
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\[
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\begin{split}
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p\left(s_t|s_{t-1}\right) &= \int p\left(s_t|s_{t-1},\varepsilon_t\right)p\left(\varepsilon_t|s_{t-1}\right)\mathrm d\varepsilon_t\\
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&= \int p\left(s_t|s_{t-1},\varepsilon_t\right)p\left(\varepsilon_t\right)\mathrm d\varepsilon_t
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\end{split}
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\]
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Nonlinear filter (II)}
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\begin{itemize}
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\item $p\left(s_t|y_{1:t-1}\right)$ can be interpreted as our prior belief about the state variables.\newline
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\item Use Bayes theorem to update our beliefs:
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\begin{equation}\label{update}
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p\left(s_t|y_{1:t}\right) = \frac{p\left(y_t|s_t\right)p\left(s_t|y_{1:t-1}\right)}{p\left(y_t|y_{1:t-1}\right)}
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\end{equation}
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where:
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\[
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p\left(y_t|y_{1:t-1}\right) = \int p\left(y_t|s_t\right)p\left(s_t|y_{1:t-1}\right) \mathrm d s_t
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\]
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which is the conditional density required to evaluate the likelihood of the model.\newline
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\item $p\left(y_t|s_t\right)$, likelihood of $s_t$, is defined by the measurement equations.\newline
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\item[$\Rightarrow$] Iterate over (\ref{prediction}) and (\ref{update}).
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Approximated nonlinear filter (I)}
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\framesubtitle{Approximate the distribution of $s_t| y_{1:t}$}
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\begin{itemize}
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\item Suppose the distribution of $s_t| y_{1:t}$ can be accurately approximated by
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a set of nodes and weights
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$\left\{\left(s_t^i,\omega_t^i\right)\right\}_{i=1}^N$, with non negative weights
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summing up to one.\newline
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\item[$\Rightarrow$] When $N\rightarrow\infty$ $\sum_{i=1}^N\omega_i h(s_t^i)$ converges to $\mathbb E_{p(s_t|y_{1:t})}\left[h(s)\right]$.\newline
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\item Can be used to approximate the predictive densities:
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\[
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\widehat{p}\left(s_t|y_{1:t-1}\right) =
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\sum_{i=1}^N p(s_t | s_{t-1}^i)\omega_{t-1}^i
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\]
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\item Particles $\leftarrow$ random deviates from $s_t| y_{1:t}$.\newline
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\item How to sample from $s_t| y_{1:t}$?
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Approximated nonlinear filter (II)}
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\framesubtitle{Importance sampling (a)}
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\begin{itemize}
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\item Suppose we can sample from $q\left(s_t| y_{1:t}\right)$ but not from $q\left(s_t| y_{1:t}\right)$.\newline
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\item Then:
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\[
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\begin{split}
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\mathbb E_{p(s_t|y_{1:t})}\left[h(s)\right] &= \int
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\underbrace{\frac{p\left(s_t| y_{1:t}\right)}{q\left(s_t| y_{1:t}\right)}}_{\widetilde w_t(s_t)}q\left(s_t| y_{1:t}\right) h(s_t)\mathrm d s_t\\
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&= \mathbb E_{q(s_t|y_{1:t})}\left[\widetilde \omega_t(s_t)h(s_t)\right]
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\end{split}
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\]
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\item By the Bayes theorem:
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\[
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p\left(s_t | y_{1:t} \right) = \frac{p\left(y_{1:t}|s_t\right)p\left(s_t\right)}{p\left(y_{1:t}\right)}
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\]
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\item Unormalized weights:
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\[
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\widehat \omega_t(s_t) = \frac{p\left(y_{1:t}|s_t\right)p\left(s_t\right)}{q\left(s_t| y_{1:t}\right)}
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= p\left(y_{1:t}\right)\widetilde \omega_t(s_t)
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\]
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Approximated nonlinear filter (II)}
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\framesubtitle{Importance sampling (b)}
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With particles:
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\[
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\hat \omega_t^i = \frac{p\left(y_{1:t}|s_{t}^i\right)p\left(s_t^i\right)}{q\left(s_t^i| y_{1:t}\right)}
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\]
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\bigskip
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Then
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\[
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\widehat{\mathbb E}_{p(s_t|y_{1:t})}\left[h(s)\right] = \sum_{i=1}^N \widetilde{\omega}_ih(s_t^i)
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\]
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where
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\[
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\widetilde{\omega}_i = \frac{\widehat \omega_i}{\sum_{i=1}^N\widehat \omega_i}
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\]
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\end{frame}
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\begin{frame}[c]
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\frametitle{Approximated nonlinear filter (III)}
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\framesubtitle{Sequential importance sampling}
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Let's pick a proposal $q$ such that:
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\[
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q\left(s_t|y_{1:t}\right) = q\left(s_t| s_{t-1},y_t\right)q(s_{t-1}| y_{1:t-1})
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\]
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\bigskip
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Then, one can show that:
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\[
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\hat w_t(s_t) = \hat w_{t-1}(s_{t-1})\frac{p\left(y_t|s_t\right)p\left(s_t|s_{t-1}\right)}{q\left(s_t|s_{t-1}, y_t\right)}
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\]
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\bigskip\bigskip
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A particle’s current weight depends on its previous level and an incremental weight. This increment will put more weight on a particle if she is more likely with respect to its history and the current observation.
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\end{frame}
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\begin{frame}[fragile,c]
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\frametitle{Approximated nonlinear filter (III)}
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\framesubtitle{Generic particle filter algorithm}
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\begin{algorithmic}[1]
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\State $\left\{s_{0}^i, w_{0}^i\right\}_{i=1}^N \gets$ initial sample of particles.
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\For{$t \gets 1,T$}
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\ForAll{$i \gets 1,N$}
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\State $\tilde s_t^i \gets q\left(s_t|s_{t-1}^i, y_t\right)$
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\State $\hat w_t^i \gets w_{t-1}^i\frac{p\left(y_t|\tilde s_t^i\right)p\left(\tilde s_t^i| s_{t-1}^i\right)}{q\left(\tilde s_t^i|s_{t-1}^i, y_t\right)}$
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\EndFor
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\State $\tilde w_t^i \gets \frac{\hat w_t^i}{\sum_{i=1}^N\hat w_t^i}$
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\State $\left\{s_t^i, w_t^i\right\}_{i=1}^N \gets \left\{\tilde s_t^i, \tilde w_t^i\right\}_{i=1}^N$
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\EndFor
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\end{algorithmic}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Issue 1}
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\framesubtitle{Degeneracy}
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\begin{itemize}
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\item Weights degenerate as $t$ goes to infinity...\newline
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\item Up to the point where all but one particles have zero weights.\newline
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\item We don't want to sum-up a distribution with a single point.\newline
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\item[$\Rightarrow$] Resampling (kill particles with low weights and replicate good particles).
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile,c]
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\frametitle{Approximated nonlinear filter (IV)}
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\framesubtitle{Particle filter algorithm with resampling}
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\begin{algorithmic}[1]
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\State $\left\{s_{0}^i, w_{0}^i\right\}_{i=1}^N \gets$ initial sample of particles.
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\For{$t \gets 1,T$}
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\ForAll{$i \gets 1,N$}
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\State $\tilde s_t^i \gets q\left(s_t|s_{t-1}^i, y_t\right)$
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\State $\hat w_t^i \gets w_{t-1}^i\frac{p\left(y_t|\tilde s_t^i\right)p\left(\tilde s_t^i| s_{t-1}^i\right)}{q\left(\tilde s_t^i|s_{t-1}^i, y_t\right)}$
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\EndFor
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\State $\tilde w_t^i \gets \frac{\hat w_t^i}{\sum_{i=1}^N\hat w_t^i}$
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\State $N_t^{\star} \gets \left(\sum_{i=1}^N \left(\tilde w_t^i\right)^2\right)^{-1}$
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\If{$N_t^{\star}<\alpha N$}
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\State $\left\{s_t^i, w_t^i\right\}_{i=1}^N \gets $ Resampling step, with $w_t^i=\nicefrac{1}{N}$
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\Else
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\State $\left\{s_t^i, w_t^i\right\}_{i=1}^N \gets \left\{\tilde s_t^i, \tilde w_t^i\right\}_{i=1}^N$
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\EndIf
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\EndFor
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\end{algorithmic}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Set the proposal distribution}
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\framesubtitle{The blind proposal}
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\[
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q\left(s_t| s_{t-1}, y_t\right) = p\left(s_t | s_{t-1}\right)
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\]
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\bigskip
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\begin{itemize}
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\item Does not use current observation.\newline
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\item Weights recursive expression simplifies into:
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\[
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\widehat \omega_t\left(s_t\right) \propto \widetilde \omega_{t-1}\left(s_{t-1}\right) p\left(y_t | s_t\right)
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\]
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\item Conditional density of observation is:
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\[
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p\left(y_t | \tilde s_t^i\right) = \left(2\pi\right)^{-\frac{n}{2}}~|R|^{-\frac{1}{2}}e^{-\frac{1}{2}\left(y_t-g(\tilde s_t^i;\bm\theta)\right)' R^{-1} \left(y_t-g(\tilde s_t^i;\bm\theta)\right)}
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\]
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\bigskip
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\item[$\Rightarrow$] Measurement errors are mandatory!
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\end{itemize}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Issues 2 and 3}
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\framesubtitle{Likelihood}
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\[
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\begin{split}
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p\left(y_{1:T} | \bm\theta \right) &= p\left(y_1 | s_0;\bm\theta \right)p\left(s_0 |\bm\theta \right) \prod_{t=2}^{T}{p\left(y_t | y_{1:t-1};\bm\theta \right)}\\
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&\approx \sum_{i=1}^N{ \widetilde\omega_{t-1}^ip\left(y_t | s_t^i ; \bm\theta\right)}
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\end{split}
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\]
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\bigskip\bigskip
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\begin{enumerate}
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\item How to choose the initial distribution for the states ($s_0$)?\newline
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\item Non differentiability of the likelihhod because of the resampling step.\newline
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\end{enumerate}
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\end{frame}
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\begin{frame}[c]
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\frametitle{Estimation with particle filter}
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\bigskip\bigskip
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\begin{itemize}
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\item Do not use gradient based methods to estimate the mode...\newline
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||
|
||
\medskip
|
||
|
||
\item Simplex ``works'', but still issues with the hessian at the estimated mode.\newline
|
||
|
||
\medskip
|
||
|
||
\item Better to run directly the MCMC.\newline
|
||
|
||
\medskip
|
||
|
||
\item Slice? SMC...\newline
|
||
|
||
\medskip
|
||
|
||
\item Do we really need particle (or nonlinear) filters?
|
||
|
||
\end{itemize}
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\end{frame}
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\end{document}
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% Local Variables:
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% ispell-check-comments: exclusive
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% ispell-local-dictionary: "american-insane"
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% TeX-master: t
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% End:
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