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@ -5,6 +5,7 @@
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\usepackage{nicefrac}
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\usepackage{mathrsfs}
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\usepackage{etex}
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\usepackage{xcolor}
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\usepackage{pgfplots,tikz}
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\usepackage{tikz-qtree}
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@ -19,13 +20,13 @@
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\newlength\figureheight
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\newlength\figurewidth
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\definecolor{gray}{gray}{0.4}
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\begin{document}
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\title{The stochastic extended path approach}
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\author{St\'ephane Adjemian\footnote{Universit\'e du Maine} and Michel Juillard\footnote{Banque de France}}
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\date{June, 2016}
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\title{(stochastic) Extended path}
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\author{}
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\date{}
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\begin{frame}
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\titlepage{}
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@ -35,16 +36,15 @@
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\frametitle{Motivations}
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\begin{itemize}
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\item Severe nonlinearities play sometimes an important role in
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macroeconomics.
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\item In particular occasionally binding constraints: irreversible
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investment, borrowing constraint, ZLB.
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\item Nonlinearity can play an important role in
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macroeconomics.\newline
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\item Large deviations w.r.t. the steady state or occasionally binding constraints: irreversible
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investment, borrowing constraint, ZLB.\newline
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\item Usual local approximation techniques don't work when there are
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kinks.
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\item Deterministic, perfect forward, models can be solved with much
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greater accuracy than stochastic ones.
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\item Deterministic models can be solved with arbitrary accuracy, unlike stochastic ones with perturbation.\newline
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\item The extended path approach aims to keep the ability of
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deterministic methods to provide accurate account of nonlinearities.
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deterministic methods to provide accurate account of nonlinearities.
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\end{itemize}
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\end{frame}
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@ -82,16 +82,16 @@ auxiliary variables.
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\begin{itemize}
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\item Perfect foresight models, after a shock economy returns asymptotically
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to equilibirum.
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to equilibirum.\newline
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\item For a long enough simulation, one can consider that for all
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practical purpose the system is back to equilibrium.
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practical purpose the system is back to equilibrium.\newline
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\item This suggests to solve a two value boundary problem with
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initial conditions for some variables (backward looking) and
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terminal conditions for others (forward looking).
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terminal conditions for others (forward looking).\newline
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\item In practice, one can use a Newton method to the equations of
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the model stacked over all periods of the simulation.
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the model stacked over all periods of the simulation.\newline
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\item The Jacobian matrix of the stacked system is very sparse and
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this characteristic must be used to write a practical algorithm.
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this characteristic must be used to write a practical algorithm.
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\end{itemize}
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\end{frame}
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@ -102,10 +102,10 @@ auxiliary variables.
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\begin{itemize}
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\item Already proposed by Fair and Taylor (1983).
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\item Already proposed by Fair and Taylor (1983).\newline
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\item The extended path approach creates a stochastic simulation as if only
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the shocks of the current period were random.
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the shocks of the current period were random.\newline
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\item Substituting (\ref{eq:gmodel:1}) in (\ref{eq:gmodel:4}), define:
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\[
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@ -176,27 +176,20 @@ s_{t+H} &= \mathsf Q(s_{t+H-1},0)\\
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\frametitle{Extended path algorithm (discussion)}
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\begin{itemize}
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\item This approach takes full account of the \emph{deterministic}
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non linearities...
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\medskip
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non linearities...\newline
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\item ... But neglects the Jensen inequality by setting future
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innovations to zero (the expectation).
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\bigskip
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innovations to zero (the expectation).\newline
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\item We do not solve the rational expectation model! We solve a
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model where the agents believe that the economy will not be
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perturbed in the future. They observe new realizations of the
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innovations at each date but do not update this belief...
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\bigskip
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innovations in each period but do not update this belief...\newline
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\item Uncertainty about the future does not matter here.
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\item Uncertainty about the future does not matter here.\newline
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\medskip
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\item EP > First order perturbation (certainty equivalence)
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\item EP > OccBin > First order perturbation\newline
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\end{itemize}
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\end{frame}
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@ -206,9 +199,14 @@ s_{t+H} &= \mathsf Q(s_{t+H-1},0)\\
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\frametitle{Stochastic extended path}
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\begin{itemize}
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\item The strong assumption about future uncertainty can be relaxed
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by approximating the expected terms in the Euler
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equations~(\ref{eq:gmodel:2})
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equations~(\ref{eq:gmodel:2}):
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\begin{equation*}\label{eq:gmodel:2}%\tag{Euler equations}
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\mathsf F \left(y_t,x_t,s_t,\mathbb E_t \left[ \mathscr E_{t+1}\right]\right) = 0
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\end{equation*}
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\bigskip
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@ -230,12 +228,11 @@ s_{t+H} &= \mathsf Q(s_{t+H-1},0)\\
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\frametitle{Gaussian quadrature (univariate)}
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\begin{itemize}
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\item Let $X$ be a Gaussian random variable with mean zero and variance $\sigma^2_x>0$, and suppose that we need to evaluate
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$\mathbb E [\varphi(X)]$, where $\varphi$ is a continuous function.
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\item By definition we have:
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$\mathbb E [\varphi(X)]$, where $\varphi$ is a continuous function:
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{\small\[
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\mathbb E [\varphi(X)] = \frac{1}{\sigma_x\sqrt{2\pi}}\int_{-\infty}^{\infty} \varphi(x)e^{-\frac{x^2}{2\sigma^2_x}}\mathrm dx
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\]}
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\item It can be shown that this integral can be approximated by a finite sum using the following result:
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\item The integral can be approximated by a finite sum, using:
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{\small\[
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\int_{-\infty}^{\infty} \varphi(z)e^{-z^2}\mathrm dx = \sum_{i=1}^n\omega_i \varphi(z_i) {\gray + \frac{n!\sqrt{n}}{2^n}\frac{\varphi^{(2n)}(\xi)}{(2n)!}}
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\]} where $z_i$ ($i=1,\dots,n$) are the roots of an order $n$
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@ -254,39 +251,37 @@ s_{t+H} &= \mathsf Q(s_{t+H-1},0)\\
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{\small\[
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\mathbb E [\varphi(X)] = (2\pi)^{-\frac{p}{2}}\int_{\mathbb R^{p}} \varphi(\bm{x})e^{-\frac{1}{2} \bm{x}'\bm{x}}\mathrm d\bm{x}
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\]}
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\item Let $\{(\omega_i,z_i)\}_{i=1}^n$ be the weights and nodes of an order $n$ univariate Gaussian quadrature.
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\item Let $\{(\omega_i,z_i)\}_{i=1}^n$ be the weights and nodes of an order $n$ univariate Gaussian quadrature.\newline
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\item This integral can be approximated using a tensor grid:
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{\small\[
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\int_{\mathbb R^{p}} \varphi(\bm{z})e^{-\bm{z}'\bm{z}}\mathrm d\bm{z} \approx \sum_{i_1,\dots,i_p=1}^n\omega_{i_1}\dots\omega_{i_p} \varphi(z_{i_1},\dots,z_{i_p})
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\]}
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\item \textbf{Curse of dimensionality:} The number of terms in the sum grows exponentially with the number of shocks.
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\item \textbf{Curse of dimensionality 1}: The number of terms in the sum grows exponentially with the number of shocks.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Unscented transform}
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\begin{frame}[fragile]
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\frametitle{Other numerical integration rules}
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\begin{itemize}
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\item Let $X$ be a $p\times 1$ multivariate random variable with mean zero and variance $\Sigma_x$. We need to compute moments of $Y = \varphi(X)$.
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\item Let $\mathscr S_p = \{\omega_i,\bm{x}_i\}_{i=1}^{2p+1}$ be a set of deterministic weights and points:
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{\footnotesize
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\[
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\begin{array}{rclcrcl}
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\bm{x}_0 &=& 0 && \omega_0 &=& \frac{\kappa}{p+\kappa} \\
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\bm{x}_i &=& \left(\sqrt{(p+\kappa)\Sigma_x}\right)_i && \omega_i &=& \frac{1}{2(p+\kappa)}\text{, for i=1,\dots,p}\\
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\bm{x}_i &=& -\left(\sqrt{(p+\kappa)\Sigma_x}\right)_{i-p} && \omega_i &=& \frac{1}{2(p+\kappa)}\text{, for i=p+1,\dots,2p}\\
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\end{array}
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\]}
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where $\kappa$ is a real positive scaling parameter.
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\item It can be shown that the weights are positive and summing-up to one and that the first and second order ``sample'' moments of $\mathscr S_p$ are matching those of $X$.
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\item Compute the moments of $Y$ by applying the mapping $\varphi$ to $\mathscr S_p$.
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\item Exact mean and variance of $Y$ for a second order Taylor approximation of $\varphi$.
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\item By default Dynare uses multivariate Gaussian quadrature to approximate the expected terms.\newline
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\item To cope with \textbf{curse of dimensionality 1} we can consider other numerical integration approaches:\newline
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\begin{itemize}
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\item[--] Cubature (ordre 3 or 5)\\
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Set \verb+options_.ep.IntegrationAlgorithm+ to \verb+'Stroud-Cubature-3'+ or \verb+'Stroud-Cubature-5'+ \newline
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\item[--] Unscented transform\\
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Set \verb+options_.ep.IntegrationAlgorithm+ to \verb+'Unscented'+\newline
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\end{itemize}
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\item These alternatives rely on a smaller number of nodes, which do
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not grow exponentially with the number of innovations.\newline
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\item But there is another curse of dimensionality.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Forward histories (one shock, three nodes, order two SEP)}
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\framesubtitle{Curse of dimensionality 2}
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\begin{tikzpicture}
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\tikzset{grow'=right,level distance=80pt}
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\tikzset{execute at begin node=\strut}
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@ -311,11 +306,11 @@ s_{t+H} &= \mathsf Q(s_{t+H-1},0)\\
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\begin{frame}
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\frametitle{Fishbone integration}
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\begin{itemize}
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\item The curse of dimensionality can be overcome by pruning the tree of forward histories.
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\item This can be done by considering that innovations, say, at time $t+1$ and $t+2$ are unrelated variables (even if they share the same name).
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%\item Analogy with a good at different periods in an intertemporal general equilibrium model.
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\item If we have $n_u$ innovations and if agents perceive uncertainty for the next $k$ following periods, we consider an integration problem involving $n_u \times k$ unrelated variables.
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\item We use a two points Cubature rule to compute the integral (unscented transform with $\kappa=0$) $\rightarrow$ The complexity of the integration problem grows linearly with $n_u$ or $k$
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\item We can prune the tree of forward histories by removing low probability branches...\newline
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\item Or by considering that innovations, say, at time $t+1$ and $t+2$ are unrelated variables.\newline
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\item {\color{gray} Analogy with a good at different periods in an intertemporal general equilibrium model.}\newline
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\item If we have $n_u$ innovations and if agents perceive uncertainty for the next $k$ following periods, we consider an integration problem involving $n_u \times k$ unrelated variables.\newline
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\item We use a two points Cubature rule to compute the integral (unscented transform with $\kappa=0$) $\rightarrow$ The complexity of the integration problem grows linearly with $n_u$ or $k$\newline
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\end{itemize}
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\end{frame}
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@ -423,16 +418,14 @@ s_{t+H}^{i,j} &= \mathsf Q(s_{t+H-1}^{i,j},0)\\
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\end{frame}
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\begin{frame}
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\frametitle{Stochastic extended path (discussion)}
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\begin{itemize}
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\item The extended path approach takes full account of the deterministic nonlinearities of the model.
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\item It takes into account the nonlinear effects of future shocks $k$-period ahead.
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\item It neglects the effects of uncertainty in the long run. In most models this effect declines with the discount factor.
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\item The Stochastic Perfect Foresight model, that must be solved at each date, is very large.
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\item Curse of dimensionality with respect with the number of innovations and the order of approximation but \emph{not with the number of state variables}!
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\item The extended path approach takes full account of the deterministic nonlinearities of the model.\newline
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\item It also takes into account the nonlinear effects of future shocks $k$-period ahead.\newline
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\item It neglects the effects of uncertainty in the long run.\newline
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\item The Stochastic Perfect Foresight model, that must be solved at each date, is huge.\newline
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\item Curse of dimensionality with respect with the number of innovations \textbf{and} the order of approximation but \emph{not with the number of state variables}!
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\end{itemize}
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\end{frame}
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@ -446,7 +439,7 @@ s_{t+H}^{i,j} &= \mathsf Q(s_{t+H-1}^{i,j},0)\\
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\item Household's intertemporal utility is given by
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\[
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\mathbb E_t\left[\sum_{\tau = 0}^\infty
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\beta^{ẗ-\tau}\frac{c^\theta_{t+\tau}}{\theta}\right]\quad\quad\mbox{with }
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\beta^{t-\tau}\frac{c^\theta_{t+\tau}}{\theta}\right]\quad\quad\mbox{with }
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\theta \in (-\infty,0) \cup (0,1]
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\]
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\item Budget constraint is
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@ -521,6 +514,8 @@ and
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\[
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b_i = \frac{\theta\rho\left(1-\rho^i\right)}{1-\rho}
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\]
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\medskip
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Note that we abstract from the accuracy errors introduced by the numerical approximation of the integrals.
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\end{frame}
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\begin{frame}
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@ -546,33 +541,29 @@ is equal to 12.4812.
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\end{frame}
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\begin{frame}
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\frametitle{Comparing expended path and closed-form solution}
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Difference between expended path approximation, $\hat y_t$, and closed-form
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solution, $y_t$.
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\frametitle{Comparing extended path and closed-form solution}
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Difference between extended path approximation, $\hat y_t$, and closed-form
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solution, $y_t$.\newline
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\begin{itemize}
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\item Using 800 terms to approximate the infinite summation
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\item Computing over 30000 periods
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\item Using 800 terms to approximate the infinite summation\newline
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\item Computing over 30000 periods\newline
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\end{itemize}
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\begin{align*}
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\min\left(y_t-\hat y_t\right) &= 0.1726\\
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\max\left(y_t-\hat y_t\right) &= 0.1820\\
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\end{align*}
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\begin{itemize}
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\item The effect of future volatility isn't trivial
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\[
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\frac{\tilde y-\bar y}{\bar y} = 0.0144
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\]
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\item The effect of future volatility doesn't depend much on the state
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of the economy.
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\item The effect of future volatility is around 1\%, $\frac{\tilde y-\bar y}{\bar y} = 0.0144$, and doesn't depend much on the state of the economy.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Stochastic extended path}
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\begin{itemize}
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\item A $k$-order stochastic expended path approach computes the conditional
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\item A $k$-order stochastic extended path approach computes the conditional
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expectation taking into accounts the shocks over the next $k$ periods.
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\item The closed formula is
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\item The closed formula is
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\[
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\check{y}_t = \sum_{i=1}^\infty \beta^i e^{a_i+b_i\hat x_t}
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\]
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@ -584,7 +575,7 @@ solution, $y_t$.
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\frac{\theta^2\sigma^2}{2(1-\rho)^2}\left(i-2\rho\frac{1-\rho^i}{1-\rho}+\rho^2\frac{1-\rho^{2i}}{1-\rho^2}\right)
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& \mbox{ for } & i\le k\\
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\frac{\theta^2\sigma^2}{2(1-\rho)^2}\left(k-2\rho\frac{\rho^{i-k}-\rho^i}{1-\rho}+\rho^2\frac{\rho^{2(i-k)}-\rho^{2i}}{1-\rho^2}\right)
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& \mbox{ for } & i > k
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& \mbox{ for } & i > k
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\end{array}
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\right.
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\]
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@ -599,16 +590,16 @@ solution, $y_t$.
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\begin{frame}
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\frametitle{Quantitative evaluation}
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\begin{itemize}
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\item What is the ability of the stochastic extended path approach to
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capture the effect of future volatility?
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\item What part of the difference between the risky steady state and
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deterministic steady state is captured by different values of $k$?
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\item Deterministic steady state: 12.3035
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\item Risky steady state: 12.4812
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\item What is the ability of the stochastic extended path approach to capture
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the effect of future volatility? What part of the difference between the risky
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steady state and deterministic steady state is captured by different values of
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$k$?\newline
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\item Deterministic steady state: 12.3035\newline
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\item Risky steady state: 12.4812\newline
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\item The contribution of $k$ future periods
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\begin{center}
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\begin{tabular}{cc}
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$k$ & Percentage\\
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$k$ & Percentage\\
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1 & 7.4\%\\
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2 & 14.3\%\\
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9 & 50.0\%\\
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@ -616,8 +607,8 @@ capture the effect of future volatility?
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60 & 99.0\%
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\end{tabular}
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\end{center}
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\item In such a model, it is extremely costly to give full account of
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the effects of future volatility with the stochastic extended path approach.
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%\item In such a model, it is extremely costly to give full account of
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% the effects of future volatility with the stochastic extended path approach.
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\end{itemize}
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\end{frame}
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@ -628,11 +619,11 @@ capture the effect of future volatility?
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\item A very large number of periods forward (the order of stochastic
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|
extended path) is necessary to obtain an accurate figure of the
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effects of future volatility.
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effects of future volatility.\newline
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\item However, even a local approximation with a Taylor expansion of
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low order provides better information on this effect of future
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volatility.
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volatility.\newline
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\item This suggests to combine the two approaches.
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@ -684,7 +675,6 @@ s_{t+H}^i &= \mathsf Q(s_{t+H-1}^i,0)\\
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\end{tabular}
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\end{center}
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\end{frame}
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@ -782,14 +772,12 @@ where $\mu_t$ is the Lagrange multiplier associated with the constraint on inves
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|
\begin{frame}
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|
\frametitle{Conclusion and future work}
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|
\begin{itemize}
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|
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|
\item The extended path approach takes into account effects of nonlinearities.
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|
\item The stochastic extended path approach takes also partially into account nonlinear effects of future volatility.
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|
\item Possible to use an hybrid approach, using the risky steady state as terminal condition.
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|
|
\item The approach suffers from the curse of dimensionality but it can be mitigated by
|
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|
|
\begin{itemize}
|
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|
|
|
\item using monomial formulas for integration when there are several shocks
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|
|
\item exploiting embarassingly parallel nature of the algorithm
|
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|
|
|
\end{itemize}
|
|
|
|
|
\item The extended path approach takes into account effects of nonlinearities.\newline
|
|
|
|
|
\item The stochastic extended path approach takes also partially into account nonlinear effects of future volatility.\newline
|
|
|
|
|
\item Possible to use an hybrid approach.\newline
|
|
|
|
|
\item The approach suffers from the curse of dimensionality but it can be mitigated.\newline
|
|
|
|
|
\item Parallelization is possible if EP is used in estimation (SMM or PF) but not in the SEP solver.\newline
|
|
|
|
|
\item Improvements: produce IRFs with SEP, write a mex for building the stacked SEP nonlinear problem.
|
|
|
|
|
\end{itemize}
|
|
|
|
|
\end{frame}
|
|
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|
|
Stéphane Adjemian (Argos)
Stéphane Adjemian (Charybdis)