Slides for Ispra 2022 workshop.
parent
a151225fb3
commit
f38b87a8e5

@ 10,28 +10,23 @@


*_dynamic.mexmaci64


*_static.mexmaci64




*.fdb_latexmk


*.fls




matlab.mk




mod/rbc/rbc.m


mod/rbc/rbc_pea2.m


mod/rbc/locals.m




mod/rbc/*_set_auxiliary_variables.m


mod/rbc/*_dynamic.m


mod/rbc/*_static.m


mod/rbc/*_steadystate2.m


mod/rbc/+rbc/*


mod/rbc/rbc/*


mod/rbc/locals.m


mod/rbc/rbc_pea2/*


mod/rbc/rbc_pea3/*


mod/rbc/+rbc_pea2/*


mod/rbc/+rbc_pea3/*




mod/rbcii/rbc.m


mod/rbcii/rbc/*


mod/rbcii/rbc_set_auxiliary_variables.m


mod/rbcii/rbc_steadystate2.m


mod/rbcii/rbcii.m


mod/rbcii/+rbc/*


mod/rbcii/rbcii/*


mod/rbcii/rbcii_set_auxiliary_variables.m


mod/rbcii/rbcii_steadystate2.m


mod/rbcii/+rbcii/*




mod/rbcsimple/locals.m


mod/rbcsimple/results/*



@ 45,9 +40,10 @@ mod/rbciisimple/*.txt


mod/rbciisimple/*.mat


mod/rbciisimple/*.log






slides/ep.aux


slides/ep.nav


slides/ep.out


slides/ep.snm


slides/ep.toc


slides/auto/


slides/slides.aux


slides/slides.nav


slides/slides.out


slides/slides.snm


slides/slides.toc


slides/slides.vrb





@ 16,7 +16,7 @@ psi = 1.000;


delta = 0.100;


rho = 0.800;


effstar = 1.000;


sigma = 0.150;


sigma = 0.100;




model(use_dll);





@ 54,7 +54,7 @@ end;




// Write analytical steady state file (without globals)


options_.steadystate_flag = 2;


copyfile('rbcii_steady_state.m','rbcii_steadystate2.m');


copyfile('rbcii_steady_state.m','+rbcii/steadystate.m');




steady;





@ 64,12 +64,19 @@ shocks;


var EfficiencyInnovation = 1;


end;




ts = extended_path(oo_.steady_state, 1000, [], options_, M_, oo_);


ptions_.ep.stochastic.quadrature.nodes = 3;




extended_path(periods=200);


ts = Simulated_time_series;


ts.save('simulateddataep0');




extended_path(periods=200, order=1);


ts = Simulated_time_series;


ts.save('simulateddataep1');








return


extended_path(periods=100);


ts0 = Simulated_time_series;




transition;


regression;


distribution;





@ 6,6 +6,7 @@ slides.pdf: slides.tex


while ($(LATEX) slides.tex ; \


grep q "Rerun to get cross" slides.log ) do true ; \


done




clean:


rm f *.aux *.log *.out *.nav *.rel *.toc *.snm *.synctex.gz *.vrb


rm rf auto



@ 14,7 +15,7 @@ cleanall:


rm f *.pdf




push: slides.pdf


scp ep.pdf ulysses:~/www/dynare.ithaca.fr/slides/sep.pdf


scp slides.pdf ryuk:~/home/www/dynare.adjemian.eu/prose/ispra2022.pdf






.PHONY: push





@ 5,6 +5,7 @@


\usepackage{nicefrac}


\usepackage{mathrsfs}


\usepackage{etex}


\usepackage{xcolor}




\usepackage{pgfplots,tikz}


\usepackage{tikzqtree}



@ 19,13 +20,13 @@




\newlength\figureheight


\newlength\figurewidth






\definecolor{gray}{gray}{0.4}




\begin{document}


\title{The stochastic extended path approach}


\author{St\'ephane Adjemian\footnote{Universit\'e du Maine} and Michel Juillard\footnote{Banque de France}}


\date{June, 2016}


\title{(stochastic) Extended path}


\author{}


\date{}




\begin{frame}


\titlepage{}



@ 35,16 +36,15 @@


\frametitle{Motivations}




\begin{itemize}


\item Severe nonlinearities play sometimes an important role in


macroeconomics.


\item In particular occasionally binding constraints: irreversible


investment, borrowing constraint, ZLB.


\item Nonlinearity can play an important role in


macroeconomics.\newline


\item Large deviations w.r.t. the steady state or occasionally binding constraints: irreversible


investment, borrowing constraint, ZLB.\newline


\item Usual local approximation techniques don't work when there are


kinks.


\item Deterministic, perfect forward, models can be solved with much


greater accuracy than stochastic ones.


\item Deterministic models can be solved with arbitrary accuracy, unlike stochastic ones with perturbation.\newline


\item The extended path approach aims to keep the ability of


deterministic methods to provide accurate account of nonlinearities.


deterministic methods to provide accurate account of nonlinearities.


\end{itemize}


\end{frame}





@ 82,16 +82,16 @@ auxiliary variables.




\begin{itemize}


\item Perfect foresight models, after a shock economy returns asymptotically


to equilibirum.


to equilibirum.\newline


\item For a long enough simulation, one can consider that for all


practical purpose the system is back to equilibrium.


practical purpose the system is back to equilibrium.\newline


\item This suggests to solve a two value boundary problem with


initial conditions for some variables (backward looking) and


terminal conditions for others (forward looking).


terminal conditions for others (forward looking).\newline


\item In practice, one can use a Newton method to the equations of


the model stacked over all periods of the simulation.


the model stacked over all periods of the simulation.\newline


\item The Jacobian matrix of the stacked system is very sparse and


this characteristic must be used to write a practical algorithm.


this characteristic must be used to write a practical algorithm.


\end{itemize}




\end{frame}



@ 102,10 +102,10 @@ auxiliary variables.




\begin{itemize}




\item Already proposed by Fair and Taylor (1983).


\item Already proposed by Fair and Taylor (1983).\newline




\item The extended path approach creates a stochastic simulation as if only


the shocks of the current period were random.


the shocks of the current period were random.\newline




\item Substituting (\ref{eq:gmodel:1}) in (\ref{eq:gmodel:4}), define:


\[



@ 176,27 +176,20 @@ s_{t+H} &= \mathsf Q(s_{t+H1},0)\\


\frametitle{Extended path algorithm (discussion)}


\begin{itemize}


\item This approach takes full account of the \emph{deterministic}


non linearities...




\medskip


non linearities...\newline






\item ... But neglects the Jensen inequality by setting future


innovations to zero (the expectation).




\bigskip


innovations to zero (the expectation).\newline




\item We do not solve the rational expectation model! We solve a


model where the agents believe that the economy will not be


perturbed in the future. They observe new realizations of the


innovations at each date but do not update this belief...




\bigskip


innovations in each period but do not update this belief...\newline




\item Uncertainty about the future does not matter here.


\item Uncertainty about the future does not matter here.\newline




\medskip




\item EP > First order perturbation (certainty equivalence)


\item EP > OccBin > First order perturbation\newline


\end{itemize}




\end{frame}



@ 206,9 +199,14 @@ s_{t+H} &= \mathsf Q(s_{t+H1},0)\\


\frametitle{Stochastic extended path}




\begin{itemize}




\item The strong assumption about future uncertainty can be relaxed


by approximating the expected terms in the Euler


equations~(\ref{eq:gmodel:2})


equations~(\ref{eq:gmodel:2}):




\begin{equation*}\label{eq:gmodel:2}%\tag{Euler equations}


\mathsf F \left(y_t,x_t,s_t,\mathbb E_t \left[ \mathscr E_{t+1}\right]\right) = 0


\end{equation*}




\bigskip





@ 230,12 +228,11 @@ s_{t+H} &= \mathsf Q(s_{t+H1},0)\\


\frametitle{Gaussian quadrature (univariate)}


\begin{itemize}


\item Let $X$ be a Gaussian random variable with mean zero and variance $\sigma^2_x>0$, and suppose that we need to evaluate


$\mathbb E [\varphi(X)]$, where $\varphi$ is a continuous function.


\item By definition we have:


$\mathbb E [\varphi(X)]$, where $\varphi$ is a continuous function:


{\small\[


\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


\]}


\item It can be shown that this integral can be approximated by a finite sum using the following result:


\item The integral can be approximated by a finite sum, using:


{\small\[


\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)!}}


\]} where $z_i$ ($i=1,\dots,n$) are the roots of an order $n$



@ 254,39 +251,37 @@ s_{t+H} &= \mathsf Q(s_{t+H1},0)\\


{\small\[


\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}


\]}


\item Let $\{(\omega_i,z_i)\}_{i=1}^n$ be the weights and nodes of an order $n$ univariate Gaussian quadrature.


\item Let $\{(\omega_i,z_i)\}_{i=1}^n$ be the weights and nodes of an order $n$ univariate Gaussian quadrature.\newline


\item This integral can be approximated using a tensor grid:


{\small\[


\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})


\]}


\item \textbf{Curse of dimensionality:} The number of terms in the sum grows exponentially with the number of shocks.


\item \textbf{Curse of dimensionality 1}: The number of terms in the sum grows exponentially with the number of shocks.


\end{itemize}


\end{frame}






\begin{frame}


\frametitle{Unscented transform}


\begin{frame}[fragile]


\frametitle{Other numerical integration rules}


\begin{itemize}


\item Let $X$ be a $p\times 1$ multivariate random variable with mean zero and variance $\Sigma_x$. We need to compute moments of $Y = \varphi(X)$.


\item Let $\mathscr S_p = \{\omega_i,\bm{x}_i\}_{i=1}^{2p+1}$ be a set of deterministic weights and points:


{\footnotesize


\[


\begin{array}{rclcrcl}


\bm{x}_0 &=& 0 && \omega_0 &=& \frac{\kappa}{p+\kappa} \\


\bm{x}_i &=& \left(\sqrt{(p+\kappa)\Sigma_x}\right)_i && \omega_i &=& \frac{1}{2(p+\kappa)}\text{, for i=1,\dots,p}\\


\bm{x}_i &=& \left(\sqrt{(p+\kappa)\Sigma_x}\right)_{ip} && \omega_i &=& \frac{1}{2(p+\kappa)}\text{, for i=p+1,\dots,2p}\\


\end{array}


\]}


where $\kappa$ is a real positive scaling parameter.


\item It can be shown that the weights are positive and summingup to one and that the first and second order ``sample'' moments of $\mathscr S_p$ are matching those of $X$.


\item Compute the moments of $Y$ by applying the mapping $\varphi$ to $\mathscr S_p$.


\item Exact mean and variance of $Y$ for a second order Taylor approximation of $\varphi$.


\item By default Dynare uses multivariate Gaussian quadrature to approximate the expected terms.\newline


\item To cope with \textbf{curse of dimensionality 1} we can consider other numerical integration approaches:\newline


\begin{itemize}


\item[] Cubature (ordre 3 or 5)\\


Set \verb+options_.ep.IntegrationAlgorithm+ to \verb+'StroudCubature3'+ or \verb+'StroudCubature5'+ \newline


\item[] Unscented transform\\


Set \verb+options_.ep.IntegrationAlgorithm+ to \verb+'Unscented'+\newline


\end{itemize}


\item These alternatives rely on a smaller number of nodes, which do


not grow exponentially with the number of innovations.\newline


\item But there is another curse of dimensionality.


\end{itemize}


\end{frame}






\begin{frame}


\frametitle{Forward histories (one shock, three nodes, order two SEP)}


\framesubtitle{Curse of dimensionality 2}


\begin{tikzpicture}


\tikzset{grow'=right,level distance=80pt}


\tikzset{execute at begin node=\strut}



@ 311,11 +306,11 @@ s_{t+H} &= \mathsf Q(s_{t+H1},0)\\


\begin{frame}


\frametitle{Fishbone integration}


\begin{itemize}


\item The curse of dimensionality can be overcome by pruning the tree of forward histories.


\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).


%\item Analogy with a good at different periods in an intertemporal general equilibrium model.


\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.


\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 $k$


\item We can prune the tree of forward histories by removing low probability branches...\newline


\item Or by considering that innovations, say, at time $t+1$ and $t+2$ are unrelated variables.\newline


\item {\color{gray} Analogy with a good at different periods in an intertemporal general equilibrium model.}\newline


\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


\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


\end{itemize}


\end{frame}





@ 423,16 +418,14 @@ s_{t+H}^{i,j} &= \mathsf Q(s_{t+H1}^{i,j},0)\\


\end{frame}










\begin{frame}


\frametitle{Stochastic extended path (discussion)}


\begin{itemize}


\item The extended path approach takes full account of the deterministic nonlinearities of the model.


\item It takes into account the nonlinear effects of future shocks $k$period ahead.


\item It neglects the effects of uncertainty in the long run. In most models this effect declines with the discount factor.


\item The Stochastic Perfect Foresight model, that must be solved at each date, is very large.


\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}!


\item The extended path approach takes full account of the deterministic nonlinearities of the model.\newline


\item It also takes into account the nonlinear effects of future shocks $k$period ahead.\newline


\item It neglects the effects of uncertainty in the long run.\newline


\item The Stochastic Perfect Foresight model, that must be solved at each date, is huge.\newline


\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}!


\end{itemize}


\end{frame}





@ 446,7 +439,7 @@ s_{t+H}^{i,j} &= \mathsf Q(s_{t+H1}^{i,j},0)\\


\item Household's intertemporal utility is given by


\[


\mathbb E_t\left[\sum_{\tau = 0}^\infty


\beta^{ẗ\tau}\frac{c^\theta_{t+\tau}}{\theta}\right]\quad\quad\mbox{with }


\beta^{t\tau}\frac{c^\theta_{t+\tau}}{\theta}\right]\quad\quad\mbox{with }


\theta \in (\infty,0) \cup (0,1]


\]


\item Budget constraint is



@ 521,6 +514,8 @@ and


\[


b_i = \frac{\theta\rho\left(1\rho^i\right)}{1\rho}


\]


\medskip


Note that we abstract from the accuracy errors introduced by the numerical approximation of the integrals.


\end{frame}




\begin{frame}



@ 546,33 +541,29 @@ is equal to 12.4812.


\end{frame}




\begin{frame}


\frametitle{Comparing expended path and closedform solution}


Difference between expended path approximation, $\hat y_t$, and closedform


solution, $y_t$.


\frametitle{Comparing extended path and closedform solution}




Difference between extended path approximation, $\hat y_t$, and closedform


solution, $y_t$.\newline


\begin{itemize}


\item Using 800 terms to approximate the infinite summation


\item Computing over 30000 periods


\item Using 800 terms to approximate the infinite summation\newline


\item Computing over 30000 periods\newline


\end{itemize}


\begin{align*}


\min\left(y_t\hat y_t\right) &= 0.1726\\


\max\left(y_t\hat y_t\right) &= 0.1820\\


\end{align*}


\begin{itemize}


\item The effect of future volatility isn't trivial


\[


\frac{\tilde y\bar y}{\bar y} = 0.0144


\]


\item The effect of future volatility doesn't depend much on the state


of the economy.


\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.


\end{itemize}


\end{frame}




\begin{frame}


\frametitle{Stochastic extended path}


\begin{itemize}


\item A $k$order stochastic expended path approach computes the conditional


\item A $k$order stochastic extended path approach computes the conditional


expectation taking into accounts the shocks over the next $k$ periods.


\item The closed formula is


\item The closed formula is


\[


\check{y}_t = \sum_{i=1}^\infty \beta^i e^{a_i+b_i\hat x_t}


\]



@ 584,7 +575,7 @@ solution, $y_t$.


\frac{\theta^2\sigma^2}{2(1\rho)^2}\left(i2\rho\frac{1\rho^i}{1\rho}+\rho^2\frac{1\rho^{2i}}{1\rho^2}\right)


& \mbox{ for } & i\le k\\


\frac{\theta^2\sigma^2}{2(1\rho)^2}\left(k2\rho\frac{\rho^{ik}\rho^i}{1\rho}+\rho^2\frac{\rho^{2(ik)}\rho^{2i}}{1\rho^2}\right)


& \mbox{ for } & i > k


& \mbox{ for } & i > k


\end{array}


\right.


\]



@ 599,16 +590,16 @@ solution, $y_t$.


\begin{frame}


\frametitle{Quantitative evaluation}


\begin{itemize}


\item What is the ability of the stochastic extended path approach to


capture the effect of future volatility?


\item What part of the difference between the risky steady state and


deterministic steady state is captured by different values of $k$?


\item Deterministic steady state: 12.3035


\item Risky steady state: 12.4812


\item What is the ability of the stochastic extended path approach to capture


the effect of future volatility? What part of the difference between the risky


steady state and deterministic steady state is captured by different values of


$k$?\newline


\item Deterministic steady state: 12.3035\newline


\item Risky steady state: 12.4812\newline


\item The contribution of $k$ future periods


\begin{center}


\begin{tabular}{cc}


$k$ & Percentage\\


$k$ & Percentage\\


1 & 7.4\%\\


2 & 14.3\%\\


9 & 50.0\%\\



@ 616,8 +607,8 @@ capture the effect of future volatility?


60 & 99.0\%


\end{tabular}


\end{center}


\item In such a model, it is extremely costly to give full account of


the effects of future volatility with the stochastic extended path approach.


%\item In such a model, it is extremely costly to give full account of


% the effects of future volatility with the stochastic extended path approach.


\end{itemize}


\end{frame}





@ 628,11 +619,11 @@ capture the effect of future volatility?




\item A very large number of periods forward (the order of stochastic


extended path) is necessary to obtain an accurate figure of the


effects of future volatility.


effects of future volatility.\newline




\item However, even a local approximation with a Taylor expansion of


low order provides better information on this effect of future


volatility.


volatility.\newline




\item This suggests to combine the two approaches.





@ 684,7 +675,6 @@ s_{t+H}^i &= \mathsf Q(s_{t+H1}^i,0)\\


\end{tabular}


\end{center}






\end{frame}







@ 782,14 +772,12 @@ where $\mu_t$ is the Lagrange multiplier associated with the constraint on inves


\begin{frame}


\frametitle{Conclusion and future work}


\begin{itemize}


\item The extended path approach takes into account effects of nonlinearities.


\item The stochastic extended path approach takes also partially into account nonlinear effects of future volatility.


\item Possible to use an hybrid approach, using the risky steady state as terminal condition.


\item The approach suffers from the curse of dimensionality but it can be mitigated by


\begin{itemize}


\item using monomial formulas for integration when there are several shocks


\item exploiting embarassingly parallel nature of the algorithm


\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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