SolvingMicroDSOPs-Slides.tex

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%_____________ Opening slide _______________________

\begin{document}

\title[SolvingMicroDSOPs]{\textbf{Structural Estimation of Dynamic Stochastic\\ Optimizing Models of Intertemporal Choice \\ \LARGE{For Dummies!}}}
\author[Carroll]{Christopher Carroll\inst{1}}

\institute{
  \inst{1} Johns Hopkins University and NBER\\   \texttt{ccarroll@jhu.edu} \and
    }
\date{June 2012 \\ {\tiny \url{http://www.econ2.jhu.edu/people/ccarroll/SolvingMicroDSOPs-Slides.pdf}}
}


\begin{frame}[plain]
  \titlepage
\end{frame}

\section{Introduction}

\begin{frame}
\begin{itemize}
\item Efficient Solution Methods for Canonical $C$ problem
\begin{itemize}
\item CRRA utility
\item Plausible (microeconomically calibrated) uncertainty
\item Life cycle or infinite horizon
\end{itemize}
\item How To Add a Second Choice Variable
\item Method of Simulated Moments Estimation of Parameters
\end{itemize}
\end{frame}

\section{The Problem}

\begin{frame}[label=MaxProb]
\frametitle{\large\textbf{The Basic Problem at Date $t$}}

\input{./Equations/MaxProb.tex}
\begin{equation}\begin{gathered}\begin{aligned}
{\yLev}_{t}  & = {\pLev}_{t}\tranShkEmp_{t}
\end{aligned}\end{gathered}\end{equation}
\input{./Equations/ExogVars.tex}
\end{frame}

\begin{frame}[label=vrecurse]
\frametitle{\large\textbf{Bellman Equation}}

\input{./Equations/vrecurse.tex}

\begin{equation*}\begin{gathered}\begin{aligned}
   \mLev  & - \text{ `market resources' (net worth plus current income)}
\\ \pLev  & - \text{ permanent labor income }
\end{aligned}\end{gathered}\end{equation*}

\end{frame}

\section{Tricks}
\subsection{Normalization}
\begin{frame}[label=Normalize]
\frametitle{\large\textbf{Trick: Normalize the Problem}}

\input{./Equations/vNormed.tex}

where nonbold variables are bold ones normalized by $\pLev$:
\begin{equation}\begin{gathered}\begin{aligned}
\mNrm_{t}  & = \mLvl_{t}/\pLvl_{t}
\end{aligned}\end{gathered}\end{equation}

Yields $\cFunc_{t}(m)$ from which we can obtain
\begin{equation}\begin{gathered}\begin{aligned}
  \cLvl_{t}(\mLvl_{t},\pLvl_{t})  & = \cFunc_{t}(\mLvl_{t}/\pLvl_{t})\pLvl_{t}
\end{aligned}\end{gathered}\end{equation}

\end{frame}

\begin{frame}[label=Normalize]
\frametitle{\large\textbf{When Doesn't Normalization Work?}}

\begin{itemize}
\item Non-CRRA utility
\item Non-Friedman (transitory/permanent) income process
\begin{itemize}
\item e.g., AR(1)
\item But micro evidence is consistent with Friedman
\end{itemize}
\end{itemize}

\end{frame}
\subsection{View Problem from End of Period}

\begin{frame}[label=Normalize]
\frametitle{Trick: View Everything from End of Period}

Define 
\input{./Equations/vEndPrddefn.tex}
so 
\begin{equation}\begin{gathered}\begin{aligned}
  \vFunc_{t}(\mNrm_{t})  & = \max_{\cNrm_{t}}~~ \util(\cNrm_{t}) + \vFunc_{t}(\mNrm_{t}-\cNrm_{t})
\end{aligned}\end{gathered}\end{equation}
with FOC
\input{./Equations/upEqbetaOp.tex}
and Envelope relation
\input{./Equations/Envelope.tex}

\end{frame}


\subsection{Discretization of Risks}
\begin{frame}[label=DiscretizeFig]
\frametitle{Trick: Discretize the Risks}

E.g.\ use an equiprobable 7-point distribution:\medskip\medskip

\includegraphics[width=4in]{./Figures/discreteApprox.pdf}

\end{frame}

\begin{frame}[label=DiscretizeEqn]
\frametitle{Trick: Discretize the Risks}

\begin{equation}\begin{gathered}\begin{aligned}
        \vFunc^{\prime}(a_{t})  & =  \Discount \Rfree \PermGroFac_{t+1}^{-\CRRA} \left(\frac{1}{n}\right) \sum_{i=1}^{n} \util^{\prime}\left(\cFunc_{t+1}(\RNrmByG_{t+1} a_{t} + \tranShkEmp_{i})\right)
\end{aligned}\end{gathered}\end{equation}

%\input{./Equations/vDiscrete.tex}

\pause 
So for any particular $\mNrm_{T-1}$ the corresponding $\cNrm_{T-1}$ can be found
using the FOC:
\input{./Equations/upEqbetaOp.tex}


\end{frame}
\subsection{Interpolate a Consumption Rule}
\begin{frame}
\frametitle{Trick: Interpolate a Consumption Rule}

\begin{enumerate}
\item Define a grid of points $\vctr{\mNrm}$ (indexed $\mNrm[i]$)
\item Use numerical rootfinder to solve 
$\util^{\prime}(\cNrm) = \vFunc^{\prime}_{t}(\mNrm[i]-\cNrm)$
\begin{itemize}
\item The $\cNrm$ that solves this becomes $\cNrm[i]$
\end{itemize}
\item Construct interpolating function $\grave{\cFunc}$ by linear interpolation
\begin{itemize}
\item `Connect-the-dots'
\end{itemize}
\end{enumerate}

\end{frame}






\begin{frame}[label=DiscretizeEqn]
\frametitle{Trick: Interpolate a Consumption Rule}

Example: $\vctr{\mNrm}_{T-1} = \{0.,1.,2.,3.,4.\}$ (solid is `correct' soln)

\includegraphics[width=4.0in]{./Figures/PlotcTm1Simple.pdf}

\end{frame}

\begin{frame}[label=vEndPrdSlow]
\frametitle{Problem: Numerical Rootfinding is {\it Slow}}

Numerical search for values of $\cNrm_{T-1}$ satisfying
$\util^{\prime}(\cNrm) = \vFunc^{\prime}_{t}(\mNrm[i]-\cNrm)$ at, say,
6 gridpoints of $\vctr{\mNrm}_{T-1}$ may require hundreds or even thousands of
evaluations of
\begin{equation}\begin{gathered}\begin{aligned}
        \vFunc^{\prime}_{T-1}(\overbrace{{m}_{T-1}-\cNrm_{T-1}}^{\aNrm_{T-1}})  & =   \Discount_{T} \PermGroFac_{T}^{1-\CRRA}\left(\frac{1}{n}\right)\sum_{i=1}^{n}   \left( \RNrmByG_{T} a_{T-1} + \tranShkEmp_{i}\right)^{-\CRRA} \notag
\end{aligned}\end{gathered}\end{equation}

\end{frame}




\begin{comment}

\begin{frame}[label=vApprox]
\frametitle{Solution: Approximate \$\vEndPrd\$?}

\pause Given $\{\vctr{\aNrm}_{T-1},\vctr{\EndPrd}_{T-1}\}$, an approximate function $\grave{\EndPrd}_{T-1}$ 
can be constructed by linear interpolation among the points:

\includegraphics[width=4in]{./Figures/PlotOTm1RawVsInt.pdf}

\end{frame}

\begin{frame}
\frametitle{Using $\grave{\EndPrd}_{T-1}$ In Optimization}
\begin{equation*}\begin{gathered}\begin{aligned}
  \vFunc_{t}(\mNrm_{t})  & = \max_{\cNrm_{t}}~~ \util(\cNrm_{t}) + \grave{\EndPrd}_{t}(\mNrm_{t}-\cNrm_{t})
\end{aligned}\end{gathered}\end{equation*}
is {\it much} faster,  but result is bad:
\begin{center}
\includegraphics[width=3in]{./Figures/PlotComparecTm1AB}
\end{center}

\end{frame}


\begin{frame}
\frametitle{Approximate $\vFunc^{\prime}(\aNrm)$?}

Better ... but still violates first two principles:
\includegraphics[width=4in]{./Figures/PlotOPRawVSFOC.pdf}

\end{frame}


\end{comment}

\subsection{The Method of Endogenous Gridpoints}
\begin{frame}
\frametitle{Solution: The Method of Endogenous Gridpoints}

\pause 

\begin{itemize}
\item Define vector of {\it end-of-period} asset values $\vctr{\aNrm}$
\item For each $\aNrm[j]$ compute $\vFunc^{\prime}(\aNrm[j])$
\end{itemize}

\pause 

Each of these $\vFunc^{\prime}[j]$ corresponds to a unique
$\cNrm[j]$ via FOC:
\begin{equation}\begin{gathered}\begin{aligned}
  \cNrm[j]^{-\CRRA}  & = \vFunc^{\prime}(\aNrm[j])
\\ \cNrm[j]  & = \left(\vFunc^{\prime}(\aNrm[j])\right)^{-1/\CRRA}
\end{aligned}\end{gathered}\end{equation}

\pause 

But the DBC says
\begin{equation}\begin{gathered}\begin{aligned}
  \aNrm_{t}  & = \mNrm_{t} - \cNrm_{t}
\\ \mNrm[j]  & = \aNrm[j]+\cNrm[j]
\end{aligned}\end{gathered}\end{equation}

\pause 
So computing $\vFunc^{\prime}$ at a vector of $\vctr{\aNrm}$ values has produced for us the corresponding $\vctr{\cNrm}$ and $\vctr{\mNrm}$ 
values at virtually no cost!  

\pause 
\medskip 
From these we can interpolate as before to construct $\grave{\cFunc}_{t}(\mRat)$.

\end{frame}


\begin{frame}
\frametitle{Why Directly Approximating $\vFunc_{t}$ is a Bad Idea}

Principles of Approximation

\begin{itemize}
\item Hard to approximate things that approach $\infty$ for relevant $\mNrm$
\begin{itemize}
\item Not a prob for Rep Agent models: `relevant' $\mNrm$'s are $\approx$ SS
\end{itemize}
\item Hard to approximate things that are highly nonlinear 
%\item Best to approximate things that directly govern behavior
\end{itemize}


\end{frame}


\begin{frame}
\frametitle{Approximate Something That Would Be Linear in PF Case}

\medskip

Perfect Foresight Theory:
\begin{equation}\begin{gathered}\begin{aligned}
  \cFunc_{t}(\mRat)  & = (\mRat+\hEnd_{t})\MPCmin_{t} 
\end{aligned}\end{gathered}\end{equation}
for market resources $\mNrm$ and end-of-period human wealth $\hEnd$.


\medskip\medskip
\pause 

This is why it's a good idea to approximate $\cFunc_{t}$ 

\pause \medskip\medskip

Bonus: Easy to debug programs by setting $\sigma^{2} = 0$ and
testing whether numerical solution matches analytical!

\end{frame}

\subsection{Approximate Inverted Functions}
\begin{frame}%[ValFnApprox]
\frametitle{But What if You {\it Need} the Value Function?}

Perfect foresight value function:
\input{./Equations/vFuncPF.tex}
which is linear.  

\medskip\medskip
\pause If you need the value function, approximate the {\it inverted} value function to generate $\grave{\vInv}_{t}$ 
and then obtain your approximation from 
\begin{equation}\begin{gathered}\begin{aligned}
  \grave{\vFunc}_{t}  & = \util(\grave{\vInv}_{t})
\end{aligned}\end{gathered}\end{equation}


\end{frame}

\subsection{Derivatives}
\begin{frame}
\frametitle{Approximate Slope Too}

\cite{BufferStockTheory} shows that $\cFunc^{\mNrm}_{t}$ exists everywhere.
\medskip

\pause 
Define {\it consumed} function and its derivative as 
\begin{equation}\begin{gathered}\begin{aligned}
  \cEndFunc_{t}(\aNrm)  & = (\vFunc^{\prime}_{t}(\aNrm))^{-1/\CRRA}
\\ \cEndFunc_{t}^{\aNrm}(\aNrm)  & = -(1/\CRRA)\left(\vFunc^{\prime}(a)\right)^{-1-1/\CRRA} \vFunc^{\prime\prime}(\aNrm) 
\end{aligned}\end{gathered}\end{equation}

\pause 
and using chain rule it is easy to show that
\begin{equation}\begin{gathered}\begin{aligned}
 \cFunc^{\mNrm}_{t}  & = \cEndFunc^{\aNrm}_{t}/(1+\cEndFunc^{\aNrm}_{t})
\end{aligned}\end{gathered}\end{equation}

\end{frame}

\begin{frame}
\frametitle{To Implement: Modify Prior Procedures in Two Ways}
\begin{enumerate}
\item Construct $\vctr{\cFunc}^{\mNrm}_{t}$ along with $\vctr{\cFunc}_{t}$ in EGM algorithm
\item Approximate $\cFunc_{t}(m)$ using piecewise Hermite polynomial
\begin{itemize}
\item Exact match to both level and derivative at set of points
\end{itemize}
\end{enumerate}
\end{frame}


\subsection{Improving the $\aNrm$ Grid}

\begin{frame}
\frametitle{Problem: $\Alt{\cFunc}$ Below Bottom $\mNrm$ Gridpoint and Extrapolation}

Consider what happens as $a_{T-1}$ approaches $\Min{a}_{T-1}\equiv-\Min{\tranShkEmp}\RNrmByG_{T}^{-1}$,
\begin{equation*}\begin{gathered}\begin{aligned}
        \lim_{{\aNrm} \downarrow \Min{a}_{T-1}} \vFunc{T-1}^{\prime}(\aNrm) 
& =         \lim_{{\aNrm} \downarrow \Min{a}_{T-1}} \Discount \Rfree \PermGroFac_{T}^{-\CRRA} \left(\frac{1}{n}\right) \sum_{i=1}^{n} \left(  \aNrm\RNrmByG_{T}+ \tranShkEmp_{i}\right)^{-\CRRA}
\\  & = \infty
\end{aligned}\end{gathered}\end{equation*}

This means our lowest value in $\vctr{\aNrm}_{T-1}$ should be $> \Min{\aNrm}_{T-1}$.  

\medskip
Suppose we construct $\Alt{\cFunc}$ by linear interpolation:
\begin{equation*}\begin{gathered}\begin{aligned}
  \Alt{\cFunc}_{T-1}(\mRat)  & = \Alt{\cFunc}_{T-1}(\vctr{\mNrm}_{T-1}[1])+\Alt{\cFunc}_{T-1}^{\prime}(\vctr{\mNrm}_{T-1}[1])(\mNrm-\vctr{\mNrm}_{T-1}[1]) \label{eq:ExtrapLin}
\end{aligned}\end{gathered}\end{equation*}

True $\cFunc$ is strictly concave 
$\Rightarrow \exists \mRat^{-} > \Min{\mNrm}_{T-1}$  for which $\mRat^{-}-\Alt{\cFunc}_{T-1}(\mRat^{-}) < \Min{\aNrm}_{T-1}$

\end{frame}

\begin{frame}
\frametitle{Solution: Hard-Code the Bottom Point}

Theory says that
\begin{equation}\begin{gathered}\begin{aligned}
  \lim_{\mRat \downarrow \Min{\mNrm}_{T-1}} \cFunc_{T-1}(\mRat)  & = 0
\\ \lim_{\mRat \downarrow \Min{\mNrm}_{T-1}} \cFunc_{T-1}^{\mNrm}(\mRat)  & = \MPCmax_{T-1}
\end{aligned}\end{gathered}\end{equation}

\medskip 

\begin{enumerate}
\item Redefine $\vctr{\aNrm}$ {\it relative} to $\Min{\aNrm}_{T-1}$
\item Construct corresponding $\vctr{\mNrm}_{T-1}$ and $\vctr{\cNrm}_{T-1}$
\item Prepend $\Min{\mNrm}_{T-1}$ to $\vctr{\mNrm}_{T-1}$
\item Prepend $0.$ to $\vctr{\cNrm}_{T-1}$
\item Prepend $\MPCmax_{T-1}$ to $\vctr{\MPC}_{T-1}$
\end{enumerate}
then proceed as before.

\end{frame}

\begin{frame}
\frametitle{Trick: Improving the $\aNrm$ Grid}
Grid Spacing: Uniform

\includegraphics[width=4in]{./Figures/GothVInvVSGothC.pdf}

\end{frame}


\begin{frame}
\frametitle{Trick: Improving the $\aNrm$ Grid}
Grid Spacing: Same $\{\Min{\aNrm},\bar{\aNrm}\}$ But Triple Exponential $e^{e^{e^{...}}}$ Growth

\includegraphics[width=4in]{./Figures/GothVInvVSGothCEEE.pdf}

\end{frame}


\subsection{The Method of Moderation}

\begin{frame}[label=MoM]
\frametitle{The Method of Moderation}

\begin{itemize}
\item Further improves speed and accuracy of solution
\item See my talk at the conference!
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Imposing `Artificial' Borrowing Constraints}
\input{./Equations/ConstrArt.tex}

\pause 

Define $\grave{\cFunc}^{*}_{t}$ as soln to unconstrained problem.  Then
\input{./Equations/LiqCons.tex}

\end{frame}

\begin{frame}
\frametitle{Imposing `Artificial' Borrowing Constraints}

Point where constraint makes transition from binding to not is
\begin{equation*}\begin{gathered}\begin{aligned}
    \util^{\prime}(\mNrm_{T-1}^{\#})  & = \vFunc^{\prime}_{T-1}(0.)
\\  \mNrm_{T-1}^{\#}  & = \left(\vFunc^{\prime}_{T-1}(0.)\right)^{-1/\CRRA}
\end{aligned}\end{gathered}\end{equation*}
\pause\medskip

Procedure is very easy:
\begin{itemize}
\item Add $0.$ as first point in $\vctr{\aNrm}$
\item $\Rightarrow \vctr{\mNrm}[1] = \mNrm_{T-1}^{\#}$
\item Above $\mNrm_{T-1}^{\#}$, $\grave{\cFunc}_{T-1}(\mRat)$ obtained as before
\item Below $\mNrm_{T-1}^{\#}$, $\grave{\cFunc}_{T-1}(\mRat)=\mNrm$
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{Imposing `Artificial' Borrowing Constraints}
\begin{figure}
\includegraphics[width=4in]{./Figures/cVScCon.pdf}
        \caption{Constrained (solid) and Unconstrained (dashed) Consumption}
        \label{fig:cVScCon}
\end{figure}

\end{frame}

\begin{frame}%[Recursion]
\frametitle{Recursion: Period $t$ Solution Given Period $t+1$}
\begin{enumerate}
\item Construct 
\input{./Equations/vEndeq.tex}
\item Call the result $\vctr{\cNrm}_{t}$ and generate the corresponding $\vctr{\mNrm}_{t}=\vctr{\cNrm}_{t}+\vctr{\aNrm}_{t}$
\item Interpolate to create $\grave{\cNrm}_{t}(\mRat)$
\end{enumerate}

\end{frame}

\begin{frame}%[Convergence]
\frametitle{Consumption Rules $\grave{\cFunc}_{T-n}$ Converge}

\begin{figure}
        \includegraphics[width=4in]{./Figures/PlotCFuncsConverge.pdf}
        \caption{Converging $\grave{\cFunc}_{T-n}(\mNrm)$ Functions for $n=\{1,5,10,15,20\}$}
        \label{fig:PlotCFuncsConverge}
\end{figure}

\end{frame}


\section{Multiple Control Variables}
\begin{frame}
\frametitle{Portfolio Choice}

Now the consumer has a choice between a risky and a safe asset.  \pause The portfolio
return is
\input{./Equations/Rport.tex}
\pause so (setting $\PermGroFac=1$) the maximization problem is \pause 
\input{./Equations/PortProb.tex}

\end{frame}

\begin{frame}
\frametitle{Portfolio Choice}

The FOC with respect to $\cNrm_{t}$ now yields an Euler equation
\input{./Equations/EulercRiskyR.tex}
\pause
while the FOC with respect to the portfolio share yields
%\input{./Equations/FOCw.tex}

\end{frame}

\section{The Infinite Horizon}
\subsection{Convergence}
\begin{frame}
\frametitle{Convergence}

When the problem satisfies certain conditions~(\cite{BufferStockTheory}),
it defines a `converged' consumption rule with a `target' ratio $\check{\mNrm}$
that satisfies:
\begin{equation}\begin{gathered}\begin{aligned}
  \Ex_{t}[\mNrm_{t+1}/\mNrm_{t}]  & = 1 \text{~if $\mNrm_{t} = \check{\mNrm}$}
\end{aligned}\end{gathered}\end{equation}

\pause 

Define the target $\mNrm$ implied by the consumption rule $\cFunc_{t}$ as $\check{\mNrm}_{t}$.

\medskip\pause
Then a plausible metric for convergence is to define some value $\epsilon$ and to declare
the solution to have converged when
\begin{equation}\begin{gathered}\begin{aligned}
  |\check{\mNrm}_{t+1}-\check{\mNrm}_{t}|  & < \epsilon
\end{aligned}\end{gathered}\end{equation}

\end{frame}

\subsection{Tricks}
\begin{frame}
\frametitle{Trick: Coarse then Fine $\tranShkEmp$}

\begin{enumerate}
\item Start with coarse grid for $\tranShkEmp$ (say, 3 points)
\item Solve to convergence; call period of convergence $n$
\item Construct finer grid for $\tranShkEmp$ (say, 7 points)
\item Solve for period $T-n-1$ assuming $\Alt{\cFunc}_{T-n}$ 
\item Continue to convergence
\end{enumerate}

\end{frame}

\begin{frame}
\frametitle{Trick: Coarse then Fine $\vctr{\aNrm}_{T-1}$}

\begin{enumerate}
\item Start with coarse grid for $\vctr{\aNrm}$ (say, 5 gridpoints)
\item Solve to convergence; call period of convergence $n$
\item Construct finer grid for $\vctr{\aNrm}$ (say, 20 points)
\item Solve for period $T-n-1$ assuming $\Alt{\cFunc}_{T-n}$ 
\item Continue to convergence
\end{enumerate}

\end{frame}

\section{Structural Estimation}
\subsection{Life Cycle Model}

\begin{frame}
\frametitle{Life Cycle Maximization Problem}
\input{./Equations/LifeCycleMaxNormed.tex}
\input{./Equations/subjectTo.tex}
\end{frame}

\begin{frame}
\frametitle{Details follow~\cite{cagettiWprofiles}}
\begin{itemize}
\item Parameterization of Uncertainty
\item Probability of Death
\item Demographic Adjustments to $\Discount$
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Empirical Wealth Profiles}
\begin{figure}
    \includegraphics[width=3.5in]{./Figures/PlotMeanMedianSCFcollegeGrads.pdf}
    \caption{$\mNrm$ from SCF (means (dashed) and medians (solid))}
    \label{fig:MeanMedianSCF}
\end{figure}
\end{frame}

\begin{frame}
\frametitle{Simulated Moments}

Given a set of parameter values $\{\CRRA,\beth\}$:
\begin{itemize}
\item Start at age 25 with empirical $\mNrm$ data
\item Draw shocks using calibrated $\sigma^{2}_{\permShk}$,$\sigma^{2}_{\tranShkEmp}$
\item Consume according to solved $\cFunc_{t}$
\end{itemize}
\pause 
$\Rightarrow \mNrm$ distribution by age
\end{frame}

\begin{frame}
\frametitle{Choose What to Simulate}
\input{./Equations/GapEmpiricalSimulatedMedians.tex}
\end{frame}

\begin{frame}
\frametitle{Calculate Match Between Theory and Data}
\begin{equation}\begin{gathered}\begin{aligned}
\xi  & = \{\CRRA,\beth\}
\end{aligned}\end{gathered}\end{equation}
solve
\input{./Equations/StructEstim.tex}

\end{frame}
\begin{frame}
\frametitle{Bootstrap Standard Errors (\cite{horowitzBootstrap})}

Yields estimates of 
\input{./Tables/EstResults.tex}

\end{frame}

\begin{frame}
\frametitle{Contour Plot}
\begin{figure}
     \includegraphics[width=2.5in]{./Figures/PlotContourMedianStrEst.pdf}
    \caption{Point Estimate and Height of Minimized Function}
    \label{fig:PlotContourMedianStrEst}
\end{figure}

\end{frame}

\beamerdefaultoverlayspecification{<*>}

\begin{frame}[allowframebreaks]
\frametitle{\textbf{References}}
\tiny
\input handoutBibMake
\end{frame}


\end{document}