Merge pull request #7 from jabbring/dev

Updated CE5

dynamicDiscreteChoice.m

@@ -6,7 +6,7 @@
 		90153, 5000 LE Tilburg, The Netherlands. E-mail: \url{mailto:jaap@abbring.org}{jaap@abbring.org}. Web: \url{http://jaap.abbring.org/}{jaap.abbring.org}.}}
 			\author{Tobias J. Klein\thanks{Department of Econometrics \& OR, Tilburg University, P.O. Box
 		90153, 5000 LE Tilburg, The Netherlands. E-mail: \url{mailto:t.j.klein@uvt.nl}{t.j.klein@uvt.nl}. Web: \url{http://www.tobiasklein.ws/}{www.tobiasklein.ws}.}}
-			\date{October 2021}
+			\date{December 2021}
 	\begin{abstract}
 			This document supports the first computing sessions in a graduate course on dynamic discrete choice models. It is centered around basic \textsc{Matlab} code for solving, simulating, and empirically analyzing a simple model of firm entry and exit under uncertainty. This code is available from a public Github repository and can be distributed to students as starter code, for example using the Github Classroom. Exercises ask students to adapt and extend this starter code to apply different and more advanced computational and econometric methods to a wider range of models.
 \end{abstract}
@@ -190,8 +190,9 @@ of the parameters of interest (up to numerical precision)? How is the
 		\begin{itemize} 
 		\item Add a new function for nonparametrically estimating the choice probabilities $p(a|X_t,A_{t-1})\equiv\Pr(A_t=a|X_t,A_{t-1})$ over the support of $(X_t,A_{t-1})$. With enough data and a finite state space, you can simply use the conditional sample frequency. 
 		\item Add (to this same function or in a new function) a procedure for estimating $\Delta U$ by inverting the estimated choice probabilities (as proposed by \cite{res93:hotzmiller} and \cite{res94:hotzetal}). 
-		\item Extend |negLogLik| (or add functions) so that it optionally takes nonparametric estimates of $\Delta U$ as inputs, computes the corresponding estimates of the entry and exit rules, and uses these estimates and the model to forwardly simulate $U_0(x,a)$ and $U_1(x,a)$, and then $\Delta U(x,a)$, for each point $(x,a)$ in the sample. As possible objectives to be minimized, both implement a weighted distance between the nonparametric estimates of $\Delta U$ and the simulated values of $\Delta U$, and minus a log pseudo-likelihood based on the choice probabilities implied by the simulated values of $\Delta U$.
-			\item Finally, extend the script in |dynamicDiscreteChoice.m| so that it successively calls these functions to implement a two step procedure for estimating the model, as in \cite{res93:hotzmiller} and \cite{res94:hotzetal}. Analyze the numerical and statistical performance of this procedure with both of the possible objective functions implemented and compare. Discuss the theoretical relation between both approaches; see \cite{res08:pesendorferschmidtdengler}.
+		\item Add a new function for simulating a given number of state (including $\varepsilon_t$) and choice paths of some given length from each state observed in the data and for each first choice that can be made at the beginning of the path. Make use of the linearity of the flow utility in the parameters to reduce the dimensionality of the objects returned by this function and needed for constructing the objective in the next step.
+		\item Adapt |negLogLik| into an objective function that takes nonparametric estimates of $p$ or $\Delta U$ and (a relevant summary of) simulated states and choices as inputs. As possible objectives to be minimized, implement both a weighted distance between the nonparametric estimates of $\Delta U$ and simulated values of $\Delta U$, and minus a log pseudo-likelihood based on the choice probabilities implied by the simulated values of $\Delta U$. When doing so exploit linearity of the flow utilities in the parameters.
+		\item Finally, extend the script in |dynamicDiscreteChoice.m| so that it successively calls these functions to implement a two step procedure for estimating the model, as in \cite{res93:hotzmiller} and \cite{res94:hotzetal}. Analyze the numerical and statistical performance of this procedure with both of the possible objective functions implemented and compare. Discuss the theoretical relation between both approaches; see \cite{res08:pesendorferschmidtdengler}.
 		\end{itemize}
 	See the course slides for a brief and general description of this procedure and \cite{nh94:rust} for a detailed algorithm, with discussion. Note that the algorithm described in \cite{nh94:rust} is similar to that of \cite{ecta07:bajarietal} for games that we will discuss later in the course. Both build on the ideas of \cite{res94:hotzetal}, who use the special logit structure to simplify the simulation (see the discussion in \cite{ecta07:bajarietal}).