Added week 14 material

.github/workflows/dynare-6.2-matlab-r2024b-macos.yml

@@ -152,4 +152,12 @@ jobs:
             rbcSSTest;
             cd("../dynare");
             dynare rbcLogutil
-            dynare rbcCES
+            dynare rbcCES
+
+      - name: Run week 14 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("Dynare-6.2-arm64/matlab");
+            cd("progs/matlab");
+            dsge_maximum_likelihood_illustration;

.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml

@@ -181,4 +181,12 @@ jobs:
             rbcSSTest;
             cd("../dynare");
             dynare rbcLogutil
-            dynare rbcCES
+            dynare rbcCES
+
+      - name: Run week 14 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("dynare/matlab");
+            cd("progs/matlab");
+            dsge_maximum_likelihood_illustration;

.github/workflows/dynare-6.2-matlab-r2024b-windows.yml

@@ -143,4 +143,12 @@ jobs:
             rbcSSTest;
             cd("../dynare");
             dynare rbcLogutil
-            dynare rbcCES
+            dynare rbcCES
+
+      - name: Run week 14 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("D:\hostedtoolcache\windows\dynare-6.0\matlab");
+            cd("progs/matlab");
+            dsge_maximum_likelihood_illustration;

README.md

@@ -287,6 +287,7 @@ Familiarize yourself with
 ### Goals
 
 * understand the Kalman filter and its use in DSGE models
+* understand Maximum Likelihood estimation of DSGE models
 * understand the Metropolis-Hastings algorithm and its use in DSGE models
 
 

exercises/kalman_filter.tex

@@ -1,35 +1,97 @@
 \section[The Kalman filter for DSGE models]{The Kalman filter for DSGE models\label{ex:KalmanFilterDSGE}}
-Consider a DSGE model approximated with first-order perturbation techniques such that the endogenous variables $\tilde{y}_t$ are determined linearly by previous endogenous variables $\tilde{y}_{t-1}$ and current shocks $u_t$:
-$$(\tilde{y}_t - \bar{y}) = \bar{y} + g_x (\tilde{y}_{t-1} - \bar{y}) + g_u u_t$$
-where $\bar{y}$ denotes the steady-state and the shocks are Gaussian: $u_t \sim N(0,\Sigma_u)$. Assume that a subset of variables $d_t$ from $y_t$ can be observed with a measurement error $e_t$ which is orthogonal to $u_t$ and also normally distributed with mean zero and covariance matrix $\Sigma_e$:
-$$d_t = H \tilde{y}_t + e_t$$
-In the following exercises, we'll make use of the following notation: (1) a time subscript, e.g. $d_t$, denotes a particular observation, (2) a time superscript, e.g. $d^t=\{d_t,...,d_1\}$, denotes the complete history of observations up to a certain point in time $t$, (3) $\hat{y}_{t|t-1} \equiv E(y_t|d^{t-1})$ is the conditional expectation on information provided by $d^{t-1}$ and $\Sigma_{t|t-1} \equiv E\left[ (y_t-\hat{y}_{t|t-1}) (y_t-\hat{y}_{t|t-1})'|d^{t-1}\right]$ is the corresponding mean-squared-error/covariance matrix conditional on $d^{t-1}$.
+Consider a DSGE model approximated with first-order perturbation techniques such that the endogenous variables \(\tilde{y}_t\) are determined linearly by previous endogenous variables \(\tilde{y}_{t-1}\) and current shocks \(u_t\):
+\begin{equation*}
+(\tilde{y}_t - \bar{y}) = g_x (\tilde{y}_{t-1} - \bar{y}) + g_u u_t
+\end{equation*}
+where \(\bar{y}\) denotes the steady-state and the shocks are Gaussian: \(u_t \sim N(0,\Sigma_u)\).
+Assume that a subset of variables \(d_t\) from \(y_t\) can be observed with a measurement error \(e_t\),
+  which is orthogonal to \(u_t\) and also normally distributed with mean zero and covariance matrix \(\Sigma_e\):
+\begin{equation*}
+d_t = H \tilde{y}_t + e_t
+\end{equation*}
+In the following exercises, we'll make use of the following notation:
+(1) a time subscript such as \(d_t\) denotes a particular observation,
+(2) a time superscript such as \(d^t=\{d_t,\ldots ,d_1\} \), denotes the complete history of observations up to a certain point in time \(t\),
+(3) \(\hat{y}_{t|t-1} \equiv E(y_t|d^{t-1})\) is the conditional expectation on information provided by \(d^{t-1}\)
+  and \(\Sigma_{t|t-1} \equiv E\left[ (y_t-\hat{y}_{t|t-1}) (y_t-\hat{y}_{t|t-1})'|d^{t-1}\right]\) is the corresponding mean-squared-error/covariance matrix conditional on \(d^{t-1}\).
 
 \begin{enumerate}
-	\item Denote $y_t = \tilde{y}_t - \bar{y}$ as model variables in deviation from steady-state. Show how to put the model solution into a so-called linear Gaussian state-space system. What are the controls, states, innovations and noise variables? What is the probability distribution of $d_t$? Why isn't it possible to directly construct it? What do we need?
-	\item What is the basic problem the Kalman filter solves? What is the goal?
-	\item Derive the recursive algorithm in the following steps:
-	\begin{enumerate}
-		\item Assume that the initial value $y_0$ is normally distributed: $y_0 \sim N(\hat{y}_{0|-1},\Sigma_{0|-1})$ with \textbf{known} values for $\hat{y}_{0|-1}$ and $\Sigma_{0|-1}$. What does this imply for the distribution of $d_0$? Moreover, comment on the values one typically uses for $\hat{y}_{0|-1}$ and $\Sigma_{0|-1}$ when the underlying state-space model is stationary.
-		\item Based on $d_0$ and $y_0$, compute the state forecast $\hat{y}_{1|0}$ using the idea of a population regression coefficient. That is, use a regression of the unknown difference between the true state $y_1$ and our forecast of it made yesterday $\hat{y}_{1|0}$ on the new information contained in the observation $d_1$ compared to what we had predicted it to be, i.e. $(d_1 - \hat{d}_{1|0})$. In other words, derive the following formula:
-		$$\hat{y}_{1|0} \equiv E[y_1|d^0] = g_x \hat{y}_{0|-1} + K_0\left(d_0-H\hat{y}_{0|-1}\right)$$
-	    where the matrix $K_0 = g_x \Sigma_{0|-1} H'(H\Sigma_{0|-1}H' + \Sigma_e)^{-1}$ is the so-called Kalman gain. Interpret this equation.
-		\item Derive the full distribution of $y_1|d^0$.
-		\item Generalize the previous steps and outline the recursive nature of the Kalman filter. That is, show that the conditional distribution of $y_{t+1}$ given information from $t$ is given by:
-		$$y_{t+1}|d^t \sim N(\hat{y}_{t+1|t},\Sigma_{t+1|t})$$
-		Provide the closed-form expressions for the state forecast $\hat{y}_{t+1|t}$ and corresponding mean-squared-error/covariance matrix $\Sigma_{t+1|t}$.
-		\item Summarize the recursive algorithm and key objects to compute when doing the Kalman filter recursions.
-	\end{enumerate}
-	\item Derive the log-likelihood function $p(d_T,...,d_1)$ by factorizing it. Show that the Kalman filter delivers everything needed to compute the log-likelihood as a \emph{by-product}.
-	\item What does the concept of \emph{stochastic singularity} mean in the context of computing the Gaussian likelihood function using the Kalman filter?
-	\item In which sense is the Kalman filter \emph{optimal} for linear Gaussian state-space systems?
-	%\item Write a function called \texttt{dsge\_kalman\_filter.m} that computes the log-likelihood for any DSGE model given a data matrix $d^T$,values for the steady-state of observables $\bar{d}$, values for the state-space matrices $H$, $g_x$, $g_u$, and values for the covariance matrices $\Sigma_u$ and $\Sigma_e$. The output is the $T$-dimensional vector of the conditional contributions $\log p(d_t|d^{t-1})$ to the Gaussian log-likelihood function, where the first entry is based on the prediction of $y_t$ at its unconditional mean and unconditional covariance matrix.
-	%\item Write a function called \texttt{dsge\_loglikelihood.m} that computes the log-likelihood for any DSGE model by first solving the model with first-order perturbation techniques and then summing up the conditional log-likelihood contributions $\log p(d_t|d^{t-1})$ computed by above function \texttt{dsge\_kalman\_filter.m}. Note that the inputs are data matrix $d^T$ and numerical values for the parameters (both model and covariance matrix parameters).
-	%\item Compute the log-likelihood function of the Basic New Keynesian model given a feasible calibration for the model and covariance parameters. Use the simulated data provided in \texttt{BasicNewKeynesianSimdat.mat}.
+
+\item Denote \(y_t = \tilde{y}_t - \bar{y}\) as model variables in deviation from steady-state.
+Show how to put the model solution into a so-called linear Gaussian state-space system.
+What are the controls, states, innovations and noise variables?
+What is the probability distribution of \(d_t\)?
+Why isn't it possible to directly construct it? What do we need?
+
+\item What is the basic problem the Kalman filter solves? What is the goal?
+
+\item Derive the recursive algorithm in the following steps:
+
+\begin{enumerate}
+
+\item Assume that the initial value \(y_0\) is normally distributed: \(y_0 \sim N(\hat{y}_{0|-1},\Sigma_{0|-1})\) with \textbf{known} values for \(\hat{y}_{0|-1}\) and \(\Sigma_{0|-1}\).
+What does this imply for the distribution of \(d_0\)?
+Moreover, comment on the values one typically uses for \(\hat{y}_{0|-1}\) and \(\Sigma_{0|-1}\) when the underlying state-space model is stationary.
+
+\item Based on \(d_0\) and \(y_0\), compute the state forecast \(\hat{y}_{1|0}\) using the idea of a population regression coefficient.
+That is, use a regression of the unknown difference between the true state \(y_1\)
+  and our forecast of it made yesterday \(\hat{y}_{1|0}\) on the new information contained in the observation \(d_1\) compared to what we had predicted it to be, i.e.\
+  \((d_1 - \hat{d}_{1|0})\).
+In other words, derive the following formula:
+\begin{equation*}
+\hat{y}_{1|0} \equiv E[y_1|d^0] = g_x \hat{y}_{0|-1} + K_0\left(d_0-H\hat{y}_{0|-1}\right)
+\end{equation*}
+where the matrix \(K_0 = g_x \Sigma_{0|-1} H'{\left(H\Sigma_{0|-1}H' + \Sigma_e\right)}^{-1}\) is the so-called Kalman gain.
+Interpret this equation.
+
+\item Derive the full distribution of \(y_1|d^0\).
+
+\item Generalize the previous steps and outline the recursive nature of the Kalman filter.
+That is, show that the conditional distribution of \(y_{t+1}\) given information from \(t\) is given by:
+\begin{equation*}
+y_{t+1}|d^t \sim N(\hat{y}_{t+1|t},\Sigma_{t+1|t})
+\end{equation*}
+Provide the closed-form expressions for the state forecast \(\hat{y}_{t+1|t}\) and corresponding mean-squared-error/covariance matrix \(\Sigma_{t+1|t}\).
+
+\item Summarize the recursive algorithm and key objects to compute when doing the Kalman filter recursions.
+
+\end{enumerate}
+
+\item Derive the log-likelihood function \(p(d_T,\ldots ,d_1)\) by factorizing it.
+Show that the Kalman filter delivers everything needed to compute the log-likelihood as a \emph{by-product}.
+
+\item What does the concept of \emph{stochastic singularity} mean in the context of computing the Gaussian likelihood function using the Kalman filter?
+
+\item In which sense is the Kalman filter \emph{optimal} for linear Gaussian state-space systems?
+
+\item Write a function called \texttt{dsge\_kalman\_filter.m}
+  that computes the log-likelihood for any DSGE model given
+	a data matrix \(d^T\),
+	values for the steady-state of observables \(\bar{d}\),
+	values for the state-space matrices \(H\), \(g_x\), \(g_u\),
+	and values for the covariance matrices \(\Sigma_u\) and \(\Sigma_e\).
+The output is the \(T\)-dimensional vector of the conditional contributions \(\log p(d_t|d^{t-1})\) to the Gaussian log-likelihood function,
+  where the first entry is based on the prediction of \(y_t\) at its unconditional mean and unconditional covariance matrix.
+
+\item Write a function \texttt{dsge\_loglikelihood.m} that computes the log-likelihood for any DSGE model
+  which is preprocessed and solved with Dynare.
+The function should:
+\begin{itemize}
+\item solve the model with first-order perturbation techniques%using a stripped-down version of Dynare's perturbation solver given in \texttt{dsge\_perturbation.m}
+\item then compute the conditional log-likelihood contributions \(\log p(d_t|d^{t-1})\), which are computed by \texttt{dsge\_kalman\_filter.m}.
+\end{itemize}
+Note that the inputs are data matrix \(d^T\) and numerical values for the parameters (both model and covariance matrix parameters).
+
+\item Compute the log-likelihood function of either the RBC model or the New Keynesian model given a feasible calibration for the model and covariance parameters
+  using simulated data.
+
+\item Estimate the parameters of the RBC model by Maximum Likelihood using the function \texttt{dsge\_loglikelihood.m}.
+Compare the results with Dynare's results.
+
 \end{enumerate}
 
 \begin{solution}\textbf{Solution to \nameref{ex:KalmanFilterDSGE}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/kalman_filter_solution.tex}
 \fi
 \newpage