Added week 4 material

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

@@ -64,3 +64,13 @@ jobs:
             lawOfLargeNumbers;
             lawOfLargeNumbersAR1;
             centralLimitDependentData;
+
+      - name: Run week 4 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("Dynare-6.2-arm64/matlab");
+            cd("progs/matlab");
+            AR4OLS;
+            AR4ML;
+            AR1MLLaplace;

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

@@ -93,3 +93,13 @@ jobs:
             lawOfLargeNumbers;
             lawOfLargeNumbersAR1;
             centralLimitDependentData;
+
+      - name: Run week 4 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("dynare/matlab");
+            cd("progs/matlab");
+            AR4OLS;
+            AR4ML;
+            AR1MLLaplace;

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

@@ -55,3 +55,13 @@ jobs:
             lawOfLargeNumbers;
             lawOfLargeNumbersAR1;
             centralLimitDependentData;
+
+      - name: Run week 4 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("D:\hostedtoolcache\windows\dynare-6.0\matlab");
+            cd("progs/matlab");
+            AR4OLS;
+            AR4ML;
+            AR1MLLaplace;

README.md

@@ -87,7 +87,6 @@ Please feel free to use this for teaching or learning purposes; however, taking
 
 </details>
 
-<!---
 
 <details>
   <summary>Week 4: Ordinary Least Squares (OLS) and Maximum Likelihood (ML) estimation of the autoregressive process</summary>
@@ -101,13 +100,17 @@ Please feel free to use this for teaching or learning purposes; however, taking
 
 * [x] review the solutions of [last week's exercises](https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_3.pdf) and write down all your questions
 * [x] read Lütkepohl (2004); make note of all the aspects and concepts that you are not familiar with or that you find difficult to understand
-* [x] TRY (!!!) to do exercise sheet 4; particularly, create your own ARpOLS.m and ARpML.m functions
+* [x] do exercise 1; particularly, create your own ARpOLS.m function; feel free to sent it to me via Mattermost for review
+* [x] we will do exercises 2 and 3 in class
 * [x] participate in the Q&A sessions with all your questions and concerns
 * [x] for immediate help: [schedule a meeting](https://schedule.mutschler.eu)
 * [x] (optionally) checkout the short [Advanced Git Video Tutorials from GitKraken](https://www.gitkraken.com/learn/git/tutorials#advanced)
 
 </details>
 
+
+<!---
+
 <details>
   <summary>Week 5: Information Criteria, Specification Tests, and Bootstrap</summary>
 

exercises/ml_ar_p.tex

@@ -1,12 +1,12 @@
-\section[Maximum Likelihood Estimation Of Gaussian AR(p)]{Maximum Likelihood Estimation Of Gaussian AR(p)\label{ex:MaximumLikelihoodEstimationGaussianARp}}
-Consider an AR(p) model with a constant and linear trend:
+\section[Maximum Likelihood Estimation of Gaussian AR{(p)}]{Maximum Likelihood Estimation of Gaussian AR{(p)}\label{ex:MaximumLikelihoodEstimationGaussianARp}}
+Consider an AR{(p)} model with a constant and linear trend:
 \begin{align*}
 y_t = c + d\cdot t + \phi_1 y_{t-1} + \cdots + \phi_p y_{t-p} +u_{t}=Y_{t-1}\theta + u_t
 \end{align*}
   where \(Y_{t-1}=(1,t, y_{t-1},\ldots,y_{t-p})\) is the matrix of regressors,
   \(\theta = (c,d,\phi_1,\ldots,\phi_p)\) the parameter vector
-  and the error terms \(u_t\) are white noise and normally distributed,
-  i.e.\ \( u_t\sim N(0,\sigma_u^2) \) and \(E[u_t u_s]=0\) for \(t\neq s\).
+  and the error terms \(u_t\) are white noise and normally distributed, i.e.\
+  \( u_t\sim N(0,\sigma_u^2) \) and \(E[u_t u_s]=0\) for \(t\neq s\).
 If the sample distribution is known to have probability density function \(f(y_1,\ldots,y_T)\),
   an estimation with Maximum Likelihood (ML) is possible.
 To this end, we decompose the joint distribution by
@@ -27,13 +27,15 @@
 \begin{align*}
 I_a(\theta,\sigma_u^2) = \lim\limits_{T\rightarrow \infty}-\frac{1}{T} E
 \begin{pmatrix}
-\frac{\partial^2 \log l}{\partial \theta^2} & \frac{\partial^2 \log l}{\partial \theta  \partial \sigma_u^2}  \\
+\frac{\partial^2 \log l}{\partial \theta^2} & \frac{\partial^2 \log l}{\partial \theta  \partial \sigma_u^2}
+\\
 \frac{\partial^2 \log l}{\partial \sigma_u^2  \partial \theta} & \frac{\partial^2 \log l}{\partial {(\sigma_u^2)}^2}  
 \end{pmatrix}
 \end{align*}
 
 \begin{enumerate}
-\item First consider the case of \(p=1\)
+\item
+First consider the case of \(p=1\)
 \begin{enumerate}
   \item Derive the exact log-likelihood function for the \(AR(1)\) model with \(|\theta|<1\) and \(d=0\):
   \begin{align*}
@@ -42,47 +44,50 @@
   \item Why do we often look at the log-likelihood function instead of the actual likelihood function?
   \item Regard the value of the first observation as deterministic or, equivalently,
     note that its contribution to the log-likelihood disappears asymptotically.
-    Maximize analytically the conditional log-likelihood to get the ML estimators for \(\theta\) and \(\sigma_u\).
+    Maximize analytically the conditional log-likelihood to get the ML estimators for \(\theta \) and \(\sigma_u\).
     Compare these to the OLS estimators.
 \end{enumerate}
-\item Now consider the general \(AR(p)\) model.
- 	\begin{enumerate}
-    \item Write a function \texttt{logLikeARpNorm(\(x\),\(y\),\(p\),\(const\))}
-    that computes the value of the log-likelihood
-    conditional on the first \(p\) observations of a Gaussian \(AR(p)\) model, i.e.
-    \begin{align*}
-    \log l(\theta,\sigma_u)= -\frac{T-p}{2}\log(2\pi)-\frac{T-p}{2}\log(\sigma_u^2)-\sum_{t=p+1}^{T}\frac{u_t^2}{2\sigma_u^2}
-    \end{align*}
-    where \(x=(\theta',\sigma_u)'\), \(y\) denotes the data vector,
-    \(p\) the number of lags and \(const\) is equal to 1 if there is a constant,
-    and equal to 2 if there is a constant and linear trend in the model.
-    \item Write a \texttt{function ML = ARpML(\(y\),\(p\),\(const\),\( \alpha \))}
+\item
+Now consider the general \(AR(p)\) model.
+\begin{enumerate}
+  \item Write a function \texttt{logLikeARpNorm{(\(x\),\(y\),\(p\),\(const\))}}
+    that computes the log-likelihood value
+    conditional on the first \(p\) observations of a Gaussian \(AR(p)\) model:
+  \begin{align*}
+  \log l(\theta,\sigma_u)= -\frac{T-p}{2}\log(2\pi)-\frac{T-p}{2}\log(\sigma_u^2)-\sum_{t=p+1}^{T}\frac{u_t^2}{2\sigma_u^2}
+  \end{align*}
+  where \(x=(\theta',\sigma_u)'\), \(y\) denotes the data vector,
+  \(p\) the number of lags and \(const\) is equal to 1 if there is a constant,
+  and equal to 2 if there are both a constant and linear trend in the model.
+  \item Write a \texttt{function ML = ARpML{(\(y\),\(p\),\(const\),\( \alpha \))}}
     that takes as inputs a data vector \(y\), number of lags \(p\)
     and \(const=1\) if the model has a constant term
-    or \(const=2\) if the model has a constant term and linear trend.
-    \(\alpha\) denotes the significance level.
-    The function computes 
-    \begin{itemize}
-      \item the maximum likelihood estimates of an AR(p) model by numerically minimizing the negative conditional log-likelihood function using e.g. \texttt{fminunc}
-      \item the standard errors by means of the asymptotic covariance matrix, i.e.\ the inverse of the hessian of the negative log-likelihood function
+    or \(const=2\) if the model has both a constant term and linear trend.
+  \(\alpha \) denotes the significance level.
+  The function computes 
+  \begin{itemize}
+    \item the maximum likelihood estimates of an AR{(p)} model
+      by numerically minimizing the negative conditional log-likelihood function using e.g.\ \texttt{fminunc}
+    \item the standard errors by means of the asymptotic covariance matrix, i.e.\
+      the inverse of the hessian of the negative log-likelihood function
       (hint: gradient-based optimizers also output the hessian)
-    \end{itemize}
-    Save all results into a structure \enquote{ML} containing the estimates of \(\theta\),
+  \end{itemize}
+  Save all results into a structure \enquote{ML} containing the estimates of \(\theta \),
     its standard errors, t-statistics and p-values as well as the ML estimate of \(\sigma_u\).
 
-    \item Load simulated data given in the CSV file \texttt{AR4.csv} and estimate an AR(4) model with constant term.
-    Compare your results with the OLS estimators from the previous exercise.
- 	\end{enumerate}
+  \item Load simulated data given in the CSV file \texttt{AR4.csv} and estimate an AR{(4)} model with constant term.
+  Compare your results with the OLS estimators from the previous exercise.
+\end{enumerate}
 \end{enumerate}
 
 \paragraph{Readings}
 \begin{itemize}
-	\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}.
+\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
 \end{itemize}
 
 
 \begin{solution}\textbf{Solution to \nameref{ex:MaximumLikelihoodEstimationGaussianARp}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/ml_ar_p_solution.tex}
 \fi
 \newpage

exercises/ml_ar_p_laplace.tex

@@ -1,34 +1,39 @@
-\section[Maximum Likelihood Estimation Of Laplace AR(p)]{Maximum Likelihood Estimation Of Laplace AR(p)\label{ex:MaximumLikelihoodEstimationLaPlaceARp}}
-Consider the AR(1) model with constant
+\section[Maximum Likelihood Estimation of Laplace AR{(p)}]{Maximum Likelihood Estimation of Laplace AR{(p)}\label{ex:MaximumLikelihoodEstimationLaPlaceARp}}
+Consider the AR{(1)} model with constant
 \begin{align*}
 y_t = c + \phi y_{t-1} + u_t
 \end{align*}
-Assume that the error terms \(u_t\) are i.i.d. Laplace distributed with known density
+Assume that the error terms \(u_t\) are {i.i.d.}\ Laplace distributed with known density
 \begin{align*}
-f_{u_{t}}(u)=\frac{1}{2}\exp \left( -|u|\right)
+f_{u_{t}}(u) = \frac{1}{2} \exp{\left( -|u|\right)}
 \end{align*}
 Note that for the above parametrization of the Laplace distribution
   we have that \(E(u_t)=0\) and \(Var(u_t)=2\),
-  so we are only interested in estimating \(c\) and \(\phi\)
-  and not the standard deviation of \(u_t\) as it is fixed.
+  so we are only interested in estimating \(c\) and \(\phi \)
+  and not the standard deviation of \(u_t\) as it is fixed to 2.
 \begin{enumerate}
-\item Derive the log-likelihood function conditional on the first observation.
-\item Write a function that calculates the conditional log-likelihood of \(c\) and \(\phi\).
-\item Load the dataset given in the CSV file \texttt{LaPlace.csv}.
-Numerically find the maximum likelihood estimates of \(c\) and \(\phi\)
+\item
+Derive the log-likelihood function conditional on the first observation.
+\item
+Write a function that calculates the conditional log-likelihood of \(c\) and \(\phi \).
+\item
+Load the dataset given in the CSV file \texttt{LaPlace.csv}.
+Numerically find the maximum likelihood estimates of \(c\) and \(\phi \)
   by minimizing the negative conditional log-likelihood function.
-\item Compare your results with the maximum likelihood estimate under the assumption of Gaussianity.
-That is, redo the estimation by minimizing the negative Gaussian log-likelihood function.
+\item
+Compare your results with the maximum likelihood estimate under the assumption of Gaussianity.
+That is, redo the estimation by minimizing the negative Gaussian log-likelihood function
+  using the dataset given in the CSV file \texttt{LaPlace.csv}.
 \end{enumerate}
 
 \paragraph{Readings}
 \begin{itemize}
-	\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
+\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
 \end{itemize}
 
 
 \begin{solution}\textbf{Solution to \nameref{ex:MaximumLikelihoodEstimationLaPlaceARp}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/ml_ar_p_laplace_solution.tex}
 \fi
 \newpage

exercises/ml_ar_p_laplace_solution.tex

@@ -1,9 +1,12 @@
 \begin{enumerate}
-\item Computation of the conditional expectation and variance:
-	\begin{itemize}
-	\item \(E[y_{t}|y_{t-1}] = c + \phi y_{t-1} \)
-	\item \(Var[y_{t}|y_{t-1}] = var(u_t) = 2\) 
-	\end{itemize}
+
+\item
+Computation of the conditional expectation and variance:
+\begin{gather*}
+E[y_{t}|y_{t-1}] = c + \phi y_{t-1}
+\\
+Var[y_{t}|y_{t-1}] = var(u_t) = 2
+\end{gather*}
 Hence the conditional density is
 \begin{align*}
 f_t(y_{t}|y_{t-1}; c, \phi) = \frac{1}{2} \cdot e^{-|y_{t} -(c + \phi y_{t-1})|} = \frac{1}{2} \cdot e^{-|u_t|}
@@ -13,11 +16,16 @@
 \log L(y_{2}, \dots, y_{T};c, \phi) =-(T-1) \cdot \log(2) - \sum_{t=2}^{T} |u_{t}|
 \end{align*}
 
-\item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/logLikeARpLaplace.m}
-\item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/ARpMLLaPlace.m}
-      \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/AR1MLLaPlace.m}
-\item Note that the values are very close to each other.
-  Maximizing the Gaussian likelihood, even though the underlying distribution is not Gaussian,
+\item
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/logLikeARpLaplace.m}
+
+\item
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/ARpMLLaPlace.m}
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/AR1MLLaPlace.m}
+
+\item
+Note that the values are very close to each other.
+Maximizing the Gaussian likelihood, even though the underlying distribution is not Gaussian,
   is also known as pseudo-maximum likelihood or quasi-maximum likelihood.
-  It usually performs surprisingly well if you cannot pin down the underlying distribution.
+It usually performs surprisingly well if you cannot pin down the underlying distribution.
 \end{enumerate}