From 8c7b2da1ca1ce1994c21d1db35f66a5563838e1e Mon Sep 17 00:00:00 2001
From: Willi Mutschler <willi@mutschler.eu>
Date: Mon, 4 Nov 2024 06:06:13 +0100
Subject: [PATCH] Added week 4 material

---
 .../dynare-6.2-matlab-r2024b-macos.yml        |  10 ++
 .../dynare-6.2-matlab-r2024b-ubuntu.yml       |  10 ++
 .../dynare-6.2-matlab-r2024b-windows.yml      |  10 ++
 README.md                                     |   7 +-
 exercises/ml_ar_p.tex                         |  69 ++++----
 exercises/ml_ar_p_laplace.tex                 |  33 ++--
 exercises/ml_ar_p_laplace_solution.tex        |  30 ++--
 exercises/ml_ar_p_solution.tex                | 153 ++++++++++--------
 exercises/ols_ar_p.tex                        |  29 ++--
 progs/matlab/AR1MLLaplace.m                   |  10 +-
 week_4.tex                                    |  30 ++++
 11 files changed, 245 insertions(+), 146 deletions(-)
 create mode 100644 week_4.tex

diff --git a/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml b/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml
index 9744159..9a1fb96 100644
--- a/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml
+++ b/.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;
diff --git a/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml b/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml
index a87d41c..21b3859 100644
--- a/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml
+++ b/.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;
\ No newline at end of file
diff --git a/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml b/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml
index 9fb9d27..670747b 100644
--- a/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml
+++ b/.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;
\ No newline at end of file
diff --git a/README.md b/README.md
index 59c890b..20da7ec 100644
--- a/README.md
+++ b/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>
 
diff --git a/exercises/ml_ar_p.tex b/exercises/ml_ar_p.tex
index 1af57aa..b4f346a 100644
--- a/exercises/ml_ar_p.tex
+++ b/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
diff --git a/exercises/ml_ar_p_laplace.tex b/exercises/ml_ar_p_laplace.tex
index f1d6a44..cbaee44 100644
--- a/exercises/ml_ar_p_laplace.tex
+++ b/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
diff --git a/exercises/ml_ar_p_laplace_solution.tex b/exercises/ml_ar_p_laplace_solution.tex
index d612589..798554c 100644
--- a/exercises/ml_ar_p_laplace_solution.tex
+++ b/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}
\ No newline at end of file
diff --git a/exercises/ml_ar_p_solution.tex b/exercises/ml_ar_p_solution.tex
index 1837846..e1557c6 100644
--- a/exercises/ml_ar_p_solution.tex
+++ b/exercises/ml_ar_p_solution.tex
@@ -1,73 +1,88 @@
 \begin{enumerate}
-\item Let's first consider the AR(1) model
-	\begin{enumerate}
-		\item The first observation \(y_1\) is a random variable with mean and variance equal to:
-		\begin{align*}
-		E[y_1] = \mu = \frac{c}{1-\theta} \text{ and } Var[y_1] = \frac{\sigma_u^2}{1-\theta^2}
-		\end{align*}
-		Since the errors are Gaussian, $y_1$ is also Gaussian, i.e. $y_1 \sim N\left(\frac{c}{1-\theta},\frac{\sigma_u^2}{1-\theta^2}\right)$.
-		The pdf is thus:
-		\begin{align*}
-        f_1(y_1|\theta,\sigma_u^2) = \frac{1}{\sqrt{2\pi}\sqrt{\sigma_u^2/(1-\theta^2)}}\exp\left\{-\frac{1}{2}\frac{[y_1-(c/(1-\theta))]^2}{\sigma_u^2/(1-\theta^2)}\right\}
-        \end{align*}	
-		
-	    The second observation $y_2$ conditional on $y_1$ is given by $y_2 = c + \theta y_1 + u_2$.
-        Conditional on $y_1$, $y_2$ is thus the sum of a deterministic term ($c+\theta y_1$) and the $N(0,\sigma_u^2)$ variable $u_2$.
-        Hence:
-        \begin{align*}
-        y_2|y_1 \sim N(c+\theta y_1,\sigma_u^2)
-        \end{align*}
-	    and the pdf is given by:
-        \begin{align*}
-        f_2(y_2|y_1,\theta,\sigma_u^2) = \frac{1}{\sqrt{2\pi\sigma_u^2}}\exp\left\{-\frac{1}{2}\frac{[y_2-c-\theta y_1]^2}{\sigma_u^2}\right\}
-        \end{align*}	    
-	    The joint density of observations 1 and 2 is then just:
-        \begin{align*}
-        f(y_2,y_1|\theta,\sigma_u^2) = f_2(y_2|y_1,\theta,\sigma_u^2) \cdot f_1(y_1|\theta,\sigma_u^2)
-        \end{align*}	
-	    In general the value of $y_1,y_2,...,y_{t-1}$ matter for $y_t$ only through the value $y_{t-1}$
-          and the density of observation $t$ conditional on the preceding $t-1$ observations is given by
-        \begin{align*}
-        f_t(y_t|y_{t-1},\theta,\sigma_u^2) = \frac{1}{\sqrt{2\pi\sigma_u^2}}\exp\left\{-\frac{1}{2}\frac{[y_t-c-\theta y_{t-1}]^2}{\sigma_u^2}\right\}
-        \end{align*}	
-	    The likelihood of the complete sample can thus be calculated as:
-        \begin{align*}
-	    f(y_T,y_{T-1},...,y_1|\theta,\sigma_u^2)=f_1(y_1|\theta,\sigma_u^2)\cdot \prod_{t=2}^{T}f_t(y_t|y_{t-1},\theta,\sigma_u^2)        
-        \end{align*}
-	    The log-likelihood is therefore
-	    \begin{align*}
-	    \log l(\theta,\sigma_u^2)&= \log f_1(y_1|\theta,\sigma_u^2)+  \sum_{t=2}^{T}\log f_t(y_t|y_{t-1},\theta,\sigma_u^2)
-        \\
-	    &= -\frac{1}{2}\log(2\pi) -\frac{1}{2}\log(\sigma_u^2/(1-\theta^2))-\frac{(y_1-(c/(1-\theta))^2)}{2\sigma_u^2/(1-\theta^2)}
-        \\
-	    &-((T-1)/2)\log(2\pi)-((T-1)/2)\log(\sigma_u^2)-\sum_{t=2}^{T}\frac{(y_t-c-\theta y_{t-1})^2}{2\sigma_u^2}
-	    \end{align*}
-      \item Theoretically it does not matter whether we consider the log-likelihood or the actual likelihood function,
-      as the value that maximizes the likelihood also maximize the log-likelihood,
-      because the log is a monotone transformation.
-      However, it is usually easier to work with sums instead of products theoretically,
-        e.g. the LLN or CT are based on sums.
-      Computationally, working with products is typically impossible as the resulting values very quickly surpass machine precision (they quickly go to $\pm \infty$);
-        working with sums does not have this problem.
-      So from a computational perspective we will exclusively work with the log-likelihood function.
-	    \item Discarding the first observation, the conditional log-likelihood is given by:
-	    \begin{align*}
-	    \log l^c(\theta,\sigma_u^2)&= -((T-1)/2)\log(2\pi)-((T-1)/2)\log(\sigma^2_u)-\sum_{t=2}^{T}\frac{(y_t-c-\theta y_{t-1})^2}{2\sigma_u^2}
-        \\
-	    &= -((T-1)/2)\log(2\pi)-((T-1)/2)\log(\sigma^2_u)-\sum_{t=2}^{T}\frac{u_t^2}{2\sigma_u^2}
-	    \end{align*}
-	    Note that the first two sums do not depend on $\theta$;
-          thus, when maximizing $\log l^c(\theta,\sigma_u^2)$ with respect to $\theta$,
-          we are basically minimizing the squared residuals,
-          which will simply yield the OLS estimator.
-          The estimator for the variance, however, is different,
-            as we are dividing the sum of sqared residuals ($\sum_{t=2}^T u_t^2$)
-            by $T^{eff}=(T-1)$ when doing ML instead of by $T^{eff}-1$ when doing OLS.
-          Obviously, for large $T$ this does not matter.
-    \end{enumerate}
-\item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/logLikeARpNorm.m}
-\item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/ARpML.m} 		
-\item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/AR4ML.m} 
+
+\item
+Let's first consider the AR{(1)} model
+\begin{enumerate}
+  \item
+  The first observation \(y_1\) is a random variable with mean and variance equal to:
+  \begin{align*}
+  E[y_1] = \mu = \frac{c}{1-\theta} \quad\text{and}\quad Var[y_1] = \frac{\sigma_u^2}{1-\theta^2}
+  \end{align*}
+  Since the errors are Gaussian, \(y_1\) is also Gaussian, i.e.\
+  \(y_1 \sim N\left(\frac{c}{1-\theta},\frac{\sigma_u^2}{1-\theta^2}\right)\).
+  The pdf is thus:
+  \begin{align*}
+  f_1(y_1|\theta,\sigma_u^2) = \frac{1}{\sqrt{2\pi}\sqrt{\sigma_u^2/(1-\theta^2)}}\exp\left \{-\frac{1}{2}\frac{{[y_1-(c/(1-\theta))]}^2}{\sigma_u^2/(1-\theta^2)}\right \}
+  \end{align*}
+  The second observation \(y_2\) conditional on \(y_1\) is given by \(y_2 = c + \theta y_1 + u_2\).
+  Conditional on \(y_1\), \(y_2\) is thus the sum of a deterministic term (\(c+\theta y_1\))
+    and the \(N(0,\sigma_u^2)\) variable \(u_2\).
+  Hence:
+  \begin{align*}
+  y_2|y_1 \sim N(c+\theta y_1,\sigma_u^2)
+  \end{align*}
+  and the pdf is given by:
+  \begin{align*}
+  f_2(y_2|y_1,\theta,\sigma_u^2) = \frac{1}{\sqrt{2\pi\sigma_u^2}}\exp\left \{-\frac{1}{2}\frac{[y_2-c-\theta y_1]^2}{\sigma_u^2}\right \}
+  \end{align*}
+  The joint density of observations 1 and 2 is then just:
+  \begin{align*}
+  f(y_2,y_1|\theta,\sigma_u^2) = f_2(y_2|y_1,\theta,\sigma_u^2) \cdot f_1(y_1|\theta,\sigma_u^2)
+  \end{align*}
+  In general the values of \(y_1,y_2,\ldots ,y_{t-1}\) matter for \(y_t\) only through the value \(y_{t-1}\)
+    and the density of observation \(t\) conditional on the preceding \(t-1\) observations is given by
+  \begin{align*}
+  f_t(y_t|y_{t-1},\theta,\sigma_u^2) = \frac{1}{\sqrt{2\pi\sigma_u^2}}\exp\left \{-\frac{1}{2}\frac{[y_t-c-\theta y_{t-1}]^2}{\sigma_u^2}\right \}
+  \end{align*}
+  The likelihood of the complete sample can thus be calculated as:
+  \begin{align*}
+  f(y_T,y_{T-1},\ldots ,y_1|\theta,\sigma_u^2)=f_1(y_1|\theta,\sigma_u^2)\cdot \prod_{t=2}^{T}f_t(y_t|y_{t-1},\theta,\sigma_u^2)
+  \end{align*}
+  The log-likelihood is therefore
+  \begin{align*}
+  \log l(\theta,\sigma_u^2)&= \log f_1(y_1|\theta,\sigma_u^2)+  \sum_{t=2}^{T}\log f_t(y_t|y_{t-1},\theta,\sigma_u^2)
+  \\
+  &= -\frac{1}{2}\log(2\pi) -\frac{1}{2}\log(\sigma_u^2/(1-\theta^2))-\frac{(y_1-{(c/(1-\theta))}^2)}{2\sigma_u^2/(1-\theta^2)}
+  \\
+  &-((T-1)/2)\log(2\pi)-((T-1)/2)\log(\sigma_u^2)-\sum_{t=2}^{T}\frac{{(y_t-c-\theta y_{t-1})}^2}{2\sigma_u^2}
+  \end{align*}
+
+  \item
+  Theoretically it does not matter whether we consider the log-likelihood or the actual likelihood function,
+    as the value that maximizes the likelihood also maximize the log-likelihood,
+    because the log is a monotone transformation.
+  However, it is usually easier to work with sums instead of products theoretically,
+    e.g.\ the LLN or CT are based on sums.
+  Computationally, working with products is typically impossible
+    as the resulting values very quickly surpass machine precision (they quickly go to \(\pm \infty \));
+    working with sums does not have this problem.
+  So from a computational perspective we will exclusively work with the log-likelihood function.
+
+  \item
+  Discarding the first observation, the conditional log-likelihood is given by:
+  \begin{align*}
+  \log l^c(\theta,\sigma_u^2)&= -((T-1)/2)\log(2\pi)-((T-1)/2)\log(\sigma^2_u)-\sum_{t=2}^{T}\frac{{(y_t-c-\theta y_{t-1})}^2}{2\sigma_u^2}
+  \\
+  &= -((T-1)/2)\log(2\pi)-((T-1)/2)\log(\sigma^2_u)-\sum_{t=2}^{T}\frac{u_t^2}{2\sigma_u^2}
+  \end{align*}
+  Note that the first two sums do not depend on \(\theta \);
+    thus, when maximizing \(\log l^c(\theta,\sigma_u^2)\) with respect to \(\theta \),
+    we are basically minimizing the squared residuals,
+    which will simply yield the OLS estimator.
+  The estimator for the variance, however, is different,
+    as we are dividing the sum of squared residuals (\(\sum_{t=2}^T u_t^2\))
+    by \(T^{eff}=(T-1)\) when doing {ML} instead of by \(T^{eff}-1\) when doing {OLS}.
+  Obviously, for large \(T\) this does not matter.
+\end{enumerate}
+
+\item
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/logLikeARpNorm.m}
+
+\item
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/ARpML.m}
+
+\item
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/AR4ML.m}
 The estimates for the coefficients are the same,
   but slightly different for the standard deviation of the error term.
 \end{enumerate}
\ No newline at end of file
diff --git a/exercises/ols_ar_p.tex b/exercises/ols_ar_p.tex
index c423a3f..822615a 100644
--- a/exercises/ols_ar_p.tex
+++ b/exercises/ols_ar_p.tex
@@ -1,5 +1,5 @@
-\section[Ordinary Least Squares Estimation Of AR(p)]{Ordinary Least Squares Estimation Of AR(p)\label{ex:OrdinaryLeastSquaresEstimationARp}}
-Consider an AR(p) model with a constant and linear term:
+\section[Ordinary Least Squares Estimation of AR{(p)}]{Ordinary Least Squares Estimation of AR{(p)}\label{ex:OrdinaryLeastSquaresEstimationARp}}
+Consider an AR{(p)} model with a constant and linear term:
 \begin{align*}
 y_t = c + d\cdot t + \phi_1 y_{t-1} +\cdot + \phi_p y_{t-p} +u_{t}=Y_{t-1}'\theta + u_t
 \end{align*}
@@ -10,7 +10,7 @@
 \end{align*}
 Under the assumptions of stationarity and other standard regularity conditions one can derive that
 \begin{align*}
-\sqrt{T}(\hat{\theta}-\theta)\overset{d}{\rightarrow}\tilde{U}\sim N\left(0,\sigma_u^2~{plim}\left(T^{-1}\sum_{t=1}^T Y_{t-1}Y_{t-1}'\right)^{-1} \right)
+\sqrt{T} (\hat{\theta}-\theta) \overset{d}{\rightarrow} \tilde{U} \sim N\left(0,\sigma_u^2~{plim}{\left(T^{-1}\sum_{t=1}^T Y_{t-1}Y_{t-1}'\right)}^{-1}\right)
 \end{align*}
 The residual variance may be estimated consistently by
 \begin{align*}
@@ -19,26 +19,29 @@
 where \(\hat{u}_t=y_t -Y_{t-1}'\hat{\theta}\) are the OLS residuals.
 
 \begin{enumerate}
-\item Write a \texttt{function OLS = ARpOLS(\(y\),\(p\),\(const\),\(\alpha\))}
-  that takes as inputs a data vector $y$ and number of lags $p$.
+\item
+Write a \texttt{function OLS = ARpOLS{(\(y\),\(p\),\(const\),\(\alpha \))}}
+  that takes as inputs a data vector \(y\) and an integer for the number of lags \(p\).
 The input \(const\) is \(1\) if there is a constant in the model,
-  \(2\) if there is a constant and a linear trend.
-The function outputs a structure OLS,
-  which contains the OLS estimates of \(\theta\),
-  its standard errors, t-statistics and p-values given significance value \(\alpha\),
+  \(2\) if there is a constant and a linear trend,
+  and \(0\) otherwise.
+The function outputs a structure called \texttt{OLS},
+  which contains fields for the OLS estimates of \(\theta \),
+  its standard errors, t-statistics and p-values given significance value \(\alpha \),
   as well as the OLS estimate of \(\sigma_u\).
-\item Load simulated data for an AR(4) model given in the CSV file \texttt{AR4.csv}.
-Estimate an AR(4) model with a constant term using your \texttt{ARpOLS} function.
+\item
+Load (simulated) data for an AR{(4)} model given in the CSV file \texttt{AR4.csv}.
+Estimate an AR{(4)} model with a constant term using your \texttt{ARpOLS} function.
 \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:OrdinaryLeastSquaresEstimationARp}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/ols_ar_p_solution.tex}
 \fi
 \newpage
diff --git a/progs/matlab/AR1MLLaplace.m b/progs/matlab/AR1MLLaplace.m
index 05a6de0..cf65834 100644
--- a/progs/matlab/AR1MLLaplace.m
+++ b/progs/matlab/AR1MLLaplace.m
@@ -7,21 +7,21 @@
 % -------------------------------------------------------------------------
 
 clearvars; clc; close all;                 % housekeeping
-y = importdata('../../data/LaPlace.csv');  % load data
+y = importdata('../../data/Laplace.csv');  % load data
 y = y.data;                                % focus on numerical values
 p = 1;                                     % set number of lags
 const = 1;                                 % model with constant
 alph = 0.05;                               % significance level
-MLLaPlace = ARpMLLaplace(y,p,const,alph);  % estimate model using ARpMLLaplace
+MLLaplace = ARpMLLaplace(y,p,const,alph);  % estimate model using ARpMLLaplace
 
 % Display results and compare to true values
 TrueVals = [1; 0.8];
-result = table(TrueVals,MLLaPlace.thetatilde,MLLaPlace.sd_thetatilde);
+result = table(TrueVals,MLLaplace.thetatilde,MLLaplace.sd_thetatilde);
 result.Properties.VariableNames = {'True_Values','ML_Estimate','STD_ERR'};
 result.Properties.RowNames = {'c','\phi'};
 disp(result)
 
 % Compare to Gaussian ML estimates
 MLGaussian = ARpML(y,p,const,alph); % estimate model using ARpML
-disp([TrueVals MLLaPlace.thetatilde MLGaussian.thetatilde]);
-disp([MLLaPlace.sd_thetatilde MLGaussian.sd_thetatilde]);
\ No newline at end of file
+disp([TrueVals MLLaplace.thetatilde MLGaussian.thetatilde]);
+disp([MLLaplace.sd_thetatilde MLGaussian.sd_thetatilde]);
\ No newline at end of file
diff --git a/week_4.tex b/week_4.tex
new file mode 100644
index 0000000..a2751a1
--- /dev/null
+++ b/week_4.tex
@@ -0,0 +1,30 @@
+% !TEX root = week_4.tex
+\input{exercises/_common_header.tex}
+\Newassociation{solution}{Solution}{week_4_solution}
+\newif\ifDisplaySolutions\DisplaySolutionstrue%
+
+\begin{document}
+\title{Quantitative Macroeconomics\\~\\Winter 2024/25\\~\\Week 4}
+\author{Willi Mutschler\\willi@mutschler.eu}
+\date{Version: 1.0\\Latest version available on: \href{https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_4.pdf}{GitHub}}
+\maketitle\thispagestyle{empty}
+
+\newpage
+\Opensolutionfile{week_4_solution}[week_4_solution]
+\tableofcontents\thispagestyle{empty}\newpage
+
+\setcounter{page}{1}
+\input{exercises/ols_ar_p.tex}\newpage
+\input{exercises/ml_ar_p.tex}\newpage
+\input{exercises/ml_ar_p_laplace.tex}\newpage
+\printbibliography%
+\newpage
+
+\Closesolutionfile{week_4_solution}
+\ifDisplaySolutions%
+\newpage
+\appendix
+\section{Solutions}
+\input{week_4_solution}
+\fi
+\end{document}
\ No newline at end of file