From d954b4090b864c9f621c02089507e6a8a55c77d5 Mon Sep 17 00:00:00 2001
From: Willi Mutschler <willi@mutschler.eu>
Date: Fri, 25 Oct 2024 17:21:01 +0200
Subject: [PATCH] Added week 3 material

---
 .../dynare-6.2-matlab-r2024b-macos.yml        |  11 +
 .../dynare-6.2-matlab-r2024b-ubuntu.yml       |  11 +
 .../dynare-6.2-matlab-r2024b-windows.yml      |  11 +
 README.md                                     |   6 +-
 .../central_limit_theorem_dependent_data.tex  | 114 +++++----
 ..._limit_theorem_dependent_data_solution.tex | 211 +++++++++-------
 exercises/law_of_large_numbers.tex            |  48 ++--
 exercises/law_of_large_numbers_solution.tex   |  92 +++----
 exercises/properties_ar1.tex                  |  55 ++--
 exercises/properties_ar1_solution.tex         | 235 ++++++++++--------
 exercises/properties_ar1_time_trend.tex       |  22 +-
 .../properties_ar1_time_trend_solution.tex    |  86 ++++---
 week_3.tex                                    |  31 +++
 13 files changed, 561 insertions(+), 372 deletions(-)
 create mode 100644 week_3.tex

diff --git a/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml b/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml
index f2fd603..9744159 100644
--- a/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml
+++ b/.github/workflows/dynare-6.2-matlab-r2024b-macos.yml
@@ -53,3 +53,14 @@ jobs:
             definitionFrequenciesTimeSeriesData;
             whiteNoisePlots;
             plotsAR1;
+
+      - name: Run week 3 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("Dynare-6.2-arm64/matlab");
+            cd("progs/matlab");
+            acfPlots_run;
+            lawOfLargeNumbers;
+            lawOfLargeNumbersAR1;
+            centralLimitDependentData;
diff --git a/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml b/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml
index a5a0f47..a87d41c 100644
--- a/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml
+++ b/.github/workflows/dynare-6.2-matlab-r2024b-ubuntu.yml
@@ -82,3 +82,14 @@ jobs:
             definitionFrequenciesTimeSeriesData;
             whiteNoisePlots;
             plotsAR1;
+
+      - name: Run week 3 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("dynare/matlab");
+            cd("progs/matlab");
+            acfPlots_run;
+            lawOfLargeNumbers;
+            lawOfLargeNumbersAR1;
+            centralLimitDependentData;
diff --git a/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml b/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml
index 64ef52b..9fb9d27 100644
--- a/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml
+++ b/.github/workflows/dynare-6.2-matlab-r2024b-windows.yml
@@ -44,3 +44,14 @@ jobs:
             definitionFrequenciesTimeSeriesData;
             whiteNoisePlots;
             plotsAR1;
+
+      - name: Run week 3 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("D:\hostedtoolcache\windows\dynare-6.0\matlab");
+            cd("progs/matlab");
+            acfPlots_run;
+            lawOfLargeNumbers;
+            lawOfLargeNumbersAR1;
+            centralLimitDependentData;
diff --git a/README.md b/README.md
index e735438..59c890b 100644
--- a/README.md
+++ b/README.md
@@ -66,8 +66,6 @@ Please feel free to use this for teaching or learning purposes; however, taking
 
 </details>
 
-<!---
-
 
 <details>
   <summary>Week 3: Dependent time series data and the autoregressive process</summary>
@@ -81,7 +79,7 @@ Please feel free to use this for teaching or learning purposes; however, taking
 ### To Do
 
 * [x] review the solutions of [last week's exercises](https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_2.pdf) and write down all your questions.
-* [x] read Lütkepohl (2004, Sec. 2.2, 2.3, 2.5.2), Lütkepohl (2005, Appendix C) and Bjørnland and Thorsrud (2015, Ch.1 and Ch.2); make note of all the aspects and concepts that you are not familiar with or that you find difficult to understand
+* [x] read Lütkepohl (2004, Sec. 2.2, 2.3, 2.5.2) and Bjørnland and Thorsrud (2015, Ch.1 and Ch.2); make note of all the aspects and concepts that you are not familiar with or that you find difficult to understand
 * [x] prepare exercise sheet 3: do exercises 1 and 3 at home, we'll do exercises 2 and 4 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)
@@ -89,6 +87,8 @@ 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>
 
diff --git a/exercises/central_limit_theorem_dependent_data.tex b/exercises/central_limit_theorem_dependent_data.tex
index 98b9bd8..0a7754f 100644
--- a/exercises/central_limit_theorem_dependent_data.tex
+++ b/exercises/central_limit_theorem_dependent_data.tex
@@ -4,66 +4,90 @@
 Y_{t}-\mu =\phi \left(Y_{t-1}-\mu\right) +\varepsilon _{t}
 \end{align*}
 where \(\varepsilon _{t}\sim iid(0,\sigma _{\varepsilon }^{2})\) is
-  (not necessarily but in our case) normally distributed and \(|\phi |<1\).
+(not necessarily but in our case) normally distributed and \(|\phi |<1\).
+
 \begin{enumerate}
-\item Briefly state and describe the intuition of the \enquote{Lindeberg-Levy Central Limit Theorem} for iid random variables.
+\item
+Briefly state and describe the intuition of the \enquote{Lindeberg-Levy Central Limit Theorem} for iid random variables.
 What does \enquote{convergence in distribution} mean?
 Why can we not use the theorem for the \(AR(1)\) process?
-\item Show that \(Y_t\) has mean equal to \(\mu \) and finite variance equal to \(\sigma_\varepsilon^2/(1-\phi^2)\).
+
+\item
+Show that \(Y_t\) has mean equal to \(\mu \) and finite variance equal to \(\sigma_\varepsilon^2/(1-\phi^2)\).
+
 \item To derive the asymptotic distribution of the sample mean, do the following steps:
 \begin{enumerate}
-	\item Derive the asymptotic distribution of \(\frac{1}{\sqrt{T} } \sum_{t=1}^T \varepsilon_t\).
-	\item Show that
-	\begin{align*}
-		\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t = \sqrt{T}\left[(1-\phi)\left(\hat{\mu}-\mu\right) + \phi\left(\frac{Y_T - Y_0}{T}\right)\right]
-	\end{align*}
-	with \(\hat{\mu} =\frac{1}{T}\sum_{t=1}^{T}Y_{t}\).
-	\item Show that
-	\begin{align*}
-	\textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0
-	\end{align*}
-	\\\emph{Hint: Use Tchebychev's Inequality,
-	i.e.\ for a random variable \(X\) with expectation \(\mu_x\)
-	and finite variance \(\sigma_x^2\):}
-	\begin{align*}
-	\Pr(|X-\mu_x|> \delta) \leq \frac{\sigma_x^2}{\delta^2}
-	\end{align*}
-	\emph{for any small real number \(\delta>0\).}
-	\item Put your results of (a), (b) and (c) together and derive the asymptotic distribution of the sample mean.
-	That is, show that
-	\begin{align*}
-	Z_{T} =\sqrt{T}\frac{\hat{\mu} -\mu }{\sigma_Z} \overset{d}{\rightarrow} U \sim N(0,1)
-	\end{align*}
-	for \(\sigma_Z=\sqrt{\sigma_\varepsilon^2/(1-\phi)^2}\).
+  \item
+  Derive the asymptotic distribution of \(\frac{1}{\sqrt{T} } \sum_{t=1}^T \varepsilon_t\).
+
+  \item
+   Show that
+  \begin{align*}
+  \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t = \sqrt{T}\left[(1-\phi)\left(\hat{\mu}-\mu\right) + \phi\left(\frac{Y_T - Y_0}{T}\right)\right]
+  \end{align*}
+  with \(\hat{\mu} =\frac{1}{T}\sum_{t=1}^{T}Y_{t}\).
+
+  \item
+  Show that
+  \begin{align*}
+  \textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0
+  \end{align*}
+  \\
+  \emph{Hint: Use Tchebychev's Inequality,
+  i.e.\ for a random variable \(X\) with expectation \(\mu_x\)
+  and finite variance \(\sigma_x^2\):}
+  \begin{align*}
+  \Pr(|X-\mu_x|> \delta) \leq \frac{\sigma_x^2}{\delta^2}
+  \end{align*}
+  \emph{for any small real number \(\delta>0\).}
+
+  \item
+  Put your results of (a), (b) and (c) together and derive the asymptotic distribution of the sample mean.
+  That is, show that
+  \begin{align*}
+  Z_{T} =\sqrt{T}\frac{\hat{\mu} -\mu }{\sigma_Z} \overset{d}{\rightarrow} U \sim N(0,1)
+  \end{align*}
+  for \(\sigma_Z=\sqrt{\sigma_\varepsilon^2/{(1-\phi)}^2}\).
 \end{enumerate}
-\item Write a program to demonstrate the central limit theorem for the AR(1) process. To this end:
+
+\item
+ Write a program to demonstrate the central limit theorem for the AR{(1)} process. To this end:
+
 \begin{itemize}
-	\item Simulate \(B=5000\) stationary (e.g.\
-	\(\phi=0.8\)) AR(1) processes with each \(T=10000\) observations.
-	Store these in a \(T \times B\) matrix \(Y\).
-	\item Compute \(\hat{\mu}\) for each column of \(Y\).
-	\item Plot the histograms of the standardized variables according to the Lindeberg-Levy Central Limit Theorem:
-	\begin{align*}
-	\widetilde{Z}_T = \sqrt{T}\frac{\hat{\mu}-\mu}{\sigma_{\varepsilon }/\sqrt{1-\phi^2}}
-	\end{align*}
-	and of the correct standardized variables that we derived in 3(d):
-	\begin{align*}
-	Z_T = \sqrt{T}\frac{\hat{\mu}-\mu}{\sigma_{\varepsilon }/(1-\phi)}
-	\end{align*}
-	Compare the histograms to the standard normal distribution.
+
+  \item
+  Simulate \(B=5000\) stationary (e.g.\
+  \(\phi=0.8\)) AR{(1)} processes with each \(T=10000\) observations.
+  Store these in a \(T \times B\) matrix \(Y\).
+
+  \item
+  Compute \(\hat{\mu}\) for each column of \(Y\).
+
+  \item
+  Plot the histograms of the standardized variables according to the Lindeberg-Levy Central Limit Theorem:
+  \begin{align*}
+  \widetilde{Z}_T = \sqrt{T}\frac{\hat{\mu}-\mu}{\sigma_{\varepsilon }/\sqrt{1-\phi^2}}
+  \end{align*}
+  and of the correct standardized variables that we derived in 3(d):
+  \begin{align*}
+  Z_T = \sqrt{T}\frac{\hat{\mu}-\mu}{\sigma_{\varepsilon }/(1-\phi)}
+  \end{align*}
+  Compare the histograms to the standard normal distribution.
+
 \end{itemize}
+
 \end{enumerate}
 
 \paragraph{Readings}
 \begin{itemize}
-	\item \textcite{Crack.Ledoit_2010_CentralLimitTheorems}
-	\item \textcite[App. C]{Lutkepohl_2005_NewIntroductionMultiple}
-	\item \textcite[App. C]{Neusser_2016_TimeSeriesEconometrics}
-	\item \textcite[Ch. 5]{White_2001_AsymptoticTheoryEconometricians}
+\item \textcite{Crack.Ledoit_2010_CentralLimitTheorems}
+\item \textcite[App. C]{Lutkepohl_2005_NewIntroductionMultiple}
+\item \textcite[App. C]{Neusser_2016_TimeSeriesEconometrics}
+\item \textcite[Ch. 5]{White_2001_AsymptoticTheoryEconometricians}
 \end{itemize}
 
 \begin{solution}\textbf{Solution to \nameref{ex:CentralLimitTheoremDependentData}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/central_limit_theorem_dependent_data_solution.tex}
 \fi
 \newpage
diff --git a/exercises/central_limit_theorem_dependent_data_solution.tex b/exercises/central_limit_theorem_dependent_data_solution.tex
index 495e954..95db11c 100644
--- a/exercises/central_limit_theorem_dependent_data_solution.tex
+++ b/exercises/central_limit_theorem_dependent_data_solution.tex
@@ -1,5 +1,7 @@
 \begin{enumerate}
-\item We usually consider the \emph{Lindeberg-Levy Central Limit Theorem} for identically and independently (iid) random variables
+
+\item
+We usually consider the \emph{Lindeberg-Levy Central Limit Theorem} for identically and independently (iid) random variables
   with finite mean \( \mu \) and finite variance \(\sigma_Y^2\).
 Then the \emph{Lindeberg-Levy Central Limit Theorem} establishes the distribution of the sample mean \(\overline{Y}_T\) for a growing sample size:
 \begin{align*}
@@ -16,7 +18,7 @@
   i.e.\ the distribution of the sample mean.
 More formally, a sequence \(\bar{Y}_{1},\bar{Y}_{2},\ldots \) of random variables with distribution functions
 \(F_{1},F_{2},\ldots \) converges \textbf{in distribution (weakly; in law)}
-to a variable \(\mu\) with distribution function \(F\), if
+to a variable \(\mu \) with distribution function \(F\), if
 \begin{equation*}
 \lim_{T\rightarrow \infty }F_{T}(x)=F(x)
 \end{equation*}
@@ -24,113 +26,134 @@
 Typically, we use the following notation for this:
 \begin{equation*}
 \bar{Y}_{T}\overset{d}{\rightarrow }\mu
-\end{equation*}			
+\end{equation*}
 
 Unfortunately, the \emph{Lindeberg-Levy Central Limit Theorem} does not apply
-  for the AR(1) process as we have dependent and not iid data.
-For stationary AR(1) processes, we can however use similar central limit theorems
+  for the AR{(1)} process as we have dependent and not iid data.
+For stationary AR{(1)} processes, we can however use similar central limit theorems
   either for Martingale-Difference-processes or mixing processes.
-	
-\item First let's derive the expectation and variance of the AR(1) process with \(|\phi|<1\).
+
+\item
+First let's derive the expectation and variance of the AR{(1)} process with \(|\phi|<1\).
 For this, we use recursive substitution techniques given a starting value \(Y_0\):
 \begin{align*}
 Y_t = (1-\phi)(1+\phi+\phi^2+\cdots+\phi^T)\mu + \varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2} + \cdots + \phi^T \varepsilon_{t-T} + \phi^{T+1} Y_0
 \end{align*}
 Note that \(\lim_{T\rightarrow \infty} \phi^{T+1} = 0\)
-  and \(\lim_{T\rightarrow \infty} \sum_{j=0}^\infty \phi^j = \frac{1}{1-\phi}\), since $|\phi|<1$.
-The AR(1) process with \(|\phi|<1\) can therefore be equally represented by
+  and \(\lim_{T\rightarrow \infty} \sum_{j=0}^\infty \phi^j = \frac{1}{1-\phi}\), since \(|\phi|<1\).
+The AR{(1)} process with \(|\phi|<1\) can therefore be equally represented by
 \begin{align*}
 Y_t = \mu + \sum_{j=1}^\infty \phi^j \varepsilon_{t-j}
 \end{align*}
 Its expectation and variance are then equal to
 \begin{align*}
-E[Y_t] &= \mu + \sum_{j=1}^\infty \phi^j E[\varepsilon_{t-j}] = \mu\\
-Var[Y_t] &= \sum_{j=1}^\infty (\phi^j)^2 Var[\varepsilon_{t-j}] = \sum_{j=1}^\infty (\phi^2)^j \sigma_\varepsilon^2 = \frac{\sigma_\varepsilon^2}{1-\phi^2}
+E[Y_t] &= \mu + \sum_{j=1}^\infty \phi^j E[\varepsilon_{t-j}] = \mu
+\\
+Var[Y_t] &= \sum_{j=1}^\infty {(\phi^j)}^2 Var[\varepsilon_{t-j}] = \sum_{j=1}^\infty {(\phi^2)}^j \sigma_\varepsilon^2 = \frac{\sigma_\varepsilon^2}{1-\phi^2}
 \end{align*}
 where we use the white noise property of \(\varepsilon_t\).
 
-\item Let's derive the asymptotic distribution of the sample mean:
+\item
+Let's derive the asymptotic distribution of the sample mean:
+
 \begin{enumerate}
-    \item Due to the white noise assumption on \(\varepsilon_t\),
-      we can use the Lindeberg-Levy central limit theorem such that
-    \begin{equation*}
-    \sqrt{T} \left(\frac{1}{T} \sum_{t=1}^T \varepsilon_t \right) = \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t  \overset{d}{\rightarrow} U_\varepsilon \sim N(0,\sigma_\varepsilon^2)
-    \end{equation*}
-    \item Let's have a look at \(\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t\):
-    \begin{align*}
-    \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t
-    &= \frac{1}{\sqrt{T}} \sum_{t=1}^T \left[(Y_t-\mu)-\phi(Y_{t-1}-\mu)\right]\\
-    &= \frac{1}{\sqrt{T}} \left[\sum_{t=1}^T (Y_t-\mu)- \phi\sum_{t=1}^T(Y_{\color{red}{t-1}}-\mu)\right]\\
-    &= \frac{1}{\sqrt{T}} \left[\sum_{t=1}^T (Y_t-\mu)- \phi\left[\sum_{t=1}^T(Y_{\color{red}{t}}-\mu)-\color{red}{(Y_T - Y_0)}\right]\right]\\
-    &= \sqrt{T} \left[\frac{1}{T}\sum_{t=1}^T (Y_t-\mu)- \phi\left[\frac{1}{T}\sum_{t=1}^T(Y_{t}-\mu)-\left(\frac{Y_T - Y_0}{T}\right)\right]\right]\\
-    &= \sqrt{T} \left[\hat{\mu}-\mu- \phi\left[\hat{\mu}-\mu-\left(\frac{Y_T - Y_0}{T}\right)\right]\right]\\
-    &= \sqrt{T}\left[(1-\phi)\left(\hat{\mu}-\mu\right) + \phi\left(\frac{Y_T - Y_0}{T}\right)\right]
-    \end{align*}
-    \item Using the definition of the probability limit, i.e.
-    \begin{align*}
-    \textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0
-    \end{align*}
-    is shorthand notation for
-    \begin{align*}
-    \lim_{T\rightarrow \infty} Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) = 0
-    \end{align*}
-    We are going to show this by using Tchebychev's Inequality.
-    That is, according to the inequality we have:
-    \begin{align*}
-    Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) \leq \frac{1}{\delta^2} var\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right]
-    \end{align*}
-    for any \(\delta>0\).
-    Let's have a look at \(var\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right]\):
-    \begin{align*}
-    var\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right]
-    &= \frac{1}{T}\left(\frac{\phi}{1-\phi}\right)^2 Var[Y_T - Y_0]\\
-    &= \frac{1}{T}\left(\frac{\phi}{1-\phi}\right)^2 \left(Var[Y_T] + Var[Y_0] - 2 Cov[Y_T,Y_0]\right]\\
-    &= \frac{1}{T}\left(\frac{\phi}{1-\phi}\right)^2 \left[\frac{\sigma_\varepsilon^2}{1-\phi^2} + \frac{\sigma_\varepsilon^2}{1-\phi^2} - 2 Corr[Y_T,Y_0]\sqrt{\frac{\sigma_\varepsilon^2}{1-\phi^2}}\sqrt{\frac{\sigma_\varepsilon^2}{1-\phi^2}} \right]\\
-    &\leq \frac{1}{T}\left(\frac{\phi}{1-\phi}\right)^2 4 \left(\frac{\sigma_\varepsilon^2}{1-\phi^2}\right)
-    \end{align*}
-    since \(corr(Y_T,Y_0) \geq -1\).
-				
-    Thus for any \(\delta>0\), we have
-    \begin{align*}
-    Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) \leq \frac{1}{\delta^2} \frac{1}{T}\left(\frac{\phi}{1-\phi}\right)^2 4 \left(\frac{\sigma_\varepsilon^2}{1-\phi^2}\right)
-    \end{align*}
-    In the limit for \(T\rightarrow \infty\) the right hand side converges to 0; hence:
-    \begin{align*}
-    \lim_{T\rightarrow \infty} Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) = 0.
-    \end{align*}
-    or using our shorthand notation:
-    \begin{align*}
-      \textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0
-    \end{align*}
-    \item Now, let's go back to
-    \begin{align*}
-    \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t = \sqrt{T}\left[(1-\phi)\left(\hat{\mu}-\mu\right) + \phi\left(\frac{Y_T - Y_0}{T}\right)\right]
-    \end{align*}
-    Let's divide by \((1-\phi)\)
-    \begin{align*}
-    \frac{\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t}{1-\phi} & = \sqrt{T}\left(\hat{\mu}-\mu\right) + \frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)
-    \end{align*}
-				
-		For the left-hand-side we have
-		\begin{align*}
-		\frac{\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t}{1-\phi} \overset{d}{\rightarrow} \tilde{U_\varepsilon} \sim N\left(0,\frac{\sigma_\varepsilon^2}{(1-\phi)^2}\right)
-		\end{align*}
-			 
-		This is also the distribution of the right-hand side.
-    However, in the limit, the right-hand side actually simplifies
-      as we just derived that \(\textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0\).
-    Therefore:
-		\begin{align*}
-		\sqrt{T}\left(\hat{\mu}-\mu\right) \overset{d}{\rightarrow} \tilde{U} \sim N\left(0,\frac{\sigma_\varepsilon^2}{(1-\phi)^2}\right)
-		\end{align*}
-		and we're done.
-    We have just derived the distribution to which the sample mean \(\hat{\mu} = \overline{Y}_T\) of an AR(1) process converges to asymptotically.
-    Note that the required standardization is different than the Lindeberg-Levy central limit theorem would suggest.
-    The correct standardized variable to consider is:
-		\begin{align*}
-		Z_T = \sqrt{T}\frac{\hat{\mu}-\mu}{\sigma_Z} \overset{d}{\rightarrow} U \sim N(0,1)
-		\end{align*}
-    where we need to set \(\sigma_Z^2 = \frac{\sigma_\varepsilon^2}{(1-\phi)^2}\) to get convergence to the standard normal distribution.
-	\end{enumerate}
-\item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily]{progs/matlab/CentralLimitDependentData.m}
+  \item
+  Due to the white noise assumption on \(\varepsilon_t\),
+    we can use the Lindeberg-Levy central limit theorem such that
+  \begin{equation*}
+  \sqrt{T} \left(\frac{1}{T} \sum_{t=1}^T \varepsilon_t \right) = \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t  \overset{d}{\rightarrow} U_\varepsilon \sim N(0,\sigma_\varepsilon^2)
+  \end{equation*}
+  \item
+  Let's have a look at \(\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t\):
+  \begin{align*}
+  \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t
+  &= \frac{1}{\sqrt{T}} \sum_{t=1}^T \left[(Y_t-\mu)-\phi(Y_{t-1}-\mu)\right]
+  \\
+  &= \frac{1}{\sqrt{T}} \left[\sum_{t=1}^T (Y_t-\mu)- \phi\sum_{t=1}^T(Y_{\color{red}{t-1}}-\mu)\right]
+  \\
+  &= \frac{1}{\sqrt{T}} \left[\sum_{t=1}^T (Y_t-\mu)- \phi\left[\sum_{t=1}^T(Y_{\color{red}{t}}-\mu)-\color{red}{(Y_T - Y_0)}\right]\right]
+  \\
+  &= \sqrt{T} \left[\frac{1}{T}\sum_{t=1}^T (Y_t-\mu)- \phi\left[\frac{1}{T}\sum_{t=1}^T(Y_{t}-\mu)-\left(\frac{Y_T - Y_0}{T}\right)\right]\right]
+  \\
+  &= \sqrt{T} \left[\hat{\mu}-\mu- \phi\left[\hat{\mu}-\mu-\left(\frac{Y_T - Y_0}{T}\right)\right]\right]
+  \\
+  &= \sqrt{T}\left[(1-\phi)\left(\hat{\mu}-\mu\right) + \phi\left(\frac{Y_T - Y_0}{T}\right)\right]
+  \end{align*}
+
+  \item
+  Using the definition of the probability limit, i.e.\
+  \begin{align*}
+  \textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0
+  \end{align*}
+  is shorthand notation for
+  \begin{align*}
+  \lim_{T\rightarrow \infty} \Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) = 0
+  \end{align*}
+  We are going to show this by using Tchebychev's Inequality.
+  That is, according to the inequality we have:
+  \begin{align*}
+  \Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) \leq \frac{1}{\delta^2} var\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right]
+  \end{align*}
+  for any \(\delta>0\).
+  Let's have a look at \(var\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right]\):
+  \begin{multline*}
+  var\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right]
+  = \frac{1}{T}{\left(\frac{\phi}{1-\phi}\right)}^2 Var[Y_T - Y_0]
+  \\
+  = \frac{1}{T}{\left(\frac{\phi}{1-\phi}\right)}^2 \left(Var[Y_T] + Var[Y_0] - 2 Cov[Y_T,Y_0]\right)
+  \\
+  = \frac{1}{T}{\left(\frac{\phi}{1-\phi}\right)}^2 \left[\frac{\sigma_\varepsilon^2}{1-\phi^2} + \frac{\sigma_\varepsilon^2}{1-\phi^2} - 2 Corr[Y_T,Y_0]\sqrt{\frac{\sigma_\varepsilon^2}{1-\phi^2}}\sqrt{\frac{\sigma_\varepsilon^2}{1-\phi^2}} \right]
+  \\
+  \leq \frac{1}{T}{\left(\frac{\phi}{1-\phi}\right)}^2 4 \left(\frac{\sigma_\varepsilon^2}{1-\phi^2}\right)
+  \end{multline*}
+  since \(corr(Y_T,Y_0) \geq -1\).
+
+  Thus for any \(\delta>0\), we have
+  \begin{align*}
+  \Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) \leq \frac{1}{\delta^2} \frac{1}{T}{\left(\frac{\phi}{1-\phi}\right)}^2 4 \left(\frac{\sigma_\varepsilon^2}{1-\phi^2}\right)
+  \end{align*}
+  In the limit for \(T\rightarrow \infty \) the right hand side converges to 0; hence:
+  \begin{align*}
+  \lim_{T\rightarrow \infty} \Pr\left(\left|\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right|> \delta\right) = 0.
+  \end{align*}
+  or using our shorthand notation:
+  \begin{align*}
+  \textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0
+  \end{align*}
+  
+  \item
+  Now, let's go back to
+  \begin{align*}
+  \frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t = \sqrt{T}\left[(1-\phi)\left(\hat{\mu}-\mu\right) + \phi\left(\frac{Y_T - Y_0}{T}\right)\right]
+  \end{align*}
+  Let's divide by \((1-\phi)\)
+  \begin{align*}
+  \frac{\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t}{1-\phi} & = \sqrt{T}\left(\hat{\mu}-\mu\right) + \frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)
+  \end{align*}
+
+  For the left-hand-side we have
+  \begin{align*}
+  \frac{\frac{1}{\sqrt{T}} \sum_{t=1}^T \varepsilon_t}{1-\phi} \overset{d}{\rightarrow} \tilde{U_\varepsilon} \sim N\left(0,\frac{\sigma_\varepsilon^2}{{(1-\phi)}^2}\right)
+  \end{align*}
+
+  This is also the distribution of the right-hand side.
+  However, in the limit, the right-hand side actually simplifies
+    as we just derived that \(\textsl{plim}\left[\frac{\phi}{1-\phi}\left(\frac{Y_T - Y_0}{\sqrt{T}}\right)\right] = 0\).
+  Therefore:
+  \begin{align*}
+  \sqrt{T}\left(\hat{\mu}-\mu\right) \overset{d}{\rightarrow} \tilde{U} \sim N\left(0,\frac{\sigma_\varepsilon^2}{{(1-\phi)}^2}\right)
+  \end{align*}
+  and we're done.
+  We have just derived the distribution to which the sample mean \(\hat{\mu} = \overline{Y}_T\) of an AR{(1)} process converges to asymptotically.
+  Note that the required standardization is different than the Lindeberg-Levy central limit theorem would suggest.
+  The correct standardized variable to consider is:
+  \begin{align*}
+  Z_T = \sqrt{T}\frac{\hat{\mu}-\mu}{\sigma_Z} \overset{d}{\rightarrow} U \sim N(0,1)
+  \end{align*}
+  where we need to set \(\sigma_Z^2 = \frac{\sigma_\varepsilon^2}{{(1-\phi)}^2}\) to get convergence to the standard normal distribution.
+
+\end{enumerate}
+
+\item A possible script:
+\lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/CentralLimitDependentData.m}
 \end{enumerate}
\ No newline at end of file
diff --git a/exercises/law_of_large_numbers.tex b/exercises/law_of_large_numbers.tex
index d859009..1c0f0f5 100644
--- a/exercises/law_of_large_numbers.tex
+++ b/exercises/law_of_large_numbers.tex
@@ -1,29 +1,43 @@
 \section[Law Of Large Numbers]{Law Of Large Numbers\label{ex:LawOfLargeNumbers}}
 Let \(Y_{1},Y_{2},\ldots \) be an iid sequence of arbitrarily distributed random variables
-  with finite variance \(\sigma_Y ^{2}\) and expectation \(\mu\).
+  with finite variance \(\sigma_Y ^{2}\) and expectation \(\mu \).
 Define the sequence of random variables
 \begin{equation*}
 \overline{Y}_{T}=\frac{1}{T}\sum_{t=1}^{T}Y_{t}
 \end{equation*}
 \begin{enumerate}
-\item Briefly outline the intuition behind the \enquote{law of large numbers}.
+
+\item
+Briefly outline the intuition behind the \enquote{law of large numbers}.
 What are the differences between \enquote{almost-sure convergence}
   and \enquote{convergence in probability}?
+
 \item Write a program to illustrate the law of large numbers for uniformly distributed random variables
   (you may also try different distributions such as normal, gamma, geometric, student's t with finite or infinite variance).
 To this end, do the following:
 \begin{itemize}
-	\item Set \(T=10000\) and initialize the \(T \times 1\) output vector \(u\).
-	\item Choose values for the parameters of the uniform distribution.
-	Note that \(E[u] = (a+b)/2\), where \(a\) is the lower and \(b\) the upper bound of the uniform distribution.
-	\item For \(t=1,\ldots,T\) do the following computations:
-	\begin{itemize}
-		\item Draw \(t\) random variables from the uniform distribution with lower bound \(a\) and upper bound \(b\).
-		\item Compute and store the mean of the drawn values in your output vector at position \(t\).
-	\end{itemize}
-	\item Plot your output vector and add a line to indicate the theoretical mean of the uniform distribution.
+  \item
+  Set \(T=10000\) and initialize the \(T \times 1\) output vector \(u\).
+
+  \item
+  Choose values for the parameters of the uniform distribution.
+  Note that \(E[u] = (a+b)/2\), where \(a\) is the lower and \(b\) the upper bound of the uniform distribution.
+
+  \item
+  For \(t=1,\ldots,T\) do the following computations:
+  \begin{itemize}
+    \item
+    Draw \(t\) random variables from the uniform distribution with lower bound \(a\) and upper bound \(b\).
+    \item
+    Compute and store the mean of the drawn values in your output vector at position \(t\).
+  \end{itemize}
+
+  \item
+  Plot your output vector and add a line to indicate the theoretical mean of the uniform distribution.
 \end{itemize}
-\item Now suppose that the sequence \(Y_{1},Y_{2},\ldots \) is an \(AR(1)\) process:
+
+\item
+Now suppose that the sequence \(Y_{1},Y_{2},\ldots \) is an \(AR(1)\) process:
 \[Y_{t} =\phi Y_{t-1} +\varepsilon _{t}\]
 where \(\varepsilon _{t}\sim iid(0,\sigma _{\varepsilon }^{2})\) is not necessarily normally distributed and \(|\phi |<1\).
 Illustrate numerically that the law of large numbers still holds despite the intertemporal dependence.
@@ -31,14 +45,14 @@
 
 \paragraph{Readings}
 \begin{itemize}
-	\item \textcite[App. C]{Lutkepohl_2005_NewIntroductionMultiple}
-	\item \textcite[App. C]{Neusser_2016_TimeSeriesEconometrics}
-	\item \textcite{Ploberger_2010_LawLargeNumbers}
-	\item \textcite[Ch. 3]{White_2001_AsymptoticTheoryEconometricians}
+\item \textcite[App. C]{Lutkepohl_2005_NewIntroductionMultiple}
+\item \textcite[App. C]{Neusser_2016_TimeSeriesEconometrics}
+\item \textcite{Ploberger_2010_LawLargeNumbers}
+\item \textcite[Ch. 3]{White_2001_AsymptoticTheoryEconometricians}
 \end{itemize}
 
 \begin{solution}\textbf{Solution to \nameref{ex:LawOfLargeNumbers}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/law_of_large_numbers_solution.tex}
 \fi
 \newpage
diff --git a/exercises/law_of_large_numbers_solution.tex b/exercises/law_of_large_numbers_solution.tex
index 91f0c2c..3a87104 100644
--- a/exercises/law_of_large_numbers_solution.tex
+++ b/exercises/law_of_large_numbers_solution.tex
@@ -1,5 +1,7 @@
 \begin{enumerate}
-\item In probability theory, the law of large numbers (LLN) is a theorem
+
+\item
+In probability theory, the law of large numbers (LLN) is a theorem
   that describes the result of performing the same experiment a large number of times
   (repetitions, trials, experiments, iterations, sample size).
 According to the LLN, the average of the results obtained from a large number of times
@@ -8,7 +10,7 @@
 There are different laws of large numbers that differ in the underlying assumptions on the stochastic process.
 These laws are the cornerstones of asymptotic theory in statistics and econometrics.
 
-In this exercise, the LLN is about determining what happens to \(\overline{Y}_T\) as \(T\rightarrow\infty\)
+In this exercise, the LLN is about determining what happens to \(\overline{Y}_T\) as \(T\rightarrow\infty \)
   (note that \(\overline{Y}_T\) is a random variable).
 The LLN states that this series converges to the (unknown) expected value \(E[Y_t] = \mu \).
 More precisely, the strong LLN implies that at the limit, we can exactly determine \( \mu \),
@@ -17,55 +19,61 @@
 
 Econometrically speaking:
 \begin{itemize}
-    \item Strong LLN means \textbf{almost-sure convergence}:\\
-    At some point adding more observation does not matter at all for the average,
-      \(\overline{Y}_T\) will be exactly equal to the expected value \(\mu \).
-    That is, the sequence \(\overline{Y}_{1},\overline{Y}_{2},\ldots \) of random variables
-      converges \textbf{almost surely} to the variable \(\mu \), if
-    \begin{align*}
-    \Pr\left( \left \{ \lim_{T\rightarrow \infty }\overline{Y}_{T}=\mu\right \} \right) =1
-    \end{align*}
-      or simply:
-    \begin{align*}
-    \overline{Y}_{T}\overset{a.s.}{\rightarrow }\mu
-    \end{align*}
-    This definition of convergence is not very important in Quantitative Macroeconomics.
-				
-    \item Weak LLN means that the sample mean \(\overline{Y}_T\) converges in probability to the population mean \(\mu \).
-    That is, the sequence \(\overline{Y}_{1},\overline{Y}_{2},\ldots \) of random variables
-      converges \textbf{in probability} to the variable \(\mu \), if
-    \begin{align*}
-    \lim_{T\rightarrow \infty }\Pr\left( |Y_{T}-\mu|<\delta \right) =1
-    \end{align*}
-    As \(T\rightarrow \infty \), the probability is approaching 1 very closely,
-      but typically it will not be exactly equal to 1.
-    In other words, the probability that the average is \enquote{far away}
-      from the expectation \(\mu \) is zero,
-      where we measure closeness by an arbitrary small number \(\delta>0\).
-    More compact notation:
-    \begin{eqnarray*}
-    \overline{Y}_{T} &\overset{p}{\rightarrow }&\mu \\
-    \textsl{plim}~\overline{Y}_{T} &=&\mu
-    \end{eqnarray*}
-    This definition of convergence is very important in Quantitative Macroeconomics.
 
-    In Quantitative Macroeconomics, we are mainly concerned with identically distributed processes
-      that are either independent of each other (like the white noise process)
-      or that are homogenously dependent (like the VAR(1) process).
-    Given assumptions on existence and boundedness of the unconditional moments of these processes,
-      the weak LLN typically applies.
+\item
+Strong LLN means \textbf{almost-sure convergence}:\\
+At some point adding more observation does not matter at all for the average,
+  \(\overline{Y}_T\) will be exactly equal to the expected value \(\mu \).
+That is, the sequence \(\overline{Y}_{1},\overline{Y}_{2},\ldots \) of random variables
+  converges \textbf{almost surely} to the variable \(\mu \), if
+\begin{align*}
+\Pr\left( \left \{ \lim_{T\rightarrow \infty }\overline{Y}_{T}=\mu\right \} \right) =1
+\end{align*}
+or simply:
+\begin{align*}
+\overline{Y}_{T}\overset{a.s.}{\rightarrow }\mu
+\end{align*}
+This definition of convergence is not very important in Quantitative Macroeconomics.
+
+\item
+Weak LLN means that the sample mean \(\overline{Y}_T\) converges in probability to the population mean \(\mu \).
+That is, the sequence \(\overline{Y}_{1},\overline{Y}_{2},\ldots \) of random variables
+  converges \textbf{in probability} to the variable \(\mu \), if
+\begin{align*}
+\lim_{T\rightarrow \infty }\Pr\left( |Y_{T}-\mu|<\delta \right) =1
+\end{align*}
+As \(T\rightarrow \infty \), the probability is approaching 1 very closely,
+  but typically it will not be exactly equal to 1.
+In other words, the probability that the average is \enquote{far away}
+  from the expectation \(\mu \) is zero,
+  where we measure closeness by an arbitrary small number \(\delta>0\).
+More compact notation:
+\begin{eqnarray*}
+\overline{Y}_{T} &\overset{p}{\rightarrow }&\mu
+\\
+\textsl{plim}~\overline{Y}_{T} &=&\mu
+\end{eqnarray*}
+This definition of convergence is very important in Quantitative Macroeconomics.
+
+In Quantitative Macroeconomics, we are mainly concerned with identically distributed processes
+  that are either independent of each other (like the white noise process)
+  or that are homogenously dependent (like the VAR{(1)} process).
+Given assumptions on existence and boundedness of the unconditional moments of these processes,
+  the weak LLN typically applies.
 \end{itemize}
 
-\item Here is an extended illustration for several distributions: 
+\item
+Here is an extended illustration for several distributions: 
 \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/lawOfLargeNumbers.m}
 Note that for the \(t\)-distribution with infinite variance the weak LLN actually does not apply.
 
-\item Here is an extended illustration for different error term distributions: 
+\item
+Here is an extended illustration for different error term distributions: 
 \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/lawOfLargeNumbersAR1.m} 
 Note that we need to make sure that \(E[\varepsilon_t]=0\) when we simulate data.
 We see that the weak law of large numbers holds under weaker conditions than iid.
-For instance, one can show that for the stationary AR(1),
+For instance, one can show that for the stationary AR{(1)},
   necessary and sufficient conditions are:
-  \(Var[y_t]<\infty\) and \(|\gamma(k)| \rightarrow 0\) as \(k \rightarrow \infty \).
+  \(Var[y_t]<\infty \) and \(|\gamma(k)| \rightarrow 0\) as \(k \rightarrow \infty \).
 This does not hold for all considered distributions in the code.
 \end{enumerate}
\ No newline at end of file
diff --git a/exercises/properties_ar1.tex b/exercises/properties_ar1.tex
index a694e70..b9978e1 100644
--- a/exercises/properties_ar1.tex
+++ b/exercises/properties_ar1.tex
@@ -1,38 +1,53 @@
-\section[Properties of AR(1)]{Properties Of AR(1)\label{ex:PropertiesAR1}}
+\section[Properties of AR{(1)}]{Properties Of AR{(1)}\label{ex:PropertiesAR1}}
 
 Let \( \{\varepsilon_t\} \) be a white noise process with variance \(\sigma_\varepsilon^2\).
 
 \begin{enumerate}
-\item Consider the univariate first-order autoregressive process AR(1): 
+\item Consider the univariate first-order autoregressive process AR{(1)}: 
 \[y_t = c + \phi y_{t-1} + \varepsilon_{t}\]
 where \(|\phi|<1\) and \(c\in \mathbb{R}\).
 Derive the unconditional first and second moments.
 
 \emph{Hint:}
 \begin{itemize}
-	\item If \(|\phi|<1\), then \(\sum_{j=0}^{\infty} \phi^j = \frac{1}{(1-\phi)}\)
+\item If \(|\phi|<1\), then \(\sum_{j=0}^{\infty} \phi^j = \frac{1}{(1-\phi)}\)
 \end{itemize}
 
 \item Do the following simulation exercise:
 \begin{itemize}
-	\item Simulate AR(1) processes for \(t=1,\ldots,200\) with \(c=2\), \(\sigma_\varepsilon^2 = 1^2\) and different values of \( \phi=\{-0.8;0.4;0.9;1.01\} \).
-	\item Plot the corresponding autocorrelation functions (ACF) for data vector $y$ and maximum number of lags \(p^{\text{max}}=8\) using MATLAB's \texttt{autocorr(y,'NumLags',8)} function.
-	\item Write a function \texttt{acfPlots(\(y\),\(p^{\text{max}}\),\( \alpha \))} that plots the autocorrelation function (ACF) of the data vector \(y\) with maximum number of lags \(p^{\text{max}}\).
-	Your plot should also include an approximate (1-\( \alpha \))\% confidence interval around zero.
-	Your plots should look similar to MATLAB's \texttt{autocorr} command.
+\item
+Simulate AR{(1)} processes for \(t=1,\ldots,200\) with \(c=2\),
+  \(\sigma_\varepsilon^2 = 1^2\) and different values of \( \phi=\{-0.8;0.4;0.9;1.01\} \).
+
+\item Plot the corresponding autocorrelation functions (ACF) for data vector \(y\)
+  and maximum number of lags \(p^{\text{max}}=8\) using MATLAB's \texttt{autocorr{(y,{'NumLags'},8)}} function.
+
+\item Write a function \texttt{acfPlots{(\(y\),\(p^{\text{max}}\),\( \alpha \))}}
+  that plots the autocorrelation function (ACF) of the data vector \(y\)
+  with maximum number of lags \(p^{\text{max}}\).
+Your plot should also include an approximate (1-\( \alpha \))\% confidence interval around zero.
+Your plots should look similar to MATLAB's \texttt{autocorr} command.
 \end{itemize}
 	
 \emph{Hints}:
 \begin{itemize}
-	\item The empirical autocorrelation function at lag \(k\) is defined as \(\hat{\rho}_k = \hat{\gamma}_k/\hat{\gamma}_0\) where
-	  \[\hat{\gamma}_k = \frac{1}{T} \sum_{t=k+1}^T (y_t - \bar{y})(y_{t-k}-\bar{y})\] and \[\hat{\gamma}_0 = \frac{1}{T} \sum_{t=1}^T (y_t - \bar{y})(y_{t}-\bar{y})\]
-	\item You can either use a for-loop to compute the sum or use vectors: \((y - \bar{y})'(y - \bar{y})\).
-	\item The sample autocorrelation function is an estimate of the actual autocorrelation only if the process is stationary.
-	If the process is purely random, that is, all members are mutually independent and identically distributed,
-	  then \(y_t\) and \(y_{t-k}\) should be stochastically independent for any \(k\neq 0\).
-	According to the Lindeberg-Levy central limit theorem (see corresponding exercise),
-	  the normalized estimated autocorrelations are asymptotically standard normally distributed,
-	  i.e. \(\sqrt{T} \hat{\rho}_k \rightarrow U \sim N(0,1)\) and thus \(\hat{\rho}_k \rightarrow \tilde{U} \sim N(0,1/T)\).
+\item
+The empirical autocorrelation function at lag \(k\) is defined as
+  \(\hat{\rho}_k = \hat{\gamma}_k/\hat{\gamma}_0\) where
+  \[\hat{\gamma}_k = \frac{1}{T} \sum_{t=k+1}^T (y_t - \bar{y})(y_{t-k}-\bar{y})\]
+  and \[\hat{\gamma}_0 = \frac{1}{T} \sum_{t=1}^T (y_t - \bar{y})(y_{t}-\bar{y})\]
+
+\item
+You can either use a for-loop to compute the sum or use vectors: \((y - \bar{y})'(y - \bar{y})\).
+
+\item
+The sample autocorrelation function is an estimate of the actual autocorrelation only if the process is stationary.
+If the process is purely random, that is, all members are mutually independent and identically distributed,
+  then \(y_t\) and \(y_{t-k}\) should be stochastically independent for any \(k\neq 0\).
+According to the Lindeberg-Levy central limit theorem (see corresponding exercise),
+  the normalized estimated autocorrelations are asymptotically standard normally distributed, i.e.\
+  \(\sqrt{T} \hat{\rho}_k \rightarrow U \sim N(0,1)\)
+  and thus \(\hat{\rho}_k \rightarrow \tilde{U} \sim N(0,1/T)\).
 \end{itemize}
 
 \end{enumerate}
@@ -40,12 +55,12 @@
 \paragraph{Readings}
 
 \begin{itemize}
-	\item \textcite[Ch.2]{Bjornland.Thorsrud_2015_AppliedTimeSeries}
-	\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
+\item \textcite[Ch.2]{Bjornland.Thorsrud_2015_AppliedTimeSeries}
+\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
 \end{itemize}
 
 \begin{solution}\textbf{Solution to \nameref{ex:PropertiesAR1}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/properties_ar1_solution.tex}
 \fi
 \newpage
diff --git a/exercises/properties_ar1_solution.tex b/exercises/properties_ar1_solution.tex
index b68d620..91150a4 100644
--- a/exercises/properties_ar1_solution.tex
+++ b/exercises/properties_ar1_solution.tex
@@ -1,6 +1,8 @@
 \begin{enumerate}
 
-\item \textbf{Geometric sequence:} First, note that we can use the geometric sequence with or without the lag operator, i.e.
+\item
+\textbf{Geometric sequence:}
+First, note that we can use the geometric sequence with or without the lag operator, i.e.\
 \begin{align*}
 {(1-\phi)}^{-1} &= \lim\limits_{j\rightarrow \infty}(\phi^0 + \phi^1 + \phi^2 + \cdots + \phi^j) = \sum_{j=0}^\infty \phi^j
 \\
@@ -10,7 +12,7 @@
 \begin{align*}
 S_{k} =	\sum_{j=0}^{k} \phi^j =	1 + \phi^1 + \phi^2 + \phi^3 + \cdots + \phi^k
 \end{align*}
-then multiply with \(\phi\):
+then multiply with \(\phi \):
 \begin{align*}
 \phi S_k = \phi^1 + \phi^2 + \phi^3 + \cdots + \phi^{k+1}
 \end{align*}
@@ -26,115 +28,132 @@
 Next let's get a representation of the process \(y_t\) that is useful to compute the moments.
 
 We can do this in different ways:
-    \begin{itemize}
+\begin{itemize}
 
-    \item \textbf{Recursive substitution} (starting at some infinite time \(j\)):
-    \begin{align*}
-    y_t &= c + \phi y_{t-1} + \varepsilon_t\\
-        &= c + \phi \left( c + \phi y_{t-2} + \varepsilon_{t-1}\right) + \varepsilon_t\\ 
-        &= c + \phi c + \varepsilon_t + \phi \varepsilon_{t-1} + \phi^2( c+ \phi y_{t-3} + \varepsilon_{t-2} )\\
-        &\vdots\\
-        & = c + \phi c + \phi^2 c^2 + \cdots + \phi^j c^j 
-        + \varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2} + \cdots + \phi^j \varepsilon_{t-j}
-        + \phi^{j+1} y_{t-{j+1}}
-    \end{align*}
-    \(y_t\) is a linear function of an initial value \(\phi^{j+1} y_{t-{j+1}}\),
-      historical values of the white noise process \(\varepsilon_t\), and a sum of polynomials in \(c\).
-    If \(|\phi|<1\) and \(j\) becomes large,
-      then \( \phi^{j+1} y_{t-{j+1}} \rightarrow 0\),
-      thus we get a so-called \(MA(\infty)\) process:
-    \begin{align*}
-    y_t &= \underbrace{c + \phi c + \phi^2 c^2 + \ldots}_{\sum_{j=0}^\infty \phi^j c^j} + \underbrace{\varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2} + \ldots}_{\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}}
-    \\
-    &= c\sum_{j=0}^\infty \phi^j + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}    
-    = \frac{c}{1-\phi} + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
-    \end{align*}
-    
+\item
+\textbf{Recursive substitution} (starting at some infinite time \(j\)):
+\begin{align*}
+y_t &= c + \phi y_{t-1} + \varepsilon_t
+\\
+&= c + \phi \left( c + \phi y_{t-2} + \varepsilon_{t-1}\right) + \varepsilon_t
+\\ 
+&= c + \phi c + \varepsilon_t + \phi \varepsilon_{t-1} + \phi^2( c+ \phi y_{t-3} + \varepsilon_{t-2} )
+\\
+&\vdots
+\\
+& = c + \phi c + \phi^2 c^2 + \cdots + \phi^j c^j 
+  + \varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2} + \cdots + \phi^j \varepsilon_{t-j}
+  + \phi^{j+1} y_{t-{j+1}}
+\end{align*}
+\(y_t\) is a linear function of an initial value \(\phi^{j+1} y_{t-{j+1}}\),
+historical values of the white noise process \(\varepsilon_t\), and a sum of polynomials in \(c\).
+If \(|\phi|<1\) and \(j\) becomes large,
+  then \( \phi^{j+1} y_{t-{j+1}} \rightarrow 0\),
+  thus we get a so-called \(MA(\infty)\) process:
+\begin{align*}
+y_t &= \underbrace{c + \phi c + \phi^2 c^2 + \ldots}_{\sum_{j=0}^\infty \phi^j c^j} + \underbrace{\varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2} + \ldots}_{\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}}
+\\
+&= c\sum_{j=0}^\infty \phi^j + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
+= \frac{c}{1-\phi} + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
+\end{align*}
+
+\item
+\textbf{With Lag Operators}:
+works only if \(|\phi| < 1\) and \( \{y_t\} \) is bounded;
+that is, there exists a finite number \(k\) such that \(|y_t| < k\) for all \(t\).
+Then
+\begin{align*}
+(1-\phi L) y_t = c + \varepsilon_t
+\\
+{(1-\phi L)}^{-1}{(1-\phi L)} y_t =	y_t = {(1-\phi L)}^{-1} c + {(1-\phi L)}^{-1}\varepsilon_t
+\\
+\end{align*}
+Using the geometric series, we get:
+\begin{align*}
+y_t &= (1+\phi L + \phi^2 L^2+\cdots+{(\phi L)}^j) c
+  + (1+\phi L + \phi^2 L^2+\cdots+{(\phi L)}^j)\varepsilon_t
+\\
+&= \left(c + \phi c + \phi^2 c + \ldots \right) + \left(\varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2}+\ldots\right)
+= c \sum_{j=0}^\infty \phi^j + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
+\\
+&= \frac{c}{1-\phi} + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
+\end{align*}
+\end{itemize}
+If we can express an AR process as a MA process, we call this process invertible.
+Let's now compute the moments from the MA{(\(\infty \))} representation
+  and using the fact that \(\varepsilon_t\) is a white noise process:
+\begin{itemize}
+\item
+\textbf{Unconditional Mean:}
+\begin{align*}
+E[y_t] &= E\left[\frac{c}{1-\phi}\right] + E\left[\sum_{j=1}^{\infty} \phi^j \varepsilon_{t-j}\right] = \frac{c}{1-\phi} \sum_{j=1}^{\infty} \phi^j E[\varepsilon_{t-j}]
+\\
+&= \underbrace{\frac{c}{1-\phi}}_{:=\mu}
+\end{align*}
+As this process is covariance-stationary, the unconditional mean is time invariant.
+We typically denote this time-independence by using the Greek letter \(\mu \).
+
+\item \textbf{Unconditional variance:}
+\begin{align*}
+Var[y_t] &= E\left[(y_t -E[y_t])(y_t - E[y_t])\right] = E\left[\left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\right) \left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\right) \right]
+\\
+&= E\left[ \phi^0 \phi^0 \varepsilon_{t} \varepsilon_{t} + \phi^0 \phi^1 \varepsilon_{t} \varepsilon_{t-1} + \phi^0 \phi^2 \varepsilon_{t} \varepsilon_{t-2} + \phi^0 \phi^3 \varepsilon_{t} \varepsilon_{t-3} + \ldots \right.
+\\
+& \qquad~\phi^1 \phi^0 \varepsilon_{t-1} \varepsilon_{t} + \phi^1 \phi^1 \varepsilon_{t-1} \varepsilon_{t-1} + \phi^1 \phi^2 \varepsilon_{t-1} \varepsilon_{t-2} + \phi^1 \phi^3 \varepsilon_{t-1} \varepsilon_{t-3} + \ldots
+\\
+& \qquad~\phi^2 \phi^0 \varepsilon_{t-2} \varepsilon_{t} + \phi^2 \phi^1 \varepsilon_{t-2} \varepsilon_{t-1} + \phi^2 \phi^2 \varepsilon_{t-2} \varepsilon_{t-2} + \phi^2 \phi^3 \varepsilon_{t-2} \varepsilon_{t-3} + \ldots
+\\
+&\left.\qquad~\ldots\right]
+\\
+&= \phi^0 \phi^0 E[\varepsilon_{t} \varepsilon_{t}] + \phi^0 \phi^1 E[\varepsilon_{t} \varepsilon_{t-1}] + \phi^0 \phi^2 E[\varepsilon_{t} \varepsilon_{t-2}] + \phi^0 \phi^3 E[\varepsilon_{t} \varepsilon_{t-3}] + \ldots
+\\
+& \qquad~\phi^1 \phi^0 E[\varepsilon_{t-1} \varepsilon_{t}] + \phi^1 \phi^1 E[\varepsilon_{t-1} \varepsilon_{t-1}] + \phi^1 \phi^2 E[\varepsilon_{t-1} \varepsilon_{t-2}] + \phi^1 \phi^3 E[\varepsilon_{t-1} \varepsilon_{t-3}] + \ldots
+\\
+& \qquad~\phi^2 \phi^0 E[\varepsilon_{t-2} \varepsilon_{t}] + \phi^2 \phi^1 E[\varepsilon_{t-2} \varepsilon_{t-1}] + \phi^2 \phi^2 E[\varepsilon_{t-2} \varepsilon_{t-2}] + \phi^2 \phi^3 E[\varepsilon_{t-2} \varepsilon_{t-3}] + \ldots
+\\
+&\qquad~\ldots
+\end{align*}
+Note that \(\varepsilon_t \) is a white-noise process with variance \(E[\varepsilon_{t-j} \varepsilon_{t-j}]=\sigma_\varepsilon^2\) for any \(j\),
+  but zero autocovariance, i.e. \(E[\varepsilon_{t-j} \varepsilon_{t-k}]=0\) for any \(j \neq k\).
+Therefore:
+\begin{align*}
+Var[y_t] &= \phi^0 \phi^0 E[\varepsilon_{t} \varepsilon_{t}] + \phi^1 \phi^1 E[\varepsilon_{t-1} \varepsilon_{t-1}] + \phi^2 \phi^2 E[\varepsilon_{t-2} \varepsilon_{t-2}] + \phi^3 \phi^3 E[\varepsilon_{t-3} \varepsilon_{t-3}] + \ldots
+\\
+&= \sum_{j=0}^\infty {(\phi^{2})}^j E[\varepsilon_{t-j} \varepsilon_{t-j}]
+= \sum_{j=0}^\infty {(\phi^{2})}^j \sigma_\varepsilon^2
+= \sigma_\varepsilon^2 \sum_{j=0}^\infty {(\phi^{2})}^j
+\\
+=& \underbrace{\sigma_\varepsilon^2 \frac{1}{1-\phi^2}}_{:= \gamma_0}
+\end{align*}
+As the process is covariance-stationary, the unconditional variance is time invariant.
+We typically denote this time-independence by using the Greek letter \(\gamma_0\).
 
-    \item \textbf{With Lag Operators}: works only if \(|\phi| < 1\) and \(\{y_t\}\) is bounded (that is, there exists a finite number \(k\) such that \(|y_t| < k\) for all \(t\)).
-    Then
-    \begin{align*}
-    (1-\phi L) y_t = c + \varepsilon_t\\
-    (1-\phi L)^{-1}(1-\phi L) y_t =	y_t = (1-\phi L)^{-1} c + (1-\phi L)^{-1}\varepsilon_t\\
-    \end{align*}
-    Using the geometric series, we get:
-    \begin{align*}
-    y_t &= (1+\phi L + \phi^2 L^2+\cdots+(\phi L)^j) c
-        + (1+\phi L + \phi^2 L^2+\cdots+(\phi L)^j)\varepsilon_t
-\\
-        &= \left(c + \phi c + \phi^2 c + \ldots \right) + \left(\varepsilon_t + \phi \varepsilon_{t-1} + \phi^2 \varepsilon_{t-2}+\ldots\right)
-        = c \sum_{j=0}^\infty \phi^j + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
-        \\
-        &= \frac{c}{1-\phi} + \sum_{j=0}^\infty \phi^j \varepsilon_{t-j}
-    \end{align*}
-    \end{itemize}
-    If we can express an AR process as a MA process, we call this process invertible.
-    Let's now compute the moments from the MA(\(\infty\)) representation
-      and using the fact that \(\varepsilon_t\) is a white noise process:
-    \begin{itemize}
-      \item \textbf{Unconditional Mean:}
-      \begin{align*}
-      E[y_t] &= E\left[\frac{c}{1-\phi}\right] + E\left[\sum_{j=1}^{\infty} \phi^j \varepsilon_{t-j}\right] = \frac{c}{1-\phi} \sum_{j=1}^{\infty} \phi^j E[\varepsilon_{t-j}]
-      \\
-      &= \underbrace{\frac{c}{1-\phi}}_{:=\mu}
-      \end{align*}
-      As this process is covariance-stationary, the unconditional mean is time invariant.
-      We typically denote this time-independence by using the greek letter $\mu$.
-      \item \textbf{Unconditional variance:}
-      \begin{align*}
-      Var[y_t] &= E\left[(y_t -E[y_t])(y_t - E[y_t])\right] = E\left[\left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\right) \left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\right) \right]
-      \\
-      &= E\left[ \phi^0 \phi^0 \varepsilon_{t} \varepsilon_{t} + \phi^0 \phi^1 \varepsilon_{t} \varepsilon_{t-1} + \phi^0 \phi^2 \varepsilon_{t} \varepsilon_{t-2} + \phi^0 \phi^3 \varepsilon_{t} \varepsilon_{t-3} + \ldots \right.
-      \\
-      & \qquad~\phi^1 \phi^0 \varepsilon_{t-1} \varepsilon_{t} + \phi^1 \phi^1 \varepsilon_{t-1} \varepsilon_{t-1} + \phi^1 \phi^2 \varepsilon_{t-1} \varepsilon_{t-2} + \phi^1 \phi^3 \varepsilon_{t-1} \varepsilon_{t-3} + \ldots
-      \\
-      & \qquad~\phi^2 \phi^0 \varepsilon_{t-2} \varepsilon_{t} + \phi^2 \phi^1 \varepsilon_{t-2} \varepsilon_{t-1} + \phi^2 \phi^2 \varepsilon_{t-2} \varepsilon_{t-2} + \phi^2 \phi^3 \varepsilon_{t-2} \varepsilon_{t-3} + \ldots
-      \\&\left.\qquad~\ldots\right]
-      \\
-      &= \phi^0 \phi^0 E[\varepsilon_{t} \varepsilon_{t}] + \phi^0 \phi^1 E[\varepsilon_{t} \varepsilon_{t-1}] + \phi^0 \phi^2 E[\varepsilon_{t} \varepsilon_{t-2}] + \phi^0 \phi^3 E[\varepsilon_{t} \varepsilon_{t-3}] + \ldots
-      \\
-      & \qquad~\phi^1 \phi^0 E[\varepsilon_{t-1} \varepsilon_{t}] + \phi^1 \phi^1 E[\varepsilon_{t-1} \varepsilon_{t-1}] + \phi^1 \phi^2 E[\varepsilon_{t-1} \varepsilon_{t-2}] + \phi^1 \phi^3 E[\varepsilon_{t-1} \varepsilon_{t-3}] + \ldots
-      \\
-      & \qquad~\phi^2 \phi^0 E[\varepsilon_{t-2} \varepsilon_{t}] + \phi^2 \phi^1 E[\varepsilon_{t-2} \varepsilon_{t-1}] + \phi^2 \phi^2 E[\varepsilon_{t-2} \varepsilon_{t-2}] + \phi^2 \phi^3 E[\varepsilon_{t-2} \varepsilon_{t-3}] + \ldots
-      \\&\qquad~\ldots
-      \end{align*}
-      Note that \(\varepsilon_t\) is a white-noise process with variance \(E[\varepsilon_{t-j} \varepsilon_{t-j}]=\sigma_\varepsilon^2\) for any \(j\),
-      but zero autocovariance, i.e. \(E[\varepsilon_{t-j} \varepsilon_{t-k}]=0\) for any \(j \neq k\).
-      Therefore:
-      \begin{align*}
-      Var[y_t] &= \phi^0 \phi^0 E[\varepsilon_{t} \varepsilon_{t}] + \phi^1 \phi^1 E[\varepsilon_{t-1} \varepsilon_{t-1}] + \phi^2 \phi^2 E[\varepsilon_{t-2} \varepsilon_{t-2}] + \phi^3 \phi^3 E[\varepsilon_{t-3} \varepsilon_{t-3}] + \ldots
-      \\
-      &= \sum_{j=0}^\infty (\phi^{2})^j E[\varepsilon_{t-j} \varepsilon_{t-j}]
-      = \sum_{j=0}^\infty (\phi^{2})^j \sigma_\varepsilon^2
-      = \sigma_\varepsilon^2 \sum_{j=0}^\infty (\phi^{2})^j
-      \\
-      =& \underbrace{\sigma_\varepsilon^2 \frac{1}{1-\phi^2}}_{:= \gamma_0}
-      \end{align*}
-      As the process is covariance-stationary, the unconditional variance is time invariant.
-      We typically denote this time-independence by using the Greek letter \(\gamma_0\).
-      \item \textbf{Unconditional autocovariance:}
-      \begin{align*}
-      Var[y_t,y_{t-k}] &= E\left[(y_t -E[y_t])(y_t - E[y_{t-k}])\right] = E\left[\left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\right) \left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-k-j}\right) \right]
-      \\
-      &= E\left[ \phi^0 \phi^0 \varepsilon_{t} \varepsilon_{t-k} + \phi^0 \phi^1 \varepsilon_{t} \varepsilon_{t-k-1} + \phi^0 \phi^2 \varepsilon_{t} \varepsilon_{t-k-2} + \phi^0 \phi^3 \varepsilon_{t} \varepsilon_{t-k-3} + \ldots \right.
-      \\
-      & \qquad~\phi^1 \phi^0 \varepsilon_{t-1} \varepsilon_{t-k} + \phi^1 \phi^1 \varepsilon_{t-1} \varepsilon_{t-k-1} + \phi^1 \phi^2 \varepsilon_{t-1} \varepsilon_{t-k-2} + \phi^1 \phi^3 \varepsilon_{t-1} \varepsilon_{t-k-3} + \ldots
-      \\
-      & \qquad~\phi^2 \phi^0 \varepsilon_{t-2} \varepsilon_{t-k} + \phi^2 \phi^1 \varepsilon_{t-2} \varepsilon_{t-k-1} + \phi^2 \phi^2 \varepsilon_{t-2} \varepsilon_{t-k-2} + \phi^2 \phi^3 \varepsilon_{t-2} \varepsilon_{t-k-3} + \ldots
-      \\&\left.\qquad~\ldots\right]      
-      \end{align*}
-      This can be simplified due to the white noise property of \(\varepsilon_t\) to:
-      \begin{align*}
-      Var[y_t,y_{t-k}] &= \phi^k (\phi^0 E[\varepsilon_{t-k} \varepsilon_{t-k}] + \phi^2 E[\varepsilon_{t-k-1} \varepsilon_{t-k-1}]+ \phi^4 E[\varepsilon_{t-k-2} \varepsilon_{t-k-2}] + \ldots )
-      \\
-      &= \phi^k \sum_{j=0}^\infty (\phi^{2})^j \sigma_\varepsilon^2 = \phi^k \frac{\sigma_\varepsilon^2}{1-\phi^2}      
-      = \underbrace{\phi^k \gamma_0}_{:=\gamma_k}
-      \end{align*}
-      As the process is covariance-stationary, the unconditional autocovariance is only dependent on the time difference \(k\).
-      We typically denote this by using the Greek letter \(\gamma_k\).
+\item \textbf{Unconditional autocovariance:}
+\begin{multline*}
+Var[y_t,y_{t-k}] = E\left[(y_t -E[y_t])(y_t - E[y_{t-k}])\right] = E\left[\left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-j}\right) \left(\sum_{j=0}^\infty \phi^j \varepsilon_{t-k-j}\right) \right]
+\\
+= E\left[ \phi^0 \phi^0 \varepsilon_{t} \varepsilon_{t-k} + \phi^0 \phi^1 \varepsilon_{t} \varepsilon_{t-k-1} + \phi^0 \phi^2 \varepsilon_{t} \varepsilon_{t-k-2} + \phi^0 \phi^3 \varepsilon_{t} \varepsilon_{t-k-3} + \ldots \right.
+\\
+\phi^1 \phi^0 \varepsilon_{t-1} \varepsilon_{t-k} + \phi^1 \phi^1 \varepsilon_{t-1} \varepsilon_{t-k-1} + \phi^1 \phi^2 \varepsilon_{t-1} \varepsilon_{t-k-2} + \phi^1 \phi^3 \varepsilon_{t-1} \varepsilon_{t-k-3} + \ldots
+\\
+\phi^2 \phi^0 \varepsilon_{t-2} \varepsilon_{t-k} + \phi^2 \phi^1 \varepsilon_{t-2} \varepsilon_{t-k-1} + \phi^2 \phi^2 \varepsilon_{t-2} \varepsilon_{t-k-2} + \phi^2 \phi^3 \varepsilon_{t-2} \varepsilon_{t-k-3} + \ldots
+\\
+\left.\qquad~\ldots\right]
+\end{multline*}
+This can be simplified due to the white noise property of \(\varepsilon_t\) to:
+\begin{align*}
+Var[y_t,y_{t-k}] &= \phi^k (\phi^0 E[\varepsilon_{t-k} \varepsilon_{t-k}] + \phi^2 E[\varepsilon_{t-k-1} \varepsilon_{t-k-1}]+ \phi^4 E[\varepsilon_{t-k-2} \varepsilon_{t-k-2}] + \ldots )
+\\
+&= \phi^k \sum_{j=0}^\infty {(\phi^{2})}^j \sigma_\varepsilon^2 = \phi^k \frac{\sigma_\varepsilon^2}{1-\phi^2}
+= \underbrace{\phi^k \gamma_0}_{:=\gamma_k}
+\end{align*}
+As the process is covariance-stationary,
+  the unconditional autocovariance is only dependent on the time difference \(k\).
+We typically denote this by using the Greek letter \(\gamma_k\).
 \end{itemize}
-	 	
-\item Here is a possible run-script:
+
+\item
+Here is a possible run-script:
 \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/acfPlots_run.m}
 and the corresponding \texttt{acfPlots.m} function:
 \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/acfPlots.m}
diff --git a/exercises/properties_ar1_time_trend.tex b/exercises/properties_ar1_time_trend.tex
index 5bd768a..70fc10e 100644
--- a/exercises/properties_ar1_time_trend.tex
+++ b/exercises/properties_ar1_time_trend.tex
@@ -1,25 +1,31 @@
-\section[Properties AR(1) With Time Trend]{Properties AR(1) With Time Trend\label{ex:PropertiesAR1TimeTrend}}
+\section[Properties AR{(1)} With Time Trend]{Properties AR{(1)} With Time Trend\label{ex:PropertiesAR1TimeTrend}}
 
-Consider the univariate AR(1) model with a constant and time trend
+Consider the univariate AR{(1)} model with a constant and time trend
 \[ y_t = c + d\cdot t + \phi y_{t-1} + u_t\]
 where \(u_t \sim WN(0,\sigma_u^2)\), \(|\phi|<1\), \(c \in \mathbb{R}\) and \(d \in \mathbb{R}\).
 \begin{enumerate}
-	\item Compute the unconditional first and second moments, i.e.\ the unconditional mean, variance, autocovariance and autocorrelation function of \(y_t\).
-	\item Why is this process not covariance-stationary? How could one proceed to make it covariance-stationary?
+\item
+Compute the unconditional first and second moments,
+  i.e.\ the unconditional mean, variance, autocovariance and autocorrelation function of \(y_t\).
+
+\item
+Why is this process not covariance-stationary?
+How could one proceed to make it covariance-stationary?
 \end{enumerate}
+
 \textbf{Hints:}
 \begin{itemize}
-	\item If \(|\phi|<1\), then \(\sum_{j=0}^{\infty} \phi^j = \frac{1}{(1-\phi)}\)
-	\item If \(|\phi|<1\), then \(\sum_{j=0}^{\infty} j \phi^j = \frac{\phi}{(1-\phi^2)}\)
+\item If \(|\phi|<1\), then \(\sum_{j=0}^{\infty} \phi^j = \frac{1}{(1-\phi)}\)
+\item If \(|\phi|<1\), then \(\sum_{j=0}^{\infty} j \phi^j = \frac{\phi}{(1-\phi^2)}\)
 \end{itemize}
 
 \paragraph{Readings}
 \begin{itemize}
-	\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
+\item \textcite{Lutkepohl_2004_UnivariateTimeSeries}
 \end{itemize}
 
 \begin{solution}\textbf{Solution to \nameref{ex:PropertiesAR1TimeTrend}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/properties_ar1_time_trend_solution.tex}
 \fi
 \newpage
diff --git a/exercises/properties_ar1_time_trend_solution.tex b/exercises/properties_ar1_time_trend_solution.tex
index a66b8b8..b1c676e 100644
--- a/exercises/properties_ar1_time_trend_solution.tex
+++ b/exercises/properties_ar1_time_trend_solution.tex
@@ -1,8 +1,12 @@
 \begin{enumerate}
-\item First, let's get a MA representation using the lag operator or recursive substitution techniques:
+
+\item
+First, let's get a MA representation using the lag operator or recursive substitution techniques:
 \begin{align*}
-y_t &= c + d\cdot t + \phi y_{t-1} + u_t\\
-\Leftrightarrow y_t &= \frac{c + d\cdot t + u_t}{1-\phi L}\\
+y_t &= c + d\cdot t + \phi y_{t-1} + u_t
+\\
+\Leftrightarrow y_t &= \frac{c + d\cdot t + u_t}{1-\phi L}
+\\
 \Leftrightarrow y_t &= \sum_{j=0}^{\infty}\phi^j(c+d(t-j)+u_{t-j}) = \frac{c}{1-\phi} + \frac{dt}{1-\phi} - d\sum_{j=0}^{\infty}\phi^j j + \sum_{j=0}^{\infty}\phi^j u_{t-j}
 \end{align*}
 As \( |\phi|<1 \) the geometric series holds.
@@ -13,22 +17,28 @@
 \end{align*} 
 Unconditional mean:
 \begin{align*}
-E[y_{t}] &=& \frac{c}{1-\phi} + \frac{dt}{1-\phi} - d \frac{\phi}{{(1-\phi)}^2} + \underbrace{\sum_{j=0}^{\infty} \phi^j E[u_{t-j}]}_{=0 \text{,~as~}\quad u_{t} \sim iid(0, \sigma^2)} \\
+E[y_{t}] &=& \frac{c}{1-\phi} + \frac{dt}{1-\phi} - d \frac{\phi}{{(1-\phi)}^2} + \underbrace{\sum_{j=0}^{\infty} \phi^j E[u_{t-j}]}_{=0 \text{,~as~}\quad u_{t} \sim iid(0, \sigma^2)}
+\\
 &=& \frac{c}{1-\phi} + \frac{dt}{1-\phi} - d \frac{\phi}{{(1-\phi)}^2}
 \end{align*}	
 
 Unconditional variance:
 \begin{align*}
-\gamma_0 = Var[y_{t}]
-&= E[(y_{t} - E[y_{t}])^2 ] \\
-&= E[(\sum_{j=0}^{\infty} \phi^j E[u_{t-j}]) \cdot (\sum_{j=0}^{\infty} \phi^j E[u_{t-j}])] \\
-&= E[(u_{t} + \phi^1 u_{t-1} + \phi^2 u_{t-2} + \dots)(u_{t} + \phi^1 u_{t-1} + \phi^2 u_{t-2} + \dots)] \\
+\gamma_0 &= Var[y_{t}]
+= E[{(y_{t} - E[y_{t}])}^2 ]
+\\
+&= E[(\sum_{j=0}^{\infty} \phi^j E[u_{t-j}]) \cdot (\sum_{j=0}^{\infty} \phi^j E[u_{t-j}])]
+\\
+&= E[(u_{t} + \phi^1 u_{t-1} + \phi^2 u_{t-2} + \dots)(u_{t} + \phi^1 u_{t-1} + \phi^2 u_{t-2} + \dots)]
+\\
 &= E[	u_{t}^2 + 2\phi^1 u_{t}u_{t-1} + 2\phi^2 u_{t}u_{t-2} + \cdots + 
-\phi^2 u_{t-1}^2 + 2\phi^3 u_{t-1}u_{t-2} + 2\phi^4 u_{t-1}u_{t-3} + \dots  \\
-&+ \phi^4 u_{t-2}^2 + 2\phi^5 u_{t-2}u_{t-3} + 2\phi^5 u_{t-2}u_{t-4} + \dots ] \\
+\phi^2 u_{t-1}^2 + 2\phi^3 u_{t-1}u_{t-2} + 2\phi^4 u_{t-1}u_{t-3} + \dots
+\\
+&+ \phi^4 u_{t-2}^2 + 2\phi^5 u_{t-2}u_{t-3} + 2\phi^5 u_{t-2}u_{t-4} + \dots ]
+\\
 &\overset{iid}{=} E [ u_{t}^2+ \phi^2 u_{t-1}^2 + \phi^4 u_{t-2}^2 + \dots ]
 \end{align*}
-\(\text{with } Var[u_{t}] = E[u_{t}^2] - E[u_{t}]^2 = E[u_{t}^2] = \sigma^2 \text{ we get}\)
+\(\text{with } Var[u_{t}] = E[u_{t}^2] - {(E[u_{t}])}^2 = E[u_{t}^2] = \sigma^2 \text{ we get}\)
 \begin{align*}
 Var[y_{t}] = \sigma^2 ( \phi^0 + \phi^2 + \phi^4 + \dots ) = \frac{\sigma^2}{1-\phi^2}.
 \end{align*}
@@ -36,37 +46,43 @@
 Autocovariance:
 \begin{align*}
 \gamma(k)
-&= E[(y_{t}-E[y_{t}])(y_{t-k}-E[y_{t-k}])] \\
-&= E[(\sum_{j=0}^{\infty} \phi^j 
-u_{t-j})(\sum_{j=0}^{\infty} \phi^j u_{t-j-k})] \\
-&= E[(u_{t} + \phi^1 u_{t-1} + \phi^2 u_{t-2} + \cdots + 
-\phi^k u_{t-k}  + \phi^{k+1} u_{t-k-1} + \phi^{k+2} u_{t-k-2} + \dots) \\
-& 	(u_{t-k}  + \phi^{1} u_{t-k-1} + \phi^{2} u_{t-k-2} + \dots)] \\
-&= E[u_{t}u_{t-k} + \phi^1 u_{t} u_{t-k-1} + \phi^2 u_{t} 
-u_{t-k-3} + \dots   \\
-&	\phi^1 u_{t-1}u_{t-k} + \phi^2 u_{t-1}u_{t-k-1}	+ \phi^3 
-u_{t-1}u_{t-k-2} + \dots \\
-& 	\phi^2 u_{t-2}u_{t-k} + \phi^3 u_{t-2}u_{t-k-1}	+ 
-\phi^4 u_{t-2}u_{t-k-2} + \dots \\
-&	\vdots \\
-&  \phi^{k} u_{t-k}^2 + 2\phi^{k+1} u_{t-k}u_{t-k-1}	+ 	
-2\phi^{k+2} u_{t-k}u_{t-k-2} + \dots \\
-& 	\phi^{k+2} u_{t-k-1}^2 + 2\phi^{k+3} u_{t-k-1}u_{t-k-2} +
-2\phi^{k+4} u_{t-k-1}u_{t-k-3} + \dots \\ 
-& 	\phi^{k+4} u_{t-k-2}^2 + 2\phi^{k+5} u_{t-k-2}u_{t-k-3} +
-2\phi^{k+6} u_{t-k-2}u_{t-k-4} + \dots] \\
-&\overset{iid}{=}
-E[\phi^{k}u_{t-k}^2 + \phi^{k+2}u_{t-k-1}^2 + \phi^{k+4}u_{t-k-2}^2 + \dots] \\
-&= \phi^{k}\sigma^2(\phi^0 + \phi^2 + \phi^4 + \dots ) \\
+&= E[(y_{t}-E[y_{t}])(y_{t-k}-E[y_{t-k}])]
+\\
+&= E[(\sum_{j=0}^{\infty} \phi^j u_{t-j})(\sum_{j=0}^{\infty} \phi^j u_{t-j-k})]
+\\
+&= E[(u_{t} + \phi^1 u_{t-1} + \phi^2 u_{t-2} + \cdots + \phi^k u_{t-k}  + \phi^{k+1} u_{t-k-1} + \phi^{k+2} u_{t-k-2} + \dots)
+\\
+& (u_{t-k}  + \phi^{1} u_{t-k-1} + \phi^{2} u_{t-k-2} + \dots)]
+\\
+&= E[u_{t}u_{t-k} + \phi^1 u_{t} u_{t-k-1} + \phi^2 u_{t} u_{t-k-3} + \dots
+\\
+&	\phi^1 u_{t-1}u_{t-k} + \phi^2 u_{t-1}u_{t-k-1}	+ \phi^3 u_{t-1}u_{t-k-2} + \dots
+\\
+& \phi^2 u_{t-2}u_{t-k} + \phi^3 u_{t-2}u_{t-k-1}	+ \phi^4 u_{t-2}u_{t-k-2} + \dots
+\\
+&\vdots
+\\
+& \phi^{k} u_{t-k}^2 + 2\phi^{k+1} u_{t-k}u_{t-k-1}	+ 2\phi^{k+2} u_{t-k}u_{t-k-2} + \dots
+\\
+& \phi^{k+2} u_{t-k-1}^2 + 2\phi^{k+3} u_{t-k-1}u_{t-k-2} + 2\phi^{k+4} u_{t-k-1}u_{t-k-3} + \dots
+\\ 
+& \phi^{k+4} u_{t-k-2}^2 + 2\phi^{k+5} u_{t-k-2}u_{t-k-3} + 2\phi^{k+6} u_{t-k-2}u_{t-k-4} + \dots]
+\\
+&\overset{iid}{=} E[\phi^{k}u_{t-k}^2 + \phi^{k+2}u_{t-k-1}^2 + \phi^{k+4}u_{t-k-2}^2 + \dots]
+\\
+&= \phi^{k}\sigma^2(\phi^0 + \phi^2 + \phi^4 + \dots )
+\\
 &= \frac{\phi^{k}\sigma^2}{1-\phi^2}
 \end{align*}
 
-Autocorrelation: \(\rho_k  = \frac{\gamma_k}{\gamma_0}\). Therefore:
+Autocorrelation: \(\rho_k  = \frac{\gamma_k}{\gamma_0}\).
+Therefore:
 \begin{align*}
 \rho(k) = \frac{\gamma(k)}{\gamma(0)} = \frac{\frac{\phi^{k}\sigma^2}{1-\phi^2}}{\frac{\sigma^2}{1-\phi^2}} = \phi^{k}.
 \end{align*} 
 
-\item The expectation is time-dependent, hence it is not stationary.
+\item
+The expectation is time-dependent, hence it is not stationary.
 One could subtract the expectation, e.g.\ look at \(y_{t}^{demeaned} = y_{t} -E[y_t]\),
   where \(y_t^{demeaned}\) is now covariance stationary.
 The unknown coefficient \(d\) must be estimated first though.
diff --git a/week_3.tex b/week_3.tex
new file mode 100644
index 0000000..f138c37
--- /dev/null
+++ b/week_3.tex
@@ -0,0 +1,31 @@
+% !TEX root = week_3.tex
+\input{exercises/_common_header.tex}
+\Newassociation{solution}{Solution}{week_3_solution}
+\newif\ifDisplaySolutions\DisplaySolutionstrue%
+
+\begin{document}
+\title{Quantitative Macroeconomics\\~\\Winter 2024/25\\~\\Week 3}
+\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_3.pdf}{GitHub}}
+\maketitle\thispagestyle{empty}
+
+\newpage
+\Opensolutionfile{week_3_solution}[week_3_solution]
+\tableofcontents\thispagestyle{empty}\newpage
+
+\setcounter{page}{1}
+\input{exercises/properties_ar1.tex}\newpage
+\input{exercises/properties_ar1_time_trend.tex}\newpage
+\input{exercises/law_of_large_numbers.tex}\newpage
+\input{exercises/central_limit_theorem_dependent_data.tex}\newpage
+\printbibliography%
+\newpage
+
+\Closesolutionfile{week_3_solution}
+\ifDisplaySolutions%
+\newpage
+\appendix
+\section{Solutions}
+\input{week_3_solution}
+\fi
+\end{document}
\ No newline at end of file