Added week 3 material

.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;

.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;

.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;

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>
@@ -81,14 +79,16 @@ 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)
 * [x] (optionally) checkout the short [Intermediate Git Video Tutorials from GitKraken](https://www.gitkraken.com/learn/git/tutorials#intermediate)
 
 </details>
 
+<!---
+
 <details>
   <summary>Week 4: Ordinary Least Squares (OLS) and Maximum Likelihood (ML) estimation of the autoregressive process</summary>
 

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