Added week 5 material

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

@@ -83,4 +83,16 @@ jobs:
             cd("progs/matlab");
             AR4LagSelection;
             portmanteauTest;
-            bootstrapCIAR1;
+            bootstrapCIAR1;
+
+      - name: Run week 6 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("Dynare-6.2-arm64/matlab");
+            cd("progs/matlab");
+            matrixAlgebraEigenvalues;
+            matrixAlgebraKroneckerFormula;
+            matrixAlgebraRotation;
+            matrixAlgebraCholesky;
+            matrixAlgebraLyapunovComparison;

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

@@ -113,3 +113,15 @@ jobs:
             AR4LagSelection;
             portmanteauTest;
             bootstrapCIAR1;
+
+      - name: Run week 6 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("dynare/matlab");
+            cd("progs/matlab");
+            matrixAlgebraEigenvalues;
+            matrixAlgebraKroneckerFormula;
+            matrixAlgebraRotation;
+            matrixAlgebraCholesky;
+            matrixAlgebraLyapunovComparison;

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

@@ -75,3 +75,15 @@ jobs:
             AR4LagSelection;
             portmanteauTest;
             bootstrapCIAR1;
+
+      - name: Run week 6 codes
+        uses: matlab-actions/run-command@v2
+        with:
+          command: |
+            addpath("D:\hostedtoolcache\windows\dynare-6.0\matlab");
+            cd("progs/matlab");
+            matrixAlgebraEigenvalues;
+            matrixAlgebraKroneckerFormula;
+            matrixAlgebraRotation;
+            matrixAlgebraCholesky;
+            matrixAlgebraLyapunovComparison;

README.md

@@ -129,7 +129,6 @@ Please feel free to use this for teaching or learning purposes; however, taking
 </details>
 
 
-<!---
 <details>
   <summary> Week 6: Multivariate Time Series Concepts</summary>
 
@@ -140,17 +139,20 @@ Familiarize yourself with
 * important matrix concepts such as Eigenvalues, Kronecker product, orthogonality, rotation matrices, Cholesky decomposition and Lyapunov equations
 * multivariate notation and dimensions of vectors and matrices for VAR(p) models
 * autocovariances, stability and covariance-stationarity in multivariate VAR(p) models
-* the companion form
+* the companion form of a VAR(p) model
 
 ### To Do
 
-* [x] Review the solutions of [last week's exercises](https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_5.pdf) (except the bootstrap one) and write down all your questions
+* [x] Review the solutions of [last week's exercises](https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_5.pdf) and write down all your questions
 * [x] Read Kilian and Lütkepohl (2007, Ch. 2.2) and Lütkepohl (2005, Chapter 2 and Appendix A); make note of all the aspects and concepts that you are not familiar with or that you find difficult to understand
-* [x] Do exercise sheet 6 (the solutions are already available)
+* [x] Do exercise 1 of week 6
 * [x] If you have questions, get in touch with me via email or (better) [schedule a meeting](https://schedule.mutschler.eu)
 
 </details>
 
+
+<!---
+
 <details>
   <summary> Week 7: Estimating VAR model with OLS and ML; The identification problem in SVAR models</summary>
 

exercises/dimensions_and_var_one_representation.tex

@@ -1,6 +1,6 @@
-\section[Dimensions and VAR(1) representation]{Dimensions and VAR(1) representation\label{ex:DimensionsAndVAROneRepresentation}}
+\section[Dimensions and VAR{(1)} representation]{Dimensions and VAR{(1)} representation\label{ex:DimensionsAndVAROneRepresentation}}
 Let \(y_t\) be a K-dimensional covariance stationary random vector.
-Consider the VAR(p)-process
+Consider the VAR{(p)}-process
 \begin{align*}
 y_t = \nu + \sum_{i=1}^p A_i {y_{t-i}} + u_t
 \end{align*}
@@ -10,22 +10,23 @@
 \item Assume that \(E(u_t) = 0; E(u_t u_t') = \Sigma_u\) with \(\Sigma_u\) being symmetric and positive definite.
 Which additional assumptions do we need to assure that \(u_t\) is a multivariate white noise process?	
 
-\item Consider a VAR(2) model with \(K=4\) variables and a constant term.
+\item Consider a VAR{(2)} model with \(K=4\) variables and a constant term.
 How many parameters do we need to estimate?
 
-\item Show how to represent a VAR(3) model as a VAR(1) model. Hint: stack \(y_t, y_{t-1}\) and \(y_{t-2}\) into a vector.
+\item Show how to represent a VAR{(3)} model as a VAR{(1)} model.
+Hint: stack \(y_t, y_{t-1}\) and \(y_{t-2}\) into a vector.
 
-\item Write a function to compute the \enquote{companion VAR(1) form} of any VAR(p) model with constant term.
+\item Write a function to compute the \enquote{companion VAR{(1)} form} of any VAR{(p)} model with constant term.
 \end{enumerate}
 
 \paragraph{Readings}
 \begin{itemize}
-	\item \textcite[Ch.~2.2]{Kilian.Lutkepohl_2017_StructuralVectorAutoregressive}
-	\item \textcite[Ch.~2]{Lutkepohl_2005_NewIntroductionMultiple}
+\item \textcite[Ch.~2.2]{Kilian.Lutkepohl_2017_StructuralVectorAutoregressive}
+\item \textcite[Ch.~2]{Lutkepohl_2005_NewIntroductionMultiple}
 \end{itemize}
 
 \begin{solution}\textbf{Solution to \nameref{ex:DimensionsAndVAROneRepresentation}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/dimensions_and_var_one_representation_solution.tex}
 \fi
 \newpage

exercises/dimensions_and_var_one_representation_solution.tex

@@ -1,34 +1,35 @@
 \begin{enumerate}
-\item \(\nu\) snd \(u_t\) are both K-dimensional vectors: \(\nu,u_t \in \mathbb{R}^{K \times 1}\).
+\item \(\nu \) and \(u_t\) are both K-dimensional vectors: \(\nu,u_t \in \mathbb{R}^{K \times 1}\).
 \(A_i\) is a \(K \times K\) matrix.
 
 \item We must have \({\Gamma_u(h)} = Cov(u_t, u_{t-h}) = 0_{K\times K}\) for any \(h\neq0\),
   that is all autocovariances are zero
   (except for \(h=0\) which is the covariance matrix).
 Importantly: we do not need a distributional assumption!
-	
+
 \item General: \(K\) constants + \(K^2\cdot p\) autoregressive coefficients + \(K(K+1)/2\) covariance terms (due to symmetry).
-Here: \(4+4^2\cdot2+4\cdot(4+1)/2=46\).
+Here: \(4 + 4^2 \cdot 2 + 4 \cdot (4+1)/2 = 46\).
 That's a lot!
 Therefore we will try to restrict some parameters (e.g.\ set equal to zero or by using Bayesian priors)
-or consider only small VAR systems, e.g. \(K=3\) or \(p=1\), etc.
+  or consider only small VAR systems, e.g.\
+  \(K=3\) or \(p=1\), etc.
 
-\item VAR(3): \(y_t = \nu + A_1 y_{t-1} + A_2 y_{t-2} + A_3 y_{t-3} + u_t\).
+\item VAR{(3)}: \(y_t = \nu + A_1 y_{t-1} + A_2 y_{t-2} + A_3 y_{t-3} + u_t\).
 Idea: Stack \(y_t, y_{t-1}\) and \(y_{t-2}\) into a vector and note that \(y_{t-1}=y_{t-1}\) and \(y_{t-2}=y_{t-2}\).
 That is
 \begin{align*}
-\underbrace{\begin{bmatrix} y_t\\ y_{t-1} \\ y_{t-2} \end{bmatrix}}_{\widetilde{y}_t}
+\underbrace{\begin{bmatrix} y_t \\ y_{t-1} \\ y_{t-2} \end{bmatrix}}_{\widetilde{y}_t}
 = \underbrace{\begin{bmatrix} \nu \\ 0 \\ 0 \end{bmatrix}}_{\widetilde{\nu}_t}
-+ \underbrace{\begin{bmatrix} A_1 & A_2 & A_3 \\ I & 0 & 0\\ 0&I&0\end{bmatrix}}_{\widetilde{A}}
-  \underbrace{\begin{bmatrix} y_{t-1}\\ y_{t-2} \\ y_{t-3} \end{bmatrix}}_{\widetilde{y}_{t-1}}
-+ \underbrace{\begin{bmatrix} u_t \\ 0 \\ 0\end{bmatrix}}_{\widetilde{u}_t}
++ \underbrace{\begin{bmatrix} A_1 & A_2 & A_3 \\ I & 0 & 0 \\ 0 & I & 0 \end{bmatrix}}_{\widetilde{A}}
+  \underbrace{\begin{bmatrix} y_{t-1} \\ y_{t-2} \\ y_{t-3} \end{bmatrix}}_{\widetilde{y}_{t-1}}
++ \underbrace{\begin{bmatrix} u_t \\ 0 \\ 0 \end{bmatrix}}_{\widetilde{u}_t}
 \end{align*}
 where \(I\) is the \(K\)-dimensional identity matrix and \(0\) the \(K\)-dimensional zero matrix.
 Therefore: \(\widetilde{y}_t = \widetilde{A} \widetilde{y}_{t-1} + \widetilde{u}_t\).
-This is called the Companion Form.
-It is particularly useful, when checking the stability and covariance-stationarity properties of VAR(p) processes 
-as we can simply compute the Eigenvalues of \(\widetilde{A}\)
-and check whether all of them are inside the unit circle, i.e.\ between \(-1\) and \(1\).
+This is called the \emph{Companion Form}.
+It is particularly useful, when checking the stability and covariance-stationarity properties of VAR{(p)} processes 
+  as we can simply compute the Eigenvalues of \(\widetilde{A}\)
+  and check whether all of them are inside the unit circle, i.e.\ between \(-1\) and \(1\).
 No need to find the roots of the general Lag-polynomials.
 
 \item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/companionForm.m}

exercises/matrix_algebra.tex

@@ -15,7 +15,7 @@
 \item Consider the matrices D:\@ \(m\times n\), E:\@ \(n\times p\) and F:\@ \(p\times k\).
 Show that
 \[vec(DEF)=\left(F'\otimes D\right) vec(E),\]
-where \(\otimes\) is the Kronecker product and \(vec\) the vectorization operator
+where \(\otimes \) is the Kronecker product and \(vec\) the vectorization operator
 either on paper or using a symbolic toolbox.
 
 \item Show that R is an orthogonal matrix. Why is this matrix called a rotation matrix?	
@@ -30,28 +30,28 @@
 	\begin{enumerate}
 	\item the Kronecker product and vectorization
 	\item the following iterative algorithm:
-		\begin{align*}
-		\Sigma_{y,0} &= I, A_0 = A, \Sigma_{u,0} = \Sigma_{u}\\
-		\Sigma_{y,i+1} &= A_i \Sigma_{y,i} A_i' + \Sigma_{u,i}\\
-		\Sigma_{u,i+1} &= A_i \Sigma_{u,i} A_i' + \Sigma_{u,i}\\
-		A_{i+1} &= A_i A_i
-		\end{align*}
-		Write a loop until either a maximal number of iterations (say 500) is reached
-		or each element of \(\Sigma_{y,i+1}-\Sigma_{y,i}\) is less than \(10^{-25}\) in absolute terms.
+	\begin{align*}
+	\Sigma_{y,0} &= I, A_0 = A, \Sigma_{u,0} = \Sigma_{u}\\
+	\Sigma_{y,i+1} &= A_i \Sigma_{y,i} A_i' + \Sigma_{u,i}\\
+	\Sigma_{u,i+1} &= A_i \Sigma_{u,i} A_i' + \Sigma_{u,i}\\
+	A_{i+1} &= A_i A_i
+	\end{align*}
+	Write a loop until either a maximal number of iterations (say 500) is reached
+	  or each element of \(\Sigma_{y,i+1}-\Sigma_{y,i}\) is less than \(10^{-25}\) in absolute terms.
 	\item Compare both approaches for A and \(\Sigma_u\) given above.
 	\end{enumerate}
 \end{enumerate}
 
 \paragraph{Readings:}
 \begin{itemize}	
-	\item \textcite[Ch.~4.2]{Anderson.McGrattan.Hansen.EtAl_1996_MechanicsFormingEstimating}
-	\item \textcite[Ch.~6.7]{Anderson.Moore_1979_OptimalFiltering}
-	\item \textcite[App.~A]{Lutkepohl_2005_NewIntroductionMultiple}
-	\item \textcite[Ch.~4.10]{Uribe.Schmitt-Grohe_2017_OpenEconomyMacroeconomics}
+\item \textcite[Ch.~4.2]{Anderson.McGrattan.Hansen.EtAl_1996_MechanicsFormingEstimating}
+\item \textcite[Ch.~6.7]{Anderson.Moore_1979_OptimalFiltering}
+\item \textcite[App.~A]{Lutkepohl_2005_NewIntroductionMultiple}
+\item \textcite[Ch.~4.10]{Uribe.Schmitt-Grohe_2017_OpenEconomyMacroeconomics}
 \end{itemize}
 
 \begin{solution}\textbf{Solution to \nameref{ex:MatrixAlgebra}}
-\ifDisplaySolutions
+\ifDisplaySolutions%
 \input{exercises/matrix_algebra_solution.tex}
 \fi
 \newpage