Added VAR estimation exercises

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

@@ -95,4 +95,7 @@ jobs:
             matrixAlgebraKroneckerFormula;
             matrixAlgebraRotation;
             matrixAlgebraCholesky;
-            matrixAlgebraLyapunovComparison;
+            matrixAlgebraLyapunovComparison;
+            VARpDimensionsIllustration;
+            threeVariableVAROLS;
+            threeVariableVARML;

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

@@ -124,4 +124,7 @@ jobs:
             matrixAlgebraKroneckerFormula;
             matrixAlgebraRotation;
             matrixAlgebraCholesky;
-            matrixAlgebraLyapunovComparison;
+            matrixAlgebraLyapunovComparison;
+            VARpDimensionsIllustration;
+            threeVariableVAROLS;
+            threeVariableVARML;

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

@@ -86,4 +86,7 @@ jobs:
             matrixAlgebraKroneckerFormula;
             matrixAlgebraRotation;
             matrixAlgebraCholesky;
-            matrixAlgebraLyapunovComparison;
+            matrixAlgebraLyapunovComparison;
+            VARpDimensionsIllustration;
+            threeVariableVAROLS;
+            threeVariableVARML;

README.md

@@ -130,7 +130,7 @@ Please feel free to use this for teaching or learning purposes; however, taking
 
 
 <details>
-  <summary> Week 6: Multivariate Time Series Concepts</summary>
+  <summary> Week 6: Multivariate Time Series Concepts, Estimating VAR with OLS and ML</summary>
 
 ### Goals
 
@@ -140,11 +140,12 @@ Familiarize yourself with
 * 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 of a VAR(p) model
+* estimate VAR models with Ordinary Least Squares (OLS) and Maximum Likelihood (ML)
 
 ### To Do
 
 * [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] Read Kilian and Lütkepohl (2007, Ch. 2.2, 2.3) 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 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)
 
@@ -154,11 +155,10 @@ Familiarize yourself with
 <!---
 
 <details>
-  <summary> Week 7: Estimating VAR model with OLS and ML; The identification problem in SVAR models</summary>
+  <summary> Week 7: The identification problem in SVAR models</summary>
 
 ### Goals
 
-* estimate VAR models with Ordinary Least Squares (OLS) and Maximum Likelihood (ML)
 * understand the identification problem in SVAR models
 
 ### To Do

exercises/matrix_algebra_solution.tex

@@ -1,10 +1,10 @@
 \begin{enumerate}
 \item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/matrixAlgebraEigenvalues.m}
-In the univariate AR{(1)} model we would check whether the autocorrelation coefficient is between -1 and 1,
-  i.e.\ whether \(|\phi| = |A|<1\) such that \(\sum_{j=0}^\infty {(AL)}^j=1/(1-AL)\)
+In the univariate AR{(1)} model we would check whether the autocorrelation coefficient is between -1 and 1, i.e.\
+  whether \(|\phi| = |A|<1\) such that \(\sum_{j=0}^\infty {(AL)}^j=1/(1-AL)\)
   where \(L\) is the lag operator.
-In the multivariate case, we want to check the same thing,
-  i.e.| \(\sum_{j=0}^\infty {(AL)}^j={(1-AL)}^{-1}\).
+In the multivariate case, we want to check the same thing, i.e.\
+  \(\sum_{j=0}^\infty {(AL)}^j={(1-AL)}^{-1}\).
 Note that \(A\) is a square matrix and taking the power of a matrix is not a trivial task.
 One convenient way to do so, is to consider an eigenvalue decomposition (if it exists):
 \[A= Q \Lambda Q^{-1}\]

exercises/ml_var_p.tex

@@ -1,5 +1,5 @@
-\section[Maximum Likelihood Estimation of VAR(p)]{Maximum Likelihood Estimation of VAR(p)\label{ex:MaximumLikelihoodEstimationVARp}}
-Consider the VAR(p) model with constant written in the more compact form
+\section[Maximum Likelihood Estimation of VAR{(p)}]{Maximum Likelihood Estimation of VAR{(p)}\label{ex:MaximumLikelihoodEstimationVARp}}
+Consider the VAR{(p)} model with constant written in the more compact form
 \begin{align*}
 y_t = [c, A_1, \ldots, A_p] Z_{t-1} + u_t = A Z_{t-1}+ u_t
 \end{align*}
@@ -22,15 +22,15 @@
 Comment on the asymptotic distribution.
 
 \item Provide an expression for the ML estimator \(\widetilde{\Sigma_u}\) of the innovation covariance matrix.
- 	
+
 \item Consider data given in \texttt{threeVariableVAR.csv} for \(y_t = (\Delta gnp_t,i_t,\Delta p_t)'\),
-where \(gnp_t\) denotes the log of U.S. real GNP,
-\(p_t\) the corresponding GNP deflator in logs,
-and \(i_t\) the federal funds rate, averaged by quarter.
+  where \(gnp_t\) denotes the log of U.S. real GNP,
+  \(p_t\) the corresponding GNP deflator in logs,
+  and \(i_t\) the federal funds rate, averaged by quarter.
 The estimation period is restricted to 1954q4 to 2007q4. 
 \begin{itemize}
-  \item Estimate the parameters with Maximum Likelihood.
-  \item Compare your estimation results to an OLS estimation.
+\item Estimate the parameters with Maximum Likelihood.
+\item Compare your estimation results to an OLS estimation.
 \end{itemize}
 
 \end{enumerate}

exercises/ml_var_p_solution.tex

@@ -1,19 +1,22 @@
 \begin{enumerate}
-\item From the univariate case (and undergraduate econometrics),
-we know that both estimators are identical;
-hence, the asymptotic normal distribution holds as well.
+\item
+From the univariate case (and undergraduate econometrics),
+  we know that both estimators are identical;
+  hence, the asymptotic normal distribution holds as well.
 
-\item Taking the derivative of the conditional log-likelihood function with respect to \(\Sigma_u\) yields:
+\item
+Taking the derivative of the conditional log-likelihood function with respect to \(\Sigma_u\) yields:
 \begin{align*}
 \widetilde{\Sigma}_u = \frac{\hat{U}\hat{U}'}{T}
 \end{align*}
 where \(\hat{U}\) are both the ML and OLS residuals (as \(\widetilde{A}=\widehat{A}\)).
 Note that from previous exercises in the univariate case
-we have already seen that the only difference to the OLS estimator of \(\Sigma_u\)
-is given in the fact that for ML we don't correct the degrees of freedom,
-but simply divide by the \textbf{effective} sample size used in the estimation \(T\).
+  we have already seen that the only difference to the OLS estimator of \(\Sigma_u\)
+  is given in the fact that for ML we don't correct the degrees of freedom,
+  but simply divide by the \textbf{effective} sample size used in the estimation \(T\).
 
-\item See the previous exercise, as the \texttt{VARReducedForm} function also outputs the ML estimate of \(\Sigma_u\):
+\item
+See the previous exercise, as the \texttt{VARReducedForm} function also outputs the ML estimate of \(\Sigma_u\):
 \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/threeVariableVARML.m}
 
 \end{enumerate}

exercises/ols_var_p.tex

@@ -1,5 +1,5 @@
-\section[Ordinary Least Squares Estimation of VAR(p)]{Ordinary Least Squares Estimation of VAR(p)\label{ex:OrdinaryLeastSquaresEstimationVARp}}
-Consider the VAR(p) model with a constant written in the compact form
+\section[Ordinary Least Squares Estimation of VAR{(p)}]{Ordinary Least Squares Estimation of VAR{(p)}\label{ex:OrdinaryLeastSquaresEstimationVARp}}
+Consider the VAR{(p)} model with a constant written in the compact form
 \begin{align*}
 y_t = [c, A_1, \ldots, A_p] Z_{t-1} + u_t = A Z_{t-1}+ u_t
 \end{align*}
@@ -9,7 +9,7 @@
   ordinary least squares for each equation separately results in efficient estimators.
 The OLS estimator is
 \begin{align*}
-\hat{A} = \left[ \hat{c}, \hat{A_1}, \ldots ,\hat{A_p}\right] = \left(\sum_{t=1}^{T}y_t Z_{t-1}'\right)\left(\sum_{t=1}^{T}Z_{t-1}Z_{t-1}'\right)^{-1} = Y Z'{(ZZ')}^{-1}
+\hat{A} = \left[ \hat{c}, \hat{A_1}, \ldots ,\hat{A_p}\right] = \left(\sum_{t=1}^{T}y_t Z_{t-1}'\right){\left(\sum_{t=1}^{T}Z_{t-1}Z_{t-1}'\right)}^{-1} = Y Z'{(ZZ')}^{-1}
 \end{align*}
 where \(Y=[y_1, \ldots, y_T]\) and \(Z=[Z_0, \ldots, Z_{T-1}]\).
 More precisely, stacking the columns of \(A = [c, A_1,\ldots ,A_p]\) in the vector \(\alpha = vec(A)\),
@@ -19,7 +19,7 @@
 where \(\Sigma_{\hat{\alpha}} = plim(\frac{1}{T}ZZ')^{-1}\otimes \Sigma_u\), if the process is stable.
 Under fairly general assumptions this estimator has an asymptotic normal distribution.
 A sufficient condition for the consistency and asymptotic normality of \(\hat{A}\) would be
-  that \(u_t\) is a continuous iid random variable with four finite moments.
+  that \(u_t\) is a continuous white noise random variable with four finite moments.
 A consistent estimator of the innovation covariance matrix \(\Sigma_u\) is, for example,
 \begin{align*}
 \hat{\Sigma}_u = \frac{\hat{U}\hat{U}'}{T-Kp-1}
@@ -33,8 +33,8 @@
 In other words, asymptotically the usual t-statistics can be used for testing restrictions on individual coefficients
   and for setting up confidence intervals.
 \begin{enumerate}
-\item What are the dimensions of \(y_t\), \(Y\), \(u_t\), \(U\), \(c\), \(A_1\), \ldots, \(A_p\), \(A\), \(\alpha\), \(Z_{t-1}\), \(Z\), \(\Sigma_u\) and \(\Sigma_{\hat{\alpha}}\).
-\item Modify your \texttt{ARpOLS} function such that it is able to estimate VAR(p) models.
+\item What are the dimensions of \(y_t\), \(Y\), \(u_t\), \(U\), \(c\), \(A_1\), \ldots, \(A_p\), \(A\), \(\alpha \), \(Z_{t-1}\), \(Z\), \(\Sigma_u\) and \(\Sigma_{\hat{\alpha}}\).
+\item Modify your \texttt{ARpOLS} function such that it is able to estimate VAR{(p)} models.
   Save the modified function as \texttt{VARReducedForm}.
 \item Consider data given in \texttt{threeVariableVAR.csv} for \(y_t = (\Delta gnp_t,i_t,\Delta p_t)'\),
   where \(gnp_t\) denotes the log of U.S. real GNP,
@@ -43,7 +43,7 @@
 The estimation period is restricted to 1954q4 to 2007q4. 
 \begin{itemize}
 	\item Load the data and visualize it. Comment whether you think the data looks stationary.
-	\item Estimate a VAR(4) model using the \texttt{VARReducedForm} function.
+	\item Estimate a VAR{(4)} model using the \texttt{VARReducedForm} function.
 	Examine the stability of the estimated process
 	  and the significance of the estimated parameters at a \(95\% \) level.
 \end{itemize}

week_6.tex

@@ -6,7 +6,7 @@
 \begin{document}
 \title{Quantitative Macroeconomics\\~\\Winter 2024/25\\~\\Week 6}
 \author{Willi Mutschler\\[email protected]}
-\date{Version: 1.0\\Latest version available on: \href{https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_6.pdf}{GitHub}}
+\date{Version: 1.1\\Latest version available on: \href{https://github.com/wmutschl/Quantitative-Macroeconomics/releases/latest/download/week_6.pdf}{GitHub}}
 \maketitle\thispagestyle{empty}
 
 \newpage
@@ -17,6 +17,8 @@
 \input{exercises/matrix_algebra.tex}\newpage
 \input{exercises/understanding_multivariate_concepts.tex}\newpage
 \input{exercises/dimensions_and_var_one_representation.tex}\newpage
+\input{exercises/ols_var_p.tex}\newpage
+\input{exercises/ml_var_p.tex}\newpage
 \printbibliography%
 \newpage