Added midterm exam

data/MonPol.csv

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exam/midterm_exam_WS2324.tex

@@ -0,0 +1,219 @@
+% !TeX encoding = UTF-8
+% !TeX spellcheck = en_US
+% !TEX root = midterm_exam_WS2324.tex
+\documentclass{article}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[a4paper,top=2cm]{geometry}
+\usepackage{amssymb,amsmath,amsfonts}
+\usepackage[english]{babel}
+\usepackage[a4paper]{geometry}
+\usepackage{enumitem}
+\usepackage{booktabs}
+\usepackage{csquotes}
+\usepackage{graphicx}
+\usepackage[numbered,framed]{matlab-prettifier}
+\usepackage{url}
+%\parindent0mm
+%\parskip1.5ex plus0.5ex minus0.5ex
+\usepackage[backend=biber,style=authoryear]{biblatex}
+\addbibresource{../literature/_biblio.bib}
+\begin{document}
+
+\title{Quantitative Macroeconomics}
+\author{Midterm Exam}
+\date{Winter 2023/2024}
+\maketitle
+
+\section*{General information}
+\begin{itemize}
+	\item Answer \textbf{all} of the following \textbf{four} exercises in English.
+	\item All assignments will be given the same weight in the final grade.
+	\item Hand in your solutions before January, 10 2024 at 3pm.
+	\item The solution files should contain your executable (and commented) script files
+		as well as all additional documentation as \texttt{pdf}, not \texttt{txt}, \texttt{md}, \texttt{tex}, \texttt{doc} or \texttt{docx}.
+	Your \texttt{pdf} files may also include scans or pictures of handwritten notes.
+	\item Please e-mail ALL the solution files to \url{[email protected]}.
+	I will confirm the receipt of your work also by email (typically within the hour). If not, please resend it to me.
+	\item All students must work on their own, please also give your student ID number and the name of the module you want to earn credits for.
+	\item It is advised to regularly check Ilias and your emails in case of urgent updates.
+	\item If there are any questions, do not hesitate to contact Willi Mutschler.
+\end{itemize}
+
+\newpage
+
+
+\section[Theoretical and Simulated Moments]{Theoretical and Simulated Moments\label{ex:TheoreticalAndSimulatedMoments}}
+\begin{enumerate}
+ 	\item Derive the unconditional first two moments (i.e.\ theoretical mean, covariance and autocovariance matrix) of the covariance stationary VAR(1) model: \(y_t = \nu + A y_{t-1} + u_t\)
+ 	\item Simulate \(R=100\) datasets each with \(T=100\) observations for
+ 	\begin{align*}
+ 	y_t = \begin{pmatrix}0.2 &0.3 \\-0.6 & 1.1 \end{pmatrix} y_{t-1}  + u_t
+ 	\end{align*}
+ 	provided that \(u_t \sim WN(0,\Sigma_u)\) and \(\Sigma_u = \begin{pmatrix}0.9 &  \\ 0.2 & 0.5 \end{pmatrix}\).
+	\item Compute the sample mean and sample covariance matrix for each of the \(R\) datasets. 
+	Then take the average and compare the results with part (1).
+	\item How does your choice of \(R\) or \(T\) change your results?
+
+\end{enumerate}
+
+\newpage
+
+
+\section{Properties Of Lag-Order Selection Criteria}
+Assume that the true Data-Generating-Process (DGP) follows the following VAR(4) model
+\begin{align*}
+y_t = \begin{pmatrix}
+2.4 & 1.0\\
+0 & 1.1
+\end{pmatrix}
+y_{t-1}+
+\begin{pmatrix}
+-2.15 & -0.9\\
+0 & -0.41
+\end{pmatrix}
+y_{t-2}+
+\begin{pmatrix}
+0.852 & 0.2\\
+0& 0.06
+\end{pmatrix}
+y_{t-3}+
+\begin{pmatrix}
+-0.126 & 0\\
+0 & 0.0003
+\end{pmatrix}
+y_{t-4}
++ u_t
+\end{align*}
+where \(u_t\) is a Gaussian white noise with contemporary covariance matrix \(\Sigma_u = \begin{pmatrix}
+0.9&0.2\\0.2&0.5
+\end{pmatrix}\).
+
+Perform a Monte-Carlo analysis to study both the finite-sample as well as asymptotic properties
+of the Akaike Information Criterion (AIC) and the Schwarz Information Criterion (SIC):
+\begin{align*}
+AIC(m)  &= \log(det(\tilde{\Sigma}_u(m))) + \frac{2}{T}\varphi(m)\\
+SIC(m)  &= \log(det(\tilde{\Sigma}_u(m))) + \frac{\log T}{T}\varphi(m)
+\end{align*}
+where \(\tilde{\Sigma}_u=T^{-1}\sum_{t=1}^T \hat{u}_t\hat{u}_t'\) is the residual covariance matrix estimator
+for a reduced-form VAR model of order $m$ based on OLS residuals \(\hat{u}_t\).
+The function \(\varphi(m)\) corresponds to the total number of regressors in the system of VAR equations.
+The VAR order is chosen such that the respective criterion is minimized over the possible orders \(m = 0,\ldots,p^{max}\).
+To this end, do the following.
+
+\begin{itemize}
+	\item Set the number of Monte Carlo repetitions \(R=100\) and \(p^{max}=8\).
+	\item Initialize output matrices \(aic\) and \(sic\) each of dimension \(R \times 5\).
+	\item For \(r=1,\ldots,R\) do the following:
+	\begin{itemize}
+		\item Simulate \(10100\) observations for the DGP given above and discard the first 100 observations as burn-in phase.
+		Save the remaining 10000 observations in a matrix \(Y\).
+		\item Compute the lag criteria for 5 different sample sizes \(T=\{80, 160, 240, 500, 10000\} \),
+		i.e.\ use the last \(T\) observations of your simulated data matrix \(Y\) for computations.
+		\item Save the chosen lag order in the corresponding output object at position \([r,j]\) where \(j=1,\ldots,5\) indicates the corresponding sample size.
+	\end{itemize}
+	\item Look at the frequency tables of your output objects for the different subsamples. Hint: \texttt{tabulate(aic(:,1))} displays a frequency table for the AIC criterion with sample size equal to 80.
+\end{itemize}
+Given your results, do you agree with the following (general) findings?
+\begin{enumerate}
+	\item AIC is not consistent for the true lag order, whereas SIC is consistent.
+	\item AIC never (asymptotically) selects a lag order that is lower than the true lag order.
+	\item In finite samples, we usually have \(\hat{p}_{SIC} \leq \hat{p}_{AIC}\).
+	\item In finite samples, AIC has a tendency to overestimate the lag order, SIC has a tendency to underestimate the lag order; hence, one should rely on AIC in finite samples.
+\end{enumerate}
+
+\newpage
+
+\section[Sign-identified Monetary SVAR Model for US]{Sign-identified Monetary SVAR Model for US\label{ex:SignIdentifiedMonetarySVARUS}}
+Consider data for \(y_t = (\Delta gnp_t,\Delta p_t,i_t)'\),
+  where \(gnp_t\) denotes the log of U.S. real GNP,
+  \(p_t\) the consumer price index in logs,
+  and \(i_t\) the federal funds rate.
+  Data is given in the csv file \texttt{MonPol.csv} containing the sample period of 1970Q1 to 2011Q1.
+
+We will use the following sign restrictions pattern to identify the structural shocks:
+\begin{align}
+B_0^{-1}=\begin{bmatrix}
+	+ & + & -\\
+	+ & - & -\\
+	+ & *  & +
+	\label{eq:signpattern}
+\end{bmatrix}
+\end{align}
+where \(*\) denotes an unrestricted value.
+
+\begin{enumerate}
+\item Read the short summary paper by \textcite{Wolf_2022_WhatCanWe} and provide economic intuition for the pattern given in equation \eqref{eq:signpattern}.
+\item Write a script that estimates the structural impulse response function via sign restrictions.
+To this end:
+	\begin{itemize}
+	\item Estimate a VAR(4) model with constant by using ordinary least-squares.
+	Compute the lower Cholesky decomposition of your estimated residual covariance matrix, i.e.\ \( P = chol(\hat{\Sigma}_u) \).
+	\item Set the IRF horizon \(H=20\) and the number of required sign-identified draws to \(N=1000\).
+	\item Initialize an array \texttt{IRFvals} of dimension
+	(i) number of variables \(K\) by (ii) number of shocks \(K\) by (iii) horizon \(H\) by (iv) number of draws \(N\).
+	This array will store \(n=1,\ldots,N\) sign-identified impulse response functions \(\theta_{jkhn}\) for variable \(j=1,\ldots,K\)
+	with respect to shock \(k=1,\ldots,K\) at horizon \(h=0,\ldots,H\), where \(K\) is the number of variables.
+
+	\item Initialize \texttt{n=1} and write a while loop that executes the following steps:
+		\begin{itemize}
+		\item Draw an orthogonal rotation matrix \(Q\) using the function \texttt{drawRotationMatrixQ.m} in the appendix.
+		\item Compute the implied rotated impact matrix \(\widetilde{B}_0^{-1}=PQ\).
+		\item Note that \(\widetilde{B}_0^{-1}\) is a valid impact matrix, because \(\hat{\Sigma}_u = \widetilde{B}_0^{-1} \widetilde{B}_0^{-1'}\).
+		However, we will only accept those impact matrices \(\widetilde{B}_0^{-1}\) that full-fill the sign restrictions in equation~\eqref{eq:signpattern}.
+		Therefore, use if statements to check whether \(\widetilde{B}_0^{-1}\) is a valid impact matrix:
+		\begin{itemize}
+			\item If the pattern is correct, increase \(n\) by 1 and compute the implied IRFs. Store these into \texttt{IRFvals(:,:,:,n)}.
+			\item If the pattern is incorrect, don't change the value of \(n\), discard this draw and go back to the beginning of the loop.
+			\end{itemize}
+		\item End the while loop if \texttt{n=N}.
+		\end{itemize}
+	\item Plot \textbf{ALL} \(N\) impulse response functions that full-fill the sign pattern for (i) the level of GDP, (ii) CPI inflation and (iii) the Federal Funds rate.
+	\\Hint: The graphs should look similar to figure~\ref{fig:signSVAR} in the appendix.
+	That is, in each subplot compute and plot \texttt{IRFS = squeeze(IRFvals(ivar,ishock,1:nsteps,:))}.	
+	\end{itemize}
+\end{enumerate}
+
+\newpage
+
+\section[Historical Decomposition]{Historical Decomposition}
+Reconsider the SVAR model of \textcite{Rubio-Ramirez.Waggoner.Zha_2010_StructuralVectorAutoregressions}
+(i.e. the exercise called \emph{Combining Short-Run And Long-Run Restrictions}).
+
+A historical decomposition is a tool to quantify how important a structural shock was 
+in driving the behavior of the endogenous variables in a specific time period in the past.
+It allows to track at each point in time the role of structural shocks
+in driving the VAR's endogenous variables away from their steady state.
+
+Given the notation used in the lecture, the historical decomposition for a VAR(1)
+is based on the following recursive Wold representation:
+\begin{align*}
+y_t = A^t y_0 + \sum_{j=0}^{t-1} A^j B_0^{-1} \varepsilon_{t-j}
+\end{align*}
+Therefore, the historical decomposition of say \(y_{2}\) is given as a function of present and past structural shocks
+plus the initial condition:
+\begin{align*}
+y_{2} = \underbrace{A^2 y_0}_{\text{initial}} + \underbrace{A B_0^{-1}}_{\Theta_1} \varepsilon_{1} + \underbrace{B_0^{-1}}_{\Theta_0} \varepsilon_{2}
+\end{align*}
+Write an algorithm to compute the historical decomposition of the federal funds rate with respect to the three shocks in the model.
+
+Hints:
+\begin{itemize}
+	\item Use the companion form.
+\end{itemize} 
+
+\newpage
+\appendix
+
+\printbibliography
+\newpage
+\section{Codes}
+\lstinputlisting[label=drawRotationMatrixQ,title=\lstname,style=Matlab-editor,basicstyle=\footnotesize\mlttfamily,title=\lstname]{../progs/matlab/drawRotationMatrixQ.m}
+
+\newpage
+
+\section{Figures}
+\begin{figure}[htbp]\caption[]{Sign-identified IRFs}\label{fig:signSVAR}
+	\includegraphics[width=\textwidth]{../plots/signIRFs.jpg}
+\end{figure}
+
+\end{document}

literature/_biblio.bib

@@ -318,6 +318,18 @@ @incollection{Ploberger_2010_LawLargeNumbers
   isbn = {978-0-230-23885-5 978-0-230-28083-0}
 }
 
+@article{Rubio-Ramirez.Waggoner.Zha_2010_StructuralVectorAutoregressions,
+  title = {Structural {{Vector Autoregressions}}: {{Theory}} of {{Identification}} and {{Algorithms}} for {{Inference}}},
+  author = {{Rubio-Ram{\'i}rez}, Juan F. and Waggoner, Daniel F. and Zha, Tao},
+  year = {2010},
+  month = apr,
+  journal = {Review of Economic Studies},
+  volume = {77},
+  number = {2},
+  pages = {665--696},
+  doi = {10.1111/j.1467-937X.2009.00578.x}
+}
+
 @incollection{Schorfheide_2010_BayesianMethodsMacroeconometrics,
   title = {Bayesian Methods in Macroeconometrics},
   booktitle = {Macroeconometrics and {{Time Series Analysis}}},