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<!DOCTYPE html>
<html lang="en"><head>
<script src="index_files/libs/clipboard/clipboard.min.js"></script>
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<meta name="generator" content="quarto-1.3.361">
<meta name="author" content="by Tomasz Woźniak">
<title>Lecture 8: Bayesian Structural Vector Autoregressions</title>
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<div class="slides">
<section id="title-slide" data-background-color="#9933FF" class="quarto-title-block center">
<h1 class="title">Lecture 8: Bayesian Structural Vector Autoregressions</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
by Tomasz Woźniak
</div>
</div>
</div>
</section>
<section id="section" class="slide level2" data-background-color="#9933FF">
<h2></h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="structural-vector-autoregressions">Structural Vector Autoregressions</h3>
<h3 id="identification-of-structural-vars">Identification of Structural VARs</h3>
<h3 id="dynamic-causal-effects">Dynamic Causal Effects</h3>
<h3 id="bayesian-estimation">Bayesian Estimation</h3>
<h3 id="monetary-policy-analysis-using-the-bsvars-package">Monetary Policy Analysis Using the <a href="https://cran.r-project.org/package=bsvars">bsvars</a> Package</h3>
</section>
<section id="materials" class="slide level2" data-background-color="#9933FF">
<h2>Materials</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="lecture-slides-as-a-website">Lecture Slides <a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">as a Website</a></h3>
<h3 id="quarto-document-template-for-your-own-australian-monetary-policy-analysis">Quarto <a href="https://github.com/Bayesian-Econometrics-2023/be23-lecture8/blob/main/be23-lecture8.qmd">document template</a> for your own Australian monetary policy analysis</h3>
<h3 id="github-repo-to-reproduce-the-slides-and-results">GitHub <a href="https://github.com/Bayesian-Econometrics-2023/be23-lecture8">repo</a> to reproduce the slides and results</h3>
<h3 id="tasks">Tasks</h3>
</section>
<section id="structural-vector-autoregressions-1" class="slide level2" data-background-color="#9933FF">
<h2>Structural Vector Autoregressions</h2>
</section>
<section id="structural-vector-autoregressions-2" class="slide level2">
<h2>Structural Vector Autoregressions</h2>
<ul>
<li>go-to models for the analysis of policy effects</li>
</ul>
<div>
<ul>
<li class="fragment">facilitate the analysis of <strong>dynamic causal effects</strong> of a well-isolated cause</li>
<li class="fragment">extensively used for: <em>monetary</em> and <em>fiscal</em> policy, <em>financial</em> markets, …</li>
<li class="fragment">relatively simple to work with data and provide <em>empirical evidence on the propagation of shocks</em> through economies and markets</li>
<li class="fragment">provide data-driven stylised facts to be incorporated in theoretical model</li>
<li class="fragment">require identification of the cause of the dynamic effects</li>
<li class="fragment">extendible: <em>featuring many variations in specification</em>
<ul>
<li class="fragment">non-normality</li>
<li class="fragment">heteroskedasticity</li>
<li class="fragment">time-varying parameters</li>
<li class="fragment">Bayesian</li>
</ul></li>
<li class="fragment">Proposed by <a href="https://doi.org/10.2307/1912017">Sims (1980)</a></li>
</ul>
</div>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
</section>
<section id="structural-vector-autoregressions-3" class="slide level2">
<h2>Structural Vector Autoregressions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="the-model.">The model.</h3>
<p><span class="math display">\[\begin{align}
\text{VAR equation: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \epsilon_t\\[1ex]
\text{structural equation: }&& \mathbf{B}\epsilon_t &= u_t\\[1ex]
\text{structural shocks: }&& u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\mathbf{I}_N\right)
\end{align}\]</span></p>
<div class="fragment">
<!-- -->
<h3 id="notation.">Notation.</h3>
<ul>
<li><p><span class="math inline">\(\mathbf{B}\)</span> - <span class="math inline">\(N\times N\)</span> structural matrix of contemporaneous relationships</p></li>
<li><p><span class="math inline">\(u_t\)</span> - <span class="math inline">\(N\)</span>-vector of structural shocks at time <span class="math inline">\(t\)</span></p>
<p>Isolating these shocks allows us to <em>identify dynamic effects of uncorrelated shocks</em> on variables <span class="math inline">\(y_t\)</span></p></li>
<li><p><span class="math inline">\(\epsilon_t\)</span> - <span class="math inline">\(N\)</span>-vector with VAR errors at time <span class="math inline">\(t\)</span></p></li>
<li><p>the rest as in <a href="https://bayesian-econometrics-2023.github.io/be23-lecture7/#/varp-model">Lecture 7: Bayesian VARs</a></p></li>
</ul>
</div>
</section>
<section id="structural-vector-autoregressions-4" class="slide level2">
<h2>Structural Vector Autoregressions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="the-var-errors.">The VAR errors.</h3>
<p><span class="math display">\[\begin{align}
&&&\\
\text{structural equation: }&& \epsilon_t &= \mathbf{B}^{-1}u_t\\[1ex]
\text{structural shocks: }&& \epsilon_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\Sigma\right)\\[1ex]
\text{covariance: }&& \mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime} = \Theta_0\Theta_0'
\end{align}\]</span></p>
<h3 id="notation.-1">Notation.</h3>
<ul>
<li><span class="math inline">\(\mathbf\Sigma\)</span> - <span class="math inline">\(N\times N\)</span> covariance matrix of VAR errors</li>
<li><span class="math inline">\(\Theta_0 = \mathbf{B}^{-1}\)</span> - <span class="math inline">\(N\times N\)</span> matrix of <strong>contemporaneous effects</strong></li>
</ul>
</section>
<section id="structural-vector-autoregressions-5" class="slide level2">
<h2>Structural Vector Autoregressions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p>Plug the <em>VAR equation</em> into the <em>structural equation</em> to obtain:</p>
<p><span class="math display">\[\begin{align}
\mathbf{B}y_t &= \mathbf{B}\mathbf{A}_1 y_{t-1} + \dots + \mathbf{B}\mathbf{A}_p y_{t-p} + \mathbf{B}\boldsymbol\mu_0 + u_t\\[1ex]
&\\
\end{align}\]</span></p>
<h3 id="contemporaneous-relationships.">Contemporaneous relationships.</h3>
<p>Let <span class="math inline">\(N=2\)</span></p>
<p><span class="math display">\[\begin{align}
\mathbf{B}y_t &= \begin{bmatrix}B_{11}&B_{12}\\B_{21}&B_{22}\end{bmatrix}\begin{bmatrix}y_{1t}\\y_{2t}\end{bmatrix}
\end{align}\]</span></p>
</section>
<section id="structural-vector-autoregressions-6" class="slide level2">
<h2>Structural Vector Autoregressions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p>Plug the <em>structural equation</em> for <span class="math inline">\(\epsilon_t\)</span> into the <em>VAR equation</em> to obtain:</p>
<p><span class="math display">\[\begin{align}
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \mathbf{B}^{-1}u_t\\[1ex]
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \mathbf{\Theta}_0 u_t
\end{align}\]</span></p>
<h3 id="contemporaneous-effects.">Contemporaneous effects.</h3>
<p>Let <span class="math inline">\(N=2\)</span></p>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}y_{1t}\\y_{2t}\end{bmatrix} &= \dots +
\begin{bmatrix}\Theta_{11}&\Theta_{12}\\\Theta_{21}&\Theta_{22}\end{bmatrix}\begin{bmatrix}u_{1t}\\ u_{2t}\end{bmatrix}
\end{align}\]</span></p>
<h3 id="task.">Task.</h3>
<p>What is the contemporaneous effect of the first shock on the second variable?</p>
</section>
<section id="identification-of-structural-vars-1" class="slide level2" data-background-color="#9933FF">
<h2>Identification of Structural VARs</h2>
</section>
<section id="identification-of-svars-simplified" class="slide level2">
<h2>Identification of SVARs (Simplified)</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="covariance-and-structural-relationships.">Covariance and structural relationships.</h3>
<p><span class="math display">\[\begin{align}
&\\
\mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime}\\[1ex]
\end{align}\]</span></p>
<ul>
<li><span class="math inline">\(\mathbf\Sigma\)</span> can be estimated using data easily</li>
</ul>
<div>
<ul>
<li class="fragment">The relationship presents a system of equations to be solved for <span class="math inline">\(\mathbf{B}\)</span></li>
<li class="fragment"><span class="math inline">\(\mathbf\Sigma\)</span> is a <em>symmetric</em> <span class="math inline">\(N\times N\)</span> matrix</li>
<li class="fragment"><span class="math inline">\(\mathbf\Sigma\)</span> has <span class="math inline">\(N(N+1)/2\)</span> unique elements given equations</li>
<li class="fragment"><span class="math inline">\(\mathbf{B}\)</span> is an <span class="math inline">\(N\times N\)</span> matrix with <span class="math inline">\(N^2\)</span> unique elements to estimate</li>
<li class="fragment">We cannot estimate all elements of <span class="math inline">\(\mathbf{B}\)</span> using <span class="math inline">\(N(N+1)/2\)</span> equations</li>
<li class="fragment"><span class="math inline">\(\mathbf{B}\)</span> is <text style="color:#9933FF;"><strong>not identified</strong></text></li>
</ul>
</div>
</section>
<section id="identification-of-svars-simplified-1" class="slide level2">
<h2>Identification of SVARs (Simplified)</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="covariance-and-structural-relationships.-1">Covariance and structural relationships.</h3>
<p><span class="math display">\[\begin{align}
&\\
\mathbf\Sigma &= \mathbf{B}^{-1}\mathbf{B}^{-1\prime}\\[1ex]
\end{align}\]</span></p>
<h3 id="identification.">Identification.</h3>
<ul>
<li>Only <span class="math inline">\(N(N+1)/2\)</span> elements in <span class="math inline">\(\mathbf{B}\)</span> can be estimated</li>
<li>Impose <span class="math inline">\(N(N-1)/2\)</span> restrictions on <span class="math inline">\(\mathbf{B}\)</span> to solve the equation</li>
<li>This identifies the rows of <span class="math inline">\(\mathbf{B}\)</span> (and the columns of <span class="math inline">\(\mathbf\Theta_0\)</span>) up to a sign</li>
<li>Change the sign of any number of <span class="math inline">\(\mathbf{B}\)</span> rows and <span class="math inline">\(\mathbf\Sigma\)</span> will not change</li>
<li>Often <span class="math inline">\(\mathbf{B}\)</span> is made lower-triangular</li>
</ul>
</section>
<section id="identification-of-svars-simplified-2" class="slide level2">
<h2>Identification of SVARs (Simplified)</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="covariance-and-structural-relationships.-2">Covariance and structural relationships.</h3>
<p>Let <span class="math inline">\(N=2\)</span></p>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{11}&B_{12}\\ B_{21}&B_{22}\end{bmatrix}\\[1ex]
\end{align}\]</span></p>
<ul>
<li>3 unique elements in <span class="math inline">\(\mathbf\Sigma\)</span> - 3 equations in the system</li>
<li>4 elements in <span class="math inline">\(\mathbf{B}\)</span> cannot be estimated</li>
</ul>
<h3 id="identification.-1">Identification.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{11}& 0\\ B_{21}&B_{22}\end{bmatrix}\\[1ex]
\end{align}\]</span></p>
<ul>
<li>3 equations identify 3 elements in <span class="math inline">\(\mathbf{B}\)</span></li>
</ul>
</section>
<section id="identification-of-monetary-policy-shock" class="slide level2">
<h2>Identification of Monetary Policy Shock</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p>Consider a system of four variables:</p>
<p><span class="math display">\[\begin{align}
y_t = \begin{bmatrix} \Delta rgdp_t & \pi_t & cr_t & \Delta rtwi_t \end{bmatrix}'
\end{align}\]</span></p>
<ul>
<li><span class="math inline">\(\Delta rgdp_t\)</span> - real Gross Domestic Product growth</li>
<li><span class="math inline">\(\pi_t\)</span> - Consumer Price Index inflation</li>
<li><span class="math inline">\(cr_t\)</span> - Cash Rate Target - Australian nominal interest rate</li>
<li><span class="math inline">\(\Delta rtwi_t\)</span> - real Trade-Weighted Index rate of return (exchange rate)</li>
</ul>
<h3 id="identified-system.">Identified system.</h3>
<p>A lower-triangular matrix identifies:</p>
<ul>
<li>contemporaneous relationships <span class="math inline">\(\mathbf{B}\)</span></li>
<li>contemporaneous effects <span class="math inline">\(\mathbf\Theta_0\)</span></li>
<li>structural shocks <span class="math inline">\(u_t\)</span></li>
</ul>
</section>
<section id="identification-of-monetary-policy-shock-1" class="slide level2">
<h2>Identification of Monetary Policy Shock</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="identified-system.-1">Identified system.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots +
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}\]</span></p>
<h3 id="identified-shocks.">Identified shocks.</h3>
<p><span class="math inline">\(u_t^{ad}\)</span> - aggregate demand shock is exogenous to the rest of the system</p>
<p><span class="math inline">\(u_t^{as}\)</span> - aggregate supply shock</p>
<p><span class="math inline">\(u_t^{mps}\)</span> - monetary policy shock identified via Taylor’s Rule</p>
<p><span class="math inline">\(u_t^{ex}\)</span> - currency shock</p>
</section>
<section id="identification-of-monetary-policy-shock-2" class="slide level2">
<h2>Identification of Monetary Policy Shock</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="identified-system.-2">Identified system.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots +
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}\]</span></p>
<h3 id="tasks.">Tasks.</h3>
<ul>
<li>Write out the third equation for the cash rate.</li>
<li>Let <span class="math inline">\(B_{33}>0\)</span>. What values of <span class="math inline">\(B_{31}\)</span> and <span class="math inline">\(B_{32}\)</span> does theory imply?</li>
</ul>
</section>
<section id="identification-of-monetary-policy-shock-3" class="slide level2">
<h2>Identification of Monetary Policy Shock</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="identified-system.-3">Identified system.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &= \dots +
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \end{bmatrix}
\end{align}\]</span></p>
<h3 id="monetary-policy-shock.">Monetary policy shock.</h3>
<ul>
<li>is uncorrelated with any other shock</li>
<li>consists of the unanticipated (unpredictable) part of the <em>monetary policy instrument</em>, interest rate</li>
<li>In this model, the systematic part of the monetary policy consists of:
<ul>
<li>contemporaneous relationships with GDP and inflation</li>
<li>lagged relationships with all variables</li>
</ul></li>
</ul>
</section>
<section id="identification-via-heteroskedasticity" class="slide level2">
<h2>Identification via Heteroskedasticity</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p>Suppose that:</p>
<ul>
<li>there are two covariances, <span class="math inline">\(\mathbf\Sigma_1\)</span> and <span class="math inline">\(\mathbf\Sigma_2\)</span>, associated with the sample</li>
<li>matrix <span class="math inline">\(\mathbf{B}\)</span> does not change over time</li>
<li>structural shocks are heteroskedastic with covariances <span class="math inline">\(\text{diag}\left(\boldsymbol\sigma_1^2\right)\)</span> and <span class="math inline">\(\text{diag}\left(\boldsymbol\sigma_2^2\right)\)</span></li>
</ul>
<p><span class="math display">\[\begin{align}
\mathbf\Sigma_1 &= \mathbf{B}^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}^{-1\prime}\\[1ex]
\mathbf\Sigma_2 &= \mathbf{B}^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}^{-1\prime}
\end{align}\]</span></p>
<h3 id="identification.-2">Identification.</h3>
<ul>
<li><span class="math inline">\(\mathbf\Sigma_1\)</span> and <span class="math inline">\(\mathbf\Sigma_2\)</span> contain <span class="math inline">\(N^2+N\)</span> unique elements</li>
<li>All <span class="math inline">\(N^2\)</span> elements of <span class="math inline">\(\mathbf{B}\)</span> can be estimated</li>
<li>Both <span class="math inline">\(N\)</span>-vectors <span class="math inline">\(\boldsymbol\sigma_1^2\)</span> and <span class="math inline">\(\boldsymbol\sigma_2^2\)</span> can be estimated due to additional restriction: <span class="math inline">\(E\left[\text{diag}\left(\boldsymbol\sigma_i^2\right)\right] = \mathbf{I}_N\)</span></li>
</ul>
</section>
<section id="identification-via-heteroskedasticity-1" class="slide level2">
<h2>Identification via Heteroskedasticity</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p>The setup can be generalised to conditional heteroskedasticity of structural shocks</p>
<p><span class="math display">\[\begin{align}
u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N, \text{diag}\left(\boldsymbol\sigma_t^2\right)\right)\\[1ex]
\mathbf\Sigma_t &= \mathbf{B}^{-1}\text{diag}\left(\boldsymbol\sigma_t^2\right)\mathbf{B}^{-1\prime}\\[1ex]
E\left[\text{diag}\left(\boldsymbol\sigma_t^2\right)\right] &= \mathbf{I}_N
\end{align}\]</span></p>
<h3 id="identification.-3">Identification.</h3>
<ul>
<li>Matrix <span class="math inline">\(\mathbf{B}\)</span> is identified up to its rows’ sign change and equations’ reordering</li>
<li>Structural shocks’ conditional variances <span class="math inline">\(\boldsymbol\sigma_t^2\)</span> can be estimated</li>
</ul>
<h3 id="heteroskedasticity-modeling.">Heteroskedasticity Modeling.</h3>
<p>Choose any (conditional) variance model for <span class="math inline">\(\boldsymbol\sigma_t^2\)</span> that fits the data well.</p>
</section>
<section id="dynamic-causal-effects-1" class="slide level2" data-background-color="#9933FF">
<h2>Dynamic Causal Effects</h2>
</section>
<section id="impulse-response-functions" class="slide level2">
<h2>Impulse response functions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="definition.">Definition.</h3>
<p><strong>Impulse response functions</strong> to <em>orthogonal shocks</em> computed for an empirically relevant SVAR model are considered the <text style="color:#9933FF;"><strong>dynamic causal effects</strong></text> of the underlying shocks <span class="math inline">\(u_t\)</span> on economic measurements <span class="math inline">\(y_{t+i}\)</span> <span class="math inline">\(i\)</span> periods ahead.</p>
<p><span class="math display">\[\begin{align*}
\frac{\partial y_{n.t+i}}{\partial u_{j.t}}&=\theta_{nj.i}
\end{align*}\]</span></p>
<ul>
<li><p><span class="math inline">\(\theta_{nj.i}\)</span> - response of <span class="math inline">\(n\)</span>th variable to <span class="math inline">\(j\)</span>th shock <span class="math inline">\(i\)</span> periods after shock’s occurrence</p>
<p>for <span class="math inline">\(i=0,1,\dots,h\)</span> and <span class="math inline">\(n,j=1,\dots,N\)</span></p></li>
</ul>
</section>
<section id="impulse-response-functions-1" class="slide level2">
<h2>Impulse response functions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="definition.-1">Definition.</h3>
<p><strong>Impulse response functions</strong> to <em>orthogonal shocks</em> computed for an empirically relevant SVAR model are considered the <text style="color:#9933FF;"><strong>dynamic causal effects</strong></text> of the underlying shocks <span class="math inline">\(u_t\)</span> on economic measurements <span class="math inline">\(y_{t+i}\)</span> <span class="math inline">\(i\)</span> periods ahead.</p>
<p><span class="math display">\[\begin{align*}
\frac{\partial y_{t+i}}{\partial u_t}&=\underset{N\times N}{\mathbf\Theta_i}
\end{align*}\]</span></p>
<ul>
<li><p><span class="math inline">\(\mathbf\Theta_i\)</span> - responses of all of the variables to all of the shocks <span class="math inline">\(i\)</span> periods after shocks’ occurrence</p>
<p>for <span class="math inline">\(i=0,1,\dots,h\)</span> and <span class="math inline">\(n,j=1,\dots,N\)</span></p></li>
</ul>
</section>
<section id="impulse-response-functions-2" class="slide level2">
<h2>Impulse response functions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="at-finite-horizon.">At finite horizon.</h3>
<p>Define matrices</p>
<p><span class="math display">\[
\underset{(pN\times pN)}{\mathbb{A}} = \begin{bmatrix}\mathbf{A}_1 & \mathbf{A}_2 &\dots& \mathbf{A}_p\\ &\mathbf{I}_{N(p-1)}&&\mathbf{0}_{N(p-1)\times N} \end{bmatrix}\quad\text{and}\quad
\underset{(N\times pN)}{\mathbf{J}} = \begin{bmatrix} \mathbf{I}_{N} & \mathbf{0}_{N\times N(p-1)} \end{bmatrix}
\]</span> Impulse response at horizon <span class="math inline">\(i=0,1,\dots,h\)</span> are equal to:</p>
<p><span class="math display">\[\begin{align}
\mathbf\Theta_i &= \mathbf{J}\mathbb{A}^i\mathbf{J}'\mathbf{B}^{-1}
\end{align}\]</span> where <span class="math inline">\(\mathbb{A}^0=\mathbf{I}_N\)</span>, <span class="math inline">\(\mathbb{A}^1=\mathbb{A}\)</span>, <span class="math inline">\(\mathbb{A}^2=\mathbb{A}\mathbb{A}\)</span>, …</p>
<h3 id="at-infinite-horizon.">At infinite horizon.</h3>
<p>Inform about the value of the effect in the long run.</p>
<p><span class="math display">\[\begin{align}
\mathbf\Theta_{\infty} &= \left( \mathbf{I}_N - \mathbf{A}_1 - \dots - \mathbf{A}_p \right)^{-1}\mathbf{B}^{-1}
\end{align}\]</span></p>
</section>
<section id="impulse-response-functions-3" class="slide level2">
<h2>Impulse response functions</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="bayesian-estimation.">Bayesian Estimation.</h3>
<h4 id="step-1.-estimate-the-model">Step 1. Estimate the model</h4>
<p>Obtain a sample from the posterior distribution <span class="math display">\[\left\{ \mathbf{A}^{(s)},\mathbf{B}^{(s)} \right\}_{s=1}^{S}\]</span></p>
<h4 id="step-2.-compute-impulse-responses">Step 2. Compute impulse responses</h4>
<p>For each of the <span class="math inline">\(S\)</span> draws, compute <span class="math inline">\(\mathbf\Theta_i^{(s)}\)</span> as a function of <span class="math inline">\(\mathbf{A}^{(s)}\)</span> and <span class="math inline">\(\mathbf{B}^{(s)}\)</span> and return <span class="math display">\[\left\{\mathbf\Theta_i^{(s)}\right\}_{s=1}^{S}\]</span> as a sample drew from the posterior distribution of <span class="math inline">\(\Theta_i\)</span> given data.</p>
</section>
<section id="bayesian-estimation-1" class="slide level2" data-background-color="#9933FF">
<h2>Bayesian Estimation</h2>
</section>
<section id="bayesian-estimation-2" class="slide level2">
<h2>Bayesian Estimation</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p><span class="math inline">\(\left.\right.\)</span></p>
<h4 id="gibbs-sampler-by-waggoner-zha-2003"><strong>Gibbs sampler</strong> by <a href="https://doi.org/10.1016/S0165-1889(02)00168-9">Waggoner & Zha (2003)</a></h4>
<p>facilitates estimation of Bayesian SVARs for</p>
<ul>
<li>lower-triangular and non-recursive identification patterns of exclusion restrictions</li>
<li>over-identifying (more than <span class="math inline">\(N(N − 1)/2)\)</span> exclusion restrictions</li>
<li>models identified via heteroskedasticity</li>
</ul>
<p><span class="math inline">\(\left.\right.\)</span></p>
<h4 id="further-extensions-include-svars">Further extensions include SVARs</h4>
<ul>
<li>identified through non-normal residuals</li>
<li>identified by zero and sign restrictions</li>
<li>identified using instrumental variables (Proxy SVARs)</li>
</ul>
</section>
<section id="bayesian-estimation-3" class="slide level2">
<h2>Bayesian Estimation</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="exclusion-restrictions-on-the-rows-of-mathbfb">Exclusion restrictions on the rows of <span class="math inline">\(\mathbf{B}\)</span></h3>
<p><span class="math display">\[
\underset{(1\times N)}{\mathbf{B}_{[n\cdot]}} = \underset{(1\times r_n)}{\mathbf{b}_n} \underset{(r_n\times N)}{V_n} \qquad\text{such that}\qquad
\mathbf{B} = \begin{bmatrix} \mathbf{b}_1V_1\\ \vdots \\ \mathbf{b}_NV_N \end{bmatrix}
\]</span></p>
<ul>
<li><span class="math inline">\(\mathbf{b}_n\)</span> - a <span class="math inline">\(1\times r_n\)</span> vector of unrestricted elements of <span class="math inline">\(n\)</span> row of <span class="math inline">\(\mathbf{B}\)</span></li>
<li><span class="math inline">\(V_n\)</span> - an <span class="math inline">\(r_n\times N\)</span> <em>fixed</em> matrix of ones and zeros</li>
</ul>
<h3 id="example.">Example.</h3>
<p><span class="math display">\[\mathbf{b}_n = \begin{bmatrix} b_1 & b_2\end{bmatrix}\quad V_n = \begin{bmatrix} 1&0&0\\0&0&1\end{bmatrix} \quad\rightarrow\quad \mathbf{B}_{[n\cdot]} = \begin{bmatrix} b_1&0 & b_2\end{bmatrix}
\]</span></p>
</section>
<section id="bayesian-estimation-4" class="slide level2">
<h2>Bayesian Estimation</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="the-nth-structural-equation.">The <span class="math inline">\(n\)</span>th Structural Equation.</h3>
<p><span class="math display">\[\begin{align*}
\mathbf{b}_nV_n\epsilon_t &= u_{n.t}\\
u_{n.t} &\sim N(0,1)
\end{align*}\]</span></p>
<h3 id="matrix-notation.">Matrix Notation.</h3>
<p><span class="math display">\[\begin{align*}
E V_n' \mathbf{b}_n' &= U_n\\
U_n &\sim N_T\left(\mathbf{0}_T,I_T\right)\\[2ex]
\underset{(T\times1)}{U_n} &= \begin{bmatrix} u_{n.1} & \dots & u_{n.T} \end{bmatrix}'\\
\underset{(T\times N)}{E} &\text{ - defined as before}
\end{align*}\]</span></p>
</section>
<section id="bayesian-estimation-5" class="slide level2">
<h2>Bayesian Estimation</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="likelihood-function.">Likelihood function.</h3>
<p><span class="math display">\[\begin{align*}
L(\mathbf{A},\mathbf{B}|Y,X) &\propto
|\text{det}\left( \mathbf{B} \right)|^{T}\exp\left\{ -\frac{1}{2}\sum_{n=1}^N \mathbf{b}_nV_nE'EV_n'\mathbf{b}_n' \right\}\\[1ex]
E &= Y - X\mathbf{A}
\end{align*}\]</span></p>
<h3 id="hierarchical-prior-for-mathbfb">Hierarchical prior for <span class="math inline">\(\mathbf{B}\)</span></h3>
<p><span class="math display">\[\begin{align*}
\mathbf{b}_n | \gamma_B &\sim N_{r_n}\left(\mathbf{0}_{r_n}, \gamma_B V_n\underline{S}^{-1}V_n'\right)\\[1ex]
\gamma_B &\sim IG2(\underline{s},\underline{\nu})
\end{align*}\]</span></p>
<ul>
<li><span class="math inline">\(\underline{S}\)</span> - <span class="math inline">\(N\times N\)</span> prior scale matrix</li>
<li><span class="math inline">\(\underline{s}\)</span> and <span class="math inline">\(\underline{\nu}\)</span> positive scalars of scale and shape</li>
</ul>
</section>
<section id="bayesian-estimation-6" class="slide level2">
<h2>Bayesian Estimation</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<h3 id="kernel-of-the-full-conditional-posterior-for-mathbfb">Kernel of the full conditional posterior for <span class="math inline">\(\mathbf{B}\)</span></h3>
<p><span class="math display">\[\begin{align*}
p(\mathbf{B}|Y,X,\mathbf{A}, \gamma_B)&\propto
|\text{det}\left( \mathbf{B} \right)|^{T}\exp\left\{ -\frac{1}{2}\sum_{n=1}^N \mathbf{b}_n \overline{S}_n^{-1}\mathbf{b}_n' \right\}\\[1ex]
\overline{S}_n^{-1} &= V_n\left[ \gamma_B^{-1}\underline{S}^{-1} + (Y-X\mathbf{A})'(Y-X\mathbf{A}) \right]V_n'\\[2ex]
\end{align*}\]</span></p>
<ul>
<li>This is a kernel of a <em>Generalised-Normal</em> distribution</li>
<li>A feasible Gibbs sampler was proposed by by <a href="https://doi.org/10.1016/S0165-1889(02)00168-9">Waggoner & Zha (2003)</a></li>
<li>The Gibbs sampler draws from the full conditional posterior for <span class="math inline">\(n = 1,\dots,N\)</span>: <span class="math display">\[ p(\mathbf{b}_n | \mathbf{b}_1,\dots, \mathbf{b}_{n-1},\mathbf{b}_{n+1}, \mathbf{b}_N, \mathbf{A}, \gamma_B, Y, X) \]</span></li>
</ul>
</section>
<section id="monetary-policy-analysis-using-r-package-bsvars" class="slide level2" data-background-color="#9933FF">
<h2>Monetary Policy Analysis Using R Package <a href="https://cran.r-project.org/package=bsvars">bsvars</a></h2>
</section>
<section id="bsvars-an-r-package" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars">bsvars</a> an R Package</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<img data-src="cran.png" class="r-stretch"></section>
<section id="bsvars-an-r-package-features" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars">bsvars</a> an R Package: Features</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<ul>
<li>Bayesian estimation of Structural VARs</li>
<li>identification via:
<ul>
<li>exclusion restrictions</li>
<li>heteroskedasticity</li>
<li>non-normality</li>
</ul></li>
<li>six heteroskedastic processes</li>
<li>efficient and fast Gibbs sampler</li>
<li>excellent computational speed</li>
<li>frontier econometric techniques</li>
<li>compiled code using <strong>cpp</strong> via <a href="https://www.rcpp.org"><strong>Rcpp</strong></a> and <a href="https://cran.r-project.org/package=RcppArmadillo"><strong>RcppArmadillo</strong></a></li>
<li>data analysis in <strong>R</strong></li>
</ul>
</section>
<section id="bsvars-an-r-package-features-1" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars">bsvars</a> an R Package: Features</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<img data-src="progress.png" class="r-stretch"></section>
<section id="bsvars-an-r-package-features-2" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars">bsvars</a> an R Package: Features</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<ul>
<li>simple model setup using <code>specify_*()</code></li>
<li>flexibility in setting priors, restrictions, etc.</li>
<li>one function for estimation <code>estimate()</code></li>
<li>posterior processing utility functions</li>
</ul>
<h3 id="the-simplest-workflow-using-pipe.">The simplest workflow using pipe.</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1"></a><span class="fu">library</span>(bsvars)</span>
<span id="cb1-2"><a href="#cb1-2"></a></span>
<span id="cb1-3"><a href="#cb1-3"></a><span class="fu">data</span>(us_fiscal_lsuw)</span>
<span id="cb1-4"><a href="#cb1-4"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb1-5"><a href="#cb1-5"></a></span>
<span id="cb1-6"><a href="#cb1-6"></a>us_fiscal_lsuw <span class="sc">|></span></span>
<span id="cb1-7"><a href="#cb1-7"></a> specify_bsvar_sv<span class="sc">$</span><span class="fu">new</span>(<span class="at">p =</span> <span class="dv">2</span>) <span class="sc">|></span></span>
<span id="cb1-8"><a href="#cb1-8"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">1000</span>) <span class="sc">|></span> </span>
<span id="cb1-9"><a href="#cb1-9"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">5000</span>) <span class="sc">|></span> </span>
<span id="cb1-10"><a href="#cb1-10"></a> <span class="fu">compute_impulse_responses</span>(<span class="at">horizon =</span> <span class="dv">8</span>) <span class="ot">-></span> irfs</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="australian-monetary-policy-analysis" class="slide level2">
<h2>Australian Monetary Policy Analysis</h2>
<div class="footer">
<p><a href="https://bayesian-econometrics-2023.github.io/be23-lecture8/">Bayesian Structural VARs</a></p>
</div>
<p>Based on <a href="https://doi.org/10.1111/1475-4932.12345">Turnip (2017)</a></p>
<h3 id="system-of-four-variables.">System of four variables.</h3>
<p><span class="math display">\[\begin{align}
y_t = \begin{bmatrix} \Delta rgdp_t & \pi_t & cr_t & \Delta rtwi_t \end{bmatrix}'
\end{align}\]</span></p>
<h3 id="alternative-identification-patterns.">Alternative identification patterns.</h3>
<p><span class="math display">\[\begin{align}
\textbf{lower-triangular} && \textbf{extended}\\
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&0\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix} &&
\begin{bmatrix}
B_{11}&0&0&0\\
B_{21}&B_{22}&0&0\\
B_{31}&B_{32}&B_{33}&B_{34}\\
B_{41}&B_{42}&B_{43}&B_{44}
\end{bmatrix}
\begin{bmatrix} \Delta rgdp_t \\ \pi_t \\ cr_t \\ \Delta rtwi_t \end{bmatrix}
\end{align}\]</span></p>
<ul>
<li>In the <strong>extended</strong> model, the monetary policy shock is not identified</li>
<li>Use identification via heteroskedasticity to identify it</li>
</ul>
</section>
<section id="four-variable-monetary-system" class="slide level2">
<h2>Four-Variable Monetary System</h2>
<div class="cell" data-layout-align="center" data-hash="index_cache/revealjs/data_7bf79ad4964356564d1cb4eddc351a9e">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1"></a><span class="co"># Gross domestic product (GDP); Chain volume</span></span>
<span id="cb2-2"><a href="#cb2-2"></a>rgdp_dwnld <span class="ot">=</span> readrba<span class="sc">::</span><span class="fu">read_rba</span>(<span class="at">series_id =</span> <span class="st">"GGDPCVGDP"</span>)</span>
<span id="cb2-3"><a href="#cb2-3"></a>rgdp_tmp <span class="ot">=</span> xts<span class="sc">::</span><span class="fu">xts</span>(rgdp_dwnld<span class="sc">$</span>value, rgdp_dwnld<span class="sc">$</span>date, <span class="at">tclass =</span> <span class="st">'yearqtr'</span>)</span>
<span id="cb2-4"><a href="#cb2-4"></a>drgdp <span class="ot">=</span> <span class="fu">na.omit</span>(<span class="dv">400</span> <span class="sc">*</span> <span class="fu">diff</span>(<span class="fu">log</span>(rgdp_tmp)))</span>
<span id="cb2-5"><a href="#cb2-5"></a>drgdp <span class="ot">=</span> xts<span class="sc">::</span><span class="fu">to.quarterly</span>(drgdp, <span class="at">OHLC =</span> <span class="cn">FALSE</span>)</span>
<span id="cb2-6"><a href="#cb2-6"></a></span>
<span id="cb2-7"><a href="#cb2-7"></a><span class="co"># Consumer price index; All groups; Quarterly change (in per cent)</span></span>
<span id="cb2-8"><a href="#cb2-8"></a>picpi_dwnld <span class="ot">=</span> readrba<span class="sc">::</span><span class="fu">read_rba</span>(<span class="at">series_id =</span> <span class="st">"GCPIAGSAQP"</span>)</span>
<span id="cb2-9"><a href="#cb2-9"></a>pi <span class="ot">=</span> <span class="dv">4</span> <span class="sc">*</span> xts<span class="sc">::</span><span class="fu">xts</span>(picpi_dwnld<span class="sc">$</span>value, picpi_dwnld<span class="sc">$</span>date, <span class="at">tclass =</span> <span class="st">'yearqtr'</span>)</span>