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index-speaker.html
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<!DOCTYPE html>
<html lang="en"><head>
<script src="index_files/libs/clipboard/clipboard.min.js"></script>
<script src="index_files/libs/quarto-html/tabby.min.js"></script>
<script src="index_files/libs/quarto-html/popper.min.js"></script>
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<link href="index_files/libs/quarto-html/quarto-syntax-highlighting.css" rel="stylesheet" id="quarto-text-highlighting-styles"><meta charset="utf-8">
<meta name="generator" content="quarto-1.3.361">
<meta name="author" content="by Tomasz Woźniak">
<title>Bayesian Vector Autoregressions</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
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code{white-space: pre-wrap;}
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code span { color: #003b4f; } /* Normal */
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<div class="reveal">
<div class="slides">
<section id="title-slide" data-background-color="#F500BD" class="quarto-title-block center">
<h1 class="title">Bayesian 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="#F500BD">
<h2></h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="vector-autoregressions">Vector Autoregressions</h3>
<h3 id="three-useful-distributions">Three Useful Distributions</h3>
<h3 id="bayesian-estimation">Bayesian Estimation</h3>
<h3 id="minnesota-and-dummy-observations-prior">Minnesota and Dummy Observations Prior</h3>
<h3 id="bayesian-estimation-for-hierarchical-prior">Bayesian Estimation for Hierarchical Prior</h3>
<h3 id="bayesian-forecasting-using-vars">Bayesian Forecasting using VARs</h3>
<h3 id="us-data-analysis-using-r-package-bsvarsigns">US Data Analysis Using R Package bsvarSIGNs</h3>
</section>
<section id="section-1" class="slide level2" data-background-color="#001D31">
<h2></h2>
<p><img data-src="bsvarSIGNs.png" class="absolute" style="top: 30px; right: 275px; width: 500px; "></p>
</section>
<section id="materials" class="slide level2" data-background-color="#F500BD">
<h2>Materials</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="lecture-slides-as-a-website">Lecture Slides <a href="https://bsvars.github.io/2024-10-be24-bsvarSIGNs">as a Website</a></h3>
<h3 id="a-dedicated-reading-woźniak-2016-aerev">A Dedicated Reading <a href="https://doi.org/10.1111/1467-8462.12179">Woźniak (2016, AERev)</a></h3>
<h3 id="quarto-document-template-for-your-own-australian-data-forecasting"><strong>Quarto</strong> <a href="https://github.com/bsvars/2024-10-be24-bsvarSIGNs/blob/main/be23-AUdata.qmd">document template</a> for your own Australian data forecasting</h3>
<h3 id="github-repo-to-reproduce-the-slides-and-results">GitHub <a href="https://github.com/bsvars/2024-10-be24-bsvarSIGNs">repo</a> to reproduce the slides and results</h3>
<h3 id="a-kahoot-quiz">A <a href="https://kahoot.it/">Kahoot!</a> Quiz</h3>
<h3 id="tasks">Tasks</h3>
</section>
<section id="vector-autoregressions-1" class="slide level2" data-background-color="#F500BD">
<h2>Vector Autoregressions</h2>
</section>
<section id="vector-autoregressions-2" class="slide level2">
<h2>Vector Autoregressions</h2>
<ul>
<li>go-to models for forecasting</li>
</ul>
<div>
<ul>
<li class="fragment">simple: <em>linear and Gaussian</em></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">interpretable
<ul>
<li class="fragment">Granger causality</li>
<li class="fragment">spillovers</li>
<li class="fragment">networks</li>
<li class="fragment">structural</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://bsvars.github.io/2024-10-be24-bsvarSIGNs">Bayesian VARs</a></p>
</div>
</section>
<section id="varp-model" class="slide level2">
<h2>VAR(p) Model</h2>
<h3 id="model-equations.">Model equations.</h3>
<p><span class="math display">\[\begin{align*}
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \epsilon_t\\
\epsilon_t|Y_{t-1} &\sim iidN\left(\mathbf{0}_N,\mathbf\Sigma\right)
\end{align*}\]</span> for <span class="math inline">\(t=1,\dots,T\)</span></p>
<div class="fragment">
<!-- -->
<h3 id="notation.">Notation.</h3>
<ul>
<li><span class="math inline">\(y_t\)</span> is an <span class="math inline">\(N\times 1\)</span> vector of observations at time <span class="math inline">\(t\)</span></li>
<li><span class="math inline">\(\mathbf{A}_i\)</span> - <span class="math inline">\(N\times N\)</span> matrix of autoregressive slope parameters</li>
<li><span class="math inline">\(\boldsymbol\mu_0\)</span> - <span class="math inline">\(N\times 1\)</span> vector of constant terms</li>
<li><span class="math inline">\(\epsilon_t\)</span> - <span class="math inline">\(N\times 1\)</span> vector of error terms - a multivariate white noise process</li>
<li><span class="math inline">\(Y_{t-1}\)</span> - information set collecting observations on} <span class="math inline">\(y\)</span> up to time <span class="math inline">\(t-1\)</span></li>
<li><span class="math inline">\(\mathbf\Sigma\)</span> - <span class="math inline">\(N\times N\)</span> covariance matrix of the error term</li>
</ul>
</div>
<div class="footer">
<p><a href="https://bsvars.github.io/2024-10-be24-bsvarSIGNs">Bayesian VARs</a></p>
</div>
</section>
<section id="a-bivariate-var2-model" class="slide level2">
<h2>A Bivariate VAR(2) Model</h2>
<p>Let the number of variable <span class="math inline">\(N=2\)</span> and the lag order <span class="math inline">\(p=2\)</span>.</p>
<p>Then, the model equation is:</p>
<span class="math display">\[\begin{align*}
\begin{bmatrix} y_{1.t} \\ y_{2.t} \end{bmatrix}
&= \begin{bmatrix} \mathbf{A}_{1.11} & \mathbf{A}_{1.12} \\ \mathbf{A}_{1.21} & \mathbf{A}_{1.22} \end{bmatrix} \begin{bmatrix} y_{1.t-1} \\ y_{2.t-1} \end{bmatrix}
+ \begin{bmatrix} \mathbf{A}_{2.11} & \mathbf{A}_{2.12} \\ \mathbf{A}_{2.21} & \mathbf{A}_{2.22} \end{bmatrix} \begin{bmatrix} y_{1.t-2} \\ y_{2.t-2} \end{bmatrix} + \begin{bmatrix} \boldsymbol\mu_{0.1} \\ \boldsymbol\mu_{0.2} \end{bmatrix} + \begin{bmatrix} \epsilon_{1.t} \\ \epsilon_{2.t} \end{bmatrix}\\[2ex]
\begin{bmatrix} \epsilon_{1.t} \\ \epsilon_{2.t} \end{bmatrix} &\Big|Y_{t-1} \sim iid N_2\left( \begin{bmatrix} 0\\ 0\end{bmatrix}, \begin{bmatrix}\boldsymbol\sigma_1^2 & \boldsymbol\sigma_{12} \\ \boldsymbol\sigma_{12} & \boldsymbol\sigma_2^2\end{bmatrix} \right)
\end{align*}\]</span>
<div class="fragment">
<h3 id="task">Task:</h3>
<p>Perform the matrix multiplications and write out the equations for <span class="math inline">\(y_{1.t}\)</span>.</p>
<div class="footer">
<p><a href="https://bsvars.github.io/2024-10-be24-bsvarSIGNs">Bayesian VARs</a></p>
</div>
</div>
</section>
<section id="three-useful-distributions-1" class="slide level2" data-background-color="#F500BD">
<h2>Three Useful Distributions</h2>
</section>
<section id="matrix-variate-normal-distribution" class="slide level2">
<h2>Matrix-Variate Normal Distribution</h2>
<p>A <span class="math inline">\(K\times N\)</span> matrix <span class="math inline">\(\mathbf{A}\)</span> is said to follow a <em>matrix-variate normal</em> distribution: <span class="math display">\[ \mathbf{A} \sim MN_{K\times N}\left( M, Q, P \right), \]</span> where</p>
<ul>
<li><span class="math inline">\(M\)</span> - a <span class="math inline">\(K\times N\)</span> matrix of the mean</li>
<li><span class="math inline">\(Q\)</span> - a <span class="math inline">\(N\times N\)</span> row-specific covariance matrix</li>
<li><span class="math inline">\(P\)</span> - a <span class="math inline">\(K\times K\)</span> column-specific covariance matrix</li>
</ul>
<p>if <span class="math inline">\(\text{vec}(\mathbf{A})\)</span> is multivariate normal: <span class="math display">\[ \text{vec}(\mathbf{A}) \sim N_{KN}\left( \text{vec}(M), Q\otimes P \right) \]</span></p>
<h3 id="density-function.">Density function.</h3>
<span class="math display">\[\begin{align*}
MN_{K\times N}\left( M, Q, P \right) &\propto \exp\left\{ -\frac{1}{2}\text{tr}\left[ Q^{-1}(\mathbf{A}-M)'P^{-1}(\mathbf{A}-M) \right] \right\}
\end{align*}\]</span>
<ul>
<li><span class="math inline">\(\text{tr}()\)</span> is a trace of a matrix - a sum of diagonal elements</li>
</ul>
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</section>
<section id="inverse-wishart-distribution" class="slide level2">
<h2>Inverse Wishart Distribution</h2>
<p>An <span class="math inline">\(N\times N\)</span> square symmetric and positive definite matrix <span class="math inline">\(\mathbf\Sigma\)</span> follows an <em>inverse Wishart</em> distribution: <span class="math display">\[ \mathbf\Sigma \sim IW_{N}\left( S, \nu \right) \]</span> where</p>
<ul>
<li><span class="math inline">\(S\)</span> is <span class="math inline">\(N\times N\)</span> positive definite symmetric matrix called the scale matrix</li>
<li><span class="math inline">\(\nu \geq N\)</span> denotes degrees of freedom, if its density is given by:</li>
</ul>
<h3 id="density-function.-1">Density function.</h3>
<span class="math display">\[\begin{align*}
IW_{N}\left( S, \nu \right) \propto \text{det}(\mathbf\Sigma)^{-\frac{\nu+N+1}{2}}\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf\Sigma^{-1} S \right] \right\}
\end{align*}\]</span>
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</section>
<section id="normal-inverse-wishart-distribution" class="slide level2">
<h2>Normal-Inverse Wishart Distribution</h2>
<span class="math display">\[\begin{align*}
\mathbf{A}|\mathbf\Sigma &\sim MN_{K\times N}\left( M, \mathbf\Sigma, P \right)\\
\mathbf\Sigma &\sim IW_{N}\left( S, \nu \right)
\end{align*}\]</span>
<p>then the joint distribution of <span class="math inline">\((\mathbf{A},\mathbf\Sigma)\)</span> is <em>normal-inverse Wishart</em> <span class="math display">\[
p(\mathbf{A},\mathbf\Sigma) = NIW_{K\times N}\left( M,P,S,\nu\right)
\]</span></p>
<h3 id="density-function.-2">Density function.</h3>
<span class="math display">\[\begin{align*}
NIW_{K\times N}\left( M,P,S,\nu\right) \propto &\text{det}(\mathbf{\Sigma})^{-(\nu+N+K+1)/2}\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1} S \right] \right\}\\
&\times\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1} (\mathbf{A}-M)'P^{-1}(\mathbf{A}-M) \right] \right\}
\end{align*}\]</span>
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</section>
<section id="example-error-term-distribution" class="slide level2">
<h2>Example: Error Term Distribution</h2>
<p>The model assumptions state: <span class="math display">\[\begin{align*}
\epsilon_t|Y_{t-1} &\sim iidN_N\left(\mathbf{0}_N,\mathbf\Sigma\right)
\end{align*}\]</span></p>
<p>Collect error term vectors in a <span class="math inline">\(T\times N\)</span> matrix: <span class="math display">\[\underset{(T\times N)}{E}= \begin{bmatrix}\epsilon_1 & \epsilon_2 & \dots & \epsilon_{T}\end{bmatrix}'\]</span></p>
<p>Error term matrix is matrix-variate distributed: <span class="math display">\[\begin{align*}
E|X &\sim MN_{T\times N}\left(\mathbf{0}_{T\times N},\mathbf\Sigma, I_T\right)
\end{align*}\]</span></p>
<div class="fragment fade-in">
<!-- -->
<h3 id="tasks-what-is">Tasks: what is</h3>
<ul>
<li>the covariance of <span class="math inline">\(\text{vec}(E)\)</span></li>
<li>the distribution of the first equation error terms <span class="math inline">\(\begin{bmatrix}\epsilon_{1.1} &\dots&\epsilon_{1.T}\end{bmatrix}'\)</span></li>
</ul>
</div>
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</section>
<section id="example-univariate-inverse-wishart-distribution" class="slide level2">
<h2>Example: Univariate <em>Inverse Wishart</em> Distribution</h2>
<p>The <em>inverse Wishart</em> density function is proportional to: <span class="math display">\[\begin{align*}
\text{det}(\mathbf\Sigma)^{-\frac{\nu+N+1}{2}}\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf\Sigma^{-1} S \right] \right\}
\end{align*}\]</span></p>
<p>Consider a case where:</p>
<ul>
<li><span class="math inline">\(N=1\)</span></li>
<li>the matrix <span class="math inline">\(\mathbf\Sigma\)</span> is replaced by a scalar <span class="math inline">\(\boldsymbol\sigma^2\)</span></li>
</ul>
<div class="fragment fade-in">
<!-- -->
<h3 id="task-1">Task:</h3>
<ul>
<li>write out the kernel of the density function for <span class="math inline">\(\boldsymbol\sigma^2\)</span></li>
<li>the kernel of what density it represents?</li>
</ul>
</div>
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</section>
<section id="bayesian-estimation-1" class="slide level2" data-background-color="#F500BD">
<h2>Bayesian Estimation</h2>
</section>
<section id="the-model-in-matrix-notation" class="slide level2">
<h2>The model in Matrix Notation</h2>
<h3 id="varp-model.">VAR(p) model.</h3>
<span class="math display">\[\begin{align*}
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \boldsymbol\mu_0 + \epsilon_t\\
\epsilon_t|Y_{t-1} &\sim iidN_N\left(\mathbf{0}_N,\mathbf\Sigma\right)
\end{align*}\]</span>
<h3 id="matrix-notation.">Matrix notation.</h3>
<span class="math display">\[\begin{align*}
Y &= X\mathbf{A} + E\\
E|X &\sim MN_{T\times N}\left(\mathbf{0}_{T\times N},\mathbf\Sigma, I_T\right)
\end{align*}\]</span>
<div class="fragment fade-out">
<p><span class="math display">\[
\underset{(K\times N)}{\mathbf{A}}=\begin{bmatrix} \mathbf{A}_1'\\ \vdots \\ \mathbf{A}_p' \\ \boldsymbol\mu_0' \end{bmatrix} \quad
\underset{(T\times N)}{Y}= \begin{bmatrix}y_1' \\ y_2'\\ \vdots \\ y_T'\end{bmatrix} \quad
\underset{(K\times1)}{x_t}=\begin{bmatrix} y_{t-1}\\ \vdots \\ y_{t-p}\\ 1 \end{bmatrix}\quad
\underset{(T\times K)}{X}= \begin{bmatrix}x_1' \\ x_2' \\ \vdots \\ x_{T}'\end{bmatrix} \quad
\underset{(T\times N)}{E}= \begin{bmatrix}\epsilon_1' \\ \epsilon_2' \\ \vdots \\ \epsilon_{T}'\end{bmatrix}
\]</span> where <span class="math inline">\(K=pN+1\)</span></p>
</div>
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</section>
<section id="the-model-as-predictive-density" class="slide level2">
<h2>The model as Predictive Density</h2>
<h3 id="var-model.">VAR model.</h3>
<span class="math display">\[\begin{align*}
Y &= X\mathbf{A} +E\\
E|X &\sim MN_{T\times N}\left(\mathbf{0}_{T\times N},\mathbf\Sigma, I_T\right)
\end{align*}\]</span>
<h3 id="predictive-density.">Predictive density.</h3>
<span class="math display">\[\begin{align*}
Y|X,\mathbf{A}, \mathbf{\Sigma} &\sim MN_{T\times N}\left(X\mathbf{A},\mathbf{\Sigma},I_T\right)
\end{align*}\]</span>
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</section>
<section id="likelihood-function" class="slide level2">
<h2>Likelihood Function</h2>
<h3 id="predictive-density.-1">Predictive density.</h3>
<span class="math display">\[\begin{align*}
Y|X,\mathbf{A}, \mathbf{\Sigma} &\sim MN_{T\times N}\left(X\mathbf{A},\mathbf{\Sigma},I_T\right)
\end{align*}\]</span>
<h3 id="likelihood-function.">Likelihood function.</h3>
<span class="math display">\[\begin{align*}
L\left(\mathbf{A},\mathbf{\Sigma}|Y,X\right)&\propto\text{det}(\mathbf{\Sigma})^{-\frac{T}{2}}\exp\left\{-\frac{1}{2}\text{tr}\left[\mathbf{\Sigma}^{-1}(Y-X\mathbf{A})'(Y-X\mathbf{A})\right]\right\}
\end{align*}\]</span>
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</section>
<section id="likelihood-function-as-niw" class="slide level2">
<h2>Likelihood Function as NIW</h2>
<p>Define the MLE: <span class="math inline">\(\widehat{A}=(X'X)^{-1}X'Y\)</span></p>
<p>Perform simple transformation of the likelihood</p>
<span class="math display">\[\begin{align*}
L\left(\mathbf{A},\mathbf{\Sigma}|Y,X\right)&\propto\text{det}(\mathbf{\Sigma})^{-\frac{T}{2}}\exp\left\{-\frac{1}{2}\text{tr}\left[\mathbf{\Sigma}^{-1}(Y-X\mathbf{A})'(Y-X\mathbf{A})\right]\right\}\\
&=\text{det}(\mathbf{\Sigma})^{-\frac{T}{2}}\\
&\quad\times\exp\left\{ -\frac{1}{2}\text{tr}\left[\mathbf{\Sigma}^{-1}(\mathbf{A}-\widehat{A})'X'X(\mathbf{A}-\widehat{A}) \right] \right\}\\
&\quad\times \exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1}(Y-X\widehat{A})'(Y-X\widehat{A}) \right] \right\}
\end{align*}\]</span>
<p>Under the likelihood, <span class="math inline">\((\mathbf{A},\mathbf{\Sigma})\)</span> are <em>normal-inverse Wishart</em> distributed:</p>
<span class="math display">\[\begin{align*}
L\left( \mathbf{A},\mathbf{\Sigma}|Y,X \right) &= NIW_{K\times N}\left(\widehat{A},(X'X)^{-1},(Y-X\widehat{A})'(Y-X\widehat{A}), T-N-K-1 \right)
\end{align*}\]</span>
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</section>
<section id="prior-distribution" class="slide level2">
<h2>Prior Distribution</h2>
<h3 id="construction.">Construction.</h3>
<p>A natural-conjugate prior leads to joint posterior distribution for <span class="math inline">\((\mathbf{A},\mathbf{\Sigma})\)</span> of the same form <span class="math display">\[\begin{align*}
p\left( \mathbf{A}, \mathbf{\Sigma} \right) &= p\left( \mathbf{A}| \mathbf{\Sigma} \right)p\left( \mathbf{\Sigma} \right)\\
\mathbf{A}|\mathbf{\Sigma} &\sim MN_{K\times N}\left( \underline{A},\mathbf{\Sigma},\underline{V} \right)\\
\mathbf{\Sigma} &\sim IW_N\left( \underline{S}, \underline{\nu} \right)
\end{align*}\]</span></p>
<h3 id="kernel.">Kernel.</h3>
<span class="math display">\[\begin{align*}
p\left( \mathbf{A},\mathbf{\Sigma} \right)
&\propto \text{det}(\mathbf{\Sigma})^{-\frac{N+K+\underline{\nu}+1}{2}}\\
&\quad\times\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1}(\mathbf{A}-\underline{A})'\underline{V}^{-1}(\mathbf{A}-\underline{A}) \right] \right\}\\
&\quad\times \exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1}\underline{S} \right] \right\}
\end{align*}\]</span>
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</section>
<section id="posterior-distribution" class="slide level2">
<h2>Posterior Distribution</h2>
<h3 id="bayes-rule.">Bayes Rule.</h3>
<span class="math display">\[\begin{align*}
p\left( \mathbf{A}, \mathbf{\Sigma}|Y,X \right) &\propto L(\mathbf{A},\mathbf{\Sigma}|Y,X)p\left( \mathbf{A}, \mathbf{\Sigma} \right)\\
&= L(\mathbf{A},\mathbf{\Sigma}|Y,X)p\left( \mathbf{A}| \mathbf{\Sigma} \right)p\left( \mathbf{\Sigma} \right)
\end{align*}\]</span>
<h3 id="kernel.-1">Kernel.</h3>
<span class="math display">\[\begin{align*}
p\left( \mathbf{A},\mathbf{\Sigma} |Y,X\right)
&\propto \text{det}(\mathbf{\Sigma})^{-\frac{T}{2}}\exp\left\{-\frac{1}{2}\text{tr}\left[\mathbf{\Sigma}^{-1}(Y-X\mathbf{A})'(Y-X\mathbf{A})\right]\right\}\\
& \quad\times\text{det}(\mathbf{\Sigma})^{-\frac{N+K+\underline{\nu}+1}{2}}\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1}\underline{S} \right] \right\}\\
&\quad\times\exp\left\{ -\frac{1}{2}\text{tr}\left[ \mathbf{\Sigma}^{-1}(\mathbf{A}-\underline{A})'\underline{V}^{-1}(\mathbf{A}-\underline{A}) \right] \right\}
\end{align*}\]</span>
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</section>
<section id="joint-posterior-distribution" class="slide level2">
<h2>Joint Posterior Distribution</h2>
<h3 id="conditional-and-marginal.">Conditional and marginal.</h3>
<span class="math display">\[\begin{align*}
p\left( \mathbf{A}, \mathbf{\Sigma}|Y,X \right) &= p(\mathbf{A}|Y,X,\mathbf{\Sigma})p\left( \mathbf{\Sigma}|Y,X \right)\\[2ex]
p(\mathbf{A}|Y,X,\mathbf{\Sigma}) &= MN_{K\times N}\left( \overline{A},\mathbf{\Sigma},\overline{V} \right)\\
p(\mathbf{\Sigma}|Y,X) &= IW_N\left( \overline{S}, \overline{\nu} \right)
\end{align*}\]</span>
<h3 id="posterior-parameters.">Posterior parameters.</h3>
<span class="math display">\[\begin{align*}
\overline{V}&= \left( X'X + \underline{V}^{-1}\right)^{-1} \\
\overline{A}&= \overline{V}\left( X'Y + \underline{V}^{-1}\underline{A} \right)\\
\overline{\nu}&= T+\underline{\nu}\\
\overline{S}&= \underline{S}+Y'Y + \underline{A}'\underline{V}^{-1}\underline{A} - \overline{A}'\overline{V}^{-1}\overline{A}
\end{align*}\]</span>
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</section>
<section id="joint-posterior-distribution-1" class="slide level2">
<h2>Joint Posterior Distribution</h2>
<h3 id="task.">Task.</h3>
<ul>
<li>Derive the full conditional posterior distribution of <span class="math inline">\(\mathbf{A}\)</span> given <span class="math inline">\(\mathbf{\Sigma}\)</span> and data <span class="math inline">\(Y\)</span> and <span class="math inline">\(X\)</span>, denoted by <span class="math inline">\(p(\mathbf{A}|Y,X,\mathbf{\Sigma})\)</span></li>
</ul>
<p><span class="math display">\[ \]</span></p>
<p>Hint: Proceed by:</p>
<ul>
<li>writing out the kernel of this distribution as a product of the likelihood and the prior,</li>
<li>collecting all the terms within the exponential function and under the trace operator,</li>
<li>completing the squares,</li>
<li>applying the conditioning whenever convenient.</li>
</ul>
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</section>
<section id="posterior-mean-of-mathbfa" class="slide level2">
<h2>Posterior Mean of <span class="math inline">\(\mathbf{A}\)</span></h2>
<p>Posterior mean of matrix <span class="math inline">\(\mathbf{A}\)</span> is: <span class="math display">\[\begin{align*}
\overline{A} &= \overline{V}\left( X'Y + \underline{V}^{-1}\underline{A} \right)\\[2ex]
&= \overline{V}\left( X'X\widehat{A} + \underline{V}^{-1}\underline{A} \right)\\[2ex]
&= \overline{V} X'X\widehat{A} + \overline{V}\underline{V}^{-1}\underline{A}
\end{align*}\]</span> <font color="F500BD">a linear combination</font> of the MLE <span class="math inline">\(\widehat{A}\)</span> and the prior mean <span class="math inline">\(\underline{A}\)</span></p>
<p>Note that: <span class="math display">\[
\overline{V} X'X + \overline{V}\underline{V}^{-1} = \overline{V} ( X'X + \underline{V}^{-1}) = I_K
\]</span></p>
<p><a href="https://rpsychologist.com/d3/bayes/" data-preview-link="true">Play with the posterior in an interactive graph</a></p>
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</section>
<section id="marginal-data-density" class="slide level2">
<h2>Marginal Data Density</h2>
<p>According to <em>Bayes Rule</em>, the kernel of the posterior is normalised by the <em>Marginal Data Density</em> <span class="math inline">\(p(data)\)</span>:</p>
<p><span class="math display">\[
p\left( \mathbf{A}, \mathbf{\Sigma}| data \right) = \frac{L(\mathbf{A},\mathbf{\Sigma}| data)p\left( \mathbf{A}, \mathbf{\Sigma} \right)}{p(data)}
\]</span></p>
<p>For <em>Bayesian VARs</em> the posterior is known <span class="math display">\[
p\left( \mathbf{A}, \mathbf{\Sigma}| data \right) = MNIW\left(\overline{A},\overline{V}, \overline{S}, \overline{\nu} \right)
\]</span></p>
<p>and so is the analytical formula for the <em>MDD</em>: <span class="math display">\[p(data)\]</span></p>
<p>This can be used to our advantage!</p>
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</section>
<section id="minnesota-and-dummy-observations-prior-1" class="slide level2" data-background-color="#F500BD">
<h2>Minnesota and Dummy Observations Prior</h2>
</section>
<section id="minnesota-prior" class="slide level2">
<h2>Minnesota Prior</h2>
<p><a href="https://doi.org/10.1080/07474938408800053">Sims, Litterman, Doan (1984)</a> proposed an interpretable way of setting the hyper-parameters on the NIW prior <span class="math inline">\(\underline{A}\)</span>, <span class="math inline">\(\underline{V}\)</span>, <span class="math inline">\(\underline{S}\)</span>, and <span class="math inline">\(\underline{\nu}\)</span> for macroeconomic data.</p>
<div class="fragment">
<p><span class="math display">\[ \]</span> The prior reflects the following stylised facts about macro time series:</p>
<ul>
<li>the data are unit-root non-stationary</li>
<li>the effect of more lagged variables should be smaller and smaller</li>
<li>the effect of other variables lags should be less than that of own lags</li>
</ul>
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</section>
<section id="minnesota-prior-1" class="slide level2">
<h2>Minnesota Prior</h2>
<h3 id="inverse-wishart-prior.">Inverse-Wishart prior.</h3>
<span class="math display">\[\begin{align*}
\mathbf{\Sigma} &\sim IW_N\left( \underline{S}, \underline{\nu} \right)
\end{align*}\]</span>
<p>Set</p>
<span class="math display">\[\begin{align*}
\underline{S} &= \begin{bmatrix}
\psi_1 &0 &\dots & 0 \\
0 & \psi_2 &\dots & 0\\
\vdots &\vdots&\ddots& \vdots\\
0&0&\dots&\psi_N
\end{bmatrix}\\[2ex]
\underline{\nu}&= N+2
\end{align*}\]</span>
<h3 id="hyper-parameters.">Hyper-parameters.</h3>
<p><span class="math inline">\(\psi =(\psi_1, \dots, \psi_N)\)</span> have to be chosen (or estimated)</p>
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</section>
<section id="minnesota-prior-2" class="slide level2">
<h2>Minnesota Prior</h2>
<h3 id="matrix-variate-normal-prior.">Matrix-Variate Normal prior.</h3>
<span class="math display">\[\begin{align*}
\mathbf{A}|\mathbf{\Sigma} &\sim MN_{K\times N}\left( \underline{A},\mathbf{\Sigma},\underline{V} \right)
\end{align*}\]</span>
<p>For <span class="math inline">\(\quad l = 1+\text{floor}((i-1)/N) \quad\text{and }\quad k = i - (l-1)N\)</span>, set: <span class="math display">\[\begin{align*}
\underline{A} &= \begin{bmatrix} I_N \\ \mathbf{0}_{((p-1)N +1)\times N}\end{bmatrix}&
\underline{V}_{ij} &= \left\{\begin{array} (\lambda ^ 2 / (\psi_k l^2) &\text{ for }i=j,\text{ and } i\neq pN+1 \\
\lambda^2 &\text{ for }i=j,\text{ and } i= pN+1 \\
0&\text{ for } i\neq j
\end{array}\right.
\end{align*}\]</span></p>
<h3 id="hyper-parameters.-1">Hyper-parameters.</h3>
<p><span class="math inline">\(\lambda^2\)</span> has to be chosen (or estimated)</p>
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</section>
<section id="minnesota-prior-3" class="slide level2">
<h2>Minnesota Prior</h2>
<h3 id="task.-1">Task.</h3>
<p>Consider a simple case of a model for <span class="math inline">\(N=2\)</span> observations and with <span class="math inline">\(p=1\)</span> lag.</p>
<ul>
<li>write out the <span class="math inline">\(3\times2\)</span> matrix <span class="math inline">\(\underline{A}\)</span> and the <span class="math inline">\(3\times3\)</span> matrix <span class="math inline">\(\underline{V}\)</span> for the Minnesota prior determining every of its elements.</li>
</ul>
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</section>
<section id="dummy-observations-prior" class="slide level2">
<h2>Dummy Observations Prior</h2>
<h3 id="idea.">Idea.</h3>
<ol type="1">
<li>Generate artificial data matrices with <span class="math inline">\(T_d\)</span> rows <span class="math inline">\(Y^*\)</span> and <span class="math inline">\(X^*\)</span></li>
<li>Append them to the original data matrices <span class="math inline">\(Y\)</span> and <span class="math inline">\(X\)</span> respectively.</li>
</ol>
<div class="fragment">
<!-- -->
<h3 id="implied-prior-distribution.">Implied prior distribution.</h3>
<p>Use Bayes Rule to derive the joint prior of <span class="math inline">\((\mathbf{A},\mathbf\Sigma)\)</span> given <span class="math inline">\(Y^*\)</span> and <span class="math inline">\(X^*\)</span>.</p>
<p>It is given by the MNIW distribution:</p>
<p><span class="math display">\[\begin{align*}
p\left( \mathbf{A}, \mathbf{\Sigma}|Y^*,X^* \right) &= MNIV_{K\times N}\left( \underline{A}^*,\underline{V}^*, \underline{S}^*, \underline{\nu}^* \right)
\end{align*}\]</span> <span class="math display">\[\begin{align*}
\underline{V}^*&= \left( X^{*\prime}X^* + \underline{V}^{-1}\right)^{-1} & \underline{A}^*&= \underline{V}^*\left( X^{*\prime}Y^* + \underline{V}^{-1}\underline{A} \right)\\
\underline{\nu}^*&= T_d+\underline{\nu} & \underline{S}^*&= \underline{S}+Y^{*\prime}Y^* + \underline{A}'\underline{V}^{-1}\underline{A} - \underline{A}^{*\prime}\underline{V}^{*-1}\underline{A}^*
\end{align*}\]</span></p>
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<section id="dummy-observations-prior-1" class="slide level2">
<h2>Dummy Observations Prior</h2>
<p>Let a <span class="math inline">\(p\times N\)</span> matrix <span class="math inline">\(Y_0\)</span> denote the initial observations, that is, the first <span class="math inline">\(p\)</span> observations of the available time series.</p>
<p>Let an <span class="math inline">\(N\)</span>-vector <span class="math inline">\(\bar{Y}_0\)</span> denote its columns’ means.</p>
<div class="fragment">
<!-- -->
<h3 id="sum-of-coefficients-prior.">Sum-of-coefficients prior.</h3>
<p>Generate additional <span class="math inline">\(N\)</span> rows by <span class="math display">\[
Y^+ = \text{diag}\left(\frac{\bar{Y}_0}{\mu}\right) \quad\text{ and }\quad X^+ = \begin{bmatrix}Y^+ & \dots & Y^+ & \mathbf{0}_N \end{bmatrix}
\]</span></p>
<ul>
<li><span class="math inline">\(\mu\)</span> is a hyper-parameter to be chosen (or estimated)</li>
<li>if <span class="math inline">\(\mu \rightarrow 0\)</span> the prior implies the presence of a unit root in each equation and rules out cointegration</li>
<li>if <span class="math inline">\(\mu \rightarrow\infty\)</span> the prior becomes uninformative</li>
</ul>
</div>
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</section>
<section id="dummy-observations-prior-2" class="slide level2">
<h2>Dummy Observations Prior</h2>
<h3 id="dummy-initial-observation-prior.">Dummy-initial-observation prior.</h3>
<p>Generate an additional row by <span class="math display">\[
Y^{++} = \frac{\bar{Y}_0'}{\delta} \quad\text{ and }\quad X^{++} = \begin{bmatrix} Y^{++} & \dots & Y^{++} & \frac{1}{\delta} \end{bmatrix}
\]</span></p>
<ul>
<li>hyper-parameter <span class="math inline">\(\delta\)</span> is to be chosen (or estimated)</li>
<li>if <span class="math inline">\(\delta \rightarrow 0\)</span> all the variables of the VAR are forced to be at their unconditional mean, or the system is characterized by the presence of an unspecified number of unit roots without drift (cointegration)</li>
<li>if <span class="math inline">\(\delta \rightarrow\infty\)</span> the prior becomes uninformative</li>
</ul>
<div class="fragment">
<!-- -->
<h3 id="combining-dummy-observations.">Combining dummy observations.</h3>
<p><span class="math display">\[
Y^* = \begin{bmatrix}Y^+ \\ Y^{++} \end{bmatrix}\quad\text{ and }\quad
X^* = \begin{bmatrix}X^+ \\ X^{++} \end{bmatrix}
\]</span></p>
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</section>
<section id="dummy-observations-prior-3" class="slide level2">
<h2>Dummy Observations Prior</h2>
<h3 id="task.-2">Task.</h3>
<p>Suppose that:</p>
<ul>
<li><span class="math inline">\(\bar{Y}_0 = \begin{bmatrix}1&2\end{bmatrix}'\)</span></li>
<li><span class="math inline">\(\mu = 0.5\)</span></li>
<li><span class="math inline">\(\delta = 3\)</span></li>
<li><span class="math inline">\(p = 1\)</span></li>
</ul>
<p>Write out the matrices <span class="math inline">\(Y^*\)</span> and <span class="math inline">\(X^*\)</span> of dimensions <span class="math inline">\(2\times 3\)</span> and <span class="math inline">\(3\times 3\)</span> respectively.</p>
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</section>
<section id="bayesian-estimation-for-hierarchical-prior-1" class="slide level2" data-background-color="#F500BD">
<h2>Bayesian Estimation for Hierarchical Prior</h2>
</section>
<section id="bayesian-estimation-for-hierarchical-prior-2" class="slide level2">
<h2>Bayesian Estimation for Hierarchical Prior</h2>
<p>Hyper-parameters <span class="math inline">\(\psi\)</span>, <span class="math inline">\(\lambda\)</span>, <span class="math inline">\(\mu\)</span> and <span class="math inline">\(\delta\)</span> can be fixed to values chosen by the econometrician.</p>
<h3 id="hierarcical-prior.">Hierarcical prior.</h3>
<p>A better idea is to assume priors for these hyper-parameters and estimate them as in <a href="https://doi.org/10.1162/REST_a_00483">Giannone, Lenza, Primiceri (2015)</a>.</p>
<p>Extend the existing prior to: <span class="math display">\[\begin{align*}
p\left( \mathbf{A}, \mathbf{\Sigma}|Y^*,X^*,\psi,\lambda,\mu,\delta \right) &= MNIV_{K\times N}\left( \underline{A}^*,\underline{V}^*, \underline{S}^*, \underline{\nu}^* \right)
\end{align*}\]</span> And specify: <span class="math display">\[\begin{align*}
\psi_n &\sim IG\left(0.02^2, 0.02^2\right)\\
\lambda &\sim G\left(0.2,2\right)\\
\mu &\sim G\left(1,2\right)\\
\delta &\sim G\left(1,2\right)
\end{align*}\]</span></p>
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<section id="bayesian-estimation-for-hierarchical-prior-3" class="slide level2">
<h2>Bayesian Estimation for Hierarchical Prior</h2>
<p><a href="https://doi.org/10.1162/REST_a_00483">Giannone, Lenza, Primiceri (2015)</a> propose the following estimation procedure:</p>
<h4 id="step-1-estimate-psilambdamudelta-using-a-random-walk-metropolis-hastings-sampler">Step 1: Estimate <span class="math inline">\((\psi,\lambda,\mu,\delta)\)</span> using a random-walk Metropolis-Hastings sampler</h4>
<ul>
<li>Sample these hyper-parameters marginally on <span class="math inline">\((\mathbf{A},\mathbf\Sigma)\)</span></li>
<li>extend the conditioning of <em>Marginal Data Density</em>: <span class="math display">\[ p(data|\psi,\lambda,\mu,\delta)\]</span></li>
<li>apply Bayes Rule to obtain the kernel of the posterior:</li>
</ul>
<p><span class="math display">\[ p(\psi,\lambda,\mu,\delta|data) \propto p(\psi,\lambda,\mu,\delta)p(data|\psi,\lambda,\mu,\delta)\]</span> - Use an <span class="math inline">\((N+3)\)</span>-variate Student-t distribution as the candidate generating density</p>
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</section>
<section id="bayesian-estimation-for-hierarchical-prior-4" class="slide level2">
<h2>Bayesian Estimation for Hierarchical Prior</h2>
<h4 id="step-2-for-each-draw-of-psilambdamudelta-sample-the-corresponding-draw-of-mathbfamathbfsigma">Step 2: For each draw of <span class="math inline">\((\psi,\lambda,\mu,\delta)\)</span> sample the corresponding draw of <span class="math inline">\((\mathbf{A},\mathbf{\Sigma})\)</span></h4>
<p>Use the MNIW posterior derived for the implied prior:</p>
<span class="math display">\[\begin{align*}
p\left( \mathbf{A}, \mathbf{\Sigma}|Y,X, Y^*,X^* \right) &= MNIW_{K\times N}\left( \overline{A}^*,\overline{V}^*,\overline{S}^*, \overline{\nu}^* \right)\\[2ex]
\overline{V}^*&= \left( X'X + \underline{V}^{*-1}\right)^{-1} \\
\overline{A}^*&= \overline{V}^*\left( X'Y + \underline{V}^{*-1}\underline{A}^* \right)\\
\overline{\nu}^*&= T+\underline{\nu}^*\\
\overline{S}^*&= \underline{S}^*+Y'Y + \underline{A}^{*\prime}\underline{V}^{*-1}\underline{A}^* - \overline{A}^{*\prime}\overline{V}^{*-1}\overline{A}^*
\end{align*}\]</span>
<h3 id="r-implementation.">R implementation.</h3>
<p>Package <a href="https://cran.r-project.org/package=bsvarSIGNs"><strong>bsvarSIGNs</strong></a> by <a href="https://doi.org/10.32614/CRAN.package.bsvarSIGNs">Wang & Woźniak (2024)</a> implements this algorithm.</p>
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</section>
<section id="bayesian-forecasting-using-vars-1" class="slide level2" data-background-color="#F500BD">
<h2>Bayesian Forecasting using VARs</h2>
</section>
<section id="the-objective-of-economic-forecasting" class="slide level2">
<h2>The objective of economic forecasting</h2>
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<p><span class="math inline">\(\left.\right.\)</span></p>
<p>… is to use the available data to provide a statistical characterisation of the unknown future values of quantities of interest.</p>
<p><span class="math inline">\(\left.\right.\)</span></p>
<p>The full statistical characterisation of the unknown future values of random variables is given by their <em>predictive density</em>.</p>
<p><span class="math inline">\(\left.\right.\)</span></p>
<p>Simplified outcomes in a form of statistics summarising the predictive densities are usually used in decision-making processes.</p>
<p><span class="math inline">\(\left.\right.\)</span></p>
<p>Summary statistics are also communicated to general audiences.</p>
</section>
<section id="one-period-ahead-predictive-density" class="slide level2">
<h2>One-Period Ahead Predictive Density</h2>