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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>
<script src="index_files/libs/quarto-html/tippy.umd.min.js"></script>
<link href="index_files/libs/quarto-html/tippy.css" rel="stylesheet">
<link href="index_files/libs/quarto-html/light-border.css" rel="stylesheet">
<link href="index_files/libs/quarto-html/quarto-html.min.css" rel="stylesheet" data-mode="light">
<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>index</title>
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="index_files/libs/revealjs/dist/reset.css">
<link rel="stylesheet" href="index_files/libs/revealjs/dist/reveal.css">
<style>
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
ul.task-list li input[type="checkbox"] {
width: 0.8em;
margin: 0 0.8em 0.2em -1em; /* quarto-specific, see https://github.com/quarto-dev/quarto-cli/issues/4556 */
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { display: inline-block; line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ color: #003b4f; background-color: #f1f3f5; }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span { color: #003b4f; } /* Normal */
code span.al { color: #ad0000; } /* Alert */
code span.an { color: #5e5e5e; } /* Annotation */
code span.at { color: #657422; } /* Attribute */
code span.bn { color: #ad0000; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #003b4f; } /* ControlFlow */
code span.ch { color: #20794d; } /* Char */
code span.cn { color: #8f5902; } /* Constant */
code span.co { color: #5e5e5e; } /* Comment */
code span.cv { color: #5e5e5e; font-style: italic; } /* CommentVar */
code span.do { color: #5e5e5e; font-style: italic; } /* Documentation */
code span.dt { color: #ad0000; } /* DataType */
code span.dv { color: #ad0000; } /* DecVal */
code span.er { color: #ad0000; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #ad0000; } /* Float */
code span.fu { color: #4758ab; } /* Function */
code span.im { color: #00769e; } /* Import */
code span.in { color: #5e5e5e; } /* Information */
code span.kw { color: #003b4f; } /* Keyword */
code span.op { color: #5e5e5e; } /* Operator */
code span.ot { color: #003b4f; } /* Other */
code span.pp { color: #ad0000; } /* Preprocessor */
code span.sc { color: #5e5e5e; } /* SpecialChar */
code span.ss { color: #20794d; } /* SpecialString */
code span.st { color: #20794d; } /* String */
code span.va { color: #111111; } /* Variable */
code span.vs { color: #20794d; } /* VerbatimString */
code span.wa { color: #5e5e5e; font-style: italic; } /* Warning */
</style>
<link rel="stylesheet" href="index_files/libs/revealjs/dist/theme/quarto.css">
<link href="index_files/libs/revealjs/plugin/quarto-line-highlight/line-highlight.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-menu/menu.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/reveal-menu/quarto-menu.css" rel="stylesheet">
<link href="index_files/libs/revealjs/plugin/quarto-support/footer.css" rel="stylesheet">
<style type="text/css">
.callout {
margin-top: 1em;
margin-bottom: 1em;
border-radius: .25rem;
}
.callout.callout-style-simple {
padding: 0em 0.5em;
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
display: flex;
}
.callout.callout-style-default {
border-left: solid #acacac .3rem;
border-right: solid 1px silver;
border-top: solid 1px silver;
border-bottom: solid 1px silver;
}
.callout .callout-body-container {
flex-grow: 1;
}
.callout.callout-style-simple .callout-body {
font-size: 1rem;
font-weight: 400;
}
.callout.callout-style-default .callout-body {
font-size: 0.9rem;
font-weight: 400;
}
.callout.callout-titled.callout-style-simple .callout-body {
margin-top: 0.2em;
}
.callout:not(.callout-titled) .callout-body {
display: flex;
}
.callout:not(.no-icon).callout-titled.callout-style-simple .callout-content {
padding-left: 1.6em;
}
.callout.callout-titled .callout-header {
padding-top: 0.2em;
margin-bottom: -0.2em;
}
.callout.callout-titled .callout-title p {
margin-top: 0.5em;
margin-bottom: 0.5em;
}
.callout.callout-titled.callout-style-simple .callout-content p {
margin-top: 0;
}
.callout.callout-titled.callout-style-default .callout-content p {
margin-top: 0.7em;
}
.callout.callout-style-simple div.callout-title {
border-bottom: none;
font-size: .9rem;
font-weight: 600;
opacity: 75%;
}
.callout.callout-style-default div.callout-title {
border-bottom: none;
font-weight: 600;
opacity: 85%;
font-size: 0.9rem;
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-default div.callout-content {
padding-left: 0.5em;
padding-right: 0.5em;
}
.callout.callout-style-simple .callout-icon::before {
height: 1rem;
width: 1rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 1rem 1rem;
}
.callout.callout-style-default .callout-icon::before {
height: 0.9rem;
width: 0.9rem;
display: inline-block;
content: "";
background-repeat: no-repeat;
background-size: 0.9rem 0.9rem;
}
.callout-title {
display: flex
}
.callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
.callout.no-icon::before {
display: none !important;
}
.callout.callout-titled .callout-body > .callout-content > :last-child {
margin-bottom: 0.5rem;
}
.callout.callout-titled .callout-icon::before {
margin-top: .5rem;
padding-right: .5rem;
}
.callout:not(.callout-titled) .callout-icon::before {
margin-top: 1rem;
padding-right: .5rem;
}
/* Callout Types */
div.callout-note {
border-left-color: #4582ec !important;
}
div.callout-note .callout-icon::before {
background-image: url('data:image/png;base64,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');
}
div.callout-note.callout-style-default .callout-title {
background-color: #dae6fb
}
div.callout-important {
border-left-color: #d9534f !important;
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<div class="slides">
<section id="section" class="slide level2" data-background-color="#001D31">
<h2></h2>
<p><img data-src="bsvars.org.png" class="absolute" style="top: 80px; right: 725px; width: 400px; "> <img data-src="bsvars.png" class="absolute" style="top: 80px; right: 325px; width: 400px; "> <img data-src="bsvarSIGNs.png" class="absolute" style="top: 80px; right: -75px; width: 400px; "></p>
</section>
<section id="predictive-analyses-using-r-packages-bsvars-and-bsvarsigns" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">predictive analyses using R packages bsvars and bsvarSIGNs</span></h2>
<p><span class="math display">\[ \]</span></p>
<h3 style="color:#ff68b4;" id="structural-vector-autoregressions">structural vector autoregressions</h3>
<h3 style="color:#ff68b4;" id="predictive-analyses-using-structural-vars">predictive analyses using structural VARs</h3>
<h3 style="color:#ff68b4;" id="bsvars-and-bsvarsigns-features">bsvars and bsvarSIGNs features</h3>
<h3 style="color:#ff68b4;" id="a-small-fiscal-policy-system-analysis">a small fiscal policy system analysis</h3>
</section>
<section id="materials" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">materials</span></h2>
<p><span class="math display">\[ \]</span></p>
<h3 style="color:#ff68b4;" id="lecture-slides-as-a-website">lecture slides <a href="https://bsvars.org/2024-12-F4SG/">as a website</a></h3>
<h3 style="color:#ff68b4;" id="github-repo-to-reproduce-the-slides-and-results">GitHub <a href="https://github.com/bsvars/2024-12-F4SG">repo</a> to reproduce the slides and results</h3>
<p><span class="math display">\[ \]</span></p>
<h3 style="color:#ff68b4;" id="bsvars.org-official-project-website"><a href="https://bsvars.org/">bsvars.org</a> official project website</h3>
<h3 style="color:#ff68b4;" id="bsvars-package-on-cran"><a href="https://cran.r-project.org/package=bsvars">bsvars</a> package on CRAN</h3>
<h3 style="color:#ff68b4;" id="bsvarsigns-package-on-cran"><a href="https://cran.r-project.org/package=bsvarSIGNs">bsvarSIGNs</a> package on CRAN</h3>
</section>
<section id="structural-vector-autoregressions-1" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">structural vector autoregressions</span></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">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 models</li>
<li class="fragment">extensively used for: <em>monetary</em> and <em>fiscal</em> policy, <em>financial</em> markets, …</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 hierarchical modelling</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.org">bsvars.org</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://bsvars.org">bsvars.org</a></p>
</div>
<h3 id="the-model.">the model.</h3>
<p><span class="math display">\[\begin{align}
\text{VAR equation: }&& y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \mathbf{A}_d x_{t} + \epsilon_t\\[1ex]
\text{structural equation: }&& \mathbf{B}\epsilon_t &= u_t\\[1ex]
\text{structural shocks: }&& u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N,\text{diag}\left(\boldsymbol\sigma_t^2\right)\right)
\end{align}\]</span></p>
<div class="fragment">
<!-- -->
<h3 id="notation.">notation.</h3>
<ul>
<li><span class="math inline">\(y_t\)</span> - <span class="math inline">\(N\)</span>-vector of variables at time <span class="math inline">\(t\)</span></li>
<li><span class="math inline">\(\mathbf{A}_i\)</span> and <span class="math inline">\(\mathbf{B}\)</span> - <span class="math inline">\(N\times N\)</span> matrices of autoregressive and structural coefficients</li>
<li><span class="math inline">\(\epsilon_t\)</span> and <span class="math inline">\(u_t\)</span> - <span class="math inline">\(N\)</span>-vectors of error terms and structural shocks at time <span class="math inline">\(t\)</span></li>
<li><span class="math inline">\(\boldsymbol\sigma_t^2\)</span> - <span class="math inline">\(N\)</span>-vector structural shock variances</li>
</ul>
</div>
</section>
<section id="svars-hierarchical-prior-distributions" class="slide level2">
<h2>SVARs: hierarchical prior distributions</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:45%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>normal-generalised normal prior for <span class="math inline">\(\mathbf{A}\)</span> and <span class="math inline">\(\mathbf{B}\)</span></li>
<li>3-level local-global equation-specific shrinkage</li>
<li>Minnesota prior for non-stationary time series</li>
<li>improves model-fit and forecasting precision</li>
</ul>
</div><div class="column" style="width:5%;">
</div><div class="column" style="width:45%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>normal-inverse Wishart prior for <span class="math inline">\(\mathbf{A}\)</span> and <span class="math inline">\(\mathbf{\Sigma} = (\mathbf{B}'\mathbf{B})^{-1}\)</span></li>
<li>estimated Minnesota prior shrinkage levels</li>
<li>Minnesota prior for non-stationary time series</li>
<li>improves model-fit and forecasting precision</li>
</ul>
</div>
</div>
</section>
<section id="svars-volatility-modelling" class="slide level2">
<h2>SVARs: volatility modelling</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:65%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li><p>homoskedasticity <span class="math inline">\(\boldsymbol\sigma_{n.t}^2 = 1\)</span></p></li>
<li><p>stochastic volatility</p></li>
<li><p>stationary Markov-switching heteroskedasticity</p></li>
<li><p>non-parametric Markov-switching heteroskedasticity</p></li>
<li><p>joint distribution of structural shocks</p>
<ul>
<li>normal</li>
<li>finite mixtures of normal distributions</li>
<li>non-parametric mixtures of normal distributions</li>
<li>Student-t</li>
</ul></li>
</ul>
</div><div class="column" style="width:5%;">
</div><div class="column" style="width:30%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>homoskedastic</li>
<li>normal</li>
</ul>
</div>
</div>
</section>
<section id="svars-structural-identification" class="slide level2">
<h2>SVARs: structural identification</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:45%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>exclusion restrictions</li>
<li>time-varying volatility</li>
<li>non-normality</li>
</ul>
</div><div class="column" style="width:5%;">
</div><div class="column" style="width:45%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>sign restrictions</li>
<li>exclusion restrictions</li>
<li>narrative restrictions</li>
</ul>
</div>
</div>
</section>
<section id="predictive-analyses-using-structural-vars-1" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">predictive analyses using structural VARs</span></h2>
</section>
<section id="the-objective-of-economic-forecasting" class="slide level2">
<h2>the objective of economic forecasting</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<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>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<h3 id="svar-model.">SVAR model.</h3>
<p><span class="math display">\[\begin{align*}
y_t &= \mathbf{A}_1 y_{t-1} + \dots + \mathbf{A}_p y_{t-p} + \mathbf{A}_d x_{t} + \epsilon_t\\[2ex]
\epsilon_t|Y_{t-1} &\sim iidN_N\left(\mathbf{0}_N,\mathbf\Sigma_t\right)\\[2ex]
\mathbf\Sigma_t &= \mathbf{B}^{-1} \text{diag}\left(\boldsymbol\sigma^2_t\right)\mathbf{B}^{-1\prime}
\end{align*}\]</span></p>
<h3 id="one-period-ahead-conditional-predictive-density.">one-period ahead conditional predictive density.</h3>
<p>… is implied by the model formulation: <span class="math display">\[\begin{align*}
y_{t+h}|Y_{t+h-1},\mathbf{A},\mathbf\Sigma_{t+h} &\sim N_N\left(\mathbf{A}_1 y_{t+h-1} + \dots + \mathbf{A}_p y_{t+h-p} + \mathbf{A}_d x_{t+h},\mathbf\Sigma_{t+h}\right)
\end{align*}\]</span></p>
</section>
<section id="one-period-ahead-predictive-density-1" class="slide level2">
<h2>one-period ahead predictive density</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math inline">\(\left.\right.\)</span></p>
<p>Bayesian forecasting takes into account the uncertainty w.r.t. parameter estimation by integrating it out from the predictive density.</p>
<p><span class="math display">\[\begin{align*}
&\\
p(y_{T+1}|Y,X) &= \int p(y_{T+1}|Y_{T},\mathbf{A},\mathbf\Sigma_{T+1}) p(\mathbf{A},\mathbf\Sigma_{T+1}|Y,X) d(\mathbf{A},\mathbf\Sigma_{T+1})\\ &
\end{align*}\]</span></p>
<ul>
<li><span class="math inline">\(p(y_{T+1}|Y,X)\)</span> - predictive density</li>
<li><span class="math inline">\(p(y_{T+1}|Y_{t},\mathbf{A},\mathbf\Sigma_{T+1})\)</span> - one-period-ahead conditional predictive density</li>
<li><span class="math inline">\(p(\mathbf{A},\mathbf\Sigma_{T+1}|Y,X)\)</span> - marginal posterior distribution</li>
<li>includes volatility forecasting <span class="math inline">\(\boldsymbol\sigma^2_{T+1}\)</span> to form <span class="math inline">\(\mathbf\Sigma_{T+1}\)</span></li>
</ul>
</section>
<section id="sampling-from-the-predictive-density" class="slide level2">
<h2>sampling from the predictive density</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math inline">\(\left.\right.\)</span></p>
<h4 id="step-1-sample-from-the-posterior">step 1: sample from the posterior</h4>
<p>… and obtain <span class="math inline">\(S\)</span> draws <span class="math inline">\(\left\{ \mathbf{A}^{(s)},\mathbf\Sigma_{T+1}^{(s)} \right\}_{s=1}^{S}\)</span></p>
<p><span class="math inline">\(\left.\right.\)</span></p>
<h4 id="step-2-sample-from-the-predictive-density">step 2: sample from the predictive density</h4>
<p>In order to obtain draws from <span class="math inline">\(p(y_{T+1}|Y,X)\)</span>, for each of the <span class="math inline">\(S\)</span> draws of <span class="math inline">\((\mathbf{A},\mathbf\Sigma_{T+1})\)</span> sample the corresponding draw of <span class="math inline">\(y_{T+1}\)</span>:</p>
<p>Sample <span class="math inline">\(y_{T+1}^{(s)}\)</span> from <span class="math display">\[
N_N\left(\mathbf{A}_1^{(s)} y_{T} + \dots + \mathbf{A}_p^{(s)} y_{T-p+1} + \mathbf{A}_d^{(s)} x_{T+1},\mathbf\Sigma_{T+1}^{(s)}\right)
\]</span> and obtain <span class="math inline">\(\left\{y_{T+1}^{(s)}\right\}_{s=1}^{S}\)</span></p>
</section>
<section id="impulse-response-functions" class="slide level2">
<h2>impulse response functions</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math inline">\(\left.\right.\)</span></p>
<p><strong>Impulse response functions</strong> to <em>orthogonal shocks</em> computed for an empirically relevant SVAR model are considered the <text style="color:#ff68b4;"><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://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math inline">\(\left.\right.\)</span></p>
<h3 id="as-a-difference-in-conditional-forecasts.">…as a difference in conditional forecasts.</h3>
<p><span class="math display">\[\begin{align}
\mathbf\Theta_{h[\cdot i]} &= \mathbb{E}\left[\mathbf{y}_{T+h}| u_{i.T} = 1,\mathbf{Y},\mathbf{X}\right] - \mathbb{E}\left[\mathbf{y}_{T+h}| \mathbf{Y},\mathbf{X}\right]
\end{align}\]</span></p>
<h3 id="as-the-shock-propagation-mechanism.">…as the shock propagation mechanism.</h3>
<p><span class="math display">\[\begin{align}
y_t &= \mathbf{B}^{-1}\mathbf{u}_t + \mathbf{A}_1 y_{t-1}\\
&= \mathbf{B}^{-1}\mathbf{u}_t + \mathbf{A}_1 \mathbf{B}^{-1}\mathbf{u}_{t-1} + \mathbf{A}_1^2 y_{t-2}\\
&\dots\\
&= \mathbf{\Theta}_0\mathbf{u}_t + \mathbf{\Theta}_1\mathbf{u}_{t-1} + \mathbf{\Theta}_2\mathbf{u}_{t-2} + \dots
\end{align}\]</span></p>
</section>
<section id="forecast-error-variance-decompositions" class="slide level2">
<h2>forecast error variance decompositions</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math display">\[ \]</span></p>
<p>… are a way to quantify the relative importance of the shocks in explaining the forecast error variance of the variables in the system.</p>
<p><span class="math display">\[ \]</span></p>
<h4 id="forecast-error-covariance-matrix.">forecast error covariance matrix.</h4>
<p><span class="math display">\[\begin{align}
\mathbf\Sigma_{T+h} &=
\mathbf{\Theta}_h \text{diag}\left(\boldsymbol\sigma^2_{T+h}\right)\mathbf{\Theta}_h^{\prime}
\end{align}\]</span></p>
<h4 id="forecast-error-variance.">forecast error variance.</h4>
<p>For the first variable, at the forecast horizon <span class="math inline">\(h\)</span>, for <span class="math inline">\(N=2\)</span>:</p>
<p><span class="math display">\[\begin{align}
\sum_{i=0}^{h-1}\mathbf\Sigma_{T+i[11]} &= \sum_{i=0}^{h-1}
\mathbf{\Theta}_{i[11]}^2 \sigma^2_{1.T+i} + \mathbf{\Theta}_{i[12]}^2 \sigma^2_{2.T+i}
\end{align}\]</span></p>
</section>
<section id="forecast-error-variance-decompositions-1" class="slide level2">
<h2>forecast error variance decompositions</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math display">\[ \]</span></p>
<p>… are a way to quantify the relative importance of the shocks in explaining the forecast error variance of the variables in the system.</p>
<p><span class="math display">\[ \]</span></p>
<h4 id="forecast-error-variance-decomposition.">forecast error variance decomposition.</h4>
<p>First shock’s contribution to the forecast error variance of the first variable:</p>
<p><span class="math display">\[\begin{align}
FEVD_{T+h[11]} &= \sum_{i=0}^{h-1}\frac{\mathbf{\Theta}_{i[11]}^2 \sigma^2_{1.T+i}}{
\mathbf{\Theta}_{i[11]}^2 \sigma^2_{1.T+i} + \mathbf{\Theta}_{i[12]}^2 \sigma^2_{2.T+i}}
\end{align}\]</span></p>
</section>
<section id="bsvars-and-bsvarsigns-features-1" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">bsvars and bsvarSIGNs features</span></h2>
</section>
<section id="bsvars-and-bsvarsigns-features-2" class="slide level2">
<h2>bsvars and bsvarSIGNs features</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:50%;">
<p><img data-src="bsvars_cran.png" class="absolute" style="top: 100px; width: 530px; "></p>
</div><div class="column" style="width:50%;">
<p><img data-src="bsvarSIGNs_cran.png" class="absolute" style="top: 100px; width: 530px; "></p>
</div>
</div>
</section>
<section id="bsvars-and-bsvarsigns-features-3" class="slide level2">
<h2>bsvars and bsvarSIGNs features</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math display">\[ \]</span></p>
<ul>
<li>Bayesian estimation of Structural VARs</li>
<li>coherent code structure, workflows, and objects</li>
<li>excellent computational speed</li>
<li>frontier econometric & numerical techniques</li>
<li>written in <strong>C++</strong> using <strong>Rcpp</strong> and <strong>RcppArmadillo</strong></li>
<li>data analysis in <strong>R</strong></li>
</ul>
</section>
<section id="bsvars-and-bsvarsigns-features-4" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">bsvars and bsvarSIGNs features</span></h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:50%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>package and data loading</li>
</ul>
<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 class="fu">data</span>(us_fiscal_lsuw)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<ul>
<li>simple model setup</li>
</ul>
<div class="cell">
<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>spec <span class="ot">=</span> specify_bsvar<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<ul>
<li>simple estimation</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1"></a>burn <span class="ot">=</span> <span class="fu">estimate</span>(spec, <span class="at">S =</span> <span class="dv">1000</span>)</span>
<span id="cb3-2"><a href="#cb3-2"></a>post <span class="ot">=</span> <span class="fu">estimate</span>(burn, <span class="at">S =</span> <span class="dv">10000</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column" style="width:50%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>package and data loading</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1"></a><span class="fu">library</span>(bsvarSIGNs)</span>
<span id="cb4-2"><a href="#cb4-2"></a><span class="fu">data</span>(optimism)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<ul>
<li>simple model setup</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1"></a>spec <span class="ot">=</span> specify_bsvarSIGN<span class="sc">$</span><span class="fu">new</span>(optimism)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<ul>
<li>simple estimation</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1"></a>post <span class="ot">=</span> <span class="fu">estimate</span>(spec, <span class="at">S =</span> <span class="dv">10000</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</section>
<section id="bsvars-and-bsvarsigns-features-5" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">bsvars and bsvarSIGNs features</span></h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:50%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>structural analyses</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1"></a>irfs <span class="ot">=</span> <span class="fu">compute_impulse_responses</span>(post , <span class="at">horizon =</span> <span class="dv">12</span>)</span>
<span id="cb7-2"><a href="#cb7-2"></a>fevd <span class="ot">=</span> <span class="fu">compute_variance_decompositions</span>(post, <span class="at">horizon =</span> <span class="dv">12</span>)</span>
<span id="cb7-3"><a href="#cb7-3"></a>hds <span class="ot">=</span> <span class="fu">compute_historical_decompositions</span>(post)</span>
<span id="cb7-4"><a href="#cb7-4"></a>ss <span class="ot">=</span> <span class="fu">compute_structural_shocks</span>(post)</span>
<span id="cb7-5"><a href="#cb7-5"></a>csds <span class="ot">=</span> <span class="fu">compute_conditional_sd</span>(post)</span>
<span id="cb7-6"><a href="#cb7-6"></a>sddr <span class="ot">=</span> <span class="fu">verify_identification</span>(post)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column" style="width:50%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>structural analyses</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1"></a>irfs <span class="ot">=</span> <span class="fu">compute_impulse_responses</span>(post , <span class="at">horizon =</span> <span class="dv">12</span>)</span>
<span id="cb8-2"><a href="#cb8-2"></a>fevd <span class="ot">=</span> <span class="fu">compute_variance_decompositions</span>(post, <span class="at">horizon =</span> <span class="dv">12</span>)</span>
<span id="cb8-3"><a href="#cb8-3"></a>hds <span class="ot">=</span> <span class="fu">compute_historical_decompositions</span>(post)</span>
<span id="cb8-4"><a href="#cb8-4"></a>ss <span class="ot">=</span> <span class="fu">compute_structural_shocks</span>(post)</span>
<span id="cb8-5"><a href="#cb8-5"></a>csds <span class="ot">=</span> <span class="fu">compute_conditional_sd</span>(post)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</section>
<section id="bsvars-and-bsvarsigns-features-6" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">bsvars and bsvarSIGNs features</span></h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:50%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>predictive analyses</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1"></a>fvs <span class="ot">=</span> <span class="fu">compute_fitted_values</span>(post)</span>
<span id="cb9-2"><a href="#cb9-2"></a>fore <span class="ot">=</span> <span class="fu">forecast</span>(post, <span class="at">horizon =</span> <span class="dv">12</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<ul>
<li>plots and summaries</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1"></a><span class="fu">plot</span>(irfs)</span>
<span id="cb10-2"><a href="#cb10-2"></a><span class="fu">summary</span>(irfs)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column" style="width:50%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>predictive analyses</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1"></a>fvs <span class="ot">=</span> <span class="fu">compute_fitted_values</span>(post)</span>
<span id="cb11-2"><a href="#cb11-2"></a>fore <span class="ot">=</span> <span class="fu">forecast</span>(post, <span class="at">horizon =</span> <span class="dv">12</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<ul>
<li>plots and summaries</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1"></a><span class="fu">plot</span>(irfs)</span>
<span id="cb12-2"><a href="#cb12-2"></a><span class="fu">summary</span>(irfs)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</section>
<section id="bsvars-and-bsvarsigns-features-7" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">bsvars and bsvarSIGNs features</span></h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:50%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>workflow with the pipe</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1"></a><span class="fu">library</span>(bsvars)</span>
<span id="cb13-2"><a href="#cb13-2"></a><span class="fu">data</span>(us_fiscal_lsuw)</span>
<span id="cb13-3"><a href="#cb13-3"></a></span>
<span id="cb13-4"><a href="#cb13-4"></a>us_fiscal_lsuw <span class="sc">|></span> </span>
<span id="cb13-5"><a href="#cb13-5"></a> specify_bsvar<span class="sc">$</span><span class="fu">new</span>() <span class="sc">|></span> </span>
<span id="cb13-6"><a href="#cb13-6"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">1000</span>) <span class="sc">|></span> </span>
<span id="cb13-7"><a href="#cb13-7"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">10000</span>) <span class="ot">-></span> post</span>
<span id="cb13-8"><a href="#cb13-8"></a></span>
<span id="cb13-9"><a href="#cb13-9"></a>post <span class="sc">|></span> <span class="fu">compute_impulse_responses</span>(<span class="at">horizon =</span> <span class="dv">12</span>) <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb13-10"><a href="#cb13-10"></a>post <span class="sc">|></span> <span class="fu">compute_variance_decompositions</span>(<span class="at">horizon =</span> <span class="dv">12</span>) <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb13-11"><a href="#cb13-11"></a>post <span class="sc">|></span> <span class="fu">compute_historical_decompositions</span>() <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb13-12"><a href="#cb13-12"></a>post <span class="sc">|></span> <span class="fu">compute_structural_shocks</span>() <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb13-13"><a href="#cb13-13"></a>post <span class="sc">|></span> <span class="fu">compute_conditional_sd</span>() <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb13-14"><a href="#cb13-14"></a>post <span class="sc">|></span> <span class="fu">forecast</span>(<span class="at">horizon =</span> <span class="dv">12</span>) <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb13-15"><a href="#cb13-15"></a>post <span class="sc">|></span> <span class="fu">verify_identification</span>() <span class="sc">|></span> <span class="fu">summary</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div><div class="column" style="width:50%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>workflow with the pipe</li>
</ul>
<div class="cell">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1"></a><span class="fu">library</span>(bsvarSIGNs)</span>
<span id="cb14-2"><a href="#cb14-2"></a><span class="fu">data</span>(optimism)</span>
<span id="cb14-3"><a href="#cb14-3"></a></span>
<span id="cb14-4"><a href="#cb14-4"></a>optimism <span class="sc">|></span> </span>
<span id="cb14-5"><a href="#cb14-5"></a> specify_bsvarSIGN<span class="sc">$</span><span class="fu">new</span>() <span class="sc">|></span> </span>
<span id="cb14-6"><a href="#cb14-6"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">10000</span>) <span class="ot">-></span> post</span>
<span id="cb14-7"><a href="#cb14-7"></a></span>
<span id="cb14-8"><a href="#cb14-8"></a>post <span class="sc">|></span> <span class="fu">compute_impulse_responses</span>(<span class="at">horizon =</span> <span class="dv">12</span>) <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb14-9"><a href="#cb14-9"></a>post <span class="sc">|></span> <span class="fu">compute_variance_decompositions</span>(<span class="at">horizon =</span> <span class="dv">12</span>) <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb14-10"><a href="#cb14-10"></a>post <span class="sc">|></span> <span class="fu">compute_historical_decompositions</span>() <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb14-11"><a href="#cb14-11"></a>post <span class="sc">|></span> <span class="fu">compute_structural_shocks</span>() <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb14-12"><a href="#cb14-12"></a>post <span class="sc">|></span> <span class="fu">compute_conditional_sd</span>() <span class="sc">|></span> <span class="fu">plot</span>()</span>
<span id="cb14-13"><a href="#cb14-13"></a>post <span class="sc">|></span> <span class="fu">forecast</span>(<span class="at">horizon =</span> <span class="dv">12</span>) <span class="sc">|></span> <span class="fu">plot</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</div>
</div>
</section>
<section id="bsvars-and-bsvarsigns-features-8" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">bsvars and bsvarSIGNs features</span></h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<div class="columns">
<div class="column" style="width:50%;">
<p><img data-src="bsvars.png" width="120"></p>
<ul>
<li>progress bar</li>
</ul>
<p><img data-src="bsvars_progress.png" width="500"></p>
</div><div class="column" style="width:50%;">
<p><img data-src="bsvarSIGNs.png" width="120"></p>
<ul>
<li>progress bar</li>
</ul>
<p><img data-src="bsvarSIGNs_progress.png" width="500"></p>
</div>
</div>
</section>
<section id="a-small-fiscal-policy-system-analysis-1" class="slide level2" data-background-color="#001D31">
<h2><span style="color: #ff68b4;">a small fiscal policy system analysis</span></h2>
</section>
<section id="a-small-fiscal-policy-system-analysis-2" class="slide level2">
<h2>a small fiscal policy system analysis</h2>
<div class="footer">
<p><a href="https://bsvars.org">bsvars.org</a></p>
</div>
<p><span class="math display">\[\begin{align}
y_t = \begin{bmatrix} ttr_t & gs_t & gdp_t \end{bmatrix}'
\end{align}\]</span></p>
<ul>
<li>US quarterly data from Q1 1948 to Q2 2024</li>
<li>all variables are in real, log, per person terms</li>
</ul>
<div class="cell" data-hash="index_cache/revealjs/specc_ec8e1c9fde1b2eb513020aab72ab0be1">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1"></a><span class="fu">library</span>(bsvars)</span>
<span id="cb15-2"><a href="#cb15-2"></a><span class="fu">data</span>(us_fiscal_lsuw) <span class="co"># upload dependent variables</span></span>
<span id="cb15-3"><a href="#cb15-3"></a><span class="fu">plot</span>(us_fiscal_lsuw, <span class="at">col =</span> <span class="st">"#ff68b4"</span>, <span class="at">lwd =</span> <span class="dv">4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<img data-src="index_files/figure-revealjs/specc-1.png" width="960" class="r-stretch"></section>
<section id="a-small-fiscal-policy-system-analysis-3" class="slide level2">
<h2>a small fiscal policy system analysis</h2>
<p><img data-src="bsvars.png" class="absolute" style="top: 170px; right: 10px; width: 120px; "></p>
<p><img data-src="bsvarSIGNs.png" class="absolute" style="top: 430px; right: 10px; width: 120px; "></p>
<h3 id="system-with-exclusion-restrictions.">system with exclusion restrictions.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}
B_{0.11}&0&0\\
B_{0.21}&B_{0.22}&0\\
B_{0.31}&B_{0.32}&B_{0.33}
\end{bmatrix}
\begin{bmatrix}ttr_t \\ gs_t \\ gdp_t \end{bmatrix} &= \dots +
\begin{bmatrix} u_t^{ttr} \\ u_t^{gs} \\ u_t^{gdp}\end{bmatrix}
\end{align}\]</span></p>
<h3 id="system-with-sign-restrictions.">system with sign restrictions.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}ttr_t \\ gs_t \\ gdp_t \end{bmatrix} &= \dots +
\begin{bmatrix}
+&*&+\\
*&+&*\\
*&*&+
\end{bmatrix}\begin{bmatrix} u_t^{ttr} \\ u_t^{gs} \\ u_t^{gdp}\end{bmatrix}
\end{align}\]</span></p>
</section>
<section id="specify-and-estimate-the-svar" class="slide level2">
<h2>specify and estimate the SVAR</h2>
<p><img data-src="bsvars.png" class="absolute" style="top: -20px; right: 10px; width: 120px; "></p>
<p><span class="math display">\[\begin{align}
&
\end{align}\]</span></p>
<div class="cell" data-hash="index_cache/revealjs/spec_d1d6bc6874ce46f6df4fc81bb3fa016c">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1"></a><span class="fu">data</span>(us_fiscal_ex) <span class="co"># upload exogenous variables</span></span>
<span id="cb16-2"><a href="#cb16-2"></a></span>
<span id="cb16-3"><a href="#cb16-3"></a><span class="co"># specify the model and set seed</span></span>
<span id="cb16-4"><a href="#cb16-4"></a><span class="fu">set.seed</span>(<span class="dv">123</span>)</span>
<span id="cb16-5"><a href="#cb16-5"></a>spec <span class="ot">=</span> specify_bsvar<span class="sc">$</span><span class="fu">new</span>(</span>
<span id="cb16-6"><a href="#cb16-6"></a> us_fiscal_lsuw, </span>
<span id="cb16-7"><a href="#cb16-7"></a> <span class="at">p =</span> <span class="dv">4</span>, </span>
<span id="cb16-8"><a href="#cb16-8"></a> <span class="at">exogenous =</span> us_fiscal_ex</span>
<span id="cb16-9"><a href="#cb16-9"></a>)</span>
<span id="cb16-10"><a href="#cb16-10"></a></span>
<span id="cb16-11"><a href="#cb16-11"></a><span class="co"># run the burn-in</span></span>
<span id="cb16-12"><a href="#cb16-12"></a>burn <span class="ot">=</span> <span class="fu">estimate</span>(spec, <span class="at">S =</span> <span class="dv">1000</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>**************************************************|
bsvars: Bayesian Structural Vector Autoregressions|
**************************************************|
Gibbs sampler for the SVAR model |
**************************************************|
Progress of the MCMC simulation for 1000 draws
Every draw is saved via MCMC thinning
Press Esc to interrupt the computations
**************************************************|</code></pre>
</div>
<div class="sourceCode cell-code" id="cb18"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb18-1"><a href="#cb18-1"></a><span class="co"># estimate the model</span></span>
<span id="cb18-2"><a href="#cb18-2"></a>post <span class="ot">=</span> <span class="fu">estimate</span>(burn, <span class="at">S =</span> <span class="dv">10000</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>**************************************************|
bsvars: Bayesian Structural Vector Autoregressions|
**************************************************|
Gibbs sampler for the SVAR model |
**************************************************|
Progress of the MCMC simulation for 10000 draws
Every draw is saved via MCMC thinning
Press Esc to interrupt the computations
**************************************************|</code></pre>