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index.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="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="#0056B9">
<h2></h2>
<p><img data-src="bsvars_ukr.png" class="absolute" style="top: 40px; right: 275px; width: 500px; "></p>
</section>
<section id="section-1" class="slide level2" data-background-color="#0056B9">
<h2></h2>
<!-- FFD800 -->
<p><span class="math display">\[ \]</span></p>
<p><span class="math display">\[ \]</span></p>
<h3 style="color:white;" id="bsvars-package-features"><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h3>
<h3 style="color:white;" id="bsvars-models-and-identification"><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models and identification</h3>
<h3 style="color:white;" id="bsvars-modeling-of-monetary-policy"><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h3>
</section>
<section id="section-2" class="slide level2" data-background-color="#0056B9">
<h2></h2>
<p><span class="math display">\[ \]</span></p>
<h3 style="color:white;" id="slides-as-a-website">Slides <a href="https://bsvars.github.io/2024-08-bsvars-w4UKR/" style="color:#FFD800;">as a Website</a></h3>
<h3 style="color:white;" id="github-repo-to-reproduce-the-slides-and-results">GitHub <a href="https://github.com/bsvars/2024-08-bsvars-w4UKR" style="color:#FFD800;">repo</a> to reproduce the slides and results</h3>
<h3 style="color:white;" id="r-script-to-reproduce-empirical-results"><a href="https://github.com/bsvars/2024-08-bsvars-w4UKR/blob/main/w4UKR_empirical.R" style="color:#FFD800;"><strong>R</strong> script</a> to reproduce empirical results</h3>
<p><span class="math display">\[ \]</span></p>
<h3 style="color:white;" id="bsvars-on-cran"><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> on CRAN</h3>
<h3 style="color:white;" id="bsvars.github.iobsvars"><a href="https://bsvars.github.io/bsvars/" style="color:#FFD800;">bsvars.github.io/bsvars/</a></h3>
<h3 style="color:white;" id="bsvars.github.io"><a href="https://bsvars.github.io" style="color:#FFD800;">bsvars.github.io</a></h3>
</section>
<section id="section-3" class="slide level2" data-background-color="#0056B9">
<h2></h2>
<p><img data-src="social@w4UKR.png" class="absolute" style="top: 0px; right: 200px; width: 735px; "></p>
</section>
<section id="bsvars-package-features-1" class="slide level2" data-background-color="#0056B9">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
</section>
<section id="section-4" class="slide level2">
<h2></h2>
<p><img data-src="cran.png" class="absolute" style="top: 10px; right: 30px; width: 1000px; "></p>
</section>
<section id="bsvars-package-features-2" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="bayesian-estimation-of-structural-vars">- Bayesian estimation of Structural VARs</h3>
<h3 id="volatility-3-non-normal-models-for-shocks">- 5 volatility & 3 non-normal models for shocks</h3>
<h3 id="identification-using">- identification using</h3>
<h4 id="exclusion-restrictions">- exclusion restrictions</h4>
<h4 id="heteroskedasticity">- heteroskedasticity</h4>
<h4 id="and-non-normality">- and non-normality</h4>
<h3 id="efficient-and-fast-gibbs-samplers">- efficient and fast Gibbs samplers</h3>
</section>
<section id="bsvars-package-features-3" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="excellent-computational-speed">- excellent computational speed</h3>
<h3 id="frontier-econometric-techniques">- frontier econometric techniques</h3>
<h3 id="compiled-code-using-cpp-via-rcpp-and-rcpparmadillo">- compiled code using <span style="color:#FFD800;">cpp</span> via <a href="https://cran.r-project.org/package=Rcpp" style="color:#FFD800;">Rcpp</a> and <a href="https://cran.r-project.org/package=RcppArmadillo" style="color:#FFD800;">RcppArmadillo</a></h3>
<h3 id="data-analysis-in-r">- data analysis in <a href="https://cran.r-project.org/" style="color:#FFD800;">R</a></h3>
</section>
<section id="bsvars-package-features-4" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="package-and-data-loading">- package and data loading</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(bsvars)</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></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>
<h3 id="simple-model-setup">- simple model setup</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-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>
<h3 id="simple-estimation">- simple estimation</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-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" aria-hidden="true" tabindex="-1"></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>
</section>
<section id="bsvars-package-features-5" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="structural-analyses">- structural analyses</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-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="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></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="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>hds <span class="ot">=</span> <span class="fu">compute_historical_decompositions</span>(post)</span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a>ss <span class="ot">=</span> <span class="fu">compute_structural_shocks</span>(post)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a>csds <span class="ot">=</span> <span class="fu">compute_conditional_sd</span>(post)</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></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>
<h3 id="predictive-analyses">- predictive analyses</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>fvs <span class="ot">=</span> <span class="fu">compute_fitted_values</span>(post)</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></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>
<h3 id="plots-and-summaries">- plots and summaries</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(irfs)</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></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>
</section>
<section id="bsvars-package-features-6" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="workflow-with-the-pipe">- workflow with the pipe</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(bsvars)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">data</span>(us_fiscal_lsuw)</span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a>us_fiscal_lsuw <span class="sc">|></span> </span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a> specify_bsvar<span class="sc">$</span><span class="fu">new</span>() <span class="sc">|></span> </span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">1000</span>) <span class="sc">|></span> </span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">10000</span>) <span class="ot">-></span> post</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-9"><a href="#cb7-9" aria-hidden="true" tabindex="-1"></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="cb7-10"><a href="#cb7-10" aria-hidden="true" tabindex="-1"></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="cb7-11"><a href="#cb7-11" aria-hidden="true" tabindex="-1"></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="cb7-12"><a href="#cb7-12" aria-hidden="true" tabindex="-1"></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="cb7-13"><a href="#cb7-13" aria-hidden="true" tabindex="-1"></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="cb7-14"><a href="#cb7-14" aria-hidden="true" tabindex="-1"></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="cb7-15"><a href="#cb7-15" aria-hidden="true" tabindex="-1"></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>
</section>
<section id="bsvars-package-features-7" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="specify-different-models">- specify different models</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb8"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a>spec1 <span class="ot">=</span> specify_bsvar<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw, <span class="at">p =</span> <span class="dv">4</span>) <span class="co"># SVAR(4)</span></span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a>spec2 <span class="ot">=</span> specify_bsvar_sv<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw) <span class="co"># SVAR(1)-SV</span></span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>spec3 <span class="ot">=</span> specify_bsvar_t<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw, <span class="at">p =</span> <span class="dv">2</span>) <span class="co"># SVAR(2)-t</span></span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a>spec4 <span class="ot">=</span> specify_bsvar_msh<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw, <span class="at">M =</span> <span class="dv">2</span>) <span class="co"># SVAR(1)-MSH(2)</span></span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a>spec5 <span class="ot">=</span> specify_bsvar_msh<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw, <span class="at">finiteM =</span> <span class="cn">FALSE</span>) <span class="co"># SVAR(1)-MSH(Inf)</span></span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a>spec6 <span class="ot">=</span> specify_bsvar_mix<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw, <span class="at">M =</span> <span class="dv">2</span>) <span class="co"># SVAR(1)-MIX(2)</span></span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a>spec7 <span class="ot">=</span> specify_bsvar_mix<span class="sc">$</span><span class="fu">new</span>(us_fiscal_lsuw, <span class="at">finiteM =</span> <span class="cn">FALSE</span>) <span class="co"># SVAR(1)-MIX(Inf)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<h3 id="from-now-on-its-the-same-e.g.">- from now on it’s the same, e.g.</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>spec1 <span class="sc">|></span> </span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">1000</span>) <span class="sc">|></span> </span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">10000</span>) <span class="ot">-></span> post</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a>spec2 <span class="sc">|></span> </span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">1000</span>) <span class="sc">|></span> </span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">estimate</span>(<span class="at">S =</span> <span class="dv">10000</span>) <span class="ot">-></span> post</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="bsvars-package-features-8" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> package features</h2>
<h3 id="progress-bar">- progress bar</h3>
<p><img data-src="bsvars_progress.png" class="absolute" style="top: 150px; right: 100px; width: 900px; "></p>
</section>
<section id="my-first-steps-with-bsvars" class="slide level2">
<h2>My first steps with <a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a></h2>
<h3 id="exercise.">Exercise.</h3>
<ol type="1">
<li>Download <strong>R</strong> script <a href="https://github.com/bsvars/2024-08-bsvars-w4UKR/blob/main/w4UKR_bsvars.R">w4UKR_bsvars.R</a></li>
<li>Open the file in your <strong>RStudio</strong></li>
<li>Execute the code line by line</li>
</ol>
<h4 id="you-have-just-estimated-your-first-model-using-the-bsvars-package-yay">You have just estimated your first model using the <strong>bsvars</strong> package! YAY!</h4>
<ol start="4" type="1">
<li>Modify the line specifying the model to</li>
</ol>
<pre><code>specify_bsvar_t$new(p = 4) |></code></pre>
<ol start="5" type="1">
<li>Run the code line by line</li>
</ol>
</section>
<section id="bsvars-models-and-identification-1" class="slide level2" data-background-color="#0056B9">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models and identification</h2>
</section>
<section id="bsvars-models" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="structural-var">Structural VAR</h3>
<p><span class="math display">\[\begin{align}
\text{reduced form:}&&\mathbf{y}_t &= \mathbf{A}_1\mathbf{y}_{t-1} + \dots + \mathbf{A}_p\mathbf{y}_{t-p} + \mathbf{A}_d\mathbf{d}_{t} + \boldsymbol{\varepsilon}_t \\
\text{structural form:}&&\mathbf{B}_0\boldsymbol{\varepsilon}_t &= \mathbf{u}_t
\end{align}\]</span></p>
<ul>
<li>system modelling of dependent variables <span class="math inline">\(\mathbf{y}_t\)</span></li>
<li>system dynamics captured by modeling effects of lags <span class="math inline">\(\mathbf{y}_{t-1},\dots,\mathbf{y}_{t-p}\)</span></li>
<li>deterministic terms and exogenous variables <span class="math inline">\(\mathbf{d}_{t}\)</span></li>
<li>economic structure introduced by structural matrix <span class="math inline">\(\mathbf{B}_0\)</span></li>
<li>well-isolated structural shocks <span class="math inline">\(\mathbf{u}_t\)</span></li>
</ul>
</section>
<section id="bsvars-models-1" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="structural-var-1">Structural VAR</h3>
<p><span class="math display">\[\begin{align}
\text{reduced form:}&&\mathbf{y}_t &= \mathbf{A}\mathbf{x}_t + \boldsymbol{\varepsilon}_t \\
\text{structural form:}&&\mathbf{B}_0\boldsymbol{\varepsilon}_t &= \mathbf{u}_t \\
\text{structural shocks:}&&\mathbf{u}_t\mid\mathbf{x}_t &\sim N\left( \mathbf{0}_N, \text{diag}\left(\boldsymbol{\sigma}_t^2\right) \right)
\end{align}\]</span></p>
<ul>
<li>interpretable structural specification</li>
<li>identification through
<ul>
<li>exclusion restrictions</li>
<li>heteroskedasticity</li>
<li>non-normality</li>
</ul></li>
<li>facilitates application of frontier numerical techniques</li>
</ul>
</section>
<section id="bsvars-models-2" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<h3 id="reduced-form-hierarchical-prior">Reduced form hierarchical prior</h3>
<p><span class="math display">\[\begin{align}
\text{autoregressive slopes:}&& [\mathbf{A}]_{n\cdot}'\mid\gamma_{A.n} &\sim N_{Np+1}\left( \underline{\mathbf{m}}_{n.A}, \gamma_{A.n}\underline{\Omega}_A \right) \\
\text{autoregressive shrinkage:}&&\gamma_{A.n} | s_{A.n} &\sim IG2\left(s_{A.n}, \underline{\nu}_A\right)\\
\text{local scale:}&&s_{A.n} | s_{A} &\sim G\left(s_{A}, \underline{a}_A\right)\\
\text{global scale:}&&s_{A} &\sim IG2\left(\underline{s}_{s_A}, \underline{\nu}_{s_A}\right)
\end{align}\]</span></p>
<ul>
<li>Minnesota prior mean and shrinkage decay with increasing lags</li>
<li>Flexibility in shrinkage and scale hyper-parameters</li>
<li>3-level equation-specific local-global hierarchical prior</li>
</ul>
</section>
<section id="bsvars-models-3" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<h3 id="structural-form-hierarchical-prior">Structural form hierarchical prior</h3>
<p><span class="math display">\[\begin{align}
\text{exclusion restrictions:}&& [\mathbf{B}_0]_{n\cdot} &= \mathbf{b}_n\mathbf{V}_n\\
\text{structural relations:}&& \mathbf{B}_0\mid\gamma_{B}&\sim |\det(\mathbf{B}_0)|^{\underline{\nu}_B - N}\exp\left\{-\frac{1}{2} \sum_{n=1}^{N} \gamma_{B.n}^{-1} \mathbf{b}_n\mathbf{b}_n' \right\} \\
\text{structural shrinkage:}&&\gamma_{B.n} | s_{B.n} &\sim IG2\left(s_{B.n}, \underline{\nu}_b\right)\\
\text{local scale:}&&s_{B.n} | s_{B} &\sim G\left(s_{B}, \underline{a}_B\right)\\
\text{global scale:}&&s_{B} &\sim IG2\left(\underline{s}_{s_B}, \underline{\nu}_{s_B}\right)
\end{align}\]</span></p>
<ul>
<li>Highly adaptive equation-by-equation exclusion restrictions</li>
<li>Likelihood-shape preserving prior</li>
<li>Flexibility in shrinkage and scale hyper-parameters</li>
<li>3-level equation-specific local-global hierarchical prior</li>
</ul>
</section>
<section id="bsvars-models-4" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<h3 id="volatility-models">Volatility models</h3>
<ul>
<li>Homoskedastic <span class="math inline">\(\sigma_{n.t}^2 = 1\)</span></li>
<li>Stochastic Volatility: non-centred and centred</li>
<li>Stationary Markov-switching heteroskedastisity</li>
<li>Sparse Markov-switching heteroskedastisity</li>
</ul>
<h3 id="non-normal-models">Non-normal models</h3>
<ul>
<li>Student-t shocks</li>
<li>Finite mixture of normal components</li>
<li>Sparse mixture of normal components</li>
</ul>
</section>
<section id="bsvars-models-5" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="non-centred-stochastic-volatility">Non-centred Stochastic Volatility</h3>
<p><span class="math display">\[\begin{align}
\text{conditional variance:}&&\sigma_{n.t}^2 &= \exp\left\{\omega_n h_{n.t}\right\}\\
\text{log-volatility:}&&h_{n.t} &= \rho_n h_{n.t-1} + v_{n.t}\\
\text{volatility innovation:}&&v_{n.t}&\sim N\left(0,1\right)\\
\end{align}\]</span></p>
<ul>
<li>excellent volatility forecasting performance</li>
<li>standardization around <span class="math inline">\(\sigma_{n.t}^2 = 1\)</span></li>
<li>homoskedasticity verification by testing <span class="math inline">\(\omega_n = 0\)</span></li>
</ul>
</section>
<section id="bsvars-models-6" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="centred-stochastic-volatility">Centred Stochastic Volatility</h3>
<p><span class="math display">\[\begin{align}
\text{conditional variance:}&&\sigma_{n.t}^2 &= \exp\left\{\tilde{h}_{n.t}\right\}\\
\text{log-volatility:}&&\tilde{h}_{n.t} &= \rho_n \tilde{h}_{n.t-1} + \tilde{v}_{n.t}\\
\text{volatility innovation:}&&\tilde{v}_{n.t}&\sim N\left(0,\sigma_v^2\right)\\
\end{align}\]</span></p>
<ul>
<li>excellent volatility forecasting performance</li>
<li>weak standardisation</li>
<li>no homoskedasticity verification available</li>
</ul>
</section>
<section id="bsvars-models-7" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> models</h2>
<h3 id="stochastic-volatility-conditional-variance-prior">Stochastic Volatility: conditional variance prior</h3>
<img data-src="index_files/figure-revealjs/plot_cv_prior-1.png" width="960" class="r-stretch"></section>
<section id="bsvars-identification-simplified" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<p><span class="math display">\[\begin{align}
&\\
\mathbf\Sigma &= \mathbf{B}_0^{-1}\mathbf{B}_0^{-1\prime}\\[1ex]
\end{align}\]</span></p>
<ul>
<li><span class="math inline">\(\mathbf\Sigma\)</span> can be estimated using data easily</li>
<li>The relationship presents a system of equations to be solved for <span class="math inline">\(\mathbf{B}_0\)</span></li>
<li><span class="math inline">\(\mathbf\Sigma\)</span> is a <em>symmetric</em> <span class="math inline">\(N\times N\)</span> matrix</li>
<li><span class="math inline">\(\mathbf\Sigma\)</span> has <span class="math inline">\(N(N+1)/2\)</span> unique elements (equations)</li>
<li><span class="math inline">\(\mathbf{B}_0\)</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>We cannot estimate all elements of <span class="math inline">\(\mathbf{B}_0\)</span> using <span class="math inline">\(N(N+1)/2\)</span> equations</li>
<li><span class="math inline">\(\mathbf{B}_0\)</span> is <text style="color:#0056B9;"><strong>not identified</strong></text></li>
</ul>
</section>
<section id="bsvars-identification-simplified-1" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<p><span class="math display">\[\begin{align}
&\\
\mathbf\Sigma &= \mathbf{B}_0^{-1}\mathbf{B}_0^{-1\prime}\\[1ex]
\end{align}\]</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">\(\mathbf{B}_0\)</span> can be estimated</li>
<li>Impose <span class="math inline">\(N(N-1)/2\)</span> restrictions on <span class="math inline">\(\mathbf{B}_0\)</span> to solve the equation</li>
<li>This identifies the rows of <span class="math inline">\(\mathbf{B}_0\)</span> up to a sign</li>
<li>Change the sign of any number of <span class="math inline">\(\mathbf{B}_0\)</span> rows and <span class="math inline">\(\mathbf\Sigma\)</span> will not change</li>
<li>Often <span class="math inline">\(\mathbf{B}_0\)</span> is made lower-triangular</li>
</ul>
</section>
<section id="bsvars-identification-simplified-2" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<p>Let <span class="math inline">\(N=2\)</span></p>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{0.11}&B_{0.12}\\ B_{0.21}&B_{0.22}\end{bmatrix}\\[1ex]
\end{align}\]</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">\(\mathbf{B}_0\)</span> cannot be estimated</li>
</ul>
<h3 id="identification.-1">Identification.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}\sigma_1^2&\sigma_{12}\\ \sigma_{12}&\sigma_2^2\end{bmatrix} &\qquad
\begin{bmatrix}B_{0.11}& 0\\ B_{0.21}&B_{0.22}\end{bmatrix}\\[1ex]
\end{align}\]</span></p>
<ul>
<li>3 equations identify 3 elements in <span class="math inline">\(\mathbf{B}_0\)</span></li>
</ul>
</section>
<section id="bsvars-identification-simplified-3" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<h3 id="identification-through-heteroskedasticity.">Identification through Heteroskedasticity.</h3>
<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">\(\mathbf{B}_0\)</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">\[\begin{align}
\mathbf\Sigma_1 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}_0^{-1\prime}\\[1ex]
\mathbf\Sigma_2 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}_0^{-1\prime}
\end{align}\]</span></p>
</section>
<section id="bsvars-identification-simplified-4" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<h3 id="identification-through-heteroskedasticity.-1">Identification through Heteroskedasticity.</h3>
<p><span class="math display">\[\begin{align}
\mathbf\Sigma_1 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_1^2\right)\mathbf{B}_0^{-1\prime}\\[1ex]
\mathbf\Sigma_2 &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_2^2\right)\mathbf{B}_0^{-1\prime}
\end{align}\]</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">\(\mathbf{B}_0\)</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">\(\boldsymbol\sigma_i^2 \approx \boldsymbol\imath_N\)</span></li>
</ul>
</section>
<section id="bsvars-identification-simplified-5" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<h3 id="identification-through-heteroskedasticity.-2">Identification through Heteroskedasticity.</h3>
<p>The setup can be generalised to conditional heteroskedasticity of structural shocks</p>
<p><span class="math display">\[\begin{align}
u_t |Y_{t-1} &\sim N_N\left(\mathbf{0}_N, \text{diag}\left(\boldsymbol\sigma_t^2\right)\right)\\[1ex]
\mathbf\Sigma_t &= \mathbf{B}_0^{-1}\text{diag}\left(\boldsymbol\sigma_t^2\right)\mathbf{B}_0^{-1\prime}
\end{align}\]</span></p>
<h3 id="identification.-3">Identification.</h3>
<ul>
<li>Matrix <span class="math inline">\(\mathbf{B}_0\)</span> is identified up to its rows’ sign change and equations’ reordering</li>
<li>shocks are identified if changes in their conditional variances are non-proportional</li>
<li>Structural shocks’ conditional variances <span class="math inline">\(\boldsymbol\sigma_t^2\)</span> can be estimated</li>
<li>Choose any (conditional) variance model for <span class="math inline">\(\boldsymbol\sigma_t^2\)</span> that fits the data well.</li>
</ul>
</section>
<section id="bsvars-identification-simplified-6" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<h3 id="identification-through-non-normality.">Identification through Non-normality.</h3>
<p><span class="math display">\[\begin{align}
\text{structural shocks:}&&\mathbf{u}_t\mid\mathbf{x}_t &\sim t\left( \mathbf{0}_N, \mathbf{I}_N, \nu \right)
\end{align}\]</span></p>
<ul>
<li><span class="math inline">\(\nu\)</span> - the degrees of freedom parameter is estimated</li>
<li>fat tails provide identification information</li>
<li>potential gains in forecasting precision</li>
<li>robustness to outliers</li>
<li>verify identification by checking <span class="math inline">\(\nu \rightarrow\infty\)</span></li>
</ul>
</section>
<section id="bsvars-identification-simplified-7" class="slide level2" data-background-color="#0056B9">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification (simplified)</h2>
<h3 style="color:#FFFFFF;" id="identification-through-non-normality.-1">Identification through Non-normality.</h3>
<img data-src="index_files/figure-revealjs/unnamed-chunk-11-1.png" width="960" class="r-stretch"></section>
<section id="bsvars-identification-verification" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification verification</h2>
<p><span class="math display">\[ \]</span></p>
<h3 id="savage-dickey-density-ratio.">Savage-Dickey Density Ratio.</h3>
<p>Verify the restriction through the posterior odds ratio using the SDDR: <span class="math display">\[
SDDR = \frac{\Pr[H_0 | data]}{\Pr[H_1 | data]}= \frac{p(H_0 | data)}{p(H_0 )}
\]</span></p>
<ul>
<li>suitable to verify sharp restrictions on parameters</li>
<li>is interpreted as posterior odds ratio</li>
<li>values greater than 1 provide evidence in favor of the restriction</li>
<li>simple to compute given the unrestricted model estimation output</li>
</ul>
</section>
<section id="bsvars-identification-verification-1" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification verification</h2>
<h3 id="in-non-centred-stochastic-volatility.">…in non-centred Stochastic Volatility.</h3>
<p>A structural shock is homoskedastic if <span class="math display">\[
H_0:\quad\omega_n = 0
\]</span></p>
<p><span class="math display">\[ \]</span></p>
<ul>
<li>if <span class="math inline">\(\omega_n = 0\)</span> the shock is homoskedastic with <span class="math inline">\(\sigma_{nt}^2 = 1\)</span></li>
<li>the only homoskedastic shock in the system is identified</li>
<li>two or more homoskedastic shocks have not identified through heteroskedasticity</li>
<li>heteroskedastic shocks are identified with probability 1</li>
<li>use the <code>verify_identification()</code> function</li>
</ul>
</section>
<section id="bsvars-identification-verification-2" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> identification verification</h2>
<h3 id="in-t-distributed-shocks-model.">…in t-distributed shocks model.</h3>
<p>A structural shock is homoskedastic if <span class="math display">\[
H_0:\quad\nu \rightarrow\infty
\]</span></p>
<p><span class="math display">\[ \]</span></p>
<ul>
<li>if <span class="math inline">\(\nu \rightarrow\infty\)</span> all the shocks are normal</li>
<li>when <span class="math inline">\(\nu\)</span> is finite and less than 30, the shocks are leptokurtic</li>
<li>then all the shocks are identified through non-normality</li>
<li>SDDR derivations and computation require some fancy stats</li>
<li>use the <code>verify_identification()</code> function</li>
</ul>
</section>
<section id="bsvars-hypothesis-verification" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> hypothesis verification</h2>
<h3 id="for-autoregressive-parameters.">…for autoregressive parameters.</h3>
<p>A structural shock is homoskedastic if <span class="math display">\[
H_0:\quad\mathbf{S}\text{vec}(\mathbf{A}) = \mathbf{r}
\]</span></p>
<p><span class="math display">\[ \]</span></p>
<ul>
<li>verify restrictions on the autoregressive parameters</li>
<li>can be used, e.g., to verify Granger causality</li>
<li>use the <code>verify_autoregression()</code> function</li>
</ul>
</section>
<section id="bsvars-hypothesis-verification-1" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> hypothesis verification</h2>
<h3 id="for-autoregressive-parameters.-1">…for autoregressive parameters.</h3>
<h3 id="exercise.-1">Exercise.</h3>
<ol type="1">
<li>Download the <strong>R</strong> script <a href="https://github.com/bsvars/2024-08-bsvars-w4UKR/blob/main/w4UKR_bsvars_Granger.R">w4UKR_bsvars_Granger.R</a></li>
<li>Verify the hypothesis of no Granger causality from ttr to gdp</li>
</ol>
</section>
<section id="bsvars-modeling-of-monetary-policy-1" class="slide level2" data-background-color="#0056B9">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
</section>
<section id="bsvars-modeling-of-monetary-policy-2" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="domestic-economy.">Domestic economy.</h3>
<p>Consider a system of five domestic variables:</p>
<p><span class="math display">\[\begin{align}
y_t = \begin{bmatrix} rgdp_t & cpi_t & CR_t & EX_t & aord_t \end{bmatrix}'
\end{align}\]</span></p>
<ul>
<li><p><span class="math inline">\(rgdp_t\)</span> - log real Gross Domestic Product</p></li>
<li><p><span class="math inline">\(cpi_t\)</span> - log Consumer Price Index</p></li>
<li><p><span class="math inline">\(CR_t\)</span> - Cash Rate Target - Australian nominal interest rate</p></li>
<li><p><span class="math inline">\(EX_t\)</span> - USD/AUD exchange rate</p></li>
<li><p><span class="math inline">\(aord_t\)</span> - log All Ordinaries Index</p></li>
<li><p>monthly data from August 1990 to March 2024</p></li>
<li><p>quarterly variables interpolated to monthly frequency</p></li>
</ul>
</section>
<section id="bsvars-modeling-of-monetary-policy-3" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="foreign-sector.">Foreign sector.</h3>
<p>The foreign sector includes three US variables:</p>
<ul>
<li><span class="math inline">\(rgdp_t^{(US)}\)</span> - log real Gross Domestic Product</li>
<li><span class="math inline">\(cpi_t^{(US)}\)</span> - log Consumer Price Index</li>
<li><span class="math inline">\(FFR_t\)</span> - Cash Rate Target - Australian nominal interest rate</li>
<li>monthly data from August 1990 to March 2024</li>
<li>quarterly variables interpolated to monthly frequency</li>
<li>contemporaneous and four lagged values are included in the model as exogenous variables</li>
</ul>
</section>
<section id="bsvars-modeling-of-monetary-policy-4" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="lower-triangular-system.">Lower-triangular system.</h3>
<p><span class="math display">\[\begin{align}
\begin{bmatrix}
B_{0.11}&0&0&0&0\\
B_{0.21}&B_{0.22}&0&0&0\\
B_{0.31}&B_{0.32}&B_{0.33}&0&0\\
B_{0.41}&B_{0.42}&B_{0.43}&B_{0.44}&0\\
B_{0.51}&B_{0.52}&B_{0.53}&B_{0.54}&B_{0.55}
\end{bmatrix}
\begin{bmatrix}rgdp_t \\ cpi_t \\ CR_t \\ EX_t\\ aord_t \end{bmatrix} &= \dots +
\begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \\ u_t^{aord} \end{bmatrix}
\end{align}\]</span></p>
<h3 id="identified-shocks.">Identified shocks.</h3>
<ul>
<li><span class="math inline">\(u_t^{mps}\)</span> - monetary policy shock identified via Taylor’s Rule</li>
<li><span class="math inline">\(u_t^{ex}\)</span> - currency shock</li>
</ul>
<!-- ## [bsvars](https://cran.r-project.org/package=bsvars){style="color:#FFD800;"} modeling of monetary policy -->
<!-- ### Extended system. -->
<!-- \begin{align} -->
<!-- \begin{bmatrix} -->
<!-- B_{0.11}&0&0&0&0\\ -->
<!-- B_{0.21}&B_{0.22}&0&0&0\\ -->
<!-- B_{0.31}&B_{0.32}&B_{0.33}&B_{0.34}&0\\ -->
<!-- B_{0.41}&B_{0.42}&B_{0.43}&B_{0.44}&0\\ -->
<!-- B_{0.51}&B_{0.52}&B_{0.53}&B_{0.54}&B_{0.55} -->
<!-- \end{bmatrix} -->
<!-- \begin{bmatrix}rgdp_t \\ cpi_t \\ CR_t \\ EX_t\\ aord_t \end{bmatrix} &= \dots + -->
<!-- \begin{bmatrix} u_t^{ad} \\ u_t^{as} \\ u_t^{mps} \\ u_t^{ex} \\ u_t^{aord} \end{bmatrix} -->
<!-- \end{align} -->
<!-- ### Identified shocks. -->
<!-- - $u_t^{mps}$ - identified via Taylor's Rule extended by exchange rate -->
<!-- - $u_t^{ex}$ - currency shock indistinguishable from the monetary policy shock -->
<!-- - identification through heteroskedasticity helps identifying the shocks -->
</section>
<section id="bsvars-modeling-of-monetary-policy-5" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="specify-and-estimate-a-lower-triangular-svar-sv.">Specify and estimate a lower-triangular SVAR-SV.</h3>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(bsvars)</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="fu">load</span>(<span class="st">"soe.rda"</span>)</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>soe <span class="ot">=</span> <span class="fu">as.matrix</span>(soe)</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a>TT <span class="ot">=</span> <span class="fu">nrow</span>(soe)</span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a>lag_order <span class="ot">=</span> <span class="dv">8</span></span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a>lag_exogenous <span class="ot">=</span> <span class="dv">4</span></span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a>T <span class="ot">=</span> TT <span class="sc">-</span> <span class="fu">max</span>(lag_order, lag_exogenous)</span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a>exogenous <span class="ot">=</span> <span class="fu">matrix</span>(<span class="cn">NA</span>, TT <span class="sc">-</span> lag_exogenous, <span class="dv">0</span>)</span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">0</span><span class="sc">:</span>lag_exogenous) {</span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a> exogenous <span class="ot">=</span> <span class="fu">cbind</span>(exogenous, <span class="fu">as.matrix</span>(soe[(lag_exogenous <span class="sc">-</span> i <span class="sc">+</span> <span class="dv">1</span>)<span class="sc">:</span>(TT <span class="sc">-</span> i), <span class="dv">6</span><span class="sc">:</span><span class="dv">8</span>]))</span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-15"><a href="#cb11-15" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1234</span>)</span>
<span id="cb11-16"><a href="#cb11-16" aria-hidden="true" tabindex="-1"></a>spec <span class="ot">=</span> specify_bsvar_sv<span class="sc">$</span><span class="fu">new</span>(</span>
<span id="cb11-17"><a href="#cb11-17" aria-hidden="true" tabindex="-1"></a> <span class="at">data =</span> <span class="fu">tail</span>(soe[,<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>], T),</span>
<span id="cb11-18"><a href="#cb11-18" aria-hidden="true" tabindex="-1"></a> <span class="at">p =</span> lag_order,</span>
<span id="cb11-19"><a href="#cb11-19" aria-hidden="true" tabindex="-1"></a> <span class="at">exogenous =</span> <span class="fu">tail</span>(exogenous, T)</span>
<span id="cb11-20"><a href="#cb11-20" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb11-21"><a href="#cb11-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-22"><a href="#cb11-22" aria-hidden="true" tabindex="-1"></a>burn <span class="ot">=</span> <span class="fu">estimate</span>(spec, <span class="fl">1e4</span>)</span>
<span id="cb11-23"><a href="#cb11-23" aria-hidden="true" tabindex="-1"></a>post <span class="ot">=</span> <span class="fu">estimate</span>(burn, <span class="fl">1e4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="bsvars-modeling-of-monetary-policy-6" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="svar-sv-compute-and-plot-impulse-responses.">SVAR-SV: Compute and plot impulse responses.</h3>
<div class="cell" data-layout-align="center" data-hash="index_cache/revealjs/bs00_ed5e158b8fa7d6165daa8b684ef5c1d4">
<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>post <span class="sc">|></span> <span class="fu">compute_impulse_responses</span>(<span class="at">horizon =</span> <span class="dv">60</span>) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">probability =</span> <span class="fl">0.68</span>, <span class="at">col =</span> <span class="st">"#0056B9"</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/bs00-1.png" width="768" class="r-stretch quarto-figure-center"></section>
<section id="bsvars-modeling-of-monetary-policy-7" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="svar-sv-forecast-error-variance-decompositions.">SVAR-SV: Forecast error variance decompositions.</h3>
<div class="cell" data-layout-align="center" data-hash="index_cache/revealjs/bs00fevd_05b6f100633a7a4d78c116eb14d4cbf6">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>post <span class="sc">|></span> <span class="fu">compute_variance_decompositions</span>(<span class="at">horizon =</span> <span class="dv">60</span>) <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> bsvars_grad)</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/bs00fevd-1.png" width="768" class="r-stretch quarto-figure-center"></section>
<section id="bsvars-modeling-of-monetary-policy-8" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="svar-sv-structural-shocks.">SVAR-SV: Structural shocks.</h3>
<div class="cell" data-layout-align="center" data-hash="index_cache/revealjs/bs00sss_7a0163c11618cec4b6d66cdec49d6db4">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a>post <span class="sc">|></span> <span class="fu">compute_structural_shocks</span>() <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> <span class="st">"#0056B9"</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/bs00sss-1.png" width="768" class="r-stretch quarto-figure-center"></section>
<section id="bsvars-modeling-of-monetary-policy-9" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="svar-sv-shocks-conditional-standard-deviations.">SVAR-SV: Shocks’ conditional standard deviations.</h3>
<div class="cell" data-layout-align="center" data-hash="index_cache/revealjs/bs00csd_2f0055cdb8980f0dd8e290476c3cf6e8">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a>post <span class="sc">|></span> <span class="fu">compute_conditional_sd</span>() <span class="sc">|></span> <span class="fu">plot</span>(<span class="at">col =</span> <span class="st">"#0056B9"</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/bs00csd-1.png" width="768" class="r-stretch quarto-figure-center"></section>
<section id="bsvars-modeling-of-monetary-policy-10" class="slide level2">
<h2><a href="https://cran.r-project.org/package=bsvars" style="color:#FFD800;">bsvars</a> modeling of monetary policy</h2>
<h3 id="svar-sv-homoskedasticity-verification.">SVAR-SV: Homoskedasticity verification.</h3>
<div class="cell" data-hash="index_cache/revealjs/bs00vv_0361e6f49f200d36857751c63aab5c02">
<div class="sourceCode cell-code" id="cb16"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb16-1"><a href="#cb16-1" aria-hidden="true" tabindex="-1"></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 class="cell-output cell-output-stdout">
<pre><code> **************************************************|
bsvars: Bayesian Structural Vector Autoregressions|
**************************************************|
Summary of identification verification |
H0: omega_n = 0 [homoskedasticity] |
H1: omega_n != 0 [heteroskedasticity] |