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xl:space-y-8 xl:space-x-0"><li class="flex items-center space-x-2"><img alt="avatar" loading="lazy" width="38" height="38" decoding="async" data-nimg="1" class="h-10 w-10 rounded-full" style="color:transparent" srcSet="/_next/image?url=%2Fstatic%2Fimages%2Favatar.png&w=48&q=75 1x, /_next/image?url=%2Fstatic%2Fimages%2Favatar.png&w=96&q=75 2x" src="/_next/image?url=%2Fstatic%2Fimages%2Favatar.png&w=96&q=75"/><dl class="text-sm leading-5 font-medium whitespace-nowrap"><dt class="sr-only">Name</dt><dd class="text-gray-900 dark:text-gray-100">Tails Azimuth</dd><dt class="sr-only">Twitter</dt><dd><a class="text-primary-500 hover:text-primary-600 dark:hover:text-primary-400" target="_blank" rel="noopener noreferrer" href="https://twitter.com/Twitter">@Twitter</a></dd></dl></li></ul></dd></dl><div class="divide-y divide-gray-200 xl:col-span-3 xl:row-span-2 xl:pb-0 dark:divide-gray-700"><div class="prose dark:prose-invert max-w-none pt-10 pb-8"><h1 class="content-header" id="introduction"><a class="break-words" href="#introduction" aria-hidden="true" tabindex="-1"><span class="content-header-link"><svg viewBox="0 0 20 20" xmlns="http://www.w3.org/2000/svg" class="h-5 linkicon w-5" fill="currentColor"><path d="M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"></path><path d="M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"></path></svg></span></a>Introduction</h1><p>Parsing and display of math equations is included in this blog template. Parsing of math is enabled by <code>remark-math</code> and <code>rehype-katex</code>. KaTeX and its associated font is included in <code>_document.js</code> so feel free to use it on any page. <sup><a class="break-words" href="#user-content-fn-footnote" aria-describedby="footnote-label" data-footnote-ref="true" id="user-content-fnref-footnote">1</a></sup></p><p>Inline math symbols can be included by enclosing the term between the <code>$</code> symbol.</p><p>Math code blocks are denoted by <code>$$</code>.</p><p>If you intend to use the <code>$</code> sign instead of math, you can escape it (<code>\$</code>), or specify the HTML entity (<code>&dollar;</code>) <sup><a class="break-words" href="#user-content-fn-2" aria-describedby="footnote-label" data-footnote-ref="true" id="user-content-fnref-2">2</a></sup></p><p>Inline or manually enumerated footnotes are also supported. Click on the links above to see them in action.</p><h1 class="content-header" id="deriving-the-ols-estimator"><a class="break-words" href="#deriving-the-ols-estimator" aria-hidden="true" tabindex="-1"><span class="content-header-link"><svg viewBox="0 0 20 20" xmlns="http://www.w3.org/2000/svg" class="h-5 linkicon w-5" fill="currentColor"><path d="M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"></path><path d="M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"></path></svg></span></a>Deriving the OLS Estimator</h1><p>Using matrix notation, let <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.4306em"></span><span class="mord mathnormal">n</span></span></span></span> denote the number of observations and <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6944em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> denote the number of regressors.</p><p>The vector of outcome variables <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">Y</mi></mrow><annotation encoding="application/x-tex">\mathbf{Y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6861em"></span><span class="mord mathbf" style="margin-right:.02875em">Y</span></span></span></span> is a <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6667em;vertical-align:-.0833em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:.6444em"></span><span class="mord">1</span></span></span></span> matrix,</p><div class="relative"><pre><code class="language-tex code-highlight"><span class="code-line"><span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span> = <span class="token function selector">\left</span><span class="token punctuation">[</span><span class="token function selector">\begin</span><span class="token punctuation">{</span><span class="token keyword">array</span><span class="token punctuation">}</span>
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</span><span class="code-line"> <span class="token punctuation">{</span>c<span class="token punctuation">}</span>
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</span><span class="code-line"> y_1 <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token function selector">\\</span>
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</span><span class="code-line"> y_n
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</span><span class="code-line"><span class="token function selector">\end</span><span class="token punctuation">{</span><span class="token keyword">array</span><span class="token punctuation">}</span><span class="token function selector">\right</span><span class="token punctuation">]</span>
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</span></code></pre></div><span class="katex-display" translate="no"><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">Y</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="0.16em"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><msub><mi>y</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><msub><mi>y</mi><mi>n</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{Y} = \left[\begin{array} {c} y_1 \\ . \\ . \\ . \\ y_n \end{array}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6861em"></span><span class="mord mathbf" style="margin-right:.02875em">Y</span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.25em"><span class="pstrut" style="height:8em"></span><span style="width:.667em;height:6em"><svg viewBox="0 0 667 6000" xmlns="http://www.w3.org/2000/svg" height="6.000em" width="0.667em"><path d="M403 1759 V84 H666 V0 H319 V1759 v2400 v1759 h347 v-84
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H403z M403 1759 V0 H319 V1759 v2400 v1759 h84z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3011em"><span style="top:-2.55em;margin-left:-.0359em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.1514em"><span style="top:-2.55em;margin-left:-.0359em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.25em"><span class="pstrut" style="height:8em"></span><span style="width:.667em;height:6em"><svg viewBox="0 0 667 6000" xmlns="http://www.w3.org/2000/svg" height="6.000em" width="0.667em"><path d="M347 1759 V0 H0 V84 H263 V1759 v2400 v1759 H0 v84 H347z
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M347 1759 V0 H263 V1759 v2400 v1759 h84z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span></span></span></span></span></span></span><p>The matrix of regressors <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6861em"></span><span class="mord mathbf">X</span></span></span></span> is a <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n \times k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6667em;vertical-align:-.0833em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:.6944em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> matrix (or each row is a <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7778em;vertical-align:-.0833em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:.6444em"></span><span class="mord">1</span></span></span></span> vector),</p><div class="relative"><pre><code class="code-highlight language-latex"><span class="code-line"><span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span> = <span class="token function selector">\left</span><span class="token punctuation">[</span><span class="token function selector">\begin</span><span class="token punctuation">{</span><span class="token keyword">array</span><span class="token punctuation">}</span>
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</span><span class="code-line"> <span class="token punctuation">{</span>ccccc<span class="token punctuation">}</span>
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</span><span class="code-line"> x_<span class="token punctuation">{</span>11<span class="token punctuation">}</span> <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> x_<span class="token punctuation">{</span>1k<span class="token punctuation">}</span> <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token function selector">\\</span>
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</span><span class="code-line"> x_<span class="token punctuation">{</span>n1<span class="token punctuation">}</span> <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> . <span class="token punctuation">&</span> x_<span class="token punctuation">{</span>nn<span class="token punctuation">}</span>
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</span><span class="code-line"><span class="token function selector">\end</span><span class="token punctuation">{</span><span class="token keyword">array</span><span class="token punctuation">}</span><span class="token function selector">\right</span><span class="token punctuation">]</span> =
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</span><span class="code-line"><span class="token function selector">\left</span><span class="token punctuation">[</span><span class="token function selector">\begin</span><span class="token punctuation">{</span><span class="token keyword">array</span><span class="token punctuation">}</span>
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</span><span class="code-line"> <span class="token punctuation">{</span>c<span class="token punctuation">}</span>
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</span><span class="code-line"> <span class="token function selector">\mathbf</span><span class="token punctuation">{</span>x<span class="token punctuation">}</span>'_1 <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token function selector">\\</span>
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</span><span class="code-line"> . <span class="token function selector">\\</span>
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</span><span class="code-line"> <span class="token function selector">\mathbf</span><span class="token punctuation">{</span>x<span class="token punctuation">}</span>'_n
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</span><span class="code-line"><span class="token function selector">\end</span><span class="token punctuation">{</span><span class="token keyword">array</span><span class="token punctuation">}</span><span class="token function selector">\right</span><span class="token punctuation">]</span>
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</span></code></pre></div><span class="katex-display" translate="no"><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="bold">X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable columnalign="center center center center center" columnspacing="1em" rowspacing="0.16em"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><msub><mi>x</mi><mn>11</mn></msub></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><msub><mi>x</mi><mrow><mn>1</mn><mi>k</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><msub><mi>x</mi><mrow><mi>n</mi><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd><mtd><mstyle displaystyle="false" scriptlevel="0"><msub><mi>x</mi><mrow><mi>n</mi><mi>n</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="0.16em"><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><msubsup><mi mathvariant="bold">x</mi><mn>1</mn><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><mi mathvariant="normal">.</mi></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="false" scriptlevel="0"><msubsup><mi mathvariant="bold">x</mi><mi>n</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msubsup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\mathbf{X} = \left[\begin{array} {ccccc} x_{11} & . & . & . & x_{1k} \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ x_{n1} & . & . & . & x_{nn} \end{array}\right] = \left[\begin{array} {c} \mathbf{x}'_1 \\ . \\ . \\ . \\ \mathbf{x}'_n \end{array}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6861em"></span><span class="mord mathbf">X</span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.25em"><span class="pstrut" style="height:8em"></span><span style="width:.667em;height:6em"><svg viewBox="0 0 667 6000" xmlns="http://www.w3.org/2000/svg" height="6.000em" width="0.667em"><path d="M403 1759 V84 H666 V0 H319 V1759 v2400 v1759 h347 v-84
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H403z M403 1759 V0 H319 V1759 v2400 v1759 h84z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3011em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3011em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">n</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3361em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight mathnormal" style="margin-right:.03148em">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.1514em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">nn</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.25em"><span class="pstrut" style="height:8em"></span><span style="width:.667em;height:6em"><svg viewBox="0 0 667 6000" xmlns="http://www.w3.org/2000/svg" height="6.000em" width="0.667em"><path d="M347 1759 V0 H0 V84 H263 V1759 v2400 v1759 H0 v84 H347z
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M347 1759 V0 H263 V1759 v2400 v1759 h84z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:6em;vertical-align:-2.75em"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.25em"><span class="pstrut" style="height:8em"></span><span style="width:.667em;height:6em"><svg viewBox="0 0 667 6000" xmlns="http://www.w3.org/2000/svg" height="6.000em" width="0.667em"><path d="M403 1759 V84 H666 V0 H319 V1759 v2400 v1759 h347 v-84
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H403z M403 1759 V0 H319 V1759 v2400 v1759 h84z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.41em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7519em"><span style="top:-2.4519em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.2481em"><span></span></span></span></span></span></span></span></span><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-3.01em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-1.81em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">.</span></span></span><span style="top:-.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7519em"><span style="top:-2.453em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">n</span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.247em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.25em"><span style="top:-5.25em"><span class="pstrut" style="height:8em"></span><span style="width:.667em;height:6em"><svg viewBox="0 0 667 6000" xmlns="http://www.w3.org/2000/svg" height="6.000em" width="0.667em"><path d="M347 1759 V0 H0 V84 H263 V1759 v2400 v1759 H0 v84 H347z
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M347 1759 V0 H263 V1759 v2400 v1759 h84z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.75em"><span></span></span></span></span></span></span></span></span></span></span></span><p>The vector of error terms <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">U</mi></mrow><annotation encoding="application/x-tex">\mathbf{U}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6861em"></span><span class="mord mathbf">U</span></span></span></span> is also a <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6667em;vertical-align:-.0833em"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:.6444em"></span><span class="mord">1</span></span></span></span> matrix.</p><p>At times it might be easier to use vector notation. For consistency, I will use the bold small x to denote a vector and capital letters to denote a matrix. Single observations are denoted by the subscript.</p><h2 class="content-header" id="least-squares"><a class="break-words" href="#least-squares" aria-hidden="true" tabindex="-1"><span class="content-header-link"><svg viewBox="0 0 20 20" xmlns="http://www.w3.org/2000/svg" class="h-5 linkicon w-5" fill="currentColor"><path d="M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z"></path><path d="M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z"></path></svg></span></a>Least Squares</h2><p><strong>Start</strong>:<br/><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><msubsup><mi mathvariant="bold">x</mi><mi>i</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msubsup><mi>β</mi><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">y_i = \mathbf{x}'_i \beta + u_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.625em;vertical-align:-.1944em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3117em"><span style="top:-2.55em;margin-left:-.0359em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:1.0106em;vertical-align:-.2587em"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.7519em"><span style="top:-2.4413em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.2587em"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:.05278em">β</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:.5806em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3117em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span></p><p><strong>Assumptions</strong>:</p><ol><li>Linearity (given above)</li><li><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi mathvariant="bold">U</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E(\mathbf{U}|\mathbf{X}) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.05764em">E</span><span class="mopen">(</span><span class="mord mathbf">U</span><span class="mord">∣</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:.6444em"></span><span class="mord">0</span></span></span></span> (conditional independence)</li><li>rank(<span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold">X</mi></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6861em"></span><span class="mord mathbf">X</span></span></span></span>) = <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.6944em"></span><span class="mord mathnormal" style="margin-right:.03148em">k</span></span></span></span> (no multi-collinearity i.e. full rank)</li><li><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>a</mi><mi>r</mi><mo stretchy="false">(</mo><mi mathvariant="bold">U</mi><mi mathvariant="normal">∣</mi><mi mathvariant="bold">X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup><msub><mi>I</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Var(\mathbf{U}|\mathbf{X}) = \sigma^2 I_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal">Va</span><span class="mord mathnormal" style="margin-right:.02778em">r</span><span class="mopen">(</span><span class="mord mathbf">U</span><span class="mord">∣</span><span class="mord mathbf">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:.9641em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8141em"><span style="top:-3.063em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:.07847em">I</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.1514em"><span style="top:-2.55em;margin-left:-.0785em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span> (Homoskedascity)</li></ol><p><strong>Aim</strong>:<br/>Find <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8889em;vertical-align:-.1944em"></span><span class="mord mathnormal" style="margin-right:.05278em">β</span></span></span></span> that minimises the sum of squared errors:</p><span class="katex-display" translate="no"><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Q</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>u</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo stretchy="false">(</mo><msub><mi>y</mi><mi>i</mi></msub><mo>−</mo><msubsup><mi mathvariant="bold">x</mi><mi>i</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msubsup><mi>β</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mo>=</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>−</mo><mi>X</mi><mi>β</mi><msup><mo stretchy="false">)</mo><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>−</mo><mi>X</mi><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q = \sum_{i=1}^{n}{u_i^2} = \sum_{i=1}^{n}{(y_i - \mathbf{x}'_i\beta)^2} = (Y-X\beta)'(Y-X\beta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8778em;vertical-align:-.1944em"></span><span class="mord mathnormal">Q</span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em"><span style="top:-1.8723em;margin-left:0"><span class="pstrut" style="height:3.05em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">i</span><span class="mtight mrel">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em"><span class="pstrut" style="height:3.05em"></span><span><span class="mop large-op op-symbol">∑</span></span></span><span style="top:-4.3em;margin-left:0"><span class="pstrut" style="height:3.05em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8641em"><span style="top:-2.453em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.247em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em"><span style="top:-1.8723em;margin-left:0"><span class="pstrut" style="height:3.05em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">i</span><span class="mtight mrel">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em"><span class="pstrut" style="height:3.05em"></span><span><span class="mop large-op op-symbol">∑</span></span></span><span style="top:-4.3em;margin-left:0"><span class="pstrut" style="height:3.05em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3117em"><span style="top:-2.55em;margin-left:-.0359em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222em"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-2.453em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.247em"><span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:.05278em">β</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8641em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.22222em">Y</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.05278em">Xβ</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:.22222em">Y</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathnormal" style="margin-right:.05278em">Xβ</span><span class="mclose">)</span></span></span></span></span><p><strong>Solution</strong>:<br/>Hints: <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8778em;vertical-align:-.1944em"></span><span class="mord mathnormal">Q</span></span></span></span> is a <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \times 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.7278em;vertical-align:-.0833em"></span><span class="mord">1</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222em"></span></span><span class="base"><span class="strut" style="height:.6444em"></span><span class="mord">1</span></span></span></span> scalar, by symmetry <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><msup><mi>b</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi>A</mi><mi>b</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>b</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>A</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\frac{\partial b'Ab}{\partial b} = 2Ab</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3185em;vertical-align:-.345em"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.9735em"><span style="top:-2.655em"><span class="pstrut" style="height:3em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight" style="margin-right:.05556em">∂</span><span class="mord mtight mathnormal">b</span></span></span></span><span style="top:-3.23em"><span class="pstrut" style="height:3em"></span><span class="frac-line" style="border-bottom-width:.04em"></span></span><span style="top:-3.394em"><span class="pstrut" style="height:3em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight" style="margin-right:.05556em">∂</span><span class="mord mtight"><span class="mord mtight mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8278em"><span style="top:-2.931em;margin-right:.0714em"><span class="pstrut" style="height:2.5em"></span><span class="mtight sizing reset-size3 size1"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mtight mathnormal">A</span><span class="mord mtight mathnormal">b</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.345em"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span></span><span class="base"><span class="strut" style="height:.6944em"></span><span class="mord">2</span><span class="mord mathnormal">A</span><span class="mord mathnormal">b</span></span></span></span>.</p><p>Take matrix derivative w.r.t <span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.8889em;vertical-align:-.1944em"></span><span class="mord mathnormal" style="margin-right:.05278em">β</span></span></span></span>:</p><div class="relative"><pre><code class="language-tex code-highlight"><span class="code-line"><span class="token function selector">\begin</span><span class="token punctuation">{</span><span class="token keyword">aligned</span><span class="token punctuation">}</span>
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</span><span class="code-line"> <span class="token function selector">\min</span> Q <span class="token punctuation">&</span> = <span class="token function selector">\min</span>_<span class="token punctuation">{</span><span class="token function selector">\beta</span><span class="token punctuation">}</span> <span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span> - 2<span class="token function selector">\beta</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span> +
|
||
</span><span class="code-line"> <span class="token function selector">\beta</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span><span class="token function selector">\beta</span> <span class="token function selector">\\</span>
|
||
</span><span class="code-line"> <span class="token punctuation">&</span> = <span class="token function selector">\min</span>_<span class="token punctuation">{</span><span class="token function selector">\beta</span><span class="token punctuation">}</span> - 2<span class="token function selector">\beta</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span> + <span class="token function selector">\beta</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span><span class="token function selector">\beta</span> <span class="token function selector">\\</span>
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||
</span><span class="code-line"> <span class="token function selector">\text</span><span class="token punctuation">{</span><span class="token punctuation">[</span>FOC<span class="token punctuation">]</span><span class="token punctuation">}</span>~~~0 <span class="token punctuation">&</span> = - 2<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span> + 2<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span><span class="token function selector">\hat</span><span class="token punctuation">{</span><span class="token function selector">\beta</span><span class="token punctuation">}</span> <span class="token function selector">\\</span>
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</span><span class="code-line"> <span class="token function selector">\hat</span><span class="token punctuation">{</span><span class="token function selector">\beta</span><span class="token punctuation">}</span> <span class="token punctuation">&</span> = (<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>)^<span class="token punctuation">{</span>-1<span class="token punctuation">}</span><span class="token function selector">\mathbf</span><span class="token punctuation">{</span>X<span class="token punctuation">}</span>'<span class="token function selector">\mathbf</span><span class="token punctuation">{</span>Y<span class="token punctuation">}</span> <span class="token function selector">\\</span>
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</span><span class="code-line"> <span class="token punctuation">&</span> = (<span class="token function selector">\sum</span>^<span class="token punctuation">{</span>n<span class="token punctuation">}</span> <span class="token function selector">\mathbf</span><span class="token punctuation">{</span>x<span class="token punctuation">}</span>_i <span class="token function selector">\mathbf</span><span class="token punctuation">{</span>x<span class="token punctuation">}</span>'_i)^<span class="token punctuation">{</span>-1<span class="token punctuation">}</span> <span class="token function selector">\sum</span>^<span class="token punctuation">{</span>n<span class="token punctuation">}</span> <span class="token function selector">\mathbf</span><span class="token punctuation">{</span>x<span class="token punctuation">}</span>_i y_i
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</span><span class="code-line"><span class="token function selector">\end</span><span class="token punctuation">{</span><span class="token keyword">aligned</span><span class="token punctuation">}</span>
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</span></code></pre></div><span class="katex-display" translate="no"><span class="katex" translate="no"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable columnalign="right left" columnspacing="0em" rowspacing="0.25em"><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mi>min</mi><mo></mo><mi>Q</mi></mrow></mstyle></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo></mo></mrow><mi>β</mi></munder><msup><mi mathvariant="bold">Y</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">Y</mi><mo>−</mo><mn>2</mn><msup><mi>β</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">Y</mi><mo>+</mo><msup><mi>β</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">X</mi><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow></mrow></mstyle></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo></mo></mrow><mi>β</mi></munder><mo>−</mo><mn>2</mn><msup><mi>β</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">Y</mi><mo>+</mo><msup><mi>β</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">X</mi><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mtext>[FOC] </mtext><mn>0</mn></mrow></mstyle></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">Y</mi><mo>+</mo><mn>2</mn><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">X</mi><mover accent="true"><mi>β</mi><mo>^</mo></mover></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mover accent="true"><mi>β</mi><mo>^</mo></mover></mstyle></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">X</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="bold">X</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msup><mi mathvariant="bold">Y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow></mrow></mstyle></mtd><mtd><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mover><mo>∑</mo><mi>n</mi></mover><msub><mi mathvariant="bold">x</mi><mi>i</mi></msub><msubsup><mi mathvariant="bold">x</mi><mi>i</mi><mo lspace="0em" mathvariant="normal" rspace="0em">′</mo></msubsup><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>∑</mo><mi>n</mi></mover><msub><mi mathvariant="bold">x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned} \min Q & = \min_{\beta} \mathbf{Y}'\mathbf{Y} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\ & = \min_{\beta} - 2\beta'\mathbf{X}'\mathbf{Y} + \beta'\mathbf{X}'\mathbf{X}\beta \\ \text{[FOC]}~~~0 & = - 2\mathbf{X}'\mathbf{Y} + 2\mathbf{X}'\mathbf{X}\hat{\beta} \\ \hat{\beta} & = (\mathbf{X}'\mathbf{X})^{-1}\mathbf{X}'\mathbf{Y} \\ & = (\sum^{n} \mathbf{x}_i \mathbf{x}'_i)^{-1} \sum^{n} \mathbf{x}_i y_i \end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.7936em;vertical-align:-4.6468em"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.1468em"><span style="top:-7.9582em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:.1667em"></span><span class="mord mathnormal">Q</span></span></span><span style="top:-5.93em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"></span></span><span style="top:-3.7839em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord text"><span class="mord">[FOC]</span></span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mspace nobreak"> </span><span class="mord">0</span></span></span><span style="top:-2.166em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.9579em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord mathnormal" style="margin-right:.05278em">β</span></span><span style="top:-3.2634em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-.1667em"><span class="mord">^</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.1944em"><span></span></span></span></span></span></span></span><span style="top:.1454em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.6468em"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.1468em"><span style="top:-7.9582em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.6679em"><span style="top:-2.3479em;margin-left:0"><span class="pstrut" style="height:3em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal" style="margin-right:.05278em">β</span></span></span></span><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.8882em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mord"><span class="mord mathbf" style="margin-right:.02875em">Y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:.02875em">Y</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222em"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:.05278em">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:.02875em">Y</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05278em">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf">X</span><span class="mord mathnormal" style="margin-right:.05278em">β</span></span></span><span style="top:-5.93em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.6679em"><span style="top:-2.3479em;margin-left:0"><span class="pstrut" style="height:3em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal" style="margin-right:.05278em">β</span></span></span></span><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.8882em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mord">−</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:.05278em">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:.02875em">Y</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:.05278em">β</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf">X</span><span class="mord mathnormal" style="margin-right:.05278em">β</span></span></span><span style="top:-3.7839em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span><span class="mord">−</span><span class="mord">2</span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:.02875em">Y</span><span class="mspace" style="margin-right:.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222em"></span><span class="mord">2</span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf">X</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.9579em"><span style="top:-3em"><span class="pstrut" style="height:3em"></span><span class="mord mathnormal" style="margin-right:.05278em">β</span></span><span style="top:-3.2634em"><span class="pstrut" style="height:3em"></span><span class="accent-body" style="left:-.1667em"><span class="mord">^</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.1944em"><span></span></span></span></span></span></span></span><span style="top:-2.166em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span><span class="mopen">(</span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf">X</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8641em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mord mathbf" style="margin-right:.02875em">Y</span></span></span><span style="top:.1454em"><span class="pstrut" style="height:3.6514em"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2778em"></span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em"><span style="top:-3.05em"><span class="pstrut" style="height:3.05em"></span><span><span class="mop large-op op-symbol">∑</span></span></span><span style="top:-4.3em;margin-left:0"><span class="pstrut" style="height:3.05em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.55em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3117em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.8019em"><span style="top:-2.453em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.247em"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.8641em"><span style="top:-3.113em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em"><span style="top:-3.05em"><span class="pstrut" style="height:3.05em"></span><span><span class="mop large-op op-symbol">∑</span></span></span><span style="top:-4.3em;margin-left:0"><span class="pstrut" style="height:3.05em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight"><span class="mord mtight mathnormal">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.55em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.1667em"></span><span class="mord"><span class="mord mathbf">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3117em"><span style="top:-2.55em;margin-left:0;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:.03588em">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3117em"><span style="top:-2.55em;margin-left:-.0359em;margin-right:.05em"><span class="pstrut" style="height:2.7em"></span><span class="mtight sizing reset-size6 size3"><span class="mord mtight mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.15em"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:4.6468em"><span></span></span></span></span></span></span></span></span></span></span></span><section class="footnotes" data-footnotes="true"><h2 class="content-header" id="footnote-label"><a class="break-words" href="#footnote-label" 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