Built site for gh-pages

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@@ -1 +1 @@
-6dfc9e5b
+eb958d75

asymptotics/indistribution.html

@@ -11,7 +11,7 @@
  <meta name="generator" content="quarto-1.3.450">
 
  <meta name="author" content="Paul Schrimpf">
- <meta name="dcterms.date" content="2023-10-30">
+ <meta name="dcterms.date" content="2023-10-31">
  <title>ECON 626 - Convergence in Distribution</title>
  <meta name="apple-mobile-web-app-capable" content="yes">
  <meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
@@ -426,7 +426,7 @@ <h1 class="title">Convergence in Distribution</h1>
 </div>
 </div>
 
- <p class="date">2023-10-30</p>
+ <p class="date">2023-10-31</p>
 </section>
 <section>
 <section id="convergence-in-distribution" class="title-slide slide level1 center">
@@ -786,8 +786,8 @@ <h2>Continuous Mapping Theorem: Example</h2>
 y_i = x_i'\beta_0 + \epsilon_i
 \]</span></li>
 <li>What is the asymptotic distribution of <span class="math display">\[
-M(\beta) = \frac{1}{n} \sum_{i=1} (y_i - x_i'\beta)^2
-\]</span> when <span class="math inline">\(\beta=\beta_0\)</span>? How about when <span class="math inline">\(\beta=\hat{\beta}?\)</span></li>
+M(\beta) = \left\Vert \frac{1}{n} \sum_{i=1} x_i (y_i - x_i'\beta) \right\Vert^2
+\]</span> when <span class="math inline">\(\beta=\beta_0\)</span>?</li>
 </ul>
 </section>
 <section id="i.-non-i.d.-central-limit-theorem" class="slide level2">

asymptotics/indistribution.qmd

@@ -334,9 +334,9 @@ y_i = x_i'\beta_0 + \epsilon_i
 $$
 - What is the asymptotic distribution of
 $$
-M(\beta) = \frac{1}{n} \sum_{i=1} (y_i - x_i'\beta)^2
+M(\beta) = \left\Vert \frac{1}{n} \sum_{i=1} x_i (y_i - x_i'\beta) \right\Vert^2
 $$
-when $\beta=\beta_0$? How about when $\beta=\hat{\beta}?$
+when $\beta=\beta_0$?
 
 ## i. non i.d. Central Limit Theorem
 

asymptotics/ols.html

@@ -11,7 +11,7 @@
  <meta name="generator" content="quarto-1.3.450">
 
  <meta name="author" content="Paul Schrimpf">
- <meta name="dcterms.date" content="2023-10-16">
+ <meta name="dcterms.date" content="2023-11-01">
  <title>ECON 626 - Asymptotic Theory of Least Squares</title>
  <meta name="apple-mobile-web-app-capable" content="yes">
  <meta name="apple-mobile-web-app-status-bar-style" content="black-translucent">
@@ -426,7 +426,7 @@ <h1 class="title">Asymptotic Theory of Least Squares</h1>
 </div>
 </div>
 
- <p class="date">2023-10-16</p>
+ <p class="date">2023-11-01</p>
 </section>
 <section id="reading" class="slide level2">
 <h2>Reading</h2>
@@ -489,10 +489,11 @@ <h2>Consistency</h2>
 <ul>
 <li>For both <span class="math inline">\(Z_i = X_i'\)</span> and <span class="math inline">\(Z_i = \epsilon_i\)</span>,
 <ul>
-<li><span class="math inline">\(\Er\left[\left((X_i Z_i) - \Er[X_i Z_i'] \right) \left((X_j Z_j) - \Er[X_j Z_j]\right)\right] = 0\)</span> and</li>
+<li><span class="math inline">\(\Er\left[\left((X_i Z_i) - \Er[X_i Z_i'] \right) \left((X_j  Z_j) - \Er[X_j Z_j]\right)' 1\{\right] = 0\)</span> for <span class="math inline">\(i \neq j\)</span> and</li>
 <li><span class="math inline">\(\frac{1}{n} \max_{1 \leq i \leq n} \Er[ (X_i Z_i - \Er[X_i Z_i]) (X_i Z_i - \Er[X_i Z_i])'] \to 0\)</span></li>
 </ul></li>
-<li><span class="math inline">\(\Er[X_i \epsilon_i] = 0\)</span></li>
+<li><span class="math inline">\(\Er[X_i \epsilon_i] = 0\)</span> for all <span class="math inline">\(i\)</span></li>
+<li><span class="math inline">\(\lim_{n \to \infty} \frac{1}{n} \sum_{i=1}^n \Er[X_i X_i'] = C\)</span> is invertible</li>
 </ul></li>
 </ul>
 </section>
@@ -584,7 +585,7 @@ <h2>Heteroskedasticity</h2>
 <section id="heteroskedasticity-1" class="slide level2">
 <h2>Heteroskedasticity</h2>
 <ul>
-<li><p>With homoskedasticity, = ^{-1} ^2$</p></li>
+<li><p>With homoskedasticity, <span class="math inline">\(\Sigma = \Er[X_i X_i']^{-1} \sigma^2\)</span></p></li>
 <li><p>With heteroskedasticity, <span class="math inline">\(\Sigma = \Er[X_i X_i']^{-1} \var(X_i \epsilon_i) \Er[X_i X_i']^{-1}\)</span> and can be (with appropriate assumptions) consistently estimated by <span class="math display">\[
 \hat{\Sigma}^{robust} = (\frac{1}{n} X ' X)^{-1} \left(\frac{1}{n} \sum_{i=1}^n X_i X_i' \epsilon_i^2 \right) (\frac{1}{n} X'X)^{-1}
 \]</span></p></li>
@@ -705,8 +706,8 @@ <h2>Plot of Residuals vs Predictions</h2>
 <span id="cb4-9"><a href="#cb4-9"></a></span>
 <span id="cb4-10"><a href="#cb4-10"></a>plt <span class="op">=</span> <span class="fu">Plot</span>()</span>
 <span id="cb4-11"><a href="#cb4-11"></a>plt.layout <span class="op">=</span> <span class="fu">Config</span>()</span>
-<span id="cb4-12"><a href="#cb4-12"></a><span class="fu">plt</span>(x <span class="op">=</span> Xh<span class="op">*</span>bh, y<span class="op">=</span>eh, name<span class="op">=</span><span class="st">"Homoskedastic"</span>, mode<span class="op">=</span><span class="st">"markers"</span>)</span>
-<span id="cb4-13"><a href="#cb4-13"></a>fig <span class="op">=</span><span class="fu">plt</span>(x <span class="op">=</span> X<span class="op">*</span>b, y <span class="op">=</span> <span class="cn">e</span>, name<span class="op">=</span><span class="st">"Heteroskedastic"</span>, mode<span class="op">=</span><span class="st">"markers"</span>)</span>
+<span id="cb4-12"><a href="#cb4-12"></a><span class="fu">plt</span>(x <span class="op">=</span> <span class="fu">vec</span>(Xh<span class="op">*</span>bh), y<span class="op">=</span><span class="fu">vec</span>(eh), name<span class="op">=</span><span class="st">"Homoskedastic"</span>, mode<span class="op">=</span><span class="st">"markers"</span>, <span class="kw">type</span><span class="op">=</span><span class="st">"scatter"</span>)</span>
+<span id="cb4-13"><a href="#cb4-13"></a>fig <span class="op">=</span><span class="fu">plt</span>(x <span class="op">=</span> <span class="fu">vec</span>(X<span class="op">*</span>b), y <span class="op">=</span> <span class="fu">vec</span>(<span class="cn">e</span>), name<span class="op">=</span><span class="st">"Heteroskedastic"</span>, mode<span class="op">=</span><span class="st">"markers"</span>)</span>
 <span id="cb4-14"><a href="#cb4-14"></a>Cobweb.<span class="fu">save</span>(<span class="fu">Page</span>(fig), <span class="st">"resid.html"</span>)</span>
 <span id="cb4-15"><a href="#cb4-15"></a><span class="fu">HTML</span>(<span class="st">"&lt;iframe src=</span><span class="sc">\"</span><span class="st">resid.html</span><span class="sc">\"</span><span class="st"> width=</span><span class="sc">\"</span><span class="st">1000</span><span class="sc">\"</span><span class="st"> height=</span><span class="sc">\"</span><span class="st">650</span><span class="sc">\"</span><span class="st">&gt;&lt;/iframe&gt;</span><span class="sc">\n</span><span class="st">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </details>
@@ -941,7 +942,7 @@ <h2>Time Dependence - CLT</h2>
 \]</span></li>
 </ul>
 </section>
-<section id="time-dependence---gordins-clt" class="slide level2">
+<section id="time-dependence---gordins-clt" class="slide level2 smaller">
 <h2>Time Dependence - Gordin’s CLT</h2>
 <div class="callout callout-warning no-icon callout-titled callout-style-default">
 <div class="callout-body">

search.json

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  "title": "Asymptotic Theory of Least Squares",
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- "text": "Consistency\n\\[\n\\begin{align*}\n\\hat{\\beta} = & (X'X)^{-1} X' y \\\\\n= & (X'X)^{-1} X' (X \\beta + \\epsilon) \\\\\n= & \\beta + (X'X)^{-1} X' \\epsilon\n\\end{align*}\n\\]\nConsistent, \\(\\hat{\\beta} \\inprob \\beta\\), if\n\nUsing non-iid WLLN from convergence in distribution slides (“low level assumption”):\n\nFor both \\(Z_i = X_i'\\) and \\(Z_i = \\epsilon_i\\),\n\n\\(\\Er\\left[\\left((X_i Z_i) - \\Er[X_i Z_i'] \\right) \\left((X_j Z_j) - \\Er[X_j Z_j]\\right)\\right] = 0\\) and\n\\(\\frac{1}{n} \\max_{1 \\leq i \\leq n} \\Er[ (X_i Z_i - \\Er[X_i Z_i]) (X_i Z_i - \\Er[X_i Z_i])'] \\to 0\\)\n\n\\(\\Er[X_i \\epsilon_i] = 0\\)"
+ "text": "Consistency\n\\[\n\\begin{align*}\n\\hat{\\beta} = & (X'X)^{-1} X' y \\\\\n= & (X'X)^{-1} X' (X \\beta + \\epsilon) \\\\\n= & \\beta + (X'X)^{-1} X' \\epsilon\n\\end{align*}\n\\]\nConsistent, \\(\\hat{\\beta} \\inprob \\beta\\), if\n\nUsing non-iid WLLN from convergence in distribution slides (“low level assumption”):\n\nFor both \\(Z_i = X_i'\\) and \\(Z_i = \\epsilon_i\\),\n\n\\(\\Er\\left[\\left((X_i Z_i) - \\Er[X_i Z_i'] \\right) \\left((X_j  Z_j) - \\Er[X_j Z_j]\\right)' 1\\{\\right] = 0\\) for \\(i \\neq j\\) and\n\\(\\frac{1}{n} \\max_{1 \\leq i \\leq n} \\Er[ (X_i Z_i - \\Er[X_i Z_i]) (X_i Z_i - \\Er[X_i Z_i])'] \\to 0\\)\n\n\\(\\Er[X_i \\epsilon_i] = 0\\) for all \\(i\\)\n\\(\\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{i=1}^n \\Er[X_i X_i'] = C\\) is invertible"
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  "href": "asymptotics/ols.html#heteroskedasticity-1",
  "title": "Asymptotic Theory of Least Squares",
  "section": "Heteroskedasticity",
- "text": "Heteroskedasticity\n\nWith homoskedasticity, = ^{-1} ^2$\nWith heteroskedasticity, \\(\\Sigma = \\Er[X_i X_i']^{-1} \\var(X_i \\epsilon_i) \\Er[X_i X_i']^{-1}\\) and can be (with appropriate assumptions) consistently estimated by \\[\n\\hat{\\Sigma}^{robust} = (\\frac{1}{n} X ' X)^{-1} \\left(\\frac{1}{n} \\sum_{i=1}^n X_i X_i' \\epsilon_i^2 \\right) (\\frac{1}{n} X'X)^{-1}\n\\]\nEven with homoskedasticity, there is little downside to using \\(\\hat{\\Sigma}^{robust}\\), so always used in practice"
+ "text": "Heteroskedasticity\n\nWith homoskedasticity, \\(\\Sigma = \\Er[X_i X_i']^{-1} \\sigma^2\\)\nWith heteroskedasticity, \\(\\Sigma = \\Er[X_i X_i']^{-1} \\var(X_i \\epsilon_i) \\Er[X_i X_i']^{-1}\\) and can be (with appropriate assumptions) consistently estimated by \\[\n\\hat{\\Sigma}^{robust} = (\\frac{1}{n} X ' X)^{-1} \\left(\\frac{1}{n} \\sum_{i=1}^n X_i X_i' \\epsilon_i^2 \\right) (\\frac{1}{n} X'X)^{-1}\n\\]\nEven with homoskedasticity, there is little downside to using \\(\\hat{\\Sigma}^{robust}\\), so always used in practice"
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  "title": "Asymptotic Theory of Least Squares",
  "section": "Plot of Residuals vs Predictions",
- "text": "Plot of Residuals vs Predictions\n\n\nCode\nn = 250\nk = 3\nXh,yh = sim(n,k)\nbh, _, _ = ols(Xh,yh)\neh = yh - Xh*bh\nX,y = sim(n,k, σ=x->(0.1 + norm(x'*ones(k) .+ 3)/3))\nb, _, _ = ols(X,y)\ne = y - X*b\n\nplt = Plot()\nplt.layout = Config()\nplt(x = Xh*bh, y=eh, name=\"Homoskedastic\", mode=\"markers\")\nfig =plt(x = X*b, y = e, name=\"Heteroskedastic\", mode=\"markers\")\nCobweb.save(Page(fig), \"resid.html\")\nHTML(\"<iframe src=\\\"resid.html\\\" width=\\\"1000\\\" height=\\\"650\\\"></iframe>\\n\")"
+ "text": "Plot of Residuals vs Predictions\n\n\nCode\nn = 250\nk = 3\nXh,yh = sim(n,k)\nbh, _, _ = ols(Xh,yh)\neh = yh - Xh*bh\nX,y = sim(n,k, σ=x->(0.1 + norm(x'*ones(k) .+ 3)/3))\nb, _, _ = ols(X,y)\ne = y - X*b\n\nplt = Plot()\nplt.layout = Config()\nplt(x = vec(Xh*bh), y=vec(eh), name=\"Homoskedastic\", mode=\"markers\", type=\"scatter\")\nfig =plt(x = vec(X*b), y = vec(e), name=\"Heteroskedastic\", mode=\"markers\")\nCobweb.save(Page(fig), \"resid.html\")\nHTML(\"<iframe src=\\\"resid.html\\\" width=\\\"1000\\\" height=\\\"650\\\"></iframe>\\n\")"
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@@ -1152,7 +1152,7 @@
  "href": "asymptotics/indistribution.html#continuous-mapping-theorem-example",
  "title": "Convergence in Distribution",
  "section": "Continuous Mapping Theorem: Example",
- "text": "Continuous Mapping Theorem: Example\n\nIn linear regression, \\[\ny_i = x_i'\\beta_0 + \\epsilon_i\n\\]\nWhat is the asymptotic distribution of \\[\nM(\\beta) = \\frac{1}{n} \\sum_{i=1} (y_i - x_i'\\beta)^2\n\\] when \\(\\beta=\\beta_0\\)? How about when \\(\\beta=\\hat{\\beta}?\\)"
+ "text": "Continuous Mapping Theorem: Example\n\nIn linear regression, \\[\ny_i = x_i'\\beta_0 + \\epsilon_i\n\\]\nWhat is the asymptotic distribution of \\[\nM(\\beta) = \\left\\Vert \\frac{1}{n} \\sum_{i=1} x_i (y_i - x_i'\\beta) \\right\\Vert^2\n\\] when \\(\\beta=\\beta_0\\)?"
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@@ -1950,6 +1950,6 @@
  "href": "slides.html",
  "title": "Slides",
  "section": "",
- "text": "Asymptotic Theory of Least Squares\n\n\n\n\n\n\n\n\n\n\n\n\nOct 16, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nConvergence in Distribution\n\n\n\n\n\n\n\n\n\n\n\n\nOct 30, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nConvergence in Probability\n\n\n\n\n\n\n\n\n\n\n\n\nOct 12, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Final - Solutions\n\n\n\n\n\n\n\n\n\n\n\n\nDec 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Midterm Review\n\n\n\n\n\n\n\n\n\n\n\n\nOct 24, 2022\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Midterm Solutions\n\n\n\n\n\n\n\n\n\n\n\n\nOct 26, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 1\n\n\n\n\n\n\n\n\n\n\n\n\nSep 14, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 2\n\n\n\n\n\n\n\n\n\n\n\n\nSep 21, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 3\n\n\n\n\n\n\n\n\n\n\n\n\nSep 29, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 4\n\n\n\n\n\n\n\n\n\n\n\n\nOct 13, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 5\n\n\n\n\n\n\n\n\n\n\n\n\nOct 20, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 6\n\n\n\n\n\n\n\n\n\n\n\n\nNov 23, 2022\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 7\n\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2022\n\n\n\n\n\n\n \n\n\n\n\nEndogeneity\n\n\n\n\n\n\n\n\n\n\n\n\nNov 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nEstimation\n\n\n\n\n\n\n\n\n\n\n\n\nSep 19, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nGeneralized Method of Moments\n\n\n\n\n\n\n\n\n\n\n\n\nOct 16, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nIdentification\n\n\n\n\n\n\n\n\n\n\n\n\nSep 24, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nInstrumental Variables Estimation\n\n\n\n\n\n\n\n\n\n\n\n\nOct 16, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nLeast Squares as a Projection\n\n\n\n\n\n\n\n\n\n\n\n\nOct 3, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nMeasure\n\n\n\n\n\n\n\n\n\n\n\n\nSep 11, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nMidterm Solutions 2023\n\n\n\n\n\n\n\n\n\n\n\n\nOct 25, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nProbability\n\n\n\n\n\n\n\n\n\n\n\n\nSep 17, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\nNo matching items"
+ "text": "Asymptotic Theory of Least Squares\n\n\n\n\n\n\n\n\n\n\n\n\nNov 1, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nConvergence in Distribution\n\n\n\n\n\n\n\n\n\n\n\n\nOct 31, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nConvergence in Probability\n\n\n\n\n\n\n\n\n\n\n\n\nOct 12, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Final - Solutions\n\n\n\n\n\n\n\n\n\n\n\n\nDec 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Midterm Review\n\n\n\n\n\n\n\n\n\n\n\n\nOct 24, 2022\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Midterm Solutions\n\n\n\n\n\n\n\n\n\n\n\n\nOct 26, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 1\n\n\n\n\n\n\n\n\n\n\n\n\nSep 14, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 2\n\n\n\n\n\n\n\n\n\n\n\n\nSep 21, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 3\n\n\n\n\n\n\n\n\n\n\n\n\nSep 29, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 4\n\n\n\n\n\n\n\n\n\n\n\n\nOct 13, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 5\n\n\n\n\n\n\n\n\n\n\n\n\nOct 20, 2023\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 6\n\n\n\n\n\n\n\n\n\n\n\n\nNov 23, 2022\n\n\n\n\n\n\n \n\n\n\n\nECON 626: Problem Set 7\n\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2022\n\n\n\n\n\n\n \n\n\n\n\nEndogeneity\n\n\n\n\n\n\n\n\n\n\n\n\nNov 16, 2022\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nEstimation\n\n\n\n\n\n\n\n\n\n\n\n\nSep 19, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nGeneralized Method of Moments\n\n\n\n\n\n\n\n\n\n\n\n\nOct 16, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nIdentification\n\n\n\n\n\n\n\n\n\n\n\n\nSep 24, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nInstrumental Variables Estimation\n\n\n\n\n\n\n\n\n\n\n\n\nOct 16, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nLeast Squares as a Projection\n\n\n\n\n\n\n\n\n\n\n\n\nOct 3, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nMeasure\n\n\n\n\n\n\n\n\n\n\n\n\nSep 11, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nMidterm Solutions 2023\n\n\n\n\n\n\n\n\n\n\n\n\nOct 25, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n \n\n\n\n\nProbability\n\n\n\n\n\n\n\n\n\n\n\n\nSep 17, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\nNo matching items"
  }
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