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   <meta name="author" content="Paul Schrimpf">
-  <meta name="dcterms.date" content="2023-10-16">
+  <meta name="dcterms.date" content="2023-12-06">
   <title>ECON 626 - Generalized Method of Moments</title>
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@@ -358,7 +358,7 @@ <h1 class="title">Generalized Method of Moments</h1>
 </div>
 </div>
 
-  <p class="date">2023-10-16</p>
+  <p class="date">2023-12-06</p>
 </section>
 <section id="reading" class="slide level2">
 <h2>Reading</h2>
@@ -464,12 +464,12 @@ <h1>Econometric Theory</h1>
 </section>
 <section id="identification" class="slide level2">
 <h2>Identification</h2>
-<section id="assumption-id" class="callout-info" data-icon="false">
-<h3>Assumption: ID</h3>
+<ul>
+<li>Assume:</li>
+</ul>
 <p><span class="math inline">\(\exists \theta_0 \in \Theta\)</span> s.t. <span class="math inline">\(\forall \epsilon&gt;0\)</span>, <span class="math display">\[
 \inf_{\theta: \norm{\theta-\theta_0} &gt; \epsilon} \norm{\Er[g(Z_i,\theta)]} &gt; \norm{\Er[g(Z_i,\theta_0)]}
 \]</span></p>
-</section>
 <ul>
 <li>Slightly easier assumption to verify is (loosely) <span class="math inline">\(g(z,\theta)\)</span> continuous and <span class="math inline">\(\theta_0\)</span> is unique minimizer of <span class="math inline">\(\norm{\Er[g(Z_i,\theta)]}\)</span>, see lemma 1 in <span class="citation" data-cites="song2021">Song (<a href="#/references" role="doc-biblioref" onclick="">2021</a>)</span></li>
 </ul>
@@ -490,31 +490,16 @@ <h2>Estimation</h2>
 </section>
 <section id="consistency" class="slide level2">
 <h2>Consistency</h2>
-<div class="callout callout-tip no-icon callout-titled callout-style-default">
-<div class="callout-body">
-<div class="callout-title">
-<p><strong>GMM Consistency</strong></p>
-</div>
-<div class="callout-content">
 <p>Suppose:</p>
 <ol type="1">
 <li><p><span class="math inline">\(\exists \theta_0 \in \Theta\)</span> s.t. <span class="math inline">\(\forall \epsilon&gt;0\)</span>, <span class="math inline">\(\inf_{\theta: \norm{\theta-\theta_0} &gt; \epsilon} \norm{\Er[g(Z_i,\theta)]} &gt; \norm{\Er[g(Z_i,\theta_0)]}\)</span></p></li>
 <li><p><span class="math inline">\(\sup_{\theta \in \Theta} \norm{\En[g(Z_i,\theta)] - \Er[g(Z_i,\theta)]} \inprob 0\)</span></p></li>
 <li><p><span class="math inline">\(S_n \inprob S\)</span></p></li>
 </ol>
 <p>Then <span class="math inline">\(\hat{\theta} \inprob \theta_0\)</span></p>
-</div>
-</div>
-</div>
 </section>
 <section id="asymptotic-normality" class="slide level2">
 <h2>Asymptotic Normality</h2>
-<div class="callout callout-tip no-icon callout-titled callout-style-default">
-<div class="callout-body">
-<div class="callout-title">
-<p><strong>GMM Asymptotic Normality</strong></p>
-</div>
-<div class="callout-content">
 <p>Suppose:</p>
 <ol type="1">
 <li><p><span class="math inline">\(\theta_0 \in int(\Theta)\)</span>, &amp; <span class="math inline">\(g(z,\theta)\)</span> is twice continuously differentiable</p></li>
@@ -524,27 +509,15 @@ <h2>Asymptotic Normality</h2>
 <p>Then, <span class="math display">\[
 \sqrt{n}(\hat{\theta} - \theta_0) \indist N(0, B_0^{-1} \Omega_0 B_0^{-1})
 \]</span></p>
-</div>
-</div>
-</div>
 </section>
 <section id="optimal-weighting-matrix" class="slide level2">
 <h2>Optimal Weighting Matrix</h2>
-<div class="callout callout-tip no-icon callout-titled callout-style-default">
-<div class="callout-body">
-<div class="callout-title">
-<p><strong>Lemma 3</strong></p>
-</div>
-<div class="callout-content">
 <p>Let <span class="math display">\[
 \begin{align*}
 M &amp; = (\Gamma'C\Gamma)^{-1} \Gamma' C \Sigma C \Gamma (\Gamma'C\Gamma)^{-1} \\
 M^* &amp; = (\Gamma' \Sigma^{-1} \Gamma)^{-1}
 \end{align*}
 \]</span> then <span class="math inline">\(M \geq M^*\)</span></p>
-</div>
-</div>
-</div>
 </section></section>
 <section id="references" class="title-slide slide level1 smaller scrollable">
 <h1>References</h1>

search.json

@@ -774,7 +774,7 @@
     "href": "gmm/gmm.html#identification",
     "title": "Generalized Method of Moments",
     "section": "Identification",
-    "text": "Identification\n\nAssumption: ID\n\\(\\exists \\theta_0 \\in \\Theta\\) s.t. \\(\\forall \\epsilon>0\\), \\[\n\\inf_{\\theta: \\norm{\\theta-\\theta_0} > \\epsilon} \\norm{\\Er[g(Z_i,\\theta)]} > \\norm{\\Er[g(Z_i,\\theta_0)]}\n\\]\n\n\nSlightly easier assumption to verify is (loosely) \\(g(z,\\theta)\\) continuous and \\(\\theta_0\\) is unique minimizer of \\(\\norm{\\Er[g(Z_i,\\theta)]}\\), see lemma 1 in Song (2021)"
+    "text": "Identification\n\nAssume:\n\n\\(\\exists \\theta_0 \\in \\Theta\\) s.t. \\(\\forall \\epsilon>0\\), \\[\n\\inf_{\\theta: \\norm{\\theta-\\theta_0} > \\epsilon} \\norm{\\Er[g(Z_i,\\theta)]} > \\norm{\\Er[g(Z_i,\\theta_0)]}\n\\]\n\nSlightly easier assumption to verify is (loosely) \\(g(z,\\theta)\\) continuous and \\(\\theta_0\\) is unique minimizer of \\(\\norm{\\Er[g(Z_i,\\theta)]}\\), see lemma 1 in Song (2021)"
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     "href": "gmm/gmm.html#consistency",
     "title": "Generalized Method of Moments",
     "section": "Consistency",
-    "text": "Consistency\n\n\n\nGMM Consistency\n\n\nSuppose:\n\n\\(\\exists \\theta_0 \\in \\Theta\\) s.t. \\(\\forall \\epsilon>0\\), \\(\\inf_{\\theta: \\norm{\\theta-\\theta_0} > \\epsilon} \\norm{\\Er[g(Z_i,\\theta)]} > \\norm{\\Er[g(Z_i,\\theta_0)]}\\)\n\\(\\sup_{\\theta \\in \\Theta} \\norm{\\En[g(Z_i,\\theta)] - \\Er[g(Z_i,\\theta)]} \\inprob 0\\)\n\\(S_n \\inprob S\\)\n\nThen \\(\\hat{\\theta} \\inprob \\theta_0\\)"
+    "text": "Consistency\nSuppose:\n\n\\(\\exists \\theta_0 \\in \\Theta\\) s.t. \\(\\forall \\epsilon>0\\), \\(\\inf_{\\theta: \\norm{\\theta-\\theta_0} > \\epsilon} \\norm{\\Er[g(Z_i,\\theta)]} > \\norm{\\Er[g(Z_i,\\theta_0)]}\\)\n\\(\\sup_{\\theta \\in \\Theta} \\norm{\\En[g(Z_i,\\theta)] - \\Er[g(Z_i,\\theta)]} \\inprob 0\\)\n\\(S_n \\inprob S\\)\n\nThen \\(\\hat{\\theta} \\inprob \\theta_0\\)"
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     "href": "gmm/gmm.html#asymptotic-normality",
     "title": "Generalized Method of Moments",
     "section": "Asymptotic Normality",
-    "text": "Asymptotic Normality\n\n\n\nGMM Asymptotic Normality\n\n\nSuppose:\n\n\\(\\theta_0 \\in int(\\Theta)\\), & \\(g(z,\\theta)\\) is twice continuously differentiable\n\\(\\sqrt{n} \\frac{\\partial}{\\partial \\theta} \\hat{Q}^{GMM}(\\theta_0) \\indist N(0,\\Omega)\\)\n\\(\\sup_{\\theta \\in int(\\Theta)} \\norm{\\frac{\\partial^2}{\\partial  \\theta \\partial \\theta'} \\hat{Q}^{GMM}(\\theta) - B(\\theta)} \\inprob 0\\) with \\(B(\\cdot)\\) continuous at \\(\\theta_0\\) and \\(B(\\theta_0) > 0\\)\n\nThen, \\[\n\\sqrt{n}(\\hat{\\theta} - \\theta_0) \\indist N(0, B_0^{-1} \\Omega_0 B_0^{-1})\n\\]"
+    "text": "Asymptotic Normality\nSuppose:\n\n\\(\\theta_0 \\in int(\\Theta)\\), & \\(g(z,\\theta)\\) is twice continuously differentiable\n\\(\\sqrt{n} \\frac{\\partial}{\\partial \\theta} \\hat{Q}^{GMM}(\\theta_0) \\indist N(0,\\Omega)\\)\n\\(\\sup_{\\theta \\in int(\\Theta)} \\norm{\\frac{\\partial^2}{\\partial  \\theta \\partial \\theta'} \\hat{Q}^{GMM}(\\theta) - B(\\theta)} \\inprob 0\\) with \\(B(\\cdot)\\) continuous at \\(\\theta_0\\) and \\(B(\\theta_0) > 0\\)\n\nThen, \\[\n\\sqrt{n}(\\hat{\\theta} - \\theta_0) \\indist N(0, B_0^{-1} \\Omega_0 B_0^{-1})\n\\]"
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     "title": "Generalized Method of Moments",
     "section": "Optimal Weighting Matrix",
-    "text": "Optimal Weighting Matrix\n\n\n\nLemma 3\n\n\nLet \\[\n\\begin{align*}\nM & = (\\Gamma'C\\Gamma)^{-1} \\Gamma' C \\Sigma C \\Gamma (\\Gamma'C\\Gamma)^{-1} \\\\\nM^* & = (\\Gamma' \\Sigma^{-1} \\Gamma)^{-1}\n\\end{align*}\n\\] then \\(M \\geq M^*\\)"
+    "text": "Optimal Weighting Matrix\nLet \\[\n\\begin{align*}\nM & = (\\Gamma'C\\Gamma)^{-1} \\Gamma' C \\Sigma C \\Gamma (\\Gamma'C\\Gamma)^{-1} \\\\\nM^* & = (\\Gamma' \\Sigma^{-1} \\Gamma)^{-1}\n\\end{align*}\n\\] then \\(M \\geq M^*\\)"
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-    "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\nDifference in Diffferences\n\n\n\n\n\n\n\n\n\n\n\n\nDec 4, 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 9, 2023\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 1, 2023\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\nECON 626: Problem Set 8\n\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2023\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\nNov 28, 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\nDifference in Diffferences\n\n\n\n\n\n\n\n\n\n\n\n\nDec 4, 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 9, 2023\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 1, 2023\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\nECON 626: Problem Set 8\n\n\n\n\n\n\n\n\n\n\n\n\nDec 6, 2023\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\nDec 6, 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\nNov 28, 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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