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problemsets/02/ps02.html

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 <title>ECON 626: Problem Set 2 – ECON 626</title>
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@@ -172,7 +172,7 @@ <h1 class="title">ECON 626: Problem Set 2</h1>
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  <div class="quarto-title-meta-heading">Published</div>
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- <p class="date">September 21, 2023</p>
+ <p class="date">September 23, 2024</p>
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@@ -190,11 +190,7 @@ <h1 class="title">ECON 626: Problem Set 2</h1>
 \]</span></p>
 <section id="problem-1" class="level1">
 <h1>Problem 1</h1>
-<p>Let <span class="math inline">\(f: \R^2 \to \R\)</span>. Assume <span class="math inline">\(f(\cdot, y): \R \to \R\)</span> is measurable for all <span class="math inline">\(y : |y - y_0| &lt; \epsilon\)</span> for some <span class="math inline">\(\epsilon&gt;0\)</span>. Also assume that <span class="math inline">\(f(x, \dot): \R \to \R\)</span> is differentiable at <span class="math inline">\(y_0\)</span> for all <span class="math inline">\(x \in
-A \subset \R\)</span> and <span class="math inline">\(\left\vert \frac{\partial f}{\partial y}(x, y_0)
-\right \vert \leq M(x)\)</span> for all <span class="math inline">\(x \in A\)</span> and <span class="math inline">\(M(x)\)</span> is integrable. Show that <span class="math display">\[
-\frac{d}{dy} \int_A f(\cdot, y) d\mu \vert_{y=y_0} = \int_A \frac{\partial f}{\partial y} f(\cdot, y_0) d\mu.
-\]</span></p>
+<p>Let <span class="math inline">\(X_{1},...,X_{n}\)</span> be independent and identically distribution and <span class="math inline">\(P(X_{1}\leq t)=F(t)\)</span> for some function <span class="math inline">\(F\)</span>. Then, write the probability <span class="math inline">\(P(\max_{1\leq i\leq n}X_{i}\leq t)\)</span> in terms of <span class="math inline">\(F\)</span>.</p>
 </section>
 <section id="problem-2" class="level1">
 <h1>Problem 2</h1>

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  "title": "ECON 626: Problem Set 2",
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- "text": "\\[\n\\def\\R{{\\mathbb{R}}}\n\\def\\Er{{\\mathrm{E}}}\n\\]\n\nProblem 1\nLet \\(f: \\R^2 \\to \\R\\). Assume \\(f(\\cdot, y): \\R \\to \\R\\) is measurable for all \\(y : |y - y_0| < \\epsilon\\) for some \\(\\epsilon>0\\). Also assume that \\(f(x, \\dot): \\R \\to \\R\\) is differentiable at \\(y_0\\) for all \\(x \\in\nA \\subset \\R\\) and \\(\\left\\vert \\frac{\\partial f}{\\partial y}(x, y_0)\n\\right \\vert \\leq M(x)\\) for all \\(x \\in A\\) and \\(M(x)\\) is integrable. Show that \\[\n\\frac{d}{dy} \\int_A f(\\cdot, y) d\\mu \\vert_{y=y_0} = \\int_A \\frac{\\partial f}{\\partial y} f(\\cdot, y_0) d\\mu.\n\\]\n\n\nProblem 2\n\nShow that if \\(E[Y |X] = 0\\), then \\(Y\\) and \\(g(X)\\) are uncorrelated for any (Borel) measurable function \\(g\\). Do you have the same conclusion if we weaken the condition \\(E[Y |X] = 0\\) to \\(E[Y |X] = a\\) for some fixed constant \\(a \\in \\R\\)?\nSuppose that \\(E[Y^2 |X] < a^2\\) for some constant \\(a > 0\\). Then, show that for any \\(b > 0\\), \\[\nP\\left(|Y − EY | > b\\right) \\leq \\frac{a^2}{b^2}.\n\\]\nShow that if \\(E[Y | \\exp(X)] = E[Y]\\), then the correlation between \\(E[Y | cos(X)]\\) and \\(cos(X)\\) is zero.\n\n\n\nProblem 3\nLet \\(X: \\Omega \\to \\R\\), \\(W: \\Omega \\to \\R\\), \\(Z: \\Omega \\to \\R\\), and \\(D: \\Omega \\to \\{0, 1\\}\\) be random variables. Let \\(Y = D X + (1-D) W\\). Suppose that \\(Y\\), \\(D\\), and \\(Z\\) are observed, but \\(X\\) and \\(W\\) are not.\n\nSuppose \\(D\\) is independent of \\(X\\), \\(W\\). Then show that \\(\\Er[X - W]\\) is identified.\nSuppose \\(Z\\) is independent of \\(X\\) and \\(W\\), and \\(\\exists E_1, E_0 \\in \\sigma(Z)\\) such that \\(P(D=1 | E_1) = 1\\) and \\(P(D=0|E_0) = 1\\). Then show that \\(\\Er[X-W]\\) is identified."
+ "text": "\\[\n\\def\\R{{\\mathbb{R}}}\n\\def\\Er{{\\mathrm{E}}}\n\\]\n\nProblem 1\nLet \\(X_{1},...,X_{n}\\) be independent and identically distribution and \\(P(X_{1}\\leq t)=F(t)\\) for some function \\(F\\). Then, write the probability \\(P(\\max_{1\\leq i\\leq n}X_{i}\\leq t)\\) in terms of \\(F\\).\n\n\nProblem 2\n\nShow that if \\(E[Y |X] = 0\\), then \\(Y\\) and \\(g(X)\\) are uncorrelated for any (Borel) measurable function \\(g\\). Do you have the same conclusion if we weaken the condition \\(E[Y |X] = 0\\) to \\(E[Y |X] = a\\) for some fixed constant \\(a \\in \\R\\)?\nSuppose that \\(E[Y^2 |X] < a^2\\) for some constant \\(a > 0\\). Then, show that for any \\(b > 0\\), \\[\nP\\left(|Y − EY | > b\\right) \\leq \\frac{a^2}{b^2}.\n\\]\nShow that if \\(E[Y | \\exp(X)] = E[Y]\\), then the correlation between \\(E[Y | cos(X)]\\) and \\(cos(X)\\) is zero.\n\n\n\nProblem 3\nLet \\(X: \\Omega \\to \\R\\), \\(W: \\Omega \\to \\R\\), \\(Z: \\Omega \\to \\R\\), and \\(D: \\Omega \\to \\{0, 1\\}\\) be random variables. Let \\(Y = D X + (1-D) W\\). Suppose that \\(Y\\), \\(D\\), and \\(Z\\) are observed, but \\(X\\) and \\(W\\) are not.\n\nSuppose \\(D\\) is independent of \\(X\\), \\(W\\). Then show that \\(\\Er[X - W]\\) is identified.\nSuppose \\(Z\\) is independent of \\(X\\) and \\(W\\), and \\(\\exists E_1, E_0 \\in \\sigma(Z)\\) such that \\(P(D=1 | E_1) = 1\\) and \\(P(D=0|E_0) = 1\\). Then show that \\(\\Er[X-W]\\) is identified."
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- "text": "Asymptotic Theory of Least Squares\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nDec 16, 2022\n\n\nPaul Schrimpf\n\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\nDec 13, 2023\n\n\nPaul Schrimpf\n\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\nOct 24, 2022\n\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\nOct 26, 2022\n\n\nPaul Schrimpf\n\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\nSep 12, 2024\n\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\nSep 21, 2023\n\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\nSep 29, 2023\n\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\nOct 13, 2023\n\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\nOct 20, 2023\n\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\nNov 9, 2023\n\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\nDec 1, 2023\n\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\nDec 6, 2022\n\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\nDec 6, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nEndogeneity\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\n\nEstimation\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nIdentification\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nMeasure\n\n\n\n\n\n\n\n\n\n\n\nSep 9, 2024\n\n\nPaul Schrimpf\n\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\nOct 25, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nProbability\n\n\n\n\n\n\n\n\n\n\n\nSep 9, 2024\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nDec 16, 2022\n\n\nPaul Schrimpf\n\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\nDec 13, 2023\n\n\nPaul Schrimpf\n\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\nOct 24, 2022\n\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\nOct 26, 2022\n\n\nPaul Schrimpf\n\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\nSep 12, 2024\n\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\nSep 23, 2024\n\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\nSep 29, 2023\n\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\nOct 13, 2023\n\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\nOct 20, 2023\n\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\nNov 9, 2023\n\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\nDec 1, 2023\n\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\nDec 6, 2022\n\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\nDec 6, 2023\n\n\n\n\n\n\n\n\n\n\n\n\nEndogeneity\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\n\nEstimation\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nIdentification\n\n\n\n\n\n\n\n\n\n\n\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\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\nSep 3, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nMeasure\n\n\n\n\n\n\n\n\n\n\n\nSep 9, 2024\n\n\nPaul Schrimpf\n\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\nOct 25, 2023\n\n\nPaul Schrimpf\n\n\n\n\n\n\n\n\n\n\n\n\nProbability\n\n\n\n\n\n\n\n\n\n\n\nSep 9, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\nNo matching items"
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