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-    "text": "Asymptotic Theory of Least Squares\n\n\n\n\n\n\n\n\n\n\n\nOct 21, 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\nOct 21, 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\nOct 21, 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\nOct 21, 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 30, 2024\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 10, 2024\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 31, 2024\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\nOct 21, 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 23, 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\nOct 21, 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\nOct 21, 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 16, 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\nOct 21, 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\nOct 28, 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\nOct 21, 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\nOct 21, 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 30, 2024\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 10, 2024\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 31, 2024\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\nOct 21, 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 23, 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\nOct 21, 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\nOct 21, 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 16, 2024\n\n\nPaul Schrimpf\n\n\n\n\n\n\nNo matching items"
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     "section": "Levy’s Continuity Theorem",
-    "text": "Levy’s Continuity Theorem\n\n\n\nLemma 2.1 (Levy’s Continuity Theorem)\n\n\n\\(X_n \\indist X\\) iff \\(\\Er[e^{i t'X_n} ] \\to \\Er[e^{i t' X} ]\\) for all \\(t \\in \\R^d\\)\n\n\n\n\nsee Döbler (2022) for a short proof\n\\(\\Er[e^{i t' X}] \\equiv \\varphi(t)\\) is the characteristic function of \\(X\\)"
+    "text": "Levy’s Continuity Theorem\n\n\n\nLemma 2.1 (Levy’s Continuity Theorem)\n\n\n\\(X_n \\indist X\\) iff \\(\\Er[e^{i t'X_n} ] \\to \\Er[e^{i t' X} ]\\) for all \\(t \\in \\R^d\\)\n\n\n\n\nsee Döbler (2022) for a short proof\n\\(\\Er[e^{i t' X}] \\equiv \\varphi(t)\\) is the characteristic function of \\(X\\)\n\n\n\nhttp://theanalysisofdata.com/probability/8_8.html\nDöbler (2022)\nhttps://terrytao.wordpress.com/2010/01/05/254a-notes-2-the-central-limit-theorem/#berry"
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     "title": "Convergence in Distribution",
     "section": "Delta Method: Example",
-    "text": "Delta Method: Example\n\nWhat is the asymptotic distribution of \\[\n\\hat{\\sigma} = \\sqrt{\\frac{1}{n}\n\\sum_{i=1}^n \\left(x_i - \\frac{1}{n} \\sum_j=1^n x_j \\right)^2}?\n\\]\n\n\nWe need some additional assumptions for this to have good answer. At a high level, we want \\(\\frac{1}{n} \\sum x_i \\inprob \\Er[x]\\), and \\(\\frac{1}{n} \\sum_{i=1}^n \\left(x_i - \\frac{1}{n} \\sum_j=1^n x_j \\right)^2 - \\sigma^2 \\indist W\\) for some known random variable \\(W\\). Sufficient more primitive assumptions would be that \\(x_i\\) is i.i.d. with mean \\(\\mu\\), variance \\(\\sigma^2>0\\), and finite fourth moment."
+    "text": "Delta Method: Example\n\nWhat is the asymptotic distribution of \\[\n\\hat{\\sigma} = \\sqrt{\\frac{1}{n}\n\\sum_{i=1}^n \\left(x_i - \\frac{1}{n} \\sum_{j=1}^n x_j \\right)^2}?\n\\]\n\n\nWe need some additional assumptions for this to have good answer. At a high level, we want \\(\\frac{1}{n} \\sum x_i \\inprob \\Er[x]\\), and \\(\\frac{1}{n} \\sum_{i=1}^n \\left(x_i - \\frac{1}{n} \\sum_j=1^n x_j \\right)^2 - \\sigma^2 \\indist W\\) for some known random variable \\(W\\). Sufficient more primitive assumptions would be that \\(x_i\\) is i.i.d. with mean \\(\\mu\\), variance \\(\\sigma^2>0\\), and finite fourth moment."
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     "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) = \\left\\Vert \\frac{1}{n} \\sum_{i=1} x_i (y_i - x_i'\\beta) \\right\\Vert^2\n\\] when \\(\\beta=\\beta_0\\)?"
+    "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}{\\sqrt{n}} \\sum_{i=1} x_i (y_i - x_i'\\beta) \\right\\Vert^2\n\\] when \\(\\beta=\\beta_0\\)?"
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     "section": "Characterizing Convergence in Distribution",
     "text": "Characterizing Convergence in Distribution\n\n\n\nTheorem 1.1\n\n\nIf \\(X_n \\indist X\\) if and only if \\(P(X_n \\leq t) \\to P(X \\leq t)\\) for all \\(t\\) where \\(P(X \\leq t)\\) is continuous\n\n\n\n\n\n\n\nTheorem 1.2\n\n\nIf \\(X_n \\indist X\\) and \\(X\\) is continuous, then \\[\n\\sup_{t \\in \\R^d} | P(X_n \\leq t) - P(X \\leq t) | \\to 0\n\\]"
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+    "title": "Convergence in Distribution",
+    "section": "Weak Berry-Esseen Theorem",
+    "text": "Weak Berry-Esseen Theorem\n\n\n\n\nWeak Berry-Esseen Theorem\n\n\nLet \\(X_i\\) be i.i.d with \\(\\Er[X]=0\\), \\(\\Er[X^2]=1\\) and \\(\\Er[|X|^3]\\) finite. Let \\(\\varphi\\) be smooth with its first three derivatives uniformly bounded, and let \\(G \\sim N(0,1)\\). Then \\[\n\\left\\vert \\Er\\left[ \\varphi\\left( \\frac{1}{\\sqrt{n}} \\sum_{i=1}^n X_i \\right) \\right] -\n\\Er\\left[\\varphi(G)\\right]\n\\right\\vert \\leq C \\frac{\\Er[|X|^3]}{\\sqrt{n}} \\sup_{x \\in \\R} |\\varphi'''(x)|\n\\]\n1\n\n\n\n\n\nThis is based on Tao (2010) https://terrytao.wordpress.com/2010/01/05/254a-notes-2-the-central-limit-theorem/#berry\nProof\nLet \\(G_i \\sim N(0,1)\\). Define \\[\nZ_{n,i} = \\frac{X_1 + \\cdots X_i + G_{i+1} + \\cdots G_n}{\\sqrt{n}},\n\\] so that $Z_{n,n} = _{i=1}^n X_i $ and \\(Z_{n,0} \\equiv G\\).\nWith this notation we have \\[\n\\Er\\left[\\varphi(Z_{n,n}) - \\varphi(Z_{n,0}) \\right] = - \\sum_{i=0}^{n-i} \\Er\\left[ \\varphi(Z_{n,i}) - \\varphi(Z_{n,i+1})\\right]\n\\] and we will show that \\(\\Er\\left[ \\varphi(Z_{n,i}) - \\varphi(Z_{n,i+1})\\right]\\) is small.\nLet \\(S_{n,i} = Z_{n,i} - \\frac{G_{i+1}}{\\sqrt{n}} = Z_{n,i+1} - \\frac{X_{i+1}}/\\sqrt{n}\\) and take a second order Taylor expansion of \\(\\varphi\\) around \\(S_{n,i}\\), giving \\[\n\\varphi(Z_{n,i}) = \\varphi(S_{n,i}) + \\varphi'(S_{n,i}) \\frac{G_{i+1}}{\\sqrt{n}} + \\frac{1}{2}\\varphi''(S_{n,i}) \\frac{G_{i+1}^2}{n} + O\\left( \\frac{|G_{i+1}|^3}{n^{3/2}} \\sup \\varphi'''(x) \\right)\n\\] and \\[\n\\varphi(Z_{n,i+i}) = \\varphi(S_{n,i}) + \\varphi'(S_{n,i}) \\frac{X_{i+1}}{\\sqrt{n}} + \\frac{1}{2}\\varphi''(S_{n,i}) \\frac{X_{i+1}^2}{n} + O\\left( \\frac{|X_{i+1}|^3}{n^{3/2}} \\sup \\varphi'''(x) \\right).\n\\]\nWe have assumed that \\(\\Er[X_i] = \\Er[G_i]\\) and \\(\\Er[X_i^2] = \\Er[G_i^2]\\), so \\[\n\\left\\vert \\Er\\left[ \\varphi(Z_{n,i}) - \\varphi(Z_{n,i+1})\\right] \\right\\vert \\leq C( \\frac{|G_{i+1}|^3 + |X_{i+1}|^3}{n^{3/2}} \\sup \\varphi'''(x) \\right)\n\\] for some constant \\(C\\) that depends on \\(\\varphi\\), but not on \\(n\\) or \\(i\\). We can conclude that \\[\n\\Er\\left[\\varphi(Z_{n,n}) - \\varphi(Z_{n,0}) \\right] \\leq n C( \\frac{|G_{i+1}|^3 + |X_{i+1}|^3}{n^{3/2}} \\sup \\varphi'''(x) \\right).\n\\]\n\nFrom Tao (2010)"
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+    "href": "asymptotics/indistribution.html#berry-esseen-theorem",
+    "title": "Convergence in Distribution",
+    "section": "Berry-Esseen Theorem",
+    "text": "Berry-Esseen Theorem\n\n\n\n\nBerry-Esseen Theorem\n\n\nIf \\(X_i\\) are i.i.d. with \\(\\Er[X] = 0\\) and \\(\\var(X)=1\\), then \\[\n\\sup_{z \\in \\R} \\left\\vert\nP\\left(\\left[\\frac{1}{\\sqrt{n}} \\sum_{i=1}^n X_i\\right] \\leq z \\right) - \\Phi(z) \\right\\vert \\leq 0.5 \\Er[|X|^3]/\\sqrt{n}\n\\] where \\(\\Phi\\) is the normal CDF.\n\n\n\n\n\n\n\n\n\nMultivariate Berry-Esseen Theorem\n\n\nIf \\(X_i \\in \\R^d\\) are i.i.d. with \\(\\Er[X] = 0\\) and \\(\\var(X)=I_d\\), then \\[\n\\sup_{A \\subset \\R^d, \\text{convex}} \\left\\vert\nP\\left(\\frac{1}{\\sqrt{n}} \\sum_{i=1}^n X_i \\in A \\right) - P(N(0,I_d) \\in A) \\right\\vert \\leq\n(42 d^{1/4} + 16) \\Er[\\Vert X \\Vert ^3]/\\sqrt{n}\n\\]\n1\n\n\n\n\n\nSee Raič (2019) for details."
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+    "title": "Convergence in Distribution",
+    "section": "Simulated Illustration of Berry-Esseen CLT",
+    "text": "Simulated Illustration of Berry-Esseen CLT\n\n\nplotting code\nusing Plots, Distributions\n\nfunction dgp(n, xhi=2)\n  p = 1/(1+xhi^2)\n  xlo = -p*xhi/(1-p)\n  hi = rand(n) .< p\n  x = ifelse.(hi, xhi, xlo)\nend\n\nfunction Ex3(xhi)\n  p = 1/(1+xhi^2)\n  xlo = -p*xhi/(1-p)\n  p*xhi^3 + (1-p)*-xlo^3\nend\n\nfunction plotcdfwithbounds(dgp,  e3, n=[10,100,1000], S=9999)\n  cmap = palette(:tab10)\n  x = range(-2.5, 2.5, length=200)\n  cdfx=x->cdf(Normal(), x)\n  fig=Plots.plot(x, cdfx, label=\"Normal\", color=\"black\", linestyle=:dash)\n  for (i,ni) in enumerate(n)\n    truedist = [mean(dgp(ni))*sqrt(ni) for _ in 1:S]\n    ecdf = x-> mean(truedist .<= x)\n    Plots.plot!(x, ecdf, label=\"n=$ni\", color=cmap[i])\n    Plots.plot!(x, cdfx.(x), ribbon = 0.5*e3/√ni, fillalpha=0.2, label=\"\", color=cmap[i])\n  end\n  xlims!(-2.5,2.5)\n  ylims!(0,1)\n  title!(\"Distribution of Scaled Sample Mean\")\n  return(fig)\nend\nxhi = 2.5\nplotcdfwithbounds(n->dgp(n,xhi), Ex3(xhi))\n#unif = n->rand(n)*2sqrt(3) .- sqrt(3)\n# e3 = mean(abs.(unif(100_000)).^3)\n#plotcdfwithbounds(unif,e3)\n\n\n\n\n\n  \n    \n  \n\n\n\n  \n    \n  \n\n\n\n  \n    \n  \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nThis simulation is constructed so that the bound is close to the actual error. For many distributions, the bound is large and slack. It is usually not as informative as it appears in this example."
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 <div class="listing-item-img-placeholder card-img-top" >&nbsp;</div>
@@ -255,7 +255,7 @@ <h3 class="no-anchor listing-title">
 <div class="metadata">
 <a href="./asymptotics/indistribution.html" class="no-external">
 <div class="listing-date">
-Oct 21, 2024
+Oct 28, 2024
 </div>
 <div class="listing-author">
 Paul Schrimpf