5. (15 points) This question involves a simple example that illustrates why o can be estimated...
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5. (15 points) This question involves a simple example that illustrates why o² can be estimated by û² when we compute the heteroskedasticity robust standard errors. Consider a simple regression model that relates the wage (y) to college education: y = Bo + Bix+u, where x = 1 for college graduates and r = 0 for non-college graduates. Var (ulx = 1) = 0² > 2 = Var(ulx = 0), so the regression errors display heteroskedasiticity. In a random sample, the first ni individuals are college graduates, and the remaining non - n₁ are noncollege graduates. (a) Show that the variance of the OLS estimate 3₁ can be written as: Var (3₁) = n₁(1)²o+no(0-24) ²02 [n₁(1)²+no(0)²]² 721 [Hint: Var (3₁) = (3-7)²0 and 2 = 14.] SST? - (b) Suppose we estimate o? by o? =u?, and o? by o = 1/ Ei=₁+1², where û is the residual from the regression. Show the estimate of Var (3₁) can be written as: Var (3₁)= = Σ₁1(x₁ - x)²¹û² SST2 [Hint: (i) substitute 2 and o into Var(₁); and (ii) = 1 5. (15 points) This question involves a simple example that illustrates why o² can be estimated by û² when we compute the heteroskedasticity robust standard errors. Consider a simple regression model that relates the wage (y) to college education: y = Bo + Bix+u, where x = 1 for college graduates and r = 0 for non-college graduates. Var (ulx = 1) = 0² > 2 = Var(ulx = 0), so the regression errors display heteroskedasiticity. In a random sample, the first ni individuals are college graduates, and the remaining non - n₁ are noncollege graduates. (a) Show that the variance of the OLS estimate 3₁ can be written as: Var (3₁) = n₁(1)²o+no(0-24) ²02 [n₁(1)²+no(0)²]² 721 [Hint: Var (3₁) = (3-7)²0 and 2 = 14.] SST? - (b) Suppose we estimate o? by o? =u?, and o? by o = 1/ Ei=₁+1², where û is the residual from the regression. Show the estimate of Var (3₁) can be written as: Var (3₁)= = Σ₁1(x₁ - x)²¹û² SST2 [Hint: (i) substitute 2 and o into Var(₁); and (ii) = 1
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a The variance of the OLS estimate can be written as Var 1 n0 n1 n0 1 n b Suppose we ... View the full answer
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Financial management theory and practice
ISBN: 978-1439078099
13th edition
Authors: Eugene F. Brigham and Michael C. Ehrhardt
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