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hw3 - ECN 725 Spring 2010 Q1 ASSIGNMENT 3 DUE April...

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1 ECN 725 ASSIGNMENT 3 S.C. AHN Spring 2010 DUE April 16 (Friday), 2:00pm Q1. (10 pts.) Consider a regression model 1 2 2 3 3 y x x u , which satisfies all SIC. Assume that regressors are stochastic. Let 2 be the OLS estimator of 2 from the regression of t y on one and 2 t x (omitting 3 t x ). Show that: 2 3 2 2, 3, 2 cov( , ) lim var( ) T o o x x p x  . Q2. (30 pts.) Consider the following model: y = 1 1 T + 2 w + u, where y, w and u = (u 1 , ... , u T ) are all T 1 vectors and 1 T is a T 1 vector of ones. The vector w is nonstochastic. And all SIC hold except that var(u t ) = 2 and cov(u t ,u s ) = 2 ( < 1) for all t s. Define P(1 T ) = 1 T (1 T 1 T ) -1 1 T and M(1 T ) = I T - P(1 T ). (1) Show that Var(u) = 2 , where = (1- )M(1 T ) + (1- + T)P(1 T ). (2) Let s 2 = SSE/(T-2), where SSE is from the OLS estimation of the model. Find E(s 2 ). (3) Let 2 be the GLS estimator of 2 . Show that 2 2 ˆ .
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