Question

Consider the multiple regression model y = X3 + €, with E(€)=0 and var(€)=oʻI. Problem 1 Gauss-Mrkov theorem (revisited). We already know that E = B and var() = '(X'X)". Consider now another unbiased estimator of 3, say b = AY. Since we are assuming that b is unbiased we reach the conclusion that AX = I (why?). The Gauss-Markov theorem claims that var(b) - var() is positive semi-definite which asks that we investigate q' var(b) - var() q. Show that this last expression is > 0 and thus the expression in the square bracket is positive semi-definite. (You must use AX = I in your proof.) Problem 2 Answer the following questions: a. Consider the multiple regression model y = X3 + €. The Gauss-Markov conditions hold. Show that hai, where hij are elements of the hat mastrix H. I h = b. Show that the error sum of squares SSE = e'e, is equal to the following expressions: SSE = Y'Y 'X'X = Y'Y - 'X'Y = Y'Y - 'X'Y c. Show that (Y -XB)(Y -XB) = (Y - X) (Y - X) + ( - )'X'X(-8). Problem 3 Use R! Please find a data set with at least 3 predictors and compute the following: 1. (X'X), (X'X)-7, X'Y 2. 3. H

Answer 1

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