# HW3 - STA131C Spring 2007 Prof W Polonik HW 3 due May 2...

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Unformatted text preview: STA131C, Spring 2007 Prof. W. Polonik HW # 3 due: May 2, 2007 in class 1. (10 pts) Recall that the variance-covariance matrix of the LSE hatwide β in a general linear regression model (under standard assumptions, i.e. mean zero, homoscedastic, uncorre- lated errors; full rank design matrix) is given by Var( hatwide β ) = σ 2 ( X T X )- 1 with X denoting the design matrix. Now consider the special case of a simple linear regression model Y i = a + b x i + ε i , i = 1 , . . . , n. Show that the variances of the LSE hatwide a and hatwide b as derived in problem 1, HW# 1 (which is Var( hatwide a ) = σ 2 parenleftBig 1 n + X 2 P n i =1 ( x i- x ) 2 parenrightBig and Var( hatwide b ) = σ 2 P n i =1 ( x i- x ) 2 ) can be concluded from the general variance-covariance matrix given above. 2. Consider the model Y i = θ x 2 i + ǫ i , i = 1 , . . . , n, where x 1 , . . . , x n are fixed known constants (not all equal), and ε 1 , . . . , ε n ∼ iid N (0 , σ 2 ) with 0 < σ 2 < ∞ unknown.unknown....
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