# HW3.pdf - Regression Analysis Homework Assignment 2 Due Nov...

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Regression Analysis: Homework Assignment 2 Due: Nov. 16 (Tue.) 1. Let X be the n × p design matrix X = 1 x 11 x 12 . . . x 1 ,p 1 1 x 21 x 22 . . . x 2 ,p 1 . . . . . . . . . . . . 1 x n 1 x n 2 . . . x n,p 1 and consider the following linear regression model y = + ϵ where ϵ N (0 , I n σ 2 ). Denote the least squares estimate of β by b = ( X X ) 1 X y , the fitted value by ˆ y = Xb = X ( X X ) 1 X y , and H = X ( X X ) 1 X . (a) (10 points) Show that n i =1 var(ˆ y i ) = p · σ 2 . (b) (20 points) Derive the distribution of ( b β ) X X ( b β ) MSE · p . (c) (20 points) Note that for a random vector z with E ( z ) = µ, Cov( z ) = V , we have E ( z Az ) = tr( AV ) + µ . Use this to show E ( MSE ) = σ 2 E ( MSR ) = σ 2 + 1 p 1 ( ) ( H 1 n 11 )( ) . 1

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2. Assume the regression model

Unformatted text preview: y = β + β 1 x 1 + β 2 x 2 + ϵ , ϵ ∼ i.i.d. N (0 , σ 2 ) for the following data. y 1 5 4 4-1 x 1 1 2 1 3 3 3 x 2 1 1 2 1 2 3 (a) (10 points) Estimate β , β 1 , β 2 and σ 2 . (b) (5 points) Obtain 95% confidence interval of the mean response at x 1 = x 2 = 2. (c) (15 points) Obtain 99% confidence interval of β 1 − β 2 . (d) (5 points) Test H : β 1 = 0, H a : β 1 ̸ = 0 using α = 0 . 05. (e) (15 points) Test H : β 1 = 2 β 2 , H a : β 1 ̸ = 2 β 2 using α = 0 . 05. 2...
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