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Ch._5_Exercise_Solutions - Sol 1 Chapter 5 Solutions 1...

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Sol - 1 Chapter 5 Solutions 1. Suppose the columns of X are orthogonal. Show that the estimate of β j , the coefficient of x j , is not dependent on which are columns of X are included in the model. Suppose we have a model that includes x j and any other columns of X . We can write the model as y U r = + α . Note that the columns of U are also orthogonal so that U U is diagonal and the diagonal element corresponding to t β j is x x . Hence the diagonal element of ( corresponding to j t j ) U U t 1 β j is 1 and since ( / x x j t j ) U U t 1 is also diagonal, we have $ β j j t j t j x y x x = independent of all the other explanatory variates 2. Show that c for the full model that includes all p explanatory variates. p p = + 1 By definition if there are k explanatory variates in the model (plus a constant term), then c k p = estimated residual sum of squares 2 $ ( ) n + + σ 2 1 . If we fit the full model with p explanatory variates, we get c n p p n p p = + + = ( - - ) 2 2 1 2 1 $ $ ( ) + 1 σ σ as required. 3. The file ch6Exercise3.txt contains a response variate y and 10 explanatory variates x x 1 ,..., 10 for 100 cases. These data were created artificially for practice. The model
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