Exercise 9.2 Consider the problem of estimating p in the regression model yi =0+1xi1+ +pxip+ei. (2) Let ri be the ordinary residual from fitting

Exercise 9.2 Consider the problem of estimating βp in the regression model yi =β0+β1xi1+· · ·+βpxip+ei. (2)

Let ri be the ordinary residual from fitting yi =α0+α1xi1+· · ·+αp−1xip−1+ei and si be the residual from fitting xip =γ0+γ1xi1+· · ·+γp−1xip−1+ei.

Show that the least squares estimate of βp is ˆξ from fitting the model ri =ξ si+ei, i = 1, . . . ,n, (3)

that the SSE from models (2) and (3) are the same, and that (ˆβ0, . . . , ˆβ

p−1) =αˆ −

ˆβ

pγˆ withαˆ =(αˆ 0, . . . ,αˆ p−1) and γˆ=(γˆ0, . . . ,γˆp−1). Discuss the usefulness of these results for computing regression estimates. (These are the key results behind the sweep operator that is often used in regression computations.) What happens to the results if ri is replaced by yi in model (3)?