Suppose that the full model is y i = 0 + 1x i1 + 2x i2 + i, i = 1, 2, . . .

Suppose that the full model is y i = β0 + β1x i1 + β2x i2 + εi, i = 1, 2, . . . , n , where x i1 and x i2 have been coded so that S11 = S22 = 1. We will also consider fi tting a subset model, say y i = β0 + β1x i1 + εi .

a. Let ˆ * β1 be the least - squares estimate of β1 from the full model. Show that Var ˆ * β σ 1 2 12 2 ( ) = − ( ) 1 r , where r12 is the correlation between x1 and x2 .

b. Let ˆ

β1 be the least - squares estimate of β1 from the subset model. Show that Var ˆ

β σ 1 2 ( ) = . Is β1 estimated more precisely from the subset model or from the full model?

c. Show that E r ˆ ( ) ββ β 1 1 12 2 = + . Under what circumstances is ˆ

β1 an unbiased estimator of β1 ?

d. Find the mean square error for the subset estimator ˆ

β1. Compare MSE ˆ ( ) β1 with Var ˆ * ( ) β1 . Under what circumstances is ˆ

β1 a preferable estimator, with respect to MSE?

You may fi nd it helpful to reread Section 10.1.2 .