Suppose the the full model is
yi = β0 + β1xi1 + β2xi2 + εi
for I=1,2,..,n where xi1 and xi2 have be coded so that S11=S22=1. Consider fitting the subset model yi = β0 + βixi1 + εi.
a.) Let*be the least square estimate of B1 from the full model. Show that2
2
Var(1) = oʻ/(1 - rîu),
0where r12 is the correlation between x1 and x2.
b.)Let ^β1 be the least square estimate of β1 from the subset model. Show thatVar(ệi)
=
02.
c.) Show that E( ˆβ1) = β1 + r12β2.
d.) Find the mean square error for the subset estimator ˆβ1