*For the statistic t Bj ( j SE ffiffiffiffi vjj p to have a t-distribution, the estimators Bj and SE must be independent. [Here,

*For the statistic t ¼ Bj ( βj SE ffiffiffiffi vjj p to have a t-distribution, the estimators Bj and SE must be independent. [Here, vjj is the jth diagonal entry of ðX0 XÞ

(1

.] The coefficient Bj is the jth element of

b, and SE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e0 e=ðn ( k ( 1Þ p is a function of the residuals

e. Because both b and e are normally distributed, it suffices to prove that their covariance is 0. Demonstrate that this is the case. [Hint: Use Cðe; bÞ ¼ E½eðb ( flÞ

0

%, and begin by showing that b ( fl ¼ ðX0 XÞ

(1 X0

".]