*9.10Multiple Linear Regression 401
Since Var(Yr)=σ^2 , we see that
Cov(Bi− 1 ,Bj− 1 )=σ^2
∑n
r= 1
CirCjr (9.10.4)
=σ^2 (CC′)ij
where (CC′)ijis the element in rowi, columnjofCC′.
If we now let Cov(B) denote the matrix of covariances — that is,
Cov(B)=
Cov(B 0 ,B 0 ) ··· Cov(B 0 ,Bk)
..
.
..
.
Cov(Bk,B 0 ) ··· Cov(Bk,Bk)
then it follows from Equation 9.10.4 that
Cov(B)=σ^2 CC′ (9.10.5)
Now
C′=
(
(X′X)−^1 X′
)′
=X
(
(X′X)−^1
)′
=X(X′X)−^1
where the last equality follows since (X′X)−^1 is symmetric (sinceX′Xis) and so is equal
to its transpose. Hence
CC′=(X′X)−^1 X′X(X′X)−^1
=(X′X)−^1
and so we can conclude from Equation 9.10.5 that
Cov(B)=σ^2 (X′X)−^1 (9.10.6)
Since Cov(Bi,Bi)=Var(Bi), it follows that the variances of the least squares estimators
are given byσ^2 multiplied by the diagonal elements of (X′X)−^1.