*9.10Multiple Linear Regression 399
Multiple Linear RegressionCompute InverseBack 1 Step1 1 1 1 1 1
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22 61 2 3 4 5 6ABCFIGURE 9.16
The variances of the least squares estimators can be obtained from the matrix (X′X)−^1.
Indeed, the values of this matrix are related to the covariances of theBi’s. Specifically, the
element in the (i+1)st row, (j+1)st column of (X′X)−^1 is equal to Cov(Bi,Bj)/σ^2.
To verify the preceding statement concerning Cov(Bi,Bj), let
C=(X′X)−^1 X′SinceXis ann×pmatrix andX′ap×nmatrix, it follows thatX′Xisp×p,asis(X′X)−^1 ,
and soCwill be ap×nmatrix. LetCijdenote the element in rowi, columnjof this
matrix. Now
B 0
..
.
Bi− 1
..
.
Bk
=B=CY=
C 11 ··· C 1 n
..
...
.
Ci 1 ··· Cin
..
...
.
Cp 1 ··· Cpn
Y 1..
.
Yn