402 Chapter 9: Regression
The quantityσ^2 can be estimated by using the sum of squares of the residuals. That is,
if we let
SSR=
∑n
i= 1
(Yi−B 0 −B 1 xi 1 −B 2 xi 2 −···−Bkxik)^2
then it can be shown that
SSr
σ^2
∼χn^2 −(k+1)
and so
E
[
SSR
σ^2
]
=n−k− 1
or
E[SSR/(n−k−1)]=σ^2
That is,SSR/(n−k−1) is an unbiased estimator ofσ^2. In addition, as in the case
of simple linear regression, SSRwill be independent of the least squares estimators
B 0 ,B 1 ,...,Bk.
REMARK
If we letridenote theith residual
ri=Yi−B 0 −B 1 xi 1 −···−Bkxik, i=1,...,n
then
r=Y−XB
where
r=
r 1
r 2
..
.
rn
Hence, we may write
SSR=
∑n
i= 1
ri^2 (9.10.7)
=r′r
=(Y−XB)′(Y−XB)
=[Y′−(XB)′](Y−XB)