Introduction to Probability and Statistics for Engineers and Scientists

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)