Page 162 - Applied Statistics with R

162 CHAPTER 9. MULTIPLE LINEAR REGRESSION

̂
The derivation of the sampling distribution of involves the multivariate normal
distribution. These brief notes from semesters past give a basic overview. These
are simply for your information, as we will not present the derivation in full here.
̂
Our goal now is to obtain the distribution of the vector,
̂
⎡ 0 ̂ ⎤
⎢ 1 ⎥
̂
= ⎢ ̂ ⎥
⎢ 2 ⎥
⎢ ⋮ ⎥
̂
⎣ −1⎦
Recall from last time that when discussing sampling distributions, we now con-
̂
sider to be a random vector, thus we use instead of the data vector .

̂
⊤
⊤
= ( ) −1
Then it is a consequence of the multivariate normal distribution that,
−1
̂
⊤
2
∼ ( , ( ) ) .
We then have

̂
E[ ] =
̂
and for any we have

̂
E[ ] = .

We also have

̂
2
⊤
Var[ ] = ( ) −1
̂
and for any we have

̂
2
Var[ ] =

where

⊤
= ( ) −1
and the elements of are denoted