at and Elementary calculations show that
and
The proof of this theorem is thus co mplete. Note that D would be zero if all
xi take the same value. Hence, at least two distinct xi values are needed for the
determination of and
It is instructive at this point to restate the foregoing results by using a more
compact vector–matrix notation. As we will see, results in vector–matrix form
facilitate calculations. Also, they permit easy generalizations when we consider
more general regression models.
In terms of observed sample values (x 1 ,y 1 ), (x 2 ,y 2 ),...,(xn,yn), we have a
system of observed regression equations
Let
and let
Equations (11.11) can be represented by the vector–matrix equation
The sum of squared residuals given by Equation (11.6) is now
Linear Models and Linear Regression 339
^ ^q^2 Q
q ^^2 2 n> 0 ;D 4 nXni 1
xix^2 > 0^ ^.
yi xiei; i 1 ; 2 ;...;n:
11 : 11 C
1 x 1
1 x 2
... ...
1 xn2
6
6
6
4
3
7
7
7
5
; yy 1
y 2
...
yn2
6
6
6
4
3
7
7
7
5
; ee 1
e 2
...
en2
6
6
6
4
3
7
7
7
5
;
q:
yCqe:
11 : 12 QeTe
yCqT
yCq:
11 : 13