396 Chapter 9: Regression
β=
β 0
β 1
..
.
βk
, e=
e 1
e 2
..
.
en
thenYis ann×1,Xann×p,βap×1, andeann×1 matrix wherep≡k+1.
The multiple regression model can now be written as
Y=Xβ+e
In addition, if we let
B=
B 0
B 1
..
.
Bk
be the matrix of least squares estimators, then the normal Equations 9.10.1 can be written as
X′XB=X′Y (9.10.2)
whereX′is the transpose ofX.
To see that Equation 9.10.2 is equivalent to the normal Equations 9.10.1, note that
X′X=
11 ··· 1
x 11 x 21 ··· xn 1
x 12 x 22 ··· xn 2
..
.
..
.
..
.
x 1 k x 2 k ··· xnk
1 x 11 x 12 ··· x 1 k
1 x 21 x 22 ··· x 2 k
..
.
..
.
..
.
..
.
1 xn 1 xn 2 ··· xnk
=
n
∑
i
xi 1
∑
i
xi 2 ···
∑
i
xik
∑
i
xi 1
∑
i
x^2 i 1
∑
i
xi 1 xi 2 ···
∑
i
xi 1 xik
..
.
..
.
..
.
..
∑.
i
xik
∑
i
xikxi 1
∑
i
xikxi 2 ···
∑
i
xik^2
and
X′Y=
∑
i
Yi
∑
i
xi 1 Yi
..
∑.
i
xikYi