Introduction to Probability and Statistics for Engineers and Scientists

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









