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5.2.8 TESTING OF THE GENERAL LINEAR HYPOTHESIS
Partial coefficient test and a test of the coefficient subset, actually can be

done through the establishment of general linear hypothesis. General linear

hypothesis is defined as follows.

H0 : K’  = m

H1 : K’   m (G.28)

Where K’ is a (k x (p+1)) matrix of coefficients defining k linear function of  to

be tested. Each row of the matrix K’ consists of the coefficients of a linear

function, and m is a (kx1) vector of constant.

Suppose ’ = (0 , 1, 2, 3) and we want to test the null hypothesis that the
composition of 1 = 2 , 1 + 2 = 2 3, dan 0 = 20. Hypotheses are equivalent to

H0 : 1 - 2 = 0

1 + 2 - 23 = 0

0 = 20

These three linear functions can be written in the form K’  = m , by defining

 0 1 1 -  0   0  




K’ = 0 1 1 - 2 and m = 0



   
 1 0 0 0   20 
The least squares estimate for K’  - m is obtained by subtituting the least
ˆ
ˆ
squares estimate β for , to obtain K’ β - m . Under normality assumptions for
ˆ
Y, and Eq. (G.5) that β is linear function of Y , then
ˆ
ˆ
E(K’ β - m) = K’ E( β ) – m = K’  - m
If H0 is true, then K’  - m = 0, and varince-covariance matrix

ˆ
ˆ
Var(K’ β - m ) = Var (K’ β )- 0
ˆ
= K’ Var ( β ) K
2
-1
= K’ ( X’X ) K  . (G.29)

~~* CHAPTER 5 THE MULTIPLE LINEAR REGRESSION MODEL *~~