Page 112 - Basic Statistics

107

ˆ
Y = H Y, so that
ˆ
Var(Y ) = H [Var (Y) ] H’
= H I H’ 
2
= H H’ 
2
= H  (G.11)
2
ˆ
Diagonal element is the variance for the predicted value Y , i = 1, 2, ... n. Value
i
prediction is used to estimate the means Y for various combinations of

independent variables is given. For the value prediction of the future

(predictive value) for various combinations of values of the independent
ˆ
2
variables is given, written Y pred, then each variance increased by  . Variance-
i
covariance matrix for this prediction is

ˆ
2
Var (Y pred ) = ( I + H )  (G.12)

e = [I - H] Y , so that
Var ( e ) = [I - H] [Var (Y) ] [I - H]’

2
= [I - H] I [I - H]’ 
2
= [I - H] [I - H]’ 
2
= [I - H]  (G.13)

Summary of the distribution of each random vectors can be expressed as
follows:

Y  N( X , I )
2
ˆ
β  N( , ( X’X )  )
2
-1
ˆ
Y  N( X, H )
2
e  N( 0, [I - H]  )
2
ˆ
Y pred  N( X, [I + H]  )
2

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