Page 110 - Basic Statistics

105

ˆ
E(Y ) = E(H Y )
= H E(Y )

= [X ( X’X ) X’] X
-1
= X ( X’X ) X’X 
-1
= X 

ˆ
Thus, if the model is correct, Y is unbiased estimate of means of Y.

E(e) =E( [I - H] Y)

= [I - H ] E(Y)

= [I – H] X

= [I X – H X] 

= [X – X ] 
= 0

Thus, e observed residuals are random variables with mean zero.

b. Variance

Discussion of the variance for a linear function of Y, is done by first

review the idea of notation and matrix algebra. For Y as random vectors, eg

variance-covariance matrix has written Var (Y), and suppose a linear function

of u, written u = a’ Y .

Var (u) = a’ [Var (Y)] a

As assumptions on estimation by ordinary least squares method, Var (Y) = I ,
2
so that

2
2
Var (u) = a’ (I ) a = a’a  . Notation a’a stating the sum of squares of the
2
coefficients of linear function, that is  a
i
Form of a linear function of u = a’ Y, extended over several vector
coefficients a simultaneously, i.e by a k x n matrix of coefficients A, becomes

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