406 Chapter 9: Regression
That is, it is an unbiased estimator. Also, using the fact that the variance of a random
variable is equal to the covariance between that random variable and itself, we see that
Var
∑k
i= 0
xiBi
=Cov
∑k
i= 0
xiBi,
∑k
j= 0
xjBj
(9.10.9)
=
∑k
i= 0
∑k
j= 0
xixjCov(Bi,Bj)
If we letxdenote the matrix
x=
x 0
x 1
..
.
xk
then, recalling that Cov(Bi,Bj)/σ^2 is the element in the (i+1)st row and (j+1)st column
of (X′X)−^1 , we can express Equation 9.10.9 as
Var
∑k
i= 0
xiBi
=x′(X′X)−^1 xσ^2 (9.10.10)
Using Equations 9.10.8 and 9.10.10, we see that
∑k
i= 0
xiBi−
∑k
i= 0
xiβi
σ
√
x′(X′X)−^1 x
∼N(0, 1)
If we now replaceσby its estimator
√
SSR/(n−k−1) we obtain, by the usual argument,
that
∑k
i= 0
xiBi−
∑k
i= 0
xiβi
√
SSR
(n−k−1)
√
x′(X′X)−^1 x
∼tn−k− 1
which gives rise to the following confidence interval estimator of
∑k
i= 0 xiβi.
Confidence Interval Estimate ofE[Y|x] =
∑k
i=0xiβi,(x^0 ≡1)
A 100(1−a) percent confidence interval estimate of
∑k
i= 0 xiβiis given by
∑k
i= 0
xibi±
√
ssr
(n−k−1)
√
x′(X′X)−^1 x ta/2,n−k− 1