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Mathematical Statistics with Applications, Seventh Edition

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www.downloadslide.com 11.12 Model Parameters: Multiple Linear Regression 617 it follows that a linear combination of the β 0 ,β 1 ,...,β k corresponding to a 0 , a 1 ,..., a k may be expressed as a ′ β = a 0 β 0 + a 1 β 1 + a 2 β 2 +···+a k β k . From now on, we will refer to such linear combinations in their matrix form. Because a ′ β is a linear combination of the model parameters, an unbiased estimator for a ′ β is given by the same linear combination of the parameter estimators. That is, by Theorem 5.12, if â ′ β = a 0 ˆβ 0 + a 1 ˆβ 1 + a 2 ˆβ 2 +···+a k ˆβ k = a ′ ˆβ, then E(a ′ ˆβ) = E(a 0 ˆβ 0 + a 1 ˆβ 1 + a 2 ˆβ 2 +···+a k ˆβ k ) = a 0 β 0 + a 1 β 1 + a 2 β 2 +···+a k β k = a ′ β. Applying the same theorem, we find the variance of a ′ ˆβ: V (a ′ ˆβ) = V (a 0 ˆβ 0 + a 1 ˆβ 1 + a 2 ˆβ 2 +···+a k ˆβ k ) = a0 2 V ( ˆβ 0 ) + a1 2 V ( ˆβ 1 ) + a2 2 V ( ˆβ 2 ) +···+ak 2 V ( ˆβ k ) + 2a 0 a 1 Cov( ˆβ 0 , ˆβ 1 ) + 2a 0 a 2 Cov( ˆβ 0 , ˆβ 2 ) +···+2a 1 a 2 Cov( ˆβ 1 , ˆβ 2 ) +···+2a k−1 a k Cov( ˆβ k−1 , ˆβ k ), where V ( ˆβ i ) = c ii σ 2 and Cov( ˆβ i , ˆβ j ) = c ij σ 2 . You may verify that V (a ′ ˆβ) is given by V (a ′ ˆβ) = [a ′ (X ′ X) −1 a]σ 2 . Finally, recalling that ˆβ 0 , ˆβ 1 , ˆβ 2 ,..., ˆβ k are normally distributed in repeated sampling (Section 11.11), it is clear that a ′ ˆβ is a linear function of normally distributed random variables and hence itself is normally distributed in repeated sampling. Because a ′ ˆβ is normally distributed with E(a ′ ˆβ) = a ′ β and V (a ′ ˆβ) = [a ′ (X ′ X) −1 a]σ 2 , we conclude that Z = a′ ˆβ − a ′ β √ V (a ′ ˆβ) = a′ ˆβ − a ′ β σ √ a ′ (X ′ X) −1 a has a standard normal distribution and could be employed to test a hypothesis H 0 : a ′ β = (a ′ β) 0 when (a ′ β) 0 is some specified value. Likewise, a 100(1 − α)% confidence interval for a ′ β is a ′ ˆβ ± z α/2 σ √ a ′ (X ′ X) −1 a. Furthermore, as we might suspect, if we substitute S for σ , the quantity T = a′ ˆβ − a ′ β S √ a ′ (X ′ X) −1 a

www.downloadslide.com 618 Chapter 11 Linear Models and Estimation by Least Squares possesses a Student’s t distribution in repeated sampling, with [n − (k + 1)] df, and provides a test statistic to test the hypothesis H 0 : a ′ β = (a ′ β) 0 . A Test for a ′ β H 0 : a ⎧ ′ β = (a ′ β) 0 . a ⎪⎨ ′ β >(a ′ β) 0 , H a : a ′ β t α , ⎪⎨ Rejection region: t < −t α , ⎪⎩ |t| > t α/2 . Here, t α is based on [n − (k + 1)] df. The corresponding 100(1 − α)% confidence interval for a ′ β is as follows. A 100(1 − α)% Confidence Interval for a ′ β a ′ ˆβ ± t α/2 S √ a ′ (X ′ X) −1 a. As earlier, the tabulated t α/2 in this formula is based on [n − (k + 1)] df. Although we usually do not think of a single β i as a linear combination of β 0 ,β 1 ,...,β k , if we choose { 1, if j = i, a j = 0, if j ≠ i, then β i = a ′ β for this choice of a. In Exercise 11.71, you will show that with this choice of a, a ′ (X ′ X) −1 a = c ii , where c ii is the element in row i and column i of (X ′ X) −1 . This fact greatly simplifies the form of both the test statistic and confidence intervals that can be used to make inferences about an individual β i . As previously indicated, one useful application of the hypothesis-testing and confidence interval techniques just presented is to the problem of estimating the mean value of Y, E(Y ), for fixed values of the independent variables x 1 , x 2 ,...,x k .In particular, if xi ∗ denotes a specific value of x i , for i = 1, 2,...,k, then E(Y ) = β 0 + β 1 x1 ∗ + β 2x2 ∗ +···+β kxk ∗ . Notice that E(Y ) is a special case of a 0 β 0 + a 1 β 1 +···+a k β k = a ′ β with a 0 = 1 and a i = x ∗ i , for i = 1, 2,...,k. Thus, an inference about E(Y ) when x i = x ∗ i , for

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