NOTES

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than another test we mean that, when {3 =1= 0, the p-value of the first test tends to be smaller, thus giving a stronger indication that {3 =1= O. 3.5e. For more information on normal probability plots, see Section 3.8 and Appendix 3A in Daniel and Wood (1980), Section 3.1 and Appendix 3A in Draper and Smith (1981), and Section 6.6 in Weisberg (1985). Daniel and Wood show how difficult it is to judge nonnormality from a normal probability plot of the residuals in small samples. For information on tests of normality, see Section 6.6 in Weisberg (1985) and Section 9.6 in D'Agostino and Stephens (1986). 3.7. Matrix notation allows convenient calculation of expectations and variances. Let

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be a random vector, that is, a vector whose components are random variables. The expectation vector of y is defined to be

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E(y) =

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The variance-covariance matrix of y is defined to be Var(y,) Cov(y) = Cov( Y2' y,) Cov( y" Y2) Var( Y2) Cov( Y" Yn ) Cov( Y2' Yn )

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Suppose A is an m X n matrix whose entries are constant numbers. Two convenient rules for calculating expectations and variances are ( a)

E ( Ay)

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AE ( y)

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Cov(Ay) = A Cov(y)A'

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These are generalizations of the familiar facts that if Y is a random variable

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LEAST-SQUARES REGRESSION

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and a is a constant number, then

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E( ay) = aE( y)

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Var(ay) = a Z Var(y) These latter equations are the special cases of (a) and (b) when m = 1 and n=l. To prove these rules in general would involve a lot of subscripts, but we can convince ourselves further by looking at another special case. Let m = 1 and n = 2. Then

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[;~]

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E( Ay)

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E( a 1 Yl + azyz) = a1E( yd + azE( yz)

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=[a 1

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Also,

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az][~i:~n=AE(Y)

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+ a z y z)

Cov( Ay) = Var( a 1 y 1

=a Var(Yl) +2a 1 ZCov(Yl'YZ) +a~Var(yz) a

= [a 1 a z] [ Var( y 1)

COV(YZ'Yl)

Cov( y l' Yz) Var(yz)

1a [

= A Cov(y)A'

Let us apply (a) and (b) to the vector of least-squares estimates. Recall that P = Ay, where A = (X'X)-IX'. Using (a) we calculate

E(P) = E(Ay) = AE(y)

= (X' X) - 1 X' ( X p) = (X' X) - 1( X' X) P

This shows that the least-squares estimates are unbiased; that is, E(~) = f3 j for all j.

NOTES

Using (b) we calculate Cov(P)

Cov( Ay)

A Cov( y)A'

= (X'X)-I X '(U 2I)X(X'X)-1

u 2 (X'X) -I( X'X)( X'X)-I u 2 ( X' X) 1

3.8a. We are using SSR to denote the sum of squares of the residuals. In other books you may find SSR used to denote the sum of squares due to regression, which is L( Yi - y)2. You may find the sum of squares of the residuals denoted by SSE, standing for the sum of squares due to error. 3.8b. The expected value of the difference between the residual sums of squares is obtained as follows. Note that SSR full = Since 0- 2 in (3.11) is an unbiased estimate of u 2, it follows that SSR full is an unbiased estimate of (n - 5)u 2 Similarly, if the reduced model is true, then SSRreduced is an unbiased estimate of (n - I)u 2 The 1 in n - 1 corresponds to the fact that the reduced model has 1 regression coefficient. The expected value of SSRreduced - SSR full is (n - I)u 2 - (n - 5)u 2 = 4u 2 when the null hypothesis is true.

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3.9. There are two approaches one can take to testing the hypothesis f3 q + 1 = ... = f3 p = O. The approach we have taken in Sections 3.8 and 3.9 is to compare the sums of squares of the residuals in the full and reduced models. Another approach is to estimate f3 q + I' ... , f3 p and see how close to 0 the estimates are. To describe the second approach, let jj = (f3 q + I' ... ,f3 p ). We want to test whether or not jj = O. The least-squares estimate &LS can be obtained as the last p - q entries in PLS. The variance-covariance matrix of &LS is the (p - q) X (p - q) matrix in the lower right corner of the variance-covariance matrix of PLS. We know from Note 3.7 that COV(PLS) = u 2 (X'X)-I. Let Va denote COV(&LS); substituting 0- 2 from (3.12), we obtain an estimate Va. A reasonable measure of how close jj is to 0 is given by &'LsVa-I&Ls. The two approaches lead to exactly the same test statistic, because it turns out that test statistic (3.l3) can be calculated as F = &'LsVa-I&Ls/(P - q). 3.10. To determine Var(p), use the fact (shown in Note 3.7) that Cov(P) = u 2 (X' X)- I. Note that Var(p) is the (j + I)th diagonal entry in Cov(P).

L(Yi -

3.11a. To show that R2 can be expressed as a function of F, let Sf = yy, the sum of squared residuals in the full model, and let Sr = L(Yi - y)2, the sum of squared residuals in the reduced model Y = f30 + e with