Linear model theory and hypothesis testing

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4.5 CANONICAL FORM FOR H Suppsse that _we wish to test H : A = 0, where A is q x p of rank q, for the full-rank model Y = X5 + 5:. Since A has 9 linearly mdependent columns, we can assume without less of generality (by relabel'mg the 13;; if necessary) that these are the last q columns; thus A. = (A1,.A2), where A; is a q x q nensingular matrix. Partitioning 5 in the same way, we have 0 = A = A151 + A252: and multiplying through by A, ' leads to B2 = -And AlB1. (4.32) This means that under the hypothesis H, the regression model takes the "canonical" form XB = (X1, X2)B = XIB1 + X2B2 = (X1 - X2A?' Al) BI = XHY, (4.33) say, where Xy is n x (p - q) of rank p - q and y = B1. The matrix Xy has linearly independent columns since XHB1 =0 # X3=0 - 3=0 +> B1=0. By expressing the hypothesized model H : E[Y] = Xay in the same form as the original model E[Y] = XB, we see that the same computer package can be used for calculating both RSS and RSSy, provided, of course, that Xy can be found easily and accurately. If Xy is not readily found, then the numerator of the F-statistic for testing / can be computed directly using the method of Section 11.11. We note that q = rank(X) - rank(XA). One very simple application of the theory above is to test H : 2 = 0; Xy is simply the first p - q columns of X. Further applications are given in Section 6.4, Chapter 8, and in Section 4.6.Suppose that we have n observations on w1, W2, . .., Wp-1 and U, giving the model Vi = Yo (1 ) + 71 (1) Yp-1Wi,p-1 + ni (i = 1, 2, ..., n1). We are now given no (> p) additional observations which can be ex- pressed in the same way, namely, Ui = Yo" ( 2 ) + y ' Wil + .. . to(? ) Yp-1Wi,p-1 + ni (i = n + 1, n2 + 2, . .., n1 + 12). Derive an F-statistic for testing the hypothesis / that the additional observations come from the same model