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Moment Generating and Characteristic Functions*

1 Distribution of Variance Estimate Next, consider the variance estimator s 2 from (4.26). Using (3.29), it satisÖes (n k) s 2 = be 0 be = e 0M e: The spectral decomposition of M (see equation (A.4)) is M = HH0 where H0H = In and is diagonal with the eigenvalues of M on the diagonal. Since M is idempotent with rank n k (see Section 3.12) it has n k eigenvalues equalling 1 and k eigenvalues equalling 0, so =  Ink 0 0 0k  : Let u = H0e  N 0; In 2  (see Exercise 5.14) and partition u = (u 0 1 ;u 0 2 ) 0 where u1  N 0; Ink 2  . Then (n k) s 2 = e 0M e = e 0H  Ink 0 0 0  H0e = u 0  Ink 0 0 0  u = u 0 1u1   2 2 nk : We see that in the normal regression model the exact distribution of s 2 is a scaled chi-square. Since be is independent of b it follows that s 2 is independent of b as well. Theorem 5.16 In the linear regression model, (n k) s 2  2   2 nk and is ind