U'U - y'y — 2p'X'y + p'X'Xp Using rules of matrix differentiation given in Appendix B, we obtain
Setting the preceding equation to zero gives
(X'X)p - X'y whence p - (X'X)—1X'y, provided the inverse exists.
CA.3 VARIANCE-COVARIANCE MATRIX OF p
From (C.3.11) we obtain p - (X'X)—1X'y Substituting y - Xp + u into the preceding expression gives p - (X'X)—1X'(Xp + u)
APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL 957
where in the last step use is made of the fact that (AB)' = B'A'.
Noting that the X's are nonstochastic, on taking expectation of (3) we obtain var-cov (p) = (X'X)-1X' E(uu')X(X'X)-1 = (X'X)-1X'a 2IX(X'X)-1 = a 2(X'X)-1
which is the result given in (C.3.13). Note that in deriving the preceding result use is made of the assumption that £(uu') = a 2I.
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