Since (X'X)-1X' is a matrix of fixed numbers, p is a linear function of Y. Hence, by definition it is a linear estimator. Recall that the PRF is y = X p + u (2)
Substituting this into (1), we obtain p = (X'X)-1X'(Xp + u) (3)
Taking expectation of (4) gives
since E(p) = p (why?) and E(u) = 0 by assumption, which shows that p is an unbiased estimator of p.
958 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL
Let p* be any other linear estimator of p, which can be written as p* = [(X'X)-1X' + C]y (6)
where C is a matrix of constants.
Substituting for y from (2) into (6), we get p* = [(X'X)-1X' + C](Xp + u) = p + CX p + (X'X)-1X'u + Cu
Now if p" is to be an unbiased estimator of p, we must have
Using (8), (7) can be written as p* — p = (X'X)-1X'u + Cu (9)
By definition, the var-cov (p ) is
E(p* - p)(p* - p)' = E[(X'X)-1X'u + Cu][(X'X)-1X'u + Cu]' (10)
Making use of the properties of matrix inversion and transposition and after algebraic simplification, we obtain var-cov (p*) = a 2(X'X)-1 + a 2CC'
which shows that the variance-covariance matrix of the alternative unbiased linear estimator p~ is equal to the variance-covariance matrix of the OLS estimator p plus a2 times CC', which is a positive semidefinite* matrix. Hence the variances of a given element of p* must necessarily be equal to or greater than the corresponding element of p, which shows that p is BLUE. Of course, if C is a null matrix, i.e., C = 0, then p~ = p, which is another way of saying that if we have found a BLUE estimator, it must be the least-squares estimator p.
*See references in App. B.
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