Advanced Econometrics Takeshi Amemiya
The Mean and Variance of 0 and a2
1.1.3 Inserting (1.1.4) into (1.2.3), we have
^-(X'X^X'y (1.2.15)
= jff + (X'X)-,X'u.
Clearly, E0 = 0 by the assumptions of Model 1. Using the second line of
(1.2.15) , we can derive the variance-covariance matrix of 0:
V0 = E(0-0K0-0)' (1.2.16)
= E(X' X)" *X 'uu' X(X' X)~1
= <t2(X'X)-1.
Using the properties of the projection matrix given in Theorem 14 of Appendix 1, we obtain
£ff2=r‘£u'Mu (1.2.17)
= T~lE tr Mini' by Theorem 6 of Appendix 1 = T~xo2 tr M = T~l(T— K)o2 by Theorems 7 and 14 of Appendix 1,
which shows that a2 is a biased estimator of a2. We define the unbiased estimator of a2 by
а2 = (Т-К)~1й'й. (1.2.18)
We shall obtain the variance of a2 later, in Section 1.3, under the additional assumption that u is normal.
The quantity Vfi can be estimated by substituting either <r2 or a2 (defined above) for the a2 that appears in the right-hand side of (1.2.16).
