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).

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