It was shown in Appendix 3A, Section 3A.2, that the least-squares estimator /32 is linear as well as unbiased (this holds true of ¡31 too). To show that these estimators are also minimum variance in the class of all linear unbiased estimators, consider the least-squares estimator /32:
X XY x ki = E(X — X)2 = Yxf (see Appendix 3A.2) (19)
which shows that /32 is a weighted average of the Y's, with ki serving as the weights.
Let us define an alternative linear estimator of ¿2 as follows:
where Wi are also weights, not necessarily equal to ki. Now £(¿2) = E WiE(Yi)
= ¿1 E Wi + ¿2 E WiXi Therefore, for ¿2 to be unbiased, we must have wi = 0 (22)
= a 2 ( wi--—2 +--) (Note the mathematical trick)
xi2 xi2 2 i xi2 xi2
because the last term in the next to the last step drops out. (Why?)
CHAPTER THREE: TWO-VARIABLE REGRESSION MODEL 105
Since the last term in (24) is constant, the variance of (fa2) can be minimized only by manipulating the first term. If we let
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