Advanced Econometrics Takeshi Amemiya

The Cramer-Rao Lower Bound for Unbiased Estimators of cr2

From (1.3.8) and (1.3.20) the Cramer-Rao lower bound for unbiased estima­tors of a2 in Model 1 with normality is equal to 2алТ~К We shall examine whether it is attained by the unbiased estimator a2 defined in Eq. (1.2.18). Using (1.2.17) and (1.3.5), we have

T-K'

Therefore it does not attain the Cramer-Rao lower bound, although the differ­ence is negligible when T is large.

We shall now show that there is a simple biased estimator of a2 that has a smaller mean squared error than the Cramer-Rao lower bound. Define the class of estimators

fi'fi

~N~’

where TV is a positive integer. Both a2 and a2, defined in (1.2.5) and (1.2.18), respectively, are special cases of (1.3.28). Using (1.2.17) and (1.3.5), we can evaluate the mean squared error of a2N as

By differentiating (1.3.29) with respect to N and equating the derivative to zero, we can find the value of N that minimizes (1.3.29) to be

N* = T-K+ 2. (1.3.30)

Inserting (1.3.30) into (1.3.29), we have

2<r4

E{g%, - a2)2 = T_K+r (1.3.31)

which is smaller than the Cramer-Rao bound if K= 1.