14Asymptotic Estimation Theory
14.1 Large Sample Efficiency
The preceding chapter introduced the idea of estimators with desirable properties in large samples. These are estimators that are consistent and asymptotically normal, henceforth abbreviated to CAN. Least squares is a member of the CAN class under standard assumptions, and naturally enough, this raises the question of efficiency. An estimator is said to be asymptotically efficient in the CAN class if for any other CAN estimator, the difference of the asymptotic variance matrices is positive semidefinite. The term asymptotic variance here refers to a matrix such as 
in (13.25), relating to the normalized error‐of‐estimate and hence well defined in the limit.
The comparison parallels that of the Gauss‐Markov theorem in Sections 8.2 and 12.5 except that in large samples the conditions for fixed and random regressors become in effect unified. For the regression model
let the linear CAN class be defined by
where 
is a 
random ...
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