Efficiency of Weight and Unweighted estimators
We will think about efficiency only if the subjects of comparison are consistent estimators. Thus, we have to compare both of them under the circumstance where they are consistent. That is, Theorem of the Weighted estimators holds with Z=X and the objective function is E[q(W,theta)|X]. Alternatively, the following scenary must be true.
SCENARY 5: a feature of the conditional distribution is correctly specified, R depends only on X, X are completely recorded and some regularity condtions hold.
Given this scenary, it can be show that, for Weighted estimator, asym variance is the same whether the missing probabilities are estimated or are known.
This result does not depend on whether or not a generalised conditional information matrix equality (GCIME) holds. Thus, if all conditions of this scenary are satisfied, the result holds even when there is heteroskedasticity in VAR of unknown form in the context of LS (see Wooldridge 2002)
At this point, we know that, according to this scenary, both Weighted and Unweighted estimators are consistent and it does not matter whether we estimate the seletion probabilities are use the known probabilities. (then the quesion remains: Should we weight or not weight?)
However, if GCIME also holds, we can show that the asymvariance of Unweighted estimator is smaller than that of Weighted estimator.
GCIME holds for conditional MLE if the conditional DENSITY is correctly specified with the true variance =1. (so both, say, population conditional mean and the density have to be correctly specified)
So, in SCENARY 5, it is likely that we will use Unweighted estimator.
In the SCENARY 2 below, we say that we will use Weighted estimator. However, if GCIME holds, we might wanna gamble with the model specification of mean function and use Unweighted estimator.