Case Analysis concluded from Wooldridge (2002) and (2003)
SCENARY 1: If R depends only on X, the covarites in the model. That is, there is no Z variable and Assumption MAR holds as (R=1|Y,X) = (R=1|X).
Other settings are special cases of this scenary, so we will not discuss it.
SCENARY 2: If R depends only on X and X is completely recorded;
Double Robustness property of Weighted estimator makes it more likely to be used in this situation. This property means that we have to correctly specify either E(Y|X) or the model of R to get consistency. Why? because (in the linear regression context??) if the model of R is correct, Weighted consistenly estimate L(Y|X), the linear projection in the population anyway. If L(Y|X) = E(Y|X), then we have consistency even if the model of R is incorrect.
The drawback is when GCIME holds and (Y|X) = E(Y|X) as Unweighted estimator is consistent and efficient. But Unweighted is not robust since if (Y|X) is not equal to E(Y|X), then it is inconsistent immediately.
SCENARY 3: If R depends only on X, X is incompletely recorded and the feature of interest is correctly specified;
Use Weighted but also see the comment below.
SCENARY 4: If R depends only on X and X is incompletely recorded;
Weighted estimator will be inconsistent as we cannot estimate R. Unweighted analysis may be better as it allow R model to depend on missing X (Ignorability Assumption holds using missing X)
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.
Both are consistent and we dont have to bother about getting the model of R correct. It is likely that we will use Unweighted estimator hoping GCIME to hold.
SCENARY 6: Same as SCENARY 5 + GCIME holds
We will use Unweighted as it is efficient estimator.