Consistency: Case Comparison
To COMPARE BETWEEN THE TWO APPROACHES
To comepare them, we have to set the situations where both of them can be used as an alternative of one another.
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). This is the most common situation where both types of analysis can be used. Also, this is quite a realistic situation since, in practice, it will be difficult to find Z which is not a subset of X and makes the independence between R and (Y,X) holds.
(1) Do not require the correct specification of E(Y|X).
(2) Allow some other variables which are not in X to alsp affect R.
(1) the model of R has to be correct.
(2) X has to be completely recorded now so that we can estimate the model of R.
(3) only Y is allowed to be missing.
(1) Y and X can be jointly missing as the conditioning variables in the model of R (X or a subset of it) can be missing.
(2) If there are some variables which are important but incompletely recorded, they can be in X. In attrition, some covariates are missing in the later waves of study, such covariates can be included using this appraoch.
(1) correct specification of the feature of interest
AS CAN BE SEEN, to use which one of them, we have to consider case by case. NOTE that MAR and NMAR is not a good criteria to divide the literature in missing data (at least for Econometricians) anymore. From the above elaboration, the circumstance where X is missing but R depends on X fits with the setting of NMAR. As can be seen, there can be the case where Unweighted M-estimator is consistent.
SCENARY 2: If R depends only on X and X is completely recorded;
Then, we should use Weighted analysis as we can allow for mispecification and some other variables to affect the missing probability.
SCENARY 3: If R depends only on X, X is incompletely recorded and the feature of interest is correctly specified;
Then, we should use Unweighted analysis because the incomplete X can be in the model of R. (As we know for sure that these incomplete X are matter for the R's model and we do not have to worry about the miscpecification.) Thus, Unweighted analysis yields consistency under weaker assumption (= more variables in the R's model to ensure the independency between Y and R)
However, there could be some restriction such that we cannot use these incomplete variables anyway (Wooldridge (2002) gives an example where the structural of the conditional expectation model refrains us from using incomplete variables). So, Weighted might be better.
SCENARY 4: If R depends only on X and X is incompletely recorded;
Now, we do not know whether, say, E(Y|X) is correctly specified or not. This case is also unclear. We might want to use Unweighted analysis and gamble about the model specification. Or, if R is not significantly dependent on those incomplete X, we may want to use Weighted analysis.