Weighted and Unweighted: Consistency
Note that, in all of this discussion, R has to be independent of Y conditional on some variables anyway. ( For Weighted analysis this is Assumption MAR but, in Unweighted analysis, it is not MAR because the conditional variables in R model can be missing)
(1) Do not require the correct specification of the feature of Y|X ( conditional mean, conditional median)
(2) Allow other variables (apart from those in X) in Z to affect R.
(3) Y and X can be jointly missing as long as Z is fully recorded and as Assumption MAR (using Z) is satisfied.
(1) Model of R must be correctly specified
(2) Z must be completely recorded.
(3) Response probability has to be positive (meaning that we cannot exclude a subsection of the population in the sampling process) ( This may imply that wage equation example is not valid here because people who dont work are excluded completely)
(1) Missing variables (except Y) can be allowed into the model of R,i.e., missing mechanism. This is because we do not have to estimate the response probability. Thus, Ignorability Assumption ( this is not MAR) of R's model tends to be weaker than that of Weighted analysis in general ( since more variables can be conditioned upon to make the independent between Y and R more plausible.)
(2) Y and X can be jointly missing even when we dont have any variable as Z.
(3) Response probability can be zero for some subset of population
(1) require correct specification of the conditional mean, conditional median or conditional distribution.