Asymptotic Properties of Weighted M-Estimators for Standard Stratified Samples

I provide a systematic treatment of the asymptotic properties of weighted M -estimators under standard stratified sampling. Simple, consistent asymptotic variance matrix estimators are proposed for a broad class of problems. When stratification is based on exogenous variables, I show that the usual, unweighted M -estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality. Hausman tests for the exogeneity of the sampling scheme, including fully robust forms, are derived.