Econometrica: Nov, 1999, Volume 67, Issue 6

Asymptotic Properties of Weighted M‐estimators for variable probability samples

https://doi.org/10.1111/1468-0262.00083
p. 1385-1406

Jeffrey M. Wooldridge

I provide a systematic treatment of the asymptotic properties of weighted M‐estimators under variable probability stratified sampling. The characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis. Simple, consistent asymptotic variance matrix estimators are proposed for a large class of problems. When stratification is based on exogenous variables, I show that the unweighted M‐estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality. When population frequencies are known, a more efficient weighting is possible. I also show how the results carry over to multinomial sampling.