Efficiency results of MLE and GMM estimation with sampling weights

This paper examines GMM and ML estimation of econometric models and the theory of Hausman tests with sampling weights. Weighted conditional GMM can be more efficient than weighted conditional MLE, an inefficient alternative to full information MLE under choice-based sampling, unless regressions have homoscedastic additive disturbances or sampling weights are independent of exogenous variables. GMM variances are necessarily smaller without sampling weights if GMM is the same as MLE or disturbances are homoscedastic, but not in general. Taking into account the dependence of sampling weights on parameters improves the efficiency of estimation.