Efficiency Results of Mle and GMM Estimation with Sampling Weights

This paper examines the theory of the estimation of econometric models and Hausman tests with sampling weights. The Manski-Lerman weighted conditional MLE is emphasized because of its popularity in econometric estimation with sampling weights. It is an inefficient alternative to full information MLE under choice-based sampling, and weighted conditional MLE can be less efficient than weighted conditional GMM, but not all efficiency results are lost. Weighted conditional MLE is still most efficient in the asymptotically linear class if sampling weights are independent of exogenous variables or linear or nonlinear regressions have homoscedastic additive disturbances. The derivation of the Hausman test and the Cramer-Rao bound are complicated by sampling weights; the covariance of an asymptotically linear estimator (such as a GMM estimator) with the score function is not an identity matrix. When weights are stochastically independent of the regressors in a model, however, of which one example is estimation of a sample mean, the MLE attains the Cramer-Rao bound, which is the standard form multiplied by the design effect from sample design. Simple random samples sometimes do and sometimes do not minimize the variances of econometric models. A simple random sample minimizes the variance of MLE, sample means, and homoscedastic linear and nonlinear regressions, but not of GMM estimators when heteroscedasticity is present. GMM variances are necessarily minimized by simple random samples if GMM is the same as MLE or disturbances are homoscedastic, but not in general. A probit model illustrates conditional GMM variances not minimized by a simple random sample and smaller than weighted conditional MLE variances. The calculation uses the theoretical expectation of the variance matrix, eliminating all sampling error from the estimation of the variances