Consistent Estimation of Regression Models with Incompletely Observed Exogenous Variables

We consider consistent estimation of regression models in which the exogenous variables are incompletely observed assuming that the response mechanism is random. In the literature on imputed data, several estimators have been proposed which are based on approximations substituted for the missing data. We discuss conditions under which these proxy variables estimators are asymptotically more efficient than the estimator based on complete observations and we show how an optimal proxy variables estimator can be obtained. For simple models, some proxy variables estimators are almost as efficient as the Gaussian maximum likelihood (ML) estimator and sometimes more efficient than the pseudo ML estimator.