Sample selection and information-theoretic alternatives to GMM

Information-theoretic alternatives to general method of moments (GMM) use over-identifying moments to estimate the data-generating distribution jointly with the parameters of interest. This paper demonstrates how these estimates can be interpreted when the sample is not a random draw from the population of interest. I make explicit the selection probability implied by the empirical likelihood and exponential tilting estimators, two commonly used estimators in this class. In addition, I propose an alternative estimator that corresponds to a logisitic selection model. The small sample properties of the estimators are demonstrated with a Monte Carlo experiment.