Inference in Moment Inequality Models with Combined Data Sources ∗

In this paper, we study the estimation and inference problems for parameters when the data set is obtained by combining data from different sources. In the case where we want to estimate the means of random variables in our data set, we consider the set of estimators that are linear combinations of certain sample averages, and find that one estimator, defined to be the Adjusted estimator, that achieves the smallest possible asymptotic variance among all estimators in this set. In particular, the Adjusted estimator has smaller asymptotic variance than two commonly used estimators, the Short estimator and the Long estimator for the mean. Based on this result, we study inference problems in moment inequality models. We implement GMS procedure with three constructions of the sample moments: the Short, the Long and the Adjusted sample moments. We show that the resulting three GMS tests control asymptotic sizes. Based on local power analysis and simulations, we recommend using GMS with the Adjusted sample moments for better power