Partial Identifcation in Moment Equality Models under Data Combination

In this paper we study identification and inference for a finite dimensional parame-ter defined by a finite number of moment equalities when the sample information comes from two separate data sets. Unlike existing work in the literature assuming the same data structure, we allow some or all moment functions to be non-additively separable. By an application of the continuous version of the monotone rearrangement inequality, we convert moment equalities corresponding to non-additively separable moment functions to moment inequalities with unknown functions. As a result, we obtain a set of moment equalities/inequalities with unknown functions characterizing the parameter of interest. Two main examples that motivate our model are: a generalized two-sample IV model and a generalized linear projection model. Via a detailed analysis of the identified set of the unknown parameter in both models, we demonstrate that incorprating moment inequalities help shrink the identified set and can help identify the sign of the parameter of interest which is not identified otherwise. Moreover using both models we illustrate how existing inference  procedures such as those in Andrews and Soares (2010) can be modified to account for the first step estimation of the unknown functions appearing in the moment inequalities.