Partial identification and inference in moment models with incomplete data

In this paper, we develop asymptotically valid inference in moment equality models with incomplete data, where the sample information is insufficient to identify the joint distribution of all the variables in the model. Examples of such models include the selection-on-observables framework, the counterfactual distribution, and parametric regressions with incomplete data. In the first two examples, the parameter of interest includes the values of the distribution and quantile functions of the individual treatment effect and the correlation coefficient of the potential outcomes. We show that the parameter of interest satisfies a semiparametric moment equality model with both point identified nuisance parameters and a partially identified copula function. We construct an asymptotically valid confidence set for the parameter of interest taking account of shape restrictions on the copula function. The critical value is constructed via a multiplier bootstrap. A simulation study is conducted to illustrate the finite sample performance of our inference procedure.