Set identification, moment restrictions, and inference

For the past 10 years, the topic of set identification has been much studied in the econometric literature. Classical inference methods have been generalized to the case in which moment inequalities and equalities define a set instead of a point. We review several instances of partial identification by focusing on examples in which the underlying economic restrictions are expressed as linear moments. This setting illustrates the fact that convex analysis helps not only for characterizing the identified set but also for inference. From this perspective, we review inference methods using convex analysis or inversion of tests and detail how geometric characterizations can be useful.