Minimal and Maximal Just-Identified Assumptions in Nonparametric Selection Models

In this paper, we show minimal just identified restrictions for a class of selection models. We deepen our knowledge about selection on observables in three directions: First, we show the model proposed by Rosenbaum and Rubin is just-identified. Second, we show that although their assumptions are not minimal, the weak selection on observables-(WSO) hypothesis we introduce does, in fact, generate a set of minimal just-identified assumptions. Finally, we show the WSO hypothesis has an immediate and sensible economic interpretation. Specifically, we show that this hypothesis can be interpreted as bounds on the information available to the analyst. Basically, identification is achieved as long as the analyst's information set is rich enough so that he is able to predict migration decision, at the decision time. As a consequence, the identification conditions become weaker as the information which is assumed to be available to the analyst becomes larger. This approach certainly has a large tradition in econometrics (Zellner, Kmenta and Dreze (1966)). In the context of selection models, similar identification strategies have been followed by Goldberger (1972) and Heckman and Robb (1985, sec.5.4.3).