Discordant Relaxations of Misspecified Models

In many set identified models, it is difficult to obtain a tractable characterization of the identified set. Therefore, empirical works often construct confidence region based on an outer set of the identified set. Because an outer set is always a superset of the identified set, this practice is often viewed as conservative yet valid. However, this paper shows that, when the model is refuted by the data, a nonempty outer set could deliver conflicting results with another outer set derived from the same underlying model structure, so that the results of outer sets could be misleading in the presence of misspecification. We provide a sufficient condition for the existence of discordant outer sets which covers models characterized by intersection bounds and the Artstein (1983) inequalities. We also derive sufficient conditions for the non-existence of discordant submodels, therefore providing a class of models for which constructing outer sets cannot lead to misleading interpretations. In the case of discordancy, we follow Masten and Poirier (2021) by developing a method to salvage misspecified models, but unlike them we focus on discrete relaxations. We consider all minimum relaxations of a refuted model which restores data-consistency. We find that the union of the identified sets of these minimum relaxations is robust to detectable misspecifications and has an intuitive empirical interpretation.