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. Furthermore, we develop a method to salvage misspecified models. We consider all minimum relaxations of a refuted model which restore data-consistency. We find that the union of the identified sets of these minimum relaxations is misspecification-robust and it has a new and intuitive empirical interpretation. Although this paper primarily focuses on discrete relaxations, our new interpretation also applies to continuous relaxations.