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.
20 April 2021
Working Paper (CWP19/21)