  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, an 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.
