  In many set-identified models, it is difficult to obtain a tractable
characterization of the identified set. Therefore, researchers often rely on
non-sharp identification conditions, and empirical results are often based on
an outer set of the identified set. This practice is often viewed as
conservative yet valid because an outer set is always a superset of the
identified set. However, this paper shows that when the model is refuted by the
data, two sets of non-sharp identification conditions derived from the same
model could lead to disjoint outer sets and conflicting empirical results. We
provide a sufficient condition for the existence of such discordancy, which
covers models characterized by conditional moment inequalities 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 that 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.
