Discordant Relaxations of Misspecified Models

TL;DR

This paper investigates the issues arising from non-sharp identification conditions in set-identified models, highlighting potential conflicts and proposing methods to salvage misspecified models through discrete relaxations.

Contribution

It introduces conditions for when discordant outer sets occur and develops a robust approach using minimum relaxations to address model misspecification.

Findings

Discordant outer sets can lead to conflicting empirical results in refuted models.

Conditions are derived for the existence and non-existence of discordant submodels.

Union of identified sets from minimum relaxations is robust to misspecification.

Abstract

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.