Don't (Fully) Exclude Me, it's Not Necessary! Identification with Semi-IVs

Abstract

This paper proposes a novel approach to identify models with a discrete endogenous variable, that I study in the general context of nonseparable models with continuous potential outcomes. I show that nonparametric identification of the potential outcome and selection equations, and thus of the individual treatment effects, can be obtained with semi-instrumental variables (semi-IVs), which are relevant but only partially excluded from the potential outcomes, i.e., excluded from one or more potential outcome equations, but not necessarily all. This contrasts with the full exclusion restriction imposed on standard instrumental variables (IVs), which is stronger than necessary for identification: IVs are only a special case of valid semi-IVs. In practice, there is a trade-off between imposing stronger exclusion restrictions, and finding semi-IVs with a larger support and stronger relevance assumptions. Since, in empirical work, the main obstacle for finding a valid IV is often the full exclusion restriction, tackling the endogeneity problem with semi-IVs instead should be an attractive alternative.

Keywords: exclusion restriction, instrumental variable, identification, nonseparable models, individual treatment effects