Don't (fully) exclude me, it's not necessary! Identification with semi-IVs

Abstract

This paper proposes a novel tool to nonparametrically identify models with a discrete endogenous variable or treatment: semi-instrumental variables (semi-IVs). A semi-IV is a variable that is relevant but only partially excluded from the potential outcomes, i.e., excluded from at least one, but not necessarily all, potential outcome equations. It follows that standard instrumental variables (IVs), which are fully excluded from all the potential outcomes, are a special (extreme) case of semi-IVs. I show that full exclusion is stronger than necessary because the same objects that are usually identified with an IV (Imbens and Angrist, 1994; Heckman and Vytlacil, 2005; Chernozhukov and Hansen, 2005) can be identified with several semi-IVs instead, provided there is (at least) one semi-IV excluded from each potential outcome. For applied work, tackling endogeneity with semi-IVs instead of IVs should be an attractive alternative, since semi-IVs are easier to find: most selection-specific costs or benefits can be valid semi-IVs, for example. The paper also provides a simple semi-IV GMM estimator for models with homogenous treatment effects and uses it to estimate the returns to education