Abstract: This paper proposes a novel identification strategy relying on quasi-instrumental variables (quasi-IVs). A quasi-IV is a relevant but possibly invalid IV because it is not completely exogenous and/or excluded. We show that a variety of models with discrete or continuous endogenous treatment, which are usually identified with an IV - quantile models with rank invariance additive models with homogenous treatment effects, and local average treatment effect models - can be identified under the joint relevance of two complementary quasi-IVs instead. To achieve identification we complement one excluded but possibly endogenous quasi-IV (e.g., "relevant proxies'' such as previous treatment choice) with one exogenous (conditional on the excluded quasi-IV) but possibly included quasi-IV (e.g., random assignment or exogenous market shocks). In practice, our identification strategy should be attractive since complementary quasi-IVs should be easier to find than standard IVs. Our approach also holds if any of the two quasi-IVs turns out to be a valid IV.
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.