We examine instrumental variables estimation in situations where the instrument is only observed for a sub-sample, which is fairly common in empirical research. Typically, researchers simply limit the analysis to the sub-sample where the instrument is non-missing. We show that when the...

Many empirical studies specify outcomes as a linear function of endogenous regressors when conducting instrumental variable (IV) estimation. We show that tests for treatment effects, selection bias, and treatment effect heterogeneity are biased if the true relationship is non-linear. These...