Traditional instrumental variable estimators do not generally estimate effects for the treated population but for the unobserved population of compliers. They do identify effects for the treated when there is one-sided perfect non-compliance. However, this property is lost when covariates are included in the model. In this case, we show that the effects for the treated are still identified but require modified estimators. We consider both average and quantile treatment effects and allow the instrument to be discrete or continuous.
Treatment effects instrumental variables non-compliance missing data