Identification of treatment effects on the treated with one-sided non-compliance

Traditional instrumental variable estimators do not generally estimate effects for the treated population but for the unobserved population of compliers. On the other hand, when there is one-sided non-compliance, they do identify effects for the treated because the populations of treated and compliers are identical in this case. 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.