Identifying treatment effects in the presence of confounded types

In this paper, I consider identification of treatment effects when the treatment is endogenous. The use of instrumental variables is a popular solution to deal with endogeneity, but this may give misleading answers when the instrument is invalid. I show that when an (unobserved) instrument is invalid due to correlation with the first stage unobserved heterogeneity, a proxy for the instrument helps partially identify not only the local average treatment effect, but also the entire potential outcomes distributions for compliers. I exploit the fact that the distribution of the observed outcome in each group defined by the treatment and the instrument is a mixture of the distributions of interest. I write the identified set in the form of conditional moment inequalities, and provide an easily implementable inference procedure. Under some tail restrictions, the potential outcomes distributions are point-identified for compliers. Finally, I illustrate my methodology on data from the National Longitudinal Survey of Young Men to estimate returns to college using college proximity as a proxy for the instrument low college cost. I find that a college degree increases the average hourly wage of the compliers by 15%–30%.