Identification of Treatment Effects with Mismeasured Imperfect Instruments

In this article, I develop a novel identification result for estimating the effect of an endogenous treatment using a proxy of an unobserved imperfect instrument. I show that the potential outcomes distributions are partially identified for the compliers. Therefore, I derive sharp bounds on the local average treatment effect. I write the identified set in the form of conditional moments inequalities, which can be implemented using existing inferential methods. I illustrate my methodology on the National Longitudinal Survey of Youth 1979 to evaluate the returns to college attendance using tuition as a proxy of the true cost of going to college. I find that the average return to college attendance for people who attend college only because the cost is low is between 29% and 78%.