Nonparametric Bounds on Treatment Effects with Imperfect Instruments

This paper extends the identification results in Nevo and Rosen(2012) to nonparametric models. We derive nonparametric bounds on the averagetreatment effect when an imperfect instrument is available. As in Nevo andRosen (2012), we assume that the correlation between the imperfect instrumentand the unobserved latent variables has the same sign as the correlationbetween the endogenous variable and the latent variables. We show that themonotone treatment selection and monotone instrumental variable restrictions,introduced by Manski and Pepper (2000, 2009), jointly imply this assumption.We introduce the concept of comonotone instrumental variable, which alsosatisfies this assumption. Moreover, we show how the assumption that theimperfect instrument is less endogenous than the treatment variable can helptighten the bounds. We also use the monotone treatment response assumption toget tighter bounds. The identified set can be written in the form ofintersection bounds, which is more conducive to inference. We illustrate ourmethodology using the National Longitudinal Survey of Young Men data toestimate returns to schooling.