This paper extends the identification results in Nevo and Rosen
(2012) to nonparametric models. We derive nonparametric bounds on the average
treatment effect when an imperfect instrument is available. As in Nevo and
Rosen (2012), we assume that the correlation between the imperfect instrument
and the unobserved latent variables has the same sign as the correlation
between the endogenous variable and the latent variables. We show that the
monotone 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 also
satisfies this assumption. Moreover, we show how the assumption that the
imperfect instrument is less endogenous than the treatment variable can help
tighten the bounds. We also use the monotone treatment response assumption to
get tighter bounds. The identified set can be written in the form of
intersection bounds, which is more conducive to inference. We illustrate our
methodology using the National Longitudinal Survey of Young Men data to
estimate returns to schooling.