Nonparametric Bounds on Treatment Effects with Imperfect Instruments

Abstract

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.