We present a simple way to estimate the eﬀects of changes in a vector of observable variables X on a limited dependent variable Y when Y is a general nonseparable function of X and unobservables. We treat models in which Y is censored from above or below or potentially from both. The basic idea is to ﬁrst estimate the derivative of the conditional mean of Y given X at x with respect to x on the uncensored sample without correcting for the eﬀect of changes in x induced on the censored population. We then correct the derivative for the eﬀects of the selection bias. We propose nonparametric and semiparametric estimators for the derivative. As extensions, we discuss the cases of discrete regressors, measurement error in dependent variables, and endogenous regressors in a cross section and panel data context.
Altonji, Joseph G.; Ichimura, Hidehiko; and Otsu, Taisuke, "Estimating Derivatives in Nonseparable Models with Limited Dependent Variables" (2008). Cowles Foundation Discussion Papers. 1976.