Estimating derivatives in nonseparable models with limited dependent variables

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.