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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, and X is independent of the unobservables. We treat models in which Y is censored from above, below, or 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 x on the censored population. We then correct the derivative for the eﬀects of the selection bias. We discuss nonparametric and semiparametric estimators for the derivative. We also discuss the cases of discrete regressors and of endogenous regressors in both cross section and panel data contexts.
Altonji, Joseph G.; Ichimura, Hidehiko; and Otsu, Taisuke, "Estimating Derivatives in Nonseparable Models with Limited Dependent Variables" (2008). Cowles Foundation Discussion Papers. 1977.