Nonparametric censored and truncated regression

WE PROPOSE NEW ESTIMATORS of the latent regression function in nonparametric cen- sored and truncated regression models. Our estimators are computationally convenient, consisting only of two nonparametric regressions and a univariate integral. We establish consistency and asymptotic normality for an implementation based on local linear ker- nel estimators. An extension permits estimation in the presence of a general form of heteroscedasticity. Both the function m(.) and the distribution F(.) of the error e are unknown. The errors are not assumed to be symmetric. This paper provides a simple consistent estimator of m(x) + k for some constant k. Under an additional tail condition, we provide a consistent estimator of m(x) when m is the conditional mean function of the uncensored population. Also, we show that F(.) can be estimated given m(x). The proposed estimator is extended to deal with the truncated regression model, where Yi is only observed when it is not censored. We also describe extensions to deal with a general form of heteroskedasticity, in which the distribution of e could depend in unknown ways on a subset of elements of x. For any continuously distributed element Xk of x, let mk(x) = am(x)/axk. We also provide direct estimators of the derivatives mk(x) in both the censored and truncated regression models. These derivatives are interpretable as the marginal effect of a change in x on the underlying uncensored population. They can also be used to test or estimate parametric or semiparametric specifications of m(x). For example, mk(x) is constant if m(x) is linear in Xk, and mk(x) depends only on Xk if m(x) is additive in a function of Xk.