Asymptotically efficient estimation of weighted average derivatives with an interval censored variable

This paper studies the identi fication and estimation of weighted average derivatives of conditional location functionals including conditional mean and conditional quantiles in settings where either the outcome variable or a regressor is interval-valued. Building on Manski and Tamer (2002) who study nonparametric bounds for mean regression with interval data, we characterize the identifi ed set of weighted average derivatives of regression functions. Since the weighted average derivatives do not rely on parametric speci fications for the regression functions, the identi fied set is well-defi ned without any parametric assumptions. Under general conditions, the identifi ed set is compact and convex and hence admits characterization by its support function. Using this characterization, we derive the semiparametric efficiency bound of the support function when the outcome variable is interval-valued. We illustrate efficient estimation by constructing an efficient estimator of the support function for the case of mean regression with an interval censored outcome.