Smoothness Adaptive Average Derivative Estimation

Many important models, such as index models widely used in limited dependent variables, partial linear models and nonparametric demand studies utilize estimation of average derivatives (sometimes weighted) of the conditional mean function. Asymptotic results in the literature focus on situations where the ADE converges at parametric rates (as a result of averaging); this requires making stringent assumptions on smoothness of the underlying density; in practice such assumptions may be violated. We extend the existing theory by relaxing smoothness assumptions. We consider both the possibility of lack of smoothness and lack of precise knowledge of degree of smoothness and propose an estimation strategy that produces the best possible rate without a priori knowledge of degree of density smoothness. The new combined estimator is a linear combination of estimators corresponding to different bandwidth/kernel choices that minimizes the trace of the part of the estimated asymptotic mean squared error that depends on the bandwidth. Estimation of the components of the AMSE, of the optimal bandwidths, selection of the set of bandwidths and kernels are discussed. Monte Carlo results for density weighted ADE confirm good performance of the combined estimator