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Efficiency of Weighted Average Derivative Estimators and Index Models

Whitney K. Newey
Thomas M. Stoker

Abstract

Weighted average derivatives are useful parameters for semiparametric index models and nonparametric demand analysis. This paper gives efficiency results for average derivative estimators, including formulating estimators that have high efficiency. Our analysis is carried out in three steps. First, we derive the efficiency bound for weighted average derivatives of conditional location functionals, such as the conditional mean and median. Second, we derive the efficiency bound for semiparametric index models, where the location measure depends only on indices, or linear combinations of the regressors. Third, we compare the bound for index models with the asymptotic variance of weighted average derivative estimators of the index coefficients. We characterize the form of the optimal weight function when the distribution of the regressors is elliptically symmetric. In more general cases, we discuss how to combine estimators with different weight functions to achieve efficiency. We derive a general condition for approximate efficiency of pooled (minimum chi square) estimators for index model coefficients, based on weighted average derivatives. Finally, we discuss ways of selecting the type and number of weighting functions to achieve high efficiency.