Generalized Jackknife Estimators of Weighted Average Derivatives

Abstract

With the aim of improving the quality of asymptotic distributional approximations for nonlinear functionals of nonparametric estimators, this article revisits the large-sample properties of an important member of that class, namely a kernel-based weighted average derivative estimator. Asymptotic linearity of the estimator is established under weak conditions. Indeed, we show that the bandwidth conditions employed are necessary in some cases. A bias-corrected version of the estimator is proposed and shown to be asymptotically linear under yet weaker bandwidth conditions. Implementational details of the estimators are discussed, including bandwidth selection procedures. Consistency of an analog estimator of the asymptotic variance is also established. Numerical results from a simulation study and an empirical illustration are reported. To establish the results, a novel result on uniform convergence rates for kernel estimators is obtained. The online supplemental material to this article includes details on the theoretical proofs and other analytic derivations, and further results from the simulation study.

KEY WORDS:

• Bias correction
• Semiparametric estimation
• Uniform consistency

Acknowledgments

For comments and suggestions, we thank Enno Mammen, Whitney Newey, Jim Powell, Rocio Titiunik, seminar participants at Brown, CEMFI/Universidad Carlos III de Madrid, Harvard/MIT, Mannheim, Michigan, NYU, Toulouse School of Economics, UC-Davis, UCSD, UIUC, UPenn, and Yale, and conference participants at the 2010 World Congress of the Econometric Society, 2011 WEAI Conference, 2012 Winter Meetings of the Econometric Society, 2012 Tsinghua International Conference in Econometrics, and 2012 SETA conference. We also thank the Associate Editor and a referee for helpful recommendations. The first author gratefully acknowledges financial support from the National Science Foundation (SES 0921505 and SES 1122994). The third author gratefully acknowledges financial support from the National Science Foundation (SES 0920953 and SES 1124174) and the research support of CREATES (funded by the Danish National Research Foundation).