Small Bandwidth Asymptotics for Density-Weighted Average Derivatives  Matias D. Cattaneo Department of Economics, UC Berkeley Richard K. Crump Department of Economics, UC Berkeley Michael Jansson Department of Economics, UC Berkeley and CREATES May 8, 2008 Abstract. This paper proposes (apparently) novel standard error for- mulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels and the standard errors are robust in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the nite sample coverage rates of condence intervals constructed using the stan- dard errors developed in this paper coincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths. 1. Introduction Semiparametric estimators employing nonparametric kernel estimators of unknown nuisance functions have been proposed for a variety of microeconometric estimands. Under suitable, application specic, regularity conditions many such estimators en- joy the properties of p n-consistency (where n is the sample size) and asymptotic normality, the variance of the limiting distribution being consistently estimable and invariant with respect to the kernel and bandwidth of the nonparametric estimator. Achieving these properties often requires a delicate choice of the kernel and band- width of the nonparametric estimator. A prime example, and the one we focus on in this paper, is provided by the density-weighted average derivative estimator of Pow- ell, Stock, and Stoker (1989, henceforth PSS). The validity of inference procedures based on this estimator and the standard errors proposed by PSS requires that the  The authors thank Bryan Graham, Jim Powell, Tom Rothenberg, Paul Ruud, and seminar participants at Cornell, Harvard, and Penn State for comments. We thank Jasjeet Sekhon and Rocio Titiunik for providing access to the Calgrid cluster. The third author gratefully acknowledges the research support of CREATES (funded by the Danish National Research Foundation). 1