Bootstrapping Density-Weighted Average Derivatives

We investigate the properties of several bootstrap-based inference procedures for semiparametric density-weighted average derivatives. The key innovation in this paper is to employ an alternative asymptotic framework to assess the properties of these inference procedures. This theoretical approach is conceptually distinct from the traditional approach (based on asymptotic linearity of the estimator and Edgeworth expansions), and leads to di¤erent theoretical prescriptions for bootstrapbased semiparametric inference. First, we show that the conventional bootstrap-based approximations to the distribution of the estimator and its classical studentized version are both invalid in general. This result shows a fundamental lack of robustness of the associated, classical bootstrap-based inference procedures with respect to the bandwidth choice. Second, we present a new bootstrap-based inference procedure for density-weighted average derivatives that is more robustto perturbations of the bandwidth choice, and hence exhibits demonstrable superior theoretical statistical properties over the traditional bootstrap-based inference procedures. Finally, we also examine the validity and invalidity of related bootstrap-based inference procedures, and discuss additional results that may be of independent interest. Some simulation evidence is also presented. Keywords: Averaged derivatives, Bootstrap, Small bandwidth asymptotics, Robust semiparametric inference. JEL: C12, C14, C21, C24. The authors thank Joel Horowitz, Guido Imbens, Lutz Kilian, Demian Pouzo, Rocio Titiunik, seminar participants at Columbia/NYU, Duke, Harvard, LSE, Michigan, Northwestern, Rochester and USC, and conference participants at the 2010 Econometric Society World Congress for comments. We also thank the co-Editor, Yoon-Jae Whang, and two anonymous referees for their suggestions. The rst author gratefully acknowledges nancial support from the National Science Foundation (SES 0921505 and SES 1122994). The third author gratefully acknowledges nancial support from the National Science Foundation (SES 0920953 and SES 1124174) and the research support of CREATES (funded by the Danish National Research Foundation). yDepartment of Economics, University of Michigan. zCapital Markets Function, Federal Reserve Bank of New York. xDepartment of Economics, UC Berkeley and CREATES. Bootstrapping Density-Weighted Average Derivatives 1 1. Introduction The bootstrap has gained great popularity in modern econometrics and statistics.1 In semiparametric problems, where estimators of a nite-dimensional parameter of interest involve a nonparametric estimator of an unknown function, the bootstrap is attractive because of its ability to approximate the distribution of the semiparametric estimator in cases where variance estimation is di¢ cult (e.g., Chen, Linton, and van Keilegom (2003) and Cheng and Huang (2010)). Even when variance estimation is relatively straightforward, the bootstrap is potentially useful in semiparametrics because it may provide more accurate approximations to the distributions of (asymptotically) pivotal quantities such as studentized estimators, whenever it achieves asymptotic re nements similar to those well-established in parametric problems (e.g., Hall (1992)). The kernel-based density-weighted average derivative estimator of Powell, Stock, and Stoker (1989) is one of the few semiparametric estimators for which the bootstrap has been shown to o¤er asymptotic re nements. Nishiyama and Robinson (2005) recently showed that a suitably implemented version of the nonparametric bootstrap provides a distributional approximation for the classical studentized test statistic that is superior to the standard Gaussian approximation. In this paper we revisit this problem, and obtain new results that can be viewed as a cautionary tale regarding the potential for bootstrap-based inference to (.) provide improvements in moderate-sized samples(Nishiyama and Robinson (2005, p. 927)). We present simulation evidence that appears hard to reconcile with the theoretical results establishing asymptotic re nements of the bootstrap in this semiparametric context, and develop an alternative theory-based explanation of this evidence. In addition, we use our theoretical framework to derive results for alternative bootstrap-based inference procedures and to show, among other things, that there exists a valid bootstrap-based inference procedure that dominates the one proposed by Nishiyama and Robinson (2005), a theory-based prediction also borne out in our simulations. The traditional approach to evaluating the accuracy of bootstrap-based inference procedures (in parametric and semiparametric problems) relies on asymptotic linearity of estimators and employs Edgeworth expansions to elucidate the role of higher-orderterms in the distributional approximation of the associated test statistics. For the density-weighted average derivative estimator, Nishiyama and Robinson (2005) used this traditional approach to demonstrate the ability of a bootstrap-based inference procedure to deliver asymptotic re nements. In contrast, we propose in this paper to employ an alternative ( rst-order) 1For reviews, see, e.g., Politis, Romano, and Wolf (1999) and Horowitz (2001). Bootstrapping Density-Weighted Average Derivatives 2 distributional approach to examine the properties of bootstrap-based inference procedures, which retains some terms that are asymptotically negligible when the estimator is asymptotically linear but can be rst-order otherwise. This alternative approach accommodates, but does not require, certain departures from asymptotic linearity, namely those that occur when the bandwidth of the nonparametric estimator vanishes too rapidly for asymptotic linearity to hold. Thus, we refer to this approach as a small bandwidthapproach (Cattaneo, Crump and Jansson (2010, 2012)). Although similar in spirit to the Edgeworth expansion approach to improve asymptotic approximations, our small bandwidth approach is conceptually distinct and leads to di¤erent theoretical prescriptions for bootstrap-based semiparametric inference. In particular, Theorem 1 nds that the conventional bootstrap-based approximations to the distribution of the kernel-based semiparametric estimator and the associated studentized version of this estimator employing the traditional (jackknife) variance estimator are both invalid in general. On the other hand, Theorem 2 establishes consistency of the bootstrap approximation to the distribution of the semiparametric estimator when studentized by a di¤erent, bias-corrected variance estimator. This alternative variance estimator is one for which the resulting studentized statistic is asymptotically standard normal even when asymptotic linearity fails. However, and perhaps surprisingly, Theorem 3 shows that pivotality of the studentized estimator is not su¢ cient for bootstrap validity: a variance estimator is exhibited which renders the associated studentized statistic asymptotically standard normal even when asymptotic linearity fails, but nonetheless the standard bootstrap provides a valid distributional approximation for this asymptotically pivotal statistic only when asymptotic linearity holds. These results have some interesting theoretical implications. First, our ndings shed new light on the properties of the bootstrap and some of its variants in the context of semiparametric inference, documenting and highlighting in particular a fragility of traditional bootstrap-based distributional approximations for kernel-based semiparametric statistics with respect to perturbations of the bandwidth choice. (See Section 4.4 for further discussion on this point.) Second, our results also include a new bootstrap-based inference procedure for density-weighted average derivatives which is more robustto perturbations of the bandwidth choice, and hence exhibiting theoretically demonstrable superior statistical properties over the traditional bootstrap-based inference procedures. The remainder of the paper is organized as follows. Section 2 introduces the model, summarizes some theoretical results available in the literature, and provides a motivation for our work using a small-scale simulation study. Section 3 reviews our alternative approach based Bootstrapping Density-Weighted Average Derivatives 3 on the small bandwidth framework and develops the main theoretical tools needed to study the bootstrap. Section 4 includes the main results of the paper, while Section 5 concludes and discusses other contexts where our results could be applied. The Appendix contains brief mathematical proofs, but the supplemental appendix includes a detailed development of our results. 2. Setup and Motivation We assume throughout that zi = (yi; xi) 0, i = 1; : : : ; n, is a random sample of z = (y; x0)0, where y 2 R is a dependent variable and x 2 R is a continuous explanatory variable with density f( ). The density-weighted average derivative of the regression function g(x) = E[yjx] is = E[f(x)@g(x)=@x]; see, e.g., Stoker (1986). (Detailed regularity conditions are given in the following section, but omitted here to ease the discussion.) Models where this estimand is of interest include single-index limited dependent variable models, generalized partially linear models, and other related semi-linear single-index generalized additive and non-additive models. For example, suppose g( ) is of the form g(x) = G(x1 ; x2) with G( ) unknown and x partitioned as x = (x1; x 0 2) 0. Then, partitioning conformably with x as = ( 1; 0 2) 0, the index parameter is proportional to 1 with proportionality factor E[f(x) _ G1(x1 ; x2)] where _ G1(u; x2) = @G(u; x2)=@u. Powell, Stock, and Stoker (1989, henceforth PSS) noted that = 2E [y@f(x)=@x], and hence proposed the kernel-based estimator