Federal Reserve Bank of New York Staff Reports

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1 Federal Reserve Bank of New York Staff Reports Bootstrapping Density-Weighted Average Derivatives Matias D. Cattaneo Richard K. Crup Michael Jansson Staff Report no. 45 May 00 This paper presents preliinary findings and is being distributed to econoists and other interested readers solely to stiulate discussion and elicit coents. The views expressed in this paper are those of the authors and are not necessarily reflective of views the Federal Reserve Bank of New York or the Federal Reserve Syste. Any errors or oissions are the responsibility of the authors.

2 Bootstrapping Density-Weighted Average Derivatives Matias D. Cattaneo, Richard K. Crup, and Michael Jansson Federal Reserve Bank of New York Staff Reports, no. 45 May 00 JEL classification: C, C4, C, C4 Abstract Eploying the sall-bandwidth asyptotic fraework of Cattaneo, Crup, and Jansson (009), this paper studies the properties of several bootstrap-based inference procedures associated with a kernel-based estiator of density-weighted average derivatives proposed by Powell, Stock, and Stoker (989). In any cases, the validity of bootstrap-based inference procedures is found to depend crucially on whether the bandwidth sequence satisfies a particular (asyptotic linearity) condition. An exception to this rule occurs for inference procedures involving a studentized estiator that eploys a robust variance estiator derived fro the sall-bandwidth asyptotic fraework. The results of a sall-scale Monte Carlo experient are found to be consistent with the theory and indicate in particular that sensitivity with respect to the bandwidth choice can be aeliorated by using the robust variance estiator. Key words: averaged derivatives, bootstrap, sall-bandwidth asyptotics Cattaneo: University of Michigan (e-ail: cattaneo@uich.edu). Crup: Federal Reserve Bank of New York (e-ail: richard.crup@ny.frb.org). Jansson: University of California at Berkeley and Center for Research in Econoetric Analysis of Tie Series (e-ail: jansson@econ.berkeley.edu). The authors are grateful for coents fro Joel Horowitz, Lutz Kilian, Deian Pouzo, Rocio Titiunik, and seinar participants at Duke University, Harvard University, the University of Michigan, Northwestern University, and the University of Rochester. Cattaneo thanks the National Science Foundation for financial support (SES 09505). Jansson gratefully acknowledges financial support fro the National Science Foundation (SES ) and research support fro the Center for Research in Econoetric Analysis of Tie Series, funded by the Danish National Research Foundation. The views expressed in this paper are those of the authors and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve Syste.

3 Bootstrapping Density-Weighted Average Derivatives. Introduction Seiparaetric estiators involving functionals of nonparaetric estiators have been studied widely in econoetrics. In particular, considerable e ort has been devoted to characterizing conditions under which such estiators are asyptotically linear (see, e.g., Newey and McFadden (994), Chen (007), and the references therein). Moreover, although the asyptotic variance of an asyptotically linear seiparaetric estiator can in principle be obtained by eans of the pathwise derivative forula of Newey (994a), it is desirable fro a practical point of view to be able to base inference procedures on easures of dispersion that are autoatic in the sense that they can be constructed without knowledge (or derivation) of the in uence function (e.g., Newey (994b)). Perhaps the ost natural candidates for such easures of dispersion are variances and/or percentiles obtained using the bootstrap. Consistency of the nonparaetric bootstrap has been established for a large class of seiparaetric estiators by Chen, Linton, and van Keilego (003). Moreover, in the iportant special case of the density-weighted average derivative estiator of Powell, Stock, and Stoker (989, henceforth PSS), a suitably ipleented version of the nonparaetric bootstrap was shown by Nishiyaa and Robinson (005, henceforth NR) to provide asyptotic re neents. The analysis in NR is conducted within the asyptotic fraework of Nishiyaa and Robinson (000, 00). Using the alternative asyptotic fraework of Cattaneo, Crup, and Jansson (009, henceforth CCJ), this paper revisits the large saple behavior of bootstrap-based inference procedures for density-weighted average derivatives and obtains (analytical and Monte Carlo) results that could be interpreted as a cautionary tale regarding the ease with which one ight realize the potential for bootstrap-based inference to (...) provide iproveents in oderate-sized saples (NR, p. 97). Because the in uence function of an asyptotically linear seiparaetric estiator is invariant with respect to the nonparaetric estiator upon which it is based (e.g., Newey (994a, Proposition )), looking beyond the in uence function is iportant if the sensitivity of the distributional properties of an estiator or test statistic with respect to user chosen objects such as kernels or bandwidths is a concern. This can be accoplished in various ways, the traditional approach being to work under assuptions that iply asyptotic linearity and then develop asyptotic expansions (of the Edgeworth or Nagar variety) intended to Another autoatic easure of dispersion is the variance estiator of Newey (994b). When applied to the density-weighted average derivative estiator studied in this paper, the variance estiator of Newey (994b) coincides with Powell, Stock, and Stoker s (989) variance estiator whose salient properties are characterized in Lea below.

4 Bootstrapping Density-Weighted Average Derivatives elucidate the role of higher-order ters (e.g., Linton (995)). Siilarly to the Edgeworth expansions eployed by Nishiyaa and Robinson (000, 00, 005), CCJ s asyptotic distribution theory for PSS s estiator (and its studentized version) is obtained by retaining ters that are asyptotically negligible when the estiator is asyptotically linear. Unlike the traditional approach, the sall bandwidth approach taken by CCJ accoodates, but does not require, certain departures fro asyptotic linearity, naely those that occur when the bandwidth of the nonparaetric estiator vanishes too rapidly for asyptotic linearity to hold. Although siilar in spirit to the Edgeworth expansion approach to iproved asyptotic approxiations, the sall bandwidth approach of CCJ is conceptually distinct fro the approach taken by Nishiyaa and Robinson (000, 00, 005) and it is therefore of interest to explore whether the sall bandwidth approach gives rise to ethodological prescriptions that di er fro those obtained using the traditional approach. The rst ain result, Theore below, studies the validity of bootstrap-based approxiations to the distribution of PSS s estiator as well as its studentized version in the case where PSS s variance estiator is used for studentization purposes. It is shown that a necessary condition for bootstrap consistency is that the bandwidth vanishes slowly enough for asyptotic linearity to hold. Unlike NR, Theore therefore suggests that in saples of oderate size even the bootstrap approxiations to the distributions of PSS s estiator and test statistic(s) ay fail to adequately capture the extent to which these distributions are a ected by the choice of the bandwidth, a prediction which is borne out in a sall scale Monte Carlo experient reported in Section 4. The second ain result, Theore, establishes consistency of the bootstrap approxiation to the distribution of PSS s estiator studentized by eans of a variance estiator proposed by CCJ. As a consequence, Theore suggests that the fragility with respect to bandwidth choice uncovered by Theore is a property which should be attributed to PSS s variance estiator rather than the bootstrap distribution estiator. Another prediction of Theore, naely that the bootstrap approxiation to the distribution of an appropriately studentized estiator perfors well across a wide range of bandwidths, is borne out in the Monte Carlo experient of Section 4. Indeed, the range of bandwidths across which the bootstrap is found to perfor well is wider than the range across which the standard noral approxiation is found to perfor well, indicating that there is an iportant sense in which bootstrap-based inference is capable of providing iproveents in oderate-sized saples. The variance estiator used for studentization purposes in Theore is one for which the studentized estiator is asyptotically standard noral across the entire range of bandwidth

5 Bootstrapping Density-Weighted Average Derivatives 3 sequences considered in CCJ s approach. The nal ain result, Theore 3, studies the bootstrap approxiation to the distribution of PSS s estiator studentized by eans of an alternative variance estiator also proposed by CCJ and nds, perhaps surprisingly, that although the associated studentized estiator is asyptotically standard noral across the entire range of bandwidth sequences considered in CCJ s approach, consistency of the bootstrap requires that the bandwidth vanishes slowly enough for asyptotic linearity to hold. In addition to NR, whose relation to the present work was discussed in soe detail above, the list of papers related to this paper includes Abadie and Ibens (008) and Gonçalves and Vogelsang (00). Abadie and Ibens (008) study a nearest-neighbor atching estiator of a popular estiand in the progra evaluation literature (the e ect of treatent on the treated) and deonstrate by exaple that the nonparaetric bootstrap variance estiator can be inconsistent in that case. Although the nature of the nonparaetric estiator eployed by Abadie and Ibens (008) di ers fro the kernel estiator studied herein, their inconsistency result would appear to be siilar to the equivalence between (i) and (ii) in Theore (a) below. Coparing the results of this paper with those obtained by Abadie and Ibens (008), one apparent attraction of kernel estiators (relative to nearest-neighbor estiators) is their tractability which allows to develop fairly detailed characterizations of the large-saple behavior of bootstrap procedures, including an array of (constructive) results on how to achieve bootstrap consistency even under departures fro asyptotic linearity. Gonçalves and Vogelsang (00) are concerned with autocorrelation robust inference in stationary regression odels and establish consistency of the bootstrap under the xed-b asyptotics of Kiefer and Vogelsang (005). Although the xed-b approach of Kiefer and Vogelsang (005) is very siilar in spirit to the sall bandwidth approach of CCJ, the fact that soe of the results of this paper are invalidity results about the bootstrap is indicative of an iportant di erence between the nature of the functionals being studied in Kiefer and Vogelsang (005) and CCJ, respectively. The reainder of the paper is organized as follows. Section introduces the odel, presents the statistics under consideration, and suarizes soe results available in the literature. Section 3 studies the bootstrap and obtains the ain results of the paper. Section 4 suarizes the results of a siulation study. Section 5 concludes. The Appendix contains proofs of the theoretical results.

6 Bootstrapping Density-Weighted Average Derivatives 4. Model and Existing Results Let Z n = fz i = (y i ; x 0 i) 0 : i = ; : : : ; ng be a rando saple of the rando vector z = (y; x 0 ) 0, where y R is a dependent variable and x R d is a continuous explanatory variable with a density f (). The density-weighted average derivative is given by = E g (x), g (x) = E [yjx]. It follows fro (regularity conditions and) integration by parts that = Noting this, PSS proposed the kernel-based estiator ^n = n nx i= ^f n;i (x i ), ^fn;i (x) = n nx K h d j=;j6=i n xj E where ^f n;i () is a leave-one-out estiator of f (), with K : R d! R a kernel function and h n a positive (bandwidth) sequence. To analyze inference procedures based on ^ n, soe assuptions on the distribution of z and the properties of the user-chosen ingredients K and h n are needed. Regarding the odel and kernel function, the following assuptions will be ade. Assuption M. (a) E[y 4 ] <, E [ (x) f (x)] > 0 and V [@e (x) =@x positive de nite, where (x) = V [yjx] and e (x) = f (x) g (x). h n x, y@f (x) =@x] is (b) f is (Q + ) ties di erentiable, and f and its rst (Q + ) derivatives are bounded, for soe Q. (c) g is twice di erentiable, and e and its rst two derivatives are bounded. (d) v is di erentiable, and vf and its rst derivative are bounded, where v (x) = E[y jx]. (e) li kxk! [f (x) + je (x)j] = 0, where kk is the Euclidean nor. Assuption K. (a) K is even and di erentiable, and K and its rst derivative are bounded. (b) R K _ (u) K _ (u) 0 du is positive de nite, where K _ (u) (u) =@u. R d (c) For soe P, R R d jk (u)j ( + kuk P )du + R R d k _ K (u) k( + kuk )du <, and Z R d u l u l d d K (u) du = ( ; if l = = l d = 0; 0; if (l ; : : : ; l d ) 0 Z d + and l + + l d < P. The following conditions on the bandwidth sequence h n will play a crucial role in the sequel. (Here, and elsewhere in the paper, liits are taken as n! unless otherwise noted.)

7 Bootstrapping Density-Weighted Average Derivatives 5 Condition B. (Bias) in nh d+ n ; nh n in(p;q)! 0. Condition AL. (Asyptotic Linearity) nh d+ n!. Condition AN. (Asyptotic Norality) n h d n!. PSS studied the large saple properties of ^ n and showed that if Assuptions M and K hold and if Conditions B and AL are satis ed, then ^ n is asyptotically linear with (e cient) in uence function L (z) = [@e ]; that is, where p n(^n ) = p n nx L (z i ) + o p () N (0; ), = E L (z) L (z) 0, () i= denotes weak convergence. PSS s derivation of this result exploits the fact that the estiator ^ n adits the (n-varying) U-statistic representation ^ n = ^ n (h n ) with ^n (h) = n Xn nx i= j=i+ U (z i ; z j ; h), U (z i ; z j ; h) = h (d+) _K xi h x j (y i y j ), which leads to the Hoe ding decoposition ^ n = B n + L n + W n, where with nx B n = (h n ), Ln = n L (z i ; h n ), Wn = i= n Xn nx i= j=i+ (h) = E [U (z i ; z j ; h)], L (z i ; h) = [E[U (z i ; z j ; h) jz i ] (h)], W (z i ; z j ; h) = U (z i ; z j ; h) (L (z i; h) + L (z j ; h)) (h). W (z i ; z j ; h n ), The purpose of Conditions B and AL is to ensure that the ters B n and W n in the Hoe ding decoposition are asyptotically negligible. Speci cally, because B n = O(h in(p;q) n ) under Assuptions M and K, Condition B ensures that the bias of ^ n is asyptotically negligible. Condition AL, on the other hand, ensures that the quadratic ter W n in the Hoe ding decoposition is asyptotically negligible because p nw n = O p (= p nh d+ n ) under Assuptions M and K. In other words, and as the notation suggests, Condition AL is crucial for asyptotic linearity of ^ n. While asyptotic linearity is a desirable feature fro the point of view of asyptotic e - ciency, a potential concern about distributional approxiations for ^ n based on assuptions

8 Bootstrapping Density-Weighted Average Derivatives 6 which iply asyptotic linearity is that such approxiations ignore the variability in the reainder ter W n. Thus, classical rst-order, asyptotically linear, large saple theory ay not accurately capture the nite saple behavior of ^ n in general. It therefore sees desirable to eploy inference procedures that are robust in the sense that they reain asyptotically valid at least under certain departures fro asyptotic linearity. In an attept to construct such inference procedures, CCJ generalized () and showed that if Assuptions M and K hold and if Conditions B and AN are satis ed, then where V n = n + V = n (^ n ) N (0; I d ), () n h n (d+), = E (x) f (x) Z _K (u) K _ (u) 0 du. R d Siilarly to the asyptotic linearity result of PSS, the derivation of () is based on the Hoe ding decoposition of ^ n. Instead of requiring asyptotic linearity of the estiator, this result provides an alternative rst-order asyptotic theory under weaker assuptions, which siultaneously accounts for both the linear and quadratic ters in the expansion of ^ n. A key di erence between () and () is the presence of the ter n h (d+) n in V n, which captures the variability of Wn. In particular, result () shows that while failure of Condition AL leads to a failure of asyptotic linearity, asyptotic norality of ^ n holds under the signi cantly weaker Condition AN. The result () suggests that asyptotic standard norality of studentized estiators ight be achievable also when Condition AL is replaced by Condition AN. As an estiator of the variance of ^ n, PSS considered ^V 0;n = n ^n, where ^ n = ^ n (h n ), ^ n (h) = n " nx ^L n;i (h)^l n;i (h) 0, ^Ln;i (h) = n i= nx j=;j6=i CCJ showed that this estiator adits the stochastic expansion U(z i ; z j ; h) n ^V 0;n = n [ + o p ()] + h n (d+) [ + o p ()], ^n (h) Condition AN perits failure not only of asyptotic linearity, but also of p n-consistency (when nh d+ 0). Indeed, ^ n can be inconsistent (when li n! n h d+ n < ) under Condition AN. #. n!

9 Bootstrapping Density-Weighted Average Derivatives 7 iplying in particular that it is consistent only when Condition AL is satis ed. Recognizing this lack of robustness of ^V 0;n with respect to h n, CCJ proposed and studied the two alternative estiators ^V ;n = ^V n 0;n h n (d+) ^ n (h n ) and ^V;n = n ^n ( =(d+) h n ), where ^ n (h) = h d+ n ^W n;ij (h) = U(z i ; z j ; h) n X nx i= j=i+ ^W n;ij (h) ^W n;ij (h) 0, ^Ln;i (h) + ^L n;j (h) ^n (h). The following result is adapted fro CCJ and forulated in a anner that facilitates coparison with the ain theores given below. Lea. Suppose Assuptions M and K hold and suppose Conditions B and AN are satis ed. (a) The following are equivalent: i. Condition AL is satis ed. ii. Vn ^V 0;n! p I d. iii. ^V = 0;n (^ n ) N (0; I d ). (b) If nh d+ n is convergent in R + = [0; ], then ^V = 0;n (^ n ) N (0; 0 ), where 0 = li n! (nh d+ n + 4) = (nh d+ n + )(nh d+ n + 4) =. (c) For k f; g, Vn ^V k;n! p I d and ^V = k;n (^ n ) N (0; I d ). Part (a) is a qualitative result highlighting the crucial role played by Condition AL in connection with asyptotic validity of inference procedures based on ^V 0;n : The equivalence between (i) and (iii) shows that Condition AL is necessary and su cient for the test statistic ^V = 0;n (^ n ) proposed by PSS to be asyptotically pivotal. In turn, this equivalence is a special case of part (b), which is a quantitative result that can furtherore be used to characterize the consequences of relaxing Condition AL. Speci cally, part (b) shows that

10 Bootstrapping Density-Weighted Average Derivatives 8 = also under departures fro Condition AL the statistic ^V 0;n (^ n ) can be asyptotically noral with ean zero, but with a variance atrix 0 whose value depends on the liiting value of nh d+ n. This atrix satis es I d = 0 I d (in a positive seide nite sense), and takes on the liiting values I d = and I d when li n! nh d+ n equals 0 and, respectively. By iplication, part (b) suggests that inference procedures based on the test statistic proposed by PSS will be conservative across a nontrivial range of bandwidths. In contrast, part (c) shows that studentization by eans of ^V ;n and ^V ;n achieves asyptotic pivotality across the full range of bandwidth sequences allowed by Condition AN, suggesting in particular that coverage probabilities of con dence intervals constructed using these variance estiators will be close to their noinal level across a nontrivial range of bandwidths. Monte Carlo evidence consistent with these conjectures was presented by CCJ. Notably absent fro consideration in Lea and the Monte Carlo work of CCJ are inference procedures based on resapling. In an iportant contribution, NR studied the behavior of the standard (nonparaetric) bootstrap approxiation to the distribution of PSS s test statistic and found that under bandwidth conditions slightly stronger than Condition AL bootstrap procedures are not erely valid, but actually capable of achieving asyptotic re neents. This nding leaves open the possibility that bootstrap validity, at least to rstorder, ight hold also under departures fro Condition AL. The rst ain result presented here (Theore below) shows that, although the bootstrap approxiation to the distribution of ^V 0;n (^ n ) is ore accurate than the standard noral approxiation across the = full range of bandwidth sequences allowed by Condition AN, Condition AL is necessary and su cient for rst-order validity of the standard nonparaetric bootstrap approxiation to the distribution of PSS s test statistic. This equivalence can be viewed as a bootstrap analog of Lea (a) and it therefore sees natural to ask whether bootstrap analogs of Lea (c) are available for the inference procedures proposed by CCJ. Theore establishes a partial bootstrap analog of Lea (c), naely validity of the nonparaetric bootstrap approxiation to the distribution of ^V = ;n (^ n ) across the full range of bandwidth sequences allowed by Condition AN. That this result is not erely a consequence of the asyptotic pivotality result reported in Lea (c) is deonstrated by Theore 3, which shows that notwithstanding the asyptotic pivotality = of ^V ;n (^ n ); the nonparaetric bootstrap approxiation to the distribution of the latter statistic is valid only when Condition AL holds.

11 Bootstrapping Density-Weighted Average Derivatives 9 3. The Bootstrap 3.. Setup. This paper studies two variants of the -out-of-n replaceent bootstrap with = (n)!, naely the standard nonparaetric bootstrap ((n) = n) and (replaceent) subsapling ((n)=n! 0). 3 To describe the bootstrap procedure(s), let Z n = fzi = (yi ; x 0 i ) 0 : i = ; : : : ; (n)g be a rando saple with replaceent fro the observed saple Z n. The bootstrap analogue of the estiator ^ n is given by ^ (n) = ^ (n)(h (n) ) with ^ (h) = X X i= j=i+ x U(zi ; zj ; h), U(zi ; zj ; h) = h (d+) i _K h x j (y i y j ), while the bootstrap analogues of the estiators ^ n and ^ n are ^ (n) = ^ (n) (h (n)) and ^ (n) = ^ (n) (h (n)), respectively, where ^ (h) = " X ^L ;i(h)^l ;i(h) 0, ^L ;i (h) = i= X j=;j6=i U(z i ; z j ; h) ^ (h) #, and ^ (h) = X h d+ ^W ;ij(h) = U(z i ; z j ; h) X i= j=i+ ^W ;ij(h) ^W ;ij(h) 0, ^L ;i(h) + ^L ;j(h) ^ (h). 3.. Preliinary Lea. The ain results of this paper follow fro Lea and the following lea, which will be used to characterize the large saple properties of bootstrap = analogues of the test statistics ^V k;n (^ n ), k f0; ; g. Let superscript on P, E, or V denote a probability or oent coputed under the bootstrap distribution conditional on Z n, and let p denote weak convergence in probability (e.g., Gine and Zinn (990)). Lea. Suppose Assuptions M and K hold, suppose h n! 0 and Condition AN is satis ed, and suppose (n)! and li n! (n)=n <. (a) V (n) V [^ (n)]! p I d, where V = + + n h (d+). 3 This paper eploys the terinology introduced in Horowitz (00). See also Politis, Roano, and Wolf (999).

12 Bootstrapping Density-Weighted Average Derivatives 0 (b) (n) ^ (n)! p I d and ^ (n)! p I d, where = + + n h (d+). (c) V = (n) (^ (n) (n)) p N (0; I d ). The (conditional on Z n ) Hoe ding decoposition gives ^ = L + W, where with where and X = (h ), L = L (zi ; h ), W = i= X X i= j=i+ W (z i ; z j ; h ), (h) = E [U(z i ; z j ; h)], L (z i ; h) = [E [U(z i ; z j ; h)jz i ] (h)], W (z i ; z j ; h) = U(z i ; z j ; h) Part (a) of Lea is obtained by noting that V [^ ] = V [L (z i ; h )] + V [L (z i ; h)] ^ n (h) + n L (z i ; h) + L (z j ; h) (h). V [W (zi ; zj ; h )], h (d+), V [W (zi ; zj ; h)] h (d+) ^ n (h) h (d+). The fact that V [W (zi ; zj ; h)] h (d+) iplies that the bootstrap consistently estiates the variability of the quadratic ter in the Hoe ding decoposition. On the other hand, the fact that V [^ n] > n V [L (z i ; h n )] n ^n (h n ) = ^V 0;n iplies that the bootstrap variance estiator exhibits an upward bias even greater than that of ^V 0;n, so the bootstrap variance estiator is inconsistent whenever PSS s estiator is. In their exaple of bootstrap failure for a nearest-neighbor atching estiator, Abadie and Ibens (008) found that the (average) bootstrap variance can overestiate as well as underestiate the asyptotic variance of interest. No such abiguity occurs here, as Lea (a) shows that in the present case the bootstrap variance systeatically exceeds the asyptotic variance (when Condition AL fails).

13 Bootstrapping Density-Weighted Average Derivatives The proof of Lea (b) shows that ^ ^ n (h ) + h (d+) ^ n (h ), iplying that the asyptotic behavior of ^ di ers fro that of ^ n (h ) whenever Condition AL fails. By continuity of the d-variate standard noral cdf d () and Polya s theore for weak convergence in probability (e.g., Xiong and Li (008, Theore 3.5)), Lea (c) is equivalent to the stateent that h sup P tr d V = (n) (^ (n) i (n)) t d (t)! p 0. (3) By arguing along subsequences, it can be shown that a su cient condition for (3) is given by the following (unifor) Craér-Wold-type condition: sup sup d tr d P (^ (n) (n)) t5 (t) q 0 V (n)! p 0, (4) where d = f R d : 0 = g denotes the unit sphere in R d. 4 uses the theore of Heyde and Brown (970) to verify (4). The proof of Lea (c) 3.3. Bootstrapping PSS. Theore below is concerned with the ability of the bootstrap to approxiate the distributional properties of PSS s test statistic. To anticipate the ain ndings, notice that Lea gives V[^ n ] n + n h n (d+) and ^V0;n = n ^n n + n h (d+) n, 4 In contrast to the case of unconditional joint weak convergence, it would appear to be an open question whether a pointwise Craér-Wold condition such as 3 sup 4 0 (^ (n) (n)) t5 (t) tr q d 0 V (n)! p 0, 8 d, P iplies weak convergence in probability of V = (n) (^ (n) (n)).

14 Bootstrapping Density-Weighted Average Derivatives while, in contrast, in the case of the standard nonparaetric bootstrap (when (n) = n) Lea gives V [^ n n] n + 3 h (d+) n and ^V 0;n = n ^ n n + 4 n h (d+) n, strongly indicating that Condition AL is crucial for consistency of the bootstrap. On the other hand, in the case of subsapling (when (n) =n! 0), Lea gives V [^ ] + h (d+) and ^V 0; = ^ + h (d+), suggesting that consistency of subsapling ight hold even if Condition AL fails, at least ^V = in those cases where 0;n (^ n ) converges in distribution. (By Lea (b), convergence = in distribution of ^V 0;n (^ n ) occurs when nh d+ n is convergent in R +.) The following result, which is an iediate consequence of Leas and the continuous apping theore for weak convergence in probability (e.g., Xiong and Li (008, Theore 3.)), akes the preceding heuristics precise. Theore. Suppose the assuptions of Lea hold. (a) The following are equivalent: i. Condition AL is satis ed. ii. Vn V [^ n]! p I d. iii. sup tr d P [V = n (^ n n) t] P[V = n (^ n ) t]! p 0. iv. sup tr d P = [ ^V 0;n (^ n = n) t] P[ ^V 0;n (^ n ) t]! p 0. (b) If nh d+ n is convergent in R +, then ^V = 0;n (^ n n) p N (0; 0), where 0 = li n! (nh d+ n + 8) = (nh d+ n + 6)(nh d+ n + 8) =. (c) If (n)! and (n)=n! 0 and if nh d+ n is convergent in R +, then ^V = 0;(n)(^ (n) (n)) p N (0; 0 ).

15 Bootstrapping Density-Weighted Average Derivatives 3 In an obvious way, Theore (a)-(b) can be viewed as a standard nonparaetric bootstrap analogue of Lea (a)-(b). In particular, Theore (a) shows that Condition AL is necessary and su cient for consistency of the bootstrap. This result shows that the nonparaetric bootstrap is inconsistent whenever the estiator is not asyptotically linear (when li n! nh d+ n < ), including in particular the knife-edge case nh d+ n! (0; ) where the estiator is p n-consistent and asyptotically noral. The iplication (i) ) (iv) in Theore (a) is essentially due to NR. 5 On the other hand, the result that Condition AL is necessary for bootstrap consistency would appear to be new. In Section 4, the nite saple relevance of this sensitivity with respect to bandwidth choice suggested by Theore (a) will be explored in a Monte Carlo experient. Theore (b) can be used to quantify the severity of the bootstrap inconsistency under departures fro Condition AL. The extent of the failure of the bootstrap to approxiate the asyptotic distribution of the test statistic is captured by the variance atrix 0, which satis es 3I d =4 0 I d and takes on the liiting values 3I d =4 and I d when li n! nh d+ n equals 0 and, respectively. Interestingly, coparing Theore (b) with Lea (b), the = nonparaetric bootstrap approxiation to the distribution of ^V 0;n (^ n ) is seen to be superior to the standard noral approxiation because 0 0 I d. As a consequence, there is a sense in which the bootstrap o ers re neents even when Condition AL fails. Theore (c) shows that a su cient condition for consistency of subsapling is convergence of nh d+ n in R +. To illustrate what can happen when the latter condition fails, suppose nh d+ n is large when n is even and sall when n is odd. Speci cally, suppose that nh d+ n! and nh d+ n+! 0. Then, if (n) is even for every n, it follows fro Theore (c) that ^V 0;(n)(^ = (n) (n)) p N (0; I d ), whereas, by Lea (b), ^V = 0;n+(^ n+ ) N (0; I d =). This exaple is intentionally extree, but the qualitative essage that consistency of subsapling can fail when li n! nh d+ n does not exist is valid ore generally. Indeed, Theore (c) adits the following partial converse: If nh d+ n is not convergent in R +, then there exists 5 The results of NR are obtained under slightly stronger assuptions than those of Lea and require nh d+3 n =(log n) 9! :

16 Bootstrapping Density-Weighted Average Derivatives 4 a sequence (n) such that ((n)!, (n)=n! 0, and) sup tr d P = [ ^V 0;(n)(^ (n) = (n)) t] P[ ^V 0;n (^ n ) t] 9 p 0. In other words, eploying critical values obtained by eans of subsapling does not autoatically robustify an inference procedure based on PSS s statistic. = = 3.4. Bootstrapping CCJ. Because both ^V ;n (^ n ) and ^V ;n (^ n ) are asyptotically standard noral under the assuptions of Lea, folklore suggests that the bootstrap should be capable of consistently estiating their distributions. In the case of the statistic studentized by eans of ^V ;n, this conjecture turns out to be correct, essentially because it follows fro Lea that ^V ; = ^ h (d+) ^ + + n h (d+) V [^ ]. More precisely, an application of Lea and the continuous apping theore for weak convergence in probability yields the following result. Theore. If the assuptions of Lea hold, (n)!, and if li n! (n) =n <, = then ^V ;(n) (^ (n) (n)) p N (0; I d ). Theore deonstrates by exaple that even if Condition AL fails it is possible, by proper choice of variance estiator, to achieve consistency of the nonparaetric bootstrap estiator of the distribution of a studentized version of PSS s estiator. The theory presented here does not allow to deterine whether the bootstrap approxiation enjoys any advantages over the standard noral approxiation, but Monte Carlo evidence reported in Section 4 suggests that bootstrap-based inference does have attractive sall saple properties. In the case of subsapling, consistency of the approxiation to the distribution of ^V = ;n (^ n ) is unsurprising in light of its asyptotic pivotality, and it is natural to expect = an analogous result holds for ^V ;n (^ n ). On the other hand, it follows fro Lea that ^V ;n = n ^ n =(d+) h n n + n h (d+) n V [^ n] n h n (d+),

17 Bootstrapping Density-Weighted Average Derivatives 5 suggesting that Condition AL will be of crucial iportance for bootstrap consistency in the = case of ^V ;n (^ n ). Theore 3. Suppose the assuptions of Lea hold. (a) If nh d+ n is convergent in R +, then ^V = ;n (^ n n) p N (0; ), where = li n! (nh d+ n + 4) = (nh d+ n + 6)(nh d+ n + 4) =. In particular, ^V = ;n (^ n n) p N (0; I d ) if and only if Condition AL is satis ed. (b) If (n)! and (n)=n! 0, then ^V = ;(n) (^ (n) (n)) p N (0; I d ). Theore 3 and the arguents on which it is based is of interest for at least two reasons. First, while there is no shortage of exaples of bootstrap failure in the literature, it sees surprising that the bootstrap fails to approxiate the distribution of the asyptotically = pivotal statistic ^V ;n (^ n ) whenever Condition AL is violated. 6 Second, a variation on the idea underlying the construction of ^V ;n can be used to construct a test statistic whose bootstrap distribution validly approxiates the distribution of PSS s statistic under the assuptions of Lea. Speci cally, because it follows fro Leas that V [^ n(3 =(d+) h n )] n + n h (d+) n V[^ n ] and ^V ;n ^V 0;n, it can be shown that if the assuptions of Lea hold, then sup P = [ ^V ;n (^ n(3 =(d+) h n ) n(3 =(d+) = h n )) t] P[ ^V 0;n (^ n ) t]! p 0, tr d even if nh d+ n does not converge. Adittedly, this construction is ainly of theoretical interest, but it does see noteworthy that this resapling procedure works even in the case where subsapling ight fail Suary of Results. The ain results of this paper are suarized in Table. This table describes the liiting distributions of the test statistics proposed by PSS and CCJ, as well as the liiting distributions (in probability) of their bootstrap analogues. (CCJ k 6 The severity of the bootstrap failure is characterized in Theore 3(a) and easured by the variance atrix, which satis es I d 3I d =, iplying that inference based on the bootstrap approxiation = to the distribution of ^V ;n (^ n ) will be asyptotically conservative.

18 Bootstrapping Density-Weighted Average Derivatives 6 with k f; g refers to the test statistics in Lea (c).) Each panel corresponds to one test statistic, and includes 3 rows corresponding to each approxiation used (large saple distribution, standard bootstrap, and replaceent subsapling, respectively). Each colun analyzes a subset of possible bandwidth sequences, which leads to di erent approxiations in general. As shown in the table, the robust studentized test statistic using ^V ;n, denoted CCJ, is the only statistic that reains valid in all cases. For the studentized test statistic of PSS ( rst panel), both the standard bootstrap and replaceent subsapling are invalid in general, while for the robust studentized test statistic using ^V ;n, denoted CCJ, only replaceent subsapling is valid. As discussed above, the extent of the failure of the bootstrap and the direction of its bias are described in the extree case of = 0. Table also highlights that when nh d+ n is not convergent in R +, weak convergence (in probability) of any asyptotically non-pivotal test statistic (under the bootstrap distribution) is not guaranteed in general. 4. Siulations In an attept to explore whether the theory-based predictions presented above are borne out in saples of oderate size, this section reports the ain results fro a Monte Carlo experient. The siulation study uses a Tobit odel y i = ~y i (~y i > 0) with ~y i = x 0 i + " i, " i s N (0; ) independent of the vector of regressors x i R d, and () representing the indicator function. The diension of the covariates is set to d = and both coponents of are set equal to unity. The vector of regressors is generated using independent rando variables with the second coponent set to x i s N (0; ). Two data generating processes are considered for the rst coponent of x i : Model iposes x i s N (0; ) and Model iposes x i s ( 4 4)= p 8, with p a chi-squared rando variable with p degrees of freedo. For siplicity only results for the rst coponent of = ( ; ) 0 are reported. The population paraeters of interest are = =8 and 0:03906 for Model and Model, respectively. Note that Model corresponds to the one analyzed in Nishiyaa and Robinson (000, 005), while both odels were also considered in CCJ and Cattaneo, Crup, and Jansson (00). The nuber of siulations is set to S = 3; 000, the saple size for each siulation is set to n = ; 000, and the nuber of bootstrap replications for each siulation is set to B = ; 000. (See Andrews and Buchinsky (000) for a discussion of the latter choice.) The Monte Carlo experient is very coputationally deanding: each design, with a grid of 5

19 Bootstrapping Density-Weighted Average Derivatives 7 bandwidths, requires approxiately 6 days to coplete, when using a C code (with wrapper in R) parallelized in 50 CPUs (:33 Ghz). The coputer code is available upon request. The siulation study presents evidence on the perforance of the standard nonparaetric bootstrap across an appropriate grid of possible bandwidth choices. Three test statistics are considered for the bootstrap procedure: PSS = 0 (^ n q n), NR = 0 (^ n q n 0 ^V 0 ^V 0;n 0;n ^B n ), CCJ = 0 (^ n q n), 0 ^V with = (; 0) 0, and where ^B n is a bias-correction estiate. The rst test statistic (PSS ) corresponds to the bootstrap analogue of the classical, asyptotically linear, test statistic proposed by PSS. The second test statistic (NR ) corresponds to the bias-corrected statistic proposed by NR. The third test statistic (CCJ ) corresponds to the bootstrap analogue of the robust, asyptotically noral, test statistic proposed by CCJ. For ipleentation, a standard Gaussian product kernel is used for P =, and a Gaussian density-based ultiplicative kernel is used for P = 4. The bias-correction estiate ^B n is constructed using a plug-in estiator for the population bias with an initial bandwidth choice of b n = :h n, as discussed in Nishiyaa and Robinson (000, 005).. The results are suarized in Figure (P = ) and Figure (P = 4). These gures plot the epirical coverage for the three copeting 95% con dence intervals as a function of the choice of bandwidth. To facilitate the analysis two additional horizontal lines at 0:90 and at the noinal coverage rate 0:95 are included for reference. In each gure, the rst and second rows correspond to Models and, respectively. Also, for each gure, the rst colun depicts the results for the copeting con dence intervals using the standard nonparaetric bootstrap to approxiate the quantiles of interest, while the second colun does the sae but using the large saple distribution quantiles (e.g., (0:975) :96). Finally, each plot also includes three population bandwidth selectors available in the literature for density-weighted average derivatives as vertical lines. Speci cally, h P S, h NR and h CCJ denote the population optial bandwidth choices described in Powell and Stoker (996), NR and Cattaneo, Crup, and Jansson (00), respectively. The bandwidths di er in general, although h P S = h NR when d = and P =. (For a detailed discussion and coparison of these bandwidth selectors, see Cattaneo, Crup, and Jansson (00).) The ain results are consistent across all designs considered. First, it is seen that bootstrapping PSS induces a bias in the distributional approxiation for sall bandwidths, ;n

20 Bootstrapping Density-Weighted Average Derivatives 8 as predicted in Theore. Second, bootstrapping CCJ (which uses ^V ;n ) provides a closeto-correct approxiation for a range of sall bandwidth choices, as predicted by Theore. Third, by coparing these results across coluns (bootstrapping vs. Gaussian approxiations), it is seen that the bias in the distributional approxiation of PSS for sall bandwidths is saller (leading to shorter con dence intervals) than the corresponding bias introduced fro using the Gaussian approxiation (longer con dence intervals), as predicted by Theore. In addition, it is found that the range of bandwidths with close-to-correct coverage has been enlarged for both PSS and CCJ when using the bootstrap approxiation instead of the Gaussian approxiation. The bias correction proposed by Nishiyaa and Robinson (000, 005) does not see to work well when P = (Figure ), but works soewhat better when P = 4 (Figure ). 7 Based on the theoretical results developed in this paper, and the siulation evidence presented, it appears that con dence intervals based on the bootstrap distribution of CCJ perfor the best, as they are valid under quite weak conditions. In ters of bandwidth selection, the Monte Carlo experient shows that h CCJ falls clearly inside the robust range of bandwidths in all cases. Interestingly, and because bootstrapping CCJ sees to enlarge the robust range of bandwidths, the bandwidth selectors h P S and h NR also appear to be valid when coupled with the bootstrapped con dence intervals based on CCJ. 5. Conclusion Eploying the sall bandwidth asyptotic fraework of CCJ, this paper has developed theory-based predictions of nite saple behavior of a variety of bootstrap-based inference procedures associated with the kernel-based density-weighted averaged derivative estiator proposed by PSS. In iportant respects, the predictions and ethodological prescriptions eerging fro the analysis presented here di er fro those obtained using Edgeworth expansions by NR. The results of a sall-scale Monte Carlo experient were found to be consistent with the theory developed here, indicating in particular that while the properties of inference procedures eploying the variance estiator of PSS are very sensitive to bandwidth choice, this sensitivity can be aeliorated by using a robust variance estiator proposed in CCJ. 7 It sees plausible that these conclusions are sensitive to the choice of initial bandwidth b n for the construction of the estiator ^B n, but we have ade no attept to iprove on the initial bandwidth choice advocated by Nishiyaa and Robinson (000, 005).

21 Bootstrapping Density-Weighted Average Derivatives 9 For any R d ; let ~ U ij;n () = 0 [U (z i ; z j ; h n ) T ;n () = T 3;n () = T 4;n () = n n 3 n 4 X i<jn X i<j<kn X i<j<k<ln as well as their bootstrap analogues T ;() = T 3;() = T 4;() = 3 4 X i<j X i<j<k X i<j<k<l 6. Appendix ~U ij;n (), T ;n () = (h n )] and de ne the n-varying U-statistics n X i<jn ~U ij;n (), ~U ij;n () U ~ ik;n () + U ~ ij;n () U ~ jk;n () + U ~ ik;n () U ~ jk;n (), 3 ~U ij;n () U ~ kl;n () + U ~ ik;n () U ~ jl;n () + U ~ il;n () U ~ jk;n (), 3 ~U ij;(), T ;() = X i<j ~U ij;(), ~U ij;() U ~ ik; () + U ~ ij;() U ~ jk; () + U ~ ik; () U ~ jk; (), 3 ~U ij;() U ~ kl; () + U ~ ik; () U ~ jl; () + U ~ il; () U ~ jk; (), 3 where ~ U ij;() = 0 [U(z i ; z j ; h ) (h )]. (Here, and elsewhere in the Appendix, the dependence of (n) on n has been suppressed.) The proof of Lea uses four technical leas, proofs of which are available upon request. The rst lea is a siple algebraic result relating ^ n and ^ n (and their bootstrap analogues) to T ;n ; T ;n ; T 3;n ; and T 4;n (and their bootstrap analogues). Lea A-. If the assuptions of Lea hold and if R d, then (a) 0 ^n (h n ) = 4 [ + o ()] n T ;n () + 4[ + o ()]T 3;n () 4T ;n (), (b) h (d+) n 0 ^n (h n ) = [ + o ()]T ;n () T ;n () [ + o ()]T 3;n () + [ + o ()]T 4;n (), (c) 0 ^ (h ) = 4 [ + o ()] T ;() + 4[ + o ()]T 3;() 4T ;(), (d) h (d+) 0 ^ (h ) = [+o ()]T ;() T ;() [+o ()]T 3;()+[+o ()]T 4;(). The next lea, which follows by standard properties of (n-varying) U-statistics (e.g., NR and CCJ), gives soe asyptotic properties of T ;n ; T ;n ; T 3;n ; and T 4;n (and their boot-

22 Bootstrapping Density-Weighted Average Derivatives 0 strap analogues). Let n = = in ; nh d+ n. Lea A-. If the assuptions of Lea hold and if R d, then (a) T ;n () = o p ( p n ), (b) T ;n () = E[ ~ U ij;n () ] + o p (h (d+) n ), (c) T 3;n () = E[(E[ ~ U ij;n ()jz i ]) ] + o p ( n ), (d) T 4;n () = o p ( n ), (e) h d+ n E[ U ~ ij;n () ]! 0 and E[(E[ U ~ ij;n ()jz i ]) ]! 0 =4, (f) T;() = o p ( p ), (g) T;() = E [ U ~ ij;() ] + o p (h (d+) ), (h) T 3;() = E [(E[ ~ U ij;()jz n ; z i ]) ] + o p ( ), (i) T 4;() = o p ( ), (j) h d+ E [ ~ U ij;() ]! p 0 and E [(E[ ~ U ij; () jz n ; z i ]) ] 0 ^n (h )=4! p 0. The next lea, which can be established by expanding sus and using siple bounding arguents, is used to establish a pointwise version of (4). Lea A-3. If the assuptions of Lea hold and if R d, then (a) E[(E[ U ~ ij;()jz n ; zi ]) 4 ] = O( + h 3 ), (b) E[ U ~ ij;() 4 ] = O(h (3d+4) ), (c) E[(E[ U ~ ij;() jz n ; zi ]) ] = O( h (3d+4) + h (d+4) ), (d) E[(E[ ~ U ij;() ~ U ik; ()jz n; z j ; z k ]) ] = O(h (d+4) + h (3d+4) ), (e) E[(E[E[ ~ U ij;()jz n ; z i ] ~ U ij;()jz n ; z j ]) ] = O( + h (d+4) + 3 h (3d+4) ). Finally, the following lea about quadratic fors is used to deduce (4) fro its pointwise counterpart. Lea A-4. There exist constants C and J (only dependent on d) and a collection l ; : : : ; l J d such that, for every d d atrix M, sup d ( 0 M) C JX j= l 0 jml j. Proof of Lea. By the properties of the (conditional on Z n ) Hoe ding decopo-

23 Bootstrapping Density-Weighted Average Derivatives sition, E[L (z i ; h)jz n ] = 0 and E[W (z i ; z j ; h)jz n ; z i ] = 0, so V [^ ] = V [L (z i ; h )] + where, using Leas A- and A-, n V [L (zi ; h )] = n Also, for any R d ; it can be shown that ^n (h ) = + n V [W (zi ; zj ; h )], h (d+) + o p ( ). 0 V [W (z i ; z j ; h )] = h (d+) n h 0 ^n (h ) + o p ()i n 3 n 0 ^n (h ). n Therefore, using Leas A- and A-, copleting the proof of part (a). Next, using Leas A- and A-, V [W (z i ; z j ; h )] = h (d+) + o p ( ), 0 ^ (h ) = 4[ + o ()] T ;() + 4[ + o ()]T 3;() 4T ;() establishing part (b). = 0 ^n (h ) + 4 h (d+) 0 + o p ( ) = 0 + o p ( ), Finally, to establish part (c), the theore of Heyde and Brown (970) is eployed to prove the following condition, which is equivalent to (4) in view of part (a): sup sup d tr d P 4 0 (^ ) q 0 V [^ ] 3 t5 (t)! p 0. For any d, 0^ q 0 0 V [^ ] = X i= Y i; (),

24 Bootstrapping Density-Weighted Average Derivatives where, de ning L i; () = 0 L (z i ; h ) and W ij; () = 0 W (z j ; z i ; h ); Y i; () = " Xi q L 0 V [^ i; () + j= ] # Wij; (). For any n; Y i; () ; F i;n is a artingale di erence sequence, where F i;n = (Z n ; z ; : : : ; z i ) : Therefore, by the theore of Heyde and Brown (970), there exists a constant C such that 3 sup sup d tr d P 4 0 (^ q ) t5 (t) 0 V [^ ] 8 < X C sup E Yi; () 4 X + E 4 E Yi; () F : i d i= i= ;n! 39 = 5 ; =5. Moreover, by Lea A-4, 8 < X sup E Yi; () 4 + E 4 : d i= X i= E Y i; () F i ;n! 39 = 5 ;! p 0 if (and only if) the following hold for every d : X i= E Y i; () 4! p 0 (5) and E 4 X i= E Y i; () F i ;n! 3 5! p 0. (6) The proof of part (c) will be copleted by xing d and verifying (5) (6) : First, using ( 0 V [^ ]) = O p ( ) and basic inequalities, it can be shown that (5) holds if R ; = X E L i; () 4! 0 i=

25 Bootstrapping Density-Weighted Average Derivatives 3 and R ; = 6 X E 4 i=! Xi 4 3 Wij; () 5! 0. j= Both conditions are satis ed because, using Lea A-3, R ; = O E[(E[ U ~ ij; () jz n ; zi ]) 4 ] = O + h = O + h d! 0 and h R ; = O 4 E ~U ij; () 4i + 3 E[(E[ U ~ ij; () jz n ; zi ]) ] = O 4 h (3d+4) + 3 h (d+4) = O h d +! 0. Next, consider (6). Because ( 0 V [^ ]) = " X X i= it su ces to show that R 3; = 6 i= 0 E Yi; () Fi 4 + E 4 R 4; = 6 E R 5; = 4 i=! Xi Wij; () j= X Xi j= i= j= # X Xi E W ij; () F i 4 E i= j= k= 3 F i 5 ;n Xi E Wij; () A j= E L i; () W ij; () jf i ;n, ;n E! 3 Wij; () 5! 0, X Xi j X E Wij; () Wik; () Fi ;n! 3 5! 0, 4 X Xi E! 3 L i; () Wij; () jz n ; zj 5! 0. i= j=

26 Bootstrapping Density-Weighted Average Derivatives 4 By siple calculations and Lea A-3, R 5; = O R 3; = O 4 E[Wij; () 4 ] = O = O 4 h (3d+4) 4 E[ U ~ ij; () 4 ] = O h d! 0, h R 4; = O E E[Wij; () Wik; () jz n ; zj ; zk] i = O E E[ U ~ ij; () U ~ ik; () jz n ; zj ; zk] = O = O h (d+4) + 4 h (3d+4) h E E E E i L i; () Wij; () jz n ; zj h = O h d + h d! 0, i E[ U ~ ij; () jz n ; zi ] U ~ ij; () jz n ; zj = O + h d + h d! 0, = O + h (d+4) + 4 h (3d+4) as was to be shown.

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