Robust Data-Driven Inference for Density-Weighted Average Derivatives

Transcription

1 Supplemetary materials for this article are available olie. Please click the JASA lik at Robust Data-Drive Iferece for Desity-Weighte Average Derivatives Matias D. CATTANEO, Richar K. CRUMP, a Michael JANSSON This paper presets a ovel ata-rive bawith selector compatible with the small bawith asymptotics evelope i Cattaeo, Crump, a Jasso (009) for esity-weighte average erivatives. The ew bawith selector is of the plug-i variety, a is obtaie base o a mea square error expasio of the estimator of iterest. A extesive Mote Carlo experimet shows a remarkable improvemet i performace whe the bawith-epeet robust iferece proceures propose by Cattaeo, Crump, a Jasso (009) are couple with this ew ata-rive bawith selector. The resultig robust ata-rive cofiece itervals compare favorably to the alterative proceures available i the literature. The olie supplemetal material to this paper cotais further results from the simulatio stuy. KEY WORDS: Average erivative; Bawith selectio; Robust iferece; Small bawith asymptotics. Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary INTRODUCTION Semiparametric moels, which iclue both a fiite imesioal parameter of iterest a a ifiite imesioal uisace parameter, play a cetral role i moer statistical a ecoometric theory, a are potetially of great iterest i empirical work. However, the applicability of semiparametric estimators is seriously hampere by the sesitivity of their performace to seemigly a hoc choices of smoothig a tuig parameters ivolve i the estimatio proceure. Although classical large sample theory for semiparametric estimators is ow well evelope, these theoretical results are typically ivariat to the particular choice of parameters associate with the oparametric estimator employe, a usually require strog utestable assumptios (e.g., smoothess of the ifiite imesioal uisace parameter). As a cosequece, iferece proceures base o these estimators are i geeral ot robust to chages i the choice of tuig a smoothig parameters uerlyig the oparametric estimator, a to epartures from key uobservable moel assumptios. These facts suggest that classical asymptotic results for semiparametric estimators may ot always accurately capture their behavior i fiite samples, posig cosierable restrictios o the overall applicability they may have for empirical work. This paper proposes two robust ata-rive iferece proceures for the semiparametric esity-weighte average erivatives estimator of Powell, Stock, a Stoker (1989). The average erivatives is a simple yet importat semiparametric estima of iterest, which aturally arises i may statistical a ecoometric moels such as (oaitive) sigle-iex moels Matias D. Cattaeo is Assistat Professor of Ecoomics, Departmet of Ecoomics, Uiversity of Michiga, A Arbor, MI ( cattaeo@umich.eu). Richar K. Crump is Ecoomist, Capital Markets Fuctio, Feeral Reserve Bak of New York, New York, NY Michael Jasso is Associate Professor of Ecoomics, Departmet of Ecoomics, Uiversity of Califoria at Berkeley a CREATES, Berkeley, CA The authors thak Sebastia Caloico, Lutz Kilia, semiar participats at Georgetow, Michiga, Pe State a Wiscosi, a coferece participats at the 009 Lati America Meetig of the Ecoometric Society a 010 North America Witer Meetig of the Ecoometric Society for commets. We also thak the eitor, associate eitor, a a referee for commets a suggestios that improve this paper. The first author gratefully ackowleges fiacial support from the Natioal Sciece Fouatio (SES ). The thir author gratefully ackowleges fiacial support from the Natioal Sciece Fouatio (SES ) a the research support of CREATES (fue by the Daish Natioal Research Fouatio). (see, e.g., Powell 1994 a Matzki 007 forreview).thisestima has bee cosiere i a variety of empirical problems, icluig oparametric ema estimatio (Härle, Hilebra, a Jeriso 1991), policy aalysis of tax a subsiy reform (Deato a Ng 1998), a oliear pricig i labor markets (Coppejas a Sieg 005). This paper focuses o the esity-weighte average erivatives estimator ot oly because of its ow importace, but also because it amits a particular U-statistic represetatio. As iscusse i etail below, this represetatio is heavily exploite i the theoretical evelopmets presete here, which implies that the results i this paper may be extee to cover other estimators havig a similar represetatio. The mai iea is to evelop a ovel ata-rive bawith selector compatible with the small bawith asymptotic theory presete i Cattaeo, Crump, a Jasso (009). This alterative (first-orer) large sample theory ecompasses the classical large sample theory available i the literature, a also ejoys several robustess properties. I particular, (i) it provies vali iferece proceures for (small) bawith sequeces that woul reer the classical results ivali, (ii) it permits the use of a seco-orer kerel regarless of the imesio of the regressors a therefore removes strog smoothess assumptios, a (iii) it provies a limitig istributio that is i geeral ot ivariat to the particular choices of smoothig a tuig parameters, without ecessarily forcig a slower tha root- rate of covergece (where is the sample size). The key theoretical isight behi these results is to accommoate bawith sequeces that break ow the asymptotic liearity of the estimator of iterest, leaig to a more geeral first-orer asymptotic theory that is o loger ivariat to the particular choices of parameters uerlyig the prelimiary oparametric estimator. Cosequetly, it is expecte that a iferece proceure base o this alterative asymptotic theory woul (at least partially) aapt to the particular choices of these parameters. The prelimiary simulatio results i Cattaeo, Crump, a Jasso (009) show that this alterative asymptotic theory opes the possibility for the costructio of a robust iferece proceure, proviig a rage of (small) bawiths for which the appropriate test statistic ejoys approximately correct size. 010 America Statistical Associatio Joural of the America Statistical Associatio September 010, Vol. 105, No. 491, Theory a Methos DOI: /jasa.010.tm

2 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1071 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 However, the bawith selectors available i the literature tur out to be icompatible with these ew results i the sese that they woul ot eliver a bawith choice withi the robust rage. The ew ata-rive bawith selector presete here achieves this goal, thereby proviig a robust automatic (i.e., fully ata-rive) iferece proceure for the estima of iterest. These results are corroborate by a extesive Mote Carlo experimet, which shows that the asymptotic theory evelope i Cattaeo, Crump, a Jasso (009) couple with the atarive bawith selector propose here leas to remarkable improvemets i iferece whe compare to the alterative proceures available i the literature. I particular, the resultig cofiece itervals exhibit close-to-correct empirical coverage across all esigs cosiere. Amog other avatages, these ata-rive statistical proceures allow for the use of a secoorer kerel, which is believe to eliver more stable results i applicatios (see, e.g., Horowitz a Härle 1996), a appear to be cosierably more robust to the aitioal variability itrouce by the estimatio of the bawith selectors. Furthermore, these results are importat because the staar oparametric bootstrap is ot a vali alterative i geeral to the large sample theory employe i this paper (Cattaeo, Crump, a Jasso 010). Aother iterestig feature of the aalysis presete here is relate to the well-kow trae-off betwee efficiecy a robustess i statistical iferece. I particular, the ovel proceures presete here are cosierably more robust while i geeral (semiparametric) iefficiet. This feature is capture by the behavior of the ew robust cofiece itervals i the simulatio stuy, where they are see to have correct size a less bias but larger legth o average. For example, whe the classical proceure is vali (i.e., whe usig a higher-orer kerel), the efficiecy loss is fou to be arou 10% o average, while the bias of the estimator is reuce by about 60% o average. This paper cotributes to the importat literature of semiparametric iferece for weighte average erivatives. This populatio parameter was origially itrouce by Stoker (1986), a has bee itesely stuie sice the. Härle a Stoker (1989) a Härle et al. (199) stuy the asymptotic properties of geeral weighte average erivatives estimators, while Newey a Stoker (1993) iscuss their semiparametric efficiecy properties uer appropriate restrictios. The asymptotic results, however, are cosierably complicate by the fact that their represetatio requires halig stochastic eomiators a appears to be very sesitive to the choice of trimmig parameters. The esity-weighte average erivatives estimator circumvets this problem, while retaiig the esirable properties of the geeral weighte average erivative, a leas to a simple a useful semiparametric estimator. Powell, Stock, a Stoker (1989) stuy the first-orer large sample properties of this estimator a provie sufficiet (but ot ecessary) coitios for root- cosistecy a asymptotic ormality. Nishiyama a Robiso (000, 005) stuy the seco-orer large sample properties of esity-weighte average erivatives by erivig vali Egeworth expasios for the estimator cosiere i this paper (see also Robiso 1995), while Härle a Tsybakov (1993) a Powell a Stoker (1996) provie seco-orer mea square error expasios for this estimator (see also Newey, Hsieh, a Robis 004). Both types of higher-orer expasios provie simple plug-i bawith selectors targetig ifferet properties of this estimator, a are compatible with the classical large sample theory available i the literature. Ichimura a To (007) provie a recet survey with particular emphasis o implemetatio. The rest of the paper is orgaize as follows. Sectio escribes the moel a reviews the mai results available i the literature regarig first-orer large sample iferece for esity-weighte average erivatives. Sectio 3 presets the higher-orer mea square error expasio a evelops the ew (ifeasible) theoretical bawith selector, while Sectio 4 escribes how to costruct a feasible (i.e., ata-rive) bawith selector a establishes its cosistecy. Sectio 5 summarizes the results of a extesive Mote Carlo experimet. Sectio 6 iscusses how the results may be geeralize a coclues.. MODEL AND PREVIOUS RESULTS Let z i = (y i, x i ), i = 1,...,, be a raom sample from a vector z = (y, x ), where y R is a epeet variable a x = (x 1,...,x ) R is a cotiuous explaatory variable with a esity f ( ). The populatio parameter of iterest is the esityweighte average erivative give by [ θ = E f (x) x g(x) where g(x) = E[y x] eotes the populatio regressio fuctio. For example, this estima is a popular choice for the estimatio of the coefficiets (up to scale) i a sigle-iex moel with ukow lik fuctio. To see this, ote that θ β whe g(x) = τ(x β) for a ukow (lik) fuctio τ( ), a semiparametric problem that arises i a variety of cotexts, icluig iscrete choice a cesore moels. The followig assumptio collects typical regularity coitios impose o this moel. Assumptio 1. (a) E[y 4 ] <, E[σ (x)f (x)] > 0 a V[ e(x)/ x y f (x)/ x] is positive efiite, where σ (x) = V[y x] a e(x) = f (x)g(x). (b) f is (Q+1) times ifferetiable, a f a its first (Q+1) erivatives are boue, for some Q. (c) g is twice ifferetiable, a e a its first two erivatives are boue. () v is ifferetiable, a vf a its first erivative are boue, where v(x) = E[y x]. (e) lim x [f (x)+ e(x) ] = 0, where is the Eucliea orm. Assumptio 1 a itegratio by parts lea to θ = E[y f (x)/ x], which i tur motivates the aalogue estimator of Powell, Stock, a Stoker (1989) give by ˆθ = 1 ˆf,i (x) = 1 1 i=1 ], y i x ˆf,i (x i ), h j=1,j i ( 1 xj x K where ˆf,i ( ) is a leave-oe-out kerel esity estimator for some kerel fuctio K : R R a some positive (bawith) sequece h. Typical regularity coitios impose o h ),

3 107 Joural of the America Statistical Associatio, September 010 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 the kerel-base oparametric estimator are give i the followig assumptio. Assumptio. (a) K is eve a ifferetiable, a K a its first erivative are boue. (b) K(u) R K(u) u is positive efiite, where K(u) = K(u)/ u. (c) For some P, R [ K(u) (1 + u P ) + K(u) (1 + u )] u <, a for (l 1,...,l ) Z +, R u l 1 1 u l K(u) u = { 1, if l1 + +l = 0 0, if 0 < l 1 + +l < P. Powell, Stock, a Stoker (1989) showe that, uer appropriate restrictios o the bawith sequece a kerel fuctio, the estimator ˆθ is asymptotically liear with ifluece fuctio give by L(z) = [ e(x)/ x y f (x)/ x θ]. Thus, the asymptotic variace of this estimator is give by = E[L(z)L(z) ]. Moreover, although ot covere by the results i Newey a Stoker (1993), it is possible to show that L(z) is the efficiet ifluece fuctio for θ, a hece is the semiparametric efficiecy bou for this estima. The followig result escribes the exact coitios a summarizes the mai coclusio. (Limits are take as uless otherwise ote.) Result 1 (Powell, Stock, a Stoker 1989). If Assumptios 1 a hol, a if h mi(p,q) 0 a h +, the (ˆθ θ) = 1 L(z i ) + o p (1) N (0, ). i=1 Result 1 follows from otig that the estimator ˆθ amits a -varyig U-statistic represetatio give by ( ) 1 1 ˆθ = U(z i, z j ; h ), ( ) U(z i, z j ; h) = h (+1) xi x j K (y i y j ), h which leas to the Hoeffig ecompositio ˆθ = θ + L + W, where θ = E[U(z i, z j ; h )], L = 1 L(z i ; h ), ( ) 1 1 W = i=1 W(z i, z j ; h ), with L(z i ; h) = [E[U(z i, z j ; h) z i ] E[U(z i, z j ; h)]] a W(z i, z j ; h) = U(z i, z j ; h) (L(z i ; h) + L(z j ; h))/ E[U(z i, z j ; h)]. This ecompositio shows that the estimator amits a biliear form represetatio i geeral, which clearly justifies the coitios impose o the bawith sequece a the kerel fuctio: (i) coitio h mi(p,q) 0 esures that the bias of the estimator is asymptotically egligible because θ θ = O(h mi(p,q) ), a (ii) coitio h + esures that the quaratic term of the Hoeffig ecompositio is also asymptotically egligible because W = O p ( 1 h (+)/ ). Uer the same coitios, Powell, Stock, a Stoker (1989) also evelop a simple cosistet estimator for, which is give by the aalogue estimator ˆ = 1 ˆL,i ˆL,i, i=1 [ 1 ˆL,i = 1 j=1,j i U(z i, z j ; h ) ˆθ ]. Cosequetly, uer the coitios impose i Result 1,itis straightforwar to form a stuetize versio of ˆθ, leaig to a asymptotically pivotal test statistic give by ˆ 1/ (ˆθ θ) N (0, I ), with ˆ p. This test statistic may be use i the usual way to costruct a cofiece iterval for θ (or, equivaletly, to carry out the correspoig ual hypothesis test). As iscusse i Newey (1994), asymptotic liearity of a semiparametric estimator has several istict features that may be cosiere attractive from a theoretical poit of view. I particular, asymptotic liearity is a ecessary coitio for semiparametric efficiecy a leas to a limitig istributio of the statistic of iterest that is ivariat to the choice of the oparametric estimator use i the costructio of the semiparametric proceure. I other wors, regarless of the particular choice of prelimiary oparametric estimator, the limitig istributio will ot epe o the oparametric estimator wheever the semiparametric estimator amits a asymptotic liear represetatio. However, achievig a asymptotic liear represetatio of a semiparametric estimator imposes several strog moel assumptios a leas to a large sample theory that may ot accurately represet the fiite sample behavior of the estimator. I the case of ˆθ, asymptotic liearity woul require P > uless = 1, which i tur requires strog smoothess coitios (Q P). Cosequetly, classical asymptotic theory will require the use of a higher-orer kerel wheever more tha oe covariate is iclue. I aitio, classical asymptotic theory (wheever vali) leas to a limitig experimet which is ivariat to the particular choices of smoothig (K) a tuig (h ) parameters ivolve i the costructio of the estimator, a therefore it is ulikely to be able to aapt to chages i these parameters. I other wors, iferece base o classical asymptotic theory is silet with respect to the impact that these parameters may have o the fiite sample behavior of ˆθ. I a attempt to better characterize the fiite sample behavior of ˆθ, Cattaeo, Crump, a Jasso (009) show that it is possible to icrease the robustess of this estimator by cosierig a ifferet asymptotic experimet. I particular, istea of forcig asymptotic liearity of the estimator, the authors evelop a alterative first-orer asymptotic theory that accommoates weaker assumptios tha those impose i the classical firstorer asymptotic theory iscusse above. Ituitively, the iea is to characterize the (joit) asymptotic behavior of both the liear ( L ) a quaratic ( W ) terms. The followig result collects the mai fiigs. Result (Cattaeo, Crump, a Jasso 009). If Assumptios 1 a hol, a if mi(h +, 1)h mi(p,q) 0 a

4 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1073 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 h, the where (V[ˆθ ]) 1/ (ˆθ θ) N (0, I ), V[ˆθ ]= 1 [ + o(1)]+ ( ) 1 h (+) [ + o(1)], with = E[σ (x)f (x)] K(u) R K(u) u. I aitio, 1 ˆ = 1 ( ) 1 [ + o p(1)]+ h (+) [ + o p (1)]. Result shows that the coitios o the bawith sequece may be cosierably weakee without ivaliatig the limitig Gaussia istributio. I particular, wheever h is chose so that h + is boue, the limitig istributio will cease to be ivariat with respect to the uerlyig prelimiary oparametric estimator because ˆθ is o loger asymptotically liear. I particular, ote that h + κ>0retais the root cosistecy of ˆθ. I aitio, because h is allowe to be smaller tha usual, the bias of the estimator is cotrolle i a ifferet way, removig the ee for higher-orer kerels. I particular, Result remais vali eve i cases whe the estimator is ot cosistet. Fially, this result also highlights the well-kow trae-off betwee robustess a efficiecy i the cotext of semiparametric estimatio. I particular, the estimator ˆθ is semiparametric efficiet if a oly if h +, while it is possible to costruct more robust iferece proceures uer cosierably weaker coitios. It follows from Result that the feasible classical testig proceure base o ˆ 1/ (ˆθ θ) will be ivali uless h +, which correspos to the classical large sample theory case (Result 1). To solve this problem, Cattaeo, Crump, a Jasso (009) propose two alterative correctios to the staar error matrix ˆ, leaig to two optios for robust staar errors. To costruct the first robust staar error formula, the authors itrouce a simple cosistet estimator for, uer the same coitios of Result, which is give by the aalogue estimator ( ) 1 ˆ = h + 1 Ŵ,ij Ŵ,ij, Ŵ,ij = U(z i, z j ; h ) 1 ( ˆL,i + ˆL,j ) ˆθ. Thus, usig this estimator, ˆV 1, = 1 ( ) 1 ˆ h (+) yiels a cosistet staar error estimate uer small bawith asymptotics (i.e., uer the weaker coitios impose i Result, which iclue i particular those impose i Result 1). To escribe the seco robust staar error formula, let ˆ (H ) be the estimator ˆ costructe usig a bawith sequece H [e.g., ˆ = ˆ (h ) by efiitio]. The, uer the same coitios of Result, ˆV, = 1 ˆ ( 1/(+) ) h ˆ also yiels a cosistet staar error estimate uer small bawith asymptotics. Cosequetly, uer the coitios impose i Result, it is straightforwar to form a stuetize versio of ˆθ, leaig to two simple, robust a pivotal test statistics of the form ˆV 1/ k, (ˆθ θ) N (0, I ), with ˆV 1 k, V[ˆθ ] p I, k = 1,. These test statistics may also be use to costruct (asymptotically equivalet) cofiece itervals for θ uer the (weaker) coitios impose i Result, a costitute alterative proceures to the classical cofiece iterval itrouce above. These results, however, have the obvious rawback of beig epeet o the choice of h, which is urestricte beyo the rate restrictios impose i Result. A prelimiary Mote Carlo experimet reporte i Cattaeo, Crump, a Jasso (009) shows that the ew, robust staar error formulas have the potetial to eliver goo fiite sample behavior if the iitial bawith is chose to be small eough. Ufortuately, the plug-i rules available i the literature for h fail to eliver a choice of bawith that woul ejoy the robustess property itrouce by the ew asymptotic theory escribe i Result. This is ot too surprisig, sice these bawith selectors are typically costructe to balace (higher-orer) bias a variace i a way that is appropriate for the classical large sample theory. 3. MSE EXPANSION AND OPTIMAL BANDWIDTH SELECTORS This paper cosiers the mea square error expasio of ˆθ as the startig poit for the costructio of the plug-i optimal bawith selector. To erive this expasio it is ecessary to stregthe the assumptios cocerig the atageeratig process. The followig assumptio escribes these aitioal mil sufficiet coitios. Assumptio 3. (a) E[ g(x)/ x f (x)] <. (b) g is (Q + 1) times ifferetiable, a e a its first (Q + 1) erivatives are boue. (c) v is three times ifferetiable, a vf a its first three erivatives are boue. () lim x [σ(x)f (x) + σ(x)/ x f (x)]=0. Assumptio 3(a) is use to esure that the higher-orer mea square expasio is vali up to the orer eee i this paper. Assumptios 3(b) a 3(c) are i agreemet with those impose i Powell a Stoker (1996) a Nishiyama a Robiso (000, 005), while Assumptio 3() is slightly stroger tha the aalogue restrictio impose i those papers. Theorem 1. If Assumptios 1,, a 3 hol, the for s = mi(p, Q) a ḟ(x) = f (x)/ x, E[(ˆθ θ)(ˆθ θ) ] = 1 ( ) 1 ( ) 1 + h (+) + h V + h s BB + O( 1 h s ) + o( h + h s ),

5 1074 Joural of the America Statistical Associatio, September 010 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 where B = ( 1)s s! 0 l 1,...,l s l 1 + +l =s [ ] u l 1 1 u l K(u) u R [( (l 1 + +l ) ) ] E x l 1 1 x l ḟ(x) g(x) a ( [ V = K(u) K(u) u E σ (x) R x x f (x) ( )( ) ) + x g(x) x g(x) f (x)] u u. The result i Theorem 1 is similar to the oe obtaie by Härle a Tsybakov (1993) a Powell a Stoker (1996), the key ifferece beig that the aitioal term of orer O( h ) is explicitly retaie here. (Recall that Result requires h.) To motivate the ew optimal bawith selector, recall that the robust variace matrix i Result is give by the first two terms of the mea square error expasio presete i Theorem 1, which suggests cosierig the ext two terms of the expasio to costruct a optimal bawith selector. (Note that, as it is commo i the literature, this approach implicitly assumes that both B a V are ozero.) Ituitively, balacig these terms correspos to the case of h + κ<, a therefore pushes the selecte bawith to the small bawith regio. This approach may be cosiere optimal i a mea square error sese because it makes the leaig terms igore i the geeral large sample approximatio presete i Result as small as possible. To escribe the ew bawith selector, let λ R a cosier (for simplicity) a bawith that miimizes the ext two terms of E[(λ (ˆθ θ)) ]. This optimal bawith selector is give by ( (λ ) Vλ) 1/(s+) h CCJ = s(λ B), if λ Vλ > 0 ( λ ) Vλ 1/(s+) (λ B), if λ Vλ < 0. This ew theoretical bawith selector is cosistet with the small bawith asymptotics escribe i Result because (h CCJ ). I aitio, observe that 1 h s = o( h ) wheever h s+ 0, which is satisfie whe h = h CCJ. This ew bawith selector may be compare to the two competig plug-i bawith selectors available i the literature, propose by Powell a Stoker (1996) a Nishiyama a Robiso (005), a give by ( ( + )(λ h ) PS = λ) 1/(s++) s(λ B) a ( (λ h ) NR = λ) 1/(s++) (λ B), respectively. Ispectio of these bawith selectors shows that h CCJ h PS h NR, leaig to a bawith selectio of smaller orer. [Nishiyama a Robiso (000) erive a thir alterative bawith selector which is ot explicitly iscusse here because this proceure is targete to oe-sie hypothesis testig. Noetheless, ispectio of this alterative bawith selectio proceure, eote h NR00, shows that h CCJ h NR00 wheever + 8 > s. Therefore, h CCJ is of smaller orer uless strog smoothess assumptios are impose i the moel a a correspoig higher-orer kerel is employe.] 4. DATA DRIVEN BANDWIDTH SELECTORS The previous sectio escribe a ew (ifeasible) plug-i bawith selector that is compatible with the small bawith asymptotic theory itrouce i Result. I orer to implemet this selector i practice, as well as its competitors h PS a h NR, it is ecessary to costruct cosistet estimates for each of the leaig costats. These estimates woul lea to a atarive (i.e., automatic) bawith selector, eote ĥ CCJ.This sectio itrouces easy to implemet, cosistet oparametric estimators for B,, a V. (Alteratively, a straightforwar bawith selector may be costructe usig a rule-of-thumb estimator base o some a hoc istributioal assumptios.) To escribe the ata-rive plug-i bawith selectors, let b be a prelimiary positive bawith sequece, which may be ifferet for each estimator. A simple aalog estimator of was itrouce i Sectio. I particular, let ˆ (b ) be the estimator ˆ costructe usig a bawith sequece b [e.g., ˆ = ˆ (h ) by efiitio]. Note that this estimator is a -varyig U-statistic as well. Theorem 1 a the calculatios provie i Cattaeo, Crump, a Jasso (009) show that, if Assumptios 1,, a 3 hol, the ˆ (b ) = + b V + O p( b 3 + 1/ + 1 b / which gives the cosistecy of this estimator if b 0 a b. Next, cosier the costructio of cosistet estimators of B a V, the two parameters eterig the ew bawith selector h CCJ. To this e, let k be a kerel fuctio, which may be ifferet for each estimator, a may be ifferet from K. The followig assumptio collects a set of sufficiet coitios to establish cosistecy of the plug-i estimators propose i this paper for B a V. Assumptio 4. (a) f, v, a e are (s S) times ifferetiable, a f, vf, e, a their first (s S) erivatives are boue, for some S 1. (b) k is eve a M times ifferetiable, a k a its first M erivatives are boue, for some M 0. (c) For some R, R k(u) (1 + u R ) u <, a for (l 1,...,l ) Z +, R u l 1 1 u l k(u) u = ), { 1, if l1 + +l = 0 0, if 0 < l 1 + +l < R. For the bias B, a plug-i estimator is give by ˆB = ( 1)s [ s! 0 l 1,...,l s l 1 + +l =s R u l 1 1 u l K(u) u ] ˆϑ l1,...,l,,

6 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1075 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 where ˆϑ l1,...,l, = 1 ( 1) i=1 j=1,j i b (+1) ( (l 1 + +l ) k(x) x l 1 1 x l )y i, x=(xi x j )/b with k(x) = k(x)/ x. The estimator ˆϑ l1,...,l, is the sample aalog estimator of the estima E[( (l 1+ +l ) ḟ(x)/ x l 1 1 x l )y], a is also a -varyig U-statistic estimator employig a leave-oe-out kerel-base esity estimator. It is also possible to form a obvious plug-i estimator for the ew higher-orer term V. However, this estimator woul have the uappealig property of requirig the estimatio of several oparametric objects (σ (x), f (x)/ x x, g(x)/ x, f (x)). Moreover, this irect plug-i approach is likely to be less stable whe implemete because it woul require halig stochastic eomiators. Fortuately, it is possible to costruct a alterative, iirect estimator much easier to implemet i practice. This estimator is ituitively justifie as follows: the results presete above show that, uer appropriate regularity coitios, b ( ˆ (b ) ) = V +O p (b + 1/ b + 1 b / ), a therefore a estimator satisfyig = + o p (b ) woul lea to ˆV = b ( ˆ (b ) ) = V + o p (1), if b 0, b 4 0, a b Uer appropriate coitios, a estimator havig these properties is give by = ˆδ R K(u) K(u) u, ( ) 1 1 ˆδ = ( ) b k xj x i (y i y j ). b I this case, ˆδ is a sample aalog estimator of the estima E[σ (x)f (x)], which is also a -varyig U-statistic estimator employig a leave-oe-out kerel-base esity estimator. Theorem. If Assumptios 1, 3, a 4 hol, the: (i) For M s + 1, [( (l 1 + +l ) ) ] ˆϑ l1,...,l, = E x l 1 1 x l ḟ(x) y (ii) For R 3, + O p ( b mi(r,s) ˆδ = E[σ (x)f (x)]+o p ( b mi(r,s+1+s) + 1/ + 1 b (++s)/ ). + 1/ + 1 b / ). This theorem gives simple sufficiet coitios to costruct a robust ata-rive bawith selector cosistet with the small bawith asymptotics erive i Cattaeo, Crump, a Jasso (009). I particular, efie ( (λ ) ˆV λ) 1/(s+) s(λ ĥ CCJ =, if λ ˆV λ > 0 ˆB ) ( λ ) ˆV λ 1/(s+) (λ, if λ ˆV λ < 0. ˆB ) The followig corollary establishes the cosistecy of the ew bawith selector ĥ CCJ. Corollary 1. If Assumptios 1,, 3, a 4 hol with M s + 1 a R 3, a if b 0 a b max(8,++s), the for λ R such that λ B 0 a λ Vλ 0, ĥ CCJ h CCJ p 1. (The aalogous result also hols for ĥ PS a ĥ NR.) The results presete so far are silet about the selectio of the iitial bawith choice b i applicatios, beyo the rate restrictios impose by Corollary 1. A simple choice for the prelimiary bawith b may be base o some atarive bawith selector evelope for a oparametric object preset i the correspoig target estimas B, a V. Typical examples of such proceures iclue simple rule-of-thumbs, plug-i bawith selectors, a (smoothe) cross-valiatio. As show i the simulatios presete i the ext sectio, it appears that a simple ata-rive bawith selector from the literature of oparametric estimatio works well for the choice of b. Noetheless, it may be esirable to improve upo this prelimiary bawith selector i orer to obtai better fiite sample behavior. Although beyo the scope of this paper, a coceptually feasible (but computatioally emaig) iea woul be to compute seco-orer mea square error expasios for ˆϑ l1,...,l,, ˆ a ˆδ. Sice these three estimators are -varyig U-statistics, the results from Powell a Stoker (1996) may be applie to obtai a correspoig set of optimal bawith choices. These proceures will, i tur, also epe o a prelimiary bawith whe implemete empirically, which agai woul ee to be chose i some way. This iea mimics, i the cotext of semiparametric estimatio, the well-kow seco-geeratio irect plug-i bawith selector (of level ) from the literature of oparametric esity estimatio. (See, e.g., Wa a Joes 1995 for a etaile iscussio.) Although the valiity of such bawith selectors woul require stroger assumptios, by aalogy from the oparametric esity estimatio literature, they woul be expecte to improve the fiite sample properties of the bawith selector for h a, i tur, the performace of the semiparametric iferece proceure. 5. MONTE CARLO EXPERIMENT This sectio summarizes the mai fiigs from a extesive Mote Carlo experimet coucte to aalyze the fiite sample properties of the ew robust ata-rive proceures a their relative merits whe compare to the other proceures available. The olie supplemetal material iclues a larger set of results from this simulatio stuy, which shows that the fiigs reporte here are cosistet across all esigs cosiere. Followig the results reporte i Cattaeo, Crump, a Jasso (009), the Mote Carlo experimet cosiers six ifferet moels of the sigle iex form y i = τ(y i ), where y i = x i β + ε i, τ( ) is a oecreasig (lik) fuctio a ε i N (0, 1) is iepeet of the vector of regressors x i R. Three ifferet lik fuctios are cosiere: τ(y ) = y, τ(y ) = 1(y > 0), a τ(y ) = y 1(y > 0), which correspo to a liear regressio, probit, a Tobit moel, respectively. [1( ) represets the

7 1076 Joural of the America Statistical Associatio, September 010 Table 1. Mote Carlo moels y i = y i y i = 1(y i > 0) y i = y i 1(y i > 0) x 1i N (0, 1) Moel 1: θ 1 = 4π 1 Moel 3: θ 1 = 1 8π 3/ Moel 5: θ 1 = 8π 1 x 1i χ 4 4 Moel : θ 8 1 = 1 4 Moel 4: θ π 1 = Moel 6: θ 1 = Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 iicator fuctio.] The vector of regressors is geerate usig iepeet raom variables a staarize to have E[x i ]=0 a E[x i x i ]=I, with the first compoet x 1i havig either a Gaussia istributio or a chi-square istributio with 4 egrees of freeom (eote χ 4 ), while the remaiig compoets have a Gaussia istributio throughout the experimet. All the compoets of β are set equal to uity, a for simplicity oly results for the first compoet θ 1 are cosiere. Table 1 summarizes the Mote Carlo moels, reports the value of the populatio parameter of iterest, a provies the correspoig label of each moel cosiere. (Wheever uavailable i close form, the populatio parameters are compute by a umerical approximatio.) The simulatio stuy cosiers three sample sizes ( = 100, = 400, = 700), two imesios of the regressors vector ( =, = 4), a two kerel orers (P =, P = 4). The kerel fuctio K( ) is chose to be a Gaussia prouct kerel, a the prelimiary kerel fuctio k( ) is chose to be a fourth-orer Gaussia prouct kerel as require by Corollary 1. For each combiatio of parameters 10,000 replicatios are carrie out. To coserve space this sectio oly iclues the results for = a = 400. The simulatio experimet cosiers the three (ifeasible) populatio bawith choices erive i Sectio 3 (h PS, h NR, h CCJ ), a their correspoig ata-rive estimates (ĥ PS, ĥ NR, ĥ CCJ ). The three estimate bawiths are obtaie usig the results escribe i Sectio 4 with a commo iitial bawith plug-i estimate use to costruct ˆB, ˆ a ˆV.To provie a parsimoious ata-rive proceure, a estimate of the iitial bawith b is costructe as a sample average of a seco-geeratio irect plug-i level-two estimate for the (margial) esity of each imesio of the regressors vector (see, e.g., Wa a Joes 1995). Cofiece itervals for θ 1 are costructe usig the classical test statistic ˆ 1/ (ˆθ θ), eote PSS, a the two alterative robust test statistics ˆV 1/ k, (ˆθ θ), k = 1,, eote by CCJ1 a CCJ, respectively. The classical iferece proceure PSS is oly theoretically vali whe P = 4, while the robust proceures CCJ1 a CCJ are always vali across all simulatio esigs. Figures 1 a plot the empirical coverage for the three competig 95% cofiece itervals as a fuctio of the choice of bawith for each of the six moels. To facilitate the aalysis two aitioal horizotal lies at 0.90 a at the omial coverage rate 0.95 are iclue for referece, a the three populatio bawith selectors (h PS, h NR, h CCJ ) are plotte as vertical lies. (Note that h PS = h NR for the case = a P =.) These figures highlight the potetial robustess properties that the test statistics CCJ1 a CCJ may have whe usig the ew ata-rive plug-i bawith selector. I particular, the theoretical bawith selector h CCJ lays withi the robust regio for which both CCJ1 a CCJ have correct empirical coverage for a rage of bawiths. For example, this suggests that (at least) some of the variability itrouce by the estimatio of this bawith selector will ot affect the performace of the robust test statistics CCJ1 a CCJ, a property ulikely to hol for the classical proceure PSS. Table reports the empirical coverage of each possible cofiece iterval (PSS, CCJ1, CCJ) whe usig each possible populatio bawith selector (h PS, h NR, h CCJ ). Figures 3 a 4 plot correspoig kerel esity estimates for the test statistic PSS couple with either h PS a h NR, a for the test statistics CCJ1 a CCJ couple with h CCJ.Tofacilitate the compariso the esity of the staar ormal is also epicte. These figures show that the Gaussia approximatio of the robust test statistics usig the ew bawith selector is cosierably better tha the correspoig approximatio for PSS whe costructe usig either of the classical bawith selectors. I particular, the empirical istributio of the classical proceure appears to be more biase a more cocetrate tha the empirical istributios of either CCJ1 or CCJ. These fiigs highlight the well-kow trae-off betwee efficiecy a robustess previously iscusse. These results are verifie i Table 3, where the average empirical bias a average empirical iterval legth are reporte for each competig cofiece iterval whe couple with each possible populatio bawith selector. To aalyze the performace of the ew ata-rive bawith selector, a the resultig robust ata-rive cofiece itervals, Table 4 presets the empirical coverage of each possible cofiece iterval (PSS, CCJ1, CCJ) whe usig each possible estimate bawith selector (ĥ PS, ĥ NR, ĥ CCJ ). These tables provie cocrete eviece of the superior performace (i terms of achievig correct coverage) of the robust test statistics whe couple with the ew estimate bawith. Both robust cofiece itervals (CCJ1, CCJ) usig ĥ CCJ provie close-to-correct empirical coverage across all esigs, a property ot ejoye by the classical cofiece iterval (PSS) usig either ĥ PS or ĥ NR. The goo performace of CCJ1 a CCJ is maitaie ot oly whe usig a seco-orer kerel (P = ), but also whe the imesio of x is larger ( = 4), which provies simulatio eviece of the relatively low sesitivity of the ew robust ata-rive proceures to the so-calle curse of imesioality. This fiig may be (heuristically) justifie by the fact that uer the small bawith asymptotics, the limitig istributio is ot ivariat to the parameter, which i tur may lea to the aitioal robustess properties fou. I aitio, as suggeste by the superior istributioal approximatio reporte i Figures 3 a 4, the mai fiigs cotiue to hol if other omial cofiece levels are cosiere. 6. EXTENSIONS AND FINAL REMARKS This paper itrouce a ovel ata-rive plug-i bawith selector compatible with the small bawith asymptotics evelope i Cattaeo, Crump, a Jasso (009) for esity-

8 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1077 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 Figure 1. Empirical coverage rates for 95% cofiece itervals: =, P =, = 400. The olie versio of this figure is i color.

9 1078 Joural of the America Statistical Associatio, September 010 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 Figure. Empirical coverage rates for 95% cofiece itervals: =, P = 4, = 400. The olie versio of this figure is i color.

10 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1079 Table. Empirical coverage rates of 95% cofiece itervals with populatio bawith: =, = 400 BW PSS CCJ1 CCJ BW PSS CCJ1 CCJ BW PSS CCJ1 CCJ Moel 1 Moel 3 Moel 5 P = h PS h NR h CCJ P = 4 h PS h NR h CCJ Moel Moel 4 Moel 6 P = h PS h NR h CCJ P = 4 h PS h NR h CCJ Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 NOTE: Colum BW reports populatio bawiths. weighte average erivatives. This ew bawith selector is of the plug-i variety, a is obtaie base o a mea square error expasio of the estimator of iterest. A extesive Mote Carlo experimet showe a remarkable improvemet i performace of the resultig ew robust ata-rive iferece proceure. I particular, the ew cofiece itervals provie approximately correct coverage i cases where there are o vali alterative iferece proceures (i.e., usig a seco-orer kerel with at least two regressors), a also compares favorably to the alterative, classical cofiece itervals whe they are theoretically justifie. Sice these results are erive by exploitig the -varyig U- statistic represetatio of ˆθ, it is plausible that similar results coul be obtaie for other estimators havig a aalogous represetatio. For example, the class of estimas cosiere i Newey, Hsieh, a Robis (004, sectio ) have this represetatio, a therefore it seems possible that the results presete here coul be geeralize to cover that class. More geerally, as suggeste i Cattaeo, Crump, a Jasso (009), a - varyig U-statistic may be represete as a miimizer of the U-process: ˆθ = arg mi θ ( ) 1 1 Q(z i, z j ; θ, h) = U(z i, z j, h) θ, Q(z i, z j ; θ, h ), which also suggests that the results presete here may be extee to cover this class of estimators (see, e.g., Araillas- Lopéz, Hooré, a Powell 007, pp ). APPENDIX Proof of Theorem 1 To save otatio, for ay fuctio a : R R let ȧ(x) = a(x)/ x a ä(x) = a(x)/ x x. A Hoeffig ecompositio of ˆθ gives E[(ˆθ θ)( ˆθ θ) ]=V[ ˆθ ]+(E[ˆθ ] θ)(e[ˆθ ] θ) = V[ L ]+V[ W ]+h s BB + o(h s ), where the bias expasio follows immeiately by a Taylor series expasio. For V[ L ], usig itegratio by parts, E[U (z i, z j ) z i ] = ė(x R i + uh )K(u) u y i ḟ(x i + uh )K(u) u, R a therefore V[ L ]=4 1 V[E[U (z i, z j ) z i ] θ ]= 1 + O( 1 h s ). For V[ W ], by staar calculatios, ( ) 1 V[ W ]= E[U (z i, z j )U (z i, z j ) ]+O( ) ( ) 1 = h (+) K(u) K(u) T(x, uh ) x u + O( ), R with T(x, u) = (v(x) + v(x + u) g(x)g(x + u))f (x)f (x + u). The, usig a Taylor series expasio, T(x, uh ) = T 1 (x) + T (x) uh + u T 3 (x)uh + o(h ),wheret 1(x) = σ (x)f (x), T (x) = σ (x) f (x)ḟ(x) + f (x) σ (x),at 3 (x) = σ (x)f (x) f(x) + f (x) σ (x)ḟ(x) + ( v(x)/ g(x) g(x))f (x). Note that R R K(u) K(u) T 1 (x) x u = R R K(u) K(u) σ (x)f (x) x u = a, usig itegratio by parts, h K(u) K(u) (T (x) u) x u R R = h K(u) K(u) R [( ) [σ (x)f (x)ḟ(x) + f (x) σ (x)] x u] u R = 0. Fially, usig itegratio by parts a the fact that σ (x) = v(x) ġ(x)ġ(x) g(x) g(x), h K(u) K(u) (u T 3 (x)u) x u R = h K(u) K(u) R ( ) ] [u σ (x) f(x)f (x) x + ġ(x)ġ(x) f (x) x u u. R R Therefore, V[ W ]= ( ) 1h (+) + ( ) 1h V + o( h ).

11 1080 Joural of the America Statistical Associatio, September 010 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 Figure 3. Empirical Gaussia approximatio with populatio bawith: =, P =, = 400. The olie versio of this figure is i color.

12 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1081 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 Figure 4. Empirical Gaussia approximatio with populatio bawith: =, P = 4, = 400. The olie versio of this figure is i color.

13 108 Joural of the America Statistical Associatio, September 010 Table 3. Empirical average legth of 95% cofiece itervals with populatio bawith: =, = 400 BIAS PSS CCJ1 CCJ BIAS PSS CCJ1 CCJ BIAS PSS CCJ1 CCJ Moel 1 Moel 3 Moel 5 P = h PS h NR h CCJ P = 4 h PS h NR h CCJ Moel Moel 4 Moel 6 P = h PS h NR h CCJ P = 4 h PS h NR h CCJ Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 NOTE: Colum BIAS reports absolute ifferece betwee average of ˆθ (across simulatios) a θ 0. All figures times 100. Proof of Theorem For part (i), ote that ˆϑ l1,...,l, may be writte as a -varyig U- statistic (assumig without loss of geerality that s is eve), give by ( ) 1 1 ˆϑ l1,...,l, = u 1 (z i, z j ; b ), with (recall that s = l 1 + +l ) ( u 1 (z i, z j ; b) = b (+1+s) s ) k(x) x l 1 1 x l (y i y j ). x=(xi x j )/b First, chage of variables a itegratio by parts give ( E[u 1 (z i, z j ; b ) z i ]= k(u) s R x l 1 1 x l ḟ(x) y i x=xi ub s ) x l 1 1 x l ė(x) u. x=xi ub Seco, a Taylor series expasio gives E[u 1 (z i, z j ; b )]= ϑ l1,...,l + O(b mi(r,s) ). Next, lettig ˆϑ = ˆϑ l1,...,l, to save otatio, a Hoeffig ecompositio gives V[ ˆϑ ]=V[ ˆϑ 1, ]+V[ ˆϑ, ], where ˆϑ 1, = 1 [ E[u 1 (z i, z j ; b ) z i ] E[u 1 (z i, z j ; b )] ], i=1 a ( ) 1 1 [ ˆϑ, = u1 (z i, z j ; b ) E[u 1 (z i, z j ; b ) z i ] E[u 1 (z i, z j ; b ) z j ]+E[u 1 (z i, z j ; b )] ]. Fially, usig staar calculatios, V[ ˆϑ 1, ]=O( 1 ) a V[ ˆϑ, ]=O( b (++s) ), a the coclusio follows by Markov s Iequality. Table 4. Empirical coverage rates of 95% cofiece itervals with estimate bawith: =, = 400 BW PSS CCJ1 CCJ BW PSS CCJ1 CCJ BW PSS CCJ1 CCJ Moel 1 Moel 3 Moel 5 P = ĥ PS ĥ NR ĥ CCJ P = 4 ĥ PS ĥ NR ĥ CCJ Moel Moel 4 Moel 6 P = ĥ PS ĥ NR ĥ CCJ P = 4 ĥ PS ĥ NR ĥ CCJ NOTE: Colum BW reports sample mea of estimate bawiths.

14 Cattaeo, Crump, a Jasso: Robust Data-Drive Iferece for Average Derivatives 1083 Dowloae by [Uiversity of Michiga] at 08:39 15 Jauary 016 For part (ii), ote that ˆδ is also a -varyig U-statistic, give by ( ) 1 1 ˆδ = u (z i, z j ; b ), ( ) u (z i, z j ; b) = b xj x i k (y i y j ). b First, chage of variables gives E[u (z i, z j ; b ) z i ]= k(u)( y R i f (x i ub ) + v(x i ub )f (x i ub ) y i e(x i ub ) ) u. Seco, a Taylor s expasio gives E[ˆδ ]=E[σ (x)f (x)] + O(b mi(r,s+1+s) ). Next, a Hoeffig ecompositio gives V[ˆδ ]= V[ˆδ 1, ]+V[ˆδ, ],where a ˆδ, = ˆδ 1, = 1 [ E[u (z i, z j ; b ) z i ] E[u (z i, z j ; b )] ], i=1 ( ) 1 1 [ u (z i, z j ; b ) E[u (z i, z j ; b ) z i ] E[u (z i, z j ; b ) z j ]+E[u (z i, z j ; b )] ]. Fially, usig staar calculatios, V[ˆδ 1, ] = O( 1 ) a V[ˆδ, ]=O( b ), a the coclusio follows by Markov s Iequality. SUPPLEMENTAL MATERIALS Further Simulatio Results: This ocumet cotais a comprehesive set of results from the Mote Carlo experimet summarize i Sectio 5. These results iclue all combiatios of sample sizes ( = 100, = 400, = 700), imesio of regressors vector ( =, = 4), a kerel orers (P =, P = 4). (06_SmallBawithMSE_6Mar010- -Supplemetal.pf) [Receive October 009. Revise February 010.] REFERENCES Araillas-Lopéz, A., Hooré, B. E., a Powell, J. L. (007), Pairwise Differece Estimatio With Noparametric Cotrol Variables, Iteratioal Ecoomic Review, 48, [1079] Cattaeo, M. D., Crump, R. K., a Jasso, M. (009), Small Bawith Asymptotics for Desity-Weighte Average Derivatives, workig paper, UC Berkeley a Uiversity of Michiga. [ ,1079] (010), Bootstrappig Desity-Weighte Average Derivatives, workig paper, UC Berkeley a Uiversity of Michiga. [1071] Coppejas, M., a Sieg, H. (005), Kerel Estimatio of Average Derivatives a Differeces, Joural of Busiess & Ecoomic Statistics, 3, [1070] Deato, A., a Ng, S. (1998), Parametric a Noparametric Approaches to Price a Tax Reform, Joural of the America Statistical Associatio,93, [1070] Härle, W., a Stoker, T. (1989), Ivestigatig Smooth Multiple Regressio by the Metho of Average Derivatives, Joural of the America Statistical Associatio, 84, [1071] Härle, W., a Tsybakov, A. (1993), How Sesitive Are Average Derivatives? Joural of Ecoometrics, 58, [1071,1074] Härle, W., Hart, J., Marro, J., a Tsybakov, A. (199), Bawith Choice for Average Derivative Estimatio, Joural of the America Statistical Associatio, 87, [1071] Härle, W., Hilebra, W., a Jeriso, M. (1991), Empirical Eviece o the Law of Dema, Ecoometrica, 59, [1070] Horowitz, J., a W. Härle (1996), Direct Semiparametric Estimatio of Sigle-Iex Moels With Discrete Covariates, Joural of the America Statistical Associatio, 91, [1071] Ichimura, H., a To, P. E. (007), Implemetig Noparametric a Semiparametric Estimators, i Habook of Ecoometrics, Vol. 6B, es. J. Heckma a E. Leamer, New York: Elsevier, pp [1071] Matzki, R. L. (007), Noparametric Ietificatio, i Habook of Ecoometrics, Vol. 6B, es. J. Heckma a E. Leamer, New York: Elsevier, pp [1070] Newey, W. K. (1994), The Asymptotic Variace of Semiparametric Estimators, Ecoometrica, 6, [107] Newey, W. K., a Stoker, T. M. (1993), Efficiecy of Weighte Average Derivative Estimators a Iex Moels, Ecoometrica, 61, [1071,107] Newey, W. K., Hsieh, F., a Robis, J. M. (004), Twicig Kerels a a Small Bias Property of Semiparametric Estimators, Ecoometrica, 7, [1071,1079] Nishiyama, Y., a Robiso, P. M. (000), Egeworth Expasios for Semiparametric Average Derivatives, Ecoometrica, 68, [1071, 1073,1074] (005), The Bootstrap a the Egeworth Correctio for Semiparametric Average Derivatives, Ecoometrica, 73, [1071,1073, 1074] Powell, J. L. (1994), Estimatio of Semiparametric Moels, i Habook of Ecoometrics, Vol. 6, es. R. Egle a D. McFae, New York: Elsevier, pp [1070] Powell, J. L., a Stoker, T. M. (1996), Optimal Bawith Choice for Desity-Weighte Averages, Joural of Ecoometrics, 75, [1071, ] Powell, J. L., Stock, J. H., a Stoker, T. M. (1989), Semiparametric Estimatio of Iex Coefficiets, Ecoometrica, 57, [ ] Robiso, P. M. (1995), The Normal Approximatio for Semiparametric Average Derivatives, Ecoometrica, 63, [1071] Stoker, T. M. (1986), Cosistet Estimatio of Scale Coefficiets, Ecoometrica, 54, [1071] Wa, M., a Joes, M. (1995), Kerel Smoothig, Boca Rato, FL: Chapma & Hall/CRC.