Journal of Econometrics

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1 Journal of Economercs 7 ( 7 4 Conens lss avalable a ScVerse ScenceDrec Journal of Economercs ournal homepage: Inernaonal mare lns and volaly ransmsson Valenna Corrad a,, Waler Dsaso b, arcelo Fernandes c,d a Deparmen of Economcs, Unversy of Warwc, Socal Sudes Buldng, Covenry CV4 7AL, Uned Kngdom b Imperal College Busness School, Souh Kensngon Campus, London SW7 AZ, Uned Kngdom c School of Economcs and Fnance, Queen ary Unversy of London, le End, London E 4NS, Uned Kngdom d São Paulo School of Economcs, Geulo Vargas Foundaon, Rua Iapeva 474, São Paulo 33-, Brazl a r c l e n f o a b s r a c Arcle hsory: Receved December 9 Receved n revsed form Sepember Acceped 3 arch Avalable onlne 7 Aprl JEL classfcaon: C4 G5 hs paper gauges volaly ransmsson beween soc mares by esng condonal ndependence of her volaly measures. In parcular, we chec wheher he condonal densy of he volaly changes f we furher condon on he volaly of anoher mare. We employ nonparamerc mehods o esmae he condonal denses and model-free realzed measures of volaly, allowng for boh mcrosrucure nose and umps. We esablsh he asympoc normaly of he es sasc as well as he frs-order valdy of he boosrap analog. Fnally, we uncover sgnfcan volaly spllovers beween he soc mares n Chna, Japan, UK and US. Elsever B.V. All rghs reserved. Keywords: Condonal ndependence Jump-dffuson Noncausaly Quadrac varaon Realzed varance. Inroducon Even hough esng for he presence of nernaonal mare lns has a long hsory n asse prcng (see survey by Roll (989, he leraure has been ganng momenum snce he Ocober 987 crash. he man neres les on he analyss of volaly ransmsson across mares. Kng and Wadhwan (99 argue ha he srengh of nernaonal mare lns depends manly on volaly. As he laer declnes, he correlaon beween prce changes n he dfferen mares also decreases and so mare lns become weaer. In conras, nernaonal mare lns become sronger n perods of hgh volaly. he dea s ha, wh Bayesan updae of belefs abou varances, a common shoc o all mares would resul n an ncrease n he perceved varance of any common facor and hence of he correlaon. hs paper develops formal sascal ools for esng condonal ndependence beween volaly processes. We propose a nonparamerc approach n sar conras wh mos papers n he leraure. In parcular, under he assumpon ha asse prces follow a mulvarae ump-dffuson process, we show how o es Correspondng auhor. el.: ; fax: E-mal addresses: [email protected] (V. Corrad, [email protected] (W. Dsaso, [email protected], [email protected] (. Fernandes. wheher he condonal densy of asse A s negraed varance (say, over a day also depends on asse B s negraed varance. he procedure s nonparamerc n ha, apar from some mld regulary condons, we mpose no paramerc assumpon on he funconal form of he drf and dffusve erm as well as on he presence of leverage or umps. Broadly speang, our esng procedure checs wheher condonng also on asse B s negraed varance, raher han exclusvely on asse A s pas negraed varance, enals a dfferen condonal densy for asse A s negraed varance. As we do no observe daly varance, we rely on modelfree realzed measures based on nraday reurns. We focus on he condonal densy for wo reasons. I allows for nonlnear channels of volaly ransmsson n conras o he sandard pracce of carryng ou ponwse analyses based on he condonal mean of he volaly. In addon, he dsrbuon of he daly negraed varance s of parcular neres for prcng varance swap conracs (Carr e al., 5. he asympoc heory we develop proceeds n hree seps. Frs, we esablsh he asympoc normaly of he unfeasble sascs based on unobservable negraed varances usng eher ernel or local lnear smoohng. Second, we provde condons on he rae of growh of nraday observaons relave o he number of days under whch he feasble sasc, based on realzed measures, s asympocally euvalen o s unfeasble counerpar. Our /$ see fron maer Elsever B.V. All rghs reserved. do:.6/.econom..3.3

2 8 V. Corrad e al. / Journal of Economercs 7 ( 7 4 seng s general enough o allow for mcrosrucure nose as well as for umps. hrd, we esablsh he frs-order valdy of boosrap-based crcal values based on he m ou of n (henceforh, moon boosrap (Breagnolle, 983; Bcel e al., 997. Boosrap nference s ypcally more robus o varaons n he bandwdh as he laer plays roughly he same role on boh he orgnal and boosrap sascs. one Carlo smulaons ndeed reveal ha he moon boosrap wors reasonably well even n relavely small samples. We carry ou he nonparamerc analyss usng dfferen realzed measures as a means o dscern he man channels of uadrac varaon ransmsson beween soc mares. he dea s smlar o Aï-Sahala and Jacod s ( analogy of urnng he nobs of a measuremen devce runnng specrographc analyss. Noe ha we consder only wo unng parameers. he frs s he ype of he realzed measure. I essenally combnes he runcaon and power nobs of Aï-Sahala and Jacod (forhcomng, enablng us o elmnae he conrbuon of eher umps or mare mcrosrucure nose o he uadrac varaon. he second s he samplng freuency, whch we vary as anoher means o allevae mare mcrosrucure effecs. For nsance, he rpower varaon s robus o umps and hence would capure only he conrbuon of he dffusve volaly o he uadrac varaon f usng reurns a a suffcenly low freuency, say 3 mn. In urn, he realzed ernel approach lls he conrbuon of he mare mcrosrucure nose o he uadrac varaon, bu no ha of umps. We hus vew he dfferences n he es resuls usng he above realzed measures as very nformave. Suppose we fnd no sgnfcan ransmsson usng he rpower varaon, bu sgnfcan evdence of spllovers usng he realzed ernel measure. hs would mply ha he man channel of ransmsson s lely hrough umps gven ha reecon of he null occurs only f we do no exclude he ump conrbuon o he uadrac varaon. We nvesgae he lns beween nernaonal soc mares usng nraday daa from Chna, Japan, UK, and US from January o December 5. By varyng he realzed measure, we show ha he prmary channel of ransmsson whn he uadrac varaon of asse prces s hrough spllovers n he negraed varance. he emprcal evdence of volaly spllovers becomes ndeed sronger once we conrol for umps and/or mare mcrosrucure effecs. he only excepon s he ransmsson from Chna o he US, whch runs manly hrough prce umps. here s o some exen sgnfcan nerdependence beween all of he soc mare ndces we consder. Furher, he volaly ransmsson mechansms are mosly symmercal n ha we normally reec he null of condonal ndependence n boh drecons usng almos he same realzed measures. Fnally, we show ha he fnancal mares seemngly prce hese nerconnecons n vew ha here are also volaly spllovers o he opons-mpled mare volaly n he US. In oher words, nvesors form her expecaons abou he fuure volaly of he S&P 5 ndex accounng for he soc mare nformaon comng from Chna, Japan and, especally, he UK. o ensure ha he volaly spllovers we uncover are ndeed genune, we carry ou a seres of robusness checs. We frs examne wheher he evdence s spurous due o me-seres perssence. We redo he spllover analyss for he US soc mare addng an exra conrol, namely, he VIX ndex. he laer s a mare volaly ndex from he Chcago Board Opons Exchange (CBOE ha gauges he opons-mpled volaly of he S&P 5 ndex. Condonng on he VIX ndex should effecvely conrol for furher seral dependence n he negraed varance gven he srong ln beween he acual volaly and s rs-neural expecaon (Band and Perron, 6. here are no ualave changes n our man resuls; n fac, he evdence of sgnfcan ransmsson from he UK and Japan become sronger. We hen nvesgae wheher he volaly spllovers runnng from Chna and Japan o he US are ruly abou Asa-specfc shocs by furher condonng on he FSE realzed measure over he same day. he dea s ha he FSE ndex should reflec any global shoc, hough no necessarly Asa-specfc shocs ha may affec he US. We observe no ualave change n he resuls for Japan. In sar conras, we now fnd much more sgnfcave volaly ransmsson from Chna o he US. Furher condonng on he realzed measure of he FSE ndex seems o help dsenangle he Chna-specfc spllovers from he global volaly shocs. Fnally, we also carry ou a smlar analyss consderng hourly realzed measures o chec for a swfer reacon me possbly hrough umps n prces or n volaly (Kng and Wadhwan, 99. We fnd srong evdence of volaly spllovers across all mares, especally f conrollng for mare mcrosrucure nose by means of he realzed ernel approach. he only excepon s ha we fnd no sgnfcan volaly ransmsson from Japan o he UK. here are several papers n he leraure ha carry ou smlar, hough mosly paramerc, analyses of volaly ransmsson across nernaonal soc mares. Engle and Ng (988, Hamao e al. (99, Engle e al. (99, Kng e al. (994, Ln e al. (994, Karoly (995, and Wongswan (6 employ mulvarae GARCH models o show ha volaly spllovers ndeed occur across foregn exchange mares as well as nernaonal soc mares, noably, beween Japan, UK and US. In conras, Cheung and Ng (996, Hong (, Panelds and Ps (4, Senser and van D (4, and van D e al. (5 propose smple ess of noncausaly n varance based on he cross-correlaon beween leads and lags of suared GARCH-sandardzed resduals. ore recenly, Goureroux and Jasa (7 address causaly n varance (or even n hgher-order momens by means of approxmae condonal log-laplace ransforms of compound auoregressve processes. he esng sraegy of he above papers manly dffers from ours n hree respecs. Frs, hey assume a dscree-me daa generang mechansm n whch he condonal varance s a measurable funcon of pas asse reurns. In conras, we assume ha asse prces follow a mulvarae ump-dffuson process and hen es for spllovers whn he uadrac varaon. Second, her ess are sensve o msspecfcaons n he condonal mean and varance euaons, whereas he nonparamerc naure of our ess mnmzes any msspecfcaon rs. hrd, hey do no conemplae any sor of nonlnear dependence beween varances as opposed o our esng procedures, whose nonrval power agans nonlnear channels of volaly ransmsson resuls from loong a he whole volaly dsrbuon. he esng sraegy pu forh by Debold and Ylmaz (, 9 s he closes o ours. hey examne volaly spllovers by means of a VAR approach usng range-based measures of volaly. he laer belongs o he class of realzed measures robus o mare mcrosrucure nose and hence provdes a conssen nonparamerc esmaor of he uadrac varaon of he underlyng dffuson process. However, Debold and Ylmaz (9 do no sudy how he realzed esmaon of he uadrac varaon affecs he subseuen VAR analyss. In addon, her approach also dffers from ours n ha hey focus on lnear spllovers n he condonal mean of he uadrac varaon of asse prces. he remander of hs paper ensues as follows. Secon descrbes he daa generang process we assume for asse prces and dscusses he hypoheses of neres. Secon 3 esablshes he asympoc normaly of he unfeasble sasc based on negraed varances. Secon 4 derves he asympoc euvalence of feasble es sascs ha subsue realzed measures for negraed varances. Secon 5 frs esablshes he frs-order

3 V. Corrad e al. / Journal of Economercs 7 ( valdy of boosrap-based crcal values and hen examnes her fne-sample properes of he resulng es hrough one Carlo smulaons. Secon 6 nvesgaes wheher here are sgnfcan volaly spllovers across nernaonal soc mares usng daa from Chna, Japan, UK and US. Secon 7 offers some closng remars, whereas he Appendx collecs all echncal proofs.. Volaly ransmsson: seup and ssues In hs secon, we dscuss how o analyze volaly ransmsson hrough nonparamerc ess of condonal ndependence. For noaonal smplcy, we resrc aenon o esng wheher he daly varance of asse B affecs he dynamc of asse A s daly varance. I s sraghforward o consder more han wo asses, hough he usual concern wh he curse of dmensonaly apples. See also Secon 6.. for emprcal applcaons wh hree soc mares. Le p and p B, denoe he log-prces of asses A and B wh nsananeous (sochasc volaly gven by σ and σ B,, respecvely. We assume ha asse prces follow a connuous-me semmarngale process, as mpled by a very wea form of he noarbrage propery (Delbaen and Schachermayer, 994. Noe ha hs s no a very srngen reuremen gven ha semmarngales nes almos every connuous-me model n fnance (e.g., Harrson and Plsa (98, Andersen e al. (7. o fx deas, we consder a smple example n whch asse prces follow a mulvarae sochasc-volaly process wh umps: dp µ dp B, dσ = µ B, b,a (σ, σ d + dσ B, B, b,b (σ, σ B, + dw, + ρ A σ σ B dj, dj, dj 3, dj 4, σ AB, ρ B σ B, dw, ρ A σ + b,a (σ, σ dw 3, + ρ B σ B, B, b,ab (σ, σ dw 4,, ( b,ba (σ, σ B, B, b,b (σ, σ B, where (W,,..., W 4, are ndependen sandard Brownan moons and (J,,..., J 4, are ump processes. he laer s such ha dj, = A c (u N ( d, du, for =,..., 4, where N ([, ], A s a Posson measure ha couns he number of umps beween and, whose sze c (u s an d random varable n A. he ump process s such ha, over a fne me span, here s only a fne number of umps. Alhough we allow for leverage effecs beween asse prces and her own sochasc volaly hrough he correlaon coeffcens ρ A and ρ B, for he sae of smplcy, we assume away cross-leverage effecs by mposng zero correlaon beween one asse prce and he sochasc volaly of he oher asse. We also assume ha he drf componens µ and µ B, follow predcable processes. hs s no srngen for her role s asympocally neglgble n he conex of volaly ransmsson. Fnally, as sandard n mulvarae sochasc volaly models, we suppose for smplcy ha asse prces do no drecly affec volaly. I s possble o decompose he uadrac varaon process of a gven asse prce, say p A, over he me nerval [, ] no he par due o he dsconnuous ump componen p D A and he Cross-leverage effecs would only add oher possble channels of volaly ransmsson beween asses A and B. Alhough hs would brng abou addonal msspecfcaon rs n any paramerc approach o es for volaly ransmsson, does no affec n any way he nonparamerc procedure we propose. par due o he connuous dffusve componen p C A. In parcular, p A = p C + A p D A, where p D A A c(u N (ds, du and p C A corresponds o he negraed varance over he me nerval [, ], namely, IV = σ A,s ds + σ AB,s ds. Now, recall ha s σ A,s ds = b,a (σ, σ A,u B,u du ds s dj 3,u ds s b,a (σ, σ A,u B,u dw 3,u ds s b,ab (σ, σ A,u B,u dw 4,u ds and ha p C, A pc B = (σ A,sσ BA,s + σ B,s σ AB,s ds s he negraed covaraon beween p A and p B over he [, ] nerval. I s easy o see ha IV B does no affec IV f and only f ( σ AB,s = a.s. ( J 3,s s ndependen of J 4,s ( b,a (σ A,u, σ B,u = b,a(σ A,u a.s. (v b,a (σ A,u, σ B,u = b,a(σ A,u a.s. (v b,ab (σ A,u, σ B,u = a.s.. If any of he above condons fals o hold, hen IV remans dependen upon IV B, even afer condonng on s own pas values. Regardless of wheher condon ( holds, volaly nerdependence may arse even n he case ha volaly s a measurable funcon of pas asse prces due o a volaon of condon (, whch reduces o b,a (p A,u, p B,u = b,a (p A,u almos surely. I hus urns ou ha does no suffce, nor s necessary, o mpose ha he cross-varaon process p C, A pc B s zero almos surely. In prncple, s possble o es drecly wheher condons ( (v hold f one s ready o specfy he paramerc funconal forms of he drf, dffusve, and ump erms. he oucome would however depend heavly on he correc specfcaon of he daa generang process n (. o mnmze he rs of msspecfcaon, we ae a nonparamerc roue. Our goal s o chec wheher he daly varance of asse B helps predc he daly varance of asse A. We hus formulae a esng procedure ha focuses on he densy resrcons mpled by condonal ndependence: H : f IV+ IV ( A,IV ( B y IV ( A, IV ( B B,+ B,+ = f IV+ IV ( A y IV ( A where f IV+ IV ( A and f IV+ IV ( A densy of IV + gven IV ( A and a.s. for all y, (,IV ( B B,+ denoe he condonal IV ( A, IV ( B B,+, wh IV ( A (IV,..., IV A + and IV ( B B,+ (IV B,,..., IV B,+ B + sandng for vecors of dmenson A and B concernng he nformaon abou he negraed varances of asses A and B, respecvely. We allow for {, } so as o conrol for me dfferences beween he mares under consderaon. As usual, we defne he alernave hypohess as he negaon of he null hypohess. In general, he negraed varance does no follow a fneorder arov process. 3 hs means ha, o es for noncausaly For nsance, as he oyo Soc Exchange closes before he openng of he New Yor Soc Exchange, one may condon on he same day nformaon ( = raher han on nformaon from he prevous radng day ( =. 3 eddah (3 shows, for nsance, ha he CIR specfcaon enals an ARA(, process for he negraed varance.

4 V. Corrad e al. / Journal of Economercs 7 ( 7 4 n varance, one would have o le he number of condonng varables ( A and B o ncrease wh he sample sze. hs s unfeasble due o he curse of dmensonaly and hence we consder he less ambous null of condonal ndependence by fxng he number of condonng varables n ( o a fne (and, possbly, small fgure. 4 o mplemen a nonparamerc es for H, we propose a sasc ha gauges he dscrepancy beween he nonparamerc esmaes of he densy funcons ha appear n (. In parcular, our es sasc hnges on he sample counerpar of he followng negraed suare relave dsance f IV+ IV ( A,IV ( B B,+ (y x ( A, x ( B f IV+ IV ( A f IV+ IV ( A (y x ( A (y x (A π(y, x ( A, x ( B dy dx ( A dx ( B, (3 where IV (, (IV,,..., IV, + wh IV, denong asse s negraed varance over day ( = A, B. We employ a weghng scheme π(,, so as o avod he lac of precson ha afflcs condonal densy esmaon n areas of low densy of he condonng varables. he negraed suare dsance ha we adop n (3 s convenen because faclaes he dervaon of he asympoc heory. Bcel and Rosenbla (973, Aï-Sahala (996, Aï-Sahala e al. (, 9, and Amaro de aos and Fernandes (7 use smlar suared dsance measures, hough one could lewse employ enropc pseudo-dsance measures as n, e.g., Robnson (99 and Hong and Whe (4. Alernavely, one could also mplemen esng procedures based on he cumulave dsrbuon funcons, such as he Kolmogorov Smrnov and Cramér von ses ess. However, Fan (996 provdes ample evdence ha hese goodness-of-f ess have lle power agans local devaons such as bumps and global devaon a hgher freuences of he Fourer expanson of he cumulave dsrbuon funcon. We derve he lmng dsrbuon of he es sasc n (3 n Secons 3 and Asympoc heory for he unfeasble case o smplfy noaon, we denoe by Y he negraed varance of neres IV +, whereas we denoe he condonng vecors IV ( A and IV ( A, IV ( B B,+ respecvely by X ( A and X (, wh = A + B sandng for he hgher dmenson. he null hypohess of condonal ndependence n ( now reads H : f Y X ((y x ( = f Y X ( A (y x ( A for all (y, x (. (4 We employ local lnear smoohng o esmae boh he rgh- and lef-hand sdes of (4. he sample analog of (3 hen s fy X ((Y X ( fy X ( A (Y X ( A fy X ( A (Y X ( A π(y, X (, (5 where he condonal densy esmaes fy X ( and fy X ( A derve from local lnear smoohng usng dfferen ses of bandwdhs. Denoe by β (y, x ( = (β (y, x (,β (y, x (,...,β (y, x ( wh x ( = (x,..., x he argumen ha mnmzes 4 In he emprcal applcaon n Secon 6, we add he VIX ndex as an exra conrol so as o accommodae for he non-arovan naure of he daa. Alernavely, one could adap our asympoc heory o deal wh dmenson reducon echnues as n, e.g., Hall and Yao (5 and Fan e al. (9. [K b (Y y β β (X x β (X x W h (X x, (6 = where K b (u = b K(u/b and W h (u = h W(u/h are symmerc ernels. he local lnear esmaor of he condonal densy funcon f Y X ( s gven by fy X ((y x ( = β (y, x (. he local lnear esmaor fy X ( A of he lower dmensonal condonal densy s analogous for x ( A = (x,..., x A. o esablsh he lmng behavor of he es sasc n (5, we shall rely on he followng assumpons. Assumpon A. he produc ernels W (u = = W(u and W (u = A = W(u res on a symmerc, nonnegave, connuous unvarae ernel W of second order wh bounded suppor [, ] for. he ernel W s also a leas wce dfferenable on he neror of s suppor. he symmerc ernel K s of order s (even neger and a leas wce dfferenable on he neror of s bounded suppor [, ]. Assumpon A. he densy funcons f Y X ((y x ( and f Y X ((y, x ( are r-mes connuously dfferenable n (y, x ( wh bounded dervaves and wh r s. he same condon also holds for he lower-dmensonal densy funcons f Y X ( A (y x ( A and f Y X ( A (y, x ( A. Assumpon A3. he weghng funcon π y, x ( s connuous and negrable, wh second dervaves n a compac suppor. Assumpon A4. he sochasc process Y, X ( s srcly saonary and β-mxng wh β τ = O (ρ τ, where < ρ <. Assumpons A A4 are ue sandard n he leraure on local lnear smoohng (see, e.g., Fan e al. (996 and hence we only brefly dscuss hem n wha follows. Assumpon A rules ou hgher-order ernels for W essenally o ensure he posvy of he creron funcon n (6. Assumpons A and A3 reure ha he weghng scheme and he densy funcons are boh well defned and smooh enough o adm funconal expansons. Assumpon A4 resrcs he amoun of daa dependence, reurng ha he sochasc process s absoluely regular wh geomerc decay rae. Alernavely, one could assume α-mxng condons as n Aï-Sahala e al. (9 and Gao and Hong (8, hough he condons under whch he uadrac varaon of a ump-dffuson process sasfes Assumpon A4 are ue wea (see dscusson n Corrad e al. (. See also Chen e al. ( for some advanages of he β-mxng assumpon relave o he α-mxng condon n he conex of nonparamerc densy esmaon. he scaled and cenered verson of (5 reads as Λ = Ω f Y X ((Y X ( fy X ( A (Y X ( A h/ b / fy X ( A (Y X ( A π(y, X ( h / b / µ, h / h A A b / µ, + h / A b / µ 3,, (7 where µ,, µ,, µ 3,, and Ω are conssen esmaors of he asympoc bas erms and varance, respecvely (see Appendx A for defnons.

5 V. Corrad e al. / Journal of Economercs 7 ( 7 4 o ensure he asympoc sandard normaly of Λ n (7, we mus mpose some condons on he raes a whch he bandwdhs shrn o zero n Lemmas (see Appendx B. I urns ou ha here are no bandwdh raes ha only mee he condons (v and (v n Lemma for > f K s of second order (s = and for > 3 f K s of hgher order (s 4. he man snag s ha, as aforemenoned, one canno ncrease he order of he ernel funcon W n he local-lnear smoohng. In urn, resrcng aenon o a mos wo condonng varables and o secondorder ernels, he bandwdh condons n Lemmas and hold f b = h = O /5 and h = o /m wh 3/7 < m 5 9. Noe ha b and h have opmal raes, whereas he bandwdh for he full condonng se enals some degree of undersmoohng. As expeced, he bandwdh condons n Lemmas and n general reure more undersmoohng as he dmensonaly of he condonng ses ncreases. We are now ready o sae our man resul concernng he asympoc behavor of he normalzed es sasc n (7 under boh he null and alernave hypoheses. heorem. Le Assumpons A A4 hold as well as he bandwdh condons ( (v n Lemmas and. I follows for 3 and A ha d ( Under he null hypohess H, Λ N(,. ( Under he alernave hypohess H A, Pr( h / b / Λ > ε for any ε >. Par ( of heorem provdes he means o compue he asympoc crcal values of he es, whereas ( esablshes conssency. If one resrcs aenon o he case n whch = and A =, he above resul follows almos mmedaely from Aï-Sahala e al. s (9 Corollary o heorem. Ye, even n hs smple case, s necessary o accoun for he bas componen ha arses due o he nonparamerc esmaon of he lowerdmensonal model. We now deal wh hgher dmensons (.e., > 3 by employng a Nadaraya Wason esmaor based on hgher-order ernels. ore specfcally, consder f Y X ((y x ( = f Y,X ((y, x ( f X ((x ( where = W h (X ( x ( K b (Y y, W h (X ( x ( W h (u = h p = W(u /h. Defne fy X ( A analogously A W(u = /h A. usng he produc ernel gven by W ha (u = h A A We nex modfy Assumpon A o accommodae hgher-order ernels. Assumpon A5. he ernel funcons K and W are of even neger orders (s, s, symmerc, connuous, and a leas wce dfferenable on he neror of her bounded suppor [, ]. he ernel-based es sasc s essenally analogous o he one based on local lnear smoohng: Λ = Ω h/ b / f Y X ((Y X ( fy X ( A (Y X ( A f Y X ( A (Y X ( A π(y, X ( h / + h / A b / µ 3, b / µ, h / h A A b / µ,. (8 We provde expressons for he bas and scalng erms n Appendx A. As before, o ensure he asympoc sandard normaly of Λ n (8, we mus mpose he bandwdh condons n Lemmas 4 and 5 (see Appendx B. he fac ha we can now ncrease he order of he ernel ha we employ o smooh he condonng se s very advanageous. We are now able o fnd bandwdhs ha sasfy he condons for = 4 and A = 3 as long as W s of fourh order. hs s a huge mprovemen relave o he local lnear case. Surprsngly, urns ou ha pays off o use a second-order ernel for K f we fx boh b and h A o her opmal raes. hs s he confguraon ha enals he leas undersmoohng. For nsance, n he even ha b = O /5 and h = O /9, we reure only a b of undersmoohng for he larger condonng se, vz., h = o /. As before, we have o ncrease he degree of undersmoohng as he dmensonaly of he condonng se grows: e.g., for = 3 and A =, he bes confguraon for he bandwdhs s b = O /5, h = O /, and h 3 = o 9/95. he nex resul documens he asympoc sandard normaly of he ernel-based es sasc n (8 under he null as well as he conssency of he resulng es. heorem. Le he bandwdh condons ( (v n Lemmas 4 and 5 hold as well as Assumpons A A5. I hen follows ha d ( Under he null hypohess H, Λ N(,. ( Under he alernave hypohess H A, Pr( h / b / Λ > ε for any ε >. A nonrval alernave s o develop he asympoc heory for hgher-order local polynomals wh K of order s (and a secondorder ernel W. For nsance, a local polynomal of second or hrd order would enal a bas of he same order of magnude as he ernel-based approach wh boh K and W of order s, hereby meeng he condons for heorem. However, he order of he local polynomal would affec he ernel consan n he asympoc varance of he densy esmaor even f does no affec s order of magnude (Fan and Gbels, 996. For nsance, he varance of he second- and hrd-order local polynomal esmaors s.68 mes he varance of he local lnear polynomal for a Gaussan ernel. Noe also ha local polynomal smoohng s only free of boundary bas for polynomals of odd order. heorems and form he bass for asympocally locally srcly unbased ess for he condonal ndependence null H n (4 based on local lnear and ernel smoohng, respecvely. o allevae he boundary bas ha hauns ernel smoohng, one could always ae he log of he daly varances before esng for condonal ndependence as an alernave o weghng down by means of π(y, X ( any realzed measure close o zero. 4. Accounng for he realzed measure esmaon he asympoc heory so far consders he unfeasble es sasc n (7. In hs secon, we show he asympoc euvalence of he correspondng feasble es sasc ha replaces negraed varances by realzed measures. o dscuss he mpac of esmang he negraed varance, we mus frs esablsh some noaon ha maes explc he dependence on he number of nraday observaons ha we employ o compue he realzed measure. We hus denoe he me seres of realzed measures by Y, and X (d,, where s he number of nraday observaons and d {, A }. Le β ( (y, x (d = β ( (y, x (d,...,β ( d (y, x (d denoe he argumen ha mnmzes

6 V. Corrad e al. / Journal of Economercs 7 ( 7 4 Kb (Y, y β β (X, x β d (X d, x d d = W hd (X, x. he local lnear esmaor of he condonal densy s ( f Y X (d (y x (d = β ( (y, x (d, yeldng he followng feasble es sasc Λ, = Ω, h/ b / ( f Y X ( (Y, X ( (, f Y X ( A (Y, X ( A, ( f Y X ( A (Y, X ( A, π(y,, X (, h / b / µ (, h / h A A b / µ ( +, h/ A b / µ ( 3,, (9 where µ (, dffers from µ, because employs realzed measures raher han he rue negraed varance. Le N,, = Y Y, and N,, = X, X,, for d {, A } denoe he errors semmng from he esmaon of he negraed varance. o ensure he asympoc euvalence beween he unfeasble and feasble es sascs, we mus resrc he rae a whch he momens of he esmaon errors converge o zero. We do ha by consranng he momens of he drf and dffusve funcons as well as of he nose due o mare-mcrosrucure effecs. Assumpon A6. he drf erms of ( are connuous locally bounded processes wh E µ, <, whereas he dffusve funcons are càdlàg wh E σ, < and he ump componens c ( are d wh E c ( < for some and, {A, B}. In addon, he mcrosrucure nose s d wh symmerc dsrbuon around zero and wh fne h momen for some. Assumpon A6 ensures ha he condons n Corrad e al. s ( Lemma hold and hence ha here exss a seuence a, wh a as, such ha E N,, = O for a / some and d {, A }. Noe ha hs esablshes a bound o he h momen of he absolue esmaon error ha depends on he realzed measure we employ o esmae he negraed varance. In parcular, a = for he realzed varance (Andersen e al., ; Barndorff-Nelsen and Shephard, and rpower varaon (Barndorff-Nelsen and Shephard, 4, whereas a = /3 for he wo-scale realzed varance (Zhang e al., 5 and a = for he mul-scale realzed varance (Zhang, 6; Aï-Sahala e al., and he realzed ernel esmaor (Barndorff-Nelsen e al., 8. heorem 3. Le Assumpons A A4 and A6 hold as well as he bandwdh condons ( (v n Lemmas and. Le also ( (ln / a / (h + h A + b as, for as defned n Assumpon A6. I follows for 3 ha + d ( Under he null hypohess H, Λ, N(,. ( Under he alernave hypohess H A, Pr( h / b / Λ, > ε for any ε >. As for he feasble ernel-based sasc, defne Λ, analogously o Λ n (8 bu replacng (Y, X ( wh (Y,, X (,. he nex resuls documens asympoc euvalence n he conex of ernel densy esmaon. heorem 4. Le Assumpons A A6 hold as well as he bandwdh condons ( (v n Lemmas 4 and 5. Also, le ( + (ln / a / (h + h A + b as, for as defned n Assumpon A6. I follows ha d ( Under he null hypohess H, Λ, N(,. ( Under he alernave hypohess H A, Pr( h / b / Λ, > ε for any ε >. heorems 3 and 4 esablsh ha he asympoc euvalence beween unfeasble and feasble es sascs necessaes ha he number of nraday observaons grows fas enough relave o he number of days. As usual, here s a radeoff beween usng a non-robus realzed measure wh a = a a freuency for whch mcrosrucure nose s neglgble and employng a mcrosrucure-robus realzed measure wh a = a he hghes avalable freuency. 5. Boosrap crcal values I s well nown ha he asympoc behavor of nonparamerc ess does no always enal a reasonable approxmaon n fne samples and ha her resuls may heavly depend on he bandwdh choce (see, e.g., Fan (995, and Fan and Lnon (3. In wha follows, we am o allevae such concerns by employng resamplng echnues. here are a number of ssues ha one mus bear n mnd, hough. Frs, gven he nonparamerc naure of he null hypohess, we canno rely on sandard resamplng algorhms based on eher paramerc or wld boosrap mehods (Härdle and ammen, 993; Andrews, 997; Aï-Sahala e al., 9. Second, negraed varance does no follow a arov process, rulng ou as well boosrap algorhms for arov seuences (Raarsh, 99; Paparods and Pols, ; Horowz, 3. 5 hrd, a sandard nonparamerc boosrap would also fal o mmc he lmng dsrbuon of our es sascs for hey nvolve degenerae U-sascs (Breagnolle, 983; Arcones and Gné, 99. o crcumven he above ssues, we resor o a varaon of he sandard moon boosrap by Breagnolle (983 and Bcel e al. (997. We sample ou of daly realzed measures by blocs (raher han ndvdually so as o cope wh me-seres dependence. In addon, for each boosrap sample, we compue he es sascs usng a bandwdh vecor (h, h A, b ha shrns o zero a he same as before rae, bu dependng on (raher han. hs mples dsnc orders of magnude for he bas erms n he orgnal and boosrap sascs and hence hey do no cancel ou. hs s n sar conras wh he bas cancellaon ha happens whn he conex of paramerc and wld boosrap. I noneheless remans unnecessary o compue he scalng erm correspondng o he asympoc varance of he es sasc. he (unscaled boosrap counerpars of (7 and (8 hen are respecvely Λ = h/ b/ f Y X ( (Y f Y X ( A (Y f Y X ( A (Y X ( A ( X X ( A 5 Resrcng aenon o he class of egenfuncon sochasc volaly models would acually yeld negraed varances wh a fne ARA represenaon (eddah, 3 and hence approxmaely arov (even f of hgher order.

7 V. Corrad e al. / Journal of Economercs 7 ( and Λ = π(y, X ( h / + h / A b / h/ b/ µ 3, b / f Y X ( (Y π(y, X ( h / + h / A b / µ 3, ( X b / µ, h/ h A A b / µ, f Y X ( A (Y f Y X ( A (Y X ( A X ( A µ, h/ h A A b / µ, (. ( As before, we provde n Appendx A he expressons for he bas erms n ( and (. We nex esablsh he frs-order valdy of he moon boosrap only for he unfeasble es sasc gven ha he asympoc euvalence beween he feasble and unfeasble boosrap sascs ensues along he same lnes as n heorem 3 provded ha Assumpon A6 holds. heorem 5. Le Assumpons A A4 hold and le he bandwdh condons ( (v n Lemmas and hold for (h, h A, b and n leu of (h, h A, b and, respecvely. Leng,, / yelds, for 3 and for any ε >, Pr sup v R Pr (Λ v Pr(Ω Λ v > ε under he null, whereas Pr Pr h / b / Λ > ε under he alernave. In addon, replacng Assumpon A wh A5 and leng bandwdh condons ( (v n Lemmas 4 and 5 hold ensure he frs-order valdy of he moon boosrap for he ernel-based es sasc n ( as well. I s mmedae o see ha he sample and boosrap sascs have he same lmng dsrbuon under he null, whereas hey dverge a dfferen raes under he alernave. In parcular, ( and ( dverge a a slower rae h / b / relave o her sample counerpars. In pracce, one mus deal wh he feasble boosrap ha replace (Y, X ( wh he es sascs Λ, and Λ, correspondng realzed measures (Y,, X (,. Assumpon A6 ensures ha he saemen n heorem 5 also apples f one subsues Λ, and Λ, for Λ and Λ, respecvely. he boosrap crcal values for Ω, Λ, are readly avalable from he emprcal dsrbuon of Λ, usng a large number, say B, of boosrap sascs. o chec wheher he moon bloc-boosrap ndeed enals accurae crcal values, we run a lmed one Carlo sudy. In parcular, we smulae nraday reurns from wo ndependen mean-reverng CIR processes (Cox e al., 985 and hen examne how he emprcal sze of our wo-sep esng procedure vares accordng o he bandwdh choce. We employ he CIR process no only because s a sandard model n fnance, bu also because mples a smple ARA(, process for he negraed varance (eddah, 3. For each of he one Carlo replcaons, we smulae nraday daa from dp A = κ A (µ A P A d + ς A P A dw A dp B = κ B (µ B P B d + ς B P B dw B, where W A and W B are wo ndependen Brownan moons, usng an Euler dscrezaon scheme wh a reflecon devce o ensure posvy. o enal realsc asse prce processes, we fx he parameer vecors o (κ A, µ A, ς A = (.8,.5,. and (κ B, µ B, ς B = (.,.,.3. Afer burnng he frs observaons of he sample, we employ he las nraday observaons, where and correspond respecvely o he number of nraday observaons whn a day and o he number of days. We focus on he relavely small sample szes of = 44 and {4, 6} so as o assess how mporan s he condon n heorem 3 ha calls for o grow a a faser rae han. From he nraday log-reurns, we rereve he daly realzed varances RV Ad and RV Bd for each day d =,..., and hen es for condonal ndependence of asse A s daly varance wh respec o asse B s daly varance by loong a he condonal densy of X = ln RV Ad gven Y = ln RV Ad and Z = ln RV Bd. We frs sandardze he daa by her mean and sandard devaon and hen esmae he condonal denses usng local lnear smoohng wh Gaussan ernels. o comply wh he bandwdh condons, we frs adus he rule-of-humb bandwdhs wh a Gaussan reference o he approprae rae, resulng n b = h = (4/3 /5 /5 and h = 9/5 / ln ln. We hen mulply hese bandwdhs by scalng facors κ b {/, 3/4, } and κ h {3/4,, 3/, 5/}, wh κ b < κ h, so as o assess sensvy. For smplcy, we employ a weghng scheme relyng on a sandard mulvarae normal densy: π XYZ (x, y, z = φ(x, y, z = φ(xφ(yφ(z. Gven ha he dsrbuon of he realzed varance logarhm s ypcally close o normal (Andersen e al.,, 3, such a weghng funcon eeps he focus on he bul of he daa raher han on exreme levels of volaly. o ensure a reasonable number of daly observaons n he boosrap arfcal samples, we consder =, hough furher smulaons show ha he resuls are ue robus o varaons n he boosrap sample sze. able repors he resuls for B = 3 boosrap samples usng a bloc lengh of 4 daly observaons (.e., approxmaely /4. Despe he fac ha <, we fnd ha emprcal sze s close o nomnal as long as κ h s no oo hgh relave o κ b. All n all, fxng κ b = 3/4 and κ h = yelds very encouragng resuls, hereby provdng some gudance for he bandwdh choce n pracce. 6. Spllovers across nernaonal soc mares We examne wheher here are volaly spllovers beween Chna, Japan, UK and US usng daa from her man soc mare ndces. In parcular, we collec ulra-hgh-freuency daa for he SSE B share ndex, he opx ndex, he FSE ndex, and he S&P 5 ndex from Reuers, avalable a he Secures Indusry Research Cenre of Asa-Pacfc ( Before descrbng he daa, s mporan o usfy our ndex selecon by esablshng some bacground. We adop he S&P 5 ndex o measure he movemens n he US soc mare because s one of he man bellwehers for he US economy. In addon, he CBOE also publshes a volaly ndex (VIX ha measures mare expecaons of he near-erm volaly mpled by he S&P 5 ndex opons. hs s convenen because provdes an exra conrol varable o cope wh he me-seres perssence n he daly volaly of he S&P 5 ndex. We also consder he UK soc mare, as represened by he FSE ndex, because s he man fnancal hub n Europe. As for he opx ndex, s a weghed ndex gaugng he performance of he mos lud socs wh he larges mare capalzaon on he oyo Soc Exchange (SE. here are wo connuous radng sessons on he SE, wh a call aucon-procedure deermnng her openng prces. he mornng sesson runs from 9: o :, whereas he afernoon sesson s from :3 o 5:. In vew of he me dfference, here s no overlappng radng hours beween oyo and he US soc

8 4 V. Corrad e al. / Journal of Economercs 7 ( 7 4 able Emprcal sze usng boosrap crcal values. o examne emprcal sze, we smulae nraday reurns from wo ndependen CIR processes and hen es for condonal ndependence n varance usng boosrap crcal values. We consder ess a he 5% and % levels of sgnfcance for sample szes of 4 and 6 daly realzed varances based on = 44 nraday observaons. We se he bandwdh scalng facors o κ b {/, 3/4, } and κ h {3/4,, 3/, 5/}, wh κ b < κ h. All resuls res on one Carlo replcaons and 3 boosrap arfcal samples of daly observaons. κ b 5% % κ h 3/4 3/ 5/ 3/4 3/ 5/ Panel A: = 4 / / Panel B: = 6 / / able Descrpve sascs for ndex reurns. We collec ransacons daa for he S&P 5, FSE, SSE B share, and opx ndces. he sample spans he perod rangng from January 3, o December 3, 5. We documen he man descrpve sascs for he ndex percenage reurns wh connuously compoundng a regular samplng nervals of and 3 mn. he sample does no nclude overngh reurns, so ha he frs nraday reurn refers o he openng prce ha ensues from he pre-sessonal aucon. S&P 5 FSE opx SSE B share Samplng freuency: mn ean Sandard devaon nmum axmum Sewness Kuross Zero reurns(% Samplng freuency: 3 mn ean Sandard devaon nmum axmum Sewness Kuross Zero reurns(% mares. he same apples o he Shangha Soc Exchange (SSE, whose mornng and afernoon consecuve bddng sessons run from 9:3 o :3 and from 3: o 5:. One of he parcular feaures of he Chnese soc mare s he relave mporance of ndvdual nvesors despe he fac hey face subsanal radng resrcons, e.g., a very srngen shor-sell consran (Herz, 998; Feng and Seasholes, 8. In addon, local nvesors could no own B shares before arch and, even hough hey may now purchase hem usng foregn currency, capal conrols sll resrc her ably o do so. See Allen e al. (7 and e e al. (9 for more deals on he nsuonal bacground. Our movaon o nclude he SSE B share ndex n he analyss s wofold. Frs, because prcng of radng for B shares s n US dollars, here s no room for exchange rae movemens o blur (or o cause spurous soc mare lns. Second, albe s soc mare s relavely young, dang bac only o November 99, Chna s becomng a maor player n he world economy and hence s neresng o sudy he role plays whn he conex of volaly ransmsson. he fac ha B shares are no as lud as A shares n he Shangha Soc Exchange means ha conrollng for mare-mcrosrucure nose s essenal for Chna. he sample runs from January 3, o December 3, 5 wh 3 common radng days. o compue he realzed measures of daly negraed varance, we frs compue connuously compounded reurns over regular me nervals of, 5, 5, and 3 mn. he sample does no nclude overngh reurns n ha he frs nraday reurn refers o he openng prce ha ensues from he pre-sessonal aucon, f any. Smlarly, we also exclude reurns over he lunch brea for Chna and Japan, hough hey do no affec n any way he ualave resuls (see Secon 6.. able repors he descrpve sascs for he - and 3-mn reurns. he average nraday reurn s slghly negave for every soc mare, hough relave lower for Japan and Chna. hs s o some exen surprsng n vew ha he opx and, especally, he Shangha B share ndces exhb larger sandard devaons. As usual, ndex reurns exhb subsanal excess uross, whch rapdly ncreases wh he samplng freuency. hs s especally he case for he FSE ndex, whch clmbs from a he 3-mn freuency o 3 a he -mn freuency. As for sewness, s srongly negave for he S&P 5 a he -mn freuency, hough slghly posve a he 3-mn freuency. he oppose apples o he FSE and opx ndces, namely, sewness s posve a he -mn freuency, whereas negave a he 3-mn freuency. he sewness coeffcen ncreases wh he samplng freuency from nearly zero o.9 for he SSE B shares ndex. he dfferences beween he sewness and uross of he - and 3-mn reurns are possbly due o ludy ssues. he proporon of zero reurns s ndeed much hgher a he -mn freuency, especally for he SSE B share ndex. No surprsngly, we also observe srong frsorder auocorrelaon for every soc mare ndex a he -mn freuency as well as for he 3-mn reurns on he SSE B share ndces. Furher analyss shows ha he ludy of he SSE B share ndex, as measured by he proporon of nonzero reurns, ncreases over me, especally afer arch. In wha follows, we carry ou an emprcal analyss of volaly ransmsson usng daly realzed measures. We consder he realzed varance, he rpower varaon, he wo-scale realzed varance, and he realzed ernel based on - and 5-mn reurns. In addon, we also compue he realzed varance and rpower varaon usng 5- and 3-mn reurns. As n Aï-Sahala and

9 V. Corrad e al. / Journal of Economercs 7 ( Fg.. Realzed measures of he daly varance of he ndex reurns. he plos dsplay he me seres of he realzed varance and rpower varaon based on he - and 3-mn ndex reurns as well as he realzed ernel esmae of he daly varance based on -mn ndex reurns. Inraday ndex reurns refer o connuously compounded reurns on he SSE B share, opx, S&P 5, and FSE ndces from January 3, o December 3, 5. Jacod (forhcomng, we vary he realzed measure we employ so as o emphasze dfferen aspecs of he uadrac varaon of he ndex reurns. he realzed varance based on - and 5- mn reurns essenally gauges he overall uadrac varaon, ncludng no only nformaon abou he daly varance bu also abou prce umps and mcrosrucure nose. As he samplng freuency decreases, reducng he mare mcrosrucure effecs, he realzed varance sars reflecng only he dffusve and ump conrbuons o he uadrac varaon. he rpower varaon excludes he conrbuon of prce umps o he uadrac varaon and hence provde a reasonable esmaor for he daly varance f based on 5- and 3-mn reurns. Fnally, he wo-scale and realzed ernel approaches elmnae he conrbuon of he mcrosrucure nose o he uadrac varaon, capurng only s ump and dffusve componens. Fg. plos he me seres of he realzed varance and rpower varaon based on he - and 3-mn reurns as well as he realzed ernel esmaes a he -mn freuency. I s neresng o observe ha conrollng for mare mcrosrucure nose affecs n a subsanal manner he esmaes of he daly varance, especally for he SSE B share ndex. ables 3 6 repor he es resuls for he null hypohess of condonal ndependence usng boosrap crcal values. As he null of condonal ndependence s nvaran o monoonc ransformaons, we frs sandardze he logarhms of he realzed measures by her mean and sandard devaon and hen esmae he condonal denses usng ernel-based smoohng wh Gaussan-ype ernels. 6 In parcular, we consder he sandard Gaussan ernel for K and he fourh-order ernel derved from he Gaussan densy for W regardless of he dmenson of he condonng se. he bandwdhs are as n he one Carlo sudy, wh scalng facors se o κ b = 3/4 and κ h =, hough he resuls reman ualavely he same for vrually every combnaon beween κ b {.5,.75, } and κ h {.75,,.5} as long as κ b < κ h. As before, we employ a weghng scheme based on he sandard mulvarae normal densy. o oban boosrap crcal values, we consruc B = 5 boosrap arfcal samples of sze = 5 by resamplng blocs of 4 daly observaons. able 3 documens ha here s some srong evdence of volaly spllovers runnng from he UK o he US. A he 5% sgnfcance level, we reec he null for he realzed varance and rpower varaon esmaes based on -mn reurns, and for he wo-scale realzed varance a he 5-mn freuency. In addon, we also reec he null a he % level of sgnfcance for he realzed ernel usng -mn reurns as well as for he rpower varaon a he 3-mn freuency. hs seems o ndcae ha he ransmsson channel from he UK o he US s manly hrough he negraed varance gven ha accounng for umps and/or mcrosrucure nose yelds sronger resuls. he evdence s weaer for Japan. 6 We do no employ local-lnear smoohng as n he prevous secon because we esmae n Secon 6. condonal denses gven hree sae varables. A any rae, he p-values of he ess based on local lnear smoohng are only margnally dfferen from he p-values of he ernel-based ess.

10 6 V. Corrad e al. / Journal of Economercs 7 ( 7 4 able 3 Daly volaly ransmsson o he US. We repor he oucome of he boosrap es for condonal ndependence of he S&P 5 ndex daly realzed measures and of he VIX ndex wh respec o he daly realzed measures of he FSE, opx, and SSE B share ndces. We employ he followng realzed measures based on - and 5-mn reurns: realzed varance (RV, rpower varaon (V, wo-scale realzed varance (S, and realzed ernel (RK. In addon, we also compue he realzed varance and rpower varaon usng 5- and 3-mn reurns. We frs sandardze he logarhm of he daa by her mean and sandard devaon and hen esmae he condonal denses by means of ernel smoohng. As per he weghng funcon, we employ a sandard mulvarae normal densy. o oban crcal values, we consruc B = 5 boosrap arfcal samples of sze = 5 by resamplng blocs of 4 daly observaons. S&P 5 ndex VIX ndex RV V S RK RV V S RK FSE ndex mn mn mn mn opx ndex mn mn mn mn SSE B share ndex mn mn mn mn able 4 Daly volaly ransmsson o he UK. We repor he oucome of he boosrap es for condonal ndependence of he FSE ndex daly realzed measures wh respec o he daly realzed measures of he S&P 5, opx, and SSE B share ndces. he es deals are exacly as n able 3. RV V S RK S&P 5 ndex mn mn mn.9. 3 mn.9. opx ndex mn mn mn mn SSE B share ndex mn mn mn.6. 3 mn..34 able 6 Daly volaly ransmsson o Chna. We repor he oucome of he boosrap es for condonal ndependence of he daly realzed measures of he SSE B share ndex wh respec o he daly realzed measures of he S&P 5, FSE and opx ndces. he es deals are exacly as n able 3. RV V S RK S&P 5 ndex mn mn mn..4 3 mn FSE ndex mn mn mn mn..43 opx ndex mn mn mn mn able 5 Daly volaly ransmsson o Japan. We repor he oucome of he boosrap es for condonal ndependence of he daly realzed measures of he opx ndex wh respec o he daly realzed measures of he S&P 5, FSE and SSE share B ndces. he es deals are exacly as n able 3. RV V S RK S&P 5 ndex mn mn mn mn FSE ndex mn mn mn.3. 3 mn SSE B share ndex mn mn mn mn..86 Volaly ransmsson from he opx ndex o he S&P 5 ndex s sgnfcan a he 5% level only for he realzed varance and rpower varaon a he -mn freuency. We also observe some borderlne resuls a he % level of sgnfcance for he rpower varaon and he wo-scale realzed varance usng 5-mn reurns. Fnally, we also uncover some wea evdence of volaly ransmsson from Chna o he US. In parcular, we reec he null a he 5% level of sgnfcance for he realzed varance a he - and 5-mn freuences and for he rpower varaon and realzed ernel measures based on 5-mn reurns. We hen as wheher he dffusve and ump componens of he uadrac varaon n he UK, Japan, or Chna affec he oponsmpled mare volaly as measured by he VIX ndex. A posve answer would mean ha nvesors prce hese spllovers. hs s exacly wha happens for he FSE ndex. able 3 reveals sgnfcan volaly ransmsson o he VIX ndex for almos every realzed measure we employ. In fac, he UK effec seems sronger on he rs-neural expeced volaly han on he realzed measures of he S&P 5 ndex, especally f one conrols for boh umps and mare mcrosrucure nose. In conras, we observe no

11 V. Corrad e al. / Journal of Economercs 7 ( change n he ualave resuls as wha concerns spllovers from he opx and SSE B share ndces. As per he former, we reec condonal ndependence for every realzed measure based on - mn reurns apar from he realzed ernel. As for he SSE B share ndex, we fnd evdence of sgnfcan spllovers o he VIX ndex for he realzed varance a he - and 3-mn freuences and for he rpower varaon usng 5-mn reurns. able 4 dsplays he resuls for he daly volaly ransmsson o he FSE ndex. For he S&P 5 ndex, we reec he null for every realzed measure a he -mn freuency as well as for he realzed varances based on 5- and 3-mn reurns and for he realzed ernel measure a he 5-mn freuency. We nerpre hese resuls as srong evdence of spllovers n he uadrac varaon. o denfy wheher he ransmsson s hrough he dffusve or dsconnuous par of he uadrac varaon, we urn our aenon o he ess usng rpower varaon. We ae her reecons a he -, 5-, and 5-mn freuences as a clue ha he FSE negraed varance depends on he pas S&P 5 negraed varance. We oban smlar resuls for Japan and Chna n ha accounng for umps and/or mcrosrucure nose seems o srenghen he evdence of spllovers n he uadrac varaon. able 5 reveals ha he dependence srucure beween he S&P 5 ndex and he opx ndex s approxmaely symmerc n ha sgnfcan ransmsson seems o evenuae n boh drecons for almos he same realzed measures. We ndeed reec condonal ndependence usng he rpower varaon only a he -mn freuency, whereas we reec he null f we employ eher he realzed varance usng -mn reurns or he nose-robus realzed measures a he 5-mn freuency. A smlar symmerc paern arses for he FSE ndex. We fnd sgnfcan spllovers from he UK o Japan for essenally he same realzed measures for whch we observe sgnfcan spllovers n he oppose drecon. As before, hs seems o ndcae he presence of spllovers n he negraed varance. Fnally, volaly shocs n he SSE B share ndex also affec he opx ndex accordng o he ess based on he realzed varance a he -mn freuency as well as on he rpower varaon and realzed ernel esmaes a he - and 5-mn freuences. Noce ha he evdence s sronger once we accoun for umps. hs suggess ha spllovers are manly hrough he negraed varance, wh umps o some exen concealng he volaly ransmsson from Chna o Japan. able 6 documens a somewha dfferen paern for he volaly ransmsson o Chna. We fnd sgnfcan spllovers from he S&P 5 ndex only for he realzed varance and wo-scale realzed varance a he -mn freuency and for he rpower varaon a he 5-mn freuency. he evdence s much sronger for he FSE ndex, especally f we conrol for umps and/or mcrosrucure effecs. As for he opx ndex, we unvel sgnfcan spllovers for he realzed varance and rpower varaon based on -mn reurns and for he rpower varaon and realzed ernel esmaes a he 5-mn freuency. hese fndngs sugges ha he man channel of ransmsson from Japan s also hrough he dffusve componen of he uadrac varaon, gven ha he evdence becomes sronger for he realzed measures ha are robus eher o umps or o mcrosrucure nose. Alogeher, seems ha he lns are generally symmerc n ha we normally reec he null of condonal ndependence n boh drecons for almos exacly he same realzed measures. o exemplfy he volaly ransmsson we uncover, Fg. llusraes how he condonal densy of he daly rpower varaon of he FSE ndex a he 5-mn freuency changes f we also condon on he correspondng realzed measure of he SSE B share ndex. he plo evaluaes he condonng rpower varaons a her frs uarle, medan, and hrd uarle. I s apparen ha he mprn of he Chna effec s n every uarle. Furher condonng on he frs uarle of he Shangha rpower varaon shfs he densy o he lef, whereas shfs o he rgh f evaluaed a he hrd uarle. Fnally, seems o reduce he spread of he condonal densy of he FSE rpower varaon once we evaluae he rpower varaon of he SSE B share ndex a s medan value. hs s neresng because we would mos lely fal o capure hs sor of nonlnear dsrbuonal mpacs usng he usual paramerc approaches n he leraure. 6.. Robusness analyss In wha follows, we carry ou some robusness analyss by conducng hree nspecons. Frs, we assess how pvoal s he assumpon ha, under he null hypohess, he pas negraed varance suffces o conrol for he perssence n he daa by also condonng on he pas mpled volaly (and vce-versa. Second, we nvesgae wheher he evdence we fnd supporng spllovers effecs from Japan and Chna o he US are robus o furher condonng on realzed measures of he FSE ndex. hrd, we examne how fas he volaly ransmsson occurs by loong a he uadrac varaon over shorer perods of me. In parcular, we focus on he condonal dsrbuon of he uadrac varaon over he frs hour of he radng day gven he las hour of prevous radng day and he las hour of radng on he oher soc mare Daa perssence he condonal ndependence resrcon we es does no exacly correspond o a null of noncausaly n varance gven he non-arovan characer of he daly negraed varance. In parcular, he emprcal analyss n Secon 6 conrols only for he own negraed varance n he prevous day raher han on he whole hsory of daly varances. hs rases a concern on wheher our fndngs are n fac genune or us an arfac due o hgherorder dependence n he daa. o assess robusness agans such concerns, we redo our emprcal analyss of volaly ransmsson o he US ncludng he pas VIX ndex as an addonal conrol. he laer measures he opons-mpled volaly of he S&P 5 ndex and hence should provde nformaon abou he fuure negraed varance. 7 In addon, Band and Perron (6 fnd srong evdence of fraconal conegraon beween mpled and realzed varances and so condonng on he VIX ndex should effecvely conrol for any hgh-order dependence mpled by he non-arovan naure of he negraed varance, regardless of wheher he null of condonal ndependence s rue or no. able 7 shows ha he spllover effecs we uncover are no an arfac due o perssence. Addng he VIX ndex o he condonng se does no aler much he ualave resuls. I acually srenghens he evdence of uadrac varaon ransmsson from he UK and Japan. In conras, accounng for he VIX ndex somewha affecs he resuls for he SSE B share ndex. We now reec he null of condonal ndependence for he realzed varance a every freuency as well as for he wo-scale realzed varance a he -mn freuency. Ineresngly, conrollng for umps by means of he rpower varaon measure acually weaens he evdence of ransmsson. hs seems o ndcae ha spllovers evenuae from Chna o he US manly hrough umps. Smlarly, we revs he evdence of volaly ransmsson o he VIX ndex by furher condonng on he realzed measure of he S&P 5 ndex n he prevous day. able 7 reveals ha, all n all, here s no much change n he ualave resuls. he evdence ha he realzed measure of he FSE ndex sgnfcanly affecs 7 Noe ha s no necessary o assume ha he VIX ndex s he bes forecas for fuure realzaons of he negraed varance nor ha s unbased or effcen (see, among ohers, Chrsensen and Prabhala (998.

12 8 V. Corrad e al. / Journal of Economercs 7 ( 7 4 (a Frs uarle. (b edan. (c hrd uarle. Fg.. Chna spllovers o he UK. he dashed red lne plos he condonal densy of he log of he uadrac varaon of he FSE ndex gven s pas realzaon, whereas he sold blac lne depcs a smlar condonal densy, bu furher condonng on he log of he uadrac varaon of he SSE B share ndex. We evaluae he condonng varables a her (a frs uarle, (b medan, and (c hrd uarle values. We measure uadrac varaon usng he rpower varaon esmaor a he 5-mn freuency. he ernel densy esmaon employs he same bandwdhs we use n he ess; see able 3 for deals. (For nerpreaon of he references o colour n hs fgure legend, he reader s referred o he web verson of hs arcle. on he VIX ndex s vrually he same regardless of wheher we conrol or no for he realzed measure of he S&P 5 ndex n he prevous day. As for spllovers from he opx and SSE B share ndces, beer accounng for daa perssence slghly srenghens he sascal evdence of volaly ransmsson. As before, he Chna effec s weaer for he rpower varaon measures, corroborang our ndrec evdence of ump spllovers Asa effec gven he UK os recen epsodes of hgh uncerany n he global fnancal mares are manly due o shocs n Europe and he US, wh lle acon ang place n Chna and Japan. 8 We hus chec wheher he evdence of volaly ransmsson runnng from Chna and 8 We han an anonymous referee for callng our aenon o hs ssue. Noe ha, alhough our sample does no nclude he subprme and ransalanc soveregn deb crses, covers he burs of he docom bubble n he US and s afermah. Japan o he US s genunely abou shocs n Asa or abou common shocs o he global fnancal mares. o do hs, we furher condon he dsrbuon of he daly realzed measures of he S&P 5 ndex on he correspondng realzed measure of he FSE ndex over he same radng day (up o 4:3 London me. he movaon les n he fac ha he UK soc mare should reflec any global shoc, bu no necessarly Asa-specfc shocs ha may affec he US. Examnng he volaly ransmsson from Asa (eher Chna or Japan o he US gven he UK realzed measure allows us o beer undersand he geographc naure of he volaly spllovers. Fndng sgnfcan effecs regardless of wheher we furher condon on he realzed measure of he FSE ndex ndcaes ha here are Asa-specfc shocs ha affec he uncerany n he US soc mares. Reecng he null hs s a perod of hgh volaly n he US soc mare due mosly o domesc facors.

13 V. Corrad e al. / Journal of Economercs 7 ( able 7 Daly volaly ransmsson o he US usng exra conrols. We repor he oucome of he boosrap es for condonal ndependence of he S&P 5 ndex daly realzed measures and of he VIX ndex wh respec o he daly realzed measures of he FSE, opx, and SSE B share ndces. o beer accoun for daa perssence, we furher condon he dsrbuon of he S&P 5 realzed measure on he VIX ndex and vce versa. o fler global volaly shocs, we also consder ess for whch we furher condon he dsrbuon of he S&P 5 realzed measure on he FSE realzed measure. he es deals are as n able 3. S&P 5 ndex VIX ndex RV V S RK RV V S RK FSE ndex (+ VIX ndex (+ S&P 5 ndex mn mn mn mn.9... opx ndex (+ VIX ndex (+ S&P 5 ndex mn mn mn mn opx ndex (+ FSE ndex mn mn mn mn.36.4 SSE B share ndex (+ VIX ndex (+ S&P 5 ndex mn mn mn mn SSE B share ndex (+ FSE ndex mn mn mn mn.77.8 only f dsregardng he soc mare volaly n he UK suggess here s no Asa-specfc ransmsson channel. he S&P 5 ndex realzed measure s hen reacng o global (volaly shocs ha are common o he UK and o Asa. Fnally, reecng he null only f accounng for he FSE realzed measure aess ha flerng he global effec by means of he varaon n he FSE ndex helps denfy as well spllovers due o Asa-specfc shocs. able 7 shows ha furher condonng on he FSE realzed measure does no change much he ualave resuls concernng spllovers from Japan. We sll reec he null of condonal ndependence for boh realzed varance and rpower varaon esmaes a he -mn freuency as well as for he woscale realzed varance usng 5-mn reurns. he dfferences are ha we now reec he null also for he realzed varance a he 5-mn freuency and for he rpower varaon based on 3-mn reurns. hs seems o sugges ha he varaon n he FSE ndex does no suffce o fully capure he Japan effec on he US soc mare volaly. Fnally, we observe a very neresng resul for he spllovers runnng from Chna. Once we conrol for he FSE realzed measure, he evdence of a sgnfcan Chna effec n he US soc mare volaly becomes much sronger, especally f we conrol for umps and/or mare mcrosrucure nose. All n all, hese fndngs appears o llusrae ha accounng for he realzed measure of he FSE ndex helps ndvduae Asaspecfc spllovers from global volaly shocs. Fg. 3 llusraes how accounng for he realzed measure of he SSE B share ndex alers he condonal densy of he S&P 5 realzed measure gven s pas realzaon and he FSE realzed measure over he same day. I s srng how he nformaon ha he SSE B share realzed measure conveys s almos exclusvely abou he dsperson n he dsrbuon of he uadrac varaon of he S&P 5 ndex. I does no shf o he rgh or lef dependng on he uarle we condon upon as n Fg. ; we observe across he board only a reducon n he spread of he dsrbuon Reacon me Kng and Wadhwan (99 derve an mperfecly revealng eulbrum model o explan conemporaneous ransmsson of volaly beween soc mares. her framewor poss ha prce umps wll ae place as soon as a mare reopens so as o reflec changes n boh dosyncrac and common facors snce las rade. Gven ha oher soc mare ndces also depend on he common facors, conagon wll resul n mmedae spllovers from one mare o anoher as he laer reopens for radng. hs s n sar conras wh our emprcal sudy focusng on daly uadrac varaon (raher han over a shorer nerval of me n ha we could well mss he almos nsananeous reacon ha Kng and Wadhwan (99 predc. We hus nvesgae n hs secon wheher reacon me s ndeed an ssue. o examne spllovers from he UK o he US, we condon he realzed measure of he S&P 5 ndex over s frs hour of radng on he realzed measures of he FSE ndex over he one-hour nerval mmedaely before (.e., 3:3 and 4:3 London me and of he S&P 5 ndex over he las hour of radng n he prevous day. For he ransmsson from he US o he UK, we loo a he condonal densy of he realzed measure over he frs hour of radng on he London Soc Exchange gven he realzed measures over he prevous day s las hour of radng on he New Yor Soc Exchange and on he London Soc Exchange. o es for spllovers from Asa o he UK and o he US, we chec wheher he realzed measures over he frs hour of radng n he UK/US depends on he correspondng realzed measures over he las hour of radng n Chna/Japan n ha same day gven he realzed measures over he las hour of radng n he UK/US n he prevous day. o examne spllovers runnng n he oppose

14 3 V. Corrad e al. / Journal of Economercs 7 ( 7 4 (a Frs uarle. (b edan. (c hrd uarle. Fg. 3. Chna spllovers o he US gven he UK. he dashed red lne porrays he condonal densy of he log of he uadrac varaon of he S&P 5 ndex gven s pas realzaon and he log of he uadrac varaon of he FSE ndex. he sold blac lne dsplays a smlar condonal densy, bu furher condonng on he log of he uadrac varaon of he SSE B share ndex. We evaluae he condonng varables a her (a frs uarle, (b medan, and (c hrd uarle values. We measure he daly uadrac varaon usng he realzed ernel esmaor a he -mn freuency. he ernel densy esmaon employs he same bandwdhs we use n he ess; see able 8 for deals. (For nerpreaon of he references o colour n hs fgure legend, he reader s referred o he web verson of hs arcle. drecon, we nvesgae wheher he realzed measures of he FSE ndex and of he S&P 5 ndex over he las hour of radng n he prevous day affecs he realzed measures over he frs hour of radng n Asa even afer conrollng for he laer s realzed measures over he las hour of radng n he prevous day. he same apples f esng for ransmsson from Chna o Japan gven ha he former shus before he openng of he laer. Fnally, gven he one-hour dfference beween Shangha and oyo, we es for spllovers from Japan o Chna by loong a wheher he realzed measure of he SSE B share ndex over he frs hour of radng depends on he realzed measures of he opx ndex over he one-hour perod mmedaely before he openng of he Shangha Soc Exchange (.e., from 9:3 o :3 oyo me gven he realzed measures of he SSE B share ndex over he las hour of radng n he prevous day. As before, asympoc euvalence beween he feasble and unfeasble sascs allows us o nerpre realzed varance resuls as concernng he oal uadrac varaon, ncludng he conrbuons of he negraed varance, umps, and mare mcrosrucure nose. rpower varaon purges he nfluence of prce umps, whereas he realzed ernel esmaor measures he conrbuons of he ump and dffusve componens o he uadrac varaon. Noe ha we do no employ he wo-scale realzed varance n he hourly ransmsson sudy because of he lmed number of nra-hour observaons. he measuremen error of he wo-scale esmaor converges a he slower rae a = /3 raher han a a = as n he realzed ernel approach. Gven ha we compue he hourly realzed measures usng -mn reurns (and hence = 6, he magnude of he measuremen error of he wo-scale realzed varance could well compromse nference. Panel A n able 8 reveals sgnfcan evdence of volaly ransmsson from he FSE ndex o he S&P 5 ndex only afer conrollng for mcrosrucure nose. hs suggess ha he

15 V. Corrad e al. / Journal of Economercs 7 ( able 8 Hourly volaly ransmsson. We repor he oucome of he boosrap es for condonal ndependence usng hree hourly realzed measures based on -mn reurns: realzed varance (RV, rpower varaon (V, and realzed ernel (RK. he es deals are exacly as n able 3. RV V RK Panel A: ransmsson o he S&P 5 ndex FSE ndex opx ndex SSE B share ndex Panel B: ransmsson o he FSE ndex S&P 5 ndex opx ndex SSE B share ndex Panel C: ransmsson o he opx ndex S&P 5 ndex FSE ndex SSE B share ndex Panel D: ransmsson o he SSE B share ndex S&P 5 ndex FSE ndex..4.6 opx ndex..4.5 vecor of any dmenson. I urns ou ha such a generalzaon s no so sraghforward as seems a frs glance. In parcular, one mus employ ernel-based mehods raher han local-lnear smoohng f he dmenson of he condonng se s large enough. We also conrbue o he leraure on nernaonal mare lns by nvesgang volaly ransmsson beween Chna, Japan, UK and US. Our emprcal fndngs reveal ha hese soc mares dsplay sgnfcan nerconnecon. he evdence s parcularly sronger for he realzed measures ha are robus o umps and/or mare mcrosrucure nose and hence seems ha he prncpal channel of ransmsson s hrough he negraed varance. he only excepon s for spllovers from Chna o he US, whch ae place predomnanly hrough prce umps. Fnally, Chna and Japan effecs on he US soc mare volaly are more pronounced f one furher conrols for he uadrac varaon n he UK. he FSE realzed measure hus helps demarcae Asa-specfc effecs n he US n he presence of global shocs. Acnowledgmens mare mcrosrucure nose blurs he evdence of spllovers n he ess usng he realzed varance and rpower varaon. In urn, we fnd sgnfcan spllovers from Japan o he US only f we employ he rpower varaon. he ransmsson channel seems manly hrough he negraed varance, wh he ump componen only obscurng he volaly spllovers. In addon, we fnd spllovers runnng from he SSE B share ndex o he S&P 5 ndex a he % level of sgnfcance for he realzed varance and realzed ernel measures. Gven ha he ransmsson a he daly level s va negraed varance, we nerpre he falure o reec usng he hourly rpower varaon as evdence of ump-n-volaly spllovers (Kng and Wadhwan, 99. Panel B uncovers noable spllovers from Chna and he US o he UK. In conras, we canno reec he absence of spllovers from Japan o he UK regardless of he realzed measure we employ. Panel C maes plan once more he mporance of accounng for he mare mcrosrucure mprn when esng for spllovers o he opx ndex. As before, hs s somewha conssen wh volaly spllovers hrough umps n he volaly. Fnally, Panel D esablshes ha he uadrac varaon ransmsson o he SSE B share ndex s sgnfcan a he usual levels for he realzed varance and rpower varaon esmaes for every soc mare. 7. Concluson hs paper develops formal sascal ools for nonparamerc ess of condonal ndependence beween negraed varances. Under he assumpon ha asse prces follow a mulvarae umpdffuson processes wh sochasc volaly, we show how o es wheher he condonal densy of asse A s negraed varance also depends on nformaon concernng asse B s negraed varance. Our esng procedure nvolves wo seps. In he frs sage, we esmae he negraed varances usng nraday reurns daa by means of realzed measures so as o avod msspecfcaon rss. In he second sep, we hen es for condonal ndependence beween he resulng realzed measures. Alhough asympoc crcal values are no very relable n fne samples, we show how o consruc more accurae crcal values by means of a smple boosrap procedure. Our conrbuon o he leraure on nonparamerc densybased ess s wofold. Frs, our asympoc heory specfcally accouns for he mpac of he esmaon error n he frs sep of he esng procedure. Second, we also consder a more general seup n whch he condonal densy may depend on a sae We are ndebed o Ron Gallan (edor and wo anonymous referees for her houghful commens as well as o Govann Cespa, Jean-Perre Florens, René Garca, Ludas Gras, assmo Gudoln, José Ferrera achado, chael ccracen, Nour eddah, Chrs Neely, Alesso Sancea, Pedro Sana Clara, Enrue Senana, Ross Valanov, and semnar parcpans a Banco de Porugal, Cambrdge Unversy, Cass Busness School, Cy Unversy, Federal Reserve Ban of S Lous, Fundação Geulo Vargas, Queen ary, Unversà d Padova, Unversy of Bah, Unversy of Brsol, Unversy of Cyprus, Unversy of oulouse, 4h CSDA Inernaonal Conference on Compuaonal and Fnancal Economercs (London, December, Far Eas and Souh Asa eeng of he Economerc Socey (oyo, Augus 9, Inernaonal Symposum on Rs anagemen and Dervaves (Xamen, July 9, QASS Conference on Fnancal Economercs and Realzed Volaly (London, June 9, London-Oxbrdge me Seres Worshop (Oxford, Ocober 8, Granger Cenre Conference on Boosrap and Numercal ehods n me Seres (Nongham, Sepember 8, eeng of he ESRC Research Semnar Seres on Nonlnear Economcs and Fnance (Keele, February 8, CEA@Cass Conference on easurng Dependence n Fnance (London, December 7, Brazlan me Seres and Economercs School (Gramado, Augus 7, eeng of he Brazlan Fnance Socey (São Paulo, July 7, Fnance and Economercs Annual Conference (Yor, ay 7, and London Oxford Fnancal Economercs Sudy Group (London, November 6. We also han Lous ercorell and Anhony Hall for ndly provdng he daa as well as Duda endes, João ergulhão and Flp Zes for excellen research asssance. We are also graeful for he fnancal suppor from he ESRC under he gran RES he usual dsclamer apples. Appendx A. Bas and scalng erms Le C (K K(u du and C (K ( K(uK(u+v du dv. Defne C (W, C (W, C ( W, and C ( W analogously. he bas and scalng erms ha appear n (7 are gven by µ, = C (K C (W s= W h (X ( s f X ((X ( π(y, X ( f Y,X ((Y, X ( b C (W s= X ( π(y s, X ( s W h (X ( s X (

16 3 V. Corrad e al. / Journal of Economercs 7 ( 7 4 µ, = C (K C ( W b C ( W K b (Y s Y WhA (X ( A s X ( A π(y s, X ( s= µ 3, = C (K W ( f Y,X ( A (Y, X ( A b W ( s= s K b (Y s Y WhA (X ( A s X ( A W ha (X ( A s X ( A π(y s, X ( s= f X ( A (X ( A s= s W ha (X ( A s X ( A K b (Y s Y WhA (X ( A s X ( A π(y s, X ( s= Ω = C (K C (W f Y,X ( A (Y, X ( A wh W h (u = h s= s K b (Y s Y WhA (X ( A s X ( A W ha (X ( A s X ( A π(y s, X ( s= f X ( A (X ( A s= π (Y, X ( f Y,X ((Y, X ( s W ha (X ( A s X ( A = W(u /h and WhA (u = h A A A = W(u /h A. o esmae he asympoc bas and varance of he negraed suared relave dfference sasc based on ernel smoohng, we employ smlar bas and scalng erms n (8. he only dfference s ha we replace he second-order unvarae ernel W wh he s-order ernel funcon W. For nsance, µ, = C (K C ( W b C ( W π(y, X ( f Y,X ((Y, X ( s= W h (X ( s f X ((X ( s= X ( π(y s, X ( W h (X ( s s X ( Fnally, for he boosrap es sascs n ( and (, we oban (µ,, µ,, µ 3, and ( µ,, µ,, µ 3, by replacng he sample uanes n (µ,, µ,, µ 3, and n ( µ,, µ,, µ 3, wh her boosrap counerpars, ha s, we subsue (Y, X (,, b, h, h A for (Y, X (,, b, h, h A. For nsance, µ, = C (K C (W π(y, X ( f Y,X ( (Y, X ( b C (W ( W h (Xs X ( π(y, s X ( s s=. f X ( (X ( ( W h (Xs X ( Appendx B. Proofs B.. Lemmaa s= he proof of heorem reles heavly on Lemmas 3, whereas we employ he resuls n Lemmas 4 6 n he proof of heorem. Lemma. Assume ha here are a mos hree condonng varables n he hgher dmensonal densy ( 3 and ha he bandwdhs. sasfy he followng condons: ( (ln / h A/ A h / b, ( h / b s+/, ( h 4+/ b /, (v h b s, (v h 4 b, (v h 3/ b 3/. I hen follows from Assumpons A A4 ha, under he null H, Ω h / b / π(y, X ( fy X ((Y X ( f Y X ((Y X ( fy X ( A (Y X ( A h / b / d µ N(, where Ω C (K C (W π (y, x dy dx and µ = C (K C (W π(y, x ( dy dx ( b C (W E π(y, X ( X ( = x ( dx (. Lemma. Assume ha here are a mos hree condonng varables n he hgher dmensonal densy ( 3 and ha he bandwdhs are such ha: ( (ln / h A A h b, (h / b s+/, ( h 4 A h / b /, (v h A A h b s, (v h 4 A A h b, and (v h 5 A/ A h b 3/. Assumpons A A4 hen ensures ha Ω h / b / where π(y, X ( fy X (A (Y X ( A f Y X ( A (Y X ( A fy X ( A (Y X ( A h / h A A b / µ = o p ( µ = C (K C ( W E π(y, X ( Y = y, X (A = x ( A dy dx ( A b C ( W E π(y, X ( X (A = x ( A dx (A. Lemma 3. Le he bandwdh condons ( (v n Lemmas and hold. Assumpons A A4 ensure ha, under he null H, Ω h / b / π(y, X ( ϵ Y X ((Y X ( ϵ Y X ( A (Y X ( A fy X ( A (Y X ( A = o p (, h / A b / µ 3 whereϵ Y X ( (Y X ( fy X ( (Y X ( f Y X ( (Y X ( and µ 3 = C (K W ( E π(y, X ( Y = y, X (A = x ( A dy dx ( A b W ( E π(y, X ( X (A = x ( A dx (A.

17 V. Corrad e al. / Journal of Economercs 7 ( Lemma 4. Le Assumpons A A5 hold as well as he followng bandwdhs condons: ( (ln / h A/ A h / b, ( h / b s+/, ( h s+/ b /, (v h b s, (v h s b, (v h 3/ b 3/. I hen follows ha, under he null H, Ω h/ b / π(y, X ( fy X ((Y X ( f Y X ((Y X ( h / f Y X ( A (Y X ( A b / µ d N(,, where Ω C (K C ( W π (y, x dy dx. Lemma 5. Le Assumpons A A5 hold as well as he followng bandwdhs condons: ( (ln / h A A h b, ( h / b s+/, ( h s A h / b /, (v h A A h b s, (v h s A A h b, and (v h 5 A/ A follows ha Ω h / b / π(y, X ( fy X (A (Y X ( A f Y X ( A (Y X ( A f Y X ( A (Y X ( A h / h A A b / µ = o p (. h b 3/. I hen Lemma 6. Le he bandwdh condons ( (v n Lemmas 4 and 5 hold. I hen follows from Assumpons A A5 ha, under he null H, Ω h / b / π(y, X ( ϵ Y X ((Y X ( ϵ Y X ( A (Y X ( A f Y X ( A (Y X ( A = o p (, h / A b / µ 3 where ϵ Y X ( (Y X ( fy X ( (Y X ( f Y X ( (Y X (. B.. Proof of heorem ( We frs observe ha Ω Ω π (Y, X ( = C (K C (W f Y,X ((Y, X ( π (y, x ( dy dx ( + π (Y, X ( f Y,X ((Y, X ( fy,x ((Y, X ( fy,x ((Y, X ( f Y,X ((Y, X (, s of order o p ( and hence we rea hem as asympocally euvalen n wha follows. Under he null ha f Y X ((Y X ( = f Y X ( A (Y X ( A almos surely, follows ha, up o a o p ( erm, Λ = Ω h / b / π(y, X ( f Y X ( (Y, X ( ϵ Y X ( (Y X ( h / b / µ + Ω h / b / ϵ Y X ( A (Y X ( A h / Ω h / b / π(y, X ( f Y X ( A (Y, X ( A A b / µ h A π(y, X ( f Y X ( (Y, X ( ϵ Y X ((Y X ( ϵ Y X ( A (Y X ( A h / A b / µ 3 + Ω h / b / π(y, X ( ϵ Y X ((Y X ( ϵ Y X ( A (Y X ( A f Y X ( A (Y X ( A f Y X ( A (Y X ( A Ω h / b / µ, µ h / h A A b / µ, µ + h / A b / µ 3, µ 3 = Λ (, + Λ(, + Λ( 3,, ( where Λ (, s he sum of he frs hree erms on he rgh-hand sde of (. Lemmas 3 yeld he asympoc normaly of Λ (, under he null and ensure ha Λ ( =, o p(. I hus remans o show ha Λ ( 3, s also of order o p(. We sar wh h / b / µ, µ = C (K C (W h / b / π(y, X ( fy,x ((Y, X ( f Y,X ((Y, X ( + h / b / C (W E π(y s, X ( s s= W h (X ( s s= fy,x ((Y, X ( f Y,X ((Y, X ( X ( s f X ((X ( = X ( X ( π(y s, X ( s W h (X ( s X (

18 34 V. Corrad e al. / Journal of Economercs 7 ( 7 4 W h (X, x K b (Y, y W h (X x K b (Y y = = W h (X, x K b (Y, y(x, x W h (X x K b (Y y(x x = =. (3. W h (X, x K b (Y, y(x, x W h (X x K b (Y y(x x = = Box I. + h / b / C (W E π(y s, X ( s = o p (. X ( s = X ( fx ((X ( f X ((X ( f X ((X ( fx ((X ( he las eualy follows from he fac ha he second and hrd erms are of smaller order han he frs erm, whereas he uany nf C(Y,X ( f Y,X ((Y, X ( s bounded away from zero n a compac se C(Y, X ( R + and he degenerae U-sasc π(y, X ( fy,x ((Y, X ( f Y,X ((Y, X ( fy,x ((Y, X ( f Y,X ((Y, X ( = o p (h / b /. In addon, follows along he same lnes ha h / h A A b / (µ, µ and h / A b / (µ 3, µ 3 are also of order o p (. ( Consder he followng expanson under he alernave hypohess H A Λ = Ω h / b / π(y, X ( f Y X ( A (Y, X ( A ϵ Y X ( (Y X ( h / b / µ, + Ω h / b / ϵ Y X ( A (Y X ( A h / Ω h / b / π(y, X ( f Y X ( A (Y, X ( A A b / µ, h A π(y, X ( f Y X ( A (Y, X ( A ϵ Y X ((Y X ( ϵ Y X ( A (Y X ( A h / A b / µ 3, + Ω h / b / π(y, X ( f Y X ( A (Y, X ( A f Y X ((Y X ( f Y X ( A (Y X ( A = Λ (, + Λ(, + Λ( 3, + Λ( 4,. he asympoc behavor of Λ (, for {,, 3} s he same under boh hypoheses. However, under he alernave, f Y X ( (Y X ( dffers from f Y X ( A (Y X ( A and hus Λ ( 4, s of order O p h / b, / whch ensures un asympoc power. B.3. Proof of heorem he resul ensues from Lemmas 4 6 along he same lnes as n he proof of heorem. B.4. Proof of heorem 3 ( he local lnear esmaor based on realzed measures (raher han on negraed varances reads β ( (y, x ( = β (y, x ( + H x ( W x (H x ( H x ( + H x ( W ( x H ( x H x ( W ( x Y y H x ( W x (Y y H x ( W x (H x ( H x ( W x (Y y + H x ( W ( x H ( x H x ( W x (H x ( H x ( W ( x Y y H x ( W x (Y y, where he ndex denoes relance on realzed measures and W ( x Y y H x ( W x (Y y s a column vecor whch s gven n Box I. We sar by boundng he frs erm of (3, namely, sup W h (X, x K b (Y, y C(Y,X ( = W h (X x K b (Y y = sup C(Y,X ( = K b ( Y, y N,, X, W h ( X, x =

19 + sup W C(Y,X ( h ( X, x = K b ( Y, y N,, + sup Y, C(Y,X ( W h ( X, x = = X, = K b ( Y, y N,, N,,, (4 Y, where X, (X,, X and Y, (Y,, Y. As for he frs erm on he rgh-hand sde of (4, urns ou ha sup W C(Y,X ( h ( X, x = X, = K b ( Y, y N,, V. Corrad e al. / Journal of Economercs 7 ( π(y,, X (, fy X ((Y X ( fy X ( A (Y X ( A π(y, X ( fy X ( A (Y X ( A + h / b / f ( Y X ( (Y, X (, f ( Y X ( A (Y, X ( A, π(y,, X (, fy X ( A (Y X ( A ( f Y X ( A (Y, X ( A fy X ( A (Y X ( A ( f Y X ( A (Y, X ( A, + O p h / b / = A, + B, + O p h / b / = A, + B, + o p + a / (h a / (h ( a / (h, + h A + b + h A + b + h A + b, (7 sup N,, = sup C(Y,X ( W h ( X, x X, = K b ( Y, y N,, = O p (h sup N,, and analogously he second erm on he rgh-hand sde of (4 s of order O p (b sup N,,. In vew of Assumpon A6 and Corrad e al. s ( Lemma, Pr sup a / N,, > ε Pr a / N,, > ε ε ε a / a / E N,, O(a / meanng ha sup N,, = O p, as, a /. I s sraghforward o show ha he hrd erm on he rgh-hand sde of (4 s of smaller order ha he frs and hrd erms. I hen follows ha sup f ( Y X ( (y x ( fy X ((y x ( C(Y,X ( = O p a / (h + b and, analogously, sup f ( Y X ( A (y x (A fy X ( A (y x (A C(Y,X ( A = O p a / (h (5 A + b. (6 I s now mmedae o see ha, gven bandwdh condon (v, Ω Λ ( Λ ( f = h / b / Y X ( (Y, X ( (, f Y X ( A (Y, X ( A, fy X ( A (Y X ( A where he las erm capures he conrbuon of he bas erms, namely, µ,, µ, = Op a / (h + h A + b for {,, 3}. Now, π(y,, X (, π(y, X ( sup π, sup N,, = O p a / C(Y,X (,, = and so leng π, yelds h / b / π(y,, X (, and π(y, X π(y, X ( fy X ((Y X ( fy X ( A (Y X ( A fy X ( A (Y X ( A (π, π(y, X = O p h / b / a /, (8 whch s of order o p (. I also follows from (5 and (6 ha h / b / f ( Y X ( (Y, X (, fy X ((Y, X (, fy X ( A (Y X ( A π, = O p h / b / + a (h + b (9 ( f h / b / Y X ( A (Y, X ( A, fy X ( A (Y, X ( A, fy X ( A (Y X ( A π, = O p h / b / + a (h A + b, ( whereas h / b / fy X ((Y, X (, fy X ((Y X ( fy X ( A (Y X ( A π, ( β ( Y, X, sup N,, h / b / =, fy X ( A (Y X ( A = O p h / b / + a (h + b π, (

20 36 V. Corrad e al. / Journal of Economercs 7 ( 7 4 Λ, = h/ b / Ω π(y, X ( ϵ Y X ( (Y X ( fy X (A (Y X ( A f Y X ( A (Y X ( A fy X ( A (Y X ( A + f Y X ( A (Y X ( A h/ b / Ω sup C(Y,X ( f Y X ( A (Y X ( A f Y X ( A (Y X ( A fy X ( A (Y X ( A + f Y X ( A (Y X ( A f Y X ( A (Y X ( A f Y X ( A (Y X ( A fy X (A (Y X ( A f Y X ( A (Y X ( A /. π (Y, X ( ϵ 4 Y X ( (Y X ( / Box II. h / b / sup N,,, fy X (A (Y, X ( A, fy X ( A (Y X ( A = h / b / β ( Y, X ( A fy X ( A (Y X ( A, fy X ( A (Y X ( A π, π, = O p h / b / + a (h A + b, ( and hence ha h / b / fy X ((Y X ( fy X ( A (Y X ( A fy X ( A (Y X ( A f ( Y X (d (Y, X (d, fy X ((Y, X (d fy X ( A (Y X ( A = O p + (, ln a / (h + h A + b. (3 Alogeher, (8 (3 mply ha A, = O p ( ( + / ln a +h A +b. Fnally, gven ha B, s of smaller probably (h order han A,, suffces o follow he same seps as n he proof of heorem ( o complee he proof of saemen (. ( Under he alernave H A, f Y X ((Y X ( and f Y X ( A (Y X ( A dffer n a subse of nonzero Lebesgue measure. hs mples ha he erms n (8 and (3 become of order O p h / b / a / (h +h A +b under he alernave, hough here s no change n he probably orders of (9 (. Alogeher, hs shows ha Λ ( Λ = O p h / b / a / (h + h A + b H A. hs complees he proof due o he fac ha a / b = o p (. B.5. Proof of heorem 4 (h under +h A + he resul ensues along he same lnes as n he proof of heorem 3. B.6. Proof of Lemma Under he null H, f Y X ((y x ( concdes almos surely wh f Y X ( A (y x ( A and hence Λ, Ω h/ b / π(y, X ( ϵy X ((Y X ( h / fy X ( A (Y X ( A b / µ = Ω h/ b / π(y, X ( ϵy X ((Y X ( h / b / µ + h / b / Ω π(y, X ( ϵ Y X ( (Y X ( f Y X ((Y X ( A f Y X ( A (Y X ( A f Y X ( A (Y X ( A = Λ, + Λ,, where Λ, s as n Box II. Now, n vew ha π (Y, X ( ϵ 4 Y X ( (Y X ( sup ϵ C(Y,X ( Y X ( (Y X ( π (Y, X ( ϵ Y X ( (Y X ( = O p ln h and ha b O p h / b / fy X (A(Y X ( A f Y X ( A (Y X ( A = O p h A/ A b /, Λ, = o p ( due o bandwdh condon (. As Λ, concerns only X (, we hereafer suppress he superscrp ndex from he condonng sae vecor and le m(x, y = E [K b (Y y X = x]. By he same reasonng as n he proof of heorem n Fan e al. (996, he bandwdh condons

21 V. Corrad e al. / Journal of Economercs 7 ( ( (v ensure ha I = h / b / = h / b / π(y, X ( π(y, X f Y,X (Y, X ϵy X ((Y X ( f Y X ((Y X ( W h (X τ X [K b (Y τ Y m(x, Y ] τ= + O p / ln h / b / = I + o p (. Leng now φ(, τ, = π(y, X f Y,X (Y, X W h (X τ X [K b (Y τ Y m(x τ, Y ] W h (X X [K b (Y Y m(x, Y ] and φ(, τ, = φ(, τ, + φ(,, τ + φ(τ,, + φ(τ,, + φ(,, τ + φ(, τ, yelds I = h / b / <τ< φ(, τ, + h / b / [φ(, τ, τ + φ(τ,, τ + φ(τ, τ, ] τ + h / b / φ(,, = I, + I, + I3,. As n Aï-Sahala e al. (9, we mus now demonsrae he followng saemens o conclude he proof; he only dfference s ha we mus also accoun for he hgher dmensonaly of he condonng se ( >. (a I, = ( h / b / <τ φ(, τ + o p (, where φ(, τ = φ(, τ, df(y, x. (b I, = ( h/ b / φ(+o p (, whereφ( = E[φ(], φ( = φ(, τ df(y τ, x τ, and φ(, τ = φ(,, τ + φ(, τ, + φ(τ,, + φ(τ,, τ + φ(τ, τ, + φ(, τ, τ. (c I3, = o p (. (d I also holds ha ( h/ b / φ( = h / b / C (K C (W π(y, x dy dx h / b / C (W E [π(y, X X = x] dx and ha + o( (4 Ω = lm Var = C (K C (W ( h / b / φ(, τ <τ π (y, x dy dx. (5 (e ( h / b / φ(, <τ τ N(, Ω. d B.6.. Proof of saemen (a I follows from he Hoeffdng decomposon ha I, = h / b / <τ< Φ(, τ, + ( h / b / <τ φ(, τ, (6 where Φ(, τ, = φ(, τ, φ(, τ φ(, φ(τ,. o show ha he frs erm on he rgh-hand sde of (6 s of order o p (, suffces o apply Lemma 5( n Aï-Sahala e al. (9 wh δ = /3. hs resuls n E I, s of order o( by condon (v. = O h 3/ b 3/, whch B.6.. Proof of saemen (b As before, applyng he Hoeffdng decomposon yelds h / b / I, = h / b / <τ φ(, τ = h / b / φ(, τ φ( φ(τ φ( <τ + ( h / b / + φ( φ( ( h/ b / φ(. Lemma 5( n Aï-Sahala e al. (9 wh δ = hen dcaes ha h / b / φ(, τ φ( φ(τ φ( <τ = O p h 5/4 b 5/4, whch s of order o p ( due o he bandwdh condon (v. Under Assumpon A4, he cenral lm for β-mxng processes ensures ha ( h / b / B.6.3. Proof of saemen (c I s mmedae o see ha I3, = h / b / φ( φ( = O p h b = o p (. φ(,, = O p h 3/ b 3/, whch s of order o p ( by condon (v. B.6.4. Proof of saemen (d As for (4 and (5, he resul follows along smlar lnes of he proof of clam (d n Aï-Sahala e al. (9. B.6.5. Proof of saemen (e I suffces o apply Fan and L (999 cenral lm heorem for degenerae U-sascs of absoluely regular processes o oban he desred resul (see Amaro de aos and Fernandes (7. See also Aï-Sahala e al. (9 and Gao and Hong (8 for alernave cenral lm heorems ha deal wh degenerae U-sascs of α-mxng processes. B.7. Proof of Lemma Le ψ(, τ and ψ( respecvely denoe he counerpars of φ(, τ and φ( once we subsue

22 38 V. Corrad e al. / Journal of Economercs 7 ( 7 4 ψ(, τ, = ( π(y, X f Y,X ( A (Y, X ( A Wh (X ( A τ X ( A K b (Y τ Y m(x ( A τ, Y Wh (X ( A X ( A K b (Y Y m(x ( A, Y for φ(, τ,. Applyng he same argumen we pu forh n he proof of Lemma hen yelds J = h / b / π(y, X ( f Y,X ( A (Y, X ( A fy X (A(Y X ( A f Y X ( A (Y X ( A = ( h / b / <τ ψ(, τ + ( h/ b / ψ( + o p (, whose frs erm on he rgh-hand sde sasfes he cenral lm heorem for U-sascs. In addon, ψ( = π(y, x ( W f (y, x ( A h A (x ( A x ( A K b (y y m(x ( A, y df(y, x ( df(y, x ( A = π(y, x ( W f (y, x ( A h A (x ( A x ( A K b (y y df(y, x ( df(y, x ( A π(y, x ( W f (y, x ( A h A (x ( A x ( A K b (y y m(x ( A, y df(y, x ( df(y, x ( A + π(y, x ( W f (y, x ( A h A (x ( A x ( A m (x ( A, y df(y, x ( df(x ( A = W (u h du K (v dv A A b π(y, x ( f (y, x ( A f (y, x ( dy dx ( + O h A + b s W (u h du A A π(y, x ( f (y, x ( A f (y, x ( m (x ( A, y f (x ( A dy dx ( + O(h A + b s, where he las eualy follows from a aylor expanson wh u = (x ( A x ( A /h A and v = (y y /b gven ha E K b (y y Y = y, X (A = x ( A = m(x ( A, y = f (y x ( A + O(h A + b s. In addon, ensues from π(y, x ( f (y, x ( A f (y, x ( dy dx ( = π(y, x ( f (x ( B y, x ( A dy dx ( = E π(y, x ( Y = y, X (A = x ( A dy dx ( A and π(y, x ( f (y, x ( A f (y x (A f (y, x ( f (x ( A dy dx ( = π(y, x ( f (y, x ( B x ( A dy dx ( = E π(y, x ( X (A = x ( A dx ( A, ha ( h/ b / ψ( = h / h A A b / C (K C ( W E π(y, x ( Y = y, X (A = x ( A + h / h A A b / C ( W E π(y, x ( X (A = x ( A compleng he proof. B.8. Proof of Lemma 3 Le dx ( A dx ( A dy + o(, ϕ(, τ, = ( π(y, X f Y,X ((Y, X ( f Y,X ( A (Y, X ( A W h (X τ ( X ( K b (Y τ Y m(x τ (, Y WhA (X ( A X ( A K b (Y Y m(x ( A, Y. Proceedng along he same lne as n he proof of Lemma hen yelds h / b / π(y, X ( ϵ Y X ((Y X ( ϵ Y X ( A (Y X ( A = ( h / b / <τ + f Y X ((Y X ( f Y X ( A (Y X ( A ϕ(, τ ( h/ b / ϕ( + o p (. Le now u = (x ( A x ( A /h A, v = (y y /b, and z = (x ( x ( /h. Gven ha under he null E K b (y y Y = y, X ( = x ( = ( m(x, y = m(x ( A, y, follows ha ϕ( = h A b C (K W ( π(y, x ( f (y, x ( A f (y, x ( dy dx ( + O h A + b s h A W (

23 f (x ( A π(y, x ( f (y, x ( A dy dx ( f (y, x ( m (x ( A, y + O h A + b s = h A b C (K W ( E π(y, x ( Y = y, X (A = x ( A hs means ha + O h A + b s h A W ( E π(y, x ( X (A = x ( A dx ( A + O h A + b s. ( h/ b / ϕ( = h / A b / C (K W ( E π(y, x ( Y = y, X (A = x ( A h / A whch complees he proof. B.9. Proofs of Lemmas 4 6 W ( π(y, x ( X (A = x ( A V. Corrad e al. / Journal of Economercs 7 ( dy dx ( A dy dx ( A dx ( A, We om he proofs because hey are almos exacly he same as he proofs of Lemmas 3. I ndeed suffces o apply he same lne of reasonng o derve he resuls n a sraghforward manner. B.. Proof of heorem 5 We denoe by Pr he probably dsrbuon nduced by he boosrap samplng, wh expecaon and varance operaors gven by E and Var, respecvely. In addon, we also le O p ( and o p ( denoe he orders of magnude accordng o he boosrapnduced probably law. Boh local-lnear and ernel smoohng resuls follow sraghforwardly once we prove he boosrap versons of Lemmas 3 and of Lemmas 4 6, respecvely. As he proofs are very smlar, n wha follows, we resrc aenon o he boosrap es based on local lnear smoohng. We sar wh he boosrap counerpar of Lemma. As n he laer s proof, urns ou ha Λ, = h/ b/ π(y, X ( f Y X ( (Y h / b / µ = h / b/ τ= K b (Y τ Y X ( f Y X ((Y X ( f Y X ( A (Y X ( A π(y, X ( f Y,X ( (Y, X ( ( W h (Xτ X ( h / b / µ + o p (. ( m(x, Y τ Le now φ (,, = π(y, X ( f Y,X ( (Y, X ( K b (Y Y W h (X ( K b (Y Y W ( h (X ( m(x, Y X ( ( m(x, Y. X ( ang condonal expecaon over boosrap samples gven (Y, X ( hen yelds φ (, = E φ (,, Y, X ( and so = π(y, X ( = f Y,X ( (Y, X ( K b (Y Y m(x (, Y ( W h (X X ( W ( h (X = π(y, x ( f Y,X ( (y, x ( K b (Y y m(x (, y ( W h (X X ( K b (Y Y m(x (, Y W ( h (X x ( x ( K b (Y y m(x (, y dx ( dy + o p ( h / b /, φ ( = E [φ (,, + φ (,, ] = = = = π(y, X ( f Y,X ( (Y, X W h K b (Y Y m(x (, Y π(y, x ( W h (x( f Y,X ( (y, x ( K b (y y m(x (, y (X ( x ( df Y,X ((y, x ( df Y,X ((y, x ( + o p h / b /. X ( As n saemen (d n he proof of Lemma, hen follows ha ( h/ b/ and, as /, ( h / b/ φ ( = h / b / µ + o p ( φ (, < = ( h / b/ < (φ (, E [φ (, ] + o p (. (7 In vew ha Var {( h / b / <τ (φ (, E [φ (, ]} = Ω + o p (, he frs erm on he rgh-hand sde of (7 wealy converges o N(, Ω as boh and go o nfny, hus mmcng he lmng dsrbuon of ( h / b / < φ(,. Defne nex ψ (,,, ψ (, and ψ ( analogously o φ (,,, φ (, and φ ( for X (A, wh Wh A replacng W h.

24 4 V. Corrad e al. / Journal of Economercs 7 ( 7 4 As /, s possble o show ha ( h / b / <τ ψ (, = o ( p and ( h/ b / ψ ( = h / h A A b / µ + o p (. I s also sraghforward o derve he boosrap counerpar of Lemma 3 as well. he saemen under he null hereby follows by nong ha µ, = µ, + o p (h p/ b /, whereas s mmedae o see ha Λ dverges a mos a rae O p h / b / under he alernave. Appendx C. Realzed measures Accordng o he daa generang process, one may employ dfferen realzed measures o esmae he daly negraed varance from a sample of nraday regularly-spaced-n-me observaons under very mld condons. Andersen e al. ( and Barndorff-Nelsen and Shephard ( propose he realzed varance RV,, p,+(+/ p,+/, (8 = whereas Barndorff-Nelsen and Shephard (4 suggess he rpower varaon V,, = µ 3 /3 3 p,(+3/ /3 p,(+/ /3 = p,(+/ /3, (9 where µ s he -h momen of a sandard normal dsrbuon. he former s a conssen esmaor for he uadrac varaon of he process, and hence esmaes conssenly he negraed varance only n he absence of umps. In conras, he laer enals a conssen esmae of he negraed varance even n he presence of umps. he above realzed measures mplcly assumes he absence of mare frcons. Decomposng he observed asse prce p, no he rue prce p, and a nose ϵ, arsng from generc mare frcons yelds p, = p, + ϵ,, = A, B. he resulng sample of nraday regularly-spaced-n-me observaons over days hen s p,+/ = p,+/ + ϵ,+/, = A, B (3 where ϵ,+/ s by assumpon a zero-mean geomerc α-mxng process. I s possble o esmae he negraed varance a day from he nosy prce daa {p,+/ ; =,..., ; =,..., } usng approprae realzed measures. Zhang e al. (5 nroduce he wo-scale realzed varance S,,L, = RV,,L, L RV,,, (3 where /L L RV,,L, = L l= = p,+ (+/L+l p,+ /L+l gauges he average realzed varance across /L subsamples of sze L = O /3. Smlarly, he mul-scale realzed varance pu forh by Zhang (6 and Aï-Sahala e al. ( consders a weghed average of realzed varances over dfferen samplng freuences. In parcular, L L l S,,L, = a l p,+ +L l p,+ l= L l = + RV,,, (3 L where a l s such ha l= a l = L and l= a l/l =. For nsance, f one consders L l = l, hen a l = l L l/l / /(L /L. Barndorff-Nelsen e al. (8 show ha boh he wo- and mulscale realzed volaly esmaors are asympoc euvalen o realzed measures belongng o he class of ernel-based esmaors gven by RK,,H, = L l κ γ,,l + γ,, l, (33 L l= where γ,,l = L =l (p,+(+/ p,+/ (p,+(+ l/ p,+( l/, and he ernel s such ha κ( = and κ( = κ ( = κ ( =. ore specfcally, he wo-scale realzed volaly corresponds o a realzed ernel esmaor wh κ(x = x and L = /3, whereas he mul-scale realzed volaly o a realzed ernel measure wh κ(x = 3x + x 3 and L = /. References Aï-Sahala, Y., 996. esng connuous-me models of he spo neres rae. Revew of Fnancal Sudes 9, Aï-Sahala, Y., Bcel, P.J., Soer,..,. Goodness-of-f ess for ernel regresson wh an applcaon o opon mpled volales. Journal of Economercs 5, Aï-Sahala, Y., Fan, J.-I., Peng, H., 9. 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