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1、Econometrica,Vol.77,No.1(January,2009),283306BOOTSTRAPPINGREALIZEDVOLATILITYBYSÍLVIAGONÇALVESANDNOURMEDDAHI1Weproposebootstrapmethodsforageneralclassofnonlineartransformationsofrealizedvolatilitywhichincludestherawversionofrealizedvolatilityanditslogarith
2、-mictransformationasspecialcases.Weconsidertheindependentandidenticallydis-tributed(i.i.d.)bootstrapandthewildbootstrap(WB),andprovetheirfirst-orderas-ymptoticvalidityundergeneralassumptionsonthelog-priceprocessthatallowfordriftandleverageeffects.WederiveE
3、dgeworthexpansionsinasimplermodelthatrulesouttheseeffects.Thei.i.d.bootstrapprovidesasecond-orderasymptoticrefinementwhenvolatilityisconstant,butnototherwise.TheWByieldsasecond-orderasymptoticre-finementunderstochasticvolatilityprovidedwechoosetheexternalra
4、ndomvariableusedtoconstructtheWBdataappropriately.Noneofthesemethodsprovidesthird-orderasymptoticrefinements.Bothmethodsimproveuponthefirst-orderasymptotictheoryinfinitesamples.KEYWORDS:Realizedvolatility,i.i.d.bootstrap,wildbootstrap,Edgeworthexpan-sions.1.
5、INTRODUCTIONTHEINCREASINGAVAILABILITYofhighfrequencyfinancialdatahascon-tributedtothepopularityofrealizedvolatilityasameasureofvolatilityinfi-nance.Realizedvolatilityissimpletocompute(itisequaltothesumofsquaredhighfrequencyreturns)andisaconsistentestimatoro
6、fintegratedvolatilityundergeneralconditions(seeAndersen,Bollerslev,andDiebold(2002)forasurveyofrealizedvolatility).Recently,aseriesofpapers,includingBarndorff-NielsenandShephard(henceforthBNS)(2002)andBarndorff-Nielsen,Graversen,Jacod,andShep-hard(BNGJS)(
7、2006)havedevelopedanasymptotictheoryformeasuresofvariationsuchasrealizedvolatility.Inparticular,forarathergeneralstochas-ticvolatilitymodel,theseauthorsestablishacentrallimittheorem(CLT)for1Wewouldliketothankparticipantsatthe2005NorthAmericanWinterMeeting
8、oftheEconometricSociety,theSBFSIFIIconference,Québec(April2005),theCIREQMontréalFi-nancialEconometrics(May2005),theSETAconference,Taipei(May2005),the2005CEAmeet-ings,thePrincetonChicagoHighFrequencyConference(June20
