Journal of Econometrics. Quasi-maximum likelihood estimation of volatility with high frequency data

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1 Joual of Ecoomecs Coes lss avalable a SceceDec Joual of Ecoomecs joual homeage: wwwelsevecom/locae/jecoom Quas-maxmum lkelhood esmao of volaly wh hgh fequecy daa Dacheg Xu Bedhem Cee fo Face, Pceo Uvesy, Pceo, NJ 85, USA acle fo absac Acle hsoy: Receved 8 Seembe 8 Receved evsed fom 8 Jue Acceed 7 July Avalable ole July JEL classfcao: C C C5 Keywods: Iegaed volaly Make mcosucue ose Quas-maxmum lkelhood esmao Realzed keels Sochasc volaly hs ae vesgaes he oees of he well-kow maxmum lkelhood esmao he esece of sochasc volaly ad make mcosucue ose, by exedg he classc asymoc esuls of quas-maxmum lkelhood esmao Whe yg o esmae he egaed volaly ad he vaace of ose, hs aamec aoach emas cosse, effce ad obus as a quas-esmao ude mssecfed assumos Moeove, shaes he model-fee feaue wh oaamec aleaves, fo sace ealzed keels, whle beg advaageous ove hem ems of fe samle efomace I lgh of quadac eeseao, hs esmao behaves lke a eave exoeal ealzed keel asymocally Comasos wh a vaey of mlemeaos of he ukey Hag keel ae ovded usg Moe Calo smulaos, ad a emcal sudy wh he Euo/US Dolla fuue llusaes s alcao acce Elseve BV All ghs eseved Ioduco he avalably of hgh fequecy daa s a double-edged swod fo he esmao of volaly O he oe had, faclaes ou emcal sudes of he asymoc oees of a aual esmao, Realzed Vaace RV, e he sum of squaed logeus; whle o he ohe had, alog wh he daa comes make mcosucue ose, whch dsus all he desable oees of he esmao Whou mcosucue ose, hs smle esmao s boh cosse ad effce I addo, ohe sudes have shed lgh o s ceal lm heoy, such as Jacod 99 ad Badoff-Nelse ad Shehad Howeve, he exsece of ose efees wh he esmao esecally whe he samlg fequecy aoaches zeo A commo acce s o samle sasely, say evey 5 m, dscadg a lage oo of he samle ad he fomao hee he oblem has eceved cosdeable aeo ecely Fo sace, Aï-Sahala e al 5 sugges samlg as fequely as ossble a he cos of modelg he ose hs ae assumes cosa volaly so ha ca efom he maxmum lkelhood esmao MLE I he seg of sochasc volaly, Zhag e al 5 bg fowad a oaamec esmao, wo-scale el: E-mal addess: dachegx@ceoedu Realzed Volaly SRV, whch s he fs cosse esmao he esece of ose, dese a elavely low covegece ae 6 Subsequely, Zhag 6 advocaes Mul-Scale Realzed Volaly MSRV, movg he covegece ae o, whch s he omal ae a model ca each as show by Gloe ad Jacod Recely, Badoff-Nelse e al 8 have desged vaous ealzed keels RKs ha ca be used o deal wh edogeous ose ad edogeously saced daa ad he covegece aes ae he same as ha of he MSRV I smulao, hese oaamec esmaos efom vey well wh omally seleced badwdh o keels; whle acce, esmaos wh badwdh based o ad hoc choces, o based o small samle efomace may behave bee, as llusaed Gaheal ad Oome 7 ad Bad ad Russell 8 Aohe gou of esmaos daes back o Zhou 996, who fs ales he auocovaace-based-aoach o cosa volaly cases Hase ad Lude 6 exed he Zhou esmao o he sochasc volaly models wh seally deede ose Howeve, he Zhou esmao ad s exesos ae cosse he movao ad sao of hs acle sem fom Gaheal ad Oome 7 ad Aï-Sahala ad Yu 9 Usg afcally smulaed zeo-ellgece daa, Gaheal ad Oome 7 comae a comehesve se of esmaos cludg hose lsed above ad he ad hoc modfcaos Accodg o he sudes, he MLE, hough ossbly mssecfed wh me vayg volaly, s amog he bes ems of effcecy ad obusess -76/$ see fo mae Elseve BV All ghs eseved do:6/jjecoom7

2 6 D Xu / Joual of Ecoomecs I addo, Aï-Sahala ad Yu 9 aly he MLE o aalyze he lqudy of NYSE socks he maxmum lkelhood esmaes wh he daa smulaed fom sochasc volaly models dcae good oees, alhough hs esmao s deved fom a cosa volaly assumo Relaed woks also clude Hase e al 8, whee he auhos sugges he cossecy of he MLE by examg movg aveage fles hese sudes movae us o cosde he MLE as a Quas-Maxmum Lkelhood Esmao QMLE ude mssecfed models, whch daes back o as ealy as Amemya 97, Whe 98, 98 I hese acles, he famewok of mssecfed esmao has bee bul ad s close coeco wh Kullback Leble Ifomao Ceo KLIC see Kullbacks ad Leble 95 has bee llusaed Sce hese semal woks, Domowz ad Whe 98 ad Baes ad Whe 985 have exeded he cossecy esuls of he QMLE wh d models o vaous cases cludg deede obsevaos, Quas-GMM-esmaos, ad Quas-M-esmaos Ou wok s heeby bul o he fuso of hgh fequecy daa ad mssecfed lkelhood esmao he coec model secfcao feaues sochasc volaly Howeve, hs model s eoally mssecfed o be oe of cosa volaly Ude hs assumo, we efom quas maxmum lkelhood esmao ad aalyze he esmao, whch s esseally he same as he MLE Aï-Sahala e al 5 Remakably, he coex of he coec model, he QMLE of volaly cossely esmaes he egaed volaly a he mos effce ae Also, he QMLE of ose vaace has he same asymoc dsbuo as befoe I ohe wods, he maxmum lkelhood esmaos ae obus o sochasc volaly I addo, hey ae sll obus o adom samlg evals ad o-gaussa make mcosucue oses he ae s ogazed as follows Seco evews he aamec lkelhood esmao ad ovdes he uo ad movao fo he QMLE Seco oules he classc asymoc heoy of he QMLE, ad deves a exeso o moe geeal segs Seco vesgaes he sascal oees of he QMLE, whee he cossecy, ceal lm heoy ad obusess of he QMLE ae esablshed Seco 5 comaes wh oaamec keel esmaos, ad Seco 6 uses Moe Calo smulaos o vefy he coclusos obaed fom he evous secos Seco 7 deals a emcal sudy wh he Euo/US Dolla fuue daa Seco 8 cocludes he Aedces ovdes all mahemacal oofs Revsg he MLE: he QMLE I hs seco, we ecaulae he aamec aoach ad oduce ou quas-esmao he adoal aamec mehodology ales o cases whee he ue value of he aamee of ees s a secal o he aamee sace heefoe, Aï-Sahala e al 5 have o make a assumo ha he volaly s ehe a sochasc ocess o a deemsc fuco, bu sead, a cosa ha s, he lae effce log ce ocess sasfes dx dw wh he obseved log asaco ce X coamaed by he mcosucue ose U a way such ha X X + U, whee s he obsevao me Fo smlcy, we assume ha he daa ae egulaly saced, sasfyg he obsevaos We gve he MLE a alas QMLE somemes ode o emhasze model mssecfcao ad kee he oao le wh he classc esuls of mssecfed models ae made wh [, ], whee s fxed, so ha he fll asymoc behavos ae deemed as goes o ad goes o smulaeously he sucue of he obseved log eu Y s feaues MA, whee Y X X W W + U U If we osulae ha he ose dsbuo s Gaussa, he ou log lkelhood fuco fo Ys s l, a log de log Y Y whee s gve Box I he MLE ˆ, â oves o be cosse a dffee aes fo s volaly a ad ose a eve f he ose dsbuo us ou o be ˆ A â a L N, 8a a + cum [U] whee cum [U] s he fouh cumula of he ue ose U Neveheless, amle evdece of sochasc volaly calls hs smlsc model o queso So, s aual o ask: wha s he mac of sochasc volaly o he MLE? Wll sochasc volaly chage he desed oees of he MLE, jus as mcosucue ose does o RV, o wll he MLE sll be cosse? Smulao sudes by Aï-Sahala ad Yu 9 ad Hase e al 8 seem o have suggesed ha he MLE may be a cosse esmao of he egaed volaly Iuvely, hs cojecue s lausble ha whe volaly becomes sochasc, he egaed volaly, he aamee of ees, haes o be he aveage of he volaly ocess, whch s execed o be a legmae caddae fo he esmae If cossecy s guaaeed, wha would be he covegece ae ad asymoc vaace? How would comae wh ohe aleave oaamec esmaos? Close scuy of he esmao he absece of mcosucue ose may yeld moe sghs Cosde a sochasc volaly model wh o ose: dx dw R he objecve s o esmae he egaed volaly d By desg, we msakely assume he so volaly o be cosa; heefoe, he quas-log lkelhood fuco s l, log log Y Y whee Y X X R ad Y Y, Y,,Y Aaely, he QMLE s ˆ X Y X X X Hee, he RV esmao ecus hough a dffee agume, ad s of couse he efec esmao ude he ue sochasc volaly model Howeve, he cossecy of he QMLE s o loge saghfowad he esece of ose, because hee may be o close fom avalable fo hs esmao Is asymoc vaace s fa moe comlcaed due o heeoskedascy ad auocoelao, as meoed by Hase e al 8 he emag ae goously vesgaes he asymoc behavo of he QMLE he seg of sochasc volaly ad mcosucue ose, whch leads us o he classc asymoc heoy of quas-esmaos

3 D Xu / Joual of Ecoomecs a a a + a a a + a C a A a + a Box I Asymoc heoy of quas-m-esmaos I hs seco, we a fs befly evew he cossecy of he QMLE ude mssecfed models, a heoy ally develoed Whe 98, 98 he aoale behd he heoy s elaed o Kullback Leble Ifomao Ceo KLIC Moe ecsely, suose ha we have a d adom samle, ad le gx be he ue ukow daa geeag desy, ad f X, ou ossbly mssecfed desy dexed by a aamee Whe 98 clams ha ude cea egula codos, he QMLE s cosse o whch mmzes KLIC: Ig : f, Elog[gX/f X, ] whee he execao s ake ude he ue model If he model s coecly secfed, ha s, hee exss, such ha gx f X,, he he KLIC aas s mmum a, hece hs esul s ageeme wh he cossecy of egula maxmum lkelhood esmaos Ohewse, he model s mssecfed, ad uvely, mmzes ou goace of he ue sucue I Domowz ad Whe 98, he auhos geealze hese esuls o clude deede obsevaos, ad show ha he QMLE ˆ, whch maxmzes Q,, sll coveges obably o, a maxmze of Q,, he execao of Q, ude he ue model I addo, Baes ad Whe 985 exed he heoy o geeal quas-m-esmaos, by oducg dsceacy fucos Now, we add moe adomess o he mssecfcao heoy, whch eables ou alcaos o sochasc volaly models, whee he aamee of ees self s adom he easog of he oof s smla o he egula oe gve Whe 98 ad Newey ad McFadde 99 heoem Le Q, ad Q, be wo adom fucos such ha fo each, a comac subse of R k, hey ae measuable fucos o ad, fo each, couous fucos o I addo, Q, s almos suely maxmzed a Fuhe, he followg wo codos ae sasfed as : Ufom covegece: su kq, Q, k P Idefably: fo evey >, hee exss a cosa >, such ha P Q, max Q, > 5 :k k he ay sequece of esmaos ˆ such ha Q, ˆ Q, + o coveges obably o, e, ˆ mgh deed o whe model s mssecfed su P Noe ha he defably codo hee s slghly dffee fom ha a commo seg, fo examle Va De Vaa ad Newey ad McFadde 99 I o oly eques some uqueess lke oey of he maxmze, bu also a oe omalzao such ha he maxmze ca be dsgushed asymocally Now we modfy he assumos o accommodae o he M-esmaos seg, case we eed dffee omalzaos fo dffee aamees heoem Le, ad, be adom veco-valued fucos Fo each, a comac subse of R k, hey ae measuable fuco o, ad fo each, couous fucos o I addo, hee exss a sequece of, sasfyg, almos suely, such ha as, Ufom covegece: su k,, k P 6 Idefably: Fo evey >, hee exss a cosa >, such ha, P m k, k > 7 :k k he ay sequece of esmaos ˆ such ha, ˆ o coveges obably o, e, ˆ P I he followg dscussos, we wll choose as he scoe fuco of a mssecfed model, u o a aoae omalzao, ad s caefully secfed coesodg o he ceal lm esul s gve by he ex heoem, whch s a exeso of heoem Domowz ad Whe 98 heoem Suose ha he codos of heoem ae sasfed I addo,, ad, ae couously dffeeable of ode o Also, hee exss a sequece of osve defe maces {V } such ha V, L N, Ik 8 If, s sochasc equcouous, ad, P,, ufomly fo all, he V, ˆ L N, Ik 9 he exesos of he classc asymoc heoy ave he way fo a hoough quy of asymoc oees of he QMLE Sascal oees of he QMLE hs seco shaes he seu wh mos volaly esmaos avalable he leaue Moe secfcally, we make he followg assumos

4 8 D Xu / Joual of Ecoomecs Assumo he udelyg lae log ce ocess sasfes dx dw wh he volaly ocess a osve ad locally bouded Iô semmagale Assumo he ose U s deedely ad decally dsbued, ad deede of ce ad volaly ocesses, wh mea, vaace a ad fe fouh mome he assumos of d ose ad s deedece wh ces ae o always cosse wh he emcal evdece, as oed ou Hase ad Lude 6 Aï-Sahala e al 5 have dscussed he way o modfy he log lkelhood fuco wh moe aamees, accodg o he assumed aamec sucue of he ose See also Gaheal ad Oome 7 fo a MA mlemeao As o he SRV, Aï-Sahala e al fohcomg have exeded o he case wh seally deede ose Kala ad Lo 8 have oosed a modfcao of he SRV wh edogeous ose ad heeoskedasc measueme eo Ideedely, Badoff-Nelse e al 8 have show he obusess of he ealzed keels wh esec o edogeous ose Howeve, seally deedece lus edogeous assumo ae sll usasfacoy, sce ealy he asaco ces ae ecoded wh oud-off eos L ad Myklad 7 have dscussed he obusess of he SRV wh esec o oudg eos whle Jacod e al 7 have ecely oosed a eaveagg aoach ha woks well fo a geeal class of eos cludg cea combao of oudg ad addve eos I hs ae, beg asmoous ad fo smlcy, we cosde he d whe ose ad ovde addo a heusc agume fo me-deede ose o valdae he alcaos of he QMLE acce Cossecy of he QMLE Cosde fs wha would hae he absece of ose Based o he dscusso Seco, he scoe fuco u o a oe omalzao ude mssecfed model s,, dl, Y Y d ad s oo s ˆ X Y he we choose, X X X Z R d heefoe, has a oo Because s a comac se, ad f we eque he aamee sace o be bouded away fom zeo, e, hee exs such ha >, he su,, X Y Z P Poof of he las se s well-kow, see eg Kaazas ad Sheve 99, heoem 58, hece ufom covegece obably heoem s show Idefably codo hs assumo accommodaes vually all couous me facal models See Jacod 8 hyohess L s fo moe deals vally holds, so he cossecy follows fom heoem he ae of covegece deeds o he ae of, whch s, as gve by Jacod 99 fo sace heefoe, ˆ X Z Y d O Aaely, we do o eed heoem a all hee I geeal, howeve, hs s cealy o he case Oe sao fom hs smle examle s egadg he seleco of If we add o leveage assumo, ha s o say, he volaly ocess s codoally deemsc, he s ohg bu he codoal execao of ude he ue model Whe he obseved daa ae osy, we have Y R + U U Besdes he cosa volaly assumo, we msakely assume ha he oses ae omally dsbued wh vaace a heefoe, he quas-log lkelhood fuco s exacly, hece he same lkelhood esmao ecus he fom of he QMLE ude mssecfed model Howeve, closedfom exessos ae o loge avalable, so we have o u o heoem fo cossecy Deoe, a ad we also assume ha he aamees say a comac se, whch s bouded away fom zeo As he o ose case, we choose he followg scoe fucos u o some @ logde @ logde @a Y ad + whee s as gve Box II Beg awae ha he covegece aes may be dffee fo he wo aamees, as show Seco, we choose dffee omalzaos accodgly hese omalzaos ae esseal o esue he defably codo Deoe j, ad j U j U j he dffeece bewee ad, fo sace s as gve Box III he fs em s a lea combao of magale dffeeces, whle he secod em s he sum of coss oducs ove dffee eces of egals he hd oe s a mxue of he magale a ad he ose a he es ae uely elaed o he ose I s clea ha he fs wo ems as a whole, he hd oe ad he es ae awse ucoelaed gve he oosed assumos Now we oceed wh vaace calculaos he lesso leaed fom he falue of RV dcaes ha he vaace of he ose may domae he ohes So fo he ose a, we accuaely comue he vaace of he devaves wh esec o dffee aamees Lemma Gve Assumos ad, we have 8 X < Z 9 Z dw : ; O

5 D Xu / Joual of Ecoomecs Z + a a a Z a + a a Z + a a a Z C A + a C X + Y : Z Box II {z } magale dffeece X Z X X j + 9 X X + ; j6,,+ j {z } due o he ose + a + j Z j j @ + + a {z } due o he ose Box III X X X Z j j6 X Z j j Z j O O j Lemma As o he ose a, we have a 6a + o 5 6 a + cum [U] + o 7 a 8 By vefyg he defably codo, solvg he equaos ad alyg he above lemmas, we ca ove heoem ad ae as gve, ˆ ˆ, â s as defed heoem Gve Assumos ad, he QMLE ˆ, â sasfes: â a P ad ˆ R P As execed, he cossecy of he volaly esmao sll holds, eve hough he volaly ocess becomes sochasc Ceal lm heoem of he QMLE Alhough we have show he cossecy of he QMLE, we have o exloed whehe he QMLE has he omal covegece aes, ehe have we show ayhg abou he magude of he asymoc vaace o ossble asymoc bas o aswe hese quesos, he followg lemma ad heoem ovde he ceal lm heoy Lemma Gve Assumos ad, we have he ceal lm heoem gve Box IV heoem 5 ad ae as gve, ˆ ˆ, â s as defed heoem Gve Assumos ad, he QMLE ˆ, â sasfes he ceal lm heoem gve Box V I he cosa volaly case, QMLE aas he omal effcecy, as ca be execed sce he QMLE s cosuced he same way as he MLE I geeal, QMLE acheves he omal covegece aes Howeve, hs QMLE mehod mgh o ovde a saghfowad esmao fo he egaed quacy, ha s, R d hus, as o he cosuco of cofdece evals, oe ca sead use he mehod gve by Jacod e al 7 Robusess of he QMLE Df Wha haes o ou coclusos f he udelyg X ocess has a df? Moe ecsely, suose X o be of he Iô ye, dx µ + dw whee µ s a locally bouded ad ogessvely measuable ocess Isead of aameezg he df em ad modfyg ou lkelhood esmao accodgly, we comleely goe he MN s a oao of mxed omal used Badoff-Nelse e al 8, ad L X meas X-sable covegece law

6 D Xu / Joual of Ecoomecs L X MN R R 5 6a 7 + a 8 5 a + a 6a 5 CC a + cum AA [U] a 8 Box IV Z ˆ â a C A L X MN 5a R R Box V + R a a + cum [U] CC AA esece of df, o ohe wods, mssecfy he model wh df I hs R case, he esmao s uchaged, ad we oly eed o elace R wh µ + R he oof Loosely seakg, hs wll o ale ou coclusos sce he hgh fequecy seg, he df s of ode d, whch s mahemacally eglgble wh esec o he dffusve comoe of ode d A aleave agume gve by Myklad ad Zhag 9 s o zeo ou he df by chagg obably measues Radom samlg evals Wha f he obsevaos ae made adomly? If he samlg evals bewee wo cosecuve obsevaos ae d, ad deede of he ce ocess, we may add oe moe mssecfcao ha he daa ae egulaly saced; hece we oba he same esmao as befoe Cossecy ca be esablshed decly by codog o he obsevao me I fac, hs esmao ca be vewed as he Peed Fxed Maxmum Lkelhood PFML esmao dscussed Aï-Sahala ad Myklad I lgh of hs, we may efom he Full Ifomao Maxmum Lkelhood FIML o Iegaed Ou Maxmum Lkelhood IOML esmao, ode o ulze he fomao egadg he dsbuo of he adom evals, f avalable Also, s ossble o exed he d samlg scheme o edogeous ad sochascally saced obsevaos, by cosdeg he me-chaged ocesses as Badoff-Nelse e al 8 o Myklad ad Zhag 6 he exeso s effoless lgh of he coeco bewee QMLE ad RKs selled ou Seco 5 No-Gaussa ad seal-deede mcosucue ose Wha abou he obusess wh esec o he ose? Fom Lemma ad heoem, we fd ha he dsbuo of he d ose does o affec he cossecy I ohe wods, whaeve he dsbuo of he ose s, s mssecfed o be Gaussa, ad he QMLE obaed by maxmzg gves he same esmao of a ad he same ode of covegece ae, hough he asymoc vaace may be dffee O he ohe had, f he oses ae fac me-deede, we ca combe he QMLE wh he subsamlg mehod Fo sace, f he ose self feaues a MAk sucue, we ca dvde he whole samle o k + dsjo as such ha he oses assocaed wh he adjace os each subsamle ae ucoelaed he, we ca smly aly he QMLE o each subsamle ad aggegae he esmaes by akg he aveage Jums If he ce ocess has jums, he ˆ wll covege o R + P X sead, whch cocdes wh he SRV esmao he oblem of seaag jums fom volaly hs seg s moe edous, ad may cobue lle, sce mos cases lage jums do o hae vey ofe wh a day If hey do occu, we may use he wavele mehod Fa ad Wag 7 o emove jums befoe esmao, o seaae he esmao eods by jums ad add u evey ece of egaed volaly ogehe, f he osos of jums ca be locaed accuaely 5 Comasos wh ealzed keels 5 Esmao mehods Realzed keels RKs clude a sees of oaamec esmaos desged fo volaly esmao he esece of ose Fla-o RKs ake o he followg fom X h K X X + k h X + h X 8 H whee h X h X X j X j X j X j h h 9 j s he hh samle auocovaace fuco ad he keel k s a wegh fuco he mos commo fe-lag fla-o keels ae of he modfed ukey Hag ye, defed by: k H x s x {alexale} I addo, hee ae fe-lag ealzed keels whch may assg ozeo wegh o all auocovaaces, such as he omal keel: k o x + xe x As execed, he sascal oees of RKs vay as dffee keels ad badwdhs ae seleced heefoe, eables us o make dffee choces fo secfc uoses Howeve, he choce of he badwdh may also become a bude acce, ha he omal badwdh gve by he heoy cao be esmaed vey accuaely, ad ha he ule-of-humb aoxmao of he badwdh may o efom as well as he oe seleced ad hoc ways hs oblem may become eve wose f he esmaes wee sesve o he choce of badwdh By coas, he QMLE s desged a aamec way, whch s fee of badwdh ad keel seleco, whle shag he desed model-fee feaue wh oaamec esmaos I s by aue a dffee esmao ad may o be egaded as oe of he RKs decly

7 D Xu / Joual of Ecoomecs AvaRK/AvaQMLE Omal ukeyhag Paze Cubc Asymoc Relave Effcecy Fg Rho Asymoc elave effcecy of he QMLE ad RKs 5 Asymoc behavo ad fe samle efomace I egad o asymoc effcecy, he keel ad badwdh ca be chose fo RKs o acheve he same omal covegece ae as he QMLE Neveheless, whe volaly s cosa, he asymoc vaace of fe-lag keels ca oly aoxmae he aamec vaace boud, whch, by coas, ca be obaed by he QMLE ad he omal keel wh fe lags Whe volaly s sochasc, he elave effcecy of he QMLE ad RKs deeds o he exe of heeoskedascy Moe ecsely, we have 5 AvaRK AvaQMLE 6 + k k q + k k / k + + q + k k / k 5 + whee R q du/ R u u du measues he vaably of he volaly ocess, ad k, k ad k ae cosas deemed by he seleced keel Fg los he elave effcecy of fou ycal RKs agas he QMLE Aaely, he QMLE eds o be moe favoable ha fe-lag keels as becomes lage, whle RKs ae bee whe s small he omal keel wh heoecally omal badwdh, howeve, fully domaes he QMLE asymocally exce fo he case, whe volaly s cosa Iuvely, he smalle s, he fuhe he mssecfed model devaes fom he uh A majo dawback of RKs s ha hey eque a umbe of ou-of-eod aday eus because of he cosuco of he auocovaace esmao h X Fo hs easo, felag keels, acula, ae o mlemeable emcally I acce, he feasble auocovaace esmao s mlemeed by ±h X X H X j X j X j X j h h jh+ Wh fe-lag fla-o keels, he cu-off ea he bouday s o a ssue asymocally, howeve may yeld a lage bas fe samle As he samle sze deceases, amely, H/ ceases, hs bas becomes moe evde he ex seco ales Moe Calo smulaos o demosae he edge effec 5 he asymoc vaace of he RK obaed hee eques a efeme of he edos as well as he omal badwdh 5 Quadac eeseao ad weghg maces A uve way o udesad he smlaes ad dffeeces bewee he QMLE ad RKs s o egad hem as quadac esmaos Moe secfcally, we have he followg quadac eave eeseao of he QMLE heoem 6 he QMLE ˆ, â sasfes he followg equaos: ˆ Y W ˆ, â Y â Y W ˆ, â Y he weghg maces sasfy: W, a W, a whee s gve by, ad C A Also, W, a ad W, a deed o a / ad a oly hough Smlaly, a feasble ealzed keel esmao ca be exessed as K X Y WY 5 whee W s deemed by he keel k ad badwdh H W, {+Haleale H} j W,j k {ale j aleh} {+Halejale H} H Fg los he weghg maces agas he ow ad colum dces fo he QMLE ad ukey Hag keel he lo dslays smla feaues such as he weghs o he dagoals ae ehe o vey close o wh seveal lags, ad decay o zeo beyod ha he followg heoem llusaes he mlc coeco bewee he wo esmaos ems of he asymoc behavo heoem 7 he QMLE s asymocally equvale o he omal R keel wh mlc badwdh H ˆ a / I ohe wods, fo ay K +, < <, ad ay K ale, j ale K, we have j W,,j, a k o heoem 7 also os ou ha he weghg max W, a s aoxmaely a symmec oelz max wh equal wegh alog he dagoal, bag he bouday effec he mlc badwdh deeds o he aamees of ees ad s subomal I heefoe exlas why he omal keel wh omal badwdh asymocally domaes he QMLE Fg Neveheless, he edg os of he dagoals have dffee aes, dcag dffee eames of he edge effec Aaely, he QMLE

8 D Xu / Joual of Ecoomecs QMLE of Volaly 5 6 QMLE of Nose Vaace he ose sasfes omal dsbuo wh sadad devao a 5% he jums follow a Posso ocess N deede of he ce ad volaly ocesses wh esy he jum sze volaly s J V exz, whee z N 5, he umbe of Moe Calo samle ahs s Fg shows he hsogams of he sadadzed esmaes comaed wh he asymoc dsbuos Moeove, able comaes he samle quale sascs wh he heoec bechmaks gve by he sadad Gaussa dsbuo All hese smulao esuls ecofm ou asymoc heoy ukeyhag Keel 5 6 Fg Weghg maces of he QMLE ad he RK Noe: We lo he wo weghg maces agas he colum ad ow dces he aamees ae, a 5, 65 ad /5 he badwdh H 57a / cools he weghs o he bouday a moe aual way, ad hece has bee fe samle efomace he quadac eeseao also sheds lgh o he dffeeces he ocedues of esmao he badwdh H of he RK s ehe esmaed as fs se o seleced a ad hoc way, whle he badwdh ˆ of he QMLE s auomacally udaed by he omzao algohm, o moe uvely, by eag ad he weghg max of he QMLE s heefoe moe adave I vew of hs, s aual o cosuc a oe-se aleave fo he QMLE, whch, sead of ug olea omzao, emloys a cosse lug- of ˆ fo ad hs oese volaly esmao cocdes wh oe of he ubased quadac esmaos gve by Su 6, whee s asymoc oees ae dscussed ude he cosa volaly assumo he quadac foms of ohe oaamec esmaos ae also cluded Adese e al fohcomg 6 Smulao sudes wh hgh fequecy daa 6 Asymoc behavo of he QMLE We a fs coduc Moe Calo smulaos o vefy he asymoc esuls gve heoem 5 We fx as day Wh [, ], he daa ae smulaed usg Eule scheme based o sochasc volaly models, fo sace he Heso Model wh jums volaly ocess dx µd + dw d ale + db + J V dn whee EdW db d he aval of asacos follows a Posso ocess ad he mea eaval me s s he ue values of aamees ae cosse wh hose chose by Aï- Sahala ad Yu 9 Secfcally, he df µ s, ad o add he leveage effec, s seleced o be 75 ale 5 ad he volaly of volaly s s samled fom he saoay dsbuo of he CIR ocess, ha s Gammaale /, /ale, so ha he ucodoal mea of volaly ocess s exacly, chose as 6 Comasos wh RKs Fo comaso uose, we mleme he ukey Hag keel whch s oe of he fla-o keels ha covege a he bes ae I s cosdeably effce comaed wh ohe keels Badoff-Nelse e al 8, ad does o eque oo may ou-of-eod daa As a bechmak, we fs mleme he ukey Hag keel wh he feasble badwdh, H c wh c gve by v u s u c k k + + k k k q whee a / R du, R q du/ R u u u du, k 9, k 7 ad k 7 Also, he edge effec s elmaed ha he calculao, usg fomula 9, cludes he ou-ofeod daa Hece, hs esmao would ovde us wh he bes ukey Hag keel esmae RK heoy Nex, we make he edge effec sad ou by aoxmag he auocovaaces usg, whle keeg he heoecal omal badwdh as befoe We deoe he secod esmao as RK Fally, o mmc he emcal alcaos of hese esmaos, we edo he exemes wh he smle ule-of-humb choce of he badwdh: q Ĥ 57â /RV X q whee â RV X/ ad RV X s he RV esmao based o m eus So we oba he coesodg RK ad RK he smulaos ae based o he same sochasc volaly model ad aamees as befoe wh a loge me wdow [, ] We egad [, ] as -samle eod, ad he es me eval s oly used fo feasble keels he esuls of esmao ae eoed able I s evde ha he QMLE domaes he fou mlemeaos of he ukey Hag keel ems of he RMSE As execed, RK s vey close o he QMLE, ad bee ha he ohe hee keels, ha emloys moe daa ad s equed wh heoecally omal badwdh Comag he fs wo keels, we ca also see ha he edge effec esuls a lage bas whe he samle sze s elavely small, ad become dameed as he samle sze goes u I addo, he dffeeces bewee he fs wo keels ad he las wo ae he losses due o he subomal choce of he badwdh I acce, oly RK ca be aled, whch may suffe fom lage bas ad vaace because of he wo souces of losses 7 Emcal wok wh he Euo/US Dolla fuue ces I hs seco, we ae eesed esmag he ealzed vaace of he Euo/US dolla fuues caed ou o he Chcago Mecale Exchage CME he yea 8 he coacs ae acvely aded o he -h clock, ad quoed ems of he u

9 D Xu / Joual of Ecoomecs able I hs able, we eo he fe samle quales of he feasble sadadzed QMLE, whch emloys he heoec asymoc vaace he bechmak quales ae hose fo he lm dsbuo N, No obs Mea Sdv 5% 5% 5% 95% 975% 995% a Iegaed Volaly 5 Nose Vaace Fg Hsogams of he sadadzed esmaes Noe: We lo he hsogams of he sadadzed esmaes he ue ose vaace s 5 he smulaos clude adom evals ad volaly jums he desy of he asymoc dsbuo s also loed able R hs able eos he esmaes fo ˆ, d whee ˆ s gve by he QMLE ad vaous mlemeaos of he ukey Hag keel esecvely Amog hem, RK ad RK have he heoecal badwdh, whle RK ad RK emloy ou-of-eod daa s 5 s s s s m m Bas QMLE Sdv RMSE RK Sdv Bas RMSE RK Sdv Bas RMSE RK Sdv Bas RMSE RK Sdv Bas RMSE value of he Euo as measued US dollas he hgh fequecy daa ae avalable fom ck Daa Ic he foeg exchage makes ae less acve dug he weekeds ad holdays; heefoe, we elmae he asacos o Saudays ad Sudays as well as US fedeal holdays I addo, we also exclude Jauay, he day afe haksgvg, Decembe 6, ad Decembe Fally, we make eveyday beg a 5 m Chcago me whe he elecoc adg sas so

10 D Xu / Joual of Ecoomecs Aualzed Daly Realzed Volaly of he EUR/USD FX Fuue 8 Realzed Keel QMLE Bea Seas Fe Sale Lehma Bohes Bakucy Fae Mae ad Fedde Mac Balou Iceladc Css 5 5 JAN FEB MAR APR MAY JUN JUL AUG SEP OC NOV DEC JAN Fg he EUR/USD fuue: a case sudy able Summay sascs of he log-eus of he Euo/US Dolla fuue Avg o of obs Avg feq Mea Sd e s lag d lag s 68e 9 6e as o elmae he oeal ce jums bewee he oe hou adg ga fom m o 5 m he summay sascs of he daa ae ovded able Evdely, he mcosucue ose dslays a MA sucue We aly boh he QMLE ad he ukey Hag keel wh he ule-of-humb badwdh ad lo he aualzed daly ealzed volaly Fg I s aae fom he lo ha he wo mehods gve almos decal esmaes ad he dffeeces ae sascally sgfca hs s ageeme wh ou Moe Calo smulao esul, whee he wo esmaes ae dsgushable whe he adg fequecy s as hgh as seveal secods Moeove, Fg ovdes amle evdece of sucue beaks o jums he volaly ocess sag fom Seembe 8, whch echoes he fac ha, sce he, he global facal css has eeed s mos ccal sage Fuhe, hese jums o lage movemes ae usually assocaed wh facal ews: fo sace, he goveme sezue of Fae Mae ad Fedde Mac, Lehma Bohes bakucy ealy Seembe, ad he collase of hee majo baks of Icelad ealy Ocobe, ec hese fdgs ae hade o oba by modelg lowe fequecy daa 8 Coclusos hs acle cobues o he esmao of egaed volaly by showg ha he oula MLE, as a ew quas-esmao, s cosse, effce ad obus wh esec o sochasc volaly Moeove, hs aamec esmao oly volves a omzao ocedue, whch s fee fom badwdh seleco, ad hece vey covee acce Moe eesgly, hs seemgly aoae esmao us ou o be asymocally equvale o he omal ealzed keel wh a mlcly secfed badwdh, ad doma ove aleave ealzed keels ems of he fe samle accuacy hs acle makes aohe cobuo o he aalyss of model secfcao by exedg he classc asymoc heoy of he QMLE o a sochasc aamee seg, ad by showg a examle of mssecfyg models o uose, whch gves se o facly ad feasbly esmao hs sudy may be alcable o moe cases whee sochasc volaly lagues he esmao, such as covaace esmao, o moe geeally, whee he objec of ees s sochasc Ackowledgemes I am debed o Yace Aï-Sahala fo ecouagg me o wok o hs oc ad helful suggesos heeafe I aecae he commes of he Co-Edo A Roald Galla ad wo efeees, whch led o cosdeable movemes of he ae I am also gaeful o Jaqg Fa, Pee Rehad Hase, Jea Jacod, Ygyg L, Ulch K Mülle, Pe Myklad, Ec Reaul, ad Roe Sca fo valuable commes, whch heled a lo movg he qualy of he ae Also, I would lke o hak odd Hes fo daa asssace, ad Ja L ad X Wa as well as he acas of he semas a Pceo Uvesy ad he Uvesy of Chcago, of he woksho o Facal Ecoomecs a Felds Isue, ad of he d Pceo-Humbold Face woksho fo fuful dscussos All eos ae me Aedx A Poof of heoem Fo each have 6 Q, ˆ >Q, Q, ˆ >Q, ˆ Q, > Q, >, wh obably aoachg wa, we 9 > due o ufom covegece >; hus, 8 >, wa, Q, ˆ > Q, Fuhe, fo ay >, le N : { : k k < } he fo each, N c \ s comac, ad max N c \ Q, Q, ale Q, Le : Q, We have, fo ay <, max Q, N c \ P Q, ˆ > max Q, N c \ P Q, ˆ > Q, P Q, ˆ > Q,, > P Q, ˆ > Q,, > heefoe, Pkˆ k < 6 he saemes hs oof ae well osed f ay desed measuably s guaaeed Howeve, hs measuably ssue ca be goed by edefg all coces ems of oue measue, see Newey ad McFadde 99,

11 D Xu / Joual of Ecoomecs Aedx B Poof of heoem X < Z 9 Z E he oof s smla o Va De Vaa s oof of cossecy of he M-esmaos I follows fom heoem, o alyg 8 : d ; o Q, k, k ad Q, k, k X < Z 9 Z ale M E dw : ; Aedx C Poof of heoem As he o ose case, we have Fo smlcy, s suffce o ove hs esul fo k 8 Because X < Z 9 Z,,, : d ; O 9 Plug ˆ ad mully boh sdes by V, he we have V, ˆ V, P Sce ˆ, s sochasc equcouous, ad P,,, ufomly fo all, follows fom a aalogous easog as heoem Domowz ad Whe 98 ha,, P, whch cocludes he oof Aedx D Poof of Lemma Le j ad X R We also defe M X hx, X R R he {X } aleale ad {M } aleale ae magales Also, we ca add oe moe codo ha B ale ale B, 8 [, ], ha a egula localzao ocedue always ales Fo coveece, we chage he vaables: + + a a he he vese chage of vaables s gve by, a + o a + a 6 a + + o a + 7 ad we have j j +j j+ j Oe ca check j I addo, s a cocave fuco of, ad clealy aas s maxmum a + ad mmum a he bouday I addo, by 6 ad 7 + a a + O + + O a a log + O a a + a + + O he, ca be easly show ha M : +, + O he magale oey, we have 8 X < Z Z : 9 A ; By Combg he wo, we oba 8 X < Z Z dw : O O O 9 A ; hus, we ove by Chebyshev s equaly o ove, oe ha j j, so follows fom he magale oey ha E X E X Z j j6 Z j j X X X X Z j kl Z k j> k l>k Z l Z j j k l 8 X < Z j X j Z 9 k E dw : kj j k ; ale B B j X E j Xj Z k kj k X Xj kj E j k kj ale B X X j k<j O k k Z k k Fally, we ove 5 Because fo ay j 6, j j, j+, ale, ad j > X X Z E j j j X X a j j j, j+, E j X ale a B, +, O Z So all he coclusos clamed he lemma hold

12 6 D Xu / Joual of Ecoomecs Aedx E Poof of Lemma Le,,, Deoe K, K,j, K,j,k, ad K,j,k,l he coesodg cumulas of { } he K, sce he mea of s zeo Accodg o McCullagh 987, Seco, we have X va j kl K,j,k,l + K K j,k,l + K,k K j,l whee V v,,j,k,l + K K k K j,l X j kl K,j,k,l + K,k K j,l,j,k,l X,j,k,l j kl cum, j, k, l + cov, k cov j, l : V, + V, X,j,k,l v j kl cum, j, k, l X V v, v j kl cov, k cov j, l,j,k,l NX NX a v j j, + j,+ j, j + j+, + j+,+ j+, j, + j,+ j, Recall ha Lemma of Aï-Sahala e al 5, cum, j, k, l 8 < cum [U], f j k l; s,j,k,l cum : [U], f max, j, k, l m, j, k, l + ;, ohewse whee s, j, k, l deoes he umbe of dces amog, j, k, l ha ae equal o m, j, k, l So usg hs lemma, we ca oba X X V v, cum [U] v + v,+ +,+ v +,+,+ + v, +,+ + v,+,+ o faclae calculao, we @ he dec comuao a { } 6 + o a + o a { + + }+o a + o a a 8a { } 5 + o 5a + o a a a + + O Combg, we + a 6a + o 5 a + a a V, a + V a + o a 8 I follows fom he smla calculao as above see also Aï-Sahala e al, 5, 96 V, @a, cum [U]+o a cum [U]+o cum [U]+o a cum [U]+o 8 cum [U]+o a cum [U]+o O cum [U] a 8 + o Hece, combg he above equales, we @ a 6a + o @ a + cum [U] + o a 8 hs cocludes he oof

13 D Xu / Joual of Ecoomecs Aedx F Poof of heoem We wa o ove by vefyg he codos of heoem I follows fom Lemmas ad ha o O + O + O + O O O + O O o + O + O So fa, we have show ha P, fo ay,, o ove ufom covegece obably, we eed sochasc equcouy of As agued Newey ad McFadde 99, see also Rockafella 97, heoem 8, he o-wse covegece of cocave fucos mles ufom covegece Noe ha ca be egaded as he dffeece of wo cocave fucos fom, he a slghly modfed oof of heoem 8 Rockafella 97, usg agle equales, sll gves se o ufom covegece heefoe, as Adese ad Gll 98, we ave a su k k P Nex, we show he defably @a J J + whee J J j whee J,, ad he ohe comoes of J s J a a a + J a + a 8a 6 + O 5 O he ohe had, choose K, ad le7 m K,K ad M +, + By 8, we have ale M + + m + e e a +O a 6 +O 5 6 e e a +O a +O 7 Hee ad subsequely, ca be elaced by + wh ay < < heefoe, fo ay K ale ale K, m +o; fo < K o > K, s domaed by m, ad he egao s ove a eval whch shks a he ae K/ /, so X Z Z m + o a Smlaly, fo K ale a + @a whle fo < K o > heefoe, a ad a + o + o + ae domaed by Z Z + o 6 + o 7 By calculao, we ca also oba R a + a a a 8a 5 a 8a + o 8 Hece, f we solve, we ge a a + a a + o a a R 9 We assume all hese lkelhood esmaes ae bouded almos suely, sce he aamee sace s self bouded Also, 6a + a 6 6a 5 8a R a a 6 a + + 8a 5 5a 6a 5 6a 7 + o

14 8 D Xu / Joual of Ecoomecs O he ohe had, le : be he dagam max wh he h eleme of he dagoal : R, he a + Z a a 6a d, a + > + q R + I J J a a + whe k k + a a > ad goes P m k k :k k a a X > P m k k > :k > P o < heefoe, follows ha he defably codo holds, hece by heoem, ˆ P, ad â a P hs cocludes he oof by 9 ad a X X a a 8a + a a 8a + O Z 8a d + o a a 8a + O whee he las equaly s gve by 7 Se, a, ha s, Z a O a a + O Aga, follows fom dec a + a R + 6a 8a + 6a 5 a + a R + d 6a 5 8a 6a + o As, m k k :k k Obseve ha, k k a P, so wh obably aoachg, + R m k k :k k a a + a 8a 5 a 8a + o Z + 8a d + o + a a 8a + O Aedx G Poof of Lemma he agume s vey smla o he oof of heoem gve he aedx of Badoff-Nelse e al 8, whch volves he coce of sable covegece law he deals abou have bee dscussed Jacod ad Shyaev ad Jacod 7 Assume fo coveece ha j, fo, j < o, j > Also, deoe M M M X X X X j< X j X j j X X j X X : Z j j E X whee X X X ad j j We beg wh M Pck K, ad cosde K ale ale K Whe j O +, j exoeally, fo ay < < Rewe as: M X whee f X, f x, x,,x K KX h h K x x h I follows fom heoem 7 Jacod 7 ha, M coveges sably law, ad s asymoc vaace ca be calculaed usg fomula 7 ad 7 Jacod 7: X Ava lm R, f

15 D Xu / Joual of Ecoomecs whee 8 < KX R, f : h U h h 9 h E U h ; h j ad U s ae d sadad Gaussa adom vaables Fo ay K ale ale K, dec calculao yelds j Kalej< 6 7 a By he same agume used he oof of heoem, we have Z M L X 5 MN, 6 7 a d Sce we have show Lemma ha M follows fom Lemma Badoff-Nelse e al 8 ha j, Z M + M L X 5 MN 6 7 a As o M, we a fs oce ha X j @ ad ha fo K ale ale K, j ad ha P @ 8 5 a become ufomly asymocally eglgble heefoe, by he sadad ceal lm heoem, codoal o he flao X, we have X X M P L N, a 8 5 a P Sce X R d, ad by Lemma ad Pooso 5 Badoff-Nelse e al 8, we ca coclude ha Z M L X a N, 8 5 a ad M + M ad M joly covege X- sably law Also, as o he ose a, codoal o he flao X, M L a N, 5 6a 5 he same easog as above yelds M + M + M + M Z L X 5 MN, 6 7 a d Z a + a 8 5 a + 6 6a 5 As o, sce he ose a domaes, mles ha L MN, a + cum [U] 7 a 8 Fally, he covaace of he ad ode O, hus he jo ceal lm heoem follows fom Pooso 5 Badoff-Nelse e al 8 Aedx H Poof of heoem 5 s of he Now we deve he ceal lm heoem of he esmaos Because, ˆ,, ˆ whee s bewee ˆ ad, ad by Lemma Badoff- Nelse e al 8, s clea ha we oly eed o vefy he assumos of heoem I fac, because of, we wll cosde Y Y ad ad he devaves mulled by oe omalzaos I s saghfowad o check ha hese fucos ae ehe covex o cocave, whch guaaees sochasc equcouy see he oof of heoem Combg heoem, we have he ceal lm esul gve Box VI Noe fom 9 ad ha because of he cossecy esuls R above, a a o, O, heefoe, he oeal asymoc bases ae elmaed, whch cocludes he oof Aedx I Poof of heoem 6 Fom he lkelhood fuco, we ca oba he scoe fucos: Y Y Y Y whee a I O he ohe had, he followg equales hold: a I a I a + As a esul, a Pluggg he scoe fucos gves he eeseao Clealy, he wo quadac foms ae o co-lea he secod clam s obvous I fac, we ca we, ad oly deeds o Pluggg o he eeseao s amou o elacg wh decly Aedx J Poof of heoem 7 Accodg o 8, we have ha fo K ale, j ale K,,j j j

16 5 D Xu / Joual of Ecoomecs ˆ â B 8a R 5a L X MN B, 8a C a A a 6 a + a R,,,, + a + cum [U] CC AA o o Box VI wh he dffeece exoeally small heefoe, we ca fuhe deduce ha + j,j X l lj l j a a j a + j,j a,j,j j a a j O he ohe had, dec calculaos deduce ha 5 + O a + O a a + O heefoe, follows fom he eeseao ha j W,,j + j j + e Refeeces j Aï-Sahala, Y, Myklad, P, he effec of adom ad dscee samlg whe esmag couous me dffusos Ecoomeca 7, 8 59 Aï-Sahala, Y, Myklad, P, Zhag, L, 5 How ofe o samle a couous-me ocess he esece of make mcosucue ose Revew of Facal Sudes 8, 5 6 Aï-Sahala, Y, Myklad, P, Zhag, L, 6 Ula hgh fequecy volaly esmao wh deede mcosucue ose Joual of Ecoomecs, fohcomg do:6/jjecoom8 Aï-Sahala, Y, Yu, J, 9 Hgh fequecy make mcosucue ose esmaes ad lqudy measues Aals of Aled Sascs, 57 Amemya,, 97 Regesso aalyss whe he deede vaable s ucaed omal Ecoomeca, Adese, G, Bolleslev,, Meddah, N, 9 Realzed volaly foecasg ad make mcosucue ose Joual of Ecoomecs, fohcomg do:6/jjecoom Adese, PK, Gll, RD, 98 Cox s egesso model fo coug ocesses: a lage samle sudy Aals of Sascs 9, Bad, M, Russell, J, 8 Make mcosucue ose, egaed vaace esmaos, ad he accuacy of asymoc aoxmaos Dscusso Pae Uvesy of Chcago, Gaduae School of Busess Badoff-Nelse, OE, Hase, PR, Lude, A, Shehad, N, 8 Desgg ealzed keels o measue he ex-os vaao of equy ces he esece of ose Ecoomeca 76, 8 56 Badoff-Nelse, OE, Shehad, N, Ecoomec aalyss of ealzed volaly ad s use esmag sochasc volaly models Joual of he Royal Sascal Socey Sees B 6, 5 8 Baes, C, Whe, H, 985 A ufed heoy of cosse esmao fo aamec models Ecoomec heoy, 5 78 Domowz, I, Whe, H, 98 Mssecfed models wh deede obsevaos Joual of Ecoomecs, 5 58 Fa, J, Wag, Y, 7 Mul-scale jum ad volaly aalyss fo hghfequecy facal daa Joual of he Ameca Sascal Assocao, 9 6 Gaheal, J, Oome, R, 7 Zeo-ellgece ealzed vaace esmao Wokg Pae Gloe, A, Jacod, J, Dffusos wh measueme eos: I Local asymoc omaly Euoea Sees Aled ad Idusal Mahemacs 5, 5 Hase, PR, Lage, J, Lude, A, 8 Movg aveage-based esmaos of egaed vaace Ecoomec Revews 7, 79 Hase, PR, Lude, A, 6 Realzed vaace ad make mcosucue ose Joual of Busess ad Ecoomc Sascs, 7 8 Jacod, J, 99 Lm of adom measues assocaed wh he cemes of a Bowa semmmagale Dscussg Pae Jacod, J, 7 Sascs ad hgh fequecy daa Uublshed Lecue Noes, Sémae Euoée de Sasque 7: Sascs fo Sochasc Dffeeal Equaos Models Jacod, J, 8 Asymoc oees of ealzed owe vaaos ad elaed fucoals of semmagales Sochasc Pocesses ad he Alcaos 8, Jacod, J, L, Y, Myklad, P, Podolskj, M, Vee, M, 7 Mcosucue ose he couous case: he e-aveagg aoach Wokg Pae Jacod, J, Shyaev, AN, Lm heoems fo Sochasc Pocesses, d ed Sge-Velag, New Yok Kala, I, Lo, O, 8 Esmag quadac vaao cossely he esece of edogeous ad dual measueme eo Joual of Ecoomecs 7, 7 59 Kaazas, I, Sheve, S, 99 Bowa Moo ad Sochasc Calculus, d ed I: GM, vol Sge-Velag Pess Kullbacks, S, Leble, R, 95 O fomao ad suffcecy Aals of Mahemacal Sascs, L, Y, Myklad, P, 7 Ae volaly esmaos obus wh esec o modelg assumos? Beoull, 6 6 McCullagh, P, 987 eso Mehods Sascs Chama ad Hall, Lodo, UK Myklad, P, Zhag, L, 6 ANOVA fo dffusos ad Iô ocesses Aals of Sascs, 9 96 Myklad, PA, Zhag, L, 9 Ifeece fo couous semmagales obseved a hgh fequecy Ecoomeca 77, 5 Newey, W, McFadde, D, 99 Lage samle esmao ad hyohess esg I: Hadbook of Ecoomecs, vol IV Chae 6 Rockafella,, 97 Covex Aalyss Pceo Uvesy Pess Su, Y, 6 Bes quadac ubased esmaos of egaed vaace he esece of make mcosucue ose Wokg Pae Va De Vaa, AW, Asymoc Sascs Cambdge Uvesy Pess Whe, H, 98 Nolea egesso o coss-seco daa Ecoomeca 8, 7 76 Whe, H, 98 Maxmum lkelhood esmao of mssecfed models Ecoomeca 5, 6 Zhag, L, 6 Effce esmao of sochasc volaly usg osy obsevaos: a mul-scale aoach Beoull, 9 Zhag, L, Myklad, PA, Aï-Sahala, Y, 5 A ale of wo me scales: deemg egaed volaly wh osy hgh fequecy daa Joual of he Ameca Sascal Assocao, 9 Zhou, B, 996 Hgh-fequecy daa ad volaly foeg-exchage aes Joual of Busess ad Ecoomc Sascs, 5 5