TEXTO PARA DISCUSSÃO. No Realized volatility: a review. Michael McAller Marcelo C. Medeiros

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1 TEXTO PARA DISCUSSÃO No. 53 Realzed volaly: a revew Mchael McAller Marcelo C. Mederos DEPARTAMENTO DE ECONOMIA

2 REALIZED VOLATILITY: A REVIEW Mchael McAleer School of Ecoomcs ad Commerce Uversy of Weser Ausrala Marcelo C. Mederos Deparme of Ecoomcs Pofcal Caholc Uversy of Ro de Jaero Frs draf: November 005 Ths revso: November 006 Ackowledgemes: The auhors wsh o ackowledge he sghful commes ad suggesos of wo aoymous referees, semar parcpas a Chag Ma Uversy, Thalad, ad Keo Uversy, Japa, ad very helpful dscussos wh Yace Aï- Sahala, Maabu Asa, Federco Bad, Felx Cha, Km Chrsese, Dck va Djk, Marcelo Ferades, J Gao, Peer Hase, Offer Leberma, Asger Lude, Esse Maasoum, Col McKeze, Nour Meddah, Kmo Mormue, Per Myklad, Roel Oome, Nel Shephard, ad Sogsak Srboocha. The frs auhor wshes o ackowledge he facal suppor of a Ausrala Research Coucl Dscovery Gra, ad he secod auhor wshes o hak he CNPq/Brazl for paral facal suppor. Correspodg auhor: mcm@eco.puc-ro.br

3 Absrac Ths paper revews he excg ad rapdly expadg leraure o realzed volaly. Afer preseg a geeral uvarae framework for esmag realzed volales, a smple dscree me model s preseed order o movae he ma resuls. A couous me specfcao provdes he heorecal foudao for he ma resuls hs leraure. Cases wh ad whou mcrosrucure ose are cosdered, ad s show how mcrosrucure ose ca cause severe problems erms of cosse esmao of he daly realzed volaly. Idepede ad depede ose processes are examed. The mos mpora mehods for provdg cosse esmaors are preseed, ad a crcal exposo of dffere echques s gve. The fe sample properes are dscussed comparso wh her asympoc properes. A mulvarae model s preseed o dscuss esmao of he realzed covaraces. Varous ssues relag o modellg ad forecasg realzed volales are cosdered. The ma emprcal fdgs usg uvarae ad mulvarae mehods are summarzed. Keywords ad phrases: Facal ecoomercs, Realzed volaly, Face, Rsk, Couous me processes, Quadrac varao, Forecasg, Hgh frequecy daa, Tradg rules.

4 . INTRODUCTION Gve he rapd growh facal markes ad he coual developme of ew ad more complex facal srumes, here s a ever-growg eed for heorecal ad emprcal kowledge of he volaly facal me seres. I s wdely kow ha he daly reurs of facal asses, especally of socks, are dffcul, f o mpossble, o predc, alhough he volaly of he reurs seems o be relavely easer o forecas. Therefore, s hardly surprsg ha facal ecoomercs, parcular he modelg of facal volaly, has played such a ceral role moder prcg ad rsk maageme heores. There s, however, a here problem usg models where he volaly measure plays a ceral role. The codoal varace s lae, ad hece s o drecly observable. I ca be esmaed, amog oher approaches, by he (Geeralzed) Auoregressve Codoal Heeroskedascy, or (G)ARCH, famly of models proposed by Egle (98) ad Bollerslev (986), sochasc volaly (SV) models (see, for example, Taylor (986)), or expoeally weghed movg averages (EWMA), as advocaed by he Rskmercs mehodology (Morga, 996) (see McAleer (005) for a rece exposo of a wde rage of uvarae ad mulvarae, codoal ad sochasc, models of volaly, ad Asa, McAleer ad Yu (006) for a revew of he growg leraure o mulvarae sochasc volaly models). However, as observed by Bollerslev (987), Malmse ad Teräsvra (004), ad Carero, Peña, ad Ruz (004), amog ohers, mos of he lae volaly models fal o descrbe sasfacorly several sylzed facs ha are observed facal me seres. A emprcal fac ha sadard lae volaly models fal o descrbe a adequae maer s he low, bu slowly decreasg, auocorrelaos he squared reurs ha are assocaed wh hgh excess kuross of reurs. Correcly descrbg he dyamcs of he reurs s mpora order o oba accurae forecass of he fuure volaly whch, ur, s mpora rsk aalyss ad maageme. I hs sese, he assumpo of 3

5 Gaussa sadardzed reurs has bee refued may sudes, ad heavy-aled dsrbuos have sead bee used. The search for a adequae framework for he esmao ad predco of he codoal varace of facal asses reurs has led o he aalyss of hgh frequecy raday daa. Mero (980) oed ha he varace over a fxed erval ca be esmaed arbrarly, alhough accuraely, as he sum of squared realzaos, provded he daa are avalable a a suffcely hgh samplg frequecy. More recely, Aderse ad Bollerslev (998) showed ha ex pos daly foreg exchage volaly s bes measured by aggregag 88 squared fve-mue reurs. The fve-mue frequecy s a rade-off bewee accuracy, whch s heorecally opmzed usg he hghes possble frequecy, ad mcrosrucure ose ha ca arse hrough he bd-ask bouce, asychroous radg, freque radg, ad prce dscreeess, amog oher facors (see Madhava (000) ad Bas, Glose ad Spa (005) for very useful surveys o hs ssue). Igorg he remag measureme error, whch ca be problemac, he ex pos volaly esseally becomes observable. Aderse ad Bollerslev (998), Hase ad Lude (005a), ad Pao (005) used hs ew volaly measure o evaluae he ou-ofsample forecasg performace of GARCH models. As volaly becomes observable, ca be modelled drecly, raher ha beg reaed as a lae varable. Based o he heorecal resuls of Bardorff-Nelse ad Shephard (00), Aderse, Bollerslev, Debold ad Labys (003) ad Meddah (00), several rece sudes have documeed he properes of realzed volales cosruced from hgh frequecy daa. However, as wll be dscussed laer, mcrosrucure effecs roduce a severe bas o he daly volaly esmao. Aï-Sahala, Myklad ad Zhag (005), Bad ad Russell (005a, 006b), Zhag, Myklad ad Aï-Sahala (005) ad Hase ad Lude (006b), amog ohers, have dscussed varous soluos o he cossecy problem. Hase ad Lude (006a) showed ha subsug a mperfec measure of volaly ca dsor (or eve reverse) he emprcal rakg of volaly models. 4

6 The purpose of hs paper s o provde a crcal revew of he major heorecal ad emprcal developmes he excg ad rapdly expadg leraure o realzed volaly (RV). Alhough several excelle revew papers have bee wre recely, he revew gve hs paper dffers from he ohers a umber of respecs. Some of he surveys are raher broad, ad revew volaly geeral, such as Poo ad Grager (00), Aderse, Bollerslev, Chrsofferse ad Debold (006a, 006b) ad Aderse, Bollerslev ad Debold (006b). Furhermore, mos of hese papers have o ake accou of he mcrosrucure ose. Ths revew paper focuses oly o he RV leraure ad carefully addresses he crucal problem of measureme error. Bad ad Russell (006a) have also revewed he RV leraure, wh a emphass o mcrosrucure ose. However, Bad ad Russell (006a) placed more emphass o he ose compoe ad s ecoomc deermas. Bardorff-Nelse ad Shephard (007) exesvely revewed he leraure o oparamerc esmao of volaly. I her excelle revew, he auhors have focused o he heorecal foudaos of he esmaors ha have bee proposed recely. They largely (bu o exclusvely) focused o he frcoless case wh ad whou he effecs of jumps. The purpose of hs revew paper s o fll he gap he leraure for he pracoer, o dscuss he ssues of modellg ad forecasg daly realzed volales, ad o prese he sreghs ad lmaos of he varous approaches ha are avalable he leraure. The ma fdgs he leraure are revewed, dffere modellg sraeges are suggesed, ad model evaluao s also cosdered. Fally, he mos mpora praccal applcaos are preseed. As he leraure s exesve, we have decded o o clude jumps he revew for purposes of beg cocse. We recommed Bardorff-Nelse ad Shephard (004a, 007) for he case of varous models of jump processes. The paper s orgazed as follows. I Seco we prese he geeral uvarae framework for esmag realzed volales. A smple dscree me model s preseed Oher emprcal quaes clude he b-power varao ad mul-power varao ha are parcularly useful for deecg jumps (see Bardorff-Nelse ad Shephard (004a, 005a, 005b), Bardorff-Nelse, Graverse, Jacod, ad Shephard (006), Aderse, Bollerslev ad Debold (003), Bollerslev, Kreschmer, Pgorsch ad Tauche (005), Huag ad Tauche (005), ad Tauche ad Zhou (004)), ad he raday rage-based esmaors (see Chrsese ad Podolskj (006a, 006b) ad va Djk ad Mares (006)). 5

7 Seco. o movae ad defe he basc oao, a couous me model whch gves he heorecal foudaos for he ma resuls hs leraure are preseed Seco., ad he effecs of mcrosrucure ose ad realzed volaly esmao are dscussed deal Seco.3, where boh depede ad depede ose processes are cosdered. I Seco 3 he ma soluos for acklg he problem of mcrosrucure ose are preseed. Seco 4 preses a mulvarae model, ad brefly dscusses he esmao of he realzed covaraces. The ssues of modellg ad forecasg realzed volales are cosdered Seco 5. Fally, some cocludg remarks are gve Seco 6.. THE GENERAL UNIVARIATE FRAMEWORK Ths seco preses a smple dscree me model o movae he RV esmaor. Seco. follows he srucure of Oome (00) closely. A couous me model, whch forms he bass of much of he heorecal resuls, s preseed Seco.. The dscusso sars wh he case whou ose (ha s, wh o measureme errors), he proceeds o corporae mcrosrucure ose... The Early Days A Smple Dscree Tme Model Cosder a smple dscree me model whch he daly reurs of a gve asse are ypcally characerzed as r = / h η, } T = where { η s a sequece of depedely ad ormally dsrbued radom varables wh zero mea ad u varace, η ~ NID(0,). Suppose ha, a gve radg day, he logarhmc prces are observed ck-by-ck. Cosder a grd = { τ,, } Λ coag all observao pos, ad se p,, =, K,, 0 K τ 6

8 o be he h prce observao durg day, where s he oal umber of observaos a day. Furhermore, suppose ha r h /,, η, =, where ~ NID( 0 ) η,,,, = p, p, r s he h ra-perod reur of day such ha = r r = 0, ad h = = h,. a= b Defe he formao se { }, = I, I p as he σ-algebra geeraed by all he a, b a=, b= 0 formao o he h ck day. Therefore, I, 0 s he formao se avalable pror o he sar of day. I follows ha ( r E I, 0 ) = h ad V( r ) = h I.,0 The realzed varace s defed as he sum of all avalable raday hgh frequecy squared reurs gve by = ( all) RV r = 0,. () The squared daly reur ca be wre as r = r, = r +, r, = 0 = 0 = 0 j= + r,, j 7

9 8 such ha ( ) ( ) I + I = I + I = I = + = = + = = 0,0,,,0 ) ( 0,0,, 0,0,,0 E E E E E j j all j j r r RV r r r r. If he raday reurs are ucorrelaed, he ( ) ( ) h RV r = I = I 0, (all),0 E E. As a resul, wo ubased esmaors for he average day- reur varace exs, amely he squared day- reur ad he realzed varace as (). However, ca be show ha ( ) ( ),0 0, 0,,0 ) ( V V all r h h RV I = < = I = =, as = + = = = + = I,, 0,,0,, 3 E j j h h h h η. I shor, he average daly reurs varace ca be esmaed more accuraely by summg he squared raday reurs raher ha calculag he squared daly reur. Moreover, whe reurs are observed a ay arbrary frequecy, s possble o esmae he average daly varace free of measureme error as ( ) 0 V lm 0, ) ( = I all RV.

10 The oly requreme o he dyamcs of he raday reur varace for he above o hold s ha = + c,, h where 0 c <. Ths resul movaes a umber of emprcal papers, such as Aderse ad Bollerslev (997, 998), Aderse, Bollerslev ad Lage (999), ad Mares (00, 00), amog ohers. The heorecal foudaos of he resuls descrbed hs seco are derved from a couous me framework ha s based o he heory of quadrac varaos. Seco. descrbes he couous me approach whou mcrosrucure ose, whereas he effecs of ose are cosdered Seco.3... A Couous Tme Model wh No Mcrosrucure Nose.. Basc Seup Suppose ha, alog day, he logarhmc prces of a gve asse follow a couous me dffuso process, as follows: dp( + τ ) = μ( + τ ) dτ + σ ( + τ ) dw ( + τ ), 0 τ, =,,K, () where p ( + τ) s he logarhmc prce a me + τ, μ( + τ) s he drf compoe, σ( + τ) s he saaeous volaly (or sadard devao), ad W ( + τ) s a sadard Browa moo. I addo, suppose also ha σ( + τ) s orhogoal o W ( + τ), such ha here s o leverage effec. 9

11 Aderse, Bollerslev, Debold ad Labys (003), hereafer ABDL (003), ad Bardorff- Nelse ad Shephard (00) showed ha daly reurs, defed as = p( ) p( ), are I I μ( + τ ), σ( + τ ) Gaussa codoally o { } τ= 0 τ= r, he σ-algebra (formao se) geeraed by he sample pahs of μ( + τ ) ad σ( + τ ), 0 τ, such ha r I ~ N + + μ( τ ) dτ, 0 σ ( τ ) dτ,. 0 The erm IV σ ( + τ ) dτ s kow as he egraed varace, whch s a measure = 0 of he day- ex pos volaly. The egraed varace s ypcally he objec of eres as a measure of he rue daly volaly... Dffere Samplg Schemes I pracce, prces are observed a dscree ad rregularly spaced ervals. I hs sese, here are may ways whch oe ca sample he daa. As prevous secos, suppose ha a gve day, we paro he erval [0,] subervals ad defe he grd of observao mes as = { τ,, } Λ, where = τ < τ < L <. The legh of he 0 K τ 0 0 τ = h suberval s gve by δ = τ τ. I s assumed ha he legh of each suberval, shrks o zero as he umber of raday observaos creases. The egraed varace over each of he subervals s defed as IV τ =, σ ( + τ ) dτ. τ - There are several samplg schemes ha ca be used, as follows: () The mos wdely used samplg scheme s caledar me samplg (CTS), where he ervals are equdsa caledar me, ha s, δ, = for all. For 0

12 example, he prces may be sampled every 5 or 5 mues. As he raday daa are rregularly spaced, mos cases caledar me sampled daa mus be cosruced arfcally (see Wasserfalle ad Zmmerma (985), Aderse ad Bollerslev (997), ad Dacoroga, Gecay, Müller, Olse ad Pce (00)). Hase ad Lude (006b) showed ha he prevous ck mehod s a sesble way o sample prces caledar me. For example, durg a fve-mue erval, we may observe several prces, whch case he prevous ck mehod akes he frs observao as he sampled prce. () Aoher samplg alerave s rasaco me samplg (TrTS), where prces are recorded every mh rasaco. ()The hrd samplg scheme s kow as busess me samplg (BTS), where he samplg mes are chose such ha IV IV =,. (v) The las samplg alerave s called ck me samplg (TkTS), where prces are recorded a every prce chage. A mpora dfferece amog hese dsc samplg schemes s ha he observao mes BTS are lae, whereas CTS, TrTS, ad TkTs hey are observed. The effecs of dffere samplg schemes o he esmao of he egraed varace wll be dscussed Seco The Dsrbuo of Realzed Volaly ABDL (003) showed, usg a semal resul semmargale process heory, ha he realzed varace usg all daa avalable, as defed equao (), s a cosse esmaor of he egraed varace whe here s o mcrosrucure ose, such ha RV (all) p IV. From he resuls Jacod ad Proer (998), Bardorff-Nelse ad Shephard (00) derved he asympoc dsrbuo of he realzed varace as

13 (all d ( RV IV ) N(0,) ) /, (3) IQ where he egraed quarcy, IQ, s defed as 4 IQ σ ( + τ ) dτ. = 0 (4) Bad ad Russell (005a) gave a alerave smple proof of he above resul. Furhermore, uder he assumpo of o mcrosrucure ose, Bardorff-Nelse ad Shephard (00) showed ha he egraed quarcy s cossely esmaed by he realzed quarcy, whch s defed as RQ (all) = 3 = 0 4, r, (5) ad (all d ( RV IV ) N(0,) ) /, (all) RQ 3 Bardorff-Nelse ad Shephard (005b), Meddah (00), Goçalves ad Meddah (005) ad Nelse ad Frederkse (005) suded he fe sample behavor of he lm heory gve (3). The ma cocluso s ha (3) s poorly szed, bu (all) ( RV ) (all d [ log( RV ) log( IV )] N(0,) ) /, (all) RQ 3

14 performs que well. Goçalves ad Meddah (005) aalyzed how he boosrap may mprove he lmg heory dscussed above. The auhors cocluded ha s possble o desg boosraps whch provde sgfca mprovemes over he lmg heory (3). They also showed ha he usual Edgeworh expasos, whch jusfy he order mproveme assocaed wh he boosrap, are o relable gudes o he fe sample behavor of he sascs. However, cases where he compuaoal burde mposed by he boosrap s hgh, Goçalves ad Meddah (006) showed ha usg Edgeworh expasos s superor o usg he lmg heory (3)..3. The Effecs of Mcrosrucure Nose I hs seco we dscuss he effecs of he presece of mcrosrucure ose he esmao of he egraed varace. Marke mcrosrucure ose has may sources, cludg he dscreeess of he prce (see Harrs (990, 99)), ad properes of he radg mechasm, as Black (976) ad Amhud ad Medelso (987). For addoal refereces o hs leraure, see O Hara (995), Madhava (000), Hasbrouck (004), ad Bas, Glose ad Spa (005). As Seco., cosder he grd of observao mes, Λ = { τ,, } 0 K τ. Usg smlar oao as Zhag, Myklad ad Aï-Sahala (005), hereafer ZMA (005), se p p( + τ ). Suppose also ha he logarhmc prces are observed wh ose, ha s:, * p, p, + ε, =, (6) where * p, s he lae effce (or rue) prce process ad ε, s he mcrosrucure ose. I follows ha 3

15 * * r, r, + ε, ε, = r, + ν, =, (7) where * * * r, = p, p, s he effce reur. I s clear ha r, s a auocorrelaed process, so ha (all) RV wll be a based esmaor of he lae rue daly volaly, as dscussed Seco.. Furhermore, as RV (all) = * * ( r, ) + r, ν, + = = ν, =, s sraghforward o show ha, codoally o he effce reurs: E (all) * *(all) ( RV r ) RV + E( ε ) =,, such ha (all) RV s also a based esmaor of he egraed varace. As Bad ad Russell (005a), cosder he followg assumpo regardg he ose srucure: Assumpo (ose srucure): (a) The mcrosrucure ose, ε,, has zero mea ad s a covarace saoary sochasc process. (b) The varace of ν = ε, ε, s O()., Uder Assumpo, Bad ad Russell (005a) showed ha RV (all) a. s. as. 4

16 Furhermore, cosder he followg assumpo: Assumpo (IID ose srucure): (a) The mcrosrucure ose, decally dsrbued radom varable. (b) The ose s depede of he prce process. (c) The varace of ν = ε, ε, s O()., ε,, has zero mea ad s a depede ad Uder Assumpo, was show ZMA (005) ha / (all) d 4 [ RV IV E( ε )] [ E( ε )] / N(0,),,. I praccal applcaos, eve samplg a he hghes avalable frequecy, he umber of raday observaos s fe ad he prce records are dscree. Ths roduces a bas due o dscrezao, such ha RV d (all) 4 4 ( ) + 4 E( ε ) + σ N(0,) IV + E ε,, d, 0 bas due o ose due o ose 443 due o dscrezao oal varace where d meas ha, whe mulpled by a suable facor, he covergece s dsrbuo. Recely, Zhag (006a) ad Aï-Sahala, Myklad ad Zhag (006), hereafer AMZ (006), cosdered he case where he ose s o IID, such ha Assumpo s modfed as follows: Assumpo 3 (depede ose srucure): 5

17 (a) The mcrosrucure ose, ε,, has a zero mea, saoary ad srog mxg sochasc process, wh he mxg coeffces decayg expoeally. I addo, E ( ε ) 4+κ [ ] <, for some κ > 0., (b) The ose s depede of he prce process. (c) The varace of ν, = ε, ε, s O(). Uder Assumpo 3, Zhag (006a) ad AMZ (006) showed ha RV (all) d 4 IV + E( ε, ) + 4 N(0,) 3 Ω ( parse) σ d, s 0 bas due o ose due o ose due o dscrezao oal varace where [( ) ] + ( ε Cov ε ε ) ( ε ε ) Ω = V,,0,,0, = [ ] ε., +, The mos mpora fac abou he las resul s ha, for large, he realzed varace () (all) may have o coeco o he rue reurs. O he corary, RV dverges o fy learly. I addo, Bad ad Russell (005a) ad ZMA (005) showed ha, scaled by ( ) cossely, such ha:, he realzed varace esmaes he varace of he mcrosrucure ose RV (all) ( ε ) p E. (8), As advocaed Aderse, Bollerslev, Debold ad Ebes (00), hereafer ABDE (00), ad ABDL (000a, 00, 003), oe possble soluo o he mcrosrucure bas 6

18 s o sample he reurs a arbrarly seleced lower frequeces, such as every 5 or 5 mues, sead of a every ck. Ths procedure s called sparse samplg. However, ZMA (005) showed ha hs s o a adequae soluo o he problem. Frs, defe a ew grd (sparse) Λ, wh (sparse) Λ s a subgrd of Λ. Se (sparse) sparsely equdsa sampled observao mes. Clearly, ( sparse) (sparse) = r, = RV. (9) Based o he resuls of Rooze (980), Jacob ad Proer (998), Bardorff-Nelse ad Shephard (00), ad Myklad ad Zhag (006), ZMA (005), Zhag (006a) ad ( sparse) AMZ (006) showed ha he bas due o ose s gve by E( ) Assumpos or 3: ε ad ha, uder, RV (sparse) d ( sparse) ( sparse) 4 4 IV + E( ε, ) + 4 E( ε, ) + N(0,) ( parse) σ d. s 0 bas due o ose due o ose due o dscrezao oal varace Alhough he bas s reduced whe ( <, he varace s creased due o s parse) dscrezao, leadg o he well kow bas-varace rade-off. Eve hough choosg he samplg frequecy o he bass of he fe sample mea-squared-error s opmal he case of realzed varace, alerave esmaors (dscussed below) have bee proposed ha have he poeal, whe appropraely mplemeed, o ouperform he classcal realzed varace esmaor. 7

19 3. MICROSTRUCTURE NOISE AND REALIZED VOLATILITY ESTIMATION 3.. Seleco of Frequecy ad Sparse Samplg Uder Assumpo, Bad ad Russell (005a, 006b) ad ZMA (005) proposed a mehod of selecg he opmal samplg frequecy based o he mmzao of he mea squared error (MSE), as follows: MSE ( sparse) ( sparse) ( sparse) 4 ( ) = E( ε, ) + 4 E( ε, ) (sparse) + [ 8RV E( ε ) V( ε )] + ( sparse) IQ.,, ( sparse) Thus, he opmal samplg frequecy may be approxmaed by 3 * IQ 4[ E(, )]. ε (0) Bad ad Russell (005a, 006b) cosdered equdsa samplg ervals, whereas ZMA (005) provded a more geeral formula for rregularly spaced daa. However, Bad ad Russell (005a) also cosdered opmal samplg wh depede ose, opmal samplg wh bas-correced realzed varace esmaes, ad opmal samplg wh pre-flered daa. As dscussed prevously, E( ) ε may be cossely esmaed by, equao (8)). Cosse esmao owhsadg, a mpora po ha mus be emphaszed s ha he egraed quarcy s o kow, ad hece mus be esmaed. However, he realzed quarcy, as gve equao (5), s o cosse he presece of mcrosrucure ose. Bad ad Russell (005a, 006b) adoped he soluo of compug (4) usg a sparse se of observaos, amely oe ha s sampled every 5 mues. The auhors showed hrough smulao ha such sparse samplg dd o seem RV (all) (see 8

20 o have a harmful effec o he seleco of he opmal frequecy. ZMA (005) developed a alerave soluo for esmag he egraed quarcy. Neverheless, he developme of a robus egraed quarcy esmaor appears o be a mpora opc for fuure research. 3.. Bas Correco ad Cosse Esmao 3... Subsamplg ZMA (005) proposed a subsamplg mehod order o esmae he egraed varace cossely he presece of mcrosrucure ose 3. The ma dea s o explore he fac ha, for example, e-mue reurs sarg a 9:30 could be measured usg he ervals 9:30-9:40, 9:40-9:50,, 9:3-9:4,9:4-9:5, ad so o. Formally, suppose ha he full grd, Λ = { τ,, } 0 K τ ( k ) Λ, k =, K, K, such ha, s paroed o K o-overlappg subgrds, U K Λ ( k ) k = Λ =, where Λ ( k ) Λ ( j) = φ whe k j. Se (k ) as he umber of observaos each subgrd, ad defe he RV for grd k as ( k ) ( k ) RV = r. =, () The proposal of ZMA (005) s o use he followg esmaor for he daly RV: RV K (ZMA) ( k ) ( all) = RV RV K k =, () 3 See also Aï-Sahala, Myklad ad Zhag (005) for a cosse maxmum lkelhood esmao of he cosa varace of a dffuso process wh mcrosrucure ose. 9

21 where s he umber of observaos he full grd, ad K = K k = K = K ( k ) +. The esmaor () s called he Two Tme Scales Esmaor (TTSE) of he egraed varace. ZMA (005) showed ha, uder Assumpo, (ZMA) d - 4 [ RV IV ] 8c E[ ( )] + c IQ N(0,) / 6 ε, 443 4, 3 3 due o ose due o dscrezao oal varace / where, he case of equdsa observaos, c = IQ E[ ( ε, )] / 3. I AMZ (006), a small sample refeme o he esmaor () s proposed. The fal esmaor becomes RV (ZMA,adj) (3) (ZMA) = RV. Boh of he esmaors () ad (3) are derved uder Assumpo (IID ose). I order o ake o accou possbly depede ose, Zhag (006a) ad AMZ (006) proposed a alerave esmaor ha s also based o he wo me scales dea. All he resuls are derved uder Assumpo 3 (o-iid ose). Frs, he auhors defed he average lag J realzed volaly, (AL) RV,J, whch s gve by 0

22 RV (AL), J = J J = 0 ( r r ), + J,. (4) The he auhors proposed a geeralzao of he TSSE derved ZMA (005), whch has he form RV (AMZ) ( K ) (AL) (AL) = RV, K RV, J, ( J ) (5) where J K, K = o( ), ( K ) ( J ) J + = ad =. Noe ha (4) K J J ad K as. A small sample ( K ) + becomes he TTSE ZMA (005) whe = correco s gve by ( ) RV (AMZ,adj) ( K ) (6) = (AMZ) RV ( J ). Zhag (006a) ad AMZ (006) showed ha RV (AMZ,adj) d 4 4 IV + N(0,) / 6 ξ + c 3 σ d , c due o ose due o dscrezao oal varace where c s a cosa ad ( ε ) + 3 Cov( ε, ε ) = ξ = 6 V.,,0, 3.. Kerel-Based Esmaors

23 Cossely esmag he quadrac varao uder he presece of mcrosrucure ose s, a sese, smlar o he well kow auocorrelao correcos ha are frequely used he me seres leraure o esmae he log ru varaces ad covaraces of saoary sochasc processes (see, for example, Newey ad Wes (987) ad Adrews (99)). Cosequely, s aural o adap smlar echques for he prese case. For example, Hase ad Lude (004, 006) cosdered he followg smple kerel-based esmaor: RV γ, H ( HL) (all) = RV + ˆh h= h (7) where h j= ˆ γ h = r, jr j+ h. h (8) Zhou (996) was he frs o cosder he use of kerel mehods o deal wh he problem of mcrosrucure ose hgh frequecy daa. For he case of depede ose, Zhou proposed (7) wh H =. Hase ad Lude (006b) examed he properes of Zhou s esmaor ad showed ha, alhough ubased uder Assumpo, he esmaor s o cosse. However, Hase ad Lude (006b) advocaed ha, whle cosse, Zhou s kerel mehod s able o ucover several properes of he mcrosrucure ose, ad cocluded ha he ose: () s correlaed wh he effce prce; () s me depede; () s que small he DJIA socks; ad (v) has properes ha have chaged subsaally over me.

24 Ther resuls are robus o boh CTS ad TrTS. Moreover, selecg hgher values for H does o solve he cossecy problem. However, he esmaor (7) s ubased by a upwards scalg of he emprcal auocovaraces. The h h auocovarace s scaled by h o compesae for he mssg auocovarace erms. The upward scalg has he drawback ha creases he varace of he esmaor. For hs reaso, Hase ad Lude (005b) cosder he Barle kerel ad defe he esmaor: RV H ( HL, Barle) (all) = RV ˆ h= h + γ h, H + (9) where H s deermed as H = esmaor (9) s also cosse / 9, ad γˆ h s defed as (8). However, he Recely, Bardorff-Nelse, Hase, Lude ad Shephard (006a), hereafer BHLS (006a), proposed he fla-op kerel-based esmaor 4 : RV H ( BHLS ) (all) h = RV + k ˆ h h h= H ( ˆ + γ ) γ, (0) where k(x) for x [0,] s a o-sochasc wegh fuco such ha k(0) = ad k() = 0. The auhors made several corbuos o he leraure by: () provg ha he saeme ha all kerel based RV esmaors were cosse s wrog ad proposed several cosse kerel-based esmaors; () desgg a kerel ha has a smaller varace ha he mulscale esmaor; 4 See also Su (006) for a smlar class of ubased ad cosse esmaors. 3

25 () proposg a esmaor for daa wh edogeously spaced observaos, such as ha daabases o rasacos; ad (v) cosderg he case where he mcrosrucure ose s edogeous. BHLS (006a) showed ha, f / 3 H = c, he he resulg esmaor s asympocally mxed Gaussa, covergg a rae / 6. The cosa, c, ca be opmally chose as a fuco of he kerel k(x). For example, he value of c ha mmzes he varace of he esmaor s gve by c = / 3 [ k' (0) + k' () ] [ E( ε, )] 3 k( x) dx IQ 0 / 3. BHLS (006a) also compared hree dffere kerels: () Barle where k(x) = - x; () d order where k(x) = - x - x ; ad ()Epaechkov where k(x) = x. Ther fdgs are summarzed as follows: he Barle kerel has he same asympoc dsrbuo as he TTSE of ZMA (005) ad s more effce ha he Epaechkov alerave, bu s less effce ha he d order kerel. Moreover, f k (0) = 0 ad k () = 0, he seg he esmaor s mxed ormal wh covergece rae equal o / H = c, he asympoc dsrbuo of / 4. BHLS (006a) dscussed he choce of he cosa c a smplfed framework where he varace of he effce prce s held cosa. I her paper, he auhors compared egh dffere kerels sasfyg k (0) = 0 ad k () = 0. The cubc kerel, where k(x) = 3x + x 3, has he same asympoc dsrbuo as he mulscale esmaor of AMZ (006) ad 4

26 [ cosπ ( x) ] Zhag (006a). The Tukey-Hag kerel, where k( x) = he bes opo erms of effcecy., seems o be BHLS (006a) also showed ha he fdgs above are robus o edogeous 5 ad/or depede ose, ad edogeously spaced observaos, as ck daa daabases. They also provded Moe Carlo evdece favour of her esmaors fe samples Flers I he early days of modellg RV, aoher commo alerave o aeuae he effecs of he mcrosrucure ose was o pre-fler he raday reurs. For example, Bolle ad Ider (00), a auoregressve (AR) fler was used, whle a movg average (MA) fler was cosdered Ebes (999), Maheu ad McCurdy (00), ad ABDE (00). More recely, Hase, Large ad Lude (006), hereafer HLL (006), showed ha he MA() srucure cosdered Ebes (999) ad ABDE (00) s well specfed whe he marke mcrosrucure ose s IID. Moreover, whe correcg he esmaor by a scalg facor, becomes a cosse esmaor of he egraed varace (see HLL (006) for furher deals) Alerave Esmaor Recely, Large (006) proposed a eresg esmaor of quadrac varao whch corols for mcrosrucure effecs whe he bes quoes chage by jumpg he mmum prce ck. The esmaor compares he umber of aleraos, where quoes jump back o her prevous prce, wh he umber of oher jumps. If he aleraos are ucorrelaed, he esmaor s cosse a lm heory where jumps are very freque ad small. 5 The auhors cosdered a smple form of depedece bewee he ose ad he effce prce process. 5

27 3.3. The Effecs of he Samplg Scheme As dscussed Seco., here are several ways of samplg raday reurs, ad he choce of samplg scheme ca have a srog fluece o he sascal properes of he realzed varace. Mos of he work dscussed prevously dd o drecly address he ssue of choosg he samplg scheme. The frs o corbue ha dreco was Oome (005), who examed he followg samplg aleraves: () caledar me samplg, () rasaco me samplg, () ck me samplg, ad (v) busess me samplg. Compared wh he sadard leraure, Oome (006) proposed a pure jump process for he hgh frequecy prces, whch allows for he aalyss of he followg samplg schemes: caledar me, busess me, ad rasaco me samplg. The prce process s formed by a effce margale compoe, whch s descrbed as a compoud Posso process plus he marke mcrosrucure ose ha s allowed o have a MA(q) srucure. Thus, he asse prce s modelled as he accumulao of a fe umber of jumps, each of whch represes a rasaco reur, wh he Posso process coug he umber of rasacos. The opmal samplg frequecy s derved o mmze he mea squared error, whch s flueced by he umber of rades ad he ose level. I was show ha, as he case of he dffuso-based models, he realzed varace s a based esmaor of he jump aalogue of he egraed varace whe mcrosrucure ose s prese. However, as dsc from prevous resuls, he bas does o dverge o fy as he sample frequecy creases. Cocerg he samplg schemes, he ma cocluso s ha rasaco me samplg s geerally superor o he commo pracce of caledar me samplg, as he former leads o a lower mea squared error of he realzed varace. Ths effec s proouced, especally whe he radg esy paer s volale. Oome (005) exeded he model Oome (006) order o sudy he effecs of he frs-order bas correco o dffere samplg schemes. Hs correco s le wh hose proposed by Zhou (996) ad Hase ad Lude (006b). However, he prese 6

28 resuls were derved uder a pure jump process wh IID ose sead of a dffusobased model. Oome (005) showed ha he bas correco sgfcaly reduces he bas caused by mcrosrucure effecs, ad s more effecve rasaco me ha caledar me. Moreover, for a equal umber of sampled reurs, bas-correced esmaors aas a lower mea squared error whe he reurs are sampled regularly spaced o a rasaco me scale raher ha o a caledar me scale. Grff ad Oome (006a) roduced a ew model for rasaco paers order o dsgush he effecs of ck me ad rasaco me samplg. The ma fdgs of he paper are: () ck me samplg s equvale o rasaco me samplg for hgh levels of mcrosrucure ose, ad s superor for low levels of mcrosrucure ose; ad () whe he frs-order bas correced esmaor of Zhou (996) ad Hase ad Lude (006b) s cosdered, rasaco me samplg s always preferred Comparso of Techques As show he prevous seco, here are may dffere possbles for dealg wh he problem of mcrosrucure ose he esmao of he egraed varace. Table compares he dffere mehods o esmae he egraed varace accordg o her asympoc properes. The am of he able s o o rak dffere mehods bu o summarze he ma large-sample properes of each of hem. 7

29 Table : Asympoc Properes of Mehods for Esmag he Daly Iegraed Varace Mehod Ubased 6 Cosse Prce Model Nose (Tme Depedece) Nose ad Effce Prces Realzed Varace (all avalable daa) Realzed Varace 7 (sparse samplg) Realzed Varace (opmal frequecy seleco) Bad ad Russell (005a, 006b, 006d) Realzed Varace (opmal frequecy seleco) Oome (006) Kerels Hase ad Lude (006b) Kerels Oome (005) TTSE ZMA (005) TTSE AMZ (006) Kerels BHLS (006a, 006b) Kerels: opmal badwdh seleco Bad ad Russell (006d) MA fler HLL (006) Alerao esmaor Large (006) No No Dffuso Depede/ IID Depede/ Idepede No No Dffuso - - No No Dffuso Depede/ Depede/ IID Idepede No No Pure Jump Depede Idepede Yes No Dffuso Depede/ Depede/ IID Idepede Yes No Pure Jump IID Idepede Yes Yes Dffuso IID Idepede Yes Yes Dffuso Depede Idepede Yes Yes Dffuso Depede/ Depede/ IID Idepede Yes Yes Dffuso Depede/ Depede/ IID Idepede Yes Yes Dffuso IID Idepede Yes Yes Pure Jump Depede - 6 I Table we cosder large sample bas. Some of he esmaors, such as TTSE, are based small samples bu o asympocally. 7 Sparse samplg he case of realzed varace does o ecessarly requre assumpos o he mcrosrucure ose. 8

30 I s mpora o meo ha, alhough here are may ubased esmaors, oly four are cosse. The frs cosse esmaor s he TTSE of ZMA (005) ad AMZ (005). The order of covergece of he TTSE ZMA (005) s s / 4 / 6, whle ha AMZ (005). Hase ad Lude (006b) oed ha a subsamplg verso of he kerel esmaor of Zhou (996) s also a cosse esmaor ad a formal proof was gve BHLS (006b). BHLS (006a) derved he realzed kerel cosse esmaor ha geeralzes he prevous resuls Hase ad Lude (006b), ad whch s also of order / 4. I a compao paper, BHLS (006b) showed he equvalece bewee her esmaors ad hose ZMA (005) ad AMZ (005). The hrd cosse esmaor s he modfed MA fler of HLL (006), whch s also of order / 4. However, hese esmaors dffer regardg he hypohess abou he mcrosrucure ose ad samplg schemes. The fourh oe s he alerao esmaor of Large (006). The precedg dscusso owhsadg, s mpora o oe ha, whle o beg cosse, he kerel esmaors dscussed Hase ad Lude (006b) are mpora ools for ucoverg, f oly parally, several properes of he mcrosrucure ose. From he praccal perspecve, a mpora ssue regardg he properes of a esmaor relae o fe sample or asympoc properes. Alhough hs s o sraghforward o deerme, we wsh o provde he pracoer wh some gudeles for choosg he mos covee esmaor, whch may be a esmaor ha s lsed Table or may be a combao of aleraves. I order o oba cosse esmaors, BHLS (006), ZMA (005) ad AMZ (005) requred ha he umber of auocovaraces (or subsamples) H ad he umber of H observaos,, o dverge o fy as he rao φ = 0. However, for a gve φ, he magude of he fe sample MSE of he esmaors ca be subsaally dffere from he asympoc approxmaos. Moreover, pracce, researchers are always forced o selec a value for φ (see Remark 3 Bad ad Russel (006c) for a dscusso o he 9

31 mporace of fe sample properes of egraed varace esmaors). Bad ad Russell (006d) uderook a dealed sudy of he fe sample performace of several kerel-based ad sub-samplg esmaors uder Assumpo, ad showed how o selec opmally he value of φ based o a fe sample MSE crero. The auhors foud ha for he realsc sample szes ecouered praccal applcaos, he asympoc resuls for some of he esmaors dscussed above, geeral, do o provde suffce gudace for praccal mplemeao, as hey provde usasfacory represeaos of he fe sample properes of he esmaors. I addo, he auhors showed how o opmze he fe sample properes of hese esmaors, provdg sgfca sascal ad ecoomc gas whe compared wh he subopmal esmaors. Cocerg ZMA s (005) esmaor ad he based kerel esmaors of Hase ad Lude (005b), her ma coclusos are as follows: () The fe sample MSE properes of he cosse esmaor ( RV (005), ad of he cosse Barle kerel esmaor ( RV (ZMA) ( HL, Barle) ) of ZMA ) dscussed Hase ad Lude (005b), are smlar, ad a sgfca compoe of her mea-squared error s duced by he fe sample bas. () Asympoc mehods o selec he badwdh ca be subopmal her case, especally for based kerel esmaors. As her fe sample bas vashes asympocally, asympoc mehods do o ake he fe sample bas o accou ad have a edecy o selec a excessvely small umber of badwdhs. A small H ca lead o a large bas compoe a fe sample. () Ths bas compoe ca be reduced by choosg H order o mmze he esmaor s fe sample MSE. I he case of ZMA s (005) esmaor ad he Barle kerel esmaor of Hase ad Lude (005b), he auhors proposed a smple (MSE-based) rule-of-humb o selec he rao, φ, whch s gve by: 30

32 / 3 (all) RV * * * 3 φ ( Barle, HL) = φzma = φ. IQ () (v) Whle he opmal fe sample MSE values of ZMA s (005) esmaor ad Hase ad Lude s (005b) Barle kerel esmaor are geerally smaller ha he opmal fe sample MSE value of he classcal realzed varace esmaor, he gas ha hese useful esmaors ca provde over he classcal realzed varace esmaor mgh be los or dramacally reduced by subopmally choosg he value of φ. Bad ad Russell (006d) also evaluae he fe sample behavour of he cosse fla-op kerel based esmaors proposed by BHLS (006). Alhough Bad ad Russell (006d) do o provde a expresso for he opmal rao, φ, hs case, hey coduc a dealed smulao exercse o exame he fe sample properes of hree fla-op kerels, amely he Barle, cubc, ad modfed Tukey-Hag kerels. The opmal badwdh s chose by mmzg he fe sample varace of he ubased fla-op symmerc kerels, hereby leadg o fe sample MSE opmzao. Ther ma fdgs are as follows: () Despe havg he same dsrbuo as he subsamplg esmaor of ZMA (005), he fla-op Barle kerel esmaor appears o be preferable o he former fe samples. Furhermore, he cubc fla-op kerel, whch s equvale o he mul-scale esmaor, does o seem o mprove o he fe sample performace of he fla-op Barle kerel. The fla-op Tukey-Hag kerel performs margally beer ha do he oher wo kerels. () The use of asympoc crera o selec he opmal value of H (amely, he umber of auocovarace erms) ca be more or less sasfacory depedg o he choce of kerel. I was foud ha he asympoc crera are accurae whe he cubc kerel s chose. 3

33 () Due o he lack of a subsaal bas erm ad he flaess of he varace erm as a fuco of φ, he subopmal badwdh choces do o lead o exremely large losses. (v) Alhough he cubc fla-op kerel mples a faser rae of covergece ha does he Barle fla-op kerel, he fe sample performace of he wo esmaors s almos decal. (v) The asympoc approxmaos o he fe sample dsperso of he symmerc esmaors ca be mprecse. A careful assessme of he accuracy of hese esmaors requres a closer examao of her fe sample properes. Nelse ad Frederkse (006) also evaluaed he fe sample accuracy of dffere esmaors of he egraed varace uder he presece of mcrosrucure ose ad possble jumps. The auhors cosdered hree esmaors: he realzed varace, he esmaor based o Fourer seres (Mallav ad Maco (00) ad Barucc ad Reo (00a, 00b)), ad fally, he wavele esmaor of Høg ad Lude (003). The ma cocluso of he paper s ha he Fourer esmaor s preferable whe compared o he oher wo ad, mos surprsgly, has a slghly beer fe sample performace ( erms of MSE) ha he bas-correced kerel-based esmaors as Hase ad Lude (006b). However, s sll a ope queso wha are he fe sample properes of he esmaors dscussed above uder more geeral assumpos abou he mcrosrucure ose. Aoher mpora way of selecg a esmaor for he egraed varace s o use ecoomc or facal measures. For example, oe mgh decde o choose a esmaor ha acheves greaer accuracy forecasg Value-a-Rsk hresholds deermg opmal Basel Accord capal charges. O he oher had, oe mgh proceed, as Flemg, Krby ad Osdek (00, 003), by examg he ecoomc beefs of dffere volaly measures a dyamc porfolo allocao experme (see also Seco 5. for furher dscusso). I summary, he predcve ably of dffere 3

34 esmaors mgh be used as crero o decde amog dffere aleraves (see Remark 5 Bad ad Russell (006c) for a useful dscusso). 4. THE GENERAL MULTIVARIATE FRAMEWORK There has bee growg heorecal ad emprcal eres exedg he resuls for he uvarae processes dscussed prevously o a mulvarae framework. I hs coex, wo poeerg corbuos have bee made by Bardorff-Nelse ad Shephard (004b) ad Bad ad Russell (005b). Bardorff-Nelse ad Shephard (004b) dd o cosder he presece of mcrosrucure ose, whereas he case of ose has bee cosdered Bad ad Russell (005b). Seco 4. brefly revews he resuls Bardorff-Nelse ad Shephard (004b) ad Bad ad Russell (005b). Seco 4. gves some refereces of promsg rece developmes he mulvarae coex. 4.. Realzed Covarace As (), suppose ha, alog day, he logarhmc prces of a gve se asses follow a couous me dffuso process, as follows: dp ( + τ ) = μ( + τ ) + Θ( + τ ) dw( + τ ), 0 τ, =,,K, () where p ( + τ ) s a vecor of logarhmc prces a me + τ, μ ( + τ ) s he mulvarae drf compoe, Θ ( + τ ) s he saaeous co-volaly marx, ad W ( + τ ) s he sadard mulvarae Browa moo. As before, suppose also ha Θ ( + τ ) s orhogoal o W ( + τ ). The saaeous covarace marx s Σ ( + τ ) = Θ( + τ ) Θ( + τ )', wh geerc eleme gve by Σ( )( s )( + τ ) u. Defe he realzed covarace as (ALL) ' RCov = r r. =,, (3) 33

35 Bardorff-Nelse ad Shephard (004b) showed ha / vech (ALL) d ( RCov ) vech Σ( ) d + τ τ N( 0, Π ), 0 (4) where Π s a posve defe marx (see Bardorff-Nelse ad Shephard (004b) for furher deals). Uder he presece of mcrosrucure ose, Bad ad Russell (005b) showed ha he realzed covarao esmaor gve (3) s o cosse. Bad ad Russell (005b) proposed a mehod for selecg he opmal samplg frequecy as a rade-off bewee bas ad effcecy. The opmal samplg frequecy s gve by * 3 Q( )( ) u s [ E( ( ), ( ), )] ε u ε s, (5) where Q [ Σ + τ ) Σ ( + τ ) + Σ ( τ ] ( u) ( ( s) ( u)( s) + dτ = ( u )( s) ), 0 (6) Bad ad Russell (005b) sugges esmag (6) wh a sparse samplg frequecy of 5 or 0 mues as (sparse) (sparse) (sparse) (sparse) r( u), r( s), r( u), r( s), r( u) +, r( s) +, = = Q ˆ =. ( u)( s) Bad ad Russell s (005c) resuls have bee derved uder he assumpo ha he mcrosrucure ose s a covarace-saoary zero mea vecor sochasc process ha s depede of he vecor of effce (ad uobservable) prces. 34

36 However, he esmao of egraed covaraces usg hgh-frequecy daa brgs ew mpora ssues. As poed ou by Epps (979), formao arrves a dffere frequeces for dffere asses, herefore roducg addoal mcrosrucure effecs ha are relaed o he osychrocy he process of prce formao. Eve whe here s o mcrosrucure frcos as prevously dscussed, osychroous radg roduces a dowward bas he realzed covarace esmaes whe samplg reurs caledar me a hgh frequeces. Ths s he so-called Epps effec. To accommodae hs effec, Bad ad Russell (005b) corporaed leads ad lags her esmaor. Ths s a old soluo he leraure o overcome he osychrocy of observaos (see Scholes ad Wllams (977), Dmso (979), ad Cohe, Hawa, Maer, Schwarz, ad Whcomb (983)). For wo gve asses, Bad ad Russell s (005c) lead-lag esmaor wh U lags ad L leads s gve by * (BR) RCov = r r,,()() U (), = s= L () s, (7) where r ), ( ad r ( ), are he h raday reurs for asse () ad () a day. The opmal samplg frequecy s gve by (5). As observed by Bad ad Russell (006a), a eresg opc for fuure research s he use of drec MSE-based opmzao of he lead-lag esmaor o deerme he opmal samplg frequecy as well as he choce of he umber of leads ad lags uder he presece of mcrosrucure ose. I a relaed work, Mares (005) evaluaed he MSE properes of a umber of covarace esmaors hrough smulaos based o Lo ad MacKlay s (990) osychroous rade model. 4.. Rece Exesos Recely, Hayash ad Yoshda (005, 006), Sheppard (006), ad Zhag (006b), amog ohers, roduced alerave approaches o he hgh frequecy covarace esmaor. For example, sead of usg caledar me reurs, he Hayash ad Yoshda 35

37 (HY) esmaor s based o overlappg ck-by-ck reurs. I he absece of classcal mcrosrucure frcos, bu he presece of osychroous radg, he HY esmaor s cosse ad asympocally ormally dsrbued. Sheppard (006) aalyzed he codos uder whch he realzed covarace s a ubased ad cosse esmaor of he egraed varace. The cocep of scramblg was defed by Sheppard (006) o movae a geeral famly of alerave specfcaos based o radom cesorg of reurs, whch ess he prevously suggesed correcos for mulvarae esmaors. Zhag (006b) also suded he effecs of mcrosrucure ose ad osychroous radg he esmao of he egraed covarace bewee wo asses. Zhag (006b) showed ha he bas s more proouced less lqud asses ad provded a way, as Bad ad Russell (005b), o compue he opmal samplg frequecy order o reduce he bas. Voev ad Lude (006) ad Grff ad Oome (006b) provde dealed fe sample sudes of he MSE properes of several covarace esmaors, cludg he realzed covarace, opmally-sampled realzed covarace, he HY esmaor, ad he lead-lag esmaor ( equao (6)). The auhors also provded recommedaos for praccal mplemeaos of such esmaors. Hoshkawa, Kaaa, Naga ad Nshyama (006) compared he mulvarae verso of he Fourer esmaor of Mallav ad Maco (00a), he HY esmaor, ad he classcal realzed covarace esmaor. The auhors foud ha he HY esmaor performs he bes amog he aleraves vew of he bas ad he MSE, whle he oher esmaors were show o have possbly heavy bas, mosly oward he org. 5. MODELLING AND FORECASTING REALIZED VOLATILITY 5.. Some Sylzed Facs Facal Tme Seres ad Uvarae Applcaos A well esablshed resul he facal ecoomercs me seres leraure s ha, whe GARCH ad SV lae volaly models are used, he sadardzed reurs do o have a Gaussa dsrbuo. I pracce, here s sll excess kuross, a fac ha movaes he use of heavy-aled dsrbuos. However, ABDL (000a, 000b, 00, 003) showed 36

38 ha, whe he realzed varace was used, he dsrbuo of he sadardzed exchage rae seres was almos Gaussa. Ths was also corroboraed for sock reurs ABDE (00). Furhermore, he logarhm of he realzed volales was also early Gaussa. Cocerg he dyamcs of he log-realzed varace, s well esablshed ha hs s a hghly persse, bu saoary, me seres process. I addo, here s sgfca evdece of log memory he me seres, whch has bee coveoally modelled as a ARFIMA(p,d,q) process (see ABDL (000a, 000b, 00, 003) for some examples) 8. Recely, Cors, Zumbach, Müller ad Dacoroga (00) ad Cors (003) proposed he Heerogeeous AuoRegressve Realzed Volaly (HAR-RV) model, based o he Heerogeeous ARCH (HARCH) model of Müller, Dacoroga, Davé, Olse, Puce, ad vo Wezäcker (997). The HAR-RV model s specfed as a mul-compoe volaly model wh a addve herarchcal srucure such ha he volaly s specfed as a sum of compoes over dffere horzos (see also Aderse, Bollerslev ad Debold (006a)). McAleer ad Mederos (006) exeded he HAR-RV model by proposg a flexble mulple regme smooh raso model o capure oleares ad log-rage depedece he me seres dyamcs. These resuls owhsadg, defyg he possble sources of log memory s also of parcular eres, such as case of spurous log memory, where a shor memory model may produce beer ad more precse forecass. Recely, Hyug, Poo ad Grager (005), hereafer HPG (005), dscussed he possble sources of log memory facal volaly. As ouled HPG (005), a myrad of olear shor memory models, especally models wh freque breaks, ca geerae daa wh log memory behavour. Examples of such olear models clude he break model of Grager ad Hyug (004), he volaly compoe model of Egle ad Lee (999), he regme swchg model proposed by Hamlo ad Susmel (994), ad furher dscussed Debold ad Ioue (00), ad he mulple-regme model of Mederos ad Vega (004). Hllebrad (005) also dscussed he effecs of breaks o he esmao of volaly models (see also 8 As oe of he few excepos, Carvalho, Frere, Mederos ad Souza (006) dd o foud evdece of log memory he dyamcs of realzed volales for several asses raded he Brazla sock exchage. 37

39 Hllebrad ad Mederos (006a)). Scharh ad Mederos (006) proposed a mulpleregme model based o regresso rees o descrbe he dyamcs of realzed volales of several DJIA socks. The auhors corporaed pas cumulaed daly reurs as a source of regme swches. Ther ma fdg s ha hs effec s hghly sgfca ad accous for hgh emprcal values of log memory parameer esmaes. They also showed ha he olear model sgfcaly ouperformed he cocurre log memory models (ARFIMA ad HAR-RV) a ou-of-sample experme for all 6 socks aalyzed, especally perods of hgh volaly. I each of he specfcaos dscussed above, volaly refers o shor memory bewee breaks, for each volaly compoe, ad wh each regme. More recely, Mares, va Djk ad de Pooer (004) proposed a model ha combes he log memory propery wh oleary, whch s especally mpora modellg asymmeres ad he leverage effec. They showed srog emprcal evdece favor of her proposal. Deo, Hurvch ad Lu (006) cosdered a log-memory sochasc volaly model ad Koopma, Jugbacker ad Hol (005) proposed a model combg uobserved compoes ad log-memory. I a rece work, Hllebrad ad Mederos (006b) suggesed a model ha combes log memory wh dffere ypes of oleary. Ther approach s based o a smulaeous equao framework, where volaly also drec affecs he reurs (as he GARCH--Mea model). However, s sll a ope queso as o he source of he appare log memory he realzed volaly, ad wheher he beefs of combg log memory ad olear models wll dramacally mprove he accuracy forecasg volaly (Ohassa, Russell ad Tsay (004a, 004b)). More recely, Leberma ad Phllps (006) provde some aalycal explaaos o expla he log rage depedece behavour ha has bee observed realzed volales. The auhors show ha log memory may arse from he accumulao of realzed volaly, ad dscussed how o refe he sascal ferece regardg he parameer d ARFIMA(p,d,q) models. 38

40 Aï-Sahala ad Mac (006) compare he ou-of-sample relave forecasg ably of realzed volaly a varey of coexs. Ghysels ad Sko (006) aalyze he relave predcve ably of realzed volaly wh he mxed daa samplg (MIDAS) framework of Ghysels, Saa-Clara ad Valkaov (006). Corrad, Dsaso ad Swaso (006) focused o esmag ad forecasg he codoal desy of egraed volaly ad Cors, Mk, Pgorsch ad Pgorsch (006) focused o he volaly of he realzed volaly. Aoher ssue ha should be meoed s he fac ha, eve whou he presece of mcrosrucure ose, realzed volaly s a esmaed quay raher ha he rue daly volaly (or egraed varace), ad egraed quarcy s replaced by realzed quarcy. Ths leads o he use of geeraed regressors ad geeraed varables for purposes of forecasg, wh he assocaed crcal ssues of effce esmao ad vald fereces ha arse hrough he use of based (asympoc) sadard errors (see Paga (984, 986) ad McKeze ad McAleer (997) for comprehesve dscussos). Recely, Aderse, Bollerslev ad Meddah (004, 005) have developed a geeral model-free adjusme procedure for he calculao of ubased volaly loss fucos based o realzed volaly bechmarks. The auhors have also show ha properly accoug for he measureme errors he volaly forecas evaluaos repored he exsg leraure ca lead o markedly hgher esmaes for he rue degree of reurs volaly predcably. I a rece paper, Corrad ad Dsaso (006) proposed a procedure o es for he correc specfcao of he fucoal form of he volaly process based o he class of egefuco sochasc volaly models of Meddah (00). Ther dea s o compare he momes of he realzed volaly measures wh he correspodg oes of he egraed volaly mpled by he heorecal model uder he ull hypohess. The auhors carefully ook accou of he fac ha realzed volaly s a esmaed measure, ad s hereby coamaed wh measureme errors. 5.. Mulvarae Applcaos 39

41 I a eresg paper, de Pooer, Mares ad va Djk (006) vesgae he beefs of hgh frequecy raday daa whe cosrucg mea-varace effce sock porfolos wh daly rebalacg from he dvdual cosues of he S&P 00 dex. The auhors aalyzed he ssue of deermg he opmal samplg frequecy, as judged by he performace of he esmaed porfolos. The opmal samplg frequecy rages bewee 30 ad 65 mues, ha s, much lower ha he fve-mue frequecy, whch s commoly used he leraure. The auhors also showed ha several bas-correco procedures, based o combg low ad hgh frequecy covarace marx esmaes, ad wh he addo of leads ad lags, do o subsaally affec he opmal samplg frequecy or he porfolo performace. The fdgs are also robus o he presece of rasaco coss ad o he porfolo rebalacg frequecy. I a relaed paper, Bad, Russell ad Zhou (006) evaluae he ecoomc beefs of mehods ha have bee suggesed o opmally-sample ( a MSE sese) hgh frequecy reur daa for he purpose of realzed varace ad covarace esmao he presece of marke mcrosrucure ose. The auhors compared ceray equvales derved from volaly-mg radg sraeges, relyg o opmally-sampled realzed varaces ad covaraces, o realzed varaces ad covaraces obaed by samplg every 5 mues, ad o realzed varaces ad covaraces obaed by samplg every 5 mues. They showed ha a rsk-averse vesor, who s gve he opo of choosg varace ad covarace forecass derved from MSE-based opmal samplg mehods versus forecass obaed from 5- ad 5-mue ervals (as s geerally proposed he leraure), would be wllg o pay up o abou 80 bass pos per year o acheve he level of uly ha s guaraeed by opmal samplg. They also foud ha he gas yelded by opmal samplg are ecoomcally large ad sascally sgfca. Bauer ad Vorkk (006) prese a ew marx logarhm model of he realzed covarace of sock reurs, whch uses lae facors as fucos of boh lagged volaly ad reurs. The model has several advaages ha s parsmoous, does o requre mposg paramerc resrcos, ad yelds a posve defe covarace 40