Testing and Modelling Market Microstructure Effects with an Application to the Dow Jones Industrial Average

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1 Tesng and odellng arke crosrucure Effecs wh an Applcaon o he Dow Jones Indusral Average Basel Awaran Queen ary, Unversy of London Waler Dsaso Unversy of Exeer January 2004 Prelmnary and ncomplee Valenna Corrad Queen ary, Unversy of London Absrac I s a well acceped fac ha sock reurns daa are ofen conamnaed by marke mcrosrucure effecs, such as bd-ask spreads, lqudy raos, urnover, and asymmerc nformaon. Ths s parcularly relevan when dealng wh hgh frequency daa, whch are ofen used o compue model free measures of volaly, such as realzed volaly. In hs paper we sugges wo es sascs. The frs s used o es for he null hypohess of no mcrosrucure nose. If he null s rejeced, we proceed o perform a es for he hypohess ha he mcrosrucure nose varance s ndependen of he samplng frequency a whch daa are recorded. We provde emprcal evdence based on he socks ncluded n he Dow Jones Indusral Average, for he perod Our fndngs sugges ha, whle he presence of mcrosrucure nduces a severe bas when esmang volaly usng hgh frequency daa, such a bas grows less han lnearly n he number of nraday observaons. Keywords: Bpower varaon, marke mcrosrucure, Realzed Volaly. JEL classfcaon: C22, C12, G12. The auhors graefully acknowledge fnancal suppor from he ESRC, gran code R Queen ary, Unversy of London, Deparmen of Economcs, le End, London, E14S, UK, emal: b.awaran@qmul.ac.uk. Queen ary, Unversy of London, Deparmen of Economcs, le End, London, E14S, UK, emal: v.corrad@qmul.ac.uk. Unversy of Exeer, Deparmen of Economcs, Sreaham Cour, Exeer EX4 4PU, UK, emal: w.dsaso@ex.ac.uk.

2 1 Inroducon Dang back o Black s 1986 semnal paper, s a well acceped fac ha ransacon daa occurrng n fnancal markes are ofen conamnaed by marke mcrosrucure effecs, such as bd-ask spreads, lqudy raos, urnover, and asymmerc nformaon see also e.g. Hasbrouk 1993, Ba, Russell and Tao 2000, Hasbrouck and Sepp 2001, O Hara 2003 and references heren. I s argued n hese papers ha he observed ransacon prce can be decomposed no he effcen one plus a nose due o mcrosrucure effecs. Ths fac s parcularly relevan when dealng wh hgh frequency daa, whch are ofen used o compue model free measures of volaly, such as realzed volaly see e.g. Barndorff-elsen and Shephard 2002, 2003, 2004c, Andersen, Bollerslev, Debold and Labys 2001, 2003, eddah 2002 and Andersen, Bollerslev and eddah 2004 and bpower varaon Barndorff-elsen and Shephard, 2004a,b. Alhough he relevan lm heory suggess ha volaly esmaes ge more precse as he frequency of observaons ncreases, hs s no necessarly vald n he presence of mcrosrucure nose whch s no accouned for. The effec of mcrosrucure nose on hgh frequency volaly esmaors has been recenly analyzed by Aï-Sahala, ykland and Zhang 2003, Zhang, ykland and Aï-Sahala 2003, and Band and Russell 2003a,b. These papers pon ou ha changes n ransacon prces over very small me nervals are manly composed of nose and carry lle nformaon abou he underlyng reurn volaly. Ths s because, a leas for he class of connuous semmarngale processes, volaly s of he same order of magnude as he me nerval, whle he mcrosrucure nose has a roughly consan varably. Therefore, as he me nerval shrnks o zero, he sgnal o nose rao relaed o he observed ransacon prces also ends o zero, and usng he esmaors of volaly menoned above one may run he rsk of esmang he varance of he mcrosrucure nose, raher han he underlyng reurn volaly. Hence, he need of measures of reurn volaly whch are robus o he presence of marke mcrosrucure effecs. An mporan conrbuon n hs drecon s ha of Zhang, ykland and Aï-Sahala 2003, who sugges an asympocally unbased volaly esmaor, based on subsamplng echnques. The valdy of her esmaor hnges on he chosen model for he mcrosrucure nose. Our paper complemens he papers ced above n wo drecons. Frs, we provde a es for he null hypohess of no marke mcrosrucure effec, whch s robus o he presence of possble large 1

3 and rare jumps. Second, f he null s rejeced, we es he null hypohess of correc specfcaon of he model of mcrosrucure nose of Aï-Sahala and ykland and Zhang 2003, whch assumes ha he varance of he mcrosrucure nose has a consan varance, regardless of he frequency a whch daa are sampled. The frs es sasc s based on he dfference beween wo realzed volales compued a dfferen samplng nervals, say one mnue and en mnues. Under he null, boh esmaors wll converge o he rue negraed volaly process, hough a a dfferen speed. Gven hs, by properly scalng hs dfference we have a sasc wh a normal lmng dsrbuon under he null and un asympoc power. However, such a es sasc can dverge because of he presence of eher mcrosrucure nose or jumps. To overcome hs problem, we also provde a jump robus verson of he es, whch s based on recen resuls by Barndorff-elsen and Shephard 2004a,b and Corrad and Dsaso The second es sasc s based on he dfference beween wo esmaors of he mcrosrucure nose compued agan over dfferen me nervals. Under he null model of a nose wh consan varance, by properly scalng hs dfference we oban a sasc wh a normal lmng dsrbuon. The es s conssen agans he alernave of a nose wh varance dependng on he chosen samplng frequency. Indeed, an alernave model of economc neres would be one n whch he mcrosrucure nose varance s posvely correlaed wh he me nerval. The proposed ess are hen appled o ransacon daa recorded for he socks ncluded n he Dow Jones Indusral Average DJIA for he perod , usng a fxed me span equal o fve days. The ess are compued over he dfferen fve days nervals. The emprcal analyss suggess ha whle he presence of mcrosrucure effecs nduces a severe bas when esmang volaly usng hgh frequency daa, such a bas grows less han lnearly n he number of nraday observaons. Ths paper s organzed as follows. Secon 2 descrbes he employed mehodology and derves he lmng behavor of he wo es sascs. The emprcal fndngs are repored n Secon 3 and Secon 4 conans some concludng remarks. All he proofs are conaned n he Appendx. 2

4 2 ehodology 2.1 Se-up Le X =logs, where S denoes he prce of a fnancal asse or dervave. Throughou he paper s assumed ha X follows a process of he ype X = µ d + c dq + σ dw, 1 where W s a sandard Brownan moon. As for he jump componen, Prdq = 1 = λ d, where λ s ndependen of σ 2,c s an..d. process and s assumed o be ndependen of dq. Ths specfcaon of he jump componen covers he case of large and rare jumps, analyzed by Barndorff-elsen and Shephard 2004a. Alhough we canno observe he rajecory of X, we sll have daa recorded a hgh frequency. Suppose ha he number of daly observaons s denoed by T and ha, for each day, we have nraday observaons; herefore, over a fxed me span, say T, we have a oal of T observaons. Realzed volaly s defned as RV,T, = T =1 X + X +, 0 T T. If X belongs o he class of connuous semmarngales f dq =0, a.s.,,.e. here are no jumps n he log prce process, hen see e.g. Karazas and Shreve 1991, Ch.1, as, +T Pr RV,T, σsds 2 = IV,T. 2 Here σ 2 s denoes he nsananeous volaly a me s, has E X s+h X s I s lm = σ 2 h 0 h s, where I s refers o he relevan condonng se a me s. However, f X s he sum of a connuous semmarngale componen and a jump componen, hen he saemen n 2 does no longer hold and, as poned ou by Barndorff-elsen and Shephard 2004a, +T Pr RV,T, IV + c 2, 3 = 3

5 where s a counng process and c denoes he sze of he jumps. Ineresngly, Barndorff-elsen and Shephard 2004a also sugges a measure of negraed volaly, namely bpower varaon, whch when properly scaled s a conssen esmaor of negraed volaly and s robus o he presence of rare of large jumps. 1 Bpower varaon s defned as BV,T, = T 1 =1 X + +1 ow, suppose ha we can observe S + gven by Therefore S + X + X + X +, 0 T T. only up o an error, so ha he observed prce process s = S + η +, =0, 1,...,T 1. 4 log S + Y + = logs + = X + +logη +, or 5 + ɛ +. Here ɛ + s nerpreed as a nose capurng he marke mcrosrucure effec. Smlarly o wha s cusomarly assumed n he leraure, X +/ and ɛ +/ are ndependen and E ɛ +/ =0. ow, = E = E E Y + Y + X + X + X X + E ɛ + + E ɛ + ɛ + ɛ + + E In he case of X beng a connuous semmarngale, E wll assume ha =1 E lm,, / 0 T 1 =1 ɛ + ɛ + X + X +. 6 ɛ + X + ɛ + As for bpower varaon, noe ha, by nkowsk nequaly, T 1 T 1 E Y + +1 Y + Y + Y + E =1 X + +1 X + =. X + = O 1. Also, we X + X + 1 A vable alernave o hs approach would be o follow Aï-Sahala 2004, who proposes a mehod o dsenangle he connuous Brownan and jump componens, and hen use he connuous componen o compue an esmaor of negraed volaly. Such an approach s vald for homoskedasc dffuson processes. 4

6 +E T 1 =1 ɛ + +1 ɛ + ɛ + ɛ + + E T 1 =1 ɛ + +1 ɛ + X + X + Therefore, when usng bpower varaon s no mmedae how o decompose he oal varably n negraed volaly and nose varance. everheless, s evden ha, whle bpower varaon s robus o he presence of he jump componen, s no robus o mcrosrucure effecs. As menoned n he Inroducon, our objecve s perform a es for he null hypohess of no mcrosrucure effecs over a sequence of fne me span e.g. 5 workng days perods, and, for he cases n whch he null s rejeced, proceed o es he null hypohess ha he mcrosrucure nose has consan varance, regardless of he samplng nerval over whch we compue realzed volaly Tesng he null hypohess of no mcrosrucure effecs In he sequel we shall assume ha he rue asse prce process s generaed as n 1. Defne he followng hypoheses: and H 0 : E ɛ + ɛ + H A : E ɛ + =0, for all, 7 ɛ + 0, 8 where E ɛ +/ ɛ +/ and ɛ+/ are defned respecvely n 6 and 5. Therefore he null hypohess mples ha here are no mcrosrucure effecs n he observed ransacon prces, whle he alernave s smply he negaon of he null. In order o es H 0 versus H A, we propose he followng sasc Z,,T, = =1 T T =1 Y + Y T T =1 T =1 Y + Y + Y + Y +, 9 4 where / 0as,.e. grows faser han. The numeraor n 9 can be expanded as T 10 T 2 +T T 2 +T Y + Y + σsds 2 Y + Y + σsds 2. 5 =1

7 Inspecon of 10 reveals he logc behnd he choce of he es sasc n 9; f here are no mcrosrucure effecs under he mananed assumpon of no jumps, hen boh T =1 Y+/ Y +/ and T +T =1 Y+/ Y +/ converge n probably o he rue negraed volaly, σs 2ds. Also, by he cenral lm heorems n Barndorff-elsen and Shephard 2002 and 2004c, and T =1 T =1 Y + Y + Y + Y + 2 +T σsds 2 = O p 1/2 2 +T σsds 2 = O p 1/2. Therefore, under he null hypohess, provded ha as,, / 0, T 2 +T T Y + Y + σsds 2 = o p 1. =1 Under H 0, he lmng dsrbuon of he es sasc s drven by +T Y + Y + T T =1 2 3 T T =1 Y + Y + σsds 2, 4 whch s asympocally sandard normal. Under he alernave, we expec T =1 Y+/ Y +/ o be of a larger order of magnude n probably han T,andsowe =1 Y+/ Y +/ expec he sasc o dverge. A possble problem wh he sasc above s ha sandard normal crcal values are no longer correc n he presence of jumps. In fac, n he presence of jumps, he numeraor of 9 has a non zero mean and he square of he denomnaor s no a conssen esmaor for he rue negraed quarcy.e. +T σs 4 ds. Therefore, we also sugges a sasc based whch s robus o jumps ZB,,T, = µ 2 1 T µ 4 1 T 1 =1 Y + +1 Y + Y + Y + T =1 Y + +1 Y + Y + Y + T, T 3 =1 Y + Y + +4 Y + +3 Y + +3 Y + +2 Y + +2 Y + +1 Y + +1 where µ 1 = E 0, 1. Under he null hypohess, he sasc above has a sandard normal lmng dsrbuon, regardless of he presence of possble jumps. 6

8 2.3 A smple specfcaon es for he mcrosrucure nose For all he perods n whch, accordng o eher or boh es sascs proposed n he prevous subsecon, we rejec he null hypohess of no mcrosrucure nose, may be neresng o perform a specfcaon es for he mcrosrucure error. In parcular, he hypohess of neres s ha of he mcrosrucure nose havng a consan varance, ndependen of he frequency a whch daa are recorded. In fac, mos of he recen leraure on ncorporang mcrosrucure effecs see e.g. Aï-Sahala, ykland and Zhang 2003, Zhang, ykland and Aï-Sahala 2003, and Band and Russell 2003a,b posulaes a model of nose wh consan varance. ore precsely, le E ɛ +/ ɛ +/ =2ν, and E ɛ +/ ɛ +/ =2ν,.The null and alernave hypoheses can be formulaed as follows H 0 : ν, = ν,, for all, 12 and H A : ν, <ν,. 13 Thus, he alernave of neres s ha he varance of he mcrosrucure nose s negavely correlaed wh he frequency a whch daa are recorded. The alernave hypohess s compable wh he mcrosrucure nose model oulned by Barndorff-elsen and Shephard 2004b, who specfy a wo componen model; he frs s a jump componen and he second s an error process wh he varance decayng o zero a a rae 1/. As a consequence, he bas ncurred n esmang volaly usng realzed volaly grows less han lnearly n he number of nraday observaons. 2 We propose he followng es sasc V,,T, = T 2T 2 T =1 Y + Y + 1 T T =1 Y + 2 T =1 Y + Y + 2T Y The logc behnd he sasc proposed above s he followng. Under he null hypohess, boh T =1 Y+/ Y +/ /2T and T =1 Y+/ Y +/ /2T converge o he same 2 The fac ha he varance of measuremen error s n general no ndependen of he samplng frequency has been kndly poned ou o us by Andrew Chesher. 7

9 probably lm, say ν, hough he former converges faser han he slower. Thus, 2 T =1 Y + Y + ν 2T T 1 4 T T =1 Y + Y + s asympocally neglgble and he sasc n 14 s asympoc equvalen o T 2 T =1 Y + Y + 2T 1 T T =1 Y + ν Y + whch has a sandard normal lmng dsrbuon see Theorem A1 n Zhang, ykland and Aï- Sahala Under he alernave, T =1 4, Y+/ Y +/ /2T and T =1 Y+/ Y +/ /2T converge respecvely o ν and ν, and hus he sasc dverges. In he nex subsecon, he lmng dsrbuons of he proposed es sascs are derved. 2.4 an heorecal resuls In he sequel we need he followng assumpons A1: X s generaed as n 1. A2: +T σ 4 sds <, almos surely, for any and T. 4 A3: E ɛ ɛ <. A4: T =1 ɛ+/ ɛ +/ / T = op b 1,, and T =1 ɛ+/ ɛ +/ / T = op b 1,, where, as,, b, b, 1 α, for α 0, 1 and = α. A4 : E ɛ+/ ɛ +/ / T = ν, and E ɛ+/ ɛ +/ / T = T =1 ν,, ν, /ν,, as,. T =1 oce ha A1 and A2 are cusomary n he leraure on realzed volaly; A3 requres a fne fourh momen of he marke mcrosrucure nose and seems o be rvally sasfed. Fnally, A4 allows he varably of he mcrosrucure error o decrease wh he samplng nerval. In parcular, he varance of he mcrosrucure nose s allowed o approach zero as he samplng nerval goes o zero, bu a a slow enough rae. Then, we can sae he followng Proposons. 8

10 Proposon 1 Le A1-A2 hold and assume ha λ =0for all. Under H 0, defned n 7, as, and / 0, d Z,,T, 0, 1. Le A1-A4 hold. Under H A, defned n 8, as,forε>0, lm Pr b, α 1 Z b,,t, >ε =1., Therefore, we can jus perform a one-sded es and rejec he null hypohess of no mcrosrucure effecs when we ge a value for Z,,T, larger han, say, he 95% percenle of a sandard normal. In he proposon above, he saemen under he null s robus o possble leverage effecs, bu no o possble jumps. The null lmng dsrbuon and power properes of he jump robus verson of he es for no mcrosrucure effecs es are gven n he nex Proposon. Proposon 2 Le A1-A2 hold and assume ha µ =0for all and ha σ n1sndependenofx. Under H 0, defned n 7, as, and / 0, ZB,,T, d 0, 1. Le A1-A4 hold. Under H A, defned n 8, as,forε>0, lm Pr b, α 1 ZB b,,t, >ε =1., As oulned n he prevous subsecon, every me we rejec he null hypohess n 7, accordng o eher or boh he suggesed sascs, we may be neresed n performng a specfcaon es for he mcrosrucure nose. Is properes are gven n he nex Proposon. Proposon 3 Le A1-A3 hold. Under H 0, defned n 12, as,,/ 0, V,,T, d 0, 1. 9

11 Le A1-A3 hold. Also, assume ha E 1 T ɛ T + ɛ + =2ν, =1 and E 1 T T =1 ɛ + ɛ + =2ν, are such ha ν, and ν,. Under H A, defned n 13, as,for ε>0, 1 lm Pr V,,T, >ε =1. T An applcaon of he esng procedure oulned n hs secon o he socks ncluded n he Dow Jones Indusral Average s gven n he nex Secon. 3 Emprcal evdence from he Dow Jones Indusral Average 3.1 Daa descrpon The emprcal analyss of marke mcrosrucure effecs s based on daa rereved from he Trade and Quoaon TAQ daabase a he ew York Sock Exchange. The TAQ daabase conans nraday rades and quoes for all secures lsed on he ew York Sock Exchange YSE, he Amercan Sock Exchange AEX and he asdaq aonal arke Sysem S. The daa s publshed monhly on CD-RO snce 1993 and on DVD snce June Our sample conans he DJIA socks 30 socks n oal 3 and exends from January 1, 1997 unl December 24, 2002, for a oal of 1505 radng days. 4 Also, n our emprcal example T = 5 and herefore we have a oal of 301 fve-days perods. Table 2 shows he average number of quoaons per mnue for all ndvdual socks. able presens a specrum of lqudy, rangng from as low as 3 quoaons per mnue for Uned Technologes Corp. UTX o as hgh as 94 for Inel Corp.. The wo mos lqud socks n he sample 3 I s worh menonng ha he 30 companes ncluded n he DJIA are no he same hroughou he sample perod. Woolworh, Behlehem Seel, Texaco and CBS Wesnghouse Elecrc have been replaced by Wal-ar, Johnson & Johnson, Hewle-Packard and Cgroup Travelers Group n In addon SBC Communcaons, crosof, Inel and Home Depo have replaced Unon Carbde, Chevron, Goodyear, and Sears n Our sample conans he DJIA of ndvdual frms as was n The names and he symbols of he socks ncluded n he sample are repored n Table 1. 4 Tradng days are dvded over he dfferen years as follows: 253, 252, 252, 252, 248, 252 from 1997 o oe ha here are 5 days mssng n 2001 due o Sepember 11h. The 10

12 are Inel and crosof and akes only a fracon of a second o have a fresh quoe. Lqudy has ncreased subsanally durng he sample perod and, as we approach 2002, he number of quoaons per mnue almos doubles for some socks e.g. 3 Company, Cgroup Inc. C, Home Depo Corp. HD, crosof SFT. From he orgnal daa se, whch ncludes prces recorded for every rade, we exraced 1 mnue and 10 mnues nerval daa, smlarly o Andersen and Bollerslev Provded ha here s suffcen lqudy n he marke, he 5 mnues frequency s generally acceped as he hghes frequency a whch he effec of mcrosrucure bases are no oo dsorng see Andersen, Bollerslev, Debold and Labys 2001, Andersen, Bollerslev and Lang 1999 and Ebens 1999; hence he choce of he wo menoned frequences o calculae he es sascs, n order o hghlgh he full exen of he mcrosrucure nose effecs. The prce fgures for each 1 and 10 mnues nervals are deermned as he nerpolaed average beween he precedng and he mmedaely followng quoes, weghed lnearly by her nverse relave dsance o he requred pon n me. For example, suppose ha he prce a 15:29:56 was and he nex quoe a 15:30:02 was 11.80, hen he nerpolaed prce a 15:30:00 would be exp1/3 log /3 log11.75 = From he 1 and 10 mnues prce seres we calculaed 1 and 10 mnues nradaly reurns as he dfference beween successve log prces expressed n percenages; hen R + = 100 logy + logy +, where R +/ denoes he reurn for nraday perod / on radng day,wh =0,...,T 1. The ew York Sock Exchange opens a 9:30 a.m. and closes a 4.00 p.m.. Therefore a full radng day consss of 391 resp. 40 nraday reurns calculaed over an nerval of one mnue resp. en mnues. For some socks, and n some days, quoaons arrve some me afer 9:30; n hese cases we always se he frs avalable radng prce afer 9:30 a.m o be he prce a 9:30 a.m.. all he days n our sample consss of 391 resp. 40 prce observaons; hs s arbuable o he fac ha he YSE closes early on ceran days, such as on Chrsmas Eve 5 ; for all hese nervals whou prce quoes we nser zero reurn values. Hghly lqud socks may have more han one prce a ceran pons n me for example 5 or 10 quoaons a he same me samp s normal 5 In addon o Chrsmas Eve, here are 12 oher shor days n he sample, makng a oal of 17 days. o 11

13 for ITC and SFT; when here exss more han one prce a he requred nerval, we selec he las provded quoaon. For nerpolang a prce from a mulple prce neghborhood, we selec he closes provded prce for he compuaon. 3.2 Tesng for he null of no mcrosrucure effecs Table 3, columns 2 and 3, repors he fndngs for he es based on he sasc defned n 9. ore precsely, we repor he number and he percenages of rejecons, based on a one-sded es. We frs noce ha only for sx socks he percenage of rejecon s below 20% of he cases. For welve socks he percenage of rejecon s beween 20% and 40% of he cases, and for he remanng welve socks s hgher han 40% wh a maxmum of 66%. Ths ndcaes ha, hough mcrosrucure effec plays an mporan role, s conrbuon s que varable over me. Overall, we do no fnd evdence of a clear relaonshp beween beween lqudy and mcrosrucure effec. For example, akng wo raher lqud socks lke IB and Inel, for he former we rejec he null abou 20% of he mes, whle for he laer we rejec abou 39% of he mes. In Fgure 1, we repor he plo of he es sasc over he dfferen 5 days nervals consdered, wh he doed lne represenng he 5% upper al crcal value of a sandard normal. We noce ha here are a few nsances n whch he sasc akes a large and negave values. Ths happens manly for socks characerzed by low lqudy, such as C, EK and HO. The reason for hs fndng s ha, snce hese socks are no very lqud, and herefore are no raded ofen enough, a lo of reurns over 1 mnue nerval are zero, whle are no zero over 10 mnues nerval. Hence a negave value for he es sasc. Table 3, columns 4 and 5, repors he resuls for he es based on he sasc defned n 11. We noce ha he rejecon raes are comparable wh hose obaned usng he prevous es, alhough slghly lower he only excepon beng Inel. As he laer sasc s robus o he presence of large and rare jumps, he obaned resuls seem o provde evdence n favor of he fac ha mos of he 5 days perods are no characerzed by jumps. The fac ha only a small number of days s characerzed by jumps s also confrmed by he emprcal fndngs of Andersen, Bollerslev and Debold 2003, and Huang and Tauchen The plo of he es sasc, for each 5 days nerval s nsered n Fgure 2 and dsplays a very smlar paern o he one observed n Fgure 1. As explaned n he prevous Secon, each me we rejec he hypohess of no mcrosrucure 12

14 nose effec, we perform a specfcaon es for he hypohess of a mcrosrucure nose wh consan varance ndependen of he samplng nerval. The relevan emprcal resuls are conaned n he nex subsecon. 3.3 A specfcaon es for he varably of he mcrosrucure nose For all he perods n whch we rejec he null hypohess of no marke mcrosrucure effecs, we hen es H 0 versus H A, defned respecvely n 12 and 13, usng he sasc suggesed n 14. We perform wo sequences of es, he frs one condonal on rejecng he null of no mcrosrucure usng he es sasc n 9 and he second condonal on he same oucome usng he sasc n 11. The resuls are repored n Table 4, columns 2 o 5, and he plos are gven n Fgures 3 and 4. I s mmedae o see ha he null hypohess s rejeced n almos all he cases. Ths provdes srong evdence ha whle he presence of mcrosrucure nduces a severe bas when esmang volaly usng hgh frequency daa, such a bas grows less han lnearly n he number of nraday observaons. In fac, gven 6, accepance of he null hypohess would mply ha 2 E Y + Y + = E X + X + + E ɛ + =1 =1 +1 σ 2 sds +2ν = +1 =1 σ 2 sds +2ν. ɛ + However, our fndngs srongly sugges ha ν <ν, hus ndcang ha he mcrosrucure bas grows a a slower han lnear rae as he number of nraday observaons ncreases. 4 Concludng remarks To be done. 13

15 References Aï-Sahala, Y. 2004, Dsenanglng Dffuson from Jumps, Journal of Fnancal Economcs, forhcomng. Aï-Sahala, Y., P.A. ykland and L. Zhang 2003, How Ofen o Sample a Connuous Tme Process n he Presence of arke crosrucure ose, Workng Paper, Prnceon Unversy. Andersen, T.G., and T. Bollerslev, 1997 Inraday Perodcy and Volaly Perssence n Fnancal arkes, Journal of Emprcal Fnance, 4, Andersen, T.G., T. Bollerslev, F.X. Debold and P. Labys, 2001 The Dsrbuon of Realzed Exchange Rae Volaly, Journal of he Amercan Sascal Assocaon, 96, Andersen, T.G., T. Bollerslev, F.X. Debold and P. Labys, 2003 odellng and Forecasng Realzed Volaly, Economerca, 71, Andersen, T.G., T. Bollerslev and F.X. Debold 2003, Some Lke Smooh and Some Lke Rough: Unanglng Connuous and Jump Componens n easurng, odellng and Forecasng Asse Reurn Volaly, Workng Paper, Duke Unversy. Andersen, T.G., T. Bollerslev and S. Lang 1999, Forecasng Fnancal arke Volaly. Frequency vs Forecas Horzon, Journal of Emprcal Fnance, 6, Sample Andersen, T.G., T. Bollerslev and. eddah 2004, Analyc Evaluaon of Volaly Forecass, Inernaonal Economc Revew, forhcomng. Ba, X., J.R. Russell and G. Tao 2000, Effecs on onnormaly and Dependence on he Precson of Varance Esmaes Usng Hgh Frequency Daa, Workng Paper, Unversy of Chcago, Graduae School of Busness. Band, F.., and J.R. Russell 2003a, crosrucure ose, Realzed Volaly, and Opmal Samplng, Workng Paper, Unversy of Chcago, Graduae School of Busness. Band, F.., and J.R. Russell 2003b, Volaly or ose? Graduae School of Busness. Workng Paper, Unversy of Chcago, Barndorff-elsen, O.E., and. Shephard 2002, Economerc Analyss of Realzed Volaly and Is Use n Esmang Sochasc Volaly odels, Journal of he Royal Sascal Socey, Seres B, 64, Barndorff-elsen, O.E., and. Shephard 2003, Realzed Power Varaon and Sochasc Volaly, Bernoull, 9, Barndorff-elsen, O.E., and. Shephard 2004a, Power and Bpower Varaon wh Sochasc Volaly and Jumps, Journal of Fnancal Economercs, forhcomng. 14

16 Barndorff-elsen, O.E., and. Shephard 2004b, Economercs of Tesng for Jumps n Fnancal Economcs Usng Bpower Varaon, Workng Paper, Unversy of Oxford. Barndorff-elsen, O.E., and. Shephard 2004c, A Feasble Cenral Lm Theory for Realzed Volaly under Leverage, Workng Paper, Unversy of Oxford. Black, F. 1986, ose, Journal of Fnance, XLI, Corrad, V., and W. Dsaso 2004, Specfcaon Tess for Daly Inegraed Volaly, n he Presence of Possble Jumps, Workng Paper, Queen ary, Unversy of London. Ebens, H. 1999, Realzed Sock Volaly, Workng Paper, John Hopkns Unversy. Hasbrouck, J. 1993, Assessng he Qualy of a Secury arke. A ew Approach o Transacon Cos easuremen, Revew of Fnancal Sudes, 6, Hasbrouck, J., and D. Sepp 2001, Common Facors n Prces, Order Flows, and Lqudy, Journal of Fnancal Economcs, 59, Huang, X., and G. Tauchen 2003, The Relave Conrbuon of Jumps o Toal Prce Varaon, Workng Paper, Duke Unversy. Karazsas, I., and S.E. Shreve 1991, Brownan oon and Sochasc Calculus, Sprnger and Verlag, ew York. eddah,. 2002, A Theorecal Comparson beween Inegraed and Realzed Volaly, Journal of Appled Economercs, 17, O Hara,. 2003, Presdenal Address: Lqudy and Prce Dscovery, Journal of Fnance, LVIII, Zhang L., P.A. ykland and Y. Aï-Sahala 2003, A Tale of Two Tme Scales: Deermnng Inegraed Volaly wh osy Hgh Frequency Daa, Workng Paper, Carnege ellon Unversy. 15

17 Appendx Proof of Proposon 1. The sasc defned n 9 can be expanded no Z,,T, = T T =1 T T =1 Y T T =1 Y T T =1 Y + Y + Y + Y + +T σ 2 s ds Y + +T Y + 4 σsds Theorems 1 n Barndorff-elsen and Shephard 2002, 2004c esablsh ha, as, +T +T T RV,T,,T σsds 2 d 0, 2 σsds 4. Therefore he frs erm n 15 s o p 1, gven ha, as,, / 0. As a consequence, he lmng dsrbuon of Z,,T, under H 0 wll be deermned by he second componen of 15. By a smlar argumen as before, snce, as, he saemen follows mmedaely. T 1 3 T Y + =1 Y + 4 +T Pr σs 4 ds, Under he alernave hypohess, s possble o expand he numeraors of he componens of 15 respecvely as T T X + and =1 ] +T σsds 2 T T =1 X + X + X + + ɛ + + ɛ + ɛ + ɛ + +2 X + +2 X + X + X + ɛ + ɛ + ɛ + ɛ + 16

18 ] +T σs 2 ds. Smlarly = T Y + =1 T [ X + =1 +4 X + Y + 4 X + X ɛ + 3 ɛ + ɛ + ɛ X + +4 X + X + X + ɛ + ɛ + ɛ + ɛ + 3 ]. Therefore he sasc wll dverge o plus nfny and he es wll be conssen f, as,, T =1 ɛ + ɛ + T =1 ɛ + ɛ + T, 4 =1 ɛ + ɛ + whch holds under Assumpon A4. Proof of Proposon 2. From Barndorff-elsen and Shephard 2004a, under he null hypohess, as, T 1 =1 µ 2 1 Y + +1 Y + Y + Y + +T Pr σ 2 sds, µ 4 1 T 3 =1 Y + +4 Y + +3 Y + +3 Y + +2 Y + +2 Y + +1 Y + +1 and from Theorem 1 n Barndorff-elsen and Shephard 2004b +T T µ 2 1 BV,T,,T σsds 2 d 0, T Y + Pr +T σsds 4. σ 4 s ds The saemens hen come by he same argumen as above. Proof of Proposon 3. Under he null hypohess, he es sasc can be rewren as 2 T =1 Y + Y + 2 ν V,,T, = T 1 T T =1 Y + T =1 Y + Y + Y ν. 17

19 By Theorem A1 n Zhang, ykland and Aï-Sahala 2003, T =1 Y + Y + T ν = O p 1, 2 snce converges n dsrbuon. Thus, asympocally, 2 V,,T, = T 1 T T =1 Y + T =1 Y + Y + ν Y + 4 d 0, 1. Under he alernave hypohess, he sasc can be rearranged as = V,,T, T + 1 T 2 T =1 Y + Y + 2 T ν, ν, T =1 Y + 1 T Y + ν, T =1 Y + T =1 Y + Y + Y ν, oe ha he frs erm of he rgh hand sde of 16 s asympocally normal. As he denomnaor n he second erm of he rgh hand sde of 16 s O p ν,, he saemen hen follows. 18

20 Table 1: ames and Symbols of he companes ncluded n he DJIA Company 3 Company Alcoa Inc. Alra Group, Inc. Amercan Express Co. AT&T Corp. Boeng Co. Caerpllar, Inc. Cgroup Inc. Coca-Cola Co. DuPon E.I. de emours Easman Kodak Co. Exxon oble Corp. General Elecrc Co. General oors Hewle-Packard Co. Home Depo Inc. Honeywell In l. Inc. Inel Corp. Inernaonal Bus. ach. Inernaonal Paper Co. J.P. organ Chase & Co. Johnson & Johnson cdonald Corp. erck & Co.Inc. crosof Corp. Procer & Gamble Co. SBC Communcaons, Inc. Uned Technologes Corp. Wal-ar Sores, Inc. Wal Dsney Co. Tcker Symbol AA O AXP T BA CAT C KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS 19

21 Table 2: Average number of rade quoaons per mnue of DJIA socks Sock Average AA O AXP T BA CAT C KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS

22 Table 3: Resuls of he ess for no mcrosrucure effecs Sock Tes based on Z,,T, Tes based on ZB,,T, #ofrej. %ofrej. #ofrej. %ofrej AA O AXP T BA CAT C KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS

23 Table 4: Resuls of he specfcaon ess for he mcrosrucure nose Sock Tes based on V,,T,, condonal on rejecng H 0 usng Z,,T, Tes based on V,,T,, condonal on rejecng H 0 usng ZB,,T, #ofrej. %ofrej. #ofrej. %ofrej AA O AXP T BA CAT C KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS

24 C AA O AXP TT BA CAT KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS Fgure 1: Plo of he es sasc defned n 9, wh he upper 95% percenle of he sandard normal doed lne 23

25 C AA O AXP TT BA CAT KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS Fgure 2: Plo of he es sasc defned n 11, wh he 95% percenle of he sandard normal doed lne 24

26 C AA O AXP TT BA CAT KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS Fgure 3: Plo of he es sasc defned n 14, condonally on rejecons of he null hypohess n 7 usng RV,,T, wh he 5% percenle of he sandard normal doed lne 25

27 C AA O AXP TT BA CAT KO DD EK XO GE G HPQ HD HO ITC IB IP JP JJ CD RK SFT PG SBC UTX WT DIS Fgure 4: Plo of he es sasc defned n 14, condonally on rejecons of he null hypohess n 7 usng BV,,T, wh he 5% percenle of he sandard normal doed lne 26