Working Paper The dynamics of trading duration, volume and price volatility: A vector MEM model

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Xu, Yongdeng Working Paper The dynamics of rading duraion, volume and price volailiy: A vecor MEM model Cardiff Economics Working Papers, No. E3/7 Provided in Cooperaion wih: Cardiff Business School, Cardiff Universiy Suggesed Ciaion: Xu, Yongdeng (3) : The dynamics of rading duraion, volume and price volailiy: A vecor MEM model, Cardiff Economics Working Papers, No. E3/7 This Version is available a: hp://hdl.handle.ne/49/3 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 Cardiff Economics Working Papers Working Paper No. E3/7 The dynamics of rading duraion, volume and price volailiy a vecor MEM model Yongdeng Xu April 3 Cardiff Business School Aberconway Building Colum Drive Cardiff CF 3EU Unied Kingdom : +44 () f: +44 () business.cardiff.ac.uk This working paper is produced for discussion purpose only. These working papers are expeced o be publishedin due course, in revised form, and should no be quoed or cied wihou he auhor s wrien permission. Cardiff Economics Working Papers are available online from: econpapers.repec.org/paper/cdfwpaper/ and business.cardiff.ac.uk/research/academic-secions/economics/working-papers Enquiries: EconWP@cardiff.ac.uk

3 The dynamics of rading duraion, volume and price volailiy a vecor MEM model Yongdeng Xu // Absrac We propose a general form of vecor Muliplicaive Error Model (MEM) for he dynamics of duraion, volume and price volailiy. The vecor MEM relaxes he wo resricions ofen imposed by previous empirical work in marke microsrucure research, by allowing inerdependence among he variables and relaxing weak exogeneiy resricions. We furher propose a mulivariae lognormal disribuion for he vecor MEM. The model is applied o he rade and quoe daa from he New York Sock Exchange (NYSE). The empirical resuls show ha he vecor MEM capures he dynamics of he rivariae sysem successfully. We find ha imes of greaer aciviy or rades wih larger size coincide wih a higher number of informed raders presen in he marke. Bu we highligh ha i is unexpeced componen of rading duraion or rading volume ha carry he informaion conen. Moreover, our empirical resuls also sugges a significan feedback effec from price process o rading inensiy, while he persisen quoe changes and ransien quoe changes affec rading inensiy in differen direcion, confirming Hasbrouck (988,99). JEL Classificaion: C5, C3, C5 Keywords: Vecor MEM, ACD, GARCH, inraday rading process, duraion, volume, volailiy * Economic Secion, Cardiff Business School, Cardiff Universiy, CF 3EU, UK. xuy6@cardiff.ac.uk.

4 . Inroducion Microsrucure heory generally indicaes ha rading duraion and rading volume convey informaion wih respec o fundamenal asse prices, and reflec he behaviour of financial marke paricipans. Since French and Roll (986) have found evidence ha price volailiy is caused by privae informaion ha affecs prices when informed invesors rade, he empirical sudies on rade and price processes have been based on increasingly on he analysis of he dynamics of rading duraion, volume and price volailiy. However, prior research on his issue is based on a recursive framework, in which he rade and price processes are independen of each oher. In his paper, we exend he recenly developed recursive framework of Engle () and Manganelli (5) for high frequency daa o a vecor MEM model in which he rading duraion, volume and price volailiy are involved simulaneously and are inerdependen. We furher propose a mulivariae lognormal for he disribuion of he vecor model, which allows he innovaion erms o be correlaed conemporaneously. In addiion, maximum likelihood is proposed as a suiable esimaion sraegy. In his way, we can build a sysem ha incorporaes various causal and feedback effecs among hese variables. We also consruc impulse response funcions ha show how he price reacs o a perurbaion of is long-run equilibrium. The mehod is applied o a rade and quoe daase of he NYSE, and he model is esimaed using a sample of en socks. Our empirical resuls are generally consisen wih he previous findings in he empirical microsrucure lieraure (see, for example, Dufour and Engle (), Engle () and Manganelli (5)). Bu our work is novel in wo ways. Firs, we find ha duraion and duraion shocks have a significan impac on price volailiy, while only he unexpeced componens of volume are considered o carry informaion conen Duraion is defined as he ime ha elapses beween wo consecuive ransacions. In general, duraion is considered o reflec he rading sraegy of informed raders or is an indicaor of liquidiy (Easley and O Hara 99), while volume is viewed as an imporan deerminan of he srengh of a marke move and reflecs informaion abou changes in invesors expecaions (Harris and Ravid, 993).

5 wih respec o price. This generally suggess ha i is he unexpeced componens of rading characerisics raher han he rading variables hemselves ha carry informaion conen wih respec o fundamenal asse prices. In addiion, impulse response analysis shows ha ha shocks o duraion or volume are incorporaed appropriaely ino he price wihin one rading day for frequenly raded socks, bu his akes up o one week for infrequenly raded socks. Second, our empirical resuls sugges ha volailiy has a negaive impac on rading inensiy, while volailiy shock has a posiive impac on rading inensiy. We explain his by considering he persisen quoe change (volailiy) o be moivaed by informaion based reason, and ransien quoe change (volailiy shock) o be moivaed by invenory based reason. The resuls confirm Hasbrouck (988,99) s predicion ha persisen quoe changes (volailiy) reduce rading inensiy and ransien quoe changes increase rading inensiy. The remainder of his paper is organized as follows. Secion reviews he relevan lieraure; he heoreical and empirical work on he relaionship of duraion, volume and volailiy are reviewed in his secion. Secion 3 oulines he empirical moivaion and describes he model and mehodology used in he analysis. Secion 4 inroduces he high frequency daa and empirical resuls. Secion 5 concludes he paper.. Lieraure Review Theoreically, he marke microsrucure lieraure explains rading aciviy using wo ypes of model: informaion based and invenory based models. Specifically, rading occurs eiher for informaion moivaed or liquidiy moivaed reasons. Accordingly, predicions of he relaions beween duraion, volume and price volailiy differ. In empirical analysis, he operaion of he marke is cusomarily underaken by using ime-series, high-frequency daa. The dynamics of such posiive-valued variables is generally modelled by a ype of auoregressive condiional duraion (ACD) model (Engle and Russell, 998). In his secion, we firs review he relevan marke microsrucure heory and is predicion of he relaions beween duraion, volume and volailiy. And hen he ACD model of he relevan empirical findings on hese relaionships is reviewed. 3

6 . Review of marke microsrucure heory In he informaion-based model, hree ypes of raders are assumed: informed raders; uninformed raders; and marke makers. Informed raders are usually defined as corporae officers wih privae informaion, while uninformed raders are liquidiy moivaed and simply behave according o heir curren informaion. Marke makers are also assumed o be uninformed. Obviously, he differen raders have asymmeric informaion. Informed raders hope o obain profis from heir informaion so, on average, he marke makers lose ou o he informed raders. Marke makers are specialiss and can access informaion by reading he signals in he marke, such as rading inensiy and volume, and can hus recoup any losses as uninformed raders. Their aciviies are covered by he sequenial rade model (Diamond and Verrecchia, 987; Glosen and Milgrom, 985) and he sraegic rade model ((Admai and Pfleiderer, 988; Easley and O'Hara, 99; Kyle, 985).) In he sequenial rade framework, he marke maker and marke paricipans behave compeiively. Trades ake place sequenially, wih only one rader allowed o ransac a any given poin in ime. Informed raders would like o rade as much (and as ofen) as possible. So he marke maker would quickly (perhaps insanly) adjus prices o reflec his informaion. I is obvious ha rading volume is posiively (perhaps conemporaneously) correlaed wih price volailiy. The sraegic model allows he agens o ac sraegically. For example, in order o make full use of heir privae informaion, he informed raders may conceal heir rading ype by iming heir rades carefully or choosing heir rade sizes (Easley and O'Hara, 99; Kyle, 985). Uninformed raders may also learn by observing he acions of informed raders. In paricular, Admai and Pfleiderer (988) disinguish wo ypes of uninformed raders in addiion o informed raders: non-discreionary raders are similar o liquidiy raders in he previous model; while discreionary raders, while uninformed, rade sraegically. Discreionary raders choose he iming of heir rades. They usually selec he same period of ransacion in an aemp o minimize adverse selecion coss, and informed raders follow he paern inroduced by discreionary raders. In invenory based models, he rading process is effecively moivaed by he marke makers desire o keep heir invenory posiion a some specific level. Based on heir invenory posiion and uncerainy abou order flow, dealers aler heir bid and 4

7 ask prices o elici he desired imbalance of buy and sell orders hereby moderaing deviaions in order flow. The dealer s acion in he marke is simply independen of informaion. I only depends on rading coss, he dealer s previous posiion and ne demand o he dealer (Ho and Soll, 98; O'Hara and Oldfield, 986). These ypes of model generally induce paerns of various rade characerisics, such as iming, price and volume. These facors conain informaion and reflec rade behaviour in he marke.. Predicion from marke microsrucure heory Among he key variables considered, he iming of he rade plays an imporan role. I is ignored iniially, and incorporaed explicily ino marke microsrucure models by Diamond and Verrecchia (987) and Easley and O'Hara (99). Diamond and Verrecchia (987) use a raional expecaions model wih shor-sale consrains. The informed raders acions are driven by he arrival of privae informaion, while uninformed raders are assumed o rade for reasons unrelaed o he arrival of such informaion. If he news is bad, informed raders will wish o sell (or, alernaively, o shor-sell if hey do no own he sock). Given shor-sale consrains, here may be no rade. Therefore, long duraions are associaed wih bad news and should lead an adjusmen of he prices and hence o increase he reurn volailiy. This is summarized as No rade means bad news. Easley and O'Hara (99) provide a differen explanaion for he role of ime. Informed raders only rade when here is new informaion (wheher good or bad) arriving in he marke. So variaions in rading inensiy are closely relaed o he change in he paricipaion rae of informed raders. I follows ha shor rade duraion is a signal ha informed raders are paricipaing in he marke. Consequenly, he marke maker adjuss his/her prices o reflec he increased risk of rading wih informed raders, which reveals a higher volailiy and wider bid ask spreads in he marke. To summarize, No rade means no news. In he sraegic rading assumpion, he informed rader may choose o segmen large volume rades ino a greaer number of smaller-volume, informaion-based rades, and hence conceal heir ype and make full use of privae informaion. I follows ha boh rading inensiy and rading volume may provide informaion concerning he behaviour of marke paricipans. 5

8 A relaionship beween duraion and volailiy is also explained by he model of Admai and Pfleiderer (988). I is assumed ha frequen rading is associaed wih liquidiy raders, and herefore low rading means ha liquidiy (discreionary) raders are inacive, which leaves a high proporion of informed raders in he marke. This again ranslaes ino quick price adjusmen and hence high volailiy. Goodhar and O'Hara (997) ) examine he price effec of rade. Traders may learn over ime from he informaion-based model, and adjus heir speed of rading in reacion o his. For example, a large change in a marke maker s mid-quoe price may be a signal o he informed raders ha heir privae informaion has been revealed o he marke makers, assuming ha no new signal has been released subsequenly. This means ha privae informaion is no longer superior, and herefore he incenive o rade disappears, which decreases rading inensiy. However, from he invenory model perspecive, large quoe changes would immediaely arac opposie-side raders, hus increasing rading inensiy. In addiion, when uninformed raders behave sraegically (O'Hara, 995), i becomes more complex, since he uninformed will increase he probabiliy hey aach o he risk of informed rading when hey observe large absolue reurns or large rading volume. Consequenly, hey will reduce he overall rading inensiy. Hasbrouck (988,99) explains he wo effecs using he shor-run and long-run characerisics of rading behaviour. The privae informaion is persisen and long-lived; he persisen quoe change is relaed o privae informaion, and should have a negaive impac on rading inensiy. The invenory level in saionary and invenory conrol is inherenly a ransien concern, he ransien quoe change is relaed o invenory conrol, and has a posiive impac on rading inensiy. Table summarizes he relaed marke microsrucure lieraure and is predicions. 6

9 Table : Summary of he relaed marke microsrucure lieraure Model Auhors and year Main feaure Predicions Informaion-based model Sequenial rade model Glosen and Milgrom (985) All agens ac compeiively Volume is posiive correlaed wih volailiy Invenory-based model Sraegic rade model Diamond and Verrecchia (987) Kyle (985) Easley and O Hara (99) Admai and Pfleiderer (988) Parlour (998) Ho and Soll (98) O Hara and Oldfield (986) Hasbrouck (99) Shor sale consrains Incorporaing ime Informed raders ac sraegically Long-lived informaion Incorporaing ime Uninformed raders also ac sraegically Shor-lived informaion Raional expecaions Marke makers use price o balance heir invenory No rade means bad news (duraion is correlaed posiively wih volailiy) No rade means no news (duraion is correlaed negaively wih volailiy) Trade inensiy increases, he informaiveness of rades decreases Large quoe change is a risk of informed rading; liquidiy raders may leave or slow down rading aciviy Large quoe changes arac opposie-side raders, hus increasing rading inensiy 7

10 .3 Empirical sudies Empirically invesigaion of marke microsrucure predicions is subjec o he availabiliy of high-frequency ransacion daa. Saisically speaking, high-frequency daa are realizaions of poin processes; ha is, he arrival of he observaions is random. This, joinly wih oher unique feaures of financial daa (long memory; srong skewness; and kurosis) implies ha new mehods and new economeric models are needed. I was firs addressed, by Engle and Russell (998) in he conex of an ACD model for he dynamics of ransacion ime. I represens he ime duraion as produc of a (auoregressive) scale facor and non-negaive valued random process. In he ACD framework, he rade characerisics associaed wih ime are incorporaed and modelled simulaneously, so ha he marke microsrucure predicions can be evaluaed a he ransacion level. Among hem, Engle () proposed a recursive framework o represen he dynamics of duraion and volailiy. The join densiy of duraion and volailiy is expressed as he produc of he marginal densiy of he duraion imes and he condiional densiy of he volailiy, given he duraion. The resul provides evidence of he bad-news effec of long duraions, which is he reverse of he Diamond and Verrecchia (987) resul. The recursive framework of Engle () reduces he complexiy of he model, since each process is esimaed separaely, and used widely by laer empirical works. Engle and Sun (7) model he join densiy of he duraion and he ick-by-ick reurns wihin a recursive framework. They build an economeric model for esimaing he volailiy of he unobserved efficien price change. Using his model, i is easy o forecas he volailiy of reurns over an arbirary ime inerval hrough simulaion using all he observaions available. Taylor (4) models fuure marke rading duraion using various augmenaions of he basic ACD model, and confirms ha bid ask spread and ransacion volume have a significan impac on he subsequen rading inensiy. Manganelli (5) noes ha oher high-frequency daa (rading volume, bid ask spread) share similar characerisics o duraion (for example, hey are posiive-valued and persisenly clusered over ime), so ha heir dynamics can be represened using he same auoregressive process. He incorporaes he rading volume ino Engle () s model and develops a framework o model joinly duraion, volume and price volailiy. Following Engle (), he join disribuion of duraion, volume and 8

11 volailiy is decomposed ino he produc of he marginal disribuion of duraion; he marginal disribuion of volume, given duraion; and he condiional disribuion of volailiy, given duraion and volume. Furher assumpions of weak exogeneiy are made, such as ha he hree processes are independen so hey can be esimaed separaely. Manganelli (5) sudies he causal and feedback effecs among he hree variables and found ha imes of greaer aciviy coincided wih a larger fracion of informed raders being presen in he marke. However, his empirical resuls sugges ha lagged volailiy increases rading inensiy, which is in conras o Easley and O'Hara (99), bu confirms he invenory based model predicions ha large reurns arac opposie side raders and increase rading inensiy. Grammig and Wellner () noiced ha duraion and volailiy migh be inerdependen. They have exended Engle () s recursive model by formulaing an inerdependen inraday duraion and volailiy model. In his model, condiional volailiy and inraday duraion evolve simulaneously. The condiional volailiy is formulaed as a generalized auoregressive condiional heeroskedasiciy (GARCH) process, wih ime-varying parameers ha are funcions of he expeced inerday duraion. Their empirical resuls show ha lagged volailiy significanly reduces ransacion inensiy, which is consisen wih Easley and O'Hara (99). Hausch (8) analyses he reurn volailiy, rade size and rading duraion under he Mulivariae Error Model (MEM) 3 framework. Raher han using ransacion daa, Hausch (8) uses he cumulaed five-minue daa and focuses on he sudy of he underlying common facors ha joinly drive he rading processes. He finds ha he common facor capures mos causal relaions and cross-dependencies beween he individual variables. The exisence of common facors is also an indicaor of he inerdependence of he hree processes. In addiional o he ACD framework, he vecor auoregressive (VAR) model is used in he sudy of high frequency daa. For example, Bowe e al. (9) used a rivariae VAR model o analyse he inerrelaionship beween rading volume, duraion and price volailiy, which is similar o Dufour and Engle (). Bu i is also similar o he recursive model and assumes ha rade and price processes are cross-independen. Using he daa from an emerging fuures marke, hey find ha 3 MEM is an exension of ACD model. 9

12 duraion is affeced posiively by volailiy, which is consisen wih Diamond and Verrecchia (987). To summarize he empirical sudies, he recursive frameworks are generally adoped for he analysis of high frequency daa, bu his is challenged by some empirical evidence. The empirical resuls wih respec o he relaions of rade and price process as are parially conradicory and here is no uniform conclusion a presen. 3. Mehodology In his secion, we firs specify he dynamics of duraion volume and price volailiy according o he Engle () and Manganelli (5) recursive framework and discuss he saisic and economic concerns wih his framework. We hen exend he recursive framework of Engle () and Manganelli (5) o a vecor specificaion in which rading duraion, volume and price volailiy evolve simulaneously and are inerdependen. 3. Duraion, volume and price volailiy --- a recursive framework Define{ d, v, r },,, T as he hree-dimensional ime series associaed wih inraday rading duraion, rading volume and he reurn process, respecively. In paricular, duraion is defined as he ime elapsing beween consecuive rades, volume is he rade size associaed wih each ransacion and reurn is measured as he mid-quoe change. The rivariae rading process - duraion, volume and reurn volailiy - can be modelled as follows: { d, v, r } f ( d, v, r ; ) () where denoes he informaion available up o period, and is a vecor incorporaing he parameers of ineres. In he recursive model (Manganelli, 5), he join disribuion is decomposed ino he produc of hree componens: marginal densiy of duraions, he condiional densiy of volumes given duraions and he condiional densiy of he reurn volailiy given duraions and volumes. Specially, { d, v, r } g( d ; ) h( v d, ; ) k( r d, v, ; ). () d v r

13 For he dynamics of such a non-negaive valued financial poin process, Engle and Russell (998) firs propose an ACD specificaion for financial duraion. They model duraion as he produc of is condiional expecaion and he non-negaive suppored innovaion erm, d ( d; ) u, u ~ i. i. d.(, u ). (3) The ACD model is furher characerized by he assumpions ha he condiional duraion follows a GARCH-ype process and he innovaions are independenly and idenically disribued. The base (,) specificaion of is: d. (4) The logarihmic version is also specified (Bauwens and Gio, ) o ensure posiiviy of he condiional duraion, log log d log. (5) To close he model, he parameric densiy funcion for he innovaions is needed. Engle and Russell (998) iniially consider he exponenial and Weibull disribuion, which is exended laer by Grammig and Maurer (), Allen e al. (9) and Xu (a), offering more flexible densiy and hazard funcions. Following he ACD model, Manganelli (5) considers similar specificaions for volume and volailiy. Then he rivariae sysem has he following specificaions: d u u i i d ( d ; ), ~...(, u ) v d i i d ( v;, ), ~...(, ) rˆ h ( ; d, v, ), ~ i. i. d.(,) r or rˆ h ( ; d, v, ), ~ i. i. d.(, ) r (6) where r is he proxy for volailiy 4, (,, ) ˆ h are he condiional expecaions of duraion, volume and volailiy, respecively, and, s (,,..., ) is a vecor of s parameers of ineres. Manganelli (5) considers he univariae exponenial disribuion for he innovaions in his specificaion. To capure he causal and feedback effec among hese variables, he specifies he following firs order auoregressive condiional model: 4 In order o obain a price change sequence which is free of he bid-ask bounce ha affecs price, we follow Ghysels, e al. (998) and rˆ is obained as he residuals of an ARMA(,) process of reurn series. See also in Hausch (8). One advanage of using rˆ is ha i avoids he problem of exac zero values in r.

14 w ( a d a v a rˆ ) ( b b b h ), 3 3 w ( a d a v a rˆ ) ( b b b h ) a d, 3 3 h w ( a d a v a rˆ ) ( b b b h ) a d a v (7) Under he resricions of weak exogeneiy ( bij for i j ) and independence of he innovaions erms, he hree componens are esimaed separaely. This approach is generally adoped in he exising empirical lieraure (see, for example, Engle (), Dufour and Engle (), Manganelli (5) and Engle and Sun (7)). 3. Economeric concerns Following Manganelli (5), here are wo concerns regarding he recursive model. Firs, i assumes ha he specific processes are independen. To incorporae he conemporaneous informaion, Manganelli (5) specifies causaliy from duraion o volume and from duraion and volume o price volailiy. However, modelling he disribuion of price as being condiional on duraion and volume is jus one sraegy o obain heir join disribuion. As poined ou by Engle and Sun (7), i is also possible o go from he price process and model duraion condiional on is conemporaneous reurn. Theoreically, variaion in duraion and variaion in he price process would be relaed o he same news evens or he underlying informaion process. Empirical sudies also address his issue. For example, Hausch (8) finds he exisence of a common unobserved componen ha joinly drives he dynamics of he rade and price processes. This common componen explains mos of he causaliy beween he rade and he price processes, even if he conemporaneous effec of he rade variable on he price variable is conrolled. We esed his resricion in our previous paper (Xu, b) and show he exisence of cross-dependence beween he rading and price process. Therefore, he advisable approach is o allow he innovaion erms o be conemporaneous correlaed, and specify a vecor form for he dynamics of he rivariae sysem. Second, Manganelli (5) assumes weak exogeneiy, which means he condiional expecaion of one variable is a funcion only of is own pas condiional expecaion, while he pas condiional expecaions of oher variables are no aken ino consideraion. This sraegy has been adoped by mos empirical microsrucure papers (see, for example, Dufour and Engle ()). However, we argue ha his assumpion is oo resricive. When sudying he price impac of rade, various

15 specificaions of duraion and volume should be considered. For example, rade innovaion is an exclusive a manifesaion of he privae informaion of he informed rader. Engle () and Wuensche e al. (7) argue ha i is he unexpeced componens of he rade process ha carry informaional conen wih respec o he fundamenal asse price, since price change is unpredicable. And he same happens for he feedback effecs from price o rading inensiy. For example, Grammig and Wellner () find ha expeced volailiy and volailiy shocks have significan effecs on rading inensiy. Manganelli (5) conducs a robusness es on his resricion. Specifically, he regresses he residuals of he hree equaions agains pas condiional expecaions of oher variables. The resuls indicae ha he coefficiens of pas expeced variables are almos never significan, and hus he recursive model is correcly specified. However, he robusness check migh be misleading, since he dynamics of expeced variables have been disored when esimaing and predicing he expeced variables using recursive models. I is also shown by Grammig and Maurer () in a simulaion sudy ha he misspecificaion of he condiional mean has severe consequences for he expecaion of condiional duraion. We herefore exend he recursive model ino a vecor form, by allowing he hree processes o be inerdependen and relaxing weak exogeneiy. 3.3 Vecor MEM Le x ( d, v, r )', (,, h )' and ( u,, )'. Following Cipollini e al. (7); Engle (), we wrie his sysem of equaions as a rivariae vecor muliplicaive error model (MEM). The hree-dimensional vecor MEM for duraion, volume and volailiy is: x (8) where denoes he Hadmard (elemen by elemen) produc; he componens of are process-specific innovaion erms which are assumed o be cross-dependen; and has a mean vecor I wih all componens uniy and a general variance-covariance marix,i.e, ~ DI (, ). The mulivariae specificaion for is: p A x B A z l l l l l l q (9) 3

16 where z is a vecor of predeermined variables. We do no specify recursively he conemporaneous relaionship from duraion o volume and from duraion and volume o volailiy (Manganelli, 5). However, we allow he innovaion erms o be conemporaneously correlaed. By his specificaion, he condiional expecaion of one variable is a funcion no only of is own pas condiional expecaion, bu also of pas condiional expecaions of oher variables. The wo resricions imposed by he recursive model are released. The mean equaion is furher exended o be a logarihmic version o ensure he posiiviy of he individual processes wihou imposing addiional parameer resricions. p ln( ) A ln( x ) B ln( ) A ln( z ) l l l l l l q () The firs wo momen condiions of he vecor MEM are given by: Ex ( ) Var( x ) ' diag( ) diag( ) which is a posiive defined marix by consrucion, as emphasized by Engle (). () 3.4 Specificaion of A compleely parameric formulaion of he vecor MEM requires a full specificaion of he condiional disribuion of. In he ACD lieraure, Engle and Russell (998) iniially consider he exponenial and Weibull disribuion for he error, which is exended laer by Grammig and Maurer () o be a Burr disribuion, by Lunde (999)o be a generalized gamma disribuion, and recenly by Allen e al. (9) and Xu (a) o be a lognormal disribuion. Figure plos he comparison of densiy funcions implied by hese parameric disribuions. I can be seen ha only he exponenial disribuion implies a monoonically decreasing densiy funcion, while he ohers imply hump shaped densiy funcions. Xu (a) ess he specificaion of he duraion disribuions, and finds ha he lognormal ACD model is superior o he Exponenial ACD and Weibull ACD models, while is performance is similar o he Burr or Generalized Gamma ACD models. I is well known ha price volailiy is ypically lognormally disribued, while Andersen e al. () and Cizeau e al. (997), among ohers, also showed ha he lognormal disribuion fied he 4

17 Figure : A comparison of parameric densiy realized volailiy disribuion very well. So we propose o use he mulivariae lognormal disribuion for he MEM. Indeed, he mulivariae lognormal disribuion seems o be he only feasible choice in he specificaion of vecor MEM. I has a closed form condiional densiy funcion, so ha ML esimaion can be conduced. Cipollini e al. (7) consider appropriae mulivariae gamma versions bu find ha he only useful version admis only posiive correlaion, which is oo resricive. The mulivariae lognormal disribuion admis boh posiive and negaive correlaions. Moreover, Allen e al. (8) prove ha he lognormal disribuion is sufficienly flexible o provide a good approximaion o a wide range of non-negaive disribuions, and is also sufficienly accurae so as no o induce unnecessary numerical difficulies. Assume follows a mulivariae lognormal disribuion such ha ln N( v, D) 5. The densiy funcion is: 5 where v i dii o guaranee ha E( ) I 5

18 K k / / f (, D) ( ) D ( i, j exp (ln v)' D (ln v) i () where. The condiional densiy of x is hen: K K / / f ( x, ) ( ) D xi, exp (ln x ln v)' D (ln x ln v) i. (3) where by: where The log likelihood of he model is hen: T T (4) l l ln f ( x, ) K K l ln f ( x, ) ln( ) D ln( x ) i, i (ln ln )' x v D (ln x ln v ) (5) The firs and second momens of he mulivariae lognormal disribuion are given vi d ii E( ) (,,, )' e dii djj ( vivj ) dij d ij E( )( x )' e ( e ) e ij e d ij dii jj ( e )( e ) d k ij i ij vi dii and d ij is he ij h elemen of D. I is clear ha if (ln,ln,,ln ) k are independen, hen (,,, ) are also independen and vice versa. The mulivariae lognormal disribuion allows boh posiive and negaive correlaion, which is much more flexible han he mulivariae gamma disribuion (Cipollini e al., 7). The lognormal belongs o he exponenial family. The parameers are sill consisenly esimaed, even if he chosen densiy is wrong. The asympoic disribuion of he QML esimaor differs from ha of he ML esimaor. The variance-covariance marix is no he inverse of he Fisher informaion. I has he so-called sandwich form. ˆ ˆ ˆ ˆ QML N ( ) N(, I ( ) J( ) I ( )) (6) k 6

19 where ˆ ln L( ; x) I( ) E ˆ ˆ' ˆ, ln ( ˆ; ) ln ( ˆ ˆ L x L ; x) J( ) E ˆ ˆ respecively, he componens of he empirical average Hessian and he empirical average ouer produc of he gradiens evaluaed a he esimaes ˆ. are, 3.5 Impulse response funcion Following he vecor MEM, we can derive he impulse response funcions. We concenrae on he firs order model and exclude he predeermined variables. x, ln Aln x Bln. In he impulse response, we work on he impulse of ln i on he naural logarihmic of he ineresed variable ln x. The impulse responses funcion of he model (7) for is 6 : where ln x ( ) ( ), I A B A B. This process can be rewrien in such a way ha he residuals of differen equaions are uncorrelaed. For his purpose, we choose a decomposiion of he whie noise covariance marix W W ', where is a diagonal marix wih posiive diagonal elemens and W is a lower riangular marix wih uni diagonal. Thus, ln x i i, i iw. i (7) (8) (9) Then he impulse response funcion of he model (7) for is: ln x. () The sandard errors for he impulse response are compued as followings. Le [ ', ', ']' ( p) and vec ( ) d v r. If T ( ˆ ) N(, Q), hen a a T ( ˆ ) N(, G QG '), where G. ' 6 See Appendix 3: Proofs of impulse response funcion 7

20 3.6 Vecor ARMA represenaion One of he advanages of using he lognormal disribuion for he vecor MEM model is ha i has an equivalen Vecor ARMA specificaion wih an innovaion ha follows a mulivariae Gaussian disribuion. From he following log vecor MEM model, p x, () ln( ) A ln( x ) B ln( ) A ln( z ) If we ake logs of (), we obain where e ~ iid N(, ). Then, l l l l l l q. () ln( x ) ln( ) ln( ) c ln( ) e (3) ln( ) ln( x ) c e, (4) q q q q Bl ln( l ) Bl ln( x l ) Bl c Ble l l l l l. (5) Subsiuing ln( ) and q Bl ln( l ) ino Equaion (), i follows ha l where p q q (6) ln( x ) c ( A B )ln( x ) e B e A ln( z ) l l l l l l l l l c c B c. q l l I is ineresing ha he vecor MEM model is equivalen o a VARMA specificaion. In paricular, i provides a good way o adop he VARMA inference 7 o make inferences in he vecor MEM model. 4. Empirical analysis 4. Daa We use he daa from he Trades and Quoes (TAQ) daase a NYSE. The TAQ daa consiss of wo pars: he firs repors he rade daa, while he second liss he 7 See Appendix : Inference of VARMA Models 8

21 Table : Socks used in he analysis A. Frequenly raded B. Infrequenly raded TRN Triniy Indusries ABG Group ABG R Ryder Sysem Inc. OFG Orienal Finl Grp Hold Co. ARG Airgas Inc. LSB LSB Indusries Inc. FMO Federal-Mogul Corp. HTD Huningdon Life S.G. VTS VERITAS DGC INC HUN Hun Corp. quoe daa (bid and ask daa) posed by he marke maker. The daa were kindly provided by Manganelli (5). He consruced deciles of socks covering he period from Jan,998 o June 3, 999, on he basis of he 997 oal number of rades of all socks quoed on he NYSE. We randomly seleced 5 socks from he eighh decile (frequenly raded socks) and 5 from he second decile (infrequenly raded socks) covering he period from Jan,998 o June 3, 999. The ickers and names of he en socks are repored in Table : Before he analysis began, we adoped Manganelli (5) s sraegy o prepare he daa. Firs, all rades before 9:3 am or afer 4: pm were discarded. Second, duraions over nigh were compued as if he overnigh periods did no exis. For example, he ime elapsing beween 5:59:5 and 9:3:5 of he following day is only 5 seconds. We keep overnigh duraion because our samples for infrequenly raded socks are very small. Eliminaing his duraion would cause he loss of imporan daa for hese socks. Third, all ransacion daa wih zero duraion are eliminaed. These ransacions are reaed as one single ransacion, and he relaed volumes are summed. Fourh, o deal wih he impac of dividend paymens and rading hals, we simply deleed he firs observaion whose price incorporaed he dividend paymen or a rading hal. Fifh, o adjus he daa for sock splis, we simply muliplied he price and volume by he sock spli raio. Sixh, he price of each ransacion is calculaed as he average of he prevailing bid and ask quoe. To obain he prevailing quoes, we use he 5 second rule used by Lee and Ready (99) which liniks each rade o he quoe posed a leas 5 seconds before, since he quoes can be posed more quickly han rades are recorded. This procedure is sandard in microsrucure sudies. Sevenh, he reurns were compued as he difference of he log of he prices. To obain a reurn sequence ha is free of he bid-ask bounce ha affecs prices (see Campbell e al., 997, chaper 3), we follow Ghysels e al. (998) in using he residuals of an ARMA(,) model esimaed on he reurn daa. 9

22 Ecpeced Duraion (Seconds) Reurn Square e-6 e-6 3e-6 4e-6 5e-6 6e Figure : Nonparameric esimae of daily paern of ransacion duraion and reurn square. The second issue o be addressed prior o he analysis concerns he inraday paern in he daa. I is well known ha duraion, volume and volailiy exhibi srong inraday periodic componens, wih a high rading aciviy a he beginning and end of he day. To adjus for his, we make use of a mehod used by Engle (). We regress he duraions, volumes and reurns squares on a piecewise cubic spline wih knos a 9:3, :, :, :, 3:, 4:, 5:, 5:3 and 6:. The original series are hen divided by he spline forecas o obain he adjused series. Figure depics he nonparameric esimae of daily paern of duraion and reurn square for one ypical sock ARG. Generally, less frequenly raded socks do no exhibi any regular inraday paern. More frequenly raded socks ypically show he invered U paern for duraion, he L paern for reurn squares, and no regular paern for volume. Hour of Day Hour of Day Table 3 presens some summary saisics for he en socks. For he frequenly raded socks, he number of observaions range from 33,85 o 69,7 in he sample period, and he average rading duraion ranges from 87 seconds o 59 seconds. For he infrequenly raded socks, he number of observaion ranges from,74 o 7, in he sample period, wih he average rading duraion ranging from,5 seconds o 4,5 seconds. The rading volume does no show any difference beween frequenly raded socks and infrequenly raded socks. The number of rading volumes ranges

23 Table 3: Summary saisics for he socks Obs Mean LB() MLB() Duraion Volume Duraion Volume Variance TRN GAS TCB R ARG ABG OFG LSB HNN JNS Noes: LB() denoes Ljung Box saisics for order. MLB() denoes mulivariae Ljung Box saisics. from 833 o 5,95. The mulivariae Ljung Box saisics, compued according o Hosking (98)and is given by MLB( s) : n( n ) race( Cˆ Cˆ Cˆ Cˆ ) ( ks) s j j (7) j n j where k denoes he dimension of he process ( in his case k=3), s is he number of lags aken ino accoun, and C ˆ j is he j h residual auocovariance marix. I is apparen ha duraion, volume and volailiy showsrong serial auocorrelaions, and his is paricularly rue for high frequency raded socks. The large mulivariae Ljung-Box saisics in he able indicae ha he rivairae sysem reveals srong dynamic and conemporaneous dependencies. These indicaors sugges he use of vecor form MEM. We also depic he non-parameric densiy and parameric densiies implied by he exponenial and lognormal disribuions. 8 Figure 3 repors he comparison of parameric and non-parameric densiies for one ypical sock LSB. I can be seen ha he lognormal disribuion fis wih he rue densiy very well for he duraion daa. This resul is consisen wih Xu (a). For volume daa, we are surprised o find he lognormal disribuion has he bes performance. And he raw daa flucuaes closely around he lognormal disribuion. Even for volailiy, he lognormal 8 See Xu (a) and Grammig and Maurer () for he discussion of parameric and non-parameric densiy.

24 Duraion Volume Volailiy Figure 3: A comparison of parameric densiy and non-parameric densiies--lsb

25 disribuion also performs well. For breviy, he oher 9 socks have are no been repored for breviy, bu hese findings are robus across he socks. The daa we use in his paper srongly suppor he mulivariae lognormal MEM model for he dynamics of duraion, volume and price volailiy. 4. Empirical model In he empirical analysis, we are ineresed in he causal and feedback effecs among he variables. In conras o he previous recursive model, we allow rade duraion, volume and innovaions of hese variables o affec price volailiy and vice versa: he volailiy and volailiy shocks are allowed o affec rading inensiy. So we specify and esimae he following vecor MEM: x, ~ DI (, ) x ln ln ln ln A x B C where B is a diagonal marix and C is a marix where he diagonal elemens are zero. Then, a 3( a 3 ) measures he impac of duraion (volume) on price volailiy, c 3 ( c 3 ) measures he impac of duraion (volume) shocks on price volailiy, a3 measures he impac of volailiy on rading inensiy and c3 measures he impac of volailiy shocks on rading inensiy. The esimaion resuls and various diagnosics for he five frequenly raded socks are repored in Table 4 and resuls for he five infrequenly raded socks are repored in Noe: LL denoes Log likelihood funcion. BIC denoes Bayes Informaion Crierion. LB denoes Ljung-Box saisics of flied residuals and MLB denoes. mulivariae Ljung-Box saisic. The Ljung-Box saisics are compued based on lags. Criical values (6).5 =.59, (6). =6.8 (8) 3

26 Table 5. Considering he diagnosic saisics of he model, hese sugges ha he vecor MEM improves he dynamic properies of he model significanly, as we can see from he sharp drop in he Ljung-Box saisics. This is paricularly rue for he volailiy process. Moreover, he vecor MEM reduces he mulivariae Ljung-Box saisic significanly, indicaing ha he vecor MEM does a good job in capuring he mulivariae dynamics and inerdependencies beween he individual processes. For frequenly raded socks, he dynamics of he sysem are sill no capured compleely by he model. Bu his is commonly he case wih such large ime series (see, for example, Engle ()). For infrequenly raded socks, he dynamics of he sysem are capured compleely by he vecor MEM. 4

27 Table 4: Esimaion resuls and diagnosics: frequenly raded socks. ARG TRN TCB GAS R.6***.89.55*** **.8***.***.6***.68.***.7***.5*** **.3** -.3** *** *** -.7*** -.** *** -.4*** -.387*** *** ***.37*** ***.4***.78***.36*** b.939***.73***.94***.74***.9*** b.58***.638***.695***.76***.66** b.39***.3***.75***.46***.69*** 33 c -.7*** -.33*** -.*** -.338*** -.65 c -.3*** -.7*** -.35*** -.4*** c ** -.3*** c.4.5**.8***.7***.9*** 3 c ***.5*** *** 3 c3.69***.353***.6***.474***.88** LR es 9 H : c, i j ij Diagnosics LL BIC MLB 565.8*** 99.6** 8*** 684.3*** 86 LB _ d.4*** 36.3** 4.*** 5.8*** 6.37*** LB _ v 95.97*** 84.*** 8.3*** 83.37*** 355.9*** LB _ rˆ 74.3*** 38.7*** 457.*** 9.9*** 7.3*** Noe: LL denoes Log likelihood funcion. BIC denoes Bayes Informaion Crierion. LB denoes Ljung-Box saisics of flied residuals and MLB denoes. mulivariae Ljung-Box saisic. The Ljung-Box saisics are compued based on lags. Criical values (6).5 =.59, (6). =6.8 9 We esimae five differen vecor MEMs for comparison. The resuls have no repored for breviy. LR es is based on he likelihood values of resriced and unresriced models. 5

28 Table 5: Esimaion resuls and diagnosics: infrequenly raded socks. ABG HTD LSB HUN FEP.4***.9**.3.56***.56*** -.79***.5*** ***.8*** -.4.3** *** -.564*** *** **.85*** -.79*** -.9** *** *** *** *** -.48*** *** -.9** *** b.9***.98***.967***.93***.9*** b.366***.9*** ***.643*** b.68***.665***.5***.67***.38** 33 c.8*** -.6*** *** -.3*** c.4*** -.38***.6 -.8** c -.34***.549*** c.79***.95**.3.8***.49*** 3 c -.4*** *** -.6*** 3 c.99***.663***.389.7**.58** 3 H : c, i j ij LR es Diagnosics LL BIC MLB.3** *** LB _ d 36.47** 3.8** 48.37** LB _ v *** LB _ rˆ ** *** Noe: LL denoes Log likelihood funcion. BIC denoes Bayes Informaion Crierion. LB denoes Ljung-Box saisics of flied residuals and MLB denoes. mulivariae Ljung-Box saisic. The Ljung-Box saisics are compued based on lags. Criical values (6).5 =.59, (6). =6.8 6

29 In Manganelli(5) s recursive model, he assumpion of weak exogeneiy is made in he specificaion of he condiional mean. The pas expeced variables are assumed no o carry any informaion ( cij ). Manganelli (5) and Dufour and Engle () also conduc robusness ess of his resricion, in which he residuals of he hree models are regressed agains lagged expeced variables. They find ha he lagged expeced variables are insignifican. However, we find ha he mos lagged expeced variables are significan ( c ij ) in our vecor MEMs. I is paricularly rue for infrequenly raded socks. The LR ess also sugges ha he lagged expeced variables are joinly significan in almos all cases. We argue ha he robusness checks conduced by Manganelli (5) and Dufour and Engle () are misleading, since he dynamics of expeced variables has been disored by he marginal model. Therefore, he weak exogeneiy assumpion is no suppored by he empirical daa. The lagged expeced variables should be incorporaed in his rivariae sysem. 4.3 Empirical resuls Looking firs a he price volailiy ( h ) process. The coefficien of duraion ( a 3 ) and coefficien of duraion shocks ( c 3 ) in he volailiy equaion are negaive and significan in mos cases. This is consisen wih Easley and O'Hara (99), indicaing ha rades wih shor duraion or he shocks of rading inensiy are relaed o he arriving of new informaion, which reveals a higher volailiy impac. The implici applicaion is ha marke makers will associae he higher rading aciviy or rading aciviy ha is higher han is expeced level as a signal of informed rading. The volume coefficien ( a 3 ) is only significan for 4 ou of socks and he sign is unclear. However, he volume shocks coefficien ( c 3 ) are all significan and posiive. This implies ha he unexpeced componen of volume raher han he raw volume carry informaion. Implicily, marke makers will only consider rade size ha is larger han is expeced level as a signal of privae informaion, and adjus bid-ask price accordingly. The expeced large rade size is simply for liquidiy reason. The resuls parly suppor he predicion from Easley and O'Hara (987,99). This exercise of he price impac of rade is novel in wo aspecs. Firs, mos empirical marke microsrucure lieraure (see, for example,dufour and Engle () and Manganelli (5)) uses raw duraion (volume) o deermine he presence of 7

30 informed raders in he marke. We highligh ha i is he unexpeced componens of rade ha carry informaion wih respec o asse prices. Second, in conras o Manganelli (5), our findings are generally robus for less frequenly socks. There is no reason why he informed raders should avoid aking advanage of heir privae informaion if i is relaed o infrequenly raded socks. The srikingly differen resuls, wih respec o he feedback effecs from he price process o rading inensiy, are found in he duraion equaion. For he frequenly raded socks, he coefficien of volailiy ( a 3 ) is always posiive bu significan for 3 ou of 5 socks and he coefficiens on volailiy innovaion ( c 3 ) is always negaive bu significan for 4 ou of 5 socks. Following Hasbrouck (988,99), we explain his by considering he persisen quoe change (volailiy) o be informaion moivaed and ransien quoe change (volailiy shock) o be invenory moivaed. Then our resuls are consisen wih microsrucure predicions. For example, informaion moivaed large absolue quoe changes indicae a risk of informed rading and he liquidiy raders may leave or slow down he rading aciviy o avoid adverse selecion(admai and Pfleiderer, 988; Easley and O'Hara, 99), while invenory moivaed large quoe changes may arac opposie side raders and increase rading inensiy. Similar resuls can be found for infrequenly raded socks, bu he effecs are less significan. In he exising empirical microsrucure lieraure, Dufour and Engle () and Manganelli (5) find ha shor duraions follow large reurns, while Grammig and Wellner () find ha lagged volailiy significanly reduces rade inensiy. Our findings enhance he exising lieraure by incorporaing boh of hese effecs in one model. 4.4 Impulse Response Analysis From he esimaes of he MEM in equaion (8), we generae he impulse responses which race he effec of a one ime shock o one of he innovaions on he fuure values of he endogenous variables. The impulse response funcion for wo represenaive socks TRN and ABG are ploed in Figure 4 and Figure 5. This gives he effecs of a variaion on he forecas up o he h rade. Since he impulse-response funcions are ploed in ransacion ime, hey are no direcly comparable among differen socks. We use he Manganelli (5) mehod o 8

31 Figure 4: Impulse response funcion for TRN Figure 5: Impulse response funcion for ABG 9

32 approximae he calendar ime he sysem akes o reurn o is long-run equilibrium, by muliplying he number of ransacions by heir average duraion. The average duraion per rade of he wo represenaive socks is 58 seconds for TRN and 45 seconds for ABG. This implies, for example, ha a shock o he duraion of TRN is absorbed by he expeced duraion afer abou 7 rades, or, on average, afer. hours. In he case of ABG, he same shock is absorbed afer 54 ransacions, which corresponds, on average, o a period of 63.3 hours. Similar resuls hold for he oher impulse-responses, indicaing ha he more raded he sock, he faser he marke reurns o is full informaion equilibrium afer an iniial perurbaion. In paricular, his is consisen wih he (plausible) assumpion ha he more frequenly raded he sock he higher he number of informed raders. Table 6 summarizes he resuls for he oher socks, confirming ha he price volailiy of frequenly raded socks converges much faser o is long-run equilibrium afer an iniial perurbaion. In general, for frequenly raded socks, he new informaion is implicily incorporaed in he price wihin one rading day, while i akes up o a week for he new informaion o be included ino he price for infrequenly raded socked. Overall, he effec is o sugges ha he marke is reasonably efficien. This resul, in conras o Kyle (985), confirms Admai and Pfleiderer (988) and Holden and Subrahmanyam (99) s finding ha informaion is shor lived. For example, Holden and Subrahmanyam (99) show ha wih muliple informed raders here will be more aggressive rading in he early periods, causing more informaion o be revealed earlier in he process. Table 6: Time (in hours) i akes o absorb shocks o he long erm equilibrium variances ARG TRN TCB GAS R Shock o duraion Shock o volume Shock o price volailiy ABG HTD LSB HUN FEP Shock o duraion Shock o volume Shock o price volailiy The hreshold a which he shock producing he impulse response is assumed o be absorbed is a e-7 for shocks. Tha is, Table 7 repors he ime i akes for he impulse response of he variance o fall below e-7. 3