The dynamics of trading duration, volume and price volatility a vector MEM model. Yongdeng Xu * July Abstract

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1 The dynamics of rading duraion, volume and price volailiy a vecor MEM model Yongdeng Xu * July 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 he innovaion erms o be cross-dependen and relaxing weak exogeneiy resricions. We furher propose a mulivariae lognormal disribuion for he disribuion of he vecor MEM. Moivaed by he equivalence of vecor MEM and VARMA specificaion, he model can be esimaed by maximum likelihood esimaion. The empirical resuls show ha he vecor MEM successfully capures he dynamics of he rivariae sysem. We find ha ime of greaer aciviy coincides wih a higher number of informed raders presen in he marke, bu i is only valid for he frequenly raded socks. Moreover, he lagged expeced variables are significan in he MEM, challenging he weak exogeneiy assumpions blihely used in he empirical marke microsrucure lieraure. Our findings shed new ligh on microsrucure research, indicaing ha i is unexpeced componens of rading characerisics ha carry informaion conen of he price process and he acual and unexpeced componen of volailiy affecs duraion in differen direcion, confirming Hasbrouck (988,99). JEL Classificaion: C5, C3, C5 Key Words: Muliplicaive error model, vecor MEM model, ACD, GARCH, VARMA, inraday rading process * Economic Secion, Cardiff Business School, Cardiff Universiy, UK. xuy6@cardiff.ac.uk Phone: +44()

2 Inroducion The relaionship beween price and rade processes is criical for financial marke microsrucure sudies. Microsrucure heory generally explains rading aciviy in financial markes as eiher informaion based or liquidiy based. I indicaes boh rading volume and rading duraion conveys informaion of fundamenal asse price and reflecs he behaviour financial marke paricipans. In general, duraion is considered o reflec he rading sraegy of informed rader or is an indicaor of liquidiy (Easley and O'Hara 99) and volume is viewed as an imporan deerminan of he srengh of a marke move and reflecs informaion abou changes in invesors expecaions (Harris and Raviv 993). The empirical sudies on microsrucure are hereby increasingly based on he analysis of he dynamics of rading duraion, volume and price volailiy. However, prior research is limied on his issue; i eiher focuses on he impac of one or wo specific variables (Dufour and Engle ) or incorporaes he combined effec in a marginal disribuion framework (Manganelli 5). Therefore, simulaneous modelling of rading duraion, volume and price volailiy in a vecor framework should provide more reliable evidence regarding he inerpreaion of he full dynamic inerdependence among hem. This is he approach underaken in his paper. This paper conribues wofold. Firsly, we propose a more general form of vecor Muliplicaive Error Model (MEM) for he dynamics of duraion, volume and volailiy model and a join esimaion mehod. When modelling several posiive-valued indicaors joinly in as in a vecor (mulivariae) framework, prior research ofen make wo assumpions, saying he weak exogeneiy and he independence of he innovaion erms. Then he join disribuion can be expressed as he produc of marginal disribuion of one or wo indicaors and he condiional disribuion of anoher indicaor. And he opimizaion can be done equaion by equaion. This is he sraegy adoped by Manganelli (5) in a marginal disribuion model of duraion, volume and price volailiy. The equaion by equaion esimaion is very simple and gives consisen esimaors based on he assumpions of weak exogeneiy and he independence of he innovaion erms. However, recen empirical resuls show a dynamic inerdependence beween hese indicaors, suggesing ha a Duraion is defined as he ime elapses beween wo consecuive ransacion

3 vecor approach is advisable o gain beer saisic inferences and more reliable inerpreaion of heir dynamics. In his paper, we release he weak exogeneiy assumpion and specify a more general form of he vecor MEM, in which 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. We furher propose a mulivariae lognormal for he disribuion of he vecor MEM, which allows he innovaions erms o be conemporaneously correlaed. Moivaed by he equivalence of vecor MEM and VARMA specificaion, he model can be esimaed by maximum likelihood esimaion. Secondly in empirical analysis, we use Trade and Quoe daa from NYSE o model he dynamics of duraion, volume and price volailiy in a vecor MEM model. Our empirical resuls suppor he predicion from informaion based model ha he lagged volume posiively affecs he price volailiy. We find ha a ime of greaer aciviy coincides wih a higher number of informed raders presen in he marke, bu i only valid for he frequenly raded socks. Moreover, he pas expeced variables are significan in he vecor MEM model, challenging he weak exogeneiy assumpions blihely used in he empirical marke microsrucure lieraure. Our findings shed new ligh on microsrucure research and indicae ha boh duraion and duraion shocks carry price informaion, while only unexpeced componen carried mos of volume relaed price informaion. We also find he acual and unexpeced componen of volailiy affecs duraion in differen direcions. This finding confirms Hasbrouck (988,99) s predicion ha uninformed raders decrease heir rading inensiy when hey observe larger reurn. The unexpeced change in price is invenory based reason, which will arac opposie side raders and increase rading inensiy. The remainder of his paper is organized as follows. Secion reviews he relevan lieraure. Boh he heoreical and empirical work on he relaionship of duraion, volume and volailiy are reviewed is 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 he empirical resuls. Secion 5 is he conclusion. 3

4 Lieraure Reviews 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 relaion beween duraion, volume and price volailiy differ. In empirical analysis, he operaion of he marke is cusomarily underaken uilising ime series high frequency daa and he dynamics of such posiive-valued variables is generally modelled by a ype of Muliplicaive Error Model (MEM) (Engle ). In his secion, we firs review he relevan marke microsrucure heory and is predicion of he relaions beween duraion, volume and volailiy. And hen we review he MEM model of he relevan empirical findings on hese relaionships.. Review on marke microsrucure heory.. Informaion based model 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, differen raders have asymmeric informaion. Informed raders hope o obain profis from heir informaion. The marke maker loses ou o informed raders on average. Marke makers are specialiss and can access informaion by learning he signals in he marke such as rading inensiy and volumes, and can hus recoup heir losses on uninformed raders. Their aciviies are encompassed in he sequenial rade model (Glosen and Milgrom 985; Diamond and Verrecchia 987) and sraegic rade model (Kyle 985; Admai and Pfleiderer 988; Easley and O'Hara 99). In he sequenial rade framework, he marke maker and marke paricipans ac compeiively. Trade akes place sequenially, wih one rader allowed o ransac a any poin in ime. Informed raders would like o rade as much (and as ofen) as possible. The marke maker would quickly (perhaps insanly) adjus prices o reflec his informaion. I is obvious ha rading volume is posiively (perhaps 4

5 conemporaneously) correlaed wih price volailiy. The sraegic model allows he agens o ac sraegically. For example, in order o fully make use of heir privae informaion, he informed raders may conceal heir ype by iming heir rades or choosing heir rade size (Kyle 985; Easley and O'Hara 99). Uninformed raders may also learn by observing he acions of informed raders. Paricularly, 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 where discreionary raders, while uniformed, rade sraegically. The discreionary raders choose he iming of heir rades. They usually selec he same period of ransacion in order o minimize he adverse selecion cos and informed raders follow he paern inroduced by discreionary raders. These ypes of model generally induce paerns of various rade characerisics, such as iming, price, and volume. These facors conain informaion and reflec rade behavior in he marke... Invenory based models. 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 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)...3 Predicion from marke microsrucure heory Among he key variables considered, he iming of he rade also plays an imporan role. I is ignored iniially and explicily incorporaed ino marke microsrucure models by Diamond and Verrecchia (987) and Easley and O'Hara (99). However, he role of ime and is relaionship wih oher facors is conradicory in he lieraure. Diamond and Verrecchia (987) use a raional expecaions model wih shor sale consrains. The informed rader s acions are driven by he arrival of privae informaion, while uniformed raders are assumed o rade for reasons unrelaed o he arrival of informaion. If he news is bad, informed raders will wish o sell (or alernaively shor sell i if hey do no own he sock). Given shor-sale consrains 5

6 here may be no rade. Therefore, long duraions are associaed wih bad news and should lead o adjusing of he prices and hence increase he reurn volailiy. I 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 he variaion in rading inensiy is closely relaed o he change in he paricipaion rae of informed raders. I follows ha shor rade duraion is a signal of informed raders paricipaing in he marke. Consequenly, he marke maker adjuss his 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 larger number of smaller volume informaion-based rades, and hence conceal heir ype and ge full use of privae informaion. I follows ha boh rading inensiy, and rading volume may provide informaion, concerning he behaviors of marke paricipans. In his regard, he implici effec of volume, which we underake in his paper, is incorporaed. A relaion 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. From he informaion based model, raders may learn over ime 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 hereafer. This means ha privae informaion is no longer superior. Therefore, he incenive o rade disappears, which decreases he rading inensiy. However, from he invenory model perspecive, large quoe changes would arac opposie-side raders immediaely, hus increasing he rading inensiy. In addiion, when uniformed 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 6

7 will reduce he rading inensiy. The overall effec on rading inensiy is herefore unclear. 7

8 Table : Summary of he relaed marke microsrucure lieraure Model Auhors and year Main feaure Predicion 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) Shor sale consrains, Incorporaing ime Informed raders ac sraegically. Incorporaing ime Uninformed raders also ac sraegically. Raional expecaions Marke makers use price o balance heir invenory No rade means bad news (Duraion is posiive correlaed wih volailiy) No rade means no news (Duraion is negaively correlaed wih volailiy) Trade inensiy increases, he informaiveness of rades decreases (Duraion is posiive correlaes wih volailiy) Large quoe change increases rading inensiy. 8

9 . Review on Muliplicaive Error models.. Univariae MEM/ACD models and he relaionship beween duraion and volailiy In he economeric modelling of he dynamics of such posiive-valued variables, he so-called Muliplicaive Error Model (MEM) has been proposed, iniially by Engle (). The basic idea is o model he posiive-valued indicaor in erms of he produc of a (condiional auoregressive) scale facor and an innovaion process wih nonnegaive suppor (i.e. GARCH-like process). The well known univariae MEM model is he auoregressive condiional duraion (ACD) model of Engle and Russell (998) for financial duraions. In he ACD framework, he rade characerisics associaed wih ime can be incorporaed and modelled a he same ime, so ha he marke microsrucure predicion can be evaluaed a a ransacion level. Among hem, Engle () proposed an ACD-GARCH specificaion o represen he dynamics of duraion and volailiy. The join densiy 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. Grammig and Wellner () exend he Engle () model by formulaing a model o allow he inerdependence of inraday volailiy and rade duraion. In his model, condiional volailiy and inraday duraion evolve simulaneously. The condiional volailiy is formulaed as a GARCH process wih ime-varying parameers ha are funcions of he expeced iner-day duraion. Their empirical resuls shows ha lagged volailiy significanly reduces ransacion inensiy, which is consisen wih Easley and O'Hara (99). In Engle and Sun (7), hey build an economeric model for esimaing he volailiy of he unobserved efficien price change. They model he join densiy of he duraion and he ick by ick reurns wihin an ACD-GARCH framework. Using his model, i is easy o forecas he volailiy of reurns over an arbirary ime inerval hrough simulaion using all he observaions available. 9

10 .. Vecor MEM models and he relaionship beween duraion, volume and price volailiy Alhough he amoun of research on he ACD model and he relaionship beween duraion and volailiy is vas, few papers address he join disribuion of duraion, volume and volailiy and heir relaionship. Manganelli (5) develops an economeric framework o model he duraion, volume and volailiy joinly in a vecor MEM model. The vecor MEM is a sraighforward exension of he univariae linear MEM/ACD model. The join disribuion of duraion, volume and 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 ha he hree processes are independen so hey can be esimaed separaely. The resuling model belongs o he class of economeric reduced form models. Manganelli (5) sudied boh he effec of pas rades on curren duraion and ha of pas duraion on curren rade and found ha imes of greaer aciviy coincide wih a larger fracion of informed raders presen in he marke. Hausch (8) analyses he reurn volailiy, rade size and rading duraion under he Mulivariae Error Model (MEM) framework. Raher han allowing for cross-dependen innovaions as in he vecor MEM model, he assumes ha he innovaions are independen bu are driven by a common laen auoregressive facor a he same ime. This model can accoun for (informaion) driven facor dynamics as well as idiosyncraic effecs in he processes of rading duraion, volume and price volailiy. He finds ha he common facor capures mos causal relaions and cross-dependencies beween he individual variables. This suggess a more parsimonious specificaion of high-dimensional rading processes. In addiional o he vecor MEM, Bowe, Hyde e al. (9) use a rivariae VAR model o analyse he inerrelaionship beween he rading volume, duraion and price volailiy, which is similar o Dufour and Engle (). Their model also assumes he rade and price processes are cross-independen. Using he daa from an emerging fuures marke, hey find duraion is posiively affeced by volailiy. Their resuls do no suppor he mixure of disribuion hypohesis (MDH). Insead, hey sugges volailiy and conemporaneous volume are negaively correlaed. Their resuls sugges ha high (low) volume rading periods are due o he involvemen of

11 uninformed (informed) raders in conras o previous finding in equiy markes. This resul is consisen wih he view ha liquidiy raders reduce price volailiy. 3 Mehodology 3. Economeric issue 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 - can be modelled as follows: where { d, v, r } f ( d, v, r ; ) () denoes he informaion available up o period and is a vecor incorporaing he parameers of ineres. In he marginal disribuion 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 he condiional densiy of he volailiy given duraions and volumes. { d, v, r } g( d ; ) h( v d, ; ) k( r d, v, ; ) () d v r Manganelli (5) specifies he following univariae MEM model for duraion, volume and price volailiy: d u u i i d ( d ; ), ~...(, u ) v d i i d ( v;, ), ~...(, ) rˆ h ( ; d, v, ), ~ i. i. d.(,) r ˆ ( r;,, ), ~...(, ) or r h d v i i d (3) where r is he proxy for volailiy, (,, ) ˆ h are he condiional expecaions of, duraion, volume and volailiy, respecively,,,..., ) is a vecor of s ( s s In empirical analysis, rˆ is obained as he residuals of an ARMA(,) process of reurn series. See Dacorogna (), Hausch (8). One advanage of using rˆ is ha i avoids he problem of exac zero values in r.

12 parameers of ineres. The innovaion erms are uncorrelaed wih each oher by consrucion. The firs order model has he following form: w a d a v a rˆ b 3 w a d a v a rˆ b a d 3 (4) h w a d a v a rˆ b h a d a v Under an assumpion of weak exogeneiy, 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), Engle and Sun (7)). Following Manganelli (5), here are wo concerns regarding he marginal disribuion model. Firsly, i assumes ha he specific processes are independen. To incorporae he conemporaneous informaion, Manganelli specifies causaliy from duraion o volume and from duraion and volume o price volailiy. However, modelling he disribuion of price as condiional on duraion and volume is jus one sraegy o obain heir join disribuion. As poined byengle 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. We have showed in our previous chaper he exisence of cross dependence beween he rading and price process. Recen empirical sudies also address his issue. For example, Hausch (8) finds he exisence of a common unobserved componen which joinly drives he dynamics of he rade and price processes. And his common componen explains mos of he causaliy beween he rade and he price process, even if he conemporaneous effec of he rade variable on he price variable is conrolled. Effecively, he exisence of a common facor provides he evidence of cross-dependence beween he innovaion erms. Therefore, he bes approach is o allow he innovaion erms o be conemporaneous correlaed, raher han specifying a marginal disribuion model. Secondly, he Manganelli (5) marginal disribuion model assumes weak exogeneiy, which means he condiional expecaion of one variable is a funcion only of is own pas condiional expecaion, and he pas condiional expecaions of oher variables are assumed no o carry any informaion. This sraegy has been adoped by mos empirical microsrucure papers (See, for example, Dufour and Engle

13 ()). However, we argue ha his assumpion is oo resricive. Boh pas and pas (un)expeced variables could carry informaional conen and have explanaory power for oher variables in he sysem. For example, rade innovaion is exclusively a manifesaion of he privae informaion of he informed rader. Engle () and Wuensche, Grammig e al. (7) even 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. Manganelli (5)conducs a robusness es on his resricion. Specifically, he regresses he residuals of he hree equaions agains oher pas expeced variables. The resuls indicae ha he coefficiens of pas expeced variables are almos never significan and hereby he marginal disribuion 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 lagged expeced variables using marginal disribuion 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. Therefore, he use a vecor specificaion and join esimaion mehod, he rivariae sysem, is advisable. 3. Model Specificaion Le s define x ( d, v, r )', (,, h )' and ( u,, )'. We can wrie his sysem of equaions as a rivariae vecor muliplicaive error model (MEM). Following Cipollini, Engle e al. (7) he hree-dimensional MEM for duraion, volume and volailiy is x (5) where denoes he Hadmard (elemen by elemen) produc, he componens of are process-specific innovaion erms which are assumed o be cross-dependen. has a mean vecor wih all componens uniy and 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 (6) 3

14 where z is a vecor of predeermined variables, for example z ˆ and ( d, v, r )' A is a lower riangle marix wih all elemens on he diagonal equal o zero. The mean equaion can be furher exended o be a logarihmic version o ensures he posiiveness 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 (7) In his vecor MEM model, 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. Furhermore, he innovaion erms are conemporaneously correlaed. The wo resricions imposed by marginal disribuion model are released. 3.3 Specificaion of A compleely parameric formulaion of he vecor MEM requires a full specificaion of he condiional disribuion of. As a firs sep one may aemp o generalize he univariae Gamma disribuion o a suiable mulivariae version. Bu his is frusraed by he limiaions of he mulivariae Gamma disribuion. Cipollini, Engle e al. (7) find ha he only useful version of his disribuion admis only posiive correlaion and he correlaion is sricly linked o he corresponding variance. The specificaion of in a vecor MEM process is an open quesion. In he lieraure, an indirec or non-parameric specificaion of is ofen adoped. Hausch (8) proposes a sochasic MEM (SMEM) model, in which he assumes he innovaion erms are independen bu driven by an unobserved laen common componen. Cipollini, Engle e al. (7) propose using copula funcions o esimae he parameers of he expeced variables and of he correlaions of he innovaion processes. The model can be esimaed by maximum likelihood; however, differen choices of copulas may lead o differen resuls and here is no heoreical guide for he choice of copula funcion. Cipollini, Engle e al. (9) also sugges a semi parameric formulaion relying on he firs wo momens of he innovaions. Then a GMM-based approach can be used. 4

15 3.3. Mulivariae lognormal disribuion The lognormal disribuion is very commonly used in reliabiliy analysis, bu is of less ineres in financial duraion models. Recenly, Allen, Lazarov e al. (9) and Xu () jusify he usefulness of Lognormal disribuion in specifying financial duraion. Xu () compares he performance of lognormal ACD model wih an alernaive specificaion of he ACD model. The empirical resuls show ha lognormal ACD model is always superior o Exponenial ACD and Weibull ACD models while is performance is similar o he Burr or Generalized Gamma ACD models. Moreover, Allen, Chan 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. Since rading volume exhibis similar disribuion properies o rading duraion, i is feasible o assume a lognormal disribuion for rading volume. In he volailiy lieraure, i is well known ha he price volailiy in lognormally disribued (for example, Andersen, Bollerslev e al. () and Cizeau, Liu e al. (997) among ohers also show ha he lognormal disribuion fis very well he realized volailiy disribuion). Thereby, we specify a mulivariae lognormal disribuion for he vecor MEM model. where Assume follows a mulivariae lognormal disribuion ha ln N( v, D), vi dii o guaranee ha E( ) I. The densiy funcion is: k / / f ( D) ( ) D ( * * 3 ) *exp (ln v)' D (ln v) where. The condiional densiy of x is hen: k / / 3 f ( x, ) ( ) D ( x * x * x ) by: *exp (ln x ln v)' D (ln x ln v) The firs and second momens of he mulivariae lognormal disribuion are given (8) (9) 5

16 where 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 vi dii and d ij is he ijh elemen of D. I is clear ha if (ln,ln,,ln ) k are independen, hen ij (,,, ) are also independen and vice verse. The mulivariae lognormal disribuion allows boh posiive and negaive correlaion, which is much more fruiful han mulivariae Gamma disribuion (Cipollini, Engle e al. 7) 3.3. VARMA represenaion One of he advanages in using he lognormal disribuion for he vecor MEM model is ha i has an equivalen VARMA specificaion wih an innovaion ha follows a mulivariae Gaussian disribuion. Le s see how i works. From he following log vecor MEM model x p ln( ) A ln( x ) B ln( ) A ln( z ) l l l l l l q k () () If we ake logs of (), we obain ln( x ) ln( ) ln( ) c ln( ) e () where e ~ D(, ) Then, ln( ) ln( x ) c e (3) q q q q Bl ln( l ) Bl ln( x l ) Bl c Ble l l l l l (4) Subsiuing ln( ) and q Bl ln( l ) ino (), i follows l p q q (5) ln( x ) c ( A B )ln( x ) e B e A ln( z ) l l l l l l l l l 6

17 where c c Blc q l I is ineresing ha he vecor MEM model is equivalen o a VARMA specificaion. In paricular, i provides a good way o adap he VARMA inference (See Appedix) o make inferences in he vecor MEM model. To furher jusify he performance of he Gaussian VARMA mehod in he esimaion of a vecor MEM model, we conduc a simulaion sudy. We design he following experimen. The sample size is and he number of simulaions is 5. To simplify, we assume boh A and B are diagonal marices and duraion-volume daa will be generaed from he following MEM (,) model:. d u v, u ~ i. i. d.(, ) follows a bivariae exponenial disribuion, wih corr, ). ( u ln a ln b o o a ln d b lnv a ln b ln b3 ln d The populaion parameer values are given as follows; a b., (6) a b.5, a. 85 b. 8, b 3.. The parameers values are chosen from he empirical work of Manganelli (5) and he daa are generaed according o he correlaion coefficiens: corr(, ). and corr(, ). 5 u We hen esimae he model firsly using he marginal disribuion mehod of Manganelli (5) and hen he VARMA mehod. In he marginal disribuion mehod, he wo processes are esimaed separaely, assuming no cross-dependence. In he VARMA mehod, he wo processes are esimaed joinly, assuming ha hey follow a join Gaussian disribuion. The esimaion resuls are repored in Table Table : The simulaion saisic resuls. corr(, ). corr(, ). 5 a a u Marginal disribuion VARMA Marginal disribuion VARMA Mean SD Mean SD Mean SD Mean SD u u b b b

18 Noe: SD denoes he sandard deviaion From his able, i can be seen ha he esimaors of he duraion process are no affeced by he differen esimaion mehods. Boh he marginal disribuion mehod and he VARMA mehod give unbiased esimaors and he sandard deviaions are also very similar. For he esimaion of he volume process, he resuls are quie differen. The marginal disribuion mehod esimaors are biased in general and he bias is large when he correlaion beween he marginal and condiional processes is large. The VARMA mehod esimaors perform very well. I gives unbiased esimaors, no maer wheher he cross-correlaion of he innovaion erms is low or high. There is hus srong evidence o suppor he use of he VARMA inference for inference in he vecor MEM model. 4 Empirical analysis 4. Daa We use he same daase as Engle and Paon (4). The daa were aken from he Trades and Quoes (TAQ) daase in NYSE. The TAQ daa consiss of wo pars: he firs repors he rade daa, while he second liss he quoe daa (bid and ask daa) posed by he marke maker. We randomly chose socks covering he period from Jan,998 o June 3, 999 wih he following procedure. We firs consruced deciles on he basis of he 997 oal number of rades of all socks quoed on he NYSE. Then we randomly seleced 5 socks from he eighh decile (frequenly raded socks) and 5 from he second decile (infrequenly raded socks). The ickers and names of he en socks are repored in Table 3: Table 3: Socks used in he analysis A. Frequenly raded B. Infrequenly raded TRN TRINITY INDUSTRIES ABG GROUPE AB S A ADS R RYDER SYSTEM INC OFG ORIENATAL FINL GRP HOLD CO ARG AIRGAS INC LSB LSB INDUSTRIES INC FMO FEDERAL-MOGUL CORP HTD HUNTINGDON LIFE S.G. VTS VERITAS DGC INC HUN HUNT CORP Before he analysis began, we adoped he sraegy of Manganelli (5) 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. 8

19 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 raded as one single ransacion, while 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 volumes 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 by Lee and Ready (99) which associaes 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. 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. Generally, less frequenly raded socks do no exhibi any regular inraday paern. More frequenly raded socks show he ypical invered U paern for duraion, he L paern for reurn squares, and no regular paern for volume. Table 4 presens some summary saisics for he en socks. For he frequenly raded socks, he number of observaions range from 85 o 697 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 and he average rading duraion ranges from 5 seconds o 45 seconds. The rading volume does no show any difference beween frequenly rading socks and infrequenly rading socks. The number of rading volumes ranges from 8 o 595. The mulivariae Ljung-Box saisics, compued according o Hosking (98), and is given by 9

20 MLB( s) : n( n ) race( Cˆ Cˆ Cˆ Cˆ ) s j j ks (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, Cˆ j is he j h residual auocovariance marix. The relaively large Ljung-Box saisics in able indicae ha he rading and price daa reveal srong serial dependencies. Table 4: Summary saisics for he socks. LB() denoes Ljung-Box saisics for order. MLB() denoes mulivariae Ljung-Box saisics Obs Mean LB() MLB() Duraion Volume Duraion Volume Variance TRN GAS TCB R ARG ABG OFG LSB HNN JNS Empirical resuls To keep i simple, we concenrae on a firs order model in empirical analysis. The firs order vecor MEM model is wrien as: x ~ DI (, ) (8) ln( ) Aln( x ) B ln( ) A ln( x ) (9) where wi, i,,3, denoe (3 ) vecors, and A{ a ij } and B{ b ij } for i, j,,3 is a (3 3) marices of innovaion and persisence parameers, respecively. Furhermore, A { a ij } for i, j,,3 is a (3 3) riangle marix wih where only he hree lower lef elemens can be nonzero, and { ij }, for i, j,,3 is a (3 3) marix for he covariance parameers. Our firs objecive is o evaluae he performance of he vecor MEM and empirically examine he wo resricions imposed by Manganelli (5). To do so, we

21 esimae 5 differen vecor MEMs. Model one (M) is Manganelli (5) s model, where duraion, volume and volailiy process are assumed o be independen ( ij, for i j ) and he marix B in Eq. (9) is resriced o be diagonal. The opimizaion could be done separaely for each process. Models wo o five allow he innovaion erms of he rading duraion, volume and volailiy processes o be correlaed and follow a joinly lognormal disribuion. The esimaion mehod proposed in secion 3 is used o for parameer esimaion and inference. In he mean equaion, Model (M) excludes A and resrics marix B o be diagonal. Model 3 (M3) augmens M by including A and Model 4 (M4) augmens M by releasing he resricions on marix B. Model 5 (M5) incorporaes boh effecs. The full esimaed parameers and he resuls of diagnosic saisics can be found in Appendix 3. For breviy, Error! Reference source no found. only repors diagnosic ess resuls for he socks in his secion. Since we calculaed log likelihood for all he five models, he likelihood raio (LR) es can be used o es he resricions imposed by Manganelli (5). The LR es resuls are repored in Table 6.The major empirical findings are summarized as follow: (i) The diagnosics saisics show ha Manganelli s marginal disribuion model is unsuccessful in capuring he dynamics of he sysem as is indicaed by highly significan Ljung-Box saisics for he residuals. The specificaions of vecor MEMs improve he dynamic properies of he model. 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 socks, he dynamics of he sysem are sill no compleely capured 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. Among he vecor MEM models, he more highly parameerized models, paricularly M4, capure he dynamics of he rivariae sysem bes. The likelihood and BIC value also suggess ha he overall performance of marginal disribuion model is worse han he vecor MEMs. (ii) The LR es suggesed ha he esimaed covariance marix is significanly differen from zero. This resul is surprisingly robus over all individual specificaions and clearly in line wih he noion ha he innovaion erms are correlaed. Our finding

22 is consisen wih Hausch (8) ha he underlying informaion process is joinly driving he rading and price process. In Manganelli (5) s marginal disribuion model, he causaliy from curren duraion o volume and from curren duraion and volume o price volailiy is specified o incorporae he conemporaneous informaion. However, our finding suggess ha he inclusion of curren duraion in he volume equaion and curren duraion and volume in he volailiy equaion does no reduce he conemporaneous correlaions of he innovaion erms. I is paricularly rue in Model hree, where curren duraion and volume are even insignifican. So we can conclude ha he resricion imposed by previous empirical work ha he innovaion erms are independen is no valid. The marginal disribuion model canno incorporae hese conemporaneous relaionships. (iii) In Manganelli s marginal disribuion model, he assumpion of weak exogeneiy is made in he specificaion of he condiional mean. More specifically, he pas expeced variables are assumed no o carry any informaion or he coefficien marix B in (9) which is resriced o be diagonal. 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 cross expeced variables. They find he lagged cross expeced variables are insignifican. However, we find ha he lagged expeced variables are mosly significan ( bij, i j ) in our vecor MEMs. I is paricularly rue for infrequenly raded socks. LR es also suggess he lagged expeced variables are joinly significan in mos of case. 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. This is also confirmed by Grammig and Maurer (). Moreover, he inclusion of lagged expeced variables reduces he residual auocorrelaion dramaically, as indicaed by he big drop in value of he Mulivariae Ljung-Box saisics. Acually, he bes performing models are he ones including expeced variables. Therefore, he weak exogeneiy assumpion is no suppored by he empirical daa. The lagged expeced variables should be incorporaed in his rivariae sysem.

23 Table 5: Diagnosics saisics for he esimaed resuls of differen parameerizaion of vecor MEMs. Log likelihood funcion (LL), Bayes Informaion Crierion (BIC), Ljung-Box saisics of flied residuals (LB) and mulivariae Ljung-Box saisic (MLB). The Ljung-Box saisics are compued based on lags. M M M3 M4 M5 M M M3 M4 M5 ARG TRN LL BIC MLB *** 694.*** 565.8*** 57.3*** 84 36*** 74*** 99.6** 995.3*** LB _ d 87.*** 87.*** 85.38***.4*** 3.9*** 8.95*** 3.7*** 8.3*** 36.3** 5.9*** LB _ v 74.7*** 7.9*** 53.5*** 95.97*** 95.7*** 5.*** 5.*** 74.*** 84.*** 76.3*** LB _ rˆ 93.3*** 5.8*** 73.*** 74.3*** 79.*** 37.8*** 393.8*** 89.8*** 38.7*** 36.4*** TCB GAS LL BIC MLB 586*** 34*** 38*** 8*** 8*** 3 43*** 936.*** 684.3*** 688.9*** LB _ d 45.8*** 6.3*** 37.8*** 4.*** 99.3*** 59.4*** 3.*** 3.3*** 5.8*** 6.4*** LB _ v 45.*** 47.***.*** 8.3*** 8.*** 5.4*** 59.*** 66.7*** 83.37*** 78.8*** LB _ rˆ 45.8*** 484.*** 434.8*** 457.*** 457.4*** 8.8*** 69.5*** 9.*** 9.9*** 6.7*** R FET LL BIC MLB *** 877*** 86 93*** 43.8***.7***.9** LB _ d 3.5*** 6.7*** 7.*** 6.37*** 6.9*** LB _ v 355.5*** 4.7*** 88.5*** 355.9*** 363.3*** LB _ rˆ 53.4*** 799.8*** 9.5*** 7.3*** 77.5*** 7.*** 8.8*** 77.46*** 65.36*** 6.7*** 3

24 M M M3 M4 M5 M M M3 M4 M5 ABG HTD LL BIC MLB 44.3*** 58.*** 43.4***.3**.** 9.*** 3.9** 3.8** LB _ d 37.8*** 37.7** 37.97*** 36.47** 36.48** ** 34.73** LB _ v *** 44.98*** LB _ rˆ 7.6*** 37.88*** 4.87*** *** LSB HUN LL BIC MLB 99.6*** 6.5*** 48.6*** 49.3*** 36.6*** 447.8*** 63.9*** LB _ d 47.86*** 47.*** 47.8*** 48.37** 49.58** LB _ v 37.89*** 37.83*** 36.3** 37.5*** 37.68*** LB _ rˆ 37.** 43.*** 34.7** 35.7** *** Table 6: LR es resuls ARG TRN TCB GAS R ABG HTD LSB HUN FEP Ho: b, i j ij (M vs.m4) Ho: b, i j ij (M3 vs.m5) Ho: ij, i j (M vs.m3) Noe: Criical values (6).5 =.59, (6). =6.8 4

25 4.3 Empirical resuls on he relaionships beween duraion, volume and price volailiy. Our second objecive is o empirical analyse he marke microsrucure heory and is predicions on relaionships he beween duraion, volume and price volailiy. We use he esimaed resuls of our bes parameerized vecor MEM specificaion (M4) in his analysis. We are paricularly ineresed in he price impac of rade and feedback effec from rading characerisics o rading inensiy. The esimaion resuls of M4 for he five frequenly raded socks are repored in Table 7 and resuls for he five infrequenly raded socks are repored in Table 8. Table 7: Esimaion resuls for M5: frequenly raded socks. ARG TRN TCB GAS R.6***.89.55*** *** ** -.*** *** *** *** -.88*** -.37*** *** 3.5**.57***.63***.454*** ***.4***.78***.36***.95 b.939***.73***.94***.74***.9*** b **.3***.3.3 b *** -.5***.7.44*** 3 b.7***.***.***.8***.65 b.58***.638***.695***.76***.66** b -.69*** -.353*** -.6*** -.474*** -.88** 3 b.3***.7***.35***.4***.3 3 b ** -.8*** -.7*** -.9*** 3 b.39***.3***.75***.46***.69*** Table 8: Esimaion resuls for M5: infrequenly raded socks. ABG HTD LSB HUN FEP.4***.9**.3.56***.56*** *** ** -.88***.3 -.** *** *** **.85*** 5

26 *** *** -.8*** -.68*** *** ***. b.9***.98***.967***.93***.9*** b.34*** -.549*** b.4*** ***.6*** 3 b -.8***.6***.8.3***.3*** b.366***.9*** ***.643*** b -.99*** -.663*** ** -.58** 3 b -.4***.38*** -.6.8**.94 3 b -.79*** -.95** *** -.49*** 3 b.68***.665***.5***.67***.38** Looking firs a he price volailiy ( h ) process, he coefficiens of lagged duraion ( a 3 ) in he volailiy equaion are negaive and significan in mos cases, indicaing ha rades wih longer duraion are associaed wih a lower volailiy impac. This resul is consisen wih ha of Dufour and Engle () and Manganelli (5). Of paricular ineres is he finding for he volume coefficiens ( a 3 lagged volume coefficien ( a 3 ).The ) is posiive and significan for frequenly raded socks. This suppors some of he predicions of Easley and O'Hara (99). Trades of a larger size are more likely o be execued by informed raders and hus have a greaer impac on he price of he sock. However, i is no significan for less frequenly raded sock. This is also consisen wih Manganelli (5). In he duraion process, he volailiy coefficien ( a 3 ) is negaive and only significan in ou of 5 cases for frequenly raded socks and is insignifican for less frequenly raded socks. The negaive sign suppors a predicion from invenory based model, suggesing ha large quoe changes would arac opposie-side raders hus increasing he rading inensiy. Bu i is only robus for frequenly raded socks. These resuls confirm hose of Dufour and Engle () and Manganelli (5). I is quie ineresing ha he volume coefficien ( ) is negaive and only significan for less frequenly raded socks. The evidence from infrequenly raded socks suppors he sraegy informaion rade model (Easley and O'Hara 99). 6

27 In he volume process, boh duraion and volailiy coefficiens are insignifican, which is consisen wih Manganelli (5). However, his does no indicae ha he rading volume is exogenous, as suggesed by Manganelli (5) s resuls. Our resuls sugges ha boh he coefficiens of pas condiional expecaions of duraion and he coefficiens of pas condiional expecaions of volailiy are significan for frequenly raded socks and infrequenly raded socks. The finding ha he lagged expeced variables have explanaory power and convey informaion on he conen of he price change process is new in marke microsrucure analysis. There is relaively lile lieraure o explain his; however we provide a way o explain i. Firsly, le us wrie he equivalen VARMA specificaion of vecor MEM: ln d w a b a b a3 b3 ln d ln b b b3 ln ln v w a b a b a3 b3 ln v ln b b b3 ln ln rˆ w 3 a3 b a3 b3 a b ln ˆ ln r b3 b 3 b ln Then in he volailiy equaion, he coefficien b3 can be considered o evaluae he impac of duraion shocks on price volailiy and he coefficien b3 can be considered o evaluae he effec of volume shocks on price volailiy. In he lieraure, mos empirical research uses raw duraion (volume) o proxy he privae informaion. However, heoreical research does no draw conclusion as o wheher he raw duraion (volume) or he duraion (volume) shocks possess informaion conen wih respec o he fundamenal asse value. Hasbrouck (988) and Wuensche, Grammig e al. (7) argue ha i is he unexpeced componen of rade ha conains price informaion, as change in price is unpredicable. Ye, i is well knowing ha rade duraions (volumes) are highly predicable (c.f. Engle and Russell (998)). Our empirical work confirms ha boh duraion and duraion shocks conain price informaion, while only he unexpeced componen of volume conains price informaion. In he duraion equaion, we are paricularly ineresed how he marke makers behavior in quoe adjusing process affecs he rading inensiy. This effec is evaluaed by he coefficien b3 and a3 b3. From marke microsrucure heory, quoe changes could eiher be invenory moivaed or informaion moivaed and has differen impac on rading inensiy. If i is invenory moivaed, hen large absolue quoe changes (or large volumes) may arac opposie side raders, which would 7

28 increase rading inensiy (Dufour and Engle ).If i is informaion moivaed, large absolue quoe changes indicae an 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), which would decrease he rading inensiy. Empirical sudies have no addressed his difference ye. Theoreically, Hasbrouck (988,99) has used he shor run and long run characerizaion of rading behavior o separae quoe movemen ino shor-run invenory relaed effecs and long-run informaion relaed effecs. On he one hand, since he privae informaion is persisen and long lived, he persisen quoe change is relaed o privae informaion. On he oher hand, he invenory level in saionary and invenory conrol is inherenly of ransien concern, he ransien quoe change is relaed o invenory conrol. Thereby, he coefficien b3 can be considered o evaluae he impac of ransien quoe change on rading inensiy and he coefficien a b can be considered o evaluae he effec of a persisen quoe change on 3 3 rading inensiy. Our empirical resuls indicae ha he acual and unexpeced componen of volailiy affecs duraion in differen direcion. This finding confirms Hasbrouck (988,99) s predicion ha uninformed raders decrease heir rading inensiy when hey observe larger reurn. 8

29 5 Conclusion In his paper, we propose a general vecor MEM model for he dynamics of duraion, volume and price volailiy, in which 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. We furher propose a joinly lognormal disribuion for he vecor MEM model. The vecor MEM model relaxes he wo resricions ofen imposed by previous empirical work in marke microsrucure research, by allowing he innovaion erms o be cross-dependen and removing weak exogeneiy resricions. Moivaed by he equivalence of he vecor MEM and VARMA specificaions, he model can be esimaed by maximum likelihood. In empirical analysis, we use Trade and Quoe daa from NYSE o model he dynamics of duraion, volume and price volailiy in a vecor MEM model and we ge he following empirical findings. () The marginal disribuion model is unsuccessful in capuring he dynamics of he sysem, as is indicaed by high residual auocorrelaion, and he specificaions of vecor MEM models improve he dynamics properies of he rivariae processes significanly. This is paricularly rue for he volailiy process. () The assumpion of weak exogeneiy in Manganelli (5) and Dufour and Engle () model is no valid. The robusness check conduced by Manganelli (5) is misleading since he dynamics of expeced variables have been disored by he use of a marginal model. The lagged expeced variables are significan in mos cases. Moreover, he inclusion of lagged expeced variables reduces he residual auocorrelaion dramaically. (3) In he vecor MEM model, he covariance marix of he innovaion erms is significanly differen from zero. I is robus over differen model specificaions. The marginal disribuion model ha marks condiional on curren duraion canno reduce he conemporaneous correlaion beween he innovaion erms. (4) The empirical resuls are generally consisen wih Manganelli (5). We also find ha a period of greaer aciviy coincides wih a higher number of informed raders presen in he marke, bu only valid for he frequenly raded socks. 9