The role of tme, lqudty, volume and bd-ask spread on the volatlty of the Australan equty market. Allster Keller* Bruno Rodrgues** Mawell Stevenson* * Dscplne of Fnance School of Busness The Unversty of Sydney ** Support Decson Methods School of Electrcal Engneerng Pontfcal Catholc Unversty Ro JEL Classfcaton: G1, G1, D5, N7 Abstract Tme plays an mportant role n the analyss of lqudty. In the past, an analyss of lqudty has been acheved usng multple and dsassocated varables such as traded volumes or bd-ask spreads rather than a sngle metrc of lqudty. The avalablty of ultra-hgh frequency transacton data from fnancal markets has led to the development of econometrc technques that have a greater capacty for the etracton of nformaton than those of pre-estent technologes that reled upon the dscrete samplng of data over calendar tme. The use of economc or transacton tme, as measured by duraton n the autoregressve condtonal duraton (ACD) model of Engle and Russell (1998) and Engle (000), formed the bass for the analyss of equty volatlty of stocks traded on the Australan Stock Echange (ASX). Engle (000) was replcated and etended to eght stocks at the ASX. It s shown that the ntraday seasonal patterns of duraton, traded volumes, spreads and volatltes are smlar to those ehbted by IBM at the NYSE and that duraton, as measured by the tme between trades, affected condtonal volatlty. Another varable that jontly and sgnfcantly affected volatlty wth duraton was the bd-ask spread, ether measured contemporaneously or lagged. However, the antcpated affect of traded volumes was not as evdent. Further, ths analyss was used to confrm the market-mcrostructure theores of Easley and O Hara (199) concernng the effects of tme on volatlty rather than those advocated by Damond and Vereccha (1987). The former proposed that longer duratons tend to sgnfy that no new nformaton has been mpounded nto prces, whle the latter clam they ndcated that t was the arrval of bad news n the market that caused the prce effect. 1
1. Introducton Lqudty s a frequently revsted theme n fnancal economcs. It s rarely quoted as a sngle metrc wth the standard defnton of lqudty demandng three characterstcs of a marketplace. They are the mmedacy of a trade, the ablty for an asset to trade quckly n large volumes, and the requrement that the prce mechansm be not overly responsve to the sze of traded volumes (Black, 1971; Glosten and Harrs, 1988). Lqudty s of fundamental mportance to the performance of fnancal markets and can mpact drectly upon both mplct and eplct tradng costs facng an ndvdual nvestor, as well as the proftablty of the market tself. Due to ts mportance, and for the sake of echange organsaton, regulaton and nvestment management, ths paper seeks to analyse the jont mpact of tme, lqudty, volume and bd-ask spreads on market volatlty, wth a partcular emphass on the Australan equty markets. The duraton between trades as a measure of lqudty was orgnally advanced by Engle and Russell (1998) and led to ther adopton of the Autoregressve Condtonal Duraton (ACD) model as a means to measure t. The rason d être of the model s descrbed n Engle (000) where he dstngushes between economc tme and calendar tme and clams that: Intutvely, economc tme measures the arrval rate of new nformaton that wll nfluence both volume and volatlty. The jont analyss of transacton tmes and prces mplements the standard tme deformaton models by obtanng a drect measure of the arrval rate of transactons and then measurng eactly how ths nfluences the dstrbuton of the other observables n the market. (Engle 000, p.4) The ACD model of Engle and Russell (1998) creates a sngle measure of lqudty that ncorporates all the afore-mentoned requred characterstcs. That measure of
lqudty s duraton between trades. By usng the tme between trades as a proy for lqudty, the analyss of volatlty can be naturally lnked to other potentally eplanatory market-mcrostructure varables. Goodhart and O Hara (1997) argued that one of the most puzzlng ssues n the market-mcrostructure lterature s the behavour of volatlty. In partcular, these two authors noted that the Generalsed Autoregressve Condtonal Heteroscedastc (GARCH) model, of whch the ACD s an etenson, s deal for pursung the ssue of long term (transactonal rather than calendar) persstence n volatlty. These authors suggested that t s transactons rather than volume that drve markets and the adjustment of prces and, n so dong, promoted the use and concept of economc (transacton) tme rather than dscrete calendar tme. As a result, the ACD model has become a hghly useful tool n the descrpton of the prce-volatltynformaton relatonshp (Jones, 1994). The use of economc or transacton tme (as measured by duraton) n the Autoregressve Condtonal Duraton (ACD) model of Engle and Russell (1998) and Engle (000), formed the bass of our analyss of equty volatlty for a selecton of stocks traded on the Australan Stock Echange (ASX). The study of Engle (000) was replcated and etended to eght stocks at the ASX. Frst, we establshed that the ntraday seasonal patterns of duraton, traded volumes, spreads and volatltes are smlar to those ehbted by IBM at the NYSE. We then we showed that duraton, as measured by the tme between trades, affects condtonal volatlty. Data on other varables whch jontly and sgnfcantly affected volatlty wth duraton, such as the sze of bd-ask spreads, prce and traded volumes were ncluded n the analyss. Ths analyss was used to confrm the market-mcrostructure theores concernng the effects of tme on volatlty by Easley and O Hara (199). They proposed that longer 3
duratons tend to sgnfy that no new nformaton has been mpounded nto prces, rather than the competng theory of Damond and Vereccha (1987) who advocated that longer duratons, or no trades, mpled the arrval of bad news when short-sellng was constraned. In the followng secton we dscuss the relaton between volume and volatlty. An nsttutonal dscusson of the tradng process on the Australan Stock Echange (ASX), and adjustment procedures appled to the data analysed n ths study, s dealt wth n secton 3. Secton 4 outlnes the duraton modellng of hgh frequency data, whle volatlty models and emprcal results are contaned n secton 5. We conclude n secton 6.. Duraton And The Volume-volatlty Relaton Most fnancal market studes n the past have reled upon the collecton of data at dscrete ponts n tme. The use of data that s dscretsed accordng to calendar tme may not be synchronous wth events or nformaton flows and may, thereby, lead to the erroneous measurement of varables such as volatlty. Data n calendar tme have formed the bass for data collecton n the majorty of prevous market-mcrostructure studes. Ths was partly due to the lmted avalablty of hgh-frequency data, along wth the prevalng vew at the tme that nformaton shocks to a market were unlkely or ndeed mprobable, over short tme frames. Wth the use of hgh-frequency data n ths study, all observatons were collected and the populaton ncorporated nto the analyss to mprove nferences. In effect, the statstcal nature of nformaton was analysed n real tme and ncorporated varables that conformed wth theoretcal underpnnngs of the market-mcrostructure lterature. 4
Much of the early market-mcrostructure lterature gnored the role that tme plays n the economc formulaton of prces. For nstance, the oft-cted work of Kyle (1985) modelled the prce formaton process usng aggregated data. Ignorng tme durng the nteracton between market partcpants may produce skewed results. Whle trade mbalances n statc tme may have produced theoretcally appealng outcomes, these outcomes are dffcult to reproduce n a dynamc tme settng. The nherent nformaton flow through the tmng of a trade was also gnored by Glosten and Mlgrom (1985) who also modelled trade mbalances. Damond and Verrecha (1987) concluded that a perod of calendar tme n whch no trades occur must have been a perod when bad nformaton that would dscourage tradng had shocked the system. Whle the ncluson of tme and the assocated nformaton that flows wth t was a credble contrbuton to the lterature, ther fndng was controversal. It was based upon the assumpton that the market has a short sales constrant. Informed sellers who dd not own stock were prevented from short sellng and therefore, unable to take advantage of ther nformaton. The lack of trades leads to the recognton of ths phenomenon by market makers, wth a correspondng lowerng of prces. It follows that a pattern of no trades or longer duratons s an ndcator of bad news. Followng ths artcle, Easley and O Hara (199) offered an ntutvely plausble nsght that a perod of no tradng would n fact convey the message that no news was flowng nto the system. Ths was based upon the realstc assumpton that one of the partcpants, namely the market maker, would set bd-ask spreads n response to the possblty that nformed traders were present and that they could use ther asymmetrc nformaton to act n a predatory fashon. Ths paper shows the 5
netrcable lnk between economc and calendar tme through the arrval tme and rate of nformaton flow to the market. Under the assumpton that the tme between trades s used as a proy for volume and that trade arrvals are non-trvally dstrbuted, then a GARCH-type model can be used to make the lnk between volume and volatlty. Whle not always descrbed by GARCH-type models, the lnkage between volume and volatlty has been emphassed n the lterature by authors such as Lamourou and Lastrapes (1990), Campbell, Grossman, and Wang (1993), and Gallant, Ross, and Tauchen (199). Whle epected volume may reduce bd-ask spreads, shocks to volume may wden them. As a consequence, volume plays a much more mportant role n the tradng process than was recognsed n earler market-mcrostructure studes. It follows that, as an etenson of the GARCH framework, the ACD model becomes a useful tool n modellng the relaton between market-mcrostructure varables and the understandng of volatlty. 3. The Australan Stock Echange And Data Adjustment Procedures As one of the larger stock echanges n the world, the ASX operates both straght equty and dervatve markets. Equtes are traded through the Stock Echange Automated Tradng System (SEATS) 1, a contnuous electronc order market that operates daly between appromately 10:00am and 4:00pm. Fgure 1 below depcts the operatng hours of the market and the varyng phases that t depends upon. Both openng and closng call auctons occur n batches of around ten mnutes. In the mornng, the openng call aucton runs through fve batches and may be preceded by a pre-open perod where orders may be adjusted after beng entered and cancelled f 1 At the tme of wrtng the ASX was preparng to ntroduce a new platform common to all securtes traded on that market. However, the data analysed here was determned under SEATS. 6
requred. These orders wll, however, not be eecuted untl ASX approval has been granted. A smlar process occurs n the afternoons between 4:05pm and 4:06pm when a sngle market aucton s actvated that clears the order book and determnes closng prces. Tradng s allowable after the closng call aucton wth the provso that traders are requred to report ther actvtes. Ths s a market requrement gven that the ASX hosts several stocks that are dual-lsted wth markets overseas that affect the prce formaton process. Fgure 1 Equty Tradng Hours at the ASX After Hours Adjustment (17:00-19:00) Closng Phase (16:05-17:00) Closng Sngle Prce Aucton (16:05) Pre-Open Pror to Closng (16:00-16:05) Normal Tradng (10:00-16:00) Enqury (19:00-7:00) Openng (10:00) Pre-Openng (7:00-10:00) In hs study, Engle (000) confned hs analyss to a sngle stock lsted on the New York Stock Echange (NYSE) and, n so dong, restrcted generalsaton of hs results to other equtes lsted on that market. In order to descrbe the lqudty of the ASX equty market and to allow for generalsaton of the results of our analyss, a sample of stocks was selected to represent the broader market. These stocks are defned n Table 1 below. The ASX s domnated by several stocks belongng to the bankng and 7
resource sectors. As a result, the followng bankng and resource stocks were selected. They were: CBA, MBL, and WPL. The other stocks reflect other sectors of the market and the broader Australan economy. Prce and other assocated nformaton were collected from 1 st September, 005 through untl 1 st December, 005 from the Securtes Research Centre of Australa (SIRCA). None of the stocks eperenced abnormal nformaton flows for the perod, nor dd they go e-dvdend. Table 1 Stocks used n the analyss. All stocks were consdered lqud n relaton to the overall market. Stocks Used n the Analyss Code Name Number of Observatons (before fltraton)) Number of Observatons (after fltraton)) CBA Commonwealth Bank of Australa 10938 57355 MBL Macquare Bank Lmted 93708 5109 WPL Woodsde Petroleum Lmted 9379 51953 WDC Westfeld Group 74973 448 WES Wesfarmers Lmted 7375 41754 WOW Woolworths Lmted 74814 38608 RIN Rnker Group Lmted 69903 37876 QBE QBE Insurance Group Lmted 6069 3377 To ensure accurate modellng, the same data cleansng procedures found n prevous duraton studes were adopted here. Frstly, due to the ntra-day seasonal characterstcs of the data and abnormal tradng patterns n the openng tradng perod, the openng trades were removed from the sample. Engle and Russell (1998), Engle (000), and Manganell (005) all dropped the openng trades from ther samples. Wth the removal of any trade that occurred before 10:1 am, the methodology mplemented here was consstent wth the above studes. The ASX actually operates wth a staggered openng dependent upon the alphabetcal lstng of the stock. Each trade s tme stamped n seconds from mdnght, wth the day beng subsequently comprsed of t [ 0, 0880] seconds. All subsequent trades wth the same tme stamp 8
and prce are aggregated whle any trade wth a negatve duraton s dscarded. Due to the choce of perod, there were no publc holdays durng whch the ASX temporarly ceased tradng. Table 1 shows the number of observatons that remaned after all flterng was completed. Table contans summary statstcs, whle Fgure depcts graphs of the tmeadjusted duratons for the eght stocks ncluded n ths analyss. Further adjustment of the data was carred out by fttng pecewse-lnear splnes to the trades off all stocks durng the tradng hours. Ths resulted n 1 knots for each half hour of tradng. 3 Table Summary statstcs for the ntraday duratons of the eght stocks that comprse ths study. Number of Observatons Medan Mean Std dev Mn Ma CBA 57355 1.8 3.6 8.5 11.9 47.0 MBL 5109 5.14 5.99 9.85 11.70 53.65 QBE 3377 37.5 40.68 16.31 17.98 88.35 RIN 37876 3.5 35.75 15.57 1.30 77.31 WDC 448 5.5 30.58 15.45 1.76 8.79 WES 41754 30.8 3.44 10.50 14.63 58.83 WOW 38608 31.8 33.76 13.31 13.07 71.43 WPL 51953 4.89 6.09 10.66 9.36 57.37 Adjusted Duratons Fgure Duratons for eght companes lsted on the ASX. 90 80 70 60 50 40 30 0 10 10:00:00 10:30:00 11:00:00 11:30:00 1:00:00 1:30:00 13:00:00 13:30:00 14:00:00 14:30:00 15:00:00 15:30:00 CBA MBL QBE RIN WDC WES WOW WPL A negatve duraton s an anomaly n the data as t would mply that the data was out of order. 3 Ths data adjustment s mportant for hazard functon estmaton n secton 4.4. 9
The property n all cases of shorter duratons at the start and end of the tradng day wth longer duratons n the mddle s obvous from Fgure. Table 3 and Fgure 3 contan the summary statstcs and the graphs of the returns for the eght stocks studed, respectvely. Smlarly, summary statstcs and graphs of the volume-of-trades are found n Table 4 and Fgure 4, and n Table 5 and Fgure 5 for bd-ask spreads. Table 3 Summary statstcs for the ntraday adjusted returns of the eght stocks that comprse ths study. Number of Observatons Medan Mean Std dev Mn Ma CBA 57355 0.0000604 0.0000634 0.000017 0.000043 0.0001058 MBL 5109 0.0000960 0.0001057 0.000081 0.0000710 0.0001911 QBE 3377 0.0001080 0.000110 0.000047 0.0000681 0.000190 RIN 37876 0.0001300 0.000140 0.0000386 0.00008 0.00080 WDC 448 0.0001069 0.0001086 0.000063 0.000050 0.0001935 WES 41754 0.0000744 0.0000816 0.000018 0.0000546 0.0001794 WOW 38608 0.0000883 0.0000949 0.000055 0.000061 0.0001990 WPL 51953 0.0001009 0.0001058 0.0000315 0.000066 0.00031 Adjusted returns Fgure 3 Returns for eght companes lsted on the ASX. 0.0001000 0.00019000 0.00017000 0.00015000 0.00013000 0.00011000 0.00009000 0.00007000 0.00005000 0.00003000 10:00:00 10:30:00 11:00:00 11:30:00 1:00:00 1:30:00 13:00:00 13:30:00 14:00:00 14:30:00 15:00:00 15:30:00 CBA MBL QBE RIN WDC WES WOW WPL 10
Table 3 Summary statstcs for the ntraday volumes of eght stocks that comprse ths study. Number of Observatons Medan Mean Std dev Mn Ma CBA 57,356.00 1,000.00,656.58 14,733.33 1.00,500,000.00 MBL 5,109.00 884.89 866.11 97.81 510.47 994.6 QBE 33,77.00 4,085.03 4,009.45 56.35,043.6 4,876.0 RIN 37,876.00 5,40.48 5,335.15 49.56 3,480.99 6,070.53 WDC 44,8.00 6,748.68 6,950.41 1,497.6,614.50 8,969.88 WES 41,754.00 1,360.93 1,350.96 30.10 885.19 1,819.6 WOW 38,608.00 3,740.35 5,05.93,988.11 1.00,47,46.00 WPL 51,953.00 1,875.71 1,887.88 44.03 1,37.51,364.8 Adjusted Volumes Fgure 4 Volume of trades for eght companes lsted on the ASX 8500 7500 6500 5500 4500 3500 500 1500 500 10:00:00 10:30:00 11:00:00 11:30:00 1:00:00 1:30:00 13:00:00 13:30:00 14:00:00 14:30:00 15:00:00 15:30:00 CBA MBL QBE RIN WDC WES WOW WPL Table 5 Summary statstcs for the ntraday bd-ask spreads of eght stocks that comprse ths study. Number of Observatons Medan Mean Std dev Mn Ma CBA 57355 0.0003563 0.0003599 0.0000340 0.0003044 0.0004406 MBL 5109 0.0004710 0.0004956 0.0000803 0.0003790 0.0006594 QBE 3377 0.0006660 0.0006750 0.0000558 0.0006040 0.0008640 RIN 37876 0.0007800 0.0007980 0.0000468 0.0007340 0.0009710 WDC 448 0.0006458 0.0006518 0.000039 0.000570 0.000747 WES 41754 0.0004558 0.000465 0.0000506 0.0003985 0.0007376 WOW 38608 0.000644 0.000648 0.0000306 0.0005749 0.0007566 WPL 51953 0.0004664 0.0004835 0.0000503 0.000436 0.0006817 Adjusted Spreads 11
Fgure 5 Bd-Ask spreads for eght companes lsted on the ASX. 0.00090000 0.00080000 0.00070000 0.00060000 0.00050000 0.00040000 0.00030000 10:00:00 10:30:00 11:00:00 11:30:00 1:00:00 1:30:00 13:00:00 13:30:00 14:00:00 14:30:00 15:00:00 15:30:00 CBA MBL QBE RIN WDC WES WOW WPL The V-shape pattern so obvous n Fgure s replaced below by the famlar U- shape pattern for returns (Fgure 3), and volume of trades (Fgure 4). However, ths U-shape pattern s not so obvous n the case of bd-ask spreads (Fgure 5). 4. Duraton Modellng Of Hgh Frequency Data Hgh frequency data s rregularly spaced n tme and s known statstcally as a pont process. A pont process follows a stochastc process that generates a random accumulaton of ponts along the tme as. In the case of hgh frequency fnancal data, the tmng of quotes and trades of lsted stock s vewed as a pont process, wth the assocated characterstcs of such data at any tme known as marks. For eample, prce and volume correspondng to a trade are marks. In fnancal markets, a marked pont process refers to the tme of a trade or a quote and the correspondng marks. Marked pont processes were used by Engle (000) as a framework for the analyss of the tradng process. 1
A fnancal pont process Let { t 0, t 1,, t n,.} be the tmes of the sequence of trades (or quotes) of an asset traded on a fnancal market. It follows that 0 = t 0 t 1 t n Further, f N(T) s the number of events that have occurred n the nterval [ 0,T] then t N(T) s the last observaton n the sequence of trades, 0 = t 0 t 1 t n t N(T) = T. Addtonally, let { z 0, z 1,, z n,, z N(T) } be the sequence of marks correspondng to the arrval tmes of trades, { t 0, t 1,, t n,, t N(T) }. A tme seres only vews ponts of the process marked at equdstant ponts n tme An alternatve approach s to consder the tme between consecutve observatons or duratons. Let = t t 1, where s the th duraton between trades that occur at consecutve tmes t and 1 t. Clearly, the sequence { 0, 1,, n,, N(T) } contans non-negatve elements. Followng Engle (000), the jont sequence of duratons and marks s gven by { (, z ), = 1,, T }. If I -1 s the nformaton set avalable at tme t 1, then ncluded n ths set are past duratons of fnancal trades and pre-determned marks. These marks are mcrostructure varables lke transacton volumes and quoted bd-ask spreads that are establshed covarates known to nfluence trade duraton. The th observaton has a condtonal jont densty gven by: (, ) ~ (, ~, ~ z I t 1 h z 1 z 1 ; π k ) (1) ~ where ~ 1 and z 1 represent prevous observatons of duratons and marks up to the (-1) th trade and π k belongs to the set of parameters of the densty functon. Condtoned on past duratons and marks, the jont densty functon gven by (1) can 13
be epressed n terms of the product of the margnal densty of the marks gven the duratons. That s: h(, ~, ~ ; ) ( ~, ~ ; ) (, ~, ~ z 1 z 1 π k = m 1 z 1 π n z 1 z 1; π z ) () In equaton () above, m ~, ~ z ; π ) s the margnal densty of the duraton, ( 1 1, wth parameter, π, and condtoned on the prevous duratons and marks. n, ~, ~ 1 z ; π z ) s the densty of the mark, z, wth parameter, ( z 1 π z, and condtonal on the contemporaneous and prevous duratons as well as prevous marks. The parameters, π and π z are estmated from the log-lkelhood functon, = n l π, π ) [ log m( ~, ~ z ; π ) + log n( z, ~, ~ z ; π )] (3) ( z = 1 1 1 1 1 z Engle (000) uses the property that duratons can be consdered as weakly eogenous wth respect to marks (Engle, Hendry and Rchard, 1983) to smplfy the estmaton process. Ths property allows the two parts of the lkelhood functon gven by equaton (3) to be mamsed separately. A pont process evolves wth after effects f, for any t > t0, the realzaton of events on [ t, ) depends on the sequence of trades that occur n the nterval [ t 0, t ) (Synder and Mller, 1991). Furthermore, t s defned to be condtonally orderly at tme t t 0, f the probablty of two or more trades occurrng n a suffcently short perod of tme s nfntesmal relatve to the probablty of one trade occurrng. The condtonal ntensty functon, or hazard rate, descrbes a pont process wth after effects. It defnes the nstantaneous rate of the net trade at tme t, condtonal upon no trade eventuatng untl tme t and s gven by: 14
( t, ~, ~ z ; ) = m( t t 1 1 1 λ 1 1 π m s t ~ ~ (4) ( 1 1, z 1; π ) ds s > t ~, ~ z ; π ) The survvor functon, S ( t, ~, ~ 1 z 1; π ) specfes the probablty that the tme untl the net trade arrves s greater than some tme t. Along wth the condtonal ntensty functon gven by equaton (4) and the duraton between trades,, t descrbes the condtonally orderly pont process. The Autoregressve Condtonal Duraton (ACD) Model Engle and Russell (1998) ntroduced a margnal duraton model called the autoregressve condtonal duraton (ACD) model. They defned the condtonal epected duraton as: ψ E I ) ψ ( ~, ~ z ; ), (5) = ( 1 = 1 1 θ where the θ ' s are parameters. If a multplcatve error structure s assumed where: = ψ ε, (6) and the duratons are standardsed, then the standardsed duratons, ε, are gven by: ε = (7) ψ If the ε ' s are assumed to be ndependent and dentcally dstrbuted (..d.) then: E ( ) = E ( ψ ε ) = ψ E ( ε ) = ψ, therefore, E ( ε ) = 1. It follows that : m ~, ~ z ( 1 1 m ( ψ ; π ), ; π ) = 15
wth temporal dependence n the duraton process captured by the condtonal epected duraton. Engle (000) ponts out that under the assumpton that the ε ' s are ndependent, then the log-lkelhood gven by equaton (3) s easly evaluated and the ntensty (hazard) can be computed as a functon of the densty of the standardsed duratons, ε. The hazard functon gven n equaton (4), namely λ ( t, ~, ~ 1 z 1; π ), s clearly a functon of the duratons and the marks. In order to obtan the hazard as a functon of ε, we need a hazard functon derved from the transformaton gven n equatons (6) and (7). The smplest transformaton of a non-negatve functon s the homothetc transformaton. The homothetc transformaton = ψ ε, orε =, s ψ a scaled transformaton that s also known as the accelerated tme model. In the duraton lterature ths homothetc transformaton s vewed as a change of scale of the tme varable and produces a proportonal hazard model wth the followng derved hazard functon gven by: ~ ~ 1 λ ( t,, z 1 ; π ) = λ 0 ( t, ψ ψ ; π 1 ε, ) t 1 t < t (8) λ 0 ( t, ; π ε ) s a baselne hazard functon that s a functon of tme. The hazard ψ or condtonal ntensty gven by equaton (4) s then derved by multplyng the baselne hazard functon by the parameter 1, whch s the recprocal of the epected ψ duraton. By ncorporatng the countng process, N(t), that refers to the number of trades (event arrvals) that have occurred at or pror to tme t, the derved ntensty functon gven by equaton (8) can be re-epressed as: 16
~ ~ 1 N ( t ) λ ( t, 1, z 1 ; π ) = λ 0 ( t, ; π ε ) (9) ψ ψ N ( t ) N ( t ) The baselne hazard functon measures the nstantaneous rate of arrval of the net trade based on the hstory of duratons and the magntude of the epected duraton, ψ. Because ψ enters the baselne hazard functon, the duraton as measured n economc or transactons tme wll be accelerated by a factor that depends on the magntude of the epected duraton. The smaller the epected duraton, the faster s the acceleraton of economc tme relatve to calendar tme. Wth the baselne hazard functon a functon of the accelerated tme between trades, equaton (9) has been often descrbed as an accelerated falure tme model (Engle, 000). Let p ; π ) ( ε ε be the probablty densty functon for the standardsed duratons ε wth parameters π. 4 Wth the ntensty functon n equaton (9) a scaled tme ε transformaton of a condtonally ordered pont process of the duraton between trades,, t follows that the correspondng margnal densty of the duratons can be wrtten n terms of the standardsed duratons, ε, as: m ~, ~ z ( 1 1 ; π ) = m ( ψ ; π ) 1 = p ( ; π ε ) (10) ψ ψ Incorporatng equaton (10) above, along wth the multplcatve error structure of the model gven by equaton (6), the log-lkelhood functon gven by equaton (3) can be rewrtten as: N ( t ) l ( π, π ε ) = [ log p( ; π ) logψ ] = 1 ε (11) ψ 4 p ( ε ; π ε ) requres non-negatve support due to the multplcatve error structure and the nonnegatvty of the duraton sequence. 17
Condtonal on the specfcaton of a parametrc dstrbuton for ε, mamum lkelhood estmates for π and π ε can be obtaned by numercal optmzaton. The ACD model s a hghly fleble model that allows for a varety of parameterzatons of the epected duraton, ψ, and the dstrbutons for ε. The standard form of the ACD model s attrbuted to Engle and Russell (1998) and t s ths form of the model that s utlsed for the analyss of volatlty n ths paper. 5 4.3 The Standard ACD Model The standard ACD model of Engle and Russell (1998) reles on a lnear parametersaton of equaton (5) where ψ s epressed as an autoregressve equaton that depends on past and epected duratons. The ACD (1, 1) model, where the present epected duraton, ψ, s a lnear functon of the prevous duraton and epected duraton, s gven below n equaton (1): ψ = ω + α + βψ (1) 1 1 The bndng constrants gven by ω>0, α>0 and α + β<1 ensures the condtonal duratons for all realsatons are postve and that the duratons,, are covarance statonary. Rather than specfyng a parametrc dstrbuton for the standardsed duratons to estmate parameters of the epected condton duraton equaton gven by (1) and the hazard (or condtonal ntensty functon), Engle (000) uses a sem-parametrc densty estmaton approach. He uses the parametrc eponental dstrbuton to estmate the parameters of the condtonal mean functon. Apart from beng a dstrbuton defned on postve support, t provdes quas-mamum lkelhood ε 5 For an ecellent dscusson of fleble alternatves to the standard form of the ACD model attrbuted to Engle and Russell (1998), see Pacurar (006). 18
(QML) estmators for the ACD model parameters (Engle and Russell, 1998). They are also consstent estmators. 6 However, recognsng that the choce for the dstrbuton of the error term mpacts on the hazard functon and that the flat nature of the condtonal hazard functon that results from usng the eponental dstrbuton s not approprate for most fnancal data applcatons, Engle (000) estmates the baselne hazard non-parametrcally. Hs non-parametrc approach s to use the resduals from the estmated standardsed duratons, ε, to estmate the baselne hazard, λ 0( t), functon by calculatng a sample hazard functon and smoothng t. The sample hazard functon s estmated by a Kaplan Meer estmate (essentally a k-nearest neghbour estmator) and smoothed by applyng a wder bandwdth. Once the baselne hazard functon s estmated, then equaton (9) s used to estmate the hazard for a partcular arrval from: 1 t t 1 λ ( t) = λ 0 ( ) for t 1 t < t (13) ψ ψ The hazard s the falure rate per unt of tme. It s measured as the number of falures dvded by the number of ndvduals at rsk at that unt of tme. Accordng to Engle (000), the falure rate for the smallest k standardsed duratons s k dvded by the number at rsk. For estmate of the hazard rate s gven by: n, the number of trades at t, then the k-nearest neghbour k λ ( t ) = (14) n ( t + k t k ) 6 Drost and Werker (004) show that consstent estmators are obtaned when QML estmaton s appled to dstrbutons that belong to the famly of Gamma dstrbutons of whch the eponental s a member. 19
4.4 Estmatng the Hazard Functon By way of eample, we estmate the hazard functon from the fltered trades of one of the stocks analysed n ths study and lsted n Table 1, namely, the Commonwealth Bank of Australa (CBA). Before computng the hazard usng the approach outlned n secton 4.3 above, the adjustment process requred to account for ntraday seasonal effects s dscussed. Followng both Engle and Russell (1998) and Engle (000), the data was durnally adjusted to remove any day-of-the-week effects, or ntraday seasonalty lkely to dstort the estmaton results. Ths process s based upon the assumpton that the data ehbts ntra-daly seasonalty wth hgher tradng actvty at the begnnng and the end of the tradng day (shorter duratons), and longer duratons correspondng to slower actvty outsde these perods. Ths s clearly demonstrated n Fgure for the eght stocks consdered n ths study. These tradng patterns are regarded as characterstc artefacts of the echange tself as well as the behavour of traders who, for eample, trade on overnght nformaton at the start of tradng and close ther postons at the fnsh (Bauwens 001:5). 7 An assumpton underlyng the adjustment process made by Engle and Russell (1998) was that the ntraday duratons,, can be multplcatvely decomposed nto a determnstc tme-of-day (seasonal) component at tme t, φ ( t 1 ), and a stochastc counterpart ~ that captures the dynamcs of the duratons such that ~ φ ( t ). The epected condtonal duraton gven by equaton (5) s now wrtten as: = 1 ψ E ~ I ) φ ( t ) (15) = ( 1 1 7 Not only was ths process appled to the duratons between trades, but t was also appled to the other market-mcrostructure varables analysed n ths research, namely, bd-ask spreads, returns, and average trade sze. 0
A pecewse-lnear splne was ftted to the trades of all stocks durng tradng hours wth 1 knots representng each half hour of tradng. Effectvely, the duratons were regressed on the tme-of-day and the durnally adjusted duratons obtaned from equaton (15) by takng ratos of the duratons to ther ftted values. 8 Followng the adjustment process the autocorrelaton n the data was substantally reduced. Whle the seasonal adjustment process does not affect the man propertes of duratons, some authors have noted the need for further nvestgaton to better understand ts mpact (see Metz and Teräsvrta, 006). Calculatng the sample hazard functon from equaton (14) usng a 000-nearest neghbour estmator results n the hazard functon depcted n the lower rght corner of Fgure 6. Fgure 6 Graphs of ntraday duraton, returns, spread, volume-of trades and the hazard functon for the Commonwealth Bank of Australa (CBA) 50 3600.4 300.0 40 800 1.6 DURF 30 0 SIZADJF 400 000 1600 1. 0.8 0.4 10 35000 40000 45000 50000 55000 60000 TIMEA DJ 100 35000 40000 45000 50000 55000 60000 TIMEAD J 0.0 50000 60000 70000 80000 90000 100000 HAZARD.00011.00046.4 RETNDF.00010.00009.00008.00007.00006 SPREADF.00044.0004.00040.00038.00036.00034 HAZARD1.0 1.6 1. 0.8.00005.00004 35000 40000 45000 50000 55000 60000 TIMEADJ.0003.00030 35000 40000 45000 50000 55000 60000 TIMEADJ 0.4 0 1 3 4 DURATION1 8 Alternatve procedures have been appled by others n the lterature. They nclude the use of cubc splnes by Engle and Russell (1998) and Bauwens and Got (000), quadratc functons and ndcator varables by Tsay (00) and Drost and Werker (004), whle Dufour and Engle (000) nclude durnal dummy varables n a vector autoregressve system. 1
Consstent wth the hazard functon for IBM trades reported n Engle (000), there s a sharp drop n the hazard for CBA for small duratons followed by a gradual declne for longer duratons. 5. Volatlty Models Engle (000) ntroduced a volatlty model n tck tme that was based on the decomposton of the jont densty functon of the sequence of duratons and marks gven by equaton (). He provded a sutable framework for the jont modellng of duratons between events of nterest,, and market characterstcs, z. Market mcrostructure varables such as prce, bd-ask spread and volume of trades (marks) nfluence trade duraton and convey nformaton about volatlty. The jont effect of duraton and these marks on volatlty allow for the testng of theores that predct market mcrostructure behavour. Volatlty s usually measured over fed tme ntervals. However, volatlty of an asset prce over a short between-trade nterval s lkely to be dfferent to volatlty over a longer duraton. To account for dfferences n asset prce volatlty correspondng to dfferent duraton between trades, and how these dfferences are affected by nfluental covarates or marks, Engle (000) ntroduced an ACD-GARCH model that he called the Ultra Hgh-Frequency GARCH (UHF-GARCH) model. An ACD model of the type defned n equatons (5) and (6) s used to descrbe duraton condtoned on the past nformaton set. Returns are then modelled by a GARCH model, adapted for rregularly tme-spaced data by measurng volatlty per unt of tme and condtoned on contemporary and past duratons. Usng the property that duratons are weakly eogenous (see secton 4.1, page 1), the ACD model s estmated frst. Volatlty s then estmated from the GARCH model usng epected
and contemporaneous duraton estmates from the frst stage, along wth selected covarates. 9 We defne r to be the return from the (-1) th to the th transacton. Further, defne the condtonal varance per transacton as: V = 1 ( r ) h (16) Ths varance s condtonal on contemporaneous duraton and past returns. As prevously noted, n order to adapt the varance for rregularly tme-spaced data we requre an adaptaton of equaton (16) to derve the varance per unt of tme. Followng Engle (000) ths becomes: r V 1 ( ) = σ (17) The relatonshp between the two varances gven by equatons (16) and (17) s gven by equaton (18) below: h = σ (18) The predcted transacton varance, condtonal on past returns and duratons, s as follows: E ( h ) = E ( σ ) (19) 1 1 Engle (000) models the seres, r, as an ARMA(1, 1) process wth an nnovaton term gven by e. It follows that: r r ρ, (0) 1 = + e + φe 1 1 9 It s possble that the reverse stuaton holds where volatlty has an mpact on duraton and gnorng ths mpact fals to recognse part of the comple relatonshp that ests between volatlty and duraton. We leave ths consderaton for future research endeavours. 3
and the varance of r per unt of tme s the epected value of the square of e. 5.1 The Effect Of Duraton Clusterng On Volatlty: Theoretcal Eplanatons In the theoretcal market-mcrostructure lterature, nformaton-based models are based on the dea that there are dfferent degrees of nformaton present n the market and there are two categores of traders: namely, nformed and unnformed. Unnformed traders trade for lqudty needs and are assumed to trade wth constant ntensty. Informed traders have superor nformaton and trade to take advantage of ther prvate nformaton. Informed traders reveal ther prvate nformaton by tradng and unnformed traders learn from the tradng of the nformed. Ths nformaton transfer between the two categores of trader s conveyed through such trade characterstcs as tmng, prce, volume and bd-ask spread. The nformaton-based models dscussed below endeavour to eplan the comple relatonshp that ests between these market mcrostructure varables. Tmng of trades s an mportant varable n understandng nformaton flows and hghlghts the usefulness of the ACD modellng approach. The models of Damond and Verrecha (1987) and Easley and O Hara (199) were among the frst to recognse that traders were lkely to learn from the tmng of ther trades. The presence of ether nformed traders or lqudty (unnformed) traders n the market was sgnalled by the ncdence of short or long duraton clusterng. Damond and Verrecha (1987) argued that long duraton clusterng was assocated wth bad news. Ther eplanaton reled on the assumpton that no short-sellng was permtted n the market. When bad news ht the market, nformed traders were unable to take advantage of t by short-sellng and dd not trade. Easley and O Hara (199) suggested that the sequence of trades mpled nformaton flows relatng to agents and systematc market news. Informed traders only traded when new 4
nformaton entered the market, whle lqudty traders were assumed to trade wth constant ntensty. Informaton events (ether good or bad news) were assumed to be assocated wth short duraton clusterng through the ncreased actvty of nformed traders. A dfferent eplanaton of duraton clusterng was advanced by Admat and Pflederer (1988) who eplaned t by dstngushng between two types of unnformed traders. Dscretonary traders were unnformed but able to tme ther trades, whle non-dscretonary traders were also unnformed but unable to tme ther trades. The optmal behavour for dscretonary traders was to trade frequently when the non-dscretonary traders dd n order to protect themselves aganst adverse selecton costs. Informed traders were supposed to follow the tradng patterns of the dscretonary traders. In all these models duraton clusterng occurs wth nformaton events and affects volatlty. 5. The Basc Volatlty Model The smple GARCH (1, 1) model can be used to model volatlty as a varable dependent upon both economc tme and actvty. It s modfed to capture the varance per unt of tme n order to account for the stochastc nature of economc tme. Rather than relyng on dscretely sampled data, or the aggregaton of the data at fed ntervals, each transacton perod s used: σ ω + α 1 + = e βσ (1) Whle the mean of the r seres s modelled as an ARMA(1, 1) process as detaled above, the varance gven by equaton (1) assumes that news from the last trade s captured as the square of the last nnovaton from equaton (0), and the persstence of news s unaffected by duraton. Engle (000) suggests that ths mples that the coeffcents can be taken as fed parameters. The results from the estmaton 5
Table 6 Estmaton results for equaton (1): σ ω + α e 1 + βσ =. Stock Varance Equaton Coeffcent Prob CBA C 0.8078 0.004 RESID(-1)^ 0.150 0.0000 GARCH(-1) 0.709 0.0000 MBL C 0.489 0.0000 RESID(-1)^ 0.58 0.0000 GARCH(-1) 0.7197 0.0000 QBE C.3477 0.0000 RESID(-1)^ 0.3147 0.0000 GARCH(-1) 0.3314 0.0000 RIN C 0.397 0.0000 RESID(-1)^ 0.1596 0.0000 GARCH(-1) 0.7857 0.0000 WDC C 0.7808 0.0000 RESID(-1)^ 0.134 0.0000 GARCH(-1) 0.6794 0.0000 WES C 1.555 0.000 RESID(-1)^ 0.4599 0.0067 GARCH(-1) 0.5539 0.0000 WOW C.749 0.0000 RESID(-1)^ 0.350 0.0000 GARCH(-1) 0.1739 0.0000 WPL C 1.705 0.0000 RESID(-1)^ 0.085 0.0000 GARCH(-1) 0.568 0.0000 of equaton (1) can be found n Table 6. It s a GARCH (1, 1) model where duratons and other economc varables are ecluded. Of partcular nterest n any GARCH model s the level of persstence as ndcated by the sum of the alpha and beta coeffcents. It s the presence of persstence that ndcates volatlty clusterng and the approprateness of the GARCH specfcaton for the data beng analysed. Engle (000) reports ths statstc to be less than 0.5 and regards ths value as very small for such a hgh frequency data set. 10 On the other hand, from Table 6 t can be seen that all of the sampled stocks n ths research ehbted persstence greater than 0.64 wth three of the eght securtes analysed 10 Ths was also a result found by Ghose and Kroner (1995) and Anderson and Bollerslev (1997) when analysng hgh frequency data n calendar tme. 6
havng a value greater than 0.85. The persstence s a measure of long-term volatlty. Alpha measures the news from the last perod and s reported to be around 0.3 for IBM n the Engle (000) study, but vares for the ASX stocks from around 0.1 (WDC) to 0.46 (WES). The beta s smply the last perod forecast of the varance and vares from an abnormally low value of 0.1739 for WOW to a more characterstc hgh of 0.7857 for RIN. The lmtaton of ths model s that duraton and the volatlty of returns are not necessarly related to the same nformaton events. Ths s lkely to be unrealstc. It does not nclude the possblty that the current duraton may affect the volatlty of the returns process through nformaton-based tradng. The theoretcal constructs dscussed above cannot be mplemented or tested usng the basc model presented n equaton (0). An ntutve etenson of ths model specfcaton s to augment the estng model wth economc tme varables, such as duraton and ts varants, as well as economc actvty ndcators such as the bd-ask spread and traded volume. Ths s done n the followng sectons. 5. A Volatlty Model Wth Duraton Included As a Covarate To facltate the fact that duratons and volatlty can be drven by the same news events, t s useful to nclude another ndependent varable that represents the duraton. By ntroducng duratons nto the GARCH framework the condtonal varance s gven by: σ = ω + α e + βσ + γ () 1 1 1 1 The market-mcrostructure lterature s not based upon a consensus vew of how the sequence of trades affects volatlty or ndeed, what type of nformaton can be garnered from ths sequence. Therefore, n order to provde greater support for any of the publshed theores, the basc GARCH model gven by equaton (1) has been 7
etended to offer possble etra eplanatory power. The etensons are varatons of the duraton varable n several guses. The coeffcent γ 1, n equaton () above provdes some ndcaton of the effects of duraton on the current perod s volatlty. If the theory of Easley and O Hara (199) s emprcally verfable, then short duratons followng an nformaton event would ncrease volatlty. Accordngly, t s epected that γ 1 would be sgnfcant and postve. As the duraton enters equaton () as a recprocal, then a longer duraton would ndcate no news and a reduced mpact on volatlty. Ths was certanly the case when Engle (000) analysed the volatlty of IBM at the NYSE. As shown n Table 7, ths vew s renforced n ths study. All the sampled ASX stocks were charactersed by postve and sgnfcant coeffcents. Commonwealth Bank (CBA), the most frequently traded stock n the sample, had the smallest postve and sgnfcant value for γ 1. Whle ths mples that news events mpact postvely on the volatlty for ths stock, the mpact s not as pronounced as wth the other stocks n the study. Ths s understandable gven that duraton clusterng s lkely to be more pronounced for ths stock due to a greater concentraton of lqudty tradng across perods of news or no news. The results n Table 7 go some way to confrmng the proposton of Easley and O Hara (199). As dstnct from the volatlty model specfed n equaton (1), persstence of volatlty depends on the persstence of duratons as well as the GARCH parameters n the model gven by equaton (). 5.3 A Volatlty Model Wth Three Duraton Covarates And Long-run Volatlty Varable Included By lookng at the recprocal of duraton n solaton, there may be underestmaton of the mpact that duraton (the tradng ntensty) has on varous ndependent varables that mpact on volatlty. It s possble to argue that the model gven n 8
1 Table 7 Estmaton results for equaton (): σ = ω + αe + βσ + γ. 1 1 1 Stock Varance Equaton Coeffcent Prob. Coeffcent Prob. CBA C 0.337861 0.343 WDC C -0.0538 0.0000 RESID(-1)^ 0.177545 0.0000 RESID(-1)^ 0.043598 0.0000 GARCH(-1) 0.594976 0.0003 GARCH(-1) 0.0067 0.000 1/DURS 0.8919 0.0098 1/DURS 0.619977 0.0000 MBL C -0.00561 0.6978 WES C 4.551137 0.0000 RESID(-1)^ 0.15941 0.0000 RESID(-1)^ 0.034066 0.114 GARCH(-1) 0.034408 0.0000 GARCH(-1) 0.54374 0.0714 1/DURS 0.663868 0.0000 1/DURS 0.41155 0.001 QBE C 0.039976 0.154 WOW C 0.03378 0.0999 RESID(-1)^ 0.059685 0.0000 RESID(-1)^ 0.087419 0.0000 GARCH(-1) 0.006511 0.0157 GARCH(-1) 0.00333 0.1033 1/DURS 0.64463 0.0000 1/DURS 0.774358 0.0000 RIN C -0.00406 0.776 WPL C 0.046087 0.36 RESID(-1)^ 0.104143 0.0000 RESID(-1)^ 0.133995 0.0000 GARCH(-1) 0.015734 0.0000 GARCH(-1) 0.003703 0.1987 1/DURS 0.598486 0.0000 1/DURS 0.746444 0.0000 equaton () s ms-specfed due to an nadequate number of duraton-related varables (both observed and epected) and other economc varables that affect volatlty. To overcome ths defcency we augmented to equaton () a measure of the surprse n duratons,, and an ndcator of epected trade arrvals gven by the ψ recprocal of epected duratons, ψ 1. Followng Engle (000), we also ncorporated nto the specfcaton gven by equaton (3) below, a varable, ξ 1 measures long-run volatlty. (3) σ ω α βσ γ 1 1 = + e 1 + 1 + 1 + + 3 1 + 4 ψ γ γ ξ γ ψ, that 9
ξ s the measure of long-run volatlty and s computed by eponentally smoothng the seres, r, wth a smoothng parameter equal to 0.995. Ths results n the r eponental smoothng equaton: ξ = 0.005 ( 1 ) + 0. 995 ξ 1 The results for the recprocal of duraton, gven as 1/DURS n Table 8, are consstent wth those reported n Table 7, wth all values postve and sgnfcant. Ths supports the Easley and O Hara (199) proposton that shorter (longer) duratons would ndcate news (no news) and result n a greater (lesser) mpact on volatlty. Takng ths concept one step further, DURS/EDURS, s taken to be the surprse n duratons by Engle (000) and s shown to have a negatve coeffcent n ths study, ecept for QBE. Should the duraton between the latest trade and the one precedng t be dfferent, then ths should reflect the short-run mpact of duratons. If the actual duraton s greater than what s epected, then the surprse would ndcate a reducton n volatlty. In ths case, the rato would be greater than one and translate to a reducton n volatlty as long as the coeffcent s negatve. Ths was the case for the varables analysed n ths study, ecept for QBE. Whle the coeffcent for WPL was negatve, t lacked statstcal sgnfcance. A surprse could mean that ether new nformaton has been released or that the actons of traders may be of some nterest. In partcular, the queston can be rased as to whether or not the margnal nvestor s nformed or s smply a nose (lqudty) trader. Engle (000) analysed only one stock fndng a negatve coeffcent. The results of ths study are more conclusve but requre more evdence before support can be gven to one or more of the theoretcal propostons. 1 30
Table 8 Estmaton results for equaton (3): σ = ω + αe 1 + βσ 1 + γ1 + γ + γ 3ξ 1 + γ 4ψ ψ 1 1 Stock Varance Equaton Coeffcent Prob. Stock Varance Equaton Coeffcent Prob. CBA C 0.400054 0.13 WDC C 0.37017 0.0000 RESID(-1)^ 0.7648 0.0000 RESID(-1)^ 0.8581 0.0000 GARCH(-1) 0.497933 0.0000 GARCH(-1) 0.494371 0.0000 1/DURS 0.438148 0.0000 1/DURS 0.49470 0.0000 DURS/EDURS -0.13979 0.0000 DURS/EDURS -0.06181 0.0000 LONGVOL(-1) 0.113373 0.0039 LONGVOL(-1) 0.0964 0.0000 1/EDURS 0.063631 0.8651 1/EDURS 0.068578 0.1505 MBL C 0.98417 0.0000 WES C 0.575973 0.0000 RESID(-1)^ 0.194057 0.0000 RESID(-1)^ 0.99874 0.0173 GARCH(-1) 0.483377 0.0000 GARCH(-1) 0.176 0.0000 1/DURS 0.447814 0.0000 1/DURS 0.696795 0.0000 DURS/EDURS -0.11778 0.0000 DURS/EDURS -0.036880 0.0008 LONGVOL(-1) 0.0856 0.0138 LONGVOL(-1) 0.001418 0.607 1/EDURS 6.70E-05 0.9987 1/EDURS -0.465357 0.0000 QBE C -0.14338 0.0130 WOW C 0.40000 0.0018 RESID(-1)^ 0.058131 0.0000 RESID(-1)^ 0.30000 0.0000 GARCH(-1) 0.00719 0.0011 GARCH(-1) 0.50000 0.0000 1/DURS 0.69308 0.0000 1/DURS 0.50000 0.0000 DURS/EDURS 0.09797 0.0001 DURS/EDURS -0.050000 0.0579 LONGVOL(-1) 0.0345 0.0000 LONGVOL(-1) 0.10000 0.0000 1/EDURS -0.06304 0.799 1/EDURS 0.10000 0.455 RIO C 0.67065 0.0000 WPL C 0.477850 0.1141 RESID(-1)^ 0.73693 0.0000 RESID(-1)^ 0.76570 0.0000 GARCH(-1) 0.479469 0.0000 GARCH(-1) 0.0086 0.3381 1/DURS 0.47743 0.0000 1/DURS 0.87734 0.0000 DURS/EDURS -0.199 0.0000 DURS/EDURS -0.007001 0.9630 LONGVOL(-1) 0.07366 0.0000 LONGVOL(-1) -0.09006 0.0017 1/EDURS -0.04154 0.33 1/EDURS 0.1083 0.109 The recprocal of the epected arrval rate (1/EDURS) s a reflecton of how the pror perod s forecast affects the current volatlty of returns. Ths was gven as a postve coeffcent when Engle analysed IBM, but produced med results accordng to the sgns of the coeffcents for the sample of ASX stocks. Only n the case of WES was the coeffcent sgnfcant. Of the postve coeffcents for the epected duraton, the hghest belonged to WPL and then n dmnshng order, WDC, WOW, CBA, and MBL. The stocks wth negatve coeffcents were WES, QBE, and RIO. The fact that 31
the coeffcents are not drectonally unform has mplcatons n marketmcrostructure terms. Engle (000) descrbes ths varable as the tradng ntensty whch ndcates the relaton between transacton ntensty and volatlty. In ths case the med results ndcate that hgher tradng ntensty n some stocks leads to lower volatlty and vce versa. It could also mean that the epected arrval rate s more stochastc than frst assumed n the modellng process. The long-run volatlty parameter drectly ncorporates persstence nto the model. In all cases for the ASX stocks, results are n accordance wth Engle (000) as the long-run volatlty postvely affects volatlty. By weghtng the varable by only.005 of the current trade and.995 to the pror transacton, the model mplctly assumes that volatlty s a long-run varable. 5.4 Volatlty Models Wth Addtonal Market Mcrostructure Varables Engle (000), Engle and Russell (1998), and Bauwens and Got (000) take the epectaton of the volatlty model gven n equaton (3), condtonal on past nformaton, and add market-mcrostructure varables to add further eplanatory power. Ths s a genune contrbuton as t jontly determnes the forecastng mpact of market-mcrostructure varables on volatlty, rather than followng the pror practce of concentratng on one lqudty varable at a tme. In addton to the duraton and economc varables consdered prevously n equaton (3), dummy varables for the lagged bd-ask spread and contemporaneous volume-of-trade varables are ncluded as covarates n equaton (4) below. The dummy varables take values of one when the bd-ask spread and the volume-of-trade varables are two standard devatons from ther means. 1 1 E 1 ( σ ) = ω + α e 1 + βσ 1 + γ 1 + γ + γ 3ξ 1 + γ 4ψ + γ 5ς 1 + γ 6υ (4) ψ 3