The Journal of Entrepreneural Fnance Volume 8 Issue 2 Summer 2003 Artcle 4 December 2003 The Informaton Content n Trades of Inactve Nasdaq Stocks Peter Chen Youngstown State Unversty Kasng Man Syracuse Unversty Chunch Wu Syracuse Unversty Follow ths and addtonal works at: http://dgtalcommons.pepperdne.edu/jef Recommended Ctaton Chen, Peter; Man, Kasng; and Wu, Chunch (2003) "The Informaton Content n Trades of Inactve Nasdaq Stocks," Journal of Entrepreneural Fnance and Busness Ventures: Vol. 8: Iss. 2, pp. 25-53. Avalable at: http://dgtalcommons.pepperdne.edu/jef/vol8/ss2/4 Ths Artcle s brought to you for free and open access by the Grazado School of Busness and Management at Pepperdne Dgtal Commons. It has been accepted for ncluson n The Journal of Entrepreneural Fnance by an authorzed admnstrator of Pepperdne Dgtal Commons. For more nformaton, please contact paul.stens@pepperdne.edu.
The Informaton Content n Trades of Inactve Nasdaq Stocks Peter Chen + Youngstown State Unversty, Kasng Man ++ Syracuse Unversty and Chunch Wu +++ Syracuse Unversty In ths paper we analyze the frequency and nformaton content of small Nasdaq stock trades and ther mpacts on return volatlty at the ntraday nterval. We employ an autoregressve condtonal duraton (ACD) model to estmate the ntensty of the arrval and nformaton content of trades by accountng for the determnstc nature of ntraday perodcty and rregular tradng ntervals n transacton data. We estmate and compare the prce duraton of thnly and heavly traded stocks to assess the dfferental nformaton content of stock trades. We fnd that the number of transactons s negatvely correlated wth prce duraton or postvely correlated wth return volatlty. The mpact of the number of transactons on prce duraton or volatlty s hgher for thnly traded stocks. On the other hand, the persstence of the mpact on prce duraton adjusted for ntradaly perodcty s about the same for thnly and heavly traded stocks on average. + Peter Huayu Chen s an Assstant Professor of Fnance at Youngstown State Unversty. Hs current research nterests nclude fxed ncome nvestments, market mcrostructure and credt rsk analyss. ++ Kasng Man s an Assstant Professor n Manageral Statstcs at the Martn J. Whtman School of Management, Syracuse Unversty. Hs research nterests nclude tme seres analyss and forecastng methods. +++ Chunch Wu s a Professor of Fnance at the Martn J. Whtman School of Management, Syracuse Unversty. Hs current research nterests nclude fxed ncome nvestments, market mcrostructure and entrepreneural fnance.
26 Introducton The subject of prce formaton has always been ntrgung to fnancal researchers. A vast fnancal lterature has been devoted to the study of the pattern of nformaton arrvals and how new nformaton s ncorporated nto prce. These studes range from smple event studes of the market response to news announcements, to more sophstcated nformaton flow studes analyzng how nformaton nnovatons are mpounded nto securty prces. Interest n ths ssue has been fueled by recent advances n market mcrostructure theory and the avalablty of ultra-hgh-frequency data, thanks to modern technology. The shft to hgh-frequency data analyss has posed sgnfcant challenges to emprcal studes. A major dffculty faced n hgh-frequency data studes s that transactons arrve n rregular tme ntervals. Most emprcal mcrostructure studes have employed data wth a fxed tme nterval (e.g., hourly or half-hourly) to test the mplcatons of market mcrostructure theory (see, for example, Foster and Vswanathan, 1995; Andersen and Bollerslev, 1997). Ths s because standard tme-seres econometrc technques buld on the premse of fxed tme ntervals. The selecton of the tme nterval s often arbtrary. Large heavly-traded stocks typcally have transactons every few seconds whereas small thnlytraded stocks may not have transactons every hour or day. If a short tme nterval s chosen, there wll be many ntervals wth no transactons for thnly traded stocks and heteroskedastcty of a partcular form wll be ntroduced. On the other hand, f a long nterval s chosen, we may lose most of the mcrostructure features of the data. In partcular, when transactons are averaged, the tmng relaton and characterstcs of trades wll be lost. Emprcal mcrostructure studes that examne transacton-by-transacton data (see, for example, Hasbrouck, 1991; Madhavan et al., 1997; Huang and Stoll, 1997) face a dfferent estmaton problem. Data ponts n these studes correspond to the transacton (event) tme and so they are rregularly spaced. However, these studes have typcally gnored the problem of rregular ntervals when applyng standard tme-seres econometrc technques. Assumng that data ponts are equally spaced, when n fact they are uneven, leaves out much of the mportant nformaton about trade clusterng, temporal order flow patterns and the nformaton assmlaton process. Fortunately, new econometrc methods have been developed recently to cope wth the estmaton problems of rregularly spaced data (see Engle, 2000; Dufour and Engle, 2000). Two tme-seres methods were developed to model rregularly spaced data: Tme Deformaton models (TD) and Autoregressve Condtonal Duraton (ACD) models. The TD approach uses auxlary transformatons to relate observatonal or economc tme to calendar tme. In contrast, the ACD approach drectly models the tme duraton between events (e.g., trades). The ACD model typcally adopts a dependent pont process sutable for modelng characterstcs of duraton seres such as clusterng and overdsperson. In ths paper, we employ the ACD model proposed by Engle and Russell (1998) to examne nformaton clusterng and tradng responses to nformaton at the ntraday level. There are advantages of usng ths model. Frst, ths model provdes a framework for measurng and estmatng the ntensty of transacton arrvals that s partcularly suted for the tradng process. The model accounts for the rregular tme nterval, typcally encountered n stock tradng, by treatng t as a random varable that follows a pont process. Ths treatment resolves nfrequent or nonsynchronous tradng problems n emprcal estmaton usng ntraday data of thnly traded
27 stocks. Second, the estmaton procedure s relatvely straghtforward and the model can be easly adapted to test varous mcrostructural hypotheses. The prmary objectve of ths paper s to examne the patterns of nformaton arrval of small thnly-traded versus large heavly-traded stocks, and ther mpact on prce movements. The tradng pattern and the ntensty of nformaton-based tradng of small thnly-traded stocks often devates sharply from that of large heavly-traded stocks. Asde from the sheer dfference n tradng frequency, large stocks enjoy much hgher lqudty than small stocks. Studes have shown that a good proporton of trades for large heavly-traded stocks s for lqudty purposes (see Easley et al., 1996). Thus, trades for large heavly-traded stocks may not always have hgh nformaton content. Conversely, small stocks are not as lqud and are not traded as heavly as large stocks. Fewer analysts are nterested n these stocks and so less nformaton s avalable for nvestors. Due to lack of nformaton and lqudty, there are typcally no transactons for a good porton of the open market tradng perod. However, tradng of these stocks often causes a sgnfcant prce movement. Thnly traded stocks also have hgher varatons n order flow. Trades are clustered n that the occurrence of a trade nduces another trade n a rather short tme nterval. Once these stocks are traded, there s a hgh probablty that nformed traders may trade to mnmze prce mpacts (see Admat and Pflederer, 1988). As nsders prvate nformaton s mpounded nto prce, return volatlty ncreases. Snce the number of trades s low for small nactve stocks, the nformaton content per trade may be hgher for these stocks. In ths paper we focus on the ntensty of transacton arrvals and ts effects on prce movements of thnly-traded stocks n the Nasdaq market. Prevous studes have shown that the prce dscovery process and bd-ask spread behavor of a dealer market such as Nasdaq dffer from those of a aucton market lke the NYSE (see Hasbrouck, 1995; Huang and Stoll, 1996). Dfferences n market structures and tradng mechansms cause varatons n tradng costs, order flows and the speed of nformaton transmsson. Therefore, emprcal fndngs of NYSE stocks do not necessarly characterze Nasdaq stocks. One dstnct feature of Nasdaq s that the depth of the market often vares wdely among stocks. Ths greater dsperson n tradng actvtes provdes an excellent opportunty for comparng the ntensty of trade arrvals and the extent to whch trades convey nformaton for heavly- and thnly-traded stocks. Most emprcal mcrostructure studes have not accounted for the uneven ntervals n stock trades n examnng the ssues of order flow and nformaton assmlaton. An excepton s Dufour and Engle (2000). However, ther study covers only the most actvely traded stocks at the NYSE. Unlke ther study, we examne both the actve and nactve stocks on Nasdaq. The remander of ths paper s organzed as follows. Secton I presents the emprcal model and methodology for estmatng the ntensty of trade arrvals and the effects of mcrostructure varables on the tme duraton of trades and prce changes. Secton II dscusses data and emprcal results. Fnally, Secton III summarzes the man fndngs of ths paper. I. The Model Informaton arrvals nduce trades and prce changes (see Admat and Pflederer, 1988; Easley and O Hara, 1992). To analyze nformaton flow at rregular arrval tmes, we employ the autoregressve condtonal duraton (ACD) model proposed by Engle and Russell (1998).
28 Denote the nterval between two arrval tmes, x = t - t -1, as duraton. The expectaton of the th duraton condtonal on past x s s gven by, where E x x, x,..., x ) ( x, x,..., x ; ) (1) ( 1 2 1 1 2 1 where s the vector of the parameters of the duraton process. Assumng that the stochastc process of the duraton s x (2) where s an..d. error term wth a dstrbuton whch must be specfed. Followng Engle and Russell (1998), we specfy the condtonal duraton by a general model: m j 1 q x, (3) j j j1 j j whch follows an ACD (m, q) process wth m and q referrng to the orders of the lags, and, j, ), j = 1, 2,, m and k = 1, 2,, q, are parameters to be estmated. Ths model ( k has a close connecton wth GARCH models and shares many of ther propertes. The model s convenent because t can be easly estmated usng a standard GARCH program by employng the square root of x as the dependent varable and settng the mean to zero (see Engle and Russell, 1998). In general, f duratons are condtonally exponental, the condtonal ntensty s (t x (4) N (t),...,x1) 1 N(t) 1 It can be shown that the hgher the condtonal ntensty, the hgher the volatlty of returns. There are several ways to estmate the system of (2) and (3). The smplest way s to assume that the error term follows an exponental dstrbuton and the lagged orders equal to one. Ths model s called the EACD(1,1) where E stands for the exponental dstrbuton. Another way s to assume that the condtonal dstrbuton s Webull, whch s equvalent to assumng that x s exponental where s the Webull parameter. Smlarly, we can estmate the Webull model wth the lagged orders equal to one, that s, WACD(1,1). The Webull dstrbuton functon can be wrtten as F( x ) = ( / ) x -1 exp[-( x / ) ] for, > 0 (5) When = 1, x / follows an exponental dstrbuton. The Webull dstrbuton s preferred f the data show an overdsperson wth extreme values (very short or long duratons) more lkely than the exponental dstrbuton would predct (see Dufour and Engle, 2000). Gven the
29 condtonal densty functon, we can estmate the parameters of the ACD model by maxmzng the followng log lkelhood functon: T L() = ln( / x ) + ln[ ( 11/ ) x / ] [ ( 11/ ) x / ] (6) 1 where (.) s the gamma functon, and s a column vector contanng the parameters to be estmated. Engle and Russell (1998) commented that clever optmzaton can avod repeated evaluaton of the gamma functon. Ths tactcs s useful when the sample sze s very large. The ACD model s essentally a model for ntertemporally correlated transacton (event) arrval tmes. The arrval tmes are treated as random varables followng a pont process. In the context of securty tradng, assocated wth each arrval tme are random varables such as volume, prce or bd-ask spread. These varables are defned as marks. Fnance researchers are often nterested n modelng these marks assocated wth the arrval tmes. For example, not all transactons occur because of the arrval of new nformaton. Instead, some are trggered by pure lqudty or portfolo adjustment reasons, whch are not related to changes n the expected (fundamental) value of stock. On the other hand, there are tmes when transactons occur as a result of new nformaton arrval that s not publcly observable. Market mcrostructure theory suggests that traders possessng prvate nformaton wll trade as long as ther nformaton has value. Ths results n clusterng of transactons followng an nformaton event. To examne ths hypothess, we can defne the events as a subset of the transacton arrval tmes wth specfc marks. For example, to examne the effect of nformaton events, we can select data ponts for whch prce has moved beyond the bd-ask bound. Ths process s called dependent thnnng. To dstngush nformed from unnformed trades, we modfy transacton arrval tmes nto prce arrval tmes. The basc dea s to leave out those transactons that do not sgnfcantly alter prce. The prce movements can be classfed ether as transtory or permanent movements. Defne the mdpont of the bd-ask spread or mdprce to be the current prce. Followng Engle and Russell (1998), we defne a permanent prce movement as any movement n the mdprce (mdquote) greater than or equal to $0.25 or 2 tcks. 1 Once we defne the prce arrval tmes, we can apply the ACD model to these new event arrval tmes. In ths case, we are modelng how quckly the prce s changng rather than the arrval rate of transactons. The ntensty functon s now called prce ntensty, whch measures the nstantaneous probablty of a permanent prce change. The basc formulaton of the ACD model parameterzes the condtonal ntensty of event arrvals as a functon of the tme between past events. It can be easly extended to nclude other effects such as characterstcs assocated wth past transactons or other outsde nfluences. For example, prevous studes have shown that mportant nformaton s contaned n the number of trades, and the trade sze whch s the average volume per transacton. To examne ths hypothess, we can modfy the ACD model to nclude these two varables: 1 A tck s 1/8 dollar.
30 p j1 j q x + #Trans + Volume/Trans (7) j j1 j j where the duraton s now between two consecutve prces wth a movement greater than or equal to two tcks, and the number of transactons per duraton and trade sze per transacton are added as determnants of duraton. Market mcrostructure theory contends that trades contan nformaton that affects prce movements (or volatlty). Includng the number of transactons and trade sze allows us to test ths mportant hypothess. In addton, dvdng the accumulated volume by the number of transactons yelds the average volume per transacton (or trade sze) at the nterval x. Prevous studes have ndcated that trade sze may contan nformaton. The ACD model n (7) now descrbes how quckly the prce changes, by takng nto consderaton the effects of transacton rate and trade sze. The ntensty functon becomes a measurement of the nstantaneous probablty of a prce movement called prce ntensty. It can be shown that prce duraton s nversely related to the volatlty of prce changes. In addton to transacton frequency and trade sze, we also test the ACD model wth the bd-ask spread varable. Mcrostructure theory suggests that the specalst s (or dealer s) bd-ask spread reflects the ntensty of nformed tradng. It wll be nterestng to see whether ths varable wll ncrease the explanatory power of the model. Thus, we also estmate the followng extended model: p j1 j q x + #Trans + Volume/Trans + Spread (8) j j1 j j where Spread s the bd-ask spread dvded by md-quote. It s wdely known that ntraday return volatlty exhbts sgnfcant determnstc (perodc) patterns. Snce prce duraton s the nverse of volatlty, the duraton measure s expected to contan a determnstc component. Ths determnstc component needs to be separated from the stochastc component n emprcal estmaton. The strategy followed here to elmnate the ntraday pattern s a smple seasonal adjustment approach. The tme span wthn a tradng day s dvded nto non-overlappng tme ntervals of 15 mnutes each. The mean of prce duratons wthn each nterval s computed over the entre sample perod. The adjusted prce duraton s then computed as the prce duraton dvded by the average prce duraton wthn that nterval. The adjusted prce duraton seres now has a mean approxmately equal to one. If the adjusted duraton s greater (less) than one, the duraton s greater (less) than the average duraton n that tme nterval. We estmate the ACD model usng these adjusted prce duratons, as well as the raw (unadjusted) duratons. 2 II. Data and Emprcal Estmaton Data on prce, sze, and tradng tme for Nasdaq stocks are obtaned from the TAQ database over the perod of July 1 to September 30, 1997. Trades and quotes are selected 2 We have also tred the splne method to flter the determnstc ntraday components. The results usng ths method are qute smlar.
31 strctly for Nasdaq-lsted frms, thus excludng NYSE stocks traded on Nasdaq and stocks lsted on regonal exchanges. We also exclude all preferred stocks, stock funds, stock rghts, warrants and ADRs. 3 Prevous studes (see, for example, Easley et al., 1996; Wu and Xu, 2000; Wu, 2003) have used tradng volume as a measure for defnng the actveness of stocks. Followng the nfluental paper by Easley et al. (1996), we use tradng volume to classfy the actveness of stocks for the purpose of comparng wth ther results. Tradng volume s a preferred measure for ths classfcaton because t contans the nformaton of frequency and sze of trades, both of whch are mportant ndcators of the actveness or depth of stocks. We rank all Nasdaq common stocks by the average daly tradng volume over the sample perod, and then dvde the sample nto volume decles. The frst volume decle ncludes the hghest-volume stocks and the tenth decle contans the lowest-volume stocks. To nsure enough tradng actvtes for purposes of emprcal estmaton, we choose stocks from the frst, ffth and eghth volume decles. To control for the prce effect, we construct a matched sample of stocks havng transacton prces close to each other at the begnnng of the sample perod (July 1), but at dfferent levels of tradng volume. Stocks from the three selected decles are ranked n order of ntal prce and adjacent trplets of stocks are matched. We randomly choose fve matched stocks from each of the three volume decles to perform emprcal estmaton. We choose only fve stocks from each volume decle for emprcal estmaton to allevate the computaton burden. Transacton duraton can be easly computed as the tme dfference between consecutve trades. Consecutve trades wth same tme stamp and prce are aggregated and treated as one trade. We then thn the transacton data by constructng prce duraton wth prce changes greater than or equal to two tcks. Volume s expressed n terms of the number of shares traded at each nterval. Table 1A shows the summary statstcs after dependent thnnng where any mdquote movements less than two tcks are gnored. More heavly traded stocks have lower spreads, more transactons (or shorter trade duratons) and hgher volume. Note that the daly number of transactons (or tradng frequency) n the hgh-volume group s hgher than those n the medum- and low-volume groups for all stocks except DURA. Smlarly, the daly number of transactons (or tradng frequency) n the medum-volume group s hgher than that of lowvolume group for all stocks except PSUN. In the analyss to follow, we compute the parameter estmates wth and wthout these two stocks. The averages wthout these two stocks represent the average parameter estmates of hgh and medum tradng frequency groups. After the data are thnned by prce, the prce duraton stll tends to be lower for more actvely traded stocks. On the other hand, trade sze or the average volume per transacton s about the same for both actve and nactve stock groups. Table 1B lsts the names of sample stocks. Fgure 1 shows the average prce duraton throughout a typcal tradng day for three selected stocks. The vertcal axs ndcates the prce duraton n seconds, and the horzontal axs ndcates the ntraday ntervals. We dvde each tradng day nto 25 ntervals of 15 3 To avod the problem at the market open (e.g., stale quotes, and delay of the open), data for the frst ffteen mnutes are dropped as suggested by Mller et al. (1994). Ths avods serous stale quote problems, especally for thnly traded stocks.
32 mnutes each. The average prce duraton wthn each nterval s computed over the entre sample perod. As shown, prce duraton exhbts an nverted U-shape pattern. Ths s not a surprse snce prce duraton s an nverse of prce volatlty, and ntraday prce volatlty exhbts a pronounced U shape. Prce duraton s negatvely related to tradng frequency or number of transactons. As ndcated, the prce duraton (n seconds) of ASND s much shorter than that of WIND because the former has a much greater number of daly transactons (see Table 1A). 2.1 Model Estmaton Usng Unadjusted Data We frst estmate the baselne ACD models wth no mcrostructure varables. We use the Polak-Rbere Conjugate Gradent (PRCG) to obtan the MLE estmates of the ACD parameters. The model s frst estmated usng the unadjusted prce duraton and then the adjusted duraton. The adjusted duraton s the prce duraton adjusted for the ntraday determnstc pattern. Table 2 reports the emprcal estmates for the EACD(1,1) model usng unadjusted data. As shown, most parameter estmates are statstcally sgnfcant. The ARCH and GARCH parameters, and, are postve n most cases, consstent wth the predcton and ther values fall n the theoretcal range. The results ndcate that a short prce duraton s lkely to be followed by another short prce duraton. Or equvalently, hgh prce volatlty n the current tradng nterval s lkely to brng hgh prce volatlty at the next tradng nterval. The sum of and represents the persstence of prce duraton. The results do not show a materal dfference n persstence for hgh and low tradng volume groups. Table 3 reports the estmates of the WACD(1,1) model. Agan, the estmates of and are postve n most cases and most of them are sgnfcant. The Webull parameter s hghly sgnfcant. The values of the Webull parameter are all less than one and tend to be smaller for less heavly traded stocks. The results suggest that the EACD model s not sutable because the error term does not follow exactly the exponental dstrbuton. The persstence of prce duraton s measured by the sum of and. Ignorng CBSS, the result agan does not show a materal dfference n persstence for hgh and low tradng volume groups. 4 We next test the mplcatons of market mcrostructure theores. On theoretcal grounds, Easley and O Hara (1992) predct that the number of transactons would nfluence the prce process through the nformaton-based clusterng of transactons. Admat and Pflederer (1988, 1989) predct that the number of transactons wll have no mpact on prce ntensty. Glosten and Mlgrom (1985) and Kyle (1985) predct that volume tends to be hgher as the probablty of nformed tradng ncreases. Most emprcal studes have documented a postve relatonshp between volatlty and volume for both ndvdual securtes and portfolos. Schwert (1989) and Gallant, Ross, and Tauchen (1992) fnd a postve correlaton between volatlty and tradng volume. Jones, Kaul, and Lpson (1994) show that the postve volatlty-volume relatonshp actually reflects the postve relatonshp between volatlty and the number of transactons. Based on ths fndng, they conclude that trade sze carres no nformaton beyond that contaned n the frequency of transactons. None 4 Note that although the estmate of for CBSS s sgnfcantly negatve, ths estmate mproves when adjusted prce duraton s used as dependent varable as shown n Table 6 below.
33 of these studes has addressed the ssue of uneven tradng ntervals or nfrequent tradng. In the followng, we re-examne ths ssue usng the ACD model at the ntraday level. We estmate the ACD model wth two addtonal explanatory varables: the number of transactons per duraton and average trade sze. Table 4 reports the results of estmaton. The coeffcents of the number of transactons are mostly negatve. The results suggest that the expected prce duraton tends to be shorter, or equvalently the volatlty s hgher, followng an nterval of hgh transacton rates. Ths relatonshp s much stronger for less-heavly traded stocks. Ths concluson holds regardless of whether DURA and PSUN are ncluded or not. On the other hand, the effect of trade sze s less conclusve. The sgn of the coeffcents of average volume per transacton (or trade sze) s negatve for more-heavly traded stocks but postve for less-heavly traded stocks. Table 5 reports the estmates of the ACD model when the bd-ask spread s added as an addtonal explanatory varable. The coeffcents of the number of transactons contnue to be qute sgnfcant wth a predcted negatve sgn. The coeffcents of trade sze agan have mxed sgns. The coeffcents of spreads are generally negatve, suggestng that hgher spreads generally lead to shorter prce duraton (or hgher volatlty). Excludng PSUN n the mddle group does not change the concluson. 5 2.2 Model Estmaton Usng Adjusted Data We next turn to the estmaton of the ACD model usng the adjusted data where prce duraton s adjusted for the ntradaly perodcty. Table 6 reports the estmates of the baselne WACD(1,1) model where no mcrostructure varables are ncluded. As shown, after removng the ntraday determnstc effect to retan the stochastc component of prce duraton, parameter estmates of the WACD(1,1) model become much more stable. The parameters and are now all wthn the theoretcal range wth a sum less than one. The results suggest that t s necessary to account for the ntraday perodc pattern n emprcal estmaton. Agan, the results show lttle dfference n the persstence of prce duraton between the hgh- and low-volume stocks. On average, the sum of and s qute close for the three groups. Table 7 reports the results of the WACD model wth mcrostructure varables. The coeffcents of the number of transactons are all negatve, ndcatng that the hgher the number of transactons, the shorter the prce duraton. The sze of the coeffcents (n absolute value) s much larger for less-heavly traded stocks. The average value of the coeffcent for the number of transactons s 0.51 for the lowest-volume group compared to -0.09 for the hghest-volume group. Excludng DURA and PSUN does not affect the results materally (- 0.06 for the hgh volume group). Thus, the mpact of the number of transactons (#Trans) s hgher not only for low volume group but also for low trade frequency group. Another nterestng fndng s that the coeffcents of trade sze have mxed sgns. The sgn tends to be negatve for most-heavly traded stocks. As the tradng volume decreases, the sgn becomes postve. Thus, trade sze does not necessarly decrease prce duraton (or ncrease prce volatlty). 5 Note that DURA n the frst group does not converge. Therefore, the results for the hgh-volume group also represent the results for stocks wth hgh-trade frequency.
34 Table 8 reports the results when bd-ask spread s added as an addtonal varable. Results show that the effect of spread s much hgher for less-heavly traded stocks. The coeffcents of spread are mostly negatve and sgnfcant for thnly traded stocks. On average, the absolute value of the spread coeffcent for the thnly traded group (-0.77) s much hgher than that of the heavly traded group, whch s close to zero. Furthermore, the absolute value of the coeffcent of the number of transactons s agan much hgher for the thnly traded group. The average value of ths coeffcent s -0.55 for the thnly traded group compared to - 0.08 for the heavly traded group. Excludng DURA and PSUN does not change the results materally (-0.06 for the heavly traded group). Moreover, the coeffcent of trade sze (average volume per transacton) s postve, whch contrasts wth the negatve sgn of the trade sze coeffcent for the heavly traded group. Thus, the prce ntensty appears to be qute dfferent between actve and nactve stocks. The response of prce ntensty to the arrval of nformaton, as captured by the spread and transactons, s much stronger for the thnly traded stocks than for heavly traded stocks. The results support the contenton that trades of thnly traded stocks have a larger mpact on ther duraton (or volatlty); that s, a trade of thnly traded stock tends to trgger another trade much faster. Ths result also suggests a greater trade clusterng for thnly traded stocks. 2.3 One-Step-Ahead Forecast of Prce Duratons Fgure 2 shows the one-step-ahead forecast of prce duratons n a tradng day for three selected stocks ASND (July 2), SEBL (July 18) and WIND (July 14), respectvely. These dates are chosen such that they are among the days wth more transactons after the thnnng process. The one-step-ahead forecasts are obtaned as follows. Consder the ACD(1,1) model for duraton n (3). Defne x, whch s a martngale dfference sequence wth mean 0. Rewrte the ACD (1,1) equaton n (3) as x ( ) x 1 1 (3a) The prce duraton x therefore has an ARMA(1,1) representaton, and ts forecast can be obtaned usng the usual ARMA method. We employ the adjusted duraton and compute ts one-step-ahead forecast, usng the WACD(1,1) estmates n Table 6. These forecasts are shown n the left-hand panel of Fgure 2 for three selected stocks. They are then multpled back by the ntraday perodc pattern (.e., average duraton wth a typcal shape as shown n Fgure 1) accordng to the tme of day the transacton occurs. The resultng forecast of the unadjusted duraton s shown n the rghthand panel of Fgure 2. The horzontal axs ndcates the sequence of transactons for ths partcular tradng day and the vertcal axs ndcates the duraton. Fgure 2 shows that actual duratons are n general subject to hgher fluctuaton than the forecasted duratons. As shown, the one-step-ahead forecast obtaned by multplyng the adjusted duraton forecast and the ntraday perodc component performs reasonably well for the less-heavly traded stocks. The forecasts for stock ASND, whch s n the heavly traded group, s comparatvely more stable at the level of around 1000 (seconds). Incorporatng the
35 determnstc ntraday perodc component nto the forecastng enhances the forecastng precson. The results suggest that one needs to consder the ntraday determnstc components to provde a good forecast for prce duraton. Engle and Russell (1997) establsh a relatonshp between prce duratons and volatlty. In bref, they assume the underlyng prce process s a bnomal process wth ncrements of c whch takes expected tme. They show (see eq. (22) of ther paper) that the expected varance per unt of tme s nversely related to expected duraton; n partcular, ˆ 2 2 c /. Usng ths relatonshp, we can transform our estmates of prce duraton to volatlty. 2.4 Impulse Response Functon We next employ the concept of mpulse response functon to examne the mpact of nformaton shock on prce duraton. Consder the ACD(1,1) model wth a mcrostructure varable z, x z 1 1 (9) Let x, and rewrte the ACD equaton (9) as x ( ) x 1 1 z Denote /( 1 ) as the mean, and B the back-shft operator such that By y 1. It follows that x 1 B z 1 ( ) B 1 ( ) B Let be the sum of the ARCH and GARCH parameters. The expanson 2 B B 1B 2... k B k... s useful for studyng the lastng effect of the mcrostructure varable z on duraton. k Furthermore, the lagged k term s, whch measures the mpact of one unt ncrease n the mcrostructure varable z on prce duraton x (mean adjusted) k-lags (or k-transactons) later. We refer to ths term as the mpulse response functon at lag k. It s clear that for less than 1 n magntude, the mpact goes down to zero eventually. But the mpact decreases wth a slower rate or s more persstent, when s closer to 1. Also, we refer to
36 2 k... (1 k1 ) /(1 ) (10) as the k-cumulatve mpulse response functon of a unt ncrease n z on prce duraton x (mean adjusted) after k transactons. It goes to a lmtng value of /( 1 ). Note that the k (frst) dfference n the k-th and (k-1)th cumulatve mpulse response functon gves. Table 9 reports the k-cumulatve mpulse response functons of two mcrostructure varables, number of transactons and average volume per transacton, based on the parameter estmates for adjusted duraton n Table 7. 6 For each stock, the frst row reports the estmates of,,, +,, and. The second row reports the k-cumulatve mpulse response functon of one unt ncrease n #Trans (the number of transactons per duraton) for k = 0, 1,.., 9 lags later, and wth the lmtng value (when k s nfnte) gven as the last value. Smlarly, the thrd row s the cumulatve response functons that correspond to Volume/Trans (or trade sze). The results show that an ncrease of one unt n #Trans has a hgher (negatve) mpact on prce duraton for stocks that are less actve. On the other hand, the mpact of a unt ncrease n Volume/Trans on prce duraton s mxed and has no clear pattern across stock groups. The response functon for #Trans s most nterestng and s plotted n Fgure 3 for each stock group of hgh, medum and low tradng actvtes. The graph on the top left-hand corner shows the k-cumulatve mpulse response functon (k = 0, 1,, 99) for adjusted prce duraton after a unt ncrease n #Trans. These response functons are obtaned based on the average WACD(1,1) and estmates for each stock group as reported n Table 7. As shown, the effect of a unt ncrease n #Trans has a hgher (negatve) mpact on prce duraton for stocks that are less actve. k The two graphs on the bottom show the mpulse response functon. The graph on the lower rght-hand corner s smlar but wth the value set equal to 1 for all three groups. Ths settng ams at snglng out the effect caused by the sum of the ARCH and GARCH parameters by standardzng the mpact of #Trans. For llustraton, consder the thnly traded group n the lower left-hand corner of Fgure 3. If #Trans ncreases by one unt, after one transacton (k = 1), the change n adjusted duraton s -0.51*(0.64+0.23)= -0.44 (see Table 7), whch s the second pont on the mpulse response functon for ths group. We need to dvde ths number by 100 to obtan the actual amount of reducton because the unt of adjusted duraton was multpled by 100 n Table 7. In other words, the reducton n adjusted duraton due to a trade nnovaton s 0.44% of the average prce duraton (depends on the tme of day) for ths group after one transacton. From Table 1A, we can compute the average duraton for ths group, whch s 2,308 seconds. Therefore, the reducton amounts to about 10 seconds per #Trans. Snce the average #Trans s 19 (from Table 1A) for ths group, the overall reducton n prce duraton s about 190 seconds. Smlar calculaton shows the overall reducton s about 56 seconds after k = 10 transactons. From the lower left-hand corner of Fgure 3, we agan see that the effect of a unt ncrease n #Trans has a hgher (negatve) mpact for stocks that are less actve, whch eventually goes down to zero. After standardzng the mpact of number of trades by keepng 6 Results are qualtatvely the same f we add the spread varable.
37 the same (= -1), the graph on the lower rght-hand corner shows more clearly that the persstence of mpact on the adjusted prce duraton for the heavly and thnly traded stock groups s about the same. Thus, whle trades of thnly traded stocks have a larger mpact (or a greater nformaton effect), the duraton of the mpact s close to that of heavly traded stocks on average. III. Summary In ths paper, we examne the frequency of nformaton arrvals of small thnly-traded stocks and ts mpacts on prce duraton or return volatlty at the ntraday level. We employ the autoregressve condtonal duraton (ACD) model to estmate the ntensty of nformaton arrvals and nformaton content of trades. The unque feature of ths model s ts ablty to handle hgh-frequency transacton data recorded at rregular tme ntervals. We fnd that ntraday perodcty must be consdered n the transacton data analyss. Our results show that the data adjusted for the ntraday determnstc pattern produce much more stable parameter estmates. In addton, the accuracy of forecasts s enhanced when the ntraday pattern s accounted for n the one-step-ahead forecastng. Our results show that there are dfferences n transacton and prce duratons between heavly and thnly traded stocks. The mpact of the number of transactons on adjusted prce duraton s much larger for thnly traded stocks than for heavly traded stocks. On the other hand, the persstence of the mpact on adjusted prce duraton s about the same between heavly and thnly traded stocks on average. The results show that the number of transactons has hgher explanatory power than average trade sze. We also examne the mpact of spread on prce duraton. The results show a consstent sgnfcantly negatve (postve) relatonshp between spreads and prce duraton (volatlty) only for thnly traded stocks. In addton, the effect of spreads s much stronger for thnly traded stocks, suggestng a larger mpact of asymmetrc nformaton for these stocks. Overall, we fnd that the number of trades contans most of the relevant nformaton affectng prce duraton or volatlty, and the trades of thnly traded stocks have greater mpact on prce duraton. The results suggest that the trades of thnly traded stocks contan more prvate nformaton than the trades of heavly traded stocks.
38 REFERENCES Admat, Anat R. and Paul Pflederer, 1988, A theory of ntraday patterns: Volume and prce varablty, Revew of Fnancal Studes 1, 3-40. Admat, Anat R. and Paul Pflederer, 1989, Dvde and conquer: A theory of ntraday and day-of-week mean effects, Revew of Fnancal Studes 2, 189-224. Andersen, Torben G., and Tmothy Bollerslev, 1997, Intraday perodcty and volatlty persstence n fnancal markets, Journal of Emprcal Fnance 4, 115-158. Dufour, Alfonso, and Robert F. Engle, 2000, Tme and the prce mpact of a trade, Journal of Fnance 55, 2467-2498. Easley, Davd and Maureen O Hara, 1992, Tme and the process of securty prce adjustment, Journal of Fnance 19, 69-90. Easley, Davd, Ncholas M. Kefer, Maureen O Hara and Joseph B. Paperman, 1996, Lqudty, nformaton, and nfrequently traded stocks, Journal of Fnance 51, 1405-1436. Engle, Robert F., 2000, The economcs of ultra-hgh-frequency data, Econometrca 68, 1-23. Engle, Robert F. and Jeffrey R. Russell, 1997, Forecastng the frequency of changes n quoted foregn exchange prces wth the autoregressve condtonal duraton model, Journal of Emprcal Fnance 4, 187-212. Engle, Robert F. and Jeffrey R. Russell, 1998, Autoregressve condtonal duraton: A new model for rregularly spaced transacton data, Econometrca 66, 1127-1162. Foster, F. Douglas, and S. Vswanathan, 1995, Can speculatve tradng explan the volumevolatlty relaton? Journal of Busness and Economc Statstcs 13, 379-396. Gallant, A. Ronald, Peter E. Ross, and George E. Tauchen, 1992, Stock prces and volume, Revew of Fnancal Studes 5, 199-242. Glosten Lawrence R., and P. Mlgrom, 1985, Bd ask and transacton prces n a specalst market wth heterogeneously nformed agents, Journal of Fnancal Economcs 14, 71-100. Hasbrouck, Joseph, 1991, Measurng the nformaton content of stock trades, Journal of Fnance 66, 179-207.
39 Hasbrouck, Joseph, 1995, One securty, many markets: Determnng the contrbutons to prce dscovery, Journal of Fnance 50, 1175-1199. Huang, Roger D. and Hans R. Stoll, 1997, The components of the bd-ask spread: A general approach, Revew of Fnancal Studes 10, 995-1034. Jones, M. Charles, Gautam Kaul, and Marc Lpson, 1994, Transactons, volume and volatlty, Revew of Fnancal Studes 7, 631-651. Kyle, Albert 1985, Contnuous tme auctons and nsder tradng, Econometrca 53, 1315-1336. Madhavan, Ananth, Mathew Rchardson and Mark Roomans, 1997, Why do securty prces change? A transacton-level analyss of NYSE stocks, Revew of Fnancal Studes 10, 1035-1064. Mller, Merton H., Jayaram Muthuswamy and Robert Whaley, 1994, Mean reverson of Standard & Poor s 500 ndex bass changes: Arbtrage-nduced or statstcal lluson?, Journal of Fnance 49, 479-513. Schwert, G. Wllam, 1989, Why does stock market volatlty change over tme? Journal of Fnance 44, 1115-1153. Wu, Chunch, 2003, Informaton flow, volatlty and spreads of nfrequently traded Nasdaq stocks, Quarterly Revew of Economcs and Fnance, forthcomng. Wu, Chunch and X. Eleanor Xu, 2000, Return volatlty, tradng mbalance and the nformaton content of volume, Revew of Quanttatve Fnance and Accountng 14, 131-153.
40 Table 1A Summary Statstcs Stock Symbol No. of Duratons Ave. Prce Ave. Spread /Duraton Ave. # Tran /Duraton Ave. Daly # Trans Ave. Vol/Trans Ave. Prce Duraton Ave. Daly Volume (shares) ASND 1,030 45.55 0.09 333.39 5,723.20 1,497.70 1,083.55 8,431,497 ORCL 612 47.24 0.09 304.34 3,104.27 1,335.79 1,724.23 4,131,497 NSCP 734 40.86 0.13 97.86 1,197.15 1,288.94 1,494.21 1,551,877 SBUX 395 39.97 0.13 82.49 543.06 1,304.65 2,313.90 686,025 DURA 403 39.14 0.22 45.72 307.09 1,687.19 2,446.58 577,185 CLST 519 35.88 0.27 27.62 238.91 1,626.74 1,866.45 435,837 ADTN 664 35.92 0.22 34.17 378.15 1,173.34 1,423.13 439,343 IRIDF 479 34.51 0.31 53.63 428.15 749.06 1,745.48 334,698 PSUN 463 36.72 0.46 14.34 110.66 1,737.35 1,807.68 225,440 SEBL 698 36.19 0.34 15.81 183.92 1,250.37 1,535.55 246,112 SDTI 487 38.20 0.27 18.44 149.67 1,344.45 1,952.97 223,817 APOL 463 37.31 0.29 17.18 132.57 1,428.63 1,965.89 211,208 LHSPF 311 36.16 0.31 25.31 131.19 1,285.17 2,031.17 173,928 WIND 610 41.93 0.35 15.84 161.04 1,322.36 1,704.94 220,510 CBSS 86 36.36 0.28 18.49 26.50 1,500.64 3,886.57 41,073 Ths table provdes summary statstcs for stocks n three groups classfed based on tradng volume. The frst group s the hgh-volume or heavly traded group and the thrd group s the low-volume or thnly traded group. The medum volume group s n between these two groups. The frst group can be classfed as the frequently traded group, f DURA s excluded. The second group can be classfed as the medum-frequency group f PSUN s excluded whle the thrd group can be classfed as the nfrequently traded group. The data are thnned by gnorng prce movements less than two tcks ($0.25). Duraton s the tme nterval between two trades. The duraton calculated after thnnng s called prce duraton. Prce duraton s measured n seconds. Volume s measured n number of shares. The number of duratons s the number of observatons for the duraton varable; average prce and spread are expressed n dollars; average #Trans/Duraton s the number of transacton per duraton; average daly #Trans s the mean transacton number per day; and average Vol./Trans s the average volume (n shares), or trade sze per transacton.
41 Table 1B Company Names Hgh- Volume Group Medum- Volume Group Low-Volume Group ASND ORCL NSCP SBUX DURA CLST ADTN IRIDF PSUN SEBL SDTI APOL LHSPF WIND CBSS ASCEND COMMUNICATIONS ORACLE CORP NETSCAPE COMMUNICATIONS CORP STARBUCKS CORP DURA PHARMACEUTICALS, INC. CELLSTAR CORP ADTRAN INC IRIDIUM LLC PAC SUNWEAR CA SIEBLE SYSTEMS SECURITY DYNAMICS APOLLO GROUP LERNOUT & HAUSPIE SPEECH PRODUCTS WIND RIVER SYSTEMS COMPASS BNCSHRS
42 Table 2 Estmates of the EACD(1,1) model for unadjusted prce duraton Stock ASND 8.34(4.01) 0.26(5.13) 0.41(4.08) ORCL 7.63(3.54) 0.31(4.81) 0.46(4.64) NSCP 6.66(3.09) 0.20(4.29) 0.54(4.95) SBUX 7.45(1.79) -0.03(-0.93) 0.83(8.72) DURA 0.70(1.09) 0.01(0.84) 0.97(37.07) Average 0.15 0.64 Average (wthout DURA) 0.19 0.56 CLST 1.97(3.39) 0.37(5.64) 0.63(12.68) ADTN 8.34(4.01) 0.26(5.13) 0.41(4.08) IRIDF 3.10(4.44) 0.25(5.26) 0.67(15.65) PSUN 6.27(3.48) 0.16(3.89) 0.64(8.16) SEBL 3.80(2.31) 0.06(2.82) 0.79(11.11) Average 0.22 0.63 Average (wthout PSUN) 0.24 0.63 SDTI 14.78(1.38) 0.02(0.84) 0.51(1.52) APOL 5.56(2.30) 0.07(2.28) 0.76(8.32) LHSPF 2.63(2.95) 0.59(5.41) 0.46(7.66) WIND 7.81(5.82) 0.45(6.69) 0.35(5.52) CBSS -- -- -- Average 0.28 0.52 Ths table reports the parameter estmates of the EACD(1,1) model and the t-values (n parentheses) for three stock groups descrbed n Table 1. The EACD(1,1) model for duraton s where x x 1 1 x s the duraton and s the condtonal mean of the duraton between two arrval tmes t and t -1. The prce duraton was dvded by 60 n estmaton. The parameter estmates for CBSS dd not converge. Average parameter estmates are mean estmates for each group. Averages wthout DURA or PSUN are mean parameter estmates excludng each of these two stocks whch are removed because ther tradng frequences are too low to be qualfed n the hgh and medum frequency groups. The mean estmates excludng these stocks represent the group average usng the measure of trade frequency to defne the actveness of stocks.
43 Table 3 Estmates of the WACD(1,1) model for unadjusted prce duraton Stock ASND 7.07(5.19) 0.40(7.58) 0.23(2.34) 0.91(47.99) ORCL 7.71(3.36) 0.30(3.88) 0.45(4.29) 0.94(33.37) NSCP 0.67(1.66) 0.06(3.28) 0.91(30.81) 0.81(36.10) SBUX 7.84(1.57) -0.03(-0.89) 0.83(7.49) 0.78(25.51) DURA 0.53(0.64) 0.01(0.47) 0.98(26.01) 0.71(23.58) Average 0.15 0.68 Average (wthout DURA) 0.18 0.61 CLST 1.79(2.51) 0.40(4.33) 0.61(9.13) 0.68(35.59) ADTN 8.62(2.72) 0.22(3.33) 0.42(2.81) 0.72(33.64) IRIDF 2.96(3.05) 0.29(4.02) 0.64(10.65) 0.68(31.04) PSUN 5.91(2.36) 0.21(2.71) 0.61(5.16) 0.57(29.19) SEBL 4.82(1.93) 0.12(2.38) 0.70(5.70) 0.61(37.17) Average 0.25 0.60 Average (wthout PSUN) 0.26 0.59 SDTI 16.46(1.09) 0.04(0.75) 0.45(0.96) 0.68(29.54) APOL 5.11(1.60) 0.07(1.58) 0.78(6.66) 0.65(29.48) LHSPF 2.53(2.30) 0.61(4.49) 0.44(5.82) 0.77(25.05) WIND 7.80(4.19) 0.46(4.86) 0.33(3.83) 0.67(34.95) CBSS 41.10(1.85) -0.24(-4.09) 0.60(2.13) 0.75(11.52) Average 0.19 0.52 Ths table reports the parameter estmates of the WACD(1,1) model and the t-values (n parentheses) for three stock groups descrbed n Table 1. s the parameter of the Webull dstrbuton. Estmaton s based on the lkelhood functon n equaton (6). The WACD(1,1) model for duraton s where x x 1 1 x s the duraton and s the condtonal mean of the duraton between two arrval tmes t and t -1. The prce duraton was dvded by 60 n estmaton. Average parameter estmates are mean estmates for each group. Averages wthout DURA or PSUN are mean parameter estmates excludng each of these two stocks whch are removed because ther tradng frequences are too low to be qualfed n the hgh and medum frequency groups. The mean estmates excludng these stocks represent the group average usng the measure of trade frequency to defne the actveness of stocks.
44 Table 4 Estmates of the WACD(1,1) model for unadjusted prce duraton wth number of transactons and trade sze Stock #Trans Ave Vol/ #Trans ASND 9.18 (5.21) 0.39 (6.67) 0.21 (2.43) 0.91 47.28) 0.02 (0.13) -1.11(-1.81) ORCL 5.74 (8.08) 0.32 (2.51) 0.44 (4.31) 0.94 (33.97) -0.01 (-0.01) 1.55 (1.51) NSCP 15.07(4.45) 0.39 (3.94) 0.35 (2.91) 0.81 (36.97) -4.44 (-2.91) -2.81 (-1.58) SUBX 15.46(2.73) -0.07(-1.17) 0.77 (7.87) 0.78 (28.33) 1.91 (0.75) -4.37 (-2.26) DURA 11.44(1.37) 0.05 (0.96) 0.74 (3.91) 0.71 (28.00) -5.55 (-1.08) -0.48 (-0.31) Average 11.38 0.22 0.50 0.83-1.61-1.44 Average (wthout DURA) 11.36 0.26 0.44 0.86-0.63-1.69 CLST 4.90 (3.33) 0.51 (4.60) 0.57 (8.43) 0.68 (31.45) -10.33 (-1.87) -1.07 (-1.74) ADTN 4.40 (2.38) 0.03 (0.89) 0.89 (15.93) 0.73 (35.15) -4.52 (-3.55) -0.91 (-0.95) IRIDF 2.16 (1.19) 0.35 (3.62) 0.65 (9.94) 0.68 (28.09) -4.58 (-1.72) 2.00 (0.79) PSUN 5.11 (1.69) 0.24 (2.61) 0.56 (4.50) 0.57 (28.79) -7.16 (-0.64) 1.48 (1.10) SEBL 3.73 (1.24) 0.16 (2.57) 0.64 (4.27) 0.62 (36.75) -11.11 (-1.47) 2.54 (1.95) Average 4.06 0.26 0.66 0.66-7.54 0.81 Average (wthout PSUN) 3.80 0.26 0.69 0.68-7.64 0.64 SDTI 6.72(9.89) 0.05 (0.90) 0.77 (7.97) 0.68 (30.02) -21.11 (-1.50) 2.25 (1.24) APOL 2.47 (0.99) 0.08 (1.83) 0.80 (10.19) 0.65 (29.00) -18.23 (-1.30) 3.35 (1.86) LHSPF -2.07(-2.28) 0.72 (5.11) 0.44 (6.26) 0.79 (25.52) -12.67 (-2.22) 4.72 (3.48) WIND 6.76 (2.79) 0.55 (4.32) 0.33 (3.88) 0.67 (33.46) -17.40 (-1.30) 1.38 (1.28) CBSS 41.40(5.23) -0.27(-4.15) 0.61 (6.19) 0.78 (11.39) -15.54 (-0.76) 4.82 (1.17) Average 21.06 0.23 0.59 0.714-16.99 3.30 Ths table reports the parameter estmates of the WACD(1,1) model wth the number of transacton and trade sze, and the t-values (n parentheses) for three stock groups descrbed n Table 1. s the parameter of the Webull dstrbuton. Estmaton s based on the model x 1 1 # Trans Volume / Trans where x s the duraton and s the condtonal mean of the duraton between two arrval tmes t and t -1 ; #Trans s the number of transactons per duraton and Volume/Trans s trade sze or average volume per transacton. The unadjusted duraton was dvded by 60, the number of transactons was dvded by 100, and the average volume over number of transactons was dvded by 1000. Average parameter estmates are mean estmates for each group. Averages wthout DURA or PSUN are mean parameter estmates excludng each of these two stocks whch are removed because ther tradng frequences are too low to be qualfed n the hgh and medum frequency groups. The mean estmates excludng these stocks represent the group average usng the measure of trade frequency to defne the actveness of stocks.
45 Table 5 Estmates of the WACD(1,1) model for unadjusted prce duraton wth spread, number of transactons and average trade sze Stock Spread #Trans Ave Vol/ #Trans ASND 3.80 (1.72) 0.02 (0.86) 0.91 (17.28) 0.73 (34.35) 0.01 (0.40) -4.22 (-3.03) -0.84 (-0.93) ORCL 9.52 (2.81) 0.31 (2.96) 0.48 (5.12) 0.94 (34.42) -0.42 (-3.28) -0.10 (-0.17) 0.98 (0.52) NSCP 16.99 (6.67) 0.34 (4.22) 0.44 (4.96) 0.81 (37.93) -0.38 (-6.85) -4.26 (-4.38) -1.31 (-0.96) SUBX 50.60 (4.43) 0.33 (2.98) -0.40 (-2.67) 0.79 (26.65) 0.69 (1.43) -13.09 (-2.09) -5.52 (-1.52) DURA -- -- -- -- -- -- -- Average 21.91 0.34 0.18 0.87-0.07-4.37-1.71 CLST 7.91 (3.27) 0.54 (4.88) 0.55 (8.23) 0.68 (31.91) -0.12 (-2.23) -9.94 (-2.16) -0.98 (-0.76) ADTN 3.80 (1.72) 0.02 (0.86) 0.91 (17.28) 0.73 (34.35) 0.01 (0.40) -4.22 (-3.03) -0.84 (-0.93) IRIDF 4.13 (1.40) 0.34 (3.83) 0.66 (10.58) 0.68 (33.69) -0.05 (-0.92) -4.49 (-1.87) 1.26 (0.52) PSUN 3.73 (1.06) 0.24 (2.52) 0.55 (4.77) 0.57 (28.01) 0.04 (0.56) -8.36 (-0.67) 1.51 (1.30) SEBL 6.01 (1.61) 0.17 (2.77) 0.62 (4.72) 0.62 (36.87) -0.05 (-0.80) -13.14 (-2.01) 2.64 (2.16) Average 5.12 0.26 0.66 0.66-0.03-8.03 0.72 Average (wthout PSUN) 5.46 0.27 0.69 0.68-0.05-7.95 0.52 SDTI 27.91 (1.80) 0.10 (1.28) 0.29 (0.66) 0.68 (30.91) -0.13 (-1.31) -25.21 (-1.49) 0.06 (0.05) APOL 10.99 (1.10) 0.10 (1.68) 0.72 (3.73) 0.65 (29.09) -0.23 (-1.49) -20.44 (-1.14) 3.49 (1.53) LHSPF 2.62 (0.62) 0.53 (1.84) 0.52 (2.91) 0.80 (25.34) -0.09 (-1.56) -16.63 (-4.12) 4.21 (1.84) WIND 13.07 (3.82) 0.54 (4.49) 0.34 (4.23) 0.68 (32.45) -0.15 (-2.50) -21.71 (-1.66) 1.15 (1.17) CBSS 25.91 (1.56) -0.24 (-5.18) 0.21 (1.86) 0.81 (11.98) -1.24 (-2.24) 18.01 (0.41) 4.59 (0.56) Average 16.10 0.21 0.42 0.72-0.37-13.20 2.70 Ths table reports the parameter estmates of the WACD(1,1) model wth spread, the number of transactons, and trade sze, and the t-values (n parentheses) for three stock groups descrbed n Table 1. s the parameter of the Webull dstrbuton. Estmaton s based on the model x 1 1 Spread # Trans Volume / Trans Spread s the percentage bd-ask spread and the remanng varables are as defned n Table 4. The unadjusted duraton was dvded by 60, the number of transactons was dvded by 100, the average volume over number of transactons was dvded by 1000 and the spread was multpled by 100. The estmates for DURA dd not converge. Average parameter estmates are mean estmates for each group. Averages wthout DURA or PSUN are mean parameter estmates excludng each of these two stocks whch are removed because ther tradng frequences are too low to be qualfed n the hgh and medum frequency groups. The mean estmates excludng these stocks represent the group average usng the measure of trade frequency to defne the actveness of stocks.