The Relationship between Stock Return Volatility and. Trading Volume: The case of The Philippines*


 Cory Perkins
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1 The Relaionship beween Sock Reurn Volailiy and Trading Volume: The case of The Philippines* Manabu Asai Faculy of Economics Soka Universiy Angelo Unie Economics Deparmen De La Salle Universiy Manila May 007 Absrac This paper reconsiders he relaionship beween sock reurn volailiy and rading volume. Based on he mulifacor sochasic volailiy model for sock reurn, we sugges several specificaions for he rading volume. This approach enables he unobservable informaion arrival o follow he ARMA process. We apply he model o he daa of Philippine Sock Exchange Composie Index and find ha wo facors are adequae o describe he movemens of sock reurn volailiy and variance of rading volume. We also find ha he weighs for he facors of he reurn and volume models are differen from each oher. The empirical resuls show (i) a negaive correlaion beween sock reurn volailiy and variance of rading volume, and (ii) a lack of effec of informaion arrivals on he level of rading volume. These findings are conrary o he resuls for he equiy markes of advanced counries. * The auhors would like o acknowledge he Faculy Exchange Program of Soka Universiy and De La Salle Universiy Manila for heir suppor.
2 Inroducion Many auhors including Tauchen and Pis (983), Lamoureux and Lasrapes (994), Andersen (996) and Liesenfeld (998, 00) explain he relaionship beween sock reurn volailiy and rading volume in he framework of he bivariae mixure model proposed by Tauchen and Pis (983). Tauchen and Pis (983) assume he lognormal disribuion for he laen informaion arrivals while Lamoureux and Lasrapes (994), Andersen (996) and Liesenfeld (998) employ he AR() process for he log of he laen variable. In Andersen (996) and Liesenfeld (998), he esimaed measure of volailiy persisence drops significanly compared wih he univariae specificaions for he reurn volailiy. Recenly, Liesenfeld (00) proposes he generalized mixure model and uses wo laen AR() processes. Fleming e al. (006) sugges he use of he sum of he AR() model and a whie noise for he log of he laen variable. There are wo wellknown classes of saisical models of sock reurn volailiy; one is he auoregressive condiional heeroskedasiciy (ARCH) models pioneered by Engle (98), and he oher is he sochasic volailiy (SV) model of Taylor (986). A comprehensive discussion of alernaive univariae and mulivariae, condiional and sochasic, financial volailiy models is given in McAleer (005). The model of Andersen (996) and Liesenfeld (998) can be considered as a kind of bivariae SV models. This paper reexamines he relaionship beween sock reurn volailiy and rading volume, based on recen developmens in he SV class of saisical models. Our new model ness he model proposed by Fleming e al. (006). To our knowledge, his is he firs sudy employing our proposed specificaions for modeling he variance of rading volume and he firs such sudy using Philippine daa. The remainder of he paper is organized as follows. Secion briefly reviews he bivariae mixure model of Tauchen and Pis (983) and suggess new specificaions for sock reurn and rading volume. Secion also explains he esimaion mehod. Secion 3 presens he daa and preliminary resuls. Secion 4 shows he empirical resuls and Secion 5 gives concluding remarks.
3 Model Specificaion and Economeric Mehodology. Mixure Model Tauchen and Pis (983) propose he bivariae mixure model in which he price change variance and rading volume are simulaneously direced by he informaionarrival process as he common mixing variable. By using he logprice change a day, R, and he number of shares raded a day, V, we may wrie heir model as ( ) R λ N µ, σ λ, V R R λ N ( µλσλ),, V V () Cov R V λ = 0. While µ R, σ R, µ V and σ V are parameers, he wih (, ) unobservable number of daily informaion, λ, is assumed o be a random variable. Condiional on λ, he join disribuion of R and V is he independen bivariae normal wih mixing variable λ. In he bivariae mixure model, he relaionship beween reurn variabiliy and rading volume is given by ({ µ }, ) σ µ ( λ ) R R V Cov R V = Var. () Tauchen and Pis (983) assume he lognormal disribuion for LN ( λ, λ ) λ µ σ, implying ln ~ N ( λ, λ ) λ µ σ. λ, i.e., There are several exensions of he bivariae mixure model, including Andersen (996), Lamoureux and Lasrapes (994), Liesenfeld (998), Liesenfeld (00) and Fleming e al. (006). Andersen (996), Lamoureux and Lasrapes (994) and Liesenfeld (998) assume ha ln λ follows he AR() process. In heir specificaion, he reurn process follows he AR() SV model. Empirical resuls, however, show ha he esimae for he coefficien for AR() based on he bivariae mixure model is exremely low, compared wih he univariae SV model for he reurn.
4 Liesenfeld (00) suggess he generalized mixure model and uses wo laen AR() processes in order o describe informaion. Recenly, Fleming e al. (006) sugges an alernaive model for he log of he laen variable based on he sum of an AR() process and a whie noise. In he following, we employ an alernaive approach based on he mulifacor SV models and sugges several specificaions for λ.. Modeling Sock Reurn We consider he Kfacor SV model for reurn as follows ( ) ( ) R = µ + ξ exp h, ξ N 0,, (3) wih he volailiy equaion h = ω + K α, i i= i, + i i i i i ( ) ( i K) α = φα + σ ε, ε N 0,, =,,, (4) and he imevarying mean for he mean and volailiy equaions are given by p µ = µ + ϕ R + γ hol + γ w + γ d, i i 3 i= ω = ln σ + δ hol + δ w + δ d, 3 where hol is he holiday dummy, w is he weekend dummy, and d is he dummy for he Asian currency crisis; In order o incorporae asymmeric affecs, we specify d = 0 for he precrisis, and d = for he poscrisis. ξ as K, 0 N ( 0, ),, 0 ξ = ψε ε ψ < ψ = ψ ψ K i i i i= 0, (5)
5 E ( ξ ) = 0 ( ) implying ha, V ξ =, and E ( ξ ε ) = ψ σ. If ψ < 0, a negaive shock i i i in reurn, i.e. ξ < 0, will increase fuure volailiy via α i, +. This ype of asymmeric shock is called he leverage effec. The Kfacor model (3)(5) may be considered as a special case of he discree ime approximaion of he coninuous ime SV model proposed by Chernov e al. (003). Chernov e al. (003) invesigae several wofacor coninuousime SV models wih a second SV facor exclusively dedicaed o he modeling of ail behavior. One of he conribuions of Chernov e al. (003) is o break he link beween ail hickness and volailiy persisence. i.3 Modeling Trading Volume As menioned earlier in his secion, we consider modeling rading volume he number of daily informaion λ and wrie V based on ( ) ( ) V = m + u exp v, u N 0, (6) wih condiional mean and volailiy equaions and p m = m + av + b hol + b w + b d + βλ, (7) i i 3 i= v = lnσ + c hol + c w + c d + β lnλ, (8) v 3 respecively. The erm λ capures he shorrun dynamic relaionship beween reurn volailiy and rading volume. We assume ha he informaion which drives he movemen of reurn can affec rading volume in a differen way. In order o compare several specificaions, we consider he following models of λ. Liesenfeld (998) and Liesenfeld (00) derended rading volume by he exponenial rend and wosided moving average, respecively. We do no employ such echniques as he ime rend is always insignifican when we include he Asian currency crisis dummy. I may be possible o consider an alernaive model M5;
6 Model Specificaion K M λ = exp αi i= M i λ ( α ) = exp i, i=,, K M3 K λ = exp kiαi, wih k = i= M4 K λ = exp kiαi, wih β = i= Model M comes from he mixure of disribuions hypohesis, indicaing ha flucuaions of reurn and rading volume are based on he same concenraed informaion λ. Model M i suggess ha only he ih facor of reurn has an effec on he flucuaions of rading volume. On he oher hand, he M3 and M4 specificaions assume ha several kinds of informaion affec he movemens of reurn and rading volume. When K =, models M3 and M4 correspond o he generalized mixure model of Liesenfeld (00) in he sense ha hese models ry o explain λ by combinaions of wo AR() processes, hough he specificaions are differen. I should be noed ha he M specificaion is an exension of he works of Andersen (996), Lamoureux and Lasrapes (994) and Liesenfeld (998) in he following sense. While hese auhors limied he process followed by λ o an AR(), our specificaion allows an ARMA process, as he sum of AR() processes usually follows an ARMA process (Granger and Morris (976)). Furhermore, he model of Fleming e al. (006) is a special case of M, as i is obained by seing K= and φ = ψ = 0 in M. K λ = exp kiαi + kvαv,, wih k =, i= α = φ α + σ ε, ε N(0,). V, + V V, V V, V, Compared o M, he model M5 incorporaes addiional AR() process o explain he dynamics of he rading volume. We do no deal wih M5, as i implies ha he model needs he addiional informaion in order o describe he movemen of he rading volume.
7 The combinaion of { α, α,, α } may capure he asymmeric effec and K heavyail. Regarding he asymmery, { α, α,, α } K deal wih he leverage effec for he reurn process. If he asymmeries of reurn and rading volume are caused by he same informaion, i is redundan o consider he asymmeric effec exhibied by. An alernaive bu similar specificaion is given by Liesenfeld (00). As shown by Chernov e al. (003), he Kfacor model can explain he ailfaness. Hence, M, M3 and M4 are he case. V.4 Maximum Likelihood Esimaion We may esimae he models (3)(6) for reurn and rading volume using he Mone Carlo Likelihood (MCL) mehod of Durbin and Koopman (997). The MCL mehod decomposes he likelihood funcion ino a Gaussian par and a remainder funcion for which he expecaion is evaluaed hrough simulaion. One of he feaures of he MCL mehod is he decomposiion. Since only he laer par requires Mone Carlo echniques, he MCL requires a smaller number of simulaions han he oher imporan sampling mehods. However, unlike he previous sudies ha joinly esimae he parameers of he reurn and rading volume, we separaely esimae he parameers of he models of sock reurn and rading volume. Specifically, we apply he MCL mehod developed by Sandmann and Koopman (998) and Asai (006) o he sock reurn model (3)(5). Then, given he esimaes of { α α },, K, we employ he ordinal maximum likelihood mehod for he alernaive models of rading volume, M o M4. For hese models, we use he esimaes of Sandmann and Koopman (998) as proxies for { α α },, K. We use separae esimaion of he reurn and volume models for he following reason. As repored in Liesenfeld (998), he join esimaion of he models of reurn and rading volume produces relaively low persisence inλ, indicaing ha he model is obviously inadequae for he reurn process. Therefore, in order o circumven such problem, we firs esimae he reurn model. Then, based on he esimaes of he reurn volailiy facors, we examine he model for rading volume.
8 3 Preliminary Resuls 3. Daa and Descripive saisics In his secion, we apply our proposed mehodology o invesigae he relaionship beween sock reurn volailiy and aggregae rading volume o an emerging marke economy, he Philippines. Specifically, we examine he validiy of he mixure of disribuions hypohesis using daily daa for he Philippine Sock Exchange Composie Index (PSEi). The sample period is Ocober 4, 00 o November, 006, giving T = 3035 observaions. We use he closing prices o calculae he reurn series, R. We define reurns R as 00 {ln P ln P }, where P is he closing price on day. To represen rading volume,, we use he aggregae daily number of shares raded for V 7 he componen socks of he PSEi. We scale he raw rading volume daa by 0. For he Asian currency crisis dummy, d = for July, 997 o he end of he sample period, while d = 0 for he beginning o July, 997. We presen he saisical characerisics of he empirical disribuion of reurn on he Philippine sock marke and rading volume in Table in order o deermine if he daa for he Philippine sock marke exhibi characerisics consisen wih he predicions of sandard mixure of disribuions models (see Harris, 987 for deails). Figures and plo he sock reurn and rading volume series over ime. I can be observed from Table ha boh reurns and rading volume are obviously no normally disribued. For boh series, excess kurosis is significanly differen from zero, indicaing a faailed disribuion for eiher reurn or rading volume. Boh series also exhibi significan posiive skewness. I should be noed ha magniudes of excess kurosis and skewness are much larger for rading volume. Based on he JarqueBera saisics, he null hypohesis of normaliy can be rejeced a he one percen level of significance for boh reurn and rading volume. In order o invesigae he presence of serial dependencies, we calculae he sample auocorrelaion coefficiens for reurn and volume for lags o 0, ρ(k), k =,..., 0. In addiion, we calculae he LjungBox pormaneau saisics including up o 0 lags, Q(0). For he reurn series, we find significan posiive firsorder serial correlaion while higherorder auocorrelaion coefficiens are insignifican. This is possibly due o nonsynchronous rading in he securiies ha make up a marke index (see e.g., Lo and
9 MacKinlay, 988, 990). Similarly, we find ha rading volume exhibis srong posiive firsorder auocorrelaion for he rading volume series. However, unlike he case of reurns, he higherorder sample auocorrelaions are also significan, indicaing ha rading volume exhibis a longer memory han reurn. The LjungBox saisics for squared reurns and for absolue reurns indicae ha hese series also exhibi significan auocorrelaion and hese es saisics appear o be much higher han ha for reurn. Furhermore, he LjungBox saisic for volume and squared volume sugges ha here is also srong auocorrelaion in rading volume and is squared values. Admai and Pfleiderer (988) and Foser and Viswanahan (993) offer alernaive explanaions for he exisence of posiive auocorrelaion in rading variables. Noe also ha he LjungBox saisic for rading volume is relaively much higher ha hose for squared reurns (and absolue reurns). Thus far, he disribuion characerisics of reurn and rading volume are all consisen wih he predicion of he mixure of disribuion models. Finally, we compue he sample conemporaneous correlaion coefficien beween squared reurn and volume, ρ, and beween absolue reurn and volume, ρ. R, V RV, We find, in conras o he predicion of he sandard mixure models, significan negaive conemporaneous correlaion beween reurn volailiy when absolue reurn is used as a proxy for reurn volailiy. In he case of squared volailiy and volume, he conemporaneous correlaion coefficien is also negaive bu insignifican. By he implicaion of equaion (), β in (7) migh be negaive or insignifican, alhough i is inconsisen wih he resuls for advanced counries. 3. Sock Reurn Model Alhough i is possible o simulaneously esimae he parameers of model (3)(5), we insead esimae he mean reurn equaion (3) and reurn volailiy model (4)(5) separaely for he following reasons. Firs, unlike in he model of rading volume, he mean equaion of sock reurn excludes is own volailiy erm. Second, he wo sep esimaor is convenien. Finally, i produces fas convergences. Table shows he OLS esimaes for he mean equaion using differen lags for reurn, wih Whie s heeroskedasiciyconsisen sandard errors in parenheses. We include lags of reurn in order o conrol for he auocorrelaion induced by nonsynchronous
10 rading. For all models, he esimaed coefficien of he firsorder lag, ϕ, is significan while he coefficiens for higherorder lags are all insignifican a he five percen level. Thus, we selec he case of p = and use he residual for he analysis of he volailiy equaion. Noe ha he weekend, holiday and Asian currency crisis dummies are insignifican in he mean equaion of sock reurn. Table 3(a) presens he AIC and SIC saisics for he esimaed volailiy model (4) wih K =,, 3 facors. While AIC seleced K =, SIC chose K =. On he oher hand, Table 3(b) repors he MCL esimaes for he cases of K = and K =. For boh cases, he holiday effec and he Asian currency crisis dummy are posiive and significan a he five percen level. In oher words, our resuls sugges ha reurn volailiy rises afer holidays and is relaively higher during he posasian financial crisis period. For he case of K =, he esimaes of φ and σ are close o 0.94 and 0.3, respecively, indicaing high persisence in volailiy. Compared o he index reurns of maured markes such as he S&P 500, he persisence is relaively low bu is sill a ypical value in he SV models. The esimae of ψ is insignifican, indicaing no asymmery. For he model where K =, he esimaes of φ and σ are 0.99 and 0.06, respecively, while hose of he second facor are 0.88 and 0.34, respecively. Alhough he esimaes of he firs facor sugges a possibiliy of a long memory SV model, we do no deal wih i as i is beyond he scope of his paper. Since he esimaes of φ and σ are significan, i appears ha he second facor can capure he ail behavior, similar o he analysis of Chernov e al. (003). The model of Fleming e al. (006) is inadequae, as φ = 0 is rejeced for he second facor. Finally, he esimae of ψ is negaive and significan while ha of ψ is insignifican. These resuls indicae he exisence of he asymmeric effec. Given he above resuls, we employ he esimaes from he wofacor SV model in modeling rading volume, as i capures boh asymmeric effec and ailhickness. 3.3 Trading Volume Model As a preliminary analysis, we esimae he mean equaion of model (6) wih various lag lenghs for rading volume as addiional explanaory variables in order o conrol for possible auocorrelaions. We obain he facor esimaes by applying he echnique of Sandmann and Koopman (998).
11 Table 4(a) presens he AIC and BIC saisics for he esimaed models. While AIC seleced p = 8, SIC chose p = 7. Table 4(b) shows he OLS esimaes of he parameers of he rading volume mean equaion, wih heeroskedasiciyconsisen sandard errors in parenheses. For all cases, he esimaes of he consan coefficien,, and he Asian crisis dummy, b, are significan a he five percen level, while he m 3 coefficien of he weekend dummy, b, is insignifican. While he esimae of he coefficien of he holiday dummy, b, is insignifican when p =, i is significan when p = o 0. Our general findings ha he coefficiens of he holiday and Asian crisis dummies are boh negaive and saisically significan sugges ha rading volume drops afer holidays and is relaively lower during he posasian crisis period. Turning o he esimaes of he coefficiens for lagged values of volume, we find ha when p = o 4, he coefficien of he las AR erm, a p, is significan, bu becomes insignifican when addiional lags are included. Therefore, for he succeeding analysis, we choose he value p = 7, as he SIC is consisen in he large sample. Finally, we find ha he esimae of he coefficien of reurn volailiy, β, is insignifican for all cases, indicaing ha reurn volailiy does no affec rading volume in he Philippine sock marke. 4 Empirical Resuls As menioned in he previous secion, we employ he wo facor SV model o describe he behavior of sock reurn volailiy. Based on he esimaes of his model and he resuls of our preliminary ess for he mean equaion of rading volume, we hen assess he four alernaive specificaions of rading volume, wih an AR(7) in he mean equaion. Resuls of he ML esimaion of Models MM4 for rading volume are presened in Table 5. For all cases, he holiday and weekend dummies are insignifican in he mean equaion, bu are negaive and significan in he variance equaion. These resuls sugges ha he volailiy of rading volume declines afer weekends and holidays. The coefficien esimaes for he Asian currency crisis dummy are negaive and significan in boh he mean and variance equaions, suggesing ha rading volume and volume volailiy are relaively lower afer he Asian crisis. Furhermore, he esimaes of he coefficiens β, k and k are significan, implying ha he variance of rading volume is no consan.
12 We now examine in deail he esimaes of he volailiy facors coefficiens, β, β, and. For model M, β for he mean equaion is posiive bu insignifican, k k while β for he variance equaion is negaive and significan a he five percen level. When we use one of he facors for λ, i.e., M i for i = or, he esimae of β is negaive and significan. For β, he esimae is insignifican for he firs facor, while i is posiive and significan for he second facor. These resuls imply ha model M is oo resricive o capure he movemen of rading volume. For he wo remaining specificaions, while he esimaes of β are insignifican, β for M3 and k for M4 are negaive and significan. In addiion, model M3 has he smalles AIC and SIC. Given hese, we choose o focus on he resuls of model M3. For model M3, he esimae of ha exp( 7.7 ) λ α α k is 7.7 and significan a five percen level, implying = +. Compared wih ( α α ) exp + in he reurn volailiy, α and α affec he variance of rading volume differenly. In oher words, he second facor of reurn volailiy affecs he variance of rading volume much more han he firs facor. Furhermore, he sign of β is negaive, suggesing ha he variance of rading volume increases when he reurn volailiy decreases. 5 Conclusion We reexamine he relaionship beween sock reurn volailiy and rading volume, employing he Kfacor SV model for he sock reurn. This specificaion enables he unobservable informaion o follow he ARMA process. We sugges several specificaions for he model of rading volume by using hese facors. For he Philippine Sock Exchange Composie Index, we find ha here are wo facors which direc sock reurn volailiy and variance of he rading volume. The weighs of hese facors are, however, differen for he reurn and volume models. The empirical resuls show wo feaures; (i) when he sock reurn becomes volaile, rading volume becomes less volaile, and (ii) informaion arrival does no affec he level of he rading volume. These are inconsisen wih he findings of previous sudies such as Liesenfeld (998). These feaures migh be a common phenomenon for he emerging markes. Examining he case of oher ASEAN sock markes such as Malaysia, Indonesia, Thailand and Singapore will be a fuure ask of our research.
13 References Admai, A. and P. Fleiderer (988), A Theory of Inraday Paerns: Volume and Price Variabiliy, Review of Financial Sudies,, Andersen, T.G. (996), Reurn Volailiy and Trading Volume: An Informaion Flow Inerpreaion of Sochasic Volailiy, Journal of Finance, 5, Asai, M. (006), Auoregressive Sochasic Volailiy Models wih Heavyailed Disribuions: A Comparison wih Mulifacor Volailiy Models, o appear in Journal of Empirical Finance. Chernov, M., A.R. Gallan, E. Ghysels, and G. Tauchen (003), Alernaive Models for Sock Price Dynamics, Journal of Economerics, 6, Durbin, J. and S.J. Koopman (997), Mone Carlo Maximum Likelihood Esimaion for NonGaussian Sae Space Models, Biomerika, 84, Engle, R. (98), Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of UK Inflaion, Economerica, 50, Fleming, J., C. Kirby and B. Osdiek (006), Sochasic Volailiy, Trading Volume, and he Daily Flow of Informaion, Journal of Business, 79, Foser, F. and S. Vishwanahan (993), The Effecs of Public Informaion and Compeiion on Trading Volume and Price Volailiy, Review of Financial Sudies, 6, Granger, C.W.J. and M. Morris (976), Time Series Modeling and Inerpreaion, Journal of he Royal Saisical Sociey, A, 39, Harris, L. (987), Transacion Daa Tess of he Mixure of Disribuions Hypohesis, Journal of Financial and Quaniaive Analysis,, 74. Koopman, S.J. and E.H. Uspensky (00), The Sochasic Volailiy in Mean Model: Empirical Evidence from Inernaional Sock Markes, Journal of Applied
14 Economerics, 7, Lamoureux, C.G. and W.D. Lasrapes (994), Endogenous Trading Volume and Momenum in SockReurn Volailiy, Journal of Business & Economic Saisics,, Liesenfeld, R. (998), Dynamic Bivariae Mixure Models: Modeling he Behavior of Prices and Trading Volume, Journal of Business & Economic Saisics, 6, Liesenfeld, R. (00), A generalized bivariae mixure model for sock price volailiy and rading volume, Journal of Economerics, 04, Lo, A. and A. MacKinlay (988), Sock Prices Do No Follow Random Walks: Evidence from a Simple Specificaion Tes, Review of Financial Sudies,, Lo, A. and A. MacKinlay (990), An Economeric Analysis of NonsynchronousTrading, Journal of Economerics, 45, 8. McAleer, M. (005), Auomaed Inference and Learning in Modeling Financial Volailiy, Economeric Theory,, 3 6. Sandmann, G. and S.J. Koopman (998), Esimaion of Sochasic Volailiy Models via Mone Carlo Maximum Likelihood, Journal of Economerics, 87, Taylor, S.J. (986), Modelling Financial Time Series, Chicheser: John Wiley and Sons. Tauchen, G.E. and M. Pis (983), The Price VariabiliyVolume Relaionship on Speculaive Markes, Economerica, 5,
15 Table : Summary Saisics of Reurn and Trading Volume Daa for he PSEi Saisics Reurn (%) Trading Volume (0 7 shares) Mean ** 0.909*** Minimum Maximum Sandard Deviaion Skewness 0.763*** *** Kurosis (excess).9383*** *** JarqueBera 8, ***, *** Auocorrelaions: Raw Series ρ() 0.785*** 0.789*** ρ() *** ρ(3) *** ρ(4) *** ρ(5) *** ρ(0) *** Q(0) 45.90*** 3,035.4*** Squared Series ρ() 0.098*** 0.506*** ρ() ** *** ρ(3) 0.3*** 0.47*** ρ(4) 0.043** *** ρ(5) ** 0.364*** ρ(0) 0.044** 0.609*** Q(0) 04.86*** 6,09.6*** Absolue Value Series Q(0),5.3*** Conemporaneous correlaion: ρ R, V ρ RV, *** *** indicaes significance a he 0.0 level ** indicaes significance a he 0.05 level
16 Table : OLS Esimaes of he Mean Equaion for Reurn µ γ γ γ 3 ϕ ϕ ϕ 3 ϕ 4 ϕ 5 S.E. of Regression p = p = p = 3 p = 4 p = (0.045) 0.7 (0.3596) (0.087) (0.057) (0.0346) (0.0453) (0.3607) (0.088) (0.057) (0.0344) (0.054) (0.0453) 0.89 (0.36) (0.087) (0.057) (0.0345) (0.05) (0.034) (0.0455) 0.74 (0.3600) (0.089) (0.057) (0.034) (0.053) (0.033) (0.049) (0.0455) 0.56 (0.360) (0.088) (0.0573) (0.0343) (0.05) (0.03) (0.05) (0.064) Noe: Whie s heeroskedasiciyconsisen sandard errors are given in parenheses.
17 Table 3: Esimaes of he Volailiy Equaion for Reurn (a) AIC and SIC K = K = K = 3 K = 4 K = 5 AIC * SIC 094.9* Noe: *` denoes he model seleced by AIC or BIC. (b) MCL Esimaes φ i σ i ψ i σ δ δ δ 3 K = (0.00) 0.35 (0.060) (0.065) (0.0577) (0.8) 0.57 (0.088) (0.055) K = s facor nd facor (0.0039) (0.06) 0.7 (0.346) (0.5) (0.034) 0.54 (0.0785) (0.067) Log Like (0.07) (0.033) (0.094)
18 Table 4: OLS Esimaes of he Mean Equaion for Trading Volume (a) AIC and SIC p = p = p = 3 p = 4 p = 5 AIC SIC p = 6 p = 7 p = 8 p = 9 p = 0 AIC * SIC.83.83* Noe: The AIC and SIC are no likelihoodbased, bu are he original ype. *` denoes he model seleced by AIC or BIC.
19 Table 4: OLS Esimaes of he Mean Equaion for Trading Volume (b) OLS Esimaes m a p b b b 3 β p = p = p = 3 p = 4 p = (7.7) 0.7 (0.057) (35.345) (3.36) (30.3) (3.096) (7.37) (0.079) (3.485) .8 (6.88) (3.07) (.94) 6.07 (5.86) 0.66 (0.05) (33.93) (6.034) (.84) .66 (.859) 4.79 (7.3) (0.043) (33.77) .48 (5.90) (3.7) (.87) (7.46) (0.049) (3.359) .44 (5.8) (3.353) .73 (.86) S.E. of Reg m a p b b b 3 β p = 6 p = 7 p = 8 p = 9 p = (7.89) (0.04) (3.794) (5.749) (3.63) (.566) 9.06 (7.96) (0.045) (3.606) (5.579) (3.73) .9 (.837) (8.33) (0.036) (3.53) .00 (5.56) (3.986) .90 (.835) (7.89) 0.07 (0.045) (3.03) .80 (5.593) (3.574) (.839) (7.855) (0.038) (33.07) .45 (5.599) (3.570) (.840) S.E. of Reg Noe: Whie s heeroskedasiciyconsisen sandard errors are given in parenheses. We omied he resuls for a,, ap ( ) p o save space.
20 Table 5: ML Esimaes for he Trading Volume Model m b b b 3 a 7 β M M M M3 M (8.799) (.306) (8.64) (8.03) (0.089).5449 (.4306) σ V (.48) c (0.849) c (0.070) c (0.0643) β (.649) (.389) .76 (9.906) (.635) (0.088) (3.74) (3.56) (0.93) (0.075) (0.065) (8.93) (3.49) (8.67) (8.096) (0.089).87 (.0959) (.6) (0.838) (0.075) (0.0648) (8.756) (3.730) .88 (8.338) (8.35) (0.088) ( ) (.3) (0.83) (0.076) (0.0650) (5.079) (3.339) (9.47) (8.0) 0.06 (0.084) (4.383) (3.4380) (0.8455) ( ) (0.0657) (0.033) (0.0696) (0.0465) (0.0457) k k (.484) Log Like. ( ) ( ) AIC * SIC * Noe: We omied he resuls for a,, a o save space. 6
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