Modelling the Effects of Trading Volume on Stock Return Volatility Using Conditional Heteroskedastic Models

Transcription

1 Journal of Finance and Economics, 018, Vol. 6, No. 5, Available online a hp://pubs.sciepub.com/jfe/6/5/5 Science and Educaion Publishing DOI: /jfe Modelling he Effecs of Trading Volume on Sock Reurn Volailiy Using Condiional Heeroskedasic Models Edwin Moyo 1,*, Anony Gichuhi Waiiu, Anony Ngunyi 3 1 Deparmen of Saisics, Pan African Universiy Insiue for Basic Sciences, Technology and Innovaion, Nairobi, Kenya Deparmen of Saisics and Acuarial Science, Jomo Kenyaa Universiy of Agriculure and Technology, Nairobi, Kenya 3 Deparmen of Saisics and Acuarial Science, Dedan Kimahi Universiy of Science and Technology, Nyeri, Kenya *Corresponding auhor: emoyo@aims.ac.z Received July 15, 018; Revised Augus 0, 018; Acceped Sepember 09, 018 Absrac In his sudy, we analyzed he effecs of rading volume as a proxy for he informaion arrival on sock reurn volailiy and assess wheher wih he inclusion of rading volume in condiional variance equaion, volailiy persisence disappears using he generalized auoregressive condiional heeroscedasiciy models; EGARCH and TGARCH. The analysis was done on he daily Nairobi Securiy Exchange (NSE) 0-share index and rading volume from 0/01/009 o 0/06/017 accouning for 108 observaions. The resuls of AR ()-EGARCH (1, and AR ()-TGARCH (1, models show ha he relaionship beween rading volume and sock reurns volailiy is posiive bu no saisically significan implying ha rading volume as a proxy of informaion flow can be considered generally as a poor source of volailiy in sock reurns. However, he resuls do no suppor he hypohesis ha persisence in volailiy disappears wih he inclusion of rading volume in he condiional variance equaion and his was consisen wih he Suden s -disribuion and Generalized error erm disribuion assumpion. We sugges ha he AR ()-EGARCH (1, model wihou rading volume wih suden -disribuion is a more suiable model o capure he main feaures of he sock reurns such as he volailiy clusering, he sock reurns volailiy and he leverage effec. Keywords: sock reurn, volailiy, volume, asymmeric GARCH models, leverage effec Cie This Aricle: Edwin Moyo, Anony Gichuhi Waiiu, and Anony Ngunyi, Modelling he Effecs of Trading Volume on Sock Reurn Volailiy Using Condiional Heeroskedasic Models. Journal of Finance and Economics, vol. 6, no. 5 (018): doi: /jfe Inroducion The sudy of volailiy in financial markes is of grea imporance o invesors in he managing of risk as i provides a degree of uncerainy on heir invesmen. Reference [1] ariculaes ha financial analyss and invesors in financial markes are concerned wih he unpredicabiliy on asse reurn invesmen ha are aribued o business performance insabiliy and varying marke prices. The common risk measures in financial markes are he Value a Risk (VaR) and Expeced Shorfall (ES) were he former is a more esablished saisic wihin he financial markes while he laer is increasingly becoming of research ineres. In many insances, ime series especially in naural sciences canno be modeled by a linear process. Thus, are beer modeled by nonlinear processes which include; ARCH, GARCH, TGARCH, EGARCH, PGARCH and many ohers. Financial ime series reurns ofen display volailiy clusering. Reference [] oulines he mos essenial financial ime series feaures as; hey end o have lepokuric disribuion, leverage effec, skewness and volailiy clusering. Hence, he sandard ARCH/GARCH model can model he lepokurosis, skewness and volailiy clusering. [3] shows ha he sandard model is unable o capure he dynamics of an imporan feaure of financial ime series known as he leverage effec i.e. canno model his asymmeric behaviour of sock reurns. A sock reurn is wha an invesor gains or losses on invesing in a paricular sock or porfolio which is dependen on he inheren risk in he marke ha he sock is lised. [4] ariculaes ha variaions on invesmen reurns are mosly dependen on he willingness of he invesor o ake risk ha is, he more he willingness o ake he risk, he more he reurns o he said invesor and conversely. In [5], volailiy is defined as a measure of variabiliy or dispersion abou a measure of cenral endency. Generally, in financial markes, he major concern is ofen on he spread of asse reurns. [6] ariculaes ha for any sock marke, volailiy and reurns are wo imporan facors around which he enire sock marke revolves. Volailiy is associaed wih he uncerainy of he price; however, i is really no he same

2 Journal of Finance and Economics 194 as risk. The undesirable oucome is linked o risk, whereas a sric measure of uncerainy which can be aribued o eiher a posiive or negaive oucome is volailiy [5]. For insance, a higher volailiy implies higher risk in he marke. The number of shares ha change ownership for a paricular securiy is measured by he rading volume. In he financial marke, several researchers and raders are of he view ha rading volume srongly influences movemens in prices. Reference [7] argues ha researchers have found ou ha he rading volumes conains a lo of informaion as i forms a good proxy for informaion level of invesors regarding socks a any given ime hence affecing he reacions hrough selling and buying of socks. [8] argues ha his has been consisen wih sudies by [7,9,10], who concluded ha volume and he absolue changes in price have a posiive correlaion. Reference [11] argues ha one facor ha many have considered in he predicion of sock prices is he rading volume. In addiion, he accessible amoun of new informaion abou a company on a given day can vary a securiies everyday volume depending on he expiry of opion conracs, he rading say is full or half day and oher possible facors. Among a wide range of facors influencing he rading volume, he arrival of new informaion is one facor ha corresponds he mos o a securiies fundamenal evaluaion. Reference [1] ariculaes ha he arrival of informaion assumes a criical par in sock markes as i primarily drives movemens. This informaion can be a public saemen, profi declaraion, cour ruling relaion o operaions or change in company regulaory policies. Addiionally, news arrival as an essenial driver of marke movemens is a unique propery of vecor sochasic models. Thus, implying ha news has a significan impac on he invesmen decisions of mos invesors han informaion regarding he business aciviies of lised companies such as financial saemen. Researchers have become more ineresed in he analysis of rading volume and corresponding change in prices relaing o informaional releases due o he inferences ha can be made from abnormal rading volume. However, raders focus on he rading volume because of several reasons. Theoreically, low volume in he marke means high flucuaions in prices while low price variabiliy is as a resul of high volume. Consequenly, his resuls in reducion of he price effec on large rade. Generally, broker revenue increases wih an increase in volume and due o high urnover, he marke makers have a greaer chance for profi. Alhough a fair amoun of empirical evidence exiss on he effecs of rading volume on he sock reurns volailiy for emerging sock markes in developing counries, he curren lieraure provides very few empirical sudies ha considered asymmeric GARCH models. Therefore, his sudy inends o fill his gap by analyzing he effecs of rading volume as a proxy for he arrival of informaion (hereafer, informaion arrival) on sock reurn volailiy and assess wheher wih he inclusion of rading volume in condiional variance equaion, volailiy persisence disappears using he generalized auoregressive condiional heeroscedasiciy models; EGARCH and TGARCH. The remaining pars of his paper are organized as follows. We discuss he mehodology considered in Secion. The resuls and discussion are conained in Secion 3 and lasly, we conclude he paper in Secion 4.. Mehodology The analysis was done on daily NSE 0-share price index and rading volume from 0/01/009 o /06/017 accouning for 108 observaions and was analyzed in he R sofware environmen [13]. The sock reurn is defined as: R = ln P ln P 1 ( where P and P 1 are he values of he sock index a close of he curren day and previous day respecively. RR is he logarihmic of sock reurns. The rading volume is defined as: V = ln Vol ln Vol 1 () where Vol and Vol 1 are he values of he volume of shares raded a he close of he curren day and previous day of rade respecively. VV is he logarihmic of rading volume. We le εε be he shock a ime and FF be he available informaion hrough ime. The modeling includes he esimaion of he mean and condiional variance equaions. We define he model as, ( ) R = E R F +, ~N(0, 1 R = µ +. (3) where µ is he condiional mean of R given informaion hrough ime 1 and R is he reurn a ime index. is a non-consan erm wih respec o ime and is defined as, Where, σ = σa (4) ( ) = Var R F 1 ( ) = E µ R F 1 and F 1 denoes he informaion se available a ime -1, σ is he volailiy ha evolves over ime and a ~ N( 0,1 ) iid Condiional Mean Equaion In modeling he condiional mean equaion of R, we will employ he general Box Jenkins ARMA (p, q) model defined as, p q R= µ + φir i+ + θ j1 (5) i1 = j1 = Where µ is a consan, φ i and θ j are parameers of he ARMA (p, q) model and is he disurbance erm.

3 195 Journal of Finance and Economics.. Condiional Variance Equaion To model he daily sock reurns volailiy, we used asymmeric models due o he fac ha shocks of equal magniude which may eiher be posiive or negaive are considered o have differen effecs on he volailiy in fuure. Reference [14] ariculaes ha asymmeric models are exensively moivaed by he need o disinguish beween negaive and posiive shocks and heir impac on volailiy in financial markes. In his paper, we used he sandard EGARCH (r, s) and TGARCH (r, s) models ha we discuss below:..1. Exponenial GARCH (EGARCH) Models In his model, he asymmeric responses of he imevarying variance o shocks is capured. The model ensures i a posiive variance and uses sandardized value of. σ i The EGARCH (r, s) specificaion is given by ( ) r s i + γ i i logσ = α0 + αi + βi log σ j σ i= 1 i j= 1 (6) where he asymmeric or leverage parameer is γ i. In mos empirical cases, he leverage parameer is expeced o be posiive so ha a negaive shock increases fuure volailiy or uncerainy while a posiive shock eases he impac on fuure uncerainy. When i is posiive (i.e. good news), is conribuion o he log volailiy is αi (1 + γ i ) i while if i is negaive (i.e. bad news) hen, he oal impac is αi (1 γ i ) i. If γ i is: i. γ i = 0, here is symmery i.e. no asymmeric volailiy ii. γ i < 0, hen negaive shocks (bad news) will increase he volailiy more han posiive shocks (good news). iii. γ i > 0, hen posiive shocks (good news) will increase he volailiy more han negaive shocks (bad news) The persisence ˆP of he model is given by, ˆ = s P βj j= 1 In order o analyze he effecs of he rading volume V on sock reurn volailiy, he following modificaion of he condiional variance equaion (6) is used: r i + γi i log σ = α0 + αi σ i= 1 i s + β jlog ( σ j) + δv j1 = (7) Where V is he logarihmic of he rading volume which is used as a proxy for informaion arrival while meaning of he res of he parameers are as defined in equaion (6). [15] argues ha if he proxy of informaion flow in he marke is serially correlaed o he variance hen he persisence would be significanly smaller han when V is no included and he parameer δ > 0. If he parameer δ > 0 and saisically significan, hen he proxy for informaion flow is serially correlaed o he variance and has explanaory power.... Threshold GARCH (TGARCH) Models The TGARCH (r, s) condiional variance specificaion is given by, Where r s σ = α0 + ( αi + γin i) i + β jσ j (8) i1 = j1 = N i is given by N i 1, if i < 0 = 0, if i 0 (9) where γ i is he asymmeric response parameer or leverage parameer, α i and β j are non-negaive parameers saisfying condiions similar o hose of GARCH models. If γ i =0, he model collapses o he classical GARCH (p, q) process. In his model, posiive shocks (good news) and negaive shocks (bad news) have differen effecs on he condiional variance σ ha is when i, i. i > 0, he effec on volailiy is α i. ii. i < 0, he effec on volailiy is αi + γi. The persisence ˆP of he model is given by, r s r ˆP = α j+ β j+ γik i= 1 j= 1 i= 1 where k is he expeced value of he sandardized residuals a below zero (effecively he probabiliy of being below zero), 0 k = E N 1a i = f ( a, 0,1, ) da (10) where f is he sandardized condiional densiy wih any addiional skew and shape parameers ( ). For insance, he value of k is 0.5 in he case of symmeric disribuions. In order o analyze he effecs of he rading volume V on sock reurn volailiy, he following modificaion of he condiional variance equaion (8) is used: r s σ = α0 + ( αi + γin i) i + β jσ j + δv (1 i1 = j1 = Where V is he logarihmic of he rading volume which is used as a proxy for informaion arrival o he marke while meaning of he res of he parameers are as defined in equaion (8). According o [15], he value of δ should be posiive and here should be negligible volailiy persisence.

4 Journal of Finance and Economics Disribuion Assumpions of he Error (a ) in he GARCH ype Model Reference [8] argues ha ofen non-normaliy paerns such as excess kurosis and skewness are exhibied by financial ime series. The residuals of condiional heeroscedasiciy models may generally show excess kurosis, heavy ails and skewness. In order o accoun for he skewness, excess kurosis and heavy-ails of reurn disribuions his sudy employed he use of he Suden s disribuion and he Generalized Error Disribuion (GED) Suden s -Disribuion [17] proposed ha in fiing he GARCH model for he sandardized error of he reurn series he Suden s - Disribuion can be used in order o beer capure he observed fa ails. The probabiliy densiy funcion for a random variable ha has a Suden s -disribuion wih v degrees of freedom is given by, f ( ;v) = Γ ( + ) v vπγ 1 + v X v 1 ( The densiy of he sandardized Suden s -Disribuion wih v > degrees of freedom is given by; Γ where Γ a f( a ) = 1+ v v Γ π( v ) a x v 1 ( x) e x 0 (13) = is he sandardized error, σ = is a gamma funcion, v is he parameer ha measures he hickness of he ail. The log likelihood funcion is given by equaion (14). Γ LN = N ln v Γ π( v ) N 1 a ln σ + ( v + ln 1+ = 1 σ v ( ) (14).3.. Generalized Error Disribuion The probabiliy densiy funcion of he Generalized Error Disribuion (GED) is given by, v 1x v exp v fx ( x;v ) =, < x<, 0 < v 1 λ Γ v Where Γ (.) Is he gamma funcion and v 1 Γ v λ = 3 Γ v The log likelihood funcion is given by, L N v v 1 x 1 ln ( 1 v ) ln + = N λ σ v (16) = 1 v 1 ln Γ ln σ In order o maximize he log likelihood funcion, he quasi maximum likelihood funcion esimaor will be used wih respec o he unknown parameers. This is a preferred mehodology because i is said o provide asympoic sandard errors ha are valid under nonnormaliy, is generally consisen and has a normal limiing disribuion [18]. 3. Resuls and Discussion The descripive saisics of he variables considered in his sudy are presened in Table 1. We observe ha he sock reurns series have a negaive daily mean suggesing ha hey decrease slighly over ime while he average mean daily rading volume is posiive implying ha he rading volume increase slighly wih ime. Boh he sock reurns and rading volume are righ skewed implying ha hey have an asymmeric disribuion as can be seen from he coefficien of he skewness. The values of he skewness and he kurosis are differen from zero and hree respecively. Hence, suggesing he presence of lepokuric i.e. fa ails hus implying ha he series are no normally disribued which is confirmed by he Jarque-Bera (JB) ess a 5% level of significance. Table 1. Descripive saisics of he daily sock reurns and rading volume Measure RR VV Number of observaions Mean Median Maximum Minimum Sd. deviaion Skewness Kurosis Jarque Bera P-value.e-16.e-16 Table. Augmened Dickey-Fuller (ADF) es for he daily NSE 0 share index reurn series and rading volume Tes saisic Criical value 1% 5% 10% PP RR VV

5 197 Journal of Finance and Economics Figure 1. Time plos for he daily NSE 0 Share Index Reurns Series To es for saionariy of he ime series, he Augmened Dickey Fuller (ADF) es was considered. From he compued es saisics in Table, we observe ha he compued es saisics for he daily price series es is more han he criical values and hus, he null hypohesis is no rejeced a 5% level of significance and we conclude ha he price series is no saionary (here is a uni roo) while V and R are saionary as can be observed from he visual inspecion in Figure (1b) and Figure (b). Table 3 shows he Ljung Box es for he sock reurn and rading volume series resuls a lags, 4, 6, 8 and he null hypohesis is rejeced a 5% level of significance. Therefore, he resul indicaing ha here exiss correlaion in he sock reurns and rading volumes series. Hence, deeced auocorrelaion in he sock reurn and rading volume series can be removed from he daa by fiing he simples plausible ARMA (p, q) model. Table 3. Ljung box es for he daily sock reurns series and rading volume m RR QQ mm P-value.e-16.e16.e-16.e-16 VV QQ mm P-value.e-16.e-16.e-16.e-16 Figure. Time plos for he daily NSE 0 Share Index rading volume 3.1. Esimaed Mean Equaion In his sudy, an ARMA (p, q) model was used o fi he mean reurns because i is said o provide approximaions o he condiional mean dynamics ha are flexible and parsimonious. According o [19], o deduce he order of an ARMA (p, q) model we may use he Auocorrelaion Funcion (ACF) and Parial Auocorrelaion Funcion (PACF). In his sudy, we suggesed ha he sock reurns can be modeled by an AR () process. This is consisen wih he resuls in Table 4 which suggesed he bes fiing model based on he crierion of choosing a model wih minimum AIC and BIC and larges log-likelihood funcion. Therefore, he ARMA (, 0) is seleced as he mean equaion. Table 4. Selecion crieria for ARMA (p, q) order selecion Model AIC BIC LL ARMA (0, 0) ARMA (1, 0) ARMA (1, ARMA (, 0) ARMA (0, ARMA (0, ) ARMA (, ) ARMA (1, ) ARMA (,

6 Journal of Finance and Economics 198 The es for ARCH effecs on he residuals of he AR () model resuled in he rejecion of he null hypohesis a 5% level of significance and he resuls of he ess considered are given in Table 5. The lack of fi can also be observed from he plo of ACF and PACF in Figure 3. Therefore, he implemenaion of he GARCH-ype models is valid in he modeling of he sock reurns volailiy. Table 5. ARCH effec es on residuals of he ARMA (, 0) model m Ljung Box QQ mm P-value.e-16.e-16.e-16.e-16 Lagrange Muliplier QQ mm P-value.e-16.e-16.e-16.e-16 Figure 3. Plos of he auocorrelaion funcion and parial auocorrelaion funcion for he residuals of he daily NSE 0 Share Index reurns series 3.. Esimaed Volailiy Models The resuls of he parameer esimae for he AR ()- EGARCH (1, and AR ()-TGARCH (1, wihou and wih rading volume respecively under he Suden s - disribuion and Generalized Error disribuion assumpion of he error erm disribuion are presened in Table 6 and Table 7. The p-values are given in parenheses AR ()-EGARCH (1, and AR ()-TGARCH (1, Models wihou rading volume In Table 6, he esimaes of he AR () i.e. φ 1 and φ are significan hence backing he implemenaion in modeling of he NSE sock reurns wih an AR () model. We observe ha he GARCH erm ( β 1 ) is saisically significan for AR. ()-EGARCH (1, wih GED whereas he ARCH erm ( α 1 ) and he mean parameer (μ) for boh Suden s -disribuion and GED are no saisically significan a 5% level of significance. The parameer γ 1 is posiive and significan for he AR ()-EGARCH (1, suggesing he presence of he leverage effec under he GED and Suden s disribuion. This implies a confirmaion of he fac ha good news (posiive shocks) increases volailiy more han bad news (negaive shocks) of he same magniude. This finding agrees wih earlier sudies on he NSE by [14] and [0] who modeled daily and weekly reurns using he GARCH-ype models respecively. The GARCH erm ( β 1 ) and he ARCH erm ( α 1 ) are saisically significan for AR ()-TGARCH (1, a 5% level of significance under he GED and Suden s -disribuion. The parameer γ 1 in he AR ()-TGARCH (1, model is negaive and no saisically significan a 5% level of significance suggesing ha here is no asymmery under boh error erm disribuion assumpions. Table 6. Parameer Esimaion of he AR ()-EGARCH (1, and AR ()-TGARCH (1, Models wihou rading volume Condiional disribuion AR ()- EGARCH (1, AR ()- TGARCH (1, Suden GED Suden GED μμ (0.4877) (0.4904) (0.7689) (0.345) AR ( (0.0000) 0.75 (0.0000) (0.0000) 0.7 (0.0000) AR () (0.0000) (0.0000) (0.0000) (0.0000) αα (0.0000) -.05 (0.0000) (0.0000) (0.0000) αα (0.837) (0.6360) (0.0000) (0.0000) ββ (0.4877) (0.0000) (0.0000) (0.0000) γγ (0.0000) (0.0000) (0.7470) (0.4945) shape (0.0000) (0.0000) (0.0000) (0.0000) PP Q (9) (0.1558) 0.65 (0.494) (0.3) (0.974) QQ (9).6165 (0.80) (0.9138) ( (0.0008) ARCH (7) (0.9180) (0.9048) ( (0.9303) AIC BIC LL

7 199 Journal of Finance and Economics Table 7. Parameer Esimaion of he AR ()-EGARCH (1, and AR ()-TGARCH (1, Models wih rading volume Condiional disribuion AR ()- EGARCH (1, AR ()- TGARCH (1, Suden GED Suden GED μμ (0.556) (0.543) ( (0.4397) AR ( (0.0000) (0.0000) (0.0000) 0.75 (0.0000) AR () (0.0000) (0.0000) (0.0000) (0.0000) αα (0.0000) (0.0000) (0.0000) (0.0000) αα (0.8096) (0.658) (0.0000) (0.0000) ββ (0.0000) (0.0000) (0.0000) (0.0000) γγ (0.0000) (0.0000) (0.7337) (0.4937) δδ 4.71 (0.4458) (0.3413) (0.4594) (0.3634) shape (0.0000) (0.0000) (0.0000) (0.0000) PP Q (9) ( ( (0.3580) (0.3416) QQ (9).5037 (0.80) (0.983).07 ( (0.0007) ARCH (7) (0.975) (0.918) (0.9486) (0.9393) AIC BIC LL AR ()-EGARCH (1, and AR ()-TGARCH (1, Models rading volume Table 7 gives he resuls of he parameer esimaes for he AR ()-EGARCH (1, and AR ()-TGARCH (1, wih rading volume under he GED and Suden s - disribuion. The mean parameer (μ) for all disribuion assumpions are no saisically significan a 5% level of significance. The esimaes of he AR () i.e. φ 1 and φ are significan hence approving he uilizaion of he AR () model for NSE sock reurns. The GARCH erm ( β 1 ) is saisically significan for he AR ()-EGARCH (1, and AR ()-TGARCH (1, under all disribuional assumpions whereas he ARCH erm ( α 1 ) for he AR ()-EGARCH (1, a 5% level of significance is no saisically significan. In he AR ()-EGARCH (1,, he parameer γ 1 in he model is posiive and saisically significan a 5% level of significance implying he presence of asymmery under he GED and Suden s -disribuion. This suggess ha posiive shocks (good news) resuls in more condiional variance han negaive shocks (bad news) of similar magniude indicaing ha bad news has a lesser impac on he volailiy han good news. Similarly, in he AR ()-TGARCH (1, he parameer γ 1 is negaive and no saisically significan a 5% level of significance hence, collapsing he model o a GARCH (r, s) process where good and bad news have he same impac on sock marke reurn series and hus implying ha here is no asymmery under he error erm disribuional assumpions. We observe ha in boh he AR ()-EGARCH (1, and AR ()-TGARCH (1, models respecively, he value of he parameer γ 1 slighly decreases wih he inclusion of he rading volume in he model implying ha he rading volume leads o less asymmeric volailiy on he marke. The parameer δ is posiive bu no saisically significan hence, we deduce ha he volailiy in he NSE canno be explained by he rading volume. Then, he proxy of informaion flow ha is rading volume considered in his sudy may reflec a poor source of heeroskedasiciy in he NSE sock reurns and differs wih findings of [15] as volailiy persisence remains high. This is consisen wih resuls of [1] and []. The posiive coefficien of he parameer δ indicaes a posiive relaionship beween sock reurn volailiy and rading volume. This resul is seady wih he findings of [3] who examined he relaionship beween he daily sock reurn and he rading volume in he NSE using regression analysis. The degree of persisence in he condiional variance equaions slighly increased wih he inclusion of rading volume in all he models considered and hus, his is consisen regardless of he error erm disribuion assumpion. The persisence implies ha oday s volailiy shocks have an impac on he fuure expeced volailiy. Also, he presence of he leverage effec in he NSE sock reurns is confirmed by he AR ()-EGARCH (1, process for he innovaions wih Suden s disribuion and GED. The model accuracy evaluaion was done using he Ljung-Box es for he Q (9) and Q (9) for residuals and squared residuals respecively and he ARCH (7) for all he models under he all error erm disribuion assumpions. The null hypohesis of no significan correlaions and no arch effecs is acceped for all he cases implying ha he fied models were adequae. Finally, a more suiable model o capure he main feaures of he sock reurns such as he volailiy clusering, he volailiy and he leverage effec based on he values of he AIC, BIC and LL is he AR ()-EGARCH (1, model wihou rading volume. 4. Conclusion In his sudy, we modeled he effecs of rading volume on sock reurn volailiy using he AR ()-EGARCH (1, and AR ()-TGARCH (1, models under he suden s - disribuion and GED. We observe ha including he rading volume in he AR ()-EGARCH (1, and AR ()- TGARCH (1, model slighly decreases he value of he parameer γγ 11 implying ha he rading volume leads o

8 Journal of Finance and Economics 00 less asymmeric volailiy on he marke. The parameer δδ is posiive bu no saisically significan hence, we conclude ha he rading volume does no explain volailiy in he NSE. The degree of persisence slighly increased wih he inclusion of rading volume in he condiional variance equaions of all he models considered and hus, his is consisen regardless of he error erm disribuion assumpion. The persisence implies ha volailiy shocks oday will influence he expecaion of volailiy many periods in he fuure. Therefore, he rading volume as a proxy of informaion flow can generally be considered o be a poor source of volailiy in he sock reurns. The resul agrees wih he findings as in [4] and [5]. The AR ()-EGARCH (1, model wihou rading volume was suggesed o be a more suiable model o capure he main feaures of he NSE reurn such as he volailiy clusering, he sock reurns volailiy and he leverage effec. Acknowledgemens The auhors are hankful o he Pan African Universiy Insiue for Basic Sciences, Technology and Innovaion managemen for he suppor financially and o all he people who helped in making commens on his paper. References [1] K. S. Adesina. Modelling sock marke reurn volailiy: GARCH evidence from Nigerian sock exchange. Inernaional Journal of Financial Managemen, 3(3):37, 013. [] R. Vasudevan and S. Verivel. Forecasing sock marke volailiy using GARCH models: Evidence from he Indian sock marke. Asian Journal of Research in Social Sciences and Humaniies, 6(8): , 016. [3] W. Coffie. Modelling and forecasing he condiional heeroscedasiciy of sock reurns using asymmeric models: Empirical evidence from Ghana and Nigeria. Journal of Accouning and Finance, 15(5):109, 015. [4] W. F. Sharpe. Capial asse prices: A heory of marke equilibrium under condiions of risk. The journal of finance, 19(3): 45-44, [5] A. E. M. Ahmed and S. Z. Suliman. Modeling sock marke volailiy using GARCH models evidence from Sudan. Inernaional Journal of Business and Social Science, (3), 011. [6] R. Kumar, H. Gupa, e al. Volailiy in he Indian sock marke: A case of individual securiies. Journal of Academic Research in Economics, 1(:43-54, 009. [7] G. E. Tauchen and M. Pis. The price variabiliy-volume relaionship on speculaive markes. Economerica: Journal of he Economeric Sociey, pages , [8] R. T. Mpofu. The relaionship beween rading volume and sock reurns in he JSE securiies exchange in Souh Africa. Corporae Ownership & Conrol, pages -10, 01. [9] P. K. Clark. A subordinaed sochasic process model wih finie variance for speculaive prices. Economerica: journal of he Economeric Sociey, pages , [10] R. L. Crouch. The volume of ransacions and price changes on he New York sock exchange. Financial Analyss Journal, 6(4): , [11] C. Wang. The effec of ne posiions by ype of rader on volailiy in foreign currency fuures markes. Journal of Fuures Markes, (5): , 00. [1] P. N. Van. A good news or bad news has greaer impac on he Vienamese sock marke? Munich Personal RePEc Archive, 61194, 015. [13] R Core Team. R: A Language and Environmen for Saisical Compuing. R Foundaion for Saisical Compuing, Vienna, Ausria, 016. URL hps:// [14] A. Maqsood, S. Safdar, R. Shafi, and N. J. Leli. Modeling sock marke volailiy using GARCH models: A case sudy of Nairobi Securiies Exchange (NSE). Saisics, 7: , 017. [15] C. G. Lamoureux and W. D. Lasrapes. Heeroskedasiciy in sock reurn daa: Volume versus garch effecs. The journal of finance, 45(: 1-9, [16] M. A. Thorlie, L. Song, X. Wang, and M. Amin. Modelling exchange rae volailiy using asymmeric GARCH models (evidence from Sierra Leone). Inernaional Journal of Science and Research (IJSR), 3(1: , 014. [17] T. Bollerslev. A condiionally heeroskedasic ime series model for speculaive prices and raes of reurn. The review of economics and saisics, pages , [18] T. Bollerslev and J. M. Wooldridge. Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariances. Economeric reviews, 11():143-17, 199. [19] R. S. Tsay and G. C. Tiao. Consisen esimaes of auoregressive parameers and exended sample auocorrelaion funcion for saionary and nonsaionary arma models. Journal of he American Saisical Associaion, 79(385):84-96, [0] A. Wagala, D. K. Nassiuma, A. S. Islam, and J. W. Mwangi. Volailiy modelling of he Nairobi Securiies Exchange weekly reurns using he arch-ype models. Inernaional Journal of Applied, (3), 01. [1] H. J. A. Ahmed, A. Hassan, and A. M. Nasir. The relaionship beween rading volume, volailiy and sock marke reurns: A es of mixed disribuion hypohesis for a pre and pos crisis on kuala lumpur sock exchange. Invesmen Managemen and Financial Innovaions, 3(3): , 005. [] A. F. Darra, S. Rahman, and M. Zhong. Inraday rading volume and reurn volailiy of he djia socks: A noe. Journal of Banking & Finance, 7(10): , 003. [3] K. O. Husborn, J. Nzyuko, and D. Omwansa. Relaionship beween rading volume and sock reurns a he Nairobi Securiies Exchange (NSE). Inernaional Journal of Managemen and Commerce Innovaions, 5():77-778, 018. [4] E. Girard and R. Biswas. Trading volume and marke volailiy: Developed versus emerging sock markes. Financial Review, 4(3):49-459, 007. [5] G. Gursoy, A. Yuksel, and R. Biswas. Trading volume and sock marke volailiy: evidence from emerging sock markes, invesmen managemen and financial innovaions. Financial Review, 5(4):49-459, 008.