NONLINEAR MARKET DYNAMICS BETWEEN STOCK RETURNS AND TRADING VOLUME: EMPIRICAL EVIDENCES FROM ASIAN STOCK MARKETS

Transcription

1 ANALELE ŞTIINłIFICE ALE UNIVERSITĂłII ALEXANDRU IOAN CUZA DIN IAŞI Tomul LVI ŞiinŃe Economice 009 NONLINEAR MARKET DYNAMICS BETWEEN STOCK RETURNS AND TRADING VOLUME: EMPIRICAL EVIDENCES FROM ASIAN STOCK MARKETS Wu-Jen Chuang *, Liang-Yuh Ou-Yang **, Wen-Chen Lo *** Absrac Recen empirical researches repor ha nonlinear dynamics is presen in asse reurns because of noise raders involved in he marke. This sudy examines wheher here exiss any nonlinear dynamics in Asian sock markes. We employ he smooh ransiion auoregressive model wih he percenage change in rading volume as he ransiion variable o capure he nonlinear movemen beween sock reurns and rading volume in Taiwan, Hong Kong, Singapore, and Korea sock markes. The resuls show nonlinear dynamics exis beween sock reurns and rading volume in he sock marke. Moreover, rading volume plays an imporan role for he cyclical movemens in he sock marke. Key words: Nonlinear dynamics, Cyclical behavior, Sock marke reurns, Trading volume, STAR models JEL Classificaion: C11, F30, G15 1. Inroducion Sock prices are believed o be sensiive o he relevan economic news. Based upon he capial asse pricing model (CAPM) and he arbirage pricing heory (APT), sock marke reurns can be prediced by financial and macroeconomic variables. Invesors can reward excess reurns by aking sysemaic risks, bu no earn exra premium by bearing diversifiable risk. 1 However, no saisfacory model can argue ha here is linear relaion beween sock reurns and macroeconomic facors. A number of increasing empirical evidences challenge he CAPM and APT. Firs, heoreically invesors are hough o be raional under CAPM, bu some empirical resuls show ha invesors are no raional all he ime and ha irraional invesmen behavior has influences on he price formaion of securiies. For example, De Long e al.(1990) propose a model o show ha invesors irraional beliefs have influences on he price formaion of asses. They poin ou ha he irraional beliefs would drive he sock prices furher away from he fundamenal values, and * Wu-Jen Chuang (ewjch@mail.ku.edu.w), Graduae Insiue of Money, Banking and Finance, Tamkang Universiy, Taipei, Taiwan, R.O.C. ** Liang-Yuh Ou-Yang (liangyuh@mail.ku.edu.w), Graduae Insiue of Managemen Science, Tamkang Universiy, Taipei, Taiwan, R.O.C. *** Wen-Chen Lo (wenchen@mail.sju.edu.w), Graduae Insiue of Managemen Science, Tamkang Universiy, Taipei, Taiwan, R.O.C.

2 6 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo he deviaions of sock prices become more exreme. Moreover, Lee e al. (00) employ a GARCH-M model o show ha he condiional volailiy and excess reurns of he sock marke are affeced by invesor senimen. Second, boh CAPM and APT predic asse reurns wih priced variables in a linear way. However, some sudies have already presened ha he business cycle shows nonlinear characerisics and economic variables exhibi asymmeric processes during conracion and expansion in he economy (Keynes, 1936; Teräsvira and Anderson, 199; Öcal and Osborn, 000). Recen sudies pu more aenion on he sock reurns of which dynamics are characerized by nonlinear behavior in he business cycle. For example, Leung e al. (000) predic he inernaional sock reurns by arificial neural nework mehod. Some papers examine U.S. sock marke by a Markov swiching model (Turner e al. 1989; Perez- Quiros and Timmermann, 000). Saranis (001) and McMillan (003, 004, 005,007) employ he smooh ransiion auoregressive (STAR) model o examine non-linear behavior in he inernaional sock markes. Moreover, several researches accoun for differen reasons of nonlineariies in financial markes. The main explanaions include ha heerogeneous beliefs beween informed invesors and noise raders (Brock and LeBaron, 1996; Shleifer, 000), differen invesmen horizons, geographical locaions, and risk profiles varied among invesors (Peers, 1994), marke fricions and ransacion coss (Dumas, 199; Sercu e al. 1995), and differen economic growh (Cecchei e al. 1990). Saranis (001) saes ha smooh ransiion models are no only suiable o explore he nonlinear and cyclical behavior of sock reurns, bu also appropriae o explain smooh ransiion during regime changes due o heerogeneous beliefs, varying learning speeds, and differen invesmen horizons beween invesors. Many sudies invesigae nonlinear behavior of inernaional sock prices by he STAR model (McMillan 001, 003, 004, 005, 007; Saranis, 001). 3 Bu so far, uilizing he STAR models o highligh he ineracions beween sock reurns and rading volume is few even hough pas rading volume providing some valuable informaion abou sock reurns has been recognized by financial academics(mcmillan 007). Blume e al. (1994) presen raders can learn valuable informaion of socks by pas prices and pas volume, and argue ha sock reurns and rading volume are joinly deermined by he same marke dynamics. Daar e al. (1998) uncover he relaion ha low volume firms earn higher fuure reurns, and high volume firms gain lower fuure reurns. Conrad e al. (1994) find low rading volume socks exhibi price reversal paern, bu high volume socks easily show reurn coninuaion by sudying weekly daa. Moreover, Lee and Swaminahan (000) illusrae he ineracion beween price momenum, reversal, and rading volume by momenum life cycle hypohesis and sugges invesor o make profi by he momenum invesing sraegy based on pas price and volume informaion. Shiller (000) also proposes he feedback loop heory o explain he relaionships among sock reurns, invesor senimen, and rading volume during he sock marke cycle. Furhermore, Telock (007) consrucs a sraighforward measure of media conen o proxy for invesor senimen and finds he ineracions among media conen, marke prices and rading volume. To sum up, boh heoreical models and empirical resuls menioned above demonsrae ha here is inerrelaionship beween sock reurns and volume during he sock marke cycle. Hence, he objecive of his paper is o examine nonlinear dynamics beween sock reurns and rading volume and o consider rading volume as he ransiion variable o examine he nonlinear sock marke dynamics. The remained of he paper is organized as follows. In he nex secion, we presen he specificaion and esimaions of STAR models.

3 Nonlinear Marke Dynamics Beween Sock Reurns and Trading Volume In secion 3, we show and inerpre our empirical resuls. In secion 4, we compare he ouof-sample forecasing performance of he nonlinear model wih ha of he linear model. In he final secion, we conclude our findings.. Specificaion and Esimaion of STAR Models The paper is o exploi he poenial nonlinear and cyclical behavior beween sock reurns and rading volume. We employ he STAR models which allow he ransiion variable o cause a slow change beween differen regimes o invesigae nonlinear relaion in sock marke (Teräsvira and Anderson, 199; Teräsvira, 1994). Moreover, we consider rading volume as he ransiion variable o examine he nonlinear dynamics beween socks reurns and rading volume..1. Trading volume as he ransiion variable There are some explanaions abou he role of rading volume in he marke. Prior researches inerprae rading volume as a liquidiy proxy. Based upon he liquidiy hypohesis, relaively low volume socks are less liquid, bu gain higher expeced reurn (Amihud and Mendelson, 1986; Daar e al., 1998; Brennan e al., 1998 ). Some researches find ha rading volume conains valuable informaion of socks and propose change in volume can measure abnormal aciviy. Lee and Swaminahan (000) sae ha low (high) volume sock display characerisics of value (glamour) socks. The reason why low (high) volume socks usually gain higher (lower) fuure reurns is ha invesors mispercep abou fuure earnings of hose socks. Besides, Lee and Swaminahan (000) propose he momenum life cycle hypohesis o show ha rading volume and sock reurns have inerrelaion in he sock marke cycle. Shiller (000) illusraes how invesor expecaions for fuure marke performance, relaed informaion of socks, and rading volume are driven by he mechanism of he feedback loop or he self-fulfilling prophecy. Hence, he rading volume is an imporan facor o rigger off he operaions of he marke cycle. When we ry o find ou nonlinear dynamic in he sock marke, he significan influence of rading volume on sock reurns should be considered. Tha is, in his sudy we employ he rading volume as he ransiion variable o examine he nonlinear dynamics of sock markes... Specificaion of models The STAR family of models has wo paricularly useful forms. One is he logisic STAR (LSTAR) model which describes a siuaion where differen saes of an economy have differen dynamics and he ransiion from one o he oher is smooh. The oher is he exponenial STAR (ESTAR) can explain he similar dynamic srucure of differen phases of an economy, bu he middle ground can have differen dynamics (Teräsvira and Anderson, 199) The advanage of employing STAR model in sudying financial marke is o ake consideraions ha individuals can be impossible o reac simulaneously o cerain

4 64 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo informaion. For example, when economic indicaors show bad signals, invesors in he marke would no ake he same decisions a he same ime. Some sell heir invesmens immediaely, bu some keep hem for a while. Heerogeneiies among invesors may resul from he differen risk aversion, experiences in processing informaion, invesing objecive, and ime o ge valuable informaion. The STAR model allows here are coninuous saes, no abrup srucural change, beween he exremes. Hence, STAR models are more suiable and realisic o process marke dynamics. Moreover, STAR models can explain differen saes during he marke cycle, such as, he bull and bear markes. Invesors could have differen sraegies in he bull and bear marke, so he marke dynamic could be differen in he bull and bear marke. The LSTAR model is good o describe such characerisics. Furhermore, exremes, such as peak and rough of sock markes, have similar dynamic, bu he mid-ground, periods of price going up coninuously and of price going downwards gradually, show differen dynamics. During he peak and rough of sock markes, reurns show characerisics of reversal; oherwise, reurns exhibi price momenum behavior. The ESTAR model is a suiable model o accoun for such paern. The STAR model is defined as r = α α β β ) + ε ' + x + ( ' + x ) F ( S (.1) d where r is he marke sock reurn, iid x = ' ( r, r,..., r ), 1 k 1 ' α = α, α,..., α, ( ) 1 k ' β 1 = ( β, β,..., β ), ε ~ N(0, σ ), F is he ransiion funcion, S 1 k is he ransiion d variable, and d is he delay parameer. The ransiion funcion can be a logisic form: F( S ) = {1 + exp[ ( S c)]} 1 γ, d d > 0 γ (.) where γ is he smooh parameer, measuring he ransiion speed from one regime o he oher; c indicaes he hreshold. When γ is grea han 0, he degree of auoregressive decay depends on he ransiion variable, S. When d d S is far above he hreshold, he value of ransiion funcion would approach 1; and when S is far below he hreshold, he value of d ransiion funcion approaches 0. 4 Hence, LSTAR model characerizes asymmeric processes of marke cycles. The ransiion funcion can also be an exponenial form: F( S ) = 1 exp[ γ ( S c) ], d d γ > 0 (.3) When γ is beween 0 and, he degree of auoregressive decay depends on he ransiion variable, o 0. As S d. As S d d is close o he hreshold, he value of ransiion funcion approaches S moves farher away from he hreshold, he value of ransiion funcion is 1. The ESTAR model suggess similar dynamic for low and for high values of he ransiion variable, bu differen dynamic for mid-range of he ransiion variable. We follow he modeling procedure as suppored by o Teräsvira and Anderson (199), Teräsvira (1994), and Saranis (001) o build our STAR models. Sep 1: Specify a linear AR model. We choose he lag lengh, p, of AR model o deermine he maximum value of k by Ljung-Box Q saisic for auocorrelaion.

5 Nonlinear Marke Dynamics Beween Sock Reurns and Trading Volume Sep : Tes he lineariy of STAR models for differen value of delay parameer, d, of he ransiion variable. We esimae he following auxiliary equaion. p p p ' 3 = + α x + α x s j j d + α x s x s j j d + α j j d j = 1 j= 1 j = 1 r α + u (.4) The null hypohesis for lineariy es is α α = α 0, j=1,, p. We chose d when = = j 3 j 4 j he lineariy is rejeced wih he smalles es saisic of p-value. Sep 3: Choose he appropriae model of STAR family by esing following resricions: H H H α = 0, j = 1,...,. (.5) j : p 04 4 : = p 03 3 j 4 j α = 0 / α = 0, j 1,...,. (.6) : = 0 4 α = 0 / α = α = 0, j 1,..., p (.7) j 3 j j If (.5) is rejeced we choose he LSTAR model. If (.5) is no rejeced and (.6) is rejeced, we choose he ESTAR model. If (.5) and (.6) are acceped and (.7) is rejeced, we selec he LSTAR model. 3. Empirical Resuls 3.1. The daa We invesigae weekly sock index reurns and rading volume of four Asian counries o explore heir nonlinear dynamic relaionship 5. There are Taiwan Weighed Sock Index, Hong Kong Hang Seng Index, Singapore Srais Times Index, and Korea Composie Index 6. The daa for Hong Kong, Singapore, and Korea are from he websie Yahoo! Finance 7. The daa for Taiwan Sock Index is from he Taiwan Economic Journal (TEJ) daa bank. All he index reurns are presened in naural logarihms. To avoid he problem of differen scales in sock reurns and rading volume, we employ he percenage change of rading volume as he ransiion variable o uncover he nonlinear dynamic relaionship beween sock marke reurns and rading volume. 3.. Uni roo ess and descripive saisics The uni roo ess of sock index reurns and he percenage change in rading volume are repored in Table 1. The uni roo ess for hose series show he evidences of saionariy. Table shows some descripive saisics for reurns and for percenage change in rading volume. We repor heir means, sandard deviaions, maximum, and minimum value. Because we will ake percenage change in rading volume o be he ransiion variable of he STAR models, we need o find he value of he hreshold of percenage change in rading volume, and make sure he value of he hreshold beween he maximum and minimum value. We can find percenage change in rading volume of four counries vary quie small.

6 66 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo Counry Saisic Table 1. Uni roo ess for sock reurns and for he percenage change in rading volume Uni roo ess for sock reurns Taiwan Hong Kong Singapore Korea Taiwan Hong Kong Uni roo ess for he change percenage in rading volume Singapore Korea Wd.Sy *** *** *** *** *** *** *** *** m. Dickey-F *** *** *** *** *** *** *** *** PP *** *** *** *** *** *** *** *** *** represens 1% level of significance. Wd.Sym, Dickey-F, and PP are he Weighed Symmeric es, Dickey- Fuller es, and Phillips-Perron es, respecively Tess for lineariy and selecion of STAR models The resuls for he maximum lag of he AR models and for he lineariy ess are presened in Table 3. We find ou he maximum lag of he AR models by Ljung-Box Q saisic (LB) for auocorrelaion. Mos counries we esimaing have relaively long lag lengh. Table. Descripive saisics for sock reurns and for he percenage change in rading volume Descripive saisics for sock reurns Descripive saisics for he percenage change in rading volume Counry Saisic Taiwan Hong Kong Singapore Korea Taiwan Hong Kong Singapore Korea Mean Sd.Dev Max Min When he lineariy es is rejeced a 5% level of significan, hose counries sugges nonlinear dynamics in sock markes. Resuls sugges he delay parameers of ransiion variable in mos counries are 0, excep for Hong Kong. The evidences imply ha he percenage change in rading volume a he same period can induce nonlinear dynamics of sock reurns in all of counries we invesigaing bu Hong Kong. We can argue ha here is inerrelaionship beween sock reurns and he percenage change in rading volume in he curren period in hose counries. Only he evidence of Hong Kong shows ha wo-period lagged percenage change in rading volume can resul in nonlinear dynamics of sock reurns. Table 4 repors he resuls of model specificaion. Based upon he selecion crieria menioned in secion, nonlinear models of all counries are deermined. We will employ he LSTAR model in Taiwan and Korea o explore heir asymmeric dynamics of he sock markes. Moreover, we explain nonlinear dynamics of he sock marke of Hong Kong and Singapore by he ESTAR model.

7 Counry Nonlinear Marke Dynamics Beween Sock Reurns and Trading Volume Maximum k a LB(υ) b Taiwan k=3 LB(3)= P-value= 0.66 Hong Kong k=1 LB(38)= P-value= Singapore k=40 LB(38)= P-value= 0.89 Korea k=40 LB(40)= P-value= Table 3. Tess for lineariy Minimum P-value over 0 d 8 c d= d= d= d=0 Delay parameer Noes: a: Firs, we esimae linear AR models of differen orders. The maximum value of k is seleced by Ljung-Box Q saisic for auocorrelaion. b: LB(υ) is he Ljung-Box saisic forυ order auocorrelaion in he AR model. c: Choose d wih he lowes P-value over he range 0 o 8. Counry Table 4. Specificaion of he nonlinear model a H b 04 H b 03 H 0 b Type of model Taiwan * LSTAR Hong Kong * ESTAR Singapore * ESTAR Korea * LSTAR Noes: a: The value lised on he column of H, H and H are p-value for esing model specificaions. 0 b: Equaion (6), (7) and (8) in he secion Esimaes of he nonlinear models The esimaes of STAR models are processed by he mehod of nonlinear leas squares and repored in Table 5. Moreover, we follow he suggesions of academics o esimae γ by dividing he sandard deviaion of r, σ (r), for he LSTAR model, and by dividing he variance of r, σ ( r), for he ESTAR model (Granger and Teräsvira, 1993; Teräsvira, 1994). Hence, γ is scale-free and easier o inerpre. From he Table 5, esimaes of he parameers of ransiion funcion, β, are significan and show srong evidences of nonlinear models. Mos imporan, he smoohing parameers, γ, of all models are highly significan and γ is quie small in mos counries. The resuls sugges here is slow ransiion beween regimes in hose counries. However, comparing wih oher counries, Singapore has raher quick ransiion speed beween regimes. Besides, he hresholds of all models are highly significan a he 5% level. We plo he esimaed ransiion funcions of STAR models for sock reurns in Taiwan, Hong Kong, Singapore, and Korea. From Figure 1(a) o Figure 4(b), we show he shapes of he ransiion funcions of four counries. Each poins can indicae wha value of

8 68 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo ransiion funcion has obained and how frequenly. For example, from Figure 1(a) and 1(b), we can see readily he values of ransiion funcion in Taiwan are wihin he wo hresholds mos ime. Figure (a), (b), 3(a), and 3(b) also show he ransiion funcions are normally o be one in Hong Kong and Singapore. Besides, from 4(a) and 4(b), we find values of he ransiion funcions are usually o be zero in Korea Dynamic behavior I is differen o inerpre he esimaed coefficiens of STAR model from Table 5. However, we can ge more informaion abou dynamic behavior of esimaed models by examining he characerisic roos of polynomial models. We compue he roos of he STAR models by solving he following characerisic polynomial (Teräsvira and Anderson,199) for F=0, 1. k k k j λ ( α1 j + β1 jf) λ = 0 (3.8) j= 1 When F=0, which means he lower or conracion regime in he LSTAR model and he middle range in he ESTAR model. When F=1, which explains he upper or expansion regime in he LSTAR model and he ouer (conracion or expansion) regime in he ESTAR model. The Table 6 shows he mos prominen roos of esimaed models for each regime in four counries. Taiwan: LSTAR r r Table 5. Esimaes of he nonlinear models a = - r rr rr rr r 3 + ( rr (-7.78) (.500) ( 5.818) ( 3.44) ( 4.116) (-3.9) ( 9.979) (-.139) r σ r r ) 1+ exp 1.9(1/ ( )( 0.067) 14 r s 3 r { } 1 (-.447) (-3.519) ( 3.475) ( 5.967) ( 1.81) VAR / VAR l =0.816 Hong Kong: ESTAR r = r r r 1 + ( r r r 1) ( 7.07) ( 5.98) (-9.11) (-4.11) ( ) (-5.96) ( 8.969) ( 4.067) 1 exp{ 13.00(1/ σ ( r))( s (0.637) ) } ( 4.6) ( ) VAR / VAR l =0.844 Singapore: ESTAR r = r r r r r 40 + ( r (-.074) (-.181) (.338) (-.08) ( -.588) (-3.35) (.074) (.183) r r r r 40) (-.337) (.083) (.591) (.344) exp{ 36.83(1/ σ ( r))( s (1.196)) } 1 1

9 Nonlinear Marke Dynamics Beween Sock Reurns and Trading Volume (.446) ( ) VAR / VAR l =0.845 Korea: LSTAR r = rr r rr 13 + ( r 13) ( 4.335) (-.765) (.087) (.34) (-5.467) (-1.857) { 4.78(1/ σ ( r))( s 0.55 } 1 (.068) ( 6.056) 1 + exp VAR / VAR l =0.865 Noes: a: Figures in parenhesis are -saisics of coefficiens. VAR / VAR is he raio of variances for nonlinear and linear l models. For four counries boh regimes conain complex pairs of explosive roos. This implies ha he sock markes in all counries show cyclical movemens during regimes. In Hong Kong and Singapore, he middle regime is dominaed by an explosive roo, which implies he sock marke passing hrough he hresholds very quick on he way up or down. However, he ouer regime is sable, which means he sock marke ends o say here. These argumens also can consis wih he Table 5, Figure (a), (b), 3(a) and 3(b). The ransiion parameers of Hong Kong and Singapore are large. Moreover, mos values of he ransiion funcion are near one for hese wo counries. For he LSTAR model, Taiwan and Korea show ha boh upper and lower regimes are sable. The resuls imply he sock markes of Taiwan and Korea have a endency o remain a boh conracion and expansion phases. Table 6. The mos prominen characerisic roos in regimes Counry Regimes a Mos prominen roos Modulus Taiwan L 0.960±0.0799i ±0.1000i U ±0.1885i Hong Kong M O ±0.1499i ±0.131i Singapore M O ±0.0590i ±0.0765i Korea L ±0.83i U ±0.3506i Noe: a. For he ESTAR model, M is he middle range and O is he ouer regime; for he LSTER model, L is he lower regime and U is he upper regime.

10 630 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo v a lu e o f r a n s i io n f u n c io n v a l u e o f r a m s i i o n h r e s h o ld p e r c e n a g e c h a n g e in r a d in g v o lu m e Figure 1(a) The esimaed ransiion funcion of he LSTAR model for sock reurns agains he percenage change in rading volume in Taiwan v a l u e o f r a n s i i o n f u n c io n / D e c / A u g / J u l / A p r / J a n / O c / J u l Figure 1(b) The esimaed ransiion funcion of he LSTAR model for sock reurns in Taiwan by rading volume v a l u e o f r a n s i i o n f u n c i o n h r e s h o l d p e r c e n a g e c h a n g e i n r a d in g v o l u m e Figure (a) The esimaed ransiion funcion of he ESTAR model for sock reurns agains he percenage change in rading volume in Hong Kong value of ransiion funcion 0 0 / A p r / M a r / M a r / M a r / F e b Figure (b) The esimaed ransiion funcion of he ESTAR model for sock reurns in Hong Kong v a l u e o f r a n s i i o n fu n c i o n h re s ho ld percenage change in rading volume Figure 3(a) The esimaed ransiion funcion of he ESTAR model for sock reurns agains he percenage change in rading volume in Singapore v a l ue o f r a n s i i o n f u n c io n / A u g / J u l / J u n 0 0 / M a y / A p r / M a r Figure 3(b) The esimaed ransiion funcion of he ESTAR model for sock reurns in Singapore v a lu e o f ra n s iio n fu n c io n h r e s h o ld p e r c e n a ge c h a n g e in r a d in g v o lu m e Figure 4(a) The esimaed ransiion funcion of he LSTAR model for sock reurns agains he percenage change in rading volume in Korea /M a y /M a y /M a y 0 0 /A p r /A p r /M a r /M a r /F e b Figure 4(b) The esimaed ransiion funcion of he LSTAR model for sock reurns in Korea 4. Ou-of-Sample Forecasing To evaluae he forecasing abiliies of he STAR models, we examine he ou-of sample forecasing performance by comparing nonlinear and linear models. Firs, we reesimae he STAR models by reducing 1-period daa for each counry 8. Then we make ou-of-sample forecasing for 1 o 1 periods ahead by re-esimaed parameers for STAR models and linear auoregressive (AR) models. We adop wo crieria o evaluae he accuracy of forecass. One is he roo mean squared error (RMSE). The oher is he mean absolue error (MAE). Their definiions are as followings.

11 Nonlinear Marke Dynamics Beween Sock Reurns and Trading Volume

12 63 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo 1 K f RMSE = ( r r K i = 1 + i ) a + i (4.10) r + and f i a i MAE = 1 K f a r + i r + 1 (4.11) K i = 1 r + are acual and forecas values respecively a ime +i. The forecasing period is from +1 o +K. K is 1, which is he ou-of-sample forecasing period. Table 7 summaries he resuls of ou-of-sample forecasing for all counries. We find ou values of RMSE and MAE of STAR models are smaller han hose of AR models. All resuls suppor STAR models have beer forecasing performances han AR models. 5. Conclusion This sudy examines wheher here exiss any nonlinear marke dynamics beween sock reurns and rading volume. We examine Taiwan, Hong Kong, Singapore, and Korea sock markes by employing he STAR models. The resuls are summarized as followings. Firs, four counries show nonlinear evidences in he sock marke cycle. Besides, For Taiwan and Korea, nonlinear sock marke dynamics are characerized by he LSTAR models. Sock reurns of Hong Kong and Singapore can be described by he ESTAR models. Secondly, from he characerisic polynomial of mos esimaed models displays a leas one explosive roo, which means ha sock marke cycles in mos counries are asymmeric. Moreover, from he ou-of-sample forecasing performance of nonlinear models, we also conclude nonlinear models have beer forecasing abiliies. Third, we have an imporan finding, which is rading volume as a rigger o cause nonlinear dynamic cycle of sock reurns. In sock markes invesors believe Trading volume signals prices. When rading volume goes o unusual high level, invesors would expec he highes prices o happen. Trading volume is a wide-used and informaive marke saisic. When rading volume shrinks o unusual low level, invesors predic he lowes prices o ake place. Moreover, valuable informaion provided by rading volume is useful no only for he individual sock bu also for he marke index. From he Table 5, he resuls show all hresholds are highly significan. Srong evidences suppor ha rading volume is good o be he ransiion variable inducing cyclical dynamics of sock marke. In conclusion, our sudy shows ha invesigaed counries have nonlineariies and cyclical behavior in sock marke reurns. Moreover, rading volume plays an imporan role in he cyclical movemens of he sock marke. Trading volume is a common marke indicaor for invesors in he sock markes. Invesors usually observe he change in rading volume firs and hen make heir invesmen decisions. Neverheless, quesions remain as o make invesmen sraegies based upon he change in rading volume. We hope o address such opic in our ongoing research. References Amihud, Y. and H. Mendelson (1986): Asse pricing and he bid-ask spread, Journal of Financial Economics, 17(), 3-49.

13 Nonlinear Marke Dynamics Beween Sock Reurns and Trading Volume Blume, L., D. Easley, and M. O'Hara (1994): Marke saisics and echnical analysis: he role of volume, Journal of Finance 49(1), Brennan, M., T. Chordia, and A. Subrahmanyam (1998): Alernaive facor specificaions, securiy characerisics, and cross-secion of expeced sock reurns. Journal of Financial Economics, 49(3), Brichenhall, C., H. Jessen, D. Osborn, and P. Simpson (1999): Predicing US business-cycle regimes. Journal of Business and Economic Saisics, 17(3), Brock, W. and B. LeBaron (1996): A dynamic srucural model for sock reurn volailiy and rading volume, Review of Economics and Saisics, 78(1), Cecchei, S., P-S. Lam, and N. Mark (1990): Mean reversion in equilibrium asse prices, American Economic Review, 80(3), Conrad, J., A. Hameed, and C. Niden (1994): Volume and auocovariances in shor-horizon individual securiy reurns, Journal of Finance, 49(4), Daar, V., N. Naik, and R. Radcliffe (1998): Liquidiy and sock reurns: An alernaive es, Journal of Financial Markes, 1(), De Long, B., A. Shleifer, L. Summers, and R. Waldmann (1990): Noise Trader Risk in Financial Markes, Journal of Poliical Economy, 98(4), Dumas, B. (199): Dynamic equilibrium and he real exchange rae in a spaially separaed world, Review of Financial Sudies, 5(), Granger, C. and T. Teräsvira (1993): Modeling nonlinear economic relaionships, Oxford :Oxford Universiy Press. Keynes, J. (1936): The General Theory of Employmen, Ineres and Money, London: Macmillan. Lee, C. and B. Swaminahan (000): Price Momenum and Trading Volume, Journal of Finance, 55(5), Lee, W., C. Jiang, and D. Indro (00): Sock marke volailiy, excess reurns, and he role of invesor senimen, Journal of Banking & Finance, 6(1), Leung, M., H. Daouk, and A-S. Chen (000): Forecasing sock indices: a comparison of classificaion and level esimaion models, Inernaional Journal of Forecasing, 16(), Liner, J. (1965): Securiy price, risks, and maximal gains form diversificaion, Journal of Finance, 0(4), McMillan, D. (001): Nonlinear predicabiliy of sock marke reurns: evidence from nonparameric and hreshold models, Inernaional Review of Economics and Finance, 10(4), McMillan, D. (003): Non-linear predicabiliy of UK sock marke reurns, Oxford Bullein of Economics and Saisics, 65(5), McMillan, D. (004): Nonlinear predicabiliy of shor-run deviaions in UK sock marke reurns, Economics Leers, 84(), McMillan, D. (005): Non-linear dynamics in inernaional sock marke reurns, Review of Financial Economics, 14(1), McMillan, D. (007): Non-linear forecasing of sock reurns: Does volume help?, Inernaional Journal of Forecasing, 3(1), Öcal, N. and D. Osborn (000): Business cycle non-lineariies in UK consumpion and producion, Journal of Applied Economerics, 15(1), Perez-Quiros, G. and A. Timmermann (000): Firm size and cyclical variaions in sock reurns, Journal of Finance, 55(3), Peers, E. (1994): Fracal marke analysis: applying chaos heory o invesmen and economics, New York: John Wiley.

14 634 Wu-Jen Chuang, Liang-Yuh Ou-Yang, Wen-Chen Lo Ross, S. (1976): The arbirage heory of capial asse pricing, Journal of Economic Theory, 13(3), Saranis, N. (001): Nonlineariies, cyclical behaviour and predicabiliy in sock markes: inernaional evidence, Inernaional Journal of Forecasing, 17(3), Sercu, P., R. Uppal, and C. Van Hulle (1995): The exchange rae in he presence of ransacions coss: Implicaions for ess of purchasing power pariy, Journal of Finance, 50(4), Sharpe, W. (1964): Capial asse prices: A heory of marke equilibrium under condiions of risk, Journal of Finance, 19(3), Shiller, R. (000): Irraional exuberance, New Jersey :Princeon Universiy Press. Shleifer, A. (000): Inefficien marke: an inroducion o behavioral finance, New York: Oxford Universiy Press. Teräsvira, T. (1994): Specificaion, esimaion and evaluaion of smooh ransiion auoregressive models, Journal of he American Saisical Associaion, 89(45), Teräsvira, T and H. Anderson (199): Characerizing nonlineariies in business cycles using smooh ransiion auoregressive models, Journal of Applied Economerics, 7, S119-S136. Telock, P. (007): Giving conen o invesor senimen: The role of media in he sock marke, Journal of Finance, 6(3), Turner, C., R. Sarz, and C. Nelson (1989): A Markov model of heeroskedasiciy, risk and learning in he sock marke, Journal of Financial Economics, 5(1), 3-. Noes: 1. The CAPM (Sharpe,1964 ;Liner,1965) saes ha he only sysemaic risk can be priced and ha he expeced reurn of a asse is equal o risk-free rae plus a risk premium muliplied by he asse s sysemaic risk. Ross (1976) develops he APT which implies ha here are muliple risk facors ha need o be aken ino accoun when calculaing asse reurns.. Keynes(1936) argues GNP has asymmeric processes during he business cycle. Teräsvira and Anderson (199) find nonlinear behavior of he indusrial producion growh for OECD counries. Öcal and Osborn (000) invesigae nonlineariy of UK consumpion and Indusrial producion. 3. McMillan (001,003,004,005,007) employ STAR models o examine non-linear relaionship of priced facors and index reurns, such as: UK, CAC, Nikki, DAX, Hang Seng, Kuala Lumpur and Singapore Srais; Saranis, (001) invesigaes here are poenial lineariies in he sock prices of seven major indusrial counries. 4. Saranis (001) explains ha he lower regime, F=0, is he conracion regime and he upper regime, F=1, is he expansion regime in he LSTAR model. McMillan (003) explains wo regimes under he LSTAR model as large reurns and small reurns. 5. Some economiss consider shor-frequen economic ime series daa as oo noisy o reflec he cyclical movemens of variables (Teräsvira and Anderson, 199; Birchenhall e al., 1999). However, if we process yearly or quarerly sock reurns and rading volume o find ou heir nonlinear relaionship. The resuls can no provide valuable informaion for invesors o make decisions. For example, if invesors buy socks based upon las year or las quarer daa of rading volume, i is no up-o-dae informaion for mos invesors. Hence, recen sudies concerning abou nonlinear dynamics of sock reurns employ daily sock index (McMillan,005). In our sudy, we exploi valuable informaion for mos invesors by weekly daa. 6. The sample period for Taiwan Weighed Sock Index is from 1971/1/9 o 006/11/10, Hong Kong Hang Seng Index is from 001/7/9 o 006/11/7, Singapore Srais Times Index is from 1995/1/3 o 006/11/7, and Korea Composie Index is from 1998/4/7 o 006/11/7. 7. hp://finance.yahoo.com/inlindices?e=asia 8. The re-esimaed sample period for Taiwan Weighed Sock Index is from 1971/1/9 o 006/8/18, Hong Kong Hang Seng Index is from 001/7/9 o 006/9/4, Singapore Srais Times Index is from 1995/1/3 o 006/9/4, and Korea Composie Index is from 1998/4/7 o 006/9/4.