Modeling the Conditional Heteroscedasticity and Leverage Effect in the Chinese Stock Markets

Transcription

1 9h Inernaional Congress on Modelling and Simulaion, Perh, Ausralia, 6 December 0 hp://mssanz.org.au/modsim0 Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes Z. Yin a, A.K. Tsui a and Z.Y. Zhang b a Deparmen of Economics, Naional Universy of Singapore, Singapore b School of Accouning, Finance & Economics, Edh Cowan Universy, Ausralia zhaoyong.zhang@ecu.edu.au Absrac: The Chinese sock marke has experienced an asonishing growh and unprecedened developmen since s incepion in he early 990s, emerged o be he world's second-larges by marke value by he end of 009. The Chinese sock marke is also one of he mos volaile markes, which has been called by many observers a casino. In he recen years here are several far-reaching evens ha have reshaped he Chinese sock markes. The mos noable evens include he do-com bubble in 000, China s non-radable shares reform in 005 and he global financial crisis in 008. I is noed ha he do-com bubble has caused he Chinese sock markes a sharp oscillaion since 000. Wh a shor-lived bull, he Chinese sock markes experienced a nearly five years long bear marke unil June 005 when he reform of non-radable shares was implemened, which increased he liquidy and brough he markes back o a long-erm bull run. Since he US sub-prime morgage crisis he Chinese sock markes have shown exreme insabily and severe volaily, which has become he major concern o he policy-makers and invesors. Many exising sudies have revealed ha he financial ime series daa exhib linear dependence in volaily, which indicaes he presence of heeroskedasicy, implying he exisence of volaily clusering. Alhough direc generalizaions from he univariae GARCH models are sraighforward, heir applicaions are limed by pracical issues associaed wh cumbersome compuaion and srong resricions on parameers o guaranee posive defineness of variance marixes. This sudy inends o examine he presence of heeroskedasicy and he leverage effec in he wo Chinese sock markes, and o capure he dynamics of condional correlaion beween reurns of China s sock markes and hose of he U.S. in a bivariae VC- MGARCH framework. The resuls show ha ha he leverage effec is significan in boh Shanghai and Shenzhen markes during he sample period in , and he condional correlaion beween mainland China s and he U.S. sock markes is que low and highly volaile. The resuls indicae ha ha uncerainy derived from ime-varying relaionship beween Shanghai and he U.S. sock markes is more significan han ha beween Shenzhen and he U.S. sock markes. In addion, he Chinese sock markes are found o be highly regimes persisen, hereby reducing poenial benefs induced by acively rading. These findings have imporan implicaion for invesors seeking opporuny of porfolio diversificaion. Keywords: Chinese sock marke, heeroskedasicy, leverage effec, VC-MGARCH models 338

2 Yin e al., Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes. INTRODUCTION Ever since s incepion in he early 990s, he Chinese sock marke has experienced an asonishing growh and unprecedened developmen, emerged o be he world's second-larges by marke value by he end of 009. Thanks o he las decade s inensive and exensive reforms in China's securies marke, which has improved subsanially he regulaory sysem and he marke-oriened appraisal sysem for inial public offering (IPO) as well as expanded capal supply o he marke. However, he Chinese sock marke is also one of he mos volaile markes, which has been called by many observers a casino. In he recen years here are several far-reaching evens ha have reshaped he Chinese sock markes. The mos noable evens include he do-com bubble in 000, China s non-radable shares reform in 005 and he global financial crisis in 008. The impacs of hese evens on he daily reurns of he Shanghai and Shenzhen markes can be clearly viewed in Figure. I is noed ha he do-com bubble has caused he Chinese sock markes a sharp oscillaion since 000. Wh a shor-lived bull, he Chinese sock markes experienced a nearly five years long bear marke unil June 005 when he reform of non-radable shares was implemened, which increased he liquidy and brough he markes back o a long-erm bull run. Since he US sub-prime morgage crisis he Chinese sock markes have shown exreme insabily and severe volaily, which has become he major concern o he policy-makers and invesors. Many exising sudies have revealed ha he financial ime series daa exhib linear dependence in volaily, which indicaes he presence of heeroskedasicy, implying he exisence of volaily clusering. Engle (98) proposed he ARCH (Auoregressive condional heeroskedasicy) model, which assumes imevarying variances are condional on pas informaion and uncondional variances are consan. Bollerslev (986) lae exended o he Generalized Auoregressive condional heeroscedasicy (GARCH). The GARCH(,) model is ofen sufficien for mos of financial series, hereby has effecively reduced he lag lengh in he ARCH model ha may induce cumbersome compuaion. Nelson (99) proposes he EGARCH (Exponenial GARCH) model o capure he asymmeric response o good news and bad news hrough inerpolaing absolue residuals ino he condional variances equaion and relax he non-negaivy consrains by aking he log form. The GJR-GARCH model developed by Glosen e al. (993) reas asymmeric effec as a dummy variable and is also capable of capuring leverage effec. Figure : Daily Reurns of he Sock Exchange Compose Index (a) Shanghai Alhough direc generalizaions from he univariae GARCH models are sraighforward, heir applicaions are limed by pracical issues associaed wh cumbersome compuaion and srong resricions on parameers o guaranee posive defineness of variance marixes. To ackle he compuaional complexies associaed wh he direc generalizaions, Bollerslev (990) inroduces he consan condional correlaion (CCC)- MGARCH model. In paricular, he univariae GARCH models are used o capure each reurns series and hen linked ogeher by he condional correlaion marix. I allows for more flexibily, and also is easier o inerpre. Tse and Tsui (00) develop a varying-correlaion MGARCH (VC-MGARCH) model. They assume ha he ime-varying condional-correlaion marix follows an ARMA(,) srucure, which is similar o a dynamic condional correlaion (DCC-MGARCH) model proposed by Engle (00). In his sudy we inend o invesigae he presence of heeroskedasicy and he leverage effec in he Chinese sock marke. The presence of heeroskedasicy in sock reurns affirm ha invesmen decisions in he curren period are affeced by he unexpeced volaily in he previous period. There have been a few sudies on modelling and forecasing sock marke volaily in China. Xu (999) sudies he volaily for daily spo reurns of Shanghai compose sock index in , and found ha he GARCH model is superior o ha of eher EGARCH or GJR-GARCH models, indicaing ha here is almos no so-called leverage effec in he Shanghai sock marke since volaily is mainly caused by he changes in governmenal policy. Lee e al. (00) examine he ime-series feaures of sock reurns and volaily in four of China s sock exchanges and found srong evidence of ime-varying volaily, indicaing volaily is highly persisen and predicable. Copeland and Zhang (003) also find no evidence of leverage effec in mainland China s sock markes when hey adop he EGARCH model o capure he volaily during he period in Based on he four (b) Shenzhen

3 Yin e al., Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes variable asymmeric GARCH fed in he BEKK srucure developed by Engle and Kroner (995), Li (007) concludes ha no direc linkage exiss beween mainland China s sock markes and he U.S. marke, hereby furnishing porfolio invesors wh diversificaion benefs. Tsui and Yu (999) apply his model o capure condional correlaion beween Shanghai and Shenzhen sock markes and conclude he consancy is rejeced by he informaion marix es. However, he assumpion of consan condional correlaions seems unrealisic for mos of financial series. In his sudy, we idenify wo discree regimes for each sock marke, relaively sable sae and highly volaile sae, and make probabilisic inference on he persisence of each sae, following he mehodology of Hamilon (989). In addion, we capure he dynamics of condional correlaion beween reurns of China s sock markes and hose of he U.S. in a bivariae VC-MGARCH framework o shed some ligh in how he wo markes are correlaed and wheher hey can bring diversificaion o invesors. The ime-varying-parameer models wh Markov-swching heeroskedasicy proposed by Kim (993) is adoped o capure he changing relaionship beween reurns of China s sock markes and hose of he U.S. The res of our hesis is organized as follows. Secion describes he mehodology used for his sudy, and Secion 3 analyzes he daa ses and he esimaion resuls. The las secion concludes wh implicaion drawn from our findings on equy invesmen.. METHODOLOGY AND THE MODEL I has been well documened ha for a wide range of financial daa series, ime-varying condional variances can be explained empirically hrough he auoregressive condional heeroskedasicy (ARCH) model of Engle (98). When he ime-varying condional variance has boh auoregressive and moving average componens, his leads o he generalized ARCH(p,q) (GARCH(p,q)) model of Bollerslev (986). In he seleced condional volaily model, he residual series should follow a whe noise process. Li e al. (00) provide an exensive review of recen heoreical resuls for univariae and mulivariae ime series models wh condional volaily errors. McAleer (005) reviews a wide range of univariae and mulivariae, condional and sochasic, models of financial volaily. McAleer e al. (007) discuss recen developmens in modeling univariae asymmeric volaily and McAleer e al. (008) develop he regulary condions and esablish he asympoic properies of a general model of ime-varying condional correlaions. The purpose of his secion is o brief a bivariae VC-MGARCH framework employed in his sudy. We firs define he GARCH framework for modelling condional heeroskedasicy and ime-varying condional correlaions, and hen discuss he ime-varying-parameer models wh Markov-swching heeroskedasicy proposed by Kim (993) o capure he changing relaionship beween reurns of China s sock markes and hose of he U.S. Le r be he daily reurns of he Shanghai Sock Exchange Compose Index/he Shenzhen Sock Exchange Componen Index. The condional mean equaion for each variable is effecively capured by an ARMA(,) srucure specified as follows: r = c + φ r + θ ε + ε () i i i, i i, where ε is he idenically and independenly disribued error erm. The GARCH(,) proposed by Bollerslev (986) is used for he condional variance equaion: ε = η σ η ~ i. i. d.(0,) (), σ = α + α ε + β σ (3) i0 i i, i i, where α i0 > 0, αi 0, βi 0 and αi + βi <. Parameers in he above equaions are ypically esimaed by he maximum likelihood mehod o obain Quasi-Maximum Likelihood Esimaors (QMLE) in he absence of normaly of he condional shocks ( η ). To capure he asymmeric feaure, we inroduce hree ypes of asymmeric GARCH models, i.e., he EGARCH, GJR-GARCH(,) and A-PARCH(,d,). The laer imposes Box-Cox power ransformaion on he condional volaily funcion, hus allowing for more flexibily, which can be specified as follows: σ = α + α ( ε + γ ε ) + β σ (4) d d d i0 i i, i i, i i, Where negaive γ denoes leverage effec. We have conduced quasi-maximum likelihood esimaion on d i and found he model wh d = is more robus o exreme values han wh oher value The resricion, 340

4 Yin e al., Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes α ( + γ ) + β <, is required for he A-PARCH(,,) o ensure covariance saionary. Similarly, wh A- i i i PARCH(,,), E( σ ) and E( ε ) are guaraneed for exisence, providing / π * αi + βi <. In order o capure condional correlaions beween reurns of he Shanghai Sock Exchange Compose Index/he Shenzhen Sock Exchange Componen Index and reurns of he S&P 500 Index, we follows he mehodology of Tse and Tsui (00) o model ime-varying condional correlaions in a bivariae GARCH(,) framework. In paricular, he condional correlaions formulaion in a bivariae VC-MGARCH model is ρ = (-θ -θ )ρ + θ ρ - + θ ψ - (5) where (-θ -θ )ρ is he ime-invarian condional correlaion coefficien, θ and θ are assumed o be nonnegaive and sum up o less han, and ψ - is specified as ψ = n= e, ne, n ( )( e, n e =, = n n n ) Ignoring he consan erm and assuming normaly, he condional log likelihood funcion of sample size n is, e + e ρ e e,,,, L = log( ρ ) + ( ρ ) The oal number of parameers is for a bivariae asymmeric GARCH model wh varying correlaions, and his number always exceeds ha of Bollerslev s (990) consan-correlaion model by. In fac, he CC- MGARCH model is nesed whin he VC-MGARCH model by resricing θ and θ o zero. We employ he Markov-swching variance models (equaion 7) o sudy he regimes swching and he imevarying-parameer models wh Markov-swching heeroskedasicy (equaion 8) o assess he condional heeroskedasicy of r sh and r sz. The oscillaory behavior of ime-varying volaily can be caegorized ino wo disinc regimes: relaively sable sae and highly volaile sae. Markov-swching variance models can be specified as r = c + φ r + θ ε + a r + ε, i i i, i i, i sp, ε ~ N(0, σ is ), σ = σ + ( σ σ ) S is i0 i i0, Pr[ S = S = ] = p, Pr[ S = 0 S = 0] = p 00 And he ime-varying-parameer models wh Markov-swching heeroskedasicy is (7) (6) r = c + φ r + θ ε + β r + ε, i i i, i i, sp, β = α β + v, v ε i i, ~ N (0, σ v ), i ~ N (0, σ is ), σ = σ + ( σ σ ) S is i0 i i0, Pr[ S = S = ] = p, Pr[ S = 0 S = 0] = p 00 (8) where i = sh, sz, ε is assumed o follow normal disribuion, σ denoes he variances when China s sock markes are relaively sable, and σ represens he variances when hey are suffering from huge i shocks. In paricular, σ is assumed o be smaller han i0 σ. i r is no included in he condional mean sz, equaion because of s insignifican value in he esimaion resuls. Furher, we conduced several ess by employing he efficien algorhm a la Bai and Perron (00) o idenify he exisence of muliple srucural changes and corresponding number of breaks associaed wh inerpolaing full sample of rsp, ino he condional mean equaion, The LM and Ljung-Box ess are applied o assess he appropriaeness of he wo Markov-swching models. i0 34

5 Yin e al., Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes 3. EMPIRICAL ANALYSIS 3. Daa Descripion We colleced our daa ses from Yahoo.Finance spanning from January 5, 000 o December 3, 008. The Shanghai Sock Exchange Compose Index launched on July 5, 99 is a whole marke index, including all lised A-shares and B-shares raded a he Exchange. A-shares are raded in RMB, while B-shares are raded in U.S. dollars a he Shanghai Sock Exchange and in Hong Kong dollars a he Shenzhen Sock Exchange. The index is compiled using Paasche weighed formula. Differing from he Shanghai Exchange, he Shenzhen Componen Index only selecs 40 represenaive lising companies radable shares o rack he marke s performance, hereby minimizing he inaccuracy induced by non-radable shares. The S&P 500 Index, inially published in 957, is one of he mos widely quoed and racked marke-value weighed indices, represening prices of 500 socks acively raded in eher New York Sock Exchange or NASDAQ. Since March 005, has implemened he policy ha only acively raded public shares (floa weighed) are considered for he calculaion of marke capalizaion. In subsequen secions, he Shanghai Sock Exchange Compose Index, he Shenzhen Sock Exchange Componen Index and he S&P 500 Index are abbreviaed by sh, sz and sp, respecively. The daily reurns of hose indices, r, are compued as: P r = ln( )*00 P where i = sh, sz, sp, P sands for he close price of each index adjused for dividends and spls a dae. (9) Table displays he summary saisics of he daily reurns of he hree indices. As can be seen from Table, all he series are lef-skewed and highly lepokuric. In paricular, r has he highes kurosis and also he sp skewness, indicaing ha negaive reurns are more prevalen. Such non-normal properies are also capured by he highly significan Jarque-Bera es saisics. The high Lagrange muliplier es saisics also indicae srong ARCH effecs of hese series. As such, appropriae GARCH models seem adequae o accommodae he saisical feaure of lepokurosis. Before proceeding o he specific models, we employ he augmened Dickey-Fuller (ADF) and he Efficien Modified Phillips-Perron (PP) ess o check he saionary of all he series. Our findings, available upon reques, show ha all ADF and PP es saisics are significan a he % level, hereby indicaing ha all he reurn series are saionary. 3. Empirical Resuls The esimaes of condional correlaions beween reurns in he Chinese and he U.S. sock markes based on esimaing bivariae VC-MGARCH(,) and CC- MGARCH(,) model are repored in Table. I is found ha boh VC-MGARCH(,) and CC-MGARCH(,) models saisfy he resricions imposed on he GARCH(,) model, i.e. α > 0, 0 < β < and α + β <. The insignifican consan erm in he condional correlaion equaion in he VC- MGARCH(,) is consisen wh he insignifican correlaion in he CC-MGARCH(,) where he correlaion is assumed o be ime-invarian. Tha implies here is no significan linkage beween boh he Shanghai/Shenzhen markes and he U.S. sock marke when assuming he correlaion is ime-invarian. However, his assumpion is que unreasonable. From Table, is noed ha he condional correlaion significanly follows an AR() process. The LM and Ljung-Box ess saisics for he VC-MGARCH(,) and Table : Summary Saisics for r, i = sh, sz, sp r sh r r sz sp (A) Descripive saisics Mean Sd. dev Minimum Maximum Skewness Kurosis No. of obs (B) Jarque-Bera es for normaly Jarque-Bera * * * (C) LM es for ARCH effec LM(0) * 34.59* * (*: a he % significance level) 34

6 Yin e al., Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes he CC-MGARCH(,) (no repored due o space limaion, bu available upon reques) indicae no evidence of ARCH effec or serial correlaion in boh he sandardized residuals and he sandardized squared residuals. However, he large likelihood raio es saisic demonsraes he VC-MGARCH(,) model indeed ouperforms he CC-MGARCH(,) one a any convenional level of significance. Figure displays he condional correlaion beween Shanghai/ Shenzhen and he U.S. sock markes, respecively. Alhough he linkage beween he wo markes is que low, here is sill some connecion beween he wo markes, aribued o he inegraion of global financial markes. Addionally, he correlaion varies remarkably, wh he presence of an upward or downward endency even during a shor period. Tha phenomenon can be explained by he immaury of he Chinese sock markes and governmenal influence on he markes. The low and highly volaile connecion may bring diversificaion benefs o equy porfolio. We have conduced several ess by employing he efficien algorhm a la Bai and Perron (00) o idenify he exisence of muliple srucural changes and corresponding number of breaks associaed wh inerpolaing full sample of ino he condional mean equaion. All he es saisics are insignifican a he 5% level for boh specificaions, Table : VC-MGARCH(,) and CC-MGARCH(,) indicaing ha no break has been inroduced when r is sp, inerpolaed. This finding is confirmed by he saisic value of BIC, modified Schwarz crerion (LWZ) and sequenial mehod wh all selecing zero break. We hen esimae he Markovswching variance models ** 0.087*** 0.94*** Panel B: VC-MGARCH(,) model for ε sz and ε sp SZ (0.068) (0.04) (0.044) 0.985*** (equaion 7) and he ime-varyingparameer models wh Markov- (0.0060) (0.038) (0.05) 0.05* *** 0.904*** (0.085) (0.053) (0.0099) CC-MGARCH(,) model for swching heeroskedasicy ε sz and ε sp 0.038*** *** *** (equaion 8), and he resuls are SZ (0.0063) (0.005) (0.0044) 0.03 N.A. N.A. available upon reques. I is found 0.09*** *** *** (0.06) (0.006) (0.0098) (0.00) ha all he esimaes are saisically Noe: Sandard errors are in parenheses. * indicaes he 0% significance level; ** he 5% significance significan a he 5% or % level, level; and *** he % significance level. which indicaes he exisence of wo discree regimes and Markov-swching heeroskedasicy. Boh models show a very similar resul concerning he volaily. I is found ha boh p 00 and p have a value close o uny, implying regimes persisence. This finding is consisen wh he observaion ha in 008 he Chinese marke experienced persisenly high level of volaily, while in 006 was que sable. The duraion of high volaily is abou 5 days on average, while he low volaily sae lass for 33 days on average. I is also seen ha, in conras o our expecaion, he esimaes of φ and θ are que small, far less han uny. Alhough is found posive and saisically also significan, he value of he esimae a is relaively small, implying here is sill a weak linkage beween he Chinese and he US markes and hence low diversificaion benefs for porfolio invesmen. This finding is consisen wh our casual observaion. 4. CONCLUDING REMARKS This sudy inends o examine he presence of heeroskedasicy and he leverage effec in he wo Chinese sock markes, and o capure he dynamics of condional correlaion beween reurns of China s sock markes and hose of he U.S. in a bivariae VC-MGARCH framework. We employed he imevarying-parameer models wh Markov-swching heeroskedasicy proposed by Kim (993) o capure α 0 α β θ θ Panel A: VC-MGARCH(,) model for ε and ε sp sh 0.033* 0.068** 0.96** SH (0.0) (0.09) (0.009) ** 0.04* 0.079** 0.946** (0.0705) (0.0059) (0.07) (0.04) CC-MGARCH(,) model for ε sh and ε sp 0.047** ** 0.97** SH (0.0038) (0.004) (0.0036) N.A. 0.09** 0.080** 0.99** (0.005) (0.009) (0.0098) 0.00 (0.0090) N.A. Figure : Condional correlaion beween Shanghai/Shenzhen and he U.S. markes ρ (0.086) 0.00 (0.03) 343

7 Yin e al., Modeling he Condional Heeroscedasicy and Leverage Effec in he Chinese Sock Markes he dynamic relaionship. The resuls show ha ha he leverage effec is significan in boh markes during he sample period in The ARMA(,)-A-PARCH(,) model fed wh he generalized error disribuion is found o be more suable for capuring condional volaily in mainland China s sock markes. The condional correlaion beween he Chinese and he U.S. sock markes is found o be que low and highly volaile. I is also found ha uncerainy derived from ime-varying relaionship beween Shanghai and he U.S. sock markes is more significan han ha beween Shenzhen and he U.S. sock markes. In addion, he Chinese sock markes are found o be highly regimes persisen, hereby reducing poenial benefs induced by acively rading. These findings have imporan implicaion for invesors seeking opporuny of porfolio diversificaion. REFERENCES Bai, J.S., and Perron, P. (003), Compuaion and Analysis of Muliple Srucural Change Models, Journal of Applied Economerics, 8, -. Bollerslev, T. (986), Generalized Auoregressive Condional Heeroskedasicy, Journal of Economerics, 3, Bollerslev, T. (990), Modeling he Coherence in Shor-Run Nominal Exchange Raes: A Mulivariae Generalized ARCH Model, Review of Economics and Saisics, 7, Copeland, L., and Zhang, B.Q. (003), Volaily and Volume in Chinese Sock Markes, Journal of Chinese Economic and Business Sudies, (3), Engle, R.F. (98), Auoregressive Condional Heeroskedasicy wh Esimaes of he Variance of Uned Kingdom Inflaion, Economerica, 50, Engle, R.F. (00), Dynamic Condional Correlaion-A Simple Class of Mulivariae GARCH Models, Journal of Business and Economic Saisics, 0, Engle, R.F., and Kroner, K.F. (995), Mulivariae Simulaneous Generalized ARCH, Economeric Theory,, -50. Glosen, L.R., Jagannahan, R., and Runkle, D.E. (993), On he Relaion Beween he Expeced Value and he Volaily of he Nominal Excess Reurn on Socks, Journal of Finance, 48(5), Hamilon, J. (989), A New Approach o he Economic Analysis of Nonsaionary Time Series and he Business Cycle, Economerica, 57(), Kim, C.J. (993), Sources of Moneary Growh Uncerainy and Economic Acivy: The Time-Varying- Parameer Model wh Heeroskedasic Disurbances, Review of Economics and Saisics, 75, Lee, C.F., G.M. Chen and O.M. Rui (00). Sock reurns and volaily on China sock markes, Journal of Financial Research, 4, Li, H. (007), Inernaional Linkage of he Chinese Sock Exchanges: A Mulivariae GARCH Analysis, Applied Financial Economics, 7(4), Li, W.K., S. Ling and M. McAleer (00), Recen heoreical resuls for ime series models wh GARCH errors, Journal of Economic Surveys, 6, McAleer, M. (005), Auomaed inference and learning in modeling financial volaily, Economeric Theory,, 3-6. McAleer, M., F. Chan and D. Marinova (007), An economeric analysis of asymmeric volaily: Theory and applicaion o paens, Journal of Economerics, 39, McAleer, M., F. Chan, S. Hoi and O. Lieberman (008), Generalized auoregressive condional correlaion, Economeric Theory, 4, Nelson, D.B. (99), Condional Heeroskedasicy in Asse Reurns: A New Approach, Economerica, 59(), Tse, Y.K., and Tsui, A.K.C. (00), A Mulivariae Generalized Auoregressive Condional Heeroskedasicy Model wh Time-Varying Correlaions, Journal of Business and Economic Saisics, 0, Tsui, A.K.C., and Yu, Q. (999), Consan Condional Correlaion in A Bivariae GARCH Model: Evidence from he Sock Markes of China, Mahemaic and Compuers in Simulaion, 48, Xu, J.G. (999), Modeling Shanghai Sock Marke Volaily, Annals of Operaions Research, 87, 4-5. Yahoo webse, (Available: finance.yahoo.com). 344