Volatility of Shanghai Stock Market: Using ARCH Type Models

Transcription

1 Volailiy of Shanghai Sock Marke: Using ARCH Type Models Zhang Xiaoyong Ma Chaoqun College of Business Adminisraion, Hunan Universiy, Changsha, Hunan , China Absrac: This paper is a research on he volailiies of Chinese sock marke via he ARCH ype models. The resuls show ha he ARCH ype models can capure comparaively beer he volailiy feaures of he Chinese sock marke: heeroscedasiciy, volailiy clusering, leverage effec and impac persisence. Wih a beer characerizaion of hose volailiy feaures, a solid basis for decision making can be provided o invesors and he regulaing deparmen. KeyWords: Index reurn; volailiy clusering; ARCH ype models; Heeroscedasiciy 1 Inroducion The Chinese sock marke has experienced ups and downs for over 20 years, since he esablishmens of he Shanghai Securiies Exchange (SEC) in December, 1990 and he Shenzhen Securiies Exchange in July, Enormous aenion from he regulaors, researchers and invesors has been paid o he behavior of he volailiy of he Chinese sock marke. In he meanime, he asse pricing heory has become a very acive discipline in modern finance as a resul of much research on he volailiy of he financial marke in response o he needs of asse selecion and asse pricing. Over he years, economiss have sough o consruc models o capure he volailiy feaures of he financial marke, among which he family of ARCH models are he mos successful. The ARCH model was firs proposed by Engle in Subsequenly, Bollerslev, Engle, Zakoian, Nelson, Ding and Granger proposed GARCH, ARCH-M, TARCH, EGARCH, Componen GARCH, APARCH models respecively, and hese models ogeher consiue a raher complee sysem of heories of auoregressive condiional heeroscedasiciy. This paper firs inroduces he hisory and he saus quo of he family of ARCH models, discusses he main feaures of various ARCH models and he inheren relaions beween hem, and applies hese models o research on he volailiy of he Chinese sock marke. AS a concluding par, his paper summarizes he feaures of he Chinese sock marke for invesors and he regulaory deparmen s reference in heir decision making. 2 Lieraure Overview Engle(1982) firs formulaed an ARCH model, assuming ha he reurn error follows a condiional normal disribuion wih he condiional expecaion being zero, and he condiional variance being he funcion of he square error of reurn over several periods. The properies of he ARCH model correspond o he feaures of volailiy clusering and heeroscedasiciy of he financial marke. The ARCH model disinguishes he variance from he condiional variance, and defines he condiional variance as he funcion of previous errors, hereby providing a new perspecive in solving he problem of heeroscedasiciy. When ARCH model is applied o describe cerain ime series, he requiremen ha he order q ake a large value leads o he complexiy of he model expression. Bollerslev (1986) made a furher sep, and inroduced an erm of infinie erm error ino he explanaory erm of he variance. He hus derived he GARCH model, making he ARCH model a special case of his model. The GARCH model makes model idenificaion and esimaion easier. In spie of is simpliciy, he GARCH model shed new ligh on he condiional variance, ha is, i is no only relaed o he lagged values of disurbances, bu also possibly relaed o he lagged condiional heeroscedasiciy. Laer, Engle, Lilien, and Bobbins(1987) assumed ha risk premia change wih ime, and he ime varying change of he condiional variance causes he condiional expecaion o change wih ime. The growh in variance and he change in he condiional expecaion are associaed o derive an ARCH-M model. Besides, he ARCH-M model also provides a new procedure o esimae and es he ime 686

2 varying risk compensaion. Nelson(1990),Zakoian(1990),Ding, Granger and Engle (1993),Ding and Granger (1996) proposed differen asymmeric ARCH models and applied differen mahemaical ools o express he condiional heeroscedasiciy. Nelson (1990) proposed an EGARCH model (Exponenial GARCH), which can provide a good characerizaion of he asymmery of he financial marke. In addiion, he EGARCH model has no consrains for he parameers, as he heeroscedasiciy ake he exponenial form. Zakonian (1990) proposed a TARCH model (Threshold ARCH), in which a nominal variable was inroduced ino he condiional variance equaion o capure he effecs of good news and bad news on he sock price. Ding, Granger and Engle (1993) proposed an APARCH model (Asymeric Power ARCH). The APARCH model has he characerisics of ordinary GARCH models, bu here are wo more parameers in he model, one of which is o capure he leverage effec of he sock marke. On he basis of he GARCH (1,1), Ding and Granger (1996) formulaed an asymmeric componen GARCH model, which can capure he shor-run and he long-run volailiy of he reurn series. Engle and Lee (1999) exended he Componen GARCH model o improve is characerizaion of he long-run pah dependence. In he inernaional academic circle, quie a few researchers have conduced a series of researches on volailiies of various sock markes via he family of ARCH models, for example, French, Schwer and Sambaugh (1999), Boudurha and Mark (2001), and Engle and Musafa (2002). In China, Yan Jinan and Zhnag Wei (1999) proposed a geneic algorihm o improve he deficiencies of he radiional esimaing echnique of he ARCH-M model, and heir empirical research showed ha, wih a week as he ime lengh, here exiss a posiive correlaion beween he reurns and risks of invesmens in he Shanghai sock marke. Zhang Siqi (2000) sudied he behavior of he marke reurn rae ime series of he Chinese sock marke and analysed he ime varying feaures of risk premia via he ARCH-M model. Qian Zhengming (2000), using he ARCH ype models, discussed he efficiency of he Chinese financial marke, and measured he sysemaic risk of he Chinese financial marke. There are sill furher researches wih he ARCH ype models in China. Yang Huiyao (2003) applied he APARCH model, and calculaed he VaR values of he Shanghai Composie Index wih he poserior simulaion and he condiional one-sep projecion echniques under hree assumpions of he disribuion. And he es showed ha he esimaion of VaR by he APARCH model is saisically valid, and is apparenly superior o he GARCH model. Liu Jianhua (2004) analysed he effec of informaion asymmery on he persisence of he reurn volailiy and he relaionship beween risk and he daily reurn of he sock marke wih TAR-CARCH model and daa of he daily reurn of he shanghai sock marke. Liu found ha he effec of bad news is more persisen on he reurns, and he sock marke volailiy is closely relaed wih regulaory policies. Zhang Wei, Zhang Xiaoao and Xiong Xiong (2005) ananlysed GJR-GARCH and VS-GARCH, and made correcions o VS-GARCH model. Their empirical research showed ha heir modified VS-GARCH model can beer capure he volailiy asymmery in he Chinese sock marke. Relevan researches also include Wu Shinong, Lin Shaohua and Dinghua (1999), XU Xusong (2002), and Zhang Shiying and Ke Ke (2002), ec. Bu he previous researches all have a common deficiency. They researched on a single feaure of he volailiy or researched on he volailiy wih one ARCH ype model. The resul is ha hey could no reveal he volailiy feaures in he Chinese sock marke in a sysemaic way, and hus i is difficul o provide effecive suggesions o invesors and regulaors in he Chinese sock marke. 3 Volailiy Feaures of he Chinese Sock Marke 3.1 Daa and Their Saisical Feaures The daa used in his paper are he original daa of he Shanghai Composie Index (SHZZ) and he Shenzhen Componen Index (SZCZ) from 1/2/1997 o 9/1/200, alogeher observaions. Considering ha he Chinese sock marke has adoped he mechanism of daily price movemen range beween +10% and -10%, which has a profound influence on he reurns and he volailiy, i is advisable o ake daa afer In his paper, he reurn index is defined as R = 100* log( I / I ), where 1 I is he sock reurn index a ime period, alogeher observaions. 687

3 Table 1 is he descripive saisics of he Chinese sock reurn index series. Form he able we can see ha he skew coefficiens are and for he Shanghai sock marke and he Shenzhen sock marke respecively, which indicae a righ skew, while a similar research once previously conduced on he US sock marke showed ha he skew coefficien is abou Compared wih ha daa, he skew coefficiens for he Chinese sock marke are posiive. The values of kurosis for he Shanghai and Shenzhen sock markes are and respecively, and he reurns series have spikes and fa ails. Researches show ha he kurosis of he US sock marke is abou 3.8. Compared wih he US sock marke, he kurosis of he Chinese sock marke is relaively larger, and he ails are apparenly faer. Table 1 Descripive Saisics of he Sock Reurns Series MEAN Sd SKEW KURTOSIS Jarque-Bera(p-value) SHZZ (0.00) SZCZ (0.00) 3.2 Idenificaion and Specificaion of he ARCH Type Models In previous researches, when he researchers fied he SHZZ and he SZCZ wih he ARCH ype models, mos of hem specified models separaely for he wo indexes, which led o poor resuls in ess for he models. Considering he inerplay effec of he wo indexes, we can inroduce he SZCZ reurns ino he equaion for he SHZZ o represen he dynamic influence beween he variables Saionariy Tes In regression analysis, he ime series process is mus be a saionary one, oherwise spurious regression problem will occur. In his paper, we es for saionariy wih ADF mehod and PP mehod. The es resuls are lised in Table 2. The resuls of he 2 ess show ha he uni es resuls are ADF saisic and PP saisic, boh of which are beyond he criical values. Hence he null hypohyses are rejeced, and he reurns series of he SHZZ and he SZCZ are sable. Table 2 Saionariy Tes Resuls for SHZZ and SZCZ VARIABLE ADF TEST PP TEST SHZZ (0.0001) (0.0001) SZCZ (0.0001) (0.0001) ARCH Model Among he ARCH ype models, ARCH is he mos basic one. For a regression model: ' y = xβ+ ε, if he random disurbance erm saisfies he following condiions, he model hen is an ARCH model: = h (1) ε υ 2 2 q 2 α0 αε 1-1 αqε -q α0 αε i= 1 i i = = + (2) h... Where υ is i.i.d., E( υ )=0,D( υ )=1; α 0 >0, α 0(i=1,2, q), and The ARCH model describes he disribuion of he random error erm i q α i=1 i <1. ε, given he informaion se for he previous -1 periods. The condiional variance of ε is an increasing funcion of he absolue value of he lagged errors. Therefore, a larger error follows a large error, and a smaller error follows a small error. Afer parameer esimaion and resul es, he ARCH(1) model for he Shanghai daily reurns and he volailiy are expressed as he following: SHZZ = SZCZ(-1) SHZZ(-1) 688

4 ( ) ( ) GARCH = RESID(-1)^2 ( ) ( ) Judging from he resul of he parameers esimaion, he coefficien of he residual error erm RESID(-1)^2= <1,saisfying he saionariy condiion GARCH Model In an ARCH model, he regression order q decides he ime periods where he influence of he impac persiss in he subsequen error erms. The order q needs o ake a large value, and hence causing he complexiy of he model. Under he said circumsances, Bllerslev (1986) proposed an exension o he funcional form of he condiional variance, ha is, he GARCH model. The condiional variance of he GARCH model is no only a linear funcion of he square of he lagged residual errors, bu also a linear funcion of he lagged condiional variance. The expression is: h q 2 p = α 0+ αε θ 1 i i j 1 jh i= + = -j (3) where p 0,q>0, α 0 0, θ j 0, j=1,,p, he consrain on he wide saionariy parameer q p α 1 i+ θ j 1 j< 1 i=. = Compared wih ARCH, he meris of he GARCH model are ha, a comparaively simple model can replace a high order ARCH model, and hereby making he model idenificaion and he esimaion easier. In applicaion, he order q for a GARCH model is far smaller han ha for a ARCH model. Generally, a GARCH(1,1) model can be used o model a large number of financial series. The GARCH(1,1) model for he Shanghai Index reurns and he volailiy are expressed as he following: SHZZ = SZCZ(-1) SHZZ(-1) ( ) ( ) GARCH = RESID(-1)^ GARCH(-1) ( ) ( ) ( ) Judging from he esimaion resuls for he parameers, he sum of he coefficien of he residual error erm and he coefficien of he GARCH(-1) is less han 1 ( = ), saisfying he consrains on he parameers. I implies ha he GARCH process of he SHZZ is wide sable, namely, he condiional volailiies of he Chinese sock marke saisfy he requiremen of wide saionariy GARCH-M model In financial markes, he reurn rae of he asse is closely relaed wih he risk. Engle, Lilien and Bobbins (1987) proposed ARCH-M model, and added a funcional erm of he condiional variance h o he equaion of he condiional average. h represens he expeced condiional risk, or he risk compensaion, and an increase in he variance of he reurn rae will lead o an increase in he expeced reurn rae. The equaion of he ARCH-M is: y = x β + γg( h ) + ε (4) where g( h ) is an monoonic funcion of he condiional variance h, generally aking he form of h, h or log( h ). Researches show ha when g ( h ) = log( h ), he model works beer for esimaion. The residual errors for he ARCH-M are equaions (1), and (2). If he residual errors are expressed as equaions (1), and (3), he model is an GARCH-M model. The GARCH(1,1)-M models for he SHZZ daily reurn and is volailiy are expressed as he following: SHZZ = LOG(GARCH) SZCZ(-1) SHZZ(-1) ( ) ( ) ( ) GARCH = RESID(-1)^ GARCH(-1) 689

5 ( ) ( ) ( ) The implicaion of he GARCH-M model is ha he asse reurn rae is closely relaed o he magniude of he risk. LOG(GARCH) represens he magniude of he expeced risk, and is coefficien signifies he invesor s sensiiviy degree of risk. Compared wih resuls from empirical researches in counries oher han China, Chinese invesor s sensiiviy degree is less srong, which implies ha he speculaive psychology among invesors is sronger TARCH model The assumpion for he GARCH model is ha he condiional variance is a funcion of he square of he lagged residual errors. The model implies ha he response of he condiional variance o he posiive and negaive changes in he price is symmeric, and herefore canno explain asymmeric phenomena in acual economy. Moreover, he coefficiens of he condiional variance of he GARCH model are all greaer han zero. This fac implies ha an increase in any of he lagged erms of he residual errors will resul in an increase in he condiional variance, and hus excludes he behavior of random volailiy in he condiional variance, hereby possibly causing flucuaions in esimaing he GARCH model. To remedy he deficiencies of he GARCH model, Zakonian (1990) proposed an TARCH model, expressed as he following: h q 2 2 p = α0+ αε 1 i i ϕε id 1 θ j 1 jh i= + + = -j (5) where 1 ε < 0. As a nominal variable d d is inroduced ino he model, he informaion of = 0 else he sock price rise ( ε >0) and he informaion of he sock price fall will have differen effecs on he condiional variance. When he price rises, coefficien q i=1 α i ϕε d ,. When he price falls, he effecs are q =0, he effecs can be expressed wih he i=1 α i +ϕ. When ϕ 0, i means he effecs of informaion are asymmeric. When ϕ >0, i means here exiss a leverage effec. The TARCH(1,1) models for he SHZZ daily reurn and is volailiy are expressed as he following (The hreshold value=1): SHZZ = SZCZ(-1) SHZZ(-1) ( ) ( ) GARCH = RESID(-1)^2 + ( ) ( ) RESID(-1)^2 (RESID(-1)<0) GARCH(-1) ( ) ( ) In he Equaion for he residual errors, RESID(-1)^2 (RESID(-1)<0) is he esimae of he coefficien of he leverage effec. Judging from he resul of he parameer esimaion ( >0 ), here exiss a leverage effec in he reurn of he SHZZ, which implies ha he effecs of informaion in he Chinese marke is a symmeric, ha is, he volailiies caused by he impacs of negaive reurns are greaer han he volailiies caused by he posiive impacs of equivalen magniudes. This research finding corresponds o he conclusions made in mos exisen researches EGARCH Model EGARCH was proposed by Nelson (1991) o capure he asymmery of he responses of he condiional variance o he posiive and negaive disurbances in he marke. In his case, he condiional variance h is he ani-symmeric funcion of he lagged disurbance ermε, and he expression of he condiional variance of he condiional variance is: p q ε i log( h ) = 0+ log( h ) h i α θ j j αi ϕi (6) j= i= i h i ε i 690

6 In he model, he condiional variance akes he naural logarihm form, which means h is non-negaive and he leverage effec is exponenial. When asymmeric; when -ϕ >0, he leverage effec is significan. i ϕi 0, he effecs of informaion are The EGARCH(1,1) models for he SHZZ daily reurn and is volailiy are expressed as he following: SHZZ = SZCZ(-1) SHZZ(-1) ( ) ( ) LOG(GARCH) = ABS(RESID(-1)/@SQRT(GARCH(-1))) ( ) ( ) RESID(-1)/@SQRT(GARCH(-1)) LOG(GARCH(-1)) ( ) ( ) In he condiional variance equaion, he coefficien of -RESID(-1)/@SQRT(GARCH(-1)) sands for ha of he leverage effec. In his case, i is >0, implying ha here exiss a leverage effec in he reurn of he SHZZ, namely, he effecs of informaion in he Chinese marke is a symmeric. This resul coincides wih he conclusion drawn wih he TRACH model Componen GARCH Model The Componen GARCH model proposed by Ding and Granger (1996) evolves from he GARCH(1,1). If he expression for he condiional variance of he GARCH model (1,1) represens he average deviaion of he condiional variance from a consan, hen he componen GARCH model reflecs he average deviaion rend of he condiional variance from a variable c : Where h c = α ( ε c ) + θ ( h c ) (7) c = ϖ + ρ( c ϖ ) + σ ( ε h ) (8) Expression (7) describes he shor-run componen h - c, approximaing 0 a a momenum α+ θ ; Expression (7) describes he long-run componen c, approximaing ϖ a a momenum ρ. Besides, exogenous variables can be added o boh or eiher of he wo expressions o change he shor-run or long-run volailiy levels of he series. The Componen GARCH(1,1) models for he SHZZ daily reurn and is volailiy are expressed as he following: SHZZ = SZCZ(-1) SHZZ(-1) ( ) ( ) Q = (Q(-1) ) (RESID(-1)^2 -GARCH(-1)) ( ) ( ) ( ) ( ) GARCH= Q (RESID(-1)^2-Q(-1)) (GARCH(-1)-Q(-1)) ( ) ( ) Expression Q describes he long-run componen, approximaing a a decay rae of approximaes 1, which implies ha he convergence rae is sufficien slow. Afer a monh, he residual effec of an impac will be =0.7876; afer a year s ime, he residual effec of an impac will be =0.0570, which means ha he effec of he long-run volailiy on he index reurn caused by an impac will sill remain. The GARCH expression describes he average deviaion rend of he condiional variance wih regard o a variable Q. (GARCH-Q) approximaes 0 a a decay rae ( ) = Afer one monh, he residual effec of an impac will be =0.0254, which implies ha afer one monh he shor-run volailiy caused by an impac nearly disappears The APARCH Model Ding, Granger and Engle (1993) believes ha he GARCH model neiher capures he spike and he fa-ailedness of he high frequency financial series, nor he leverage effec in he sock marke. So hey proposed an APARCH model expressed as follows: 691

7 q p δ δ δ = 0+ i( i i i ) + j j i= 1 j= 1 (9) σ α α ε γ ε β σ Where he parameers saisfy: α 0 >0, δ 0, β j 0(j=1,2,,p), α 0 0, and -1< γ i <1 (i=1,2,,p). The APARCH model enjoys he meris of he general GARCH models, bu wih 2 exra parameers, one of which, γ, is for capuring he leverage effec in he sock marke. i The APARCH(1,1) models for he SHZZ daily reurn and is volailiy are expressed as he following ( he Asymmeric =1): SHZZ = SZCZ(-1) SHZZ(-1) ( ) ( = (ABS(RESID(-1)) ( ) ( ) ( ) ( ) ( ) The esimae of γ i is >0,which shows he exisence of he leverage effec, and corroboraes he conclusion drawn wih he preceding models, TARCH and EGARCH. 3.3 Comparison of he Efficacy of he ARCH Type Models This paper measures he efficacy of he ARCH ype models, using 4 ess, he log likelihood, he AIC Crierion, he SC Crierion, and he D-W Saisic, wih he resuls abulaed in Table 3. The D-W saisic is used o measure he correlaions of he residual error series. All he ARCH ype models can pass his es, and show no significan difference. This fac implies ha all he models are raher successful in eliminaing he auocorrelaions. In erms of he principle of he maximizaion of he log likelihood, EGARCH(1,1) and APARCH work beer han oher models. The AIC Crierion and he SC Crierion are generally used o measure he goodness-of-fi in model selecion. The 2 crieria no only evaluae he goodness-of-fi of a model, bu also punish he behavior of adding parameers wihou a limi o improve he goodness-of-fi. According o he ess, he AIC and he SC values are smaller for EGARCH and APARCH Judging from he resuls in Table 3, EGARCH and APARCH are superior o oher ARCH ype models. LM ess are conduced for he residual errors of he ARCH ype models (Table 4). The LM es resuls show ha excep APARCH, here exis no ARCH phenomena for he squares of he residual series in ARCH ype model fiing. The ARCH ype models work beer o remove he ARCH effec of he original ime series. Table 3 Tes Resuls of he ARCH Type Models ARCH GARCH GARCH-M TARCH EGARCH Componen GARCH APARCH AIC SC D-W Log L

8 Table 4 LM Tes Resuls for ARCH Type Models ARCH GARCH GARCH-M TARCH EGARCH Componen GARCH APARCH F-saisic Probabiliy Obs*R-squared Probabiliy Conclusion This paper reviews he hisorical developmen, discusses he feaures, and uses he ARCH ype models o research on he volailiy of he Chinese sock marke. I will give a beer perspecive in he characerisics of he volailiy of he Chinese sock marke and provide a solid foundaion for invesors and soch marke regulaors decision making. Reference [1] Engle R. F., David M. L., and R. P. Robins. Esimaing ime varying risk premia in he erm srucure: he ARCH-M model [J]. Economerica, 1985: [2] T. Bollerslev (1986), A Generalized Auoregressive Condiional Heeroskedasiciy[J]. Journal of Economerics 31, [3] Nelson D.B(1990), ARCH models as diffusion approximaions[j], Journal of Economerics, 45,7-38. [4] Nelson B. Condiional heeroscedasiciy in asse reurns: a new approach [J]. Economerica, 1991,59: [5] R.F. Engle, D.M. Lilien, and R.P. Robins (1987), Esimaing ime varying risk premia in he erm srucure: he ARCH-M model, Economerica 55, [6] Ding Zhuanxin, Granger C.W.J, Engle R.F(1993), A long memory propery of sock marke reurns and a new model[j], Journa1 of Empirica1 Finance, 1, [7] Bollerslev T., Ghysel E. (1996), Periodic Auoregressive Condiional Heeroscedasiciy[J]. Journal of Business & Economic Saisics, 14(2), [8] Ding, Z. and Granger, C.W.J. (1996) Modeling volailiy persisence of speculaive reurns: A new approach [J]. Economerics 73,