Volatility in Malaysian Stock Market: An Empirical Study Using Fractionally Integrated Approach

Transcription

1 American Journal of Applied Sciences 5 (6): , 8 ISSN Science Publicaions Volailiy in Malaysian Sock Marke: An Empirical Sudy Using Fracionally Inegraed Approach Chin Wen Cheong Faculy of Informaion Technology, Mulimedia Universiy, 631 Cyberjaya, Selangor, Malaysia Absrac: This sudy explores he fracionally inegraed (FI) ime series analysis in Malaysian sock marke. Four proxies of laen volailiy, namely he absolue reurn, squared reurn and range-based (Parkinson and Garman and Klass) volailiies are seleced for he empirical sudies. In addiion, he well-known FI auoregressive condiional variance (ARCH) ype model is also aken ino accoun for comparison purposes. Our empirical resuls evidence he proxy of absolue reurn and ARCH-ype volailiy model provides beer performances in boh he esimaion and forecasing evaluaions. Key words: Range-based volailiy, long-range dependence, economerics, financial ime series INTRODUCTION Recen lieraures of long-range dependence [1,5] indicae ha he fracionally inegraed models are widely used in financial ime series analysis such as risk managemen, porfolio analysis and derivaive pricing. The long-range dependence processes commonly capure by he fracional inegraed operaor (1-B) d in ARFIMA model. Granger [6], Granger and Joyeux [4] and Hosking [7] employ his operaor as a filer o ransform a general ime series ino a consan plus a whie noise process. Laer, Baillie [1] inroduces he fracionally inegraed auoregressive condiional heeroscedasiciy (FIGARCH) model which relaes o financial volailiy dynamics. The phenomenon of long-range dependence asse reurns has furher improved he model specificaion in volailiy modelling which previously concenrae on shor memory basis. Moreover, hese underlying predicabiliy componens have led o new implicaions [5,8,9] in efficiency marke hypohesis (EMH) which saed ha he fuure reurns are unpredicable by using informaion on pas reurns. Anoher imporan applicaion derives from he reurn s volailiy is he measuremen of value-a-risk (VaR) in risk managemen [1,11]. The VaR implemenaion is also recommended by inernaional insiuions such as he Bank For Inernaional Selemens, he American Federal Reserve Bank and he Securiies and Exchange Commission for any derivaives marke paricipans. The Kuala Lumpur sock exchange (KLSE), which is our focus, has received grea aenions from researchers and invesors as he empirical case sudies counry and poenial invesmen alernaives in Souh Eas Asia. Cajueiro and Tabak [1] sudied he long 683 memory volailiy of KLSE from 199 o and found ha he Hurs s parameer wih he value of.68. Cheong e al. [3] invesigaed he asymmery and long-memory volailiy behaviour of he KLSE daily daa over a period of wih four sub-periods. They fied he asymmery long memory GARCH models across he periods and he resuls shown he mixure of symmery and asymmery GARCH modelling. As a whole, he above lieraures are mainly concenraed on he daily closing price indices. For furher analysis, we have seleced four volailiy proxies and direcly implemen hem in he ARFIMA models. On he oher hand, we also consider he fracionally inegraed GARCH by Baillie [1,13] for comparison in model specificaion evaluaions. A baery of saisical ess has been employed o diagnose he model specificaions. As a resul, he model wih absolue reurns shows superior in in-sample esimaion and forecasing evaluaions as compare o oher models. MATERIALS AND METHODS Daa Source: The index ransacion prices begin from 1s January unil 3 November 5 (1445 observaions) in our empirical sudy. During his period, he Malaysia sock marke was speculaed by he RM-USD un-pegged regulaion (implemened a year 6 where he RM was expeced undervalued by approximaely 6.5%), he merged of MESDAQ in KLSE beside he Main board and Second board previously sared in year, he flucuaing of perol prices, ec. We inended o sudy he sock marke volailiy and he reacions of marke paricipans respeced o good and bad evens.

2 Am. J. Applied Sci., 5 (6): , 8 ARFIMA proxies volailiy models: Absolue reurn and squared reurns are he wo volailiy proxies commonly used in empirical financial ime series analysis. Due o is availabiliy and simpliciy, mos empirical finance lieraures using his daily reurn (squared reurns/residual) as he measure of laen volailiy. Ding e al. [] claim ha he absolue reurn exhibis consisenly higher long memory behaviour han squared reurns in S and P 5. Ding and Granger [14] furher examines world wide sock markes and foreign exchange and find similar resuls. Anoher alernaive measuremen of volailiy is using he range, he logarihm difference beween he highes and lowes prices. Mos of he free disribued financial ime series, such as newspaper and websie, are provided wih he informaion of closing, opening, high and low sock prices, currencies, ineres rae, ec. We focus on he range-based volailiy proxies include he earlies work by Parkinson [15] and Garman and Klass [16] wih he assumpion of expeced reurn equal o zero. The mean reurn is no saisically differen from zero a 5% level under he -es (-saisic 1.751). The above sudies have proven ha he Parkinson [15] approach is five imes efficien han he classical average volailiy esimaor. Boh he volailiy esimaors are define as:, park = 1 4ln ( H L ) (1) maximum likelihood esimaion in he following ARFIMAX model: (1-B) d 1- I (B)) r = c o +c 1 1 r 1 + c r 1 +(1+ I (B))a, (3) where indicaes r -1 when r -1 < and is zero oherwise. While coefficien, c 1 indicaes he presence of risk premium. The condiional volailiy componen is applicable only if he esimaed volailiy performs he presence of furher volailiy componens [18]. The shock erm, a, follows a condiional ime-varying variance and he ε ~iid, N (, ) or -dis (). Fracionally Inegraed GARCH: The condiional mean equaions of KLSE sock reurns are an AR (1) model of (r ) as below: r = + 1 r -1 +a. (4) The infrequen rading ha ofen occurs in emerging marke can be adjused [19] by a firs order auoregressive model o avoid he spurious correlaion in he condiional volailiy models. For condiional variance specificaions, he a is serially uncorrelaed, bu dependen o is lagged values or he condiional variance componens as follow: a = ε,gk =.511 ( H L ).19 ( C (H L ) HL ).383C () where ε ~iid, -dis () (5) where he definiions are follow Yang and Zhang [17] wih H, L and C represen he normalized high, low and closing prices respecively. Our volailiy model appears o be fracionally inegraed and correlaed wih he lagged reurn (risk premium), negaive reurns (leverage effec) and heavyailed disribued respecively. For risk premium analysis, if he reurn-volailiy poses a posiive relaionship, we assume ha for a more volailiy (riskier) securiies, he raional marke paricipans require a greaer risk premium. On he oher hand, if he relaionship is negaive, i implies ha he marke paricipans are more favourable in saving. Whereas he asymmeric effec (leverage effec) implies ha impac of bad news increases volailiy is relaively deeper han good news. The model can be esimaed by using he 684 (B) and (B) capure he sho-run of volailiy while he fracional difference parameer d models he long run characerisics of volailiy wih he inraday informaion is as follow: (B)(1 B) d α ϕ 1 a = + 1 β(b) 1 β(b) (6) wih d 1. When <d<.5, he erm (1-B) d has an infinie binomial disribuion for non-ineger powers. Finally, Tse [] includes he new impac componen and inroduces he FIAPARCH (p,d,q): α (B)(1 B) d ϕ δ = + 1 ( a γ a ) δ (7) 1 β(b) 1 β(b)

3 Table 1: Descripive saisics of volailiy proxies Am. J. Applied Sci., 5 (6): , 8 r log r r log r log,park,park,gk log,gk Mean Maximum Minimum Sd. Dev Skewness Kurosis abs-re. sqr-re logp loggk Fig. 1: Sample auocorrelaion funcion 685

4 Am. J. Applied Sci., 5 (6): , esimaions under normaliy assumpion for boh he range-based volailiies. Long-range dependences: In Figure 1, he long-range dependences are observed in all he volailiy proxies where he sample auocorrelaion funcion indicaes significan spikes even afer long lags. All he figures also obviously show ha range-based volailiies are far more long persisence as compare o ransformedreurns volailiy. This propery can be furher examined in he model esimaions Fig. : reurn and squared-reurn where he δ and γ couple he possible varying power of sandard deviaion and leverage effec respecively. The forecasing evaluaions are base on mean squared error (MSE), mean error (ME), mean absolue error (MAE) and Mincer-Zarnowiz Regression. For Mincer-Zarnowiz regression, is he proxy of he (realized volailiy) for ime period and ˆ is he forecased condiional variance for ime. The realized volailiy is measures according o Chin e al. [3] approach. RESULTS AND DISCUSSION Descripive saisics: Table 1 repors he descripive saisics for all he volailiy proxies. For r and r indicae serious excess kurosis wih he values 15.4 and compare o a sandard normal disribuion wih kurosis 3. A parameric -disribuion is in favor o normally disribuion assumpion in he model specificaions. On he oher hand, boh he range-based volailiy esimaors (afer naural logarihm) show similar saisical propery wih normal disribuion wih uniy sandard deviaion and kurosis of 3 respecively. Therefore, his preliminary saisics sugges 686 Esimaion and diagnosic: Table repors he esimaion resuls of ARFIMAX and ARCH-ype wih he assumpion of Gaussian or suden- disribuions. Firsly, we examine he ARFIMAX (,d,1) models and find ha all he coefficien (excep r ) if he reurnvolailiy poses a posiive relaionship, we assume ha for a more volailiy (riskier) securiies, he raional marke paricipans require a greaer risk premium, are significanly differen from zero a 5% level of significan. These findings sugges ha he Malaysian sock asse reurns display a endency o be negaively correlaed wih he reurns volailiy. Or in shor, he leverage effec indicaes ha sock marke volailiy ends o rise in response o bad news and responses reverse o good news. For posiive reurn-volailiy relaionship, in fig. we conclude ha for a more volailiy (riskier) securiies, he raional marke paricipans require a greaer risk premium in heir invesmen for all he proxies excep r. The r and r are significan imply ha boh he proxies are excess kurosis and exhibi heave-ailed compare o normally disribued range-based volailiies. For parameer d ARFIMA, all he values are fall in he range of <d<.5 which indicaes he presence of long persisence volailiy. The srengh of long persisence is sronges in log,gk (.418), follows by log,park (.43), r (.83) and lowes in r (.45) respecively. I is also worh noes ha only r exhibis he ARCH-effec in he sandard residual of ARFIMA. For ARCH-ype models, he AR (1) -disribued asymmery power GARCH esimaions are repored in Table. The δs are close o sugges he condiional variance for he KLCI. On he oher hand, he γs indicae similar resuls in ARFIMA models wih leverage effec. The fracional parameer d GARCH is saisically significan wih he value.388 also indicaes idenical inference of long-range dependence volailiy. In Table, he r ARFIMA (,d,1)-garch (1,1) provides a slighly saisical improvemen over oher models based on Akaike and Schwarz informaion crierion (AIC and BIC). In Table, he Ljung-Box

5 Am. J. Applied Sci., 5 (6): , 8 Table : maximum likelihood esimaion and model diagnosic ARFIMA ARCH Type Esimaion r r log, park log,gk AR(1)-APARCH AR(1)-FIAPARCH c.177(.9).396(.39) -.675(.9) (.) c -.55(.84) -.4(.864) c 1, relag.988(.) -.7(.33).176(.45).16(.) α I.167(.).163(.) c,reneglag (.).17(.44) -.876(.) -.491(.) α I (.).465(.91).1636(.4) (.) α.639(.).1567(.17) d ARFIMA.63(.83).45(.).43(.).418(.) α II.1594(.) -.361(.438) β II.7981(.) -.857(.57) α.69(.1) 5.9(.34).66(.58).333(.58) γ.14(.15).1951(.3) α II.1884(.) (.39).(.48).14(.475) δ (.) (.) β II.6949(.).54(.).5968(.398).447(.647) d GARCH.388(.) υ.981(.).86(.) (.395) υ 5.7(.) 6.147(.) L AIC SIC Diagnosic (1) Q-(1) on a 7.9(.78) 1.758(.999) 1.31(.361) (.115) (.9) (.13) () Q-(1) on a ~ (.597) (.941) (3) ARCH es.6837(.768).473(.961) Forecasing evaluaion MSE ME MAE Theil MZ a -.31(.876).175(.836) -.343(.663) (.617) -.79(.93).65(.948) b 1.111(.44).81(.8).98(.47) 1.614(.479).743(.4).8743(.9) R NOTES: a and a ~ represen he residual and sandardized residual respecively, (1) and (): Ljung Box Serial Correlaion Tes ( Q-saisics) on a ~ and a ~ : Null hypohesis No serial correlaion, (3) LM ARCH es: Null hypohesis - No ARCH effec, (4)Forecasing evaluaion, Mincer- Zarnowiz (MZ) regression, a b ˆ = + + u, where for ime. The values in he parenheses represen he p-value. saisics indicae ha all serial correlaions in he error erms are no significan a 1% significan level. For ARCH-ype models, he asymmery FIAPARCH shows beer diagnosic resuls compare o APARCH model in he Ljung-Box sandardized residual correlaion es. For squared sandardized residuals, boh he ess exhibi no significan serial correlaions and ARCH effec a 1% level of significance. Finally, Table presens 1 one-sep-ahead daily volailiy forecass for he KLSE sock index. The fracionally inegraed ARCH-ype models show smalles MSE and MAE while he r -ARFIMA indicaes relaive lesser in erm of magniude for ME. In he regression analysis, he proxies r, r and FIAPACH show he highes R value (.3 o.63) compare o he range-based volailiy models wih insignifican bs. However, he r exhibi superior in he regressor coefficien, b, (1.111) which are neares o uniy compare o oher models. : realized volailiy wih -minue inerval and 687 ˆ : forecased condiional variance As a resul, in erms of model fiing and specificaions, we conclude ha he r, ARFIMA (,d,1)-garch model is ou-performance compare o all he condiional and range-based volailiy models. CONCLUSION This sudy invesigaes he fracionally inegraed behaviour of KLSE s volailiy over he year o year 5. During his recovery period, he sock marke exhibis he presence of long memory volailiy, presence of risk premium, significan relaion beween news (lagged reurn) and volailiy which implied ha bad news have he higher predicive power for upcoming volailiy and finally he exisence of heavyailed propery in seleced FI volailiy models. As a conclusion, he empirical resuls show ha he inclusion of ime-varying volailiy in he absolue reurn

6 Am. J. Applied Sci., 5 (6): , 8 ARFIMA model conribues a beer in-sample esimaion as well as one-day-ahead forecasing. For furher research, we are ineresed o use he esimaed volailiy in he Value-a-risk applicaion for long and shor rading analyses. ACKNOWLEDGEMENTS The auhor would like o graefully acknowledge he financial suppor from Mulimedia Universiy. REFERENCES 1. Baillie, R.T., T. Bollerslev and H.O. Mikkelsen, Fracionally inegraed auoregressive condiional heeroscedasiciy. J. Economerics, 74: Ding, Z., C.W.J. Granger and R.F. Engle, A Long Memory Propery of Sock Marke Reurns and a New Model. J. Empirical Finance, 1: Cheong C.W., S.M.N. Abu Hassan and I. Zaidi, 7. Asymmery and long memory volailiy: some empirical evidence using GARCH. Physica A: Saisical Mechanics and is Applicaions, 373: Granger, C.W.J. and R. Joyeux, 198. An inroducion o long memory ime series models and fracional differencing. J. Time Series Analysis, 1: Mandelbro, B., Fracal and scaling in finance: Disconinuiy, concenraion, risk. New York, Springer. 6. Granger, C., 198. Long memory relaionships and he aggregaion of dynamic models. J. Economerics, 14: Hosking, J., Fracional differencing. Biomerika, 68: Peers, E.E., Fracal Marke Analysis. A Wiley Finance Ediion, John Wiley and Sons, New York. 9. Mullier, U., M. Dacorogna, R. Dav, O. Pice, R. Olsen and J. Ward, Fracals and inrinsic ime-a challenge o economericians. XXXIXh Inernaional AEA Conference on Real Time Economerics, pp: Jorion, P.,. Value-a-risk: The New Benchmark for Manging Financial Risk. New York: McGraw-Hill. 11. Morgan, J.P., RiskMericsTM-Technical Documen, New York. 1. Cajueiro, D.O. and B.M. Tabak, 4. The Hurs s exponen over ime: esing he asserion ha emerging markes are becoming more efficien. Physica A, 336: Baillie, R.T., Long memory processes and fracional inegraion in economerics. J. Economerics, 66: Ding, Z. and C.W.J. Granger, Modelling volailiy persisence of speculaive reurns: a new approach. J. Economerics, 73: Parkinson, M., 198. The exreme value mehod for esimaing he variance of he rae of reurn. J. Business, 53: Garman, M. and M. Klass, 198. On he esimaion of securiy price volailiies form hisorical daa. J. Business, 53: Yang, D. and Q. Zhang,. Drif independen volailiy esimaion based on high, low, open and close prices. J. Business, 73: Corsi, F., U. Kreschemer, S. Minik and G. Pigorsch, 5. The volailiy of realized volailiy. Working paper Cener for Financial Sudies, 5/ Tse, Y.K., The Condiional Heeroscedasiciy of he Yen-Dollar Exchange Rae. J. Appl. Economerics, 193: Miller, M.H., J. Muhuswamy and R.E. Whaley, Mean reversion of Sandard and Poor 5 index basis changes: Arbirage-induced or saisical illusion. J. Finance, 49: