64 VOLATILITY CLUSTERING, NEW HEAVY-TAILED DISTRIBUTION AND THE STOCK MARKET RETURNS IN SOUTH KOREA Yoon Hong, PhD, Research Fellow Deparmen of Economics Hanyang Universiy, Souh Korea Ji-chul Lee, PhD, Assisan Professor Deparmen of Economics Dongseo Universiy, Souh Korea Guoping Ding, PhD, Assisan Professor School of Business Nanjing Universiy, China Inroducion Since he esablishmen of he Republic of Korea, he economy of Souh Korea has received grea success. The counry emerged from a very poor underdeveloped economy o one of he mos developed economies. Some Souh Korean brands become inernaionally famous wih producs sold across he naions. The sock marke is one of he crucial componens of he naion s economy. The Souh Korea Sock Exchange is he 4h larges exchange over he world wih marke capializaion is $.33 Trillion adjused US Dollars as of March 07. The marke capializaion o GDP raio is 0.64%, which indicaes he imporance of he financial marke. In his paper, we make use of he Korea Composie Sock Price Index (KOSPI). KOSPI is he index of all common socks raded on he Souh Korea Exchange. I is a very popular marke index and he represenaive sock marke index of Souh Korea. During he pas several decades, wo sylized facs for asse reurns have been frequenly presened by researchers: heavy ails and volailiy clusering. The heavy ails indicae ha hisorical disribuion of he asse reurns has heavier ails han he sandard normal disribuion, and he volailiy clusering says ha high-volailiy evens end o cluser in ime. In he following, we re-invesigae he wo sylized facs, heavy ails and volailiy clusering, bu are ineresed in he sock marke reurns on he Souh Korea Exchange. Our framework is similar as in Guo (07a, 07b), and we compare empirical performance of he wo differen ypes of heavy-ailed disribuion, Suden s, and normal reciprocal inverse Gaussian (NRIG), wihin he generalized auoregressive condiional heeroskedasiciy (GARCH) framework for he daily KOSPI reurns. We are paricularly ineresed in he risk managemen aspec. Our resuls indicae wo imporan findings: i) he daily KOSPI reurns exhibi condiional heavy ails even afer volailiy clusering effec has been accouned for; and ii) he NRIG disribuion has a beer in-sample performance han he Suden s disribuion. Lieraure Review Since he economy miracle of Souh Korea, is sock marke has also received considerable aenions from he academia. Many researchers have applied GARCH based models in analyzing he sock marke in Souh Korea. However, mos of he sudies are eiher focusing on he effecs of some special evens or he iner-linkage wih sock markes in some oher counries. Sola e al. (00) proposed a bivariae Marrkov-GARCH model o analyze volailiy links beween hree differen emerging
markes. Choudhry (000) invesigaed he day of he week effec on seven emerging Asian sock markes reurns and condiional variance (volailiy), including Souh Korea, using he GARCH model. Choudhry (000) found ha here was significan presence of he day of he week effec on boh sock reurns and volailiy, hough he resul involving boh he reurn and volailiy are no idenical in all seven cases. Lee (009) examines he volailiy spillover effecs among six Asian counry sock markes, including he sock marke in Souh Korea, using bivariae vecor auo-regression generalized auoregressive condiional heeroskedasiciy (VAR-GARCH) model. Lee (009) found ha here were saisically significan volailiy spillover effecs wihin he sock markes of hese counries. Fang (00) used sock daily daa for Souh Korea and oher four Asian counries, and showed ha currency depreciaion affeced adversely sock reurns and/or increases marke volailiy over he period of he Asian crisis 997 999. In his paper, we re-consider he GARCH framework bu focus on risk managemen of he sock marke reurns in Souh Korea. We focus on wo differen ypes of heavy-ailed disribuions: he Suden s disribuion and he normal reciprocal inverse Gaussian disribuion. In 987, Bollerslev inroduced he Suden s disribuion and he GARCH model wih he Suden s disribuion could capure dynamics of a variey of foreign exchange raes and sock price indices reurns. Poliis (004) inroduced he runcaed sandard normal disribuion ino he ARCH model and showed he empirical performance of he new ype of heavy-ailed disribuion on hree real daases. Tavares e al. (008) model he heavy ails and asymmeric effec on socks reurns volailiy ino he GARCH framework, and showed he Suden s and he sable Pareian wih (α < ) disribuion clearly ouperform he Gaussian disribuion in fiing S&P 500 reurns and FTSE reurns. Su and Hung (0) provides a comprehensive analysis of he possible influences of jump dynamics, heavy-ails, and skewness wih regard o Value a Risk (VaR) esimaes hrough he assessmen of boh accuracy and efficiency. Su and Hung (0) consider a range of sock indices across inernaional sock markes during he period of he U.S. Subprime morgage crisis, and show ha he GARCH model wih normal, generalized error disribuion (GED) and skewed normal disribuions provide accurae VaR esimaes. In his paper, we follow he model framework in Guo (07a, 07b) and are paricularly ineresed in he NRIG disribuion, a newly-developed heavy-ailed disribuion. Our focuses are on heir empirical performance in fiing he sock marke reurns in Souh Korea. The remainder of he paper is organized as follows. In Secion, we discuss GARCH models and he heavy-ailed disribuions. Secion 3 summarizes he daa. The esimaion resuls are in Secion 4. Secion 5 concludes. 65 The Models We consider a simple GARCH(,) process as: e, () 0, () where he hree posiive numbers 0, and are he parameers of he process and. The assumpion of a consan mean reurn is purely for simplificaion and reflecs ha he focus of he paper is on dynamics of reurn volailiy
66 insead of dynamics of reurns. The variable e is idenically and independenly disribued (i.i.d.). Two ypes of heavy-ailed disribuions are considered: he Suden s and he normal reciprocal inverse Gaussian (NRIG) disribuions. The densiy funcion of he sandard Suden s disribuion wih degrees of freedom is given by: ( ) ( e ) / ( ) f e ( )[( ) ], 4. (3) denoes he -field generaed by all he available informaion up where hrough ime. The NRIG is a special class of he widely-used generalized hyperbolic disribuion. The generalized hyperbolic disribuion is specified as in Prause (999): ( / ) K /( ( e ) ) f ( e,,, ) exp( ( e )), (4) / ( ( e ) / ) K ( ) where K () is he modified Bessel funcion of he hird kind and index and: 0, 0. When, we have he normalized NRIG disribuion as: f K0( ( ) ) ( ) exp( ). (5) We also consider he normal disribuion as he benchmark disribuion: e f( e ) exp( ), 4. (6) Daa and Summary Saisics We explore empirical performance of GARCH models wih heavy-ailed disribuion by using he Souh Korea sock marke reurns series. Figure is he daily KOSPI index prices. Figure. Daily KOSPI prices We colleced he sandardized KOSPI daily dividend-adjused close reurns from
Yahoo Finance for he period from July, 997 o July 4, 07, covering all he available daa in Yahoo Finance. There are in oal 4945 observaions. Figure illusraes he dynamics of he KOSPI reurns, and he figure exhibis significan volailiy clusering. 67 Figure. Daily KOSPI reurns Summary saisics of he daa are repored in Table. The daa presen he sandard se of well-known sylized facs of asse prices series: non-normaliy, limied evidence of shor-erm predicabiliy and srong evidence of predicabiliy in volailiy. All series are presened in daily percenage growh raes/reurns. The Bera Jarque es conclusively rejecs normaliy of raw reurns in all series, which confirms our assumpion ha he model seleced should accoun for he heavy-ail phenomenon. The smalles es saisic is much higher han he 5% criical value of 5.99. The marke index is negaively skewed and has fa ails. The asympoic SE of he skewness saisic under he null of normaliy is 6/T, and he SE of he kurosis saisic is 4/T, where T is he number of observaions. Almos all series exhibi saisically significan lepokurosis, suggesing ha accouning for heavy-ailedness is more pressing han skewness in modelling asse prices dynamics. Table. Summary saisics Series Obs. Mean Sd. Skewness Kurosis BJ Q(5) Q ARCH (5) Q (5) KOSPI 4945 0.04%.79% -0.06* 5.33** 76.** 8.5* 3. 3.4** Noe:* and ** denoe a skewness, kurosis, BJ or Q saisically significan a he 5% and % level respecively BJ is he Bera-Jarque saisic and is disribued as chi-squared wih degrees of freedom, Q(5) is he Ljung-Box Pormaneau saisic, Q ARCH (5) is he Ljung-Box Pormaneau saisic adjused for ARCH effecs following Diebold (986) and Q (5) is he Ljung-Box es for serial correlaion in he squared residuals. The hree Q saisics are calculaed wih 5 lags and are disribued as chi-squared wih 5 degrees of freedom. We use he Ljung-Box pormaneau, or Q, saisic wih five lags o es for serial correlaion in he daa, and adjus he Q saisic for ARCH models following Diebold (986). The resuls ha no serial correlaion is found for almos all he series confirm
our assumpion of a consan mean reurn in Equaion (). The evidence of linear dependence in he squared demeaned reurns, which is an indicaion of ARCH effecs, is significan for all he series. 68 Esimaion Resuls We sudy he GARCH(,) model wih he Suden s and he NRIG disribuions is defined by maximizing he following log-likelihood funcion of equaion: T ˆ arg max log( f (,, )). (7) Table repors esimaion resuls of he GARCH(,) model wih he wo ypes of heavy-ailed disribuion for all he KOSPI reurn series. All he parameers are significanly differen from zero. There resuls show he NRIG disribuion has beer in-sample performance. Since he wo disribuions have he same number of parameers, he Akaike informaion crierion (AIC) and he Bayesian informaion crierion (BIC) also indicae he NRIG disribuion has beer empirical performance. Table. Esimaion of he GARCH model wih heavy-ailed innovaions alpha bea /nu (/alpha) log-likelihood AIC BIC Normal 0.034** 0.93** -9694 939 9404 Suden's 0.043** 0.936** 0.7** -940 8809 887 NRIG 0.04** 0.94** 0.68** -9349 8704 87 Noe: * and ** denoe saisical significance a he 5% and % level respecively Conclusion The empirical performances of normal reciprocal inverse Gaussian and he sandard Suden s disribuions under he GARCH framework are compared in fiing he Souh Korea sock marke index reurns. Our resuls illusrae he NRIG has beer performance in capure he KOSPI reurns dynamics. Guo (07a) showed he NRIG disribuion also performs well in risk managemen of he US sock reurn series. We believe he GARCH model wih he NRIG disribuion would also perform well in risk managemen of he Souh Korea sock reurn series. In addiion, i would be ineresing o consider some oher asse classes as in Guo (07c). These are all lef for fuure research. References Bollerslev, T. (987), A condiional heeroskedasic ime series model for securiy prices and raes of reurn daa, Review of Economics and Saisics, vol. 69, pp.54-547. Choudhry, T. (000), Day of he week effec in emerging Asian sock markes: evidence from he GARCH model, Applied Financial Economics, Vol. 0 No. 3, pp. 35-4. Diebold, F. (986), Tesing for serial correlaion in he presence of ARCH, Proceedings of he Business and Economic Saisics Secion of he American Saisical Associaion, Vol. 3, pp. 33-38. Fang, W. (00), The effecs of currency depreciaion on sock reurns: evidence from
five Eas Asian economies, Applied Economics Leers, Vol. 9 No. 3, pp. 95-99. Guo, Z. (07a), Empirical Performance of GARCH Models wih Heavy-ailed Innovaions, Working paper, available a: hps://www.econsor.eu/bisream /049/6766//GUO_NRIGMar07.pdf (accessed Augus 5, 07). Guo, Z. (07b), GARCH models wih he heavy-ailed disribuions and he Hong Kong sock marke reurns, Inernaional Journal of Business and Managemen., forhcoming. Guo, Z. (07c), Models wih Shor-Term Variaions and Long-Term Dynamics in Risk Managemen of Commodiy Derivaives, working paper, available a: hps://www.econsor.eu/bisream/049/6769//.%0guo_energyfeb07.pdf (accessed Augus 5, 07). Lee, S. (009), Volailiy spillover effecs among six Asian counries, Applied Economics Leers, Vol. 6, pp. 50-508. Poliis, D. (004), A heavy-ailed disribuion for ARCH residuals wih applicaion o volailiy predicion, Annals of Economics and Finance, Vol. 3, pp. 34-56. Prause, K. (999), The generalized hyperbolic model: esimaion, financial derivaives, and risk measures, PhD Disseraion, Universiy of Freiburg, Freiburg. Sola, M., Spagnolo, F. and Spagnolo, N. (00), A es for volailiy spillovers, Economics Leers, Vol. 76, pp. 77-84. Su, J. and Hung, J. (0), Empirical analysis of jump dynamics, heavy-ails and skewness on value-a-risk esimaion, Economic Modelling, Vol. 8 No. 3, pp. 7-30. Tavares, A., Curo, J. and Tavares, G. (008), Modelling heavy ails and asymmery using ARCH-ype models wih sable Pareian disribuions, Nonlinear Dynamics, Vol. 5 No., pp. 3-43. 69 VOLATILITY CLUSTERING, NEW HEAVY-TAILED DISTRIBUTION AND THE STOCK MARKET RETURNS IN SOUTH KOREA Yoon Hong Hanyang Universiy, Souh Korea Ji-chul Lee Dongseo Universiy, Souh Korea Guoping Ding Nanjing Universiy, China Absrac As oher developed economies over he world, he sock marke plays a crucial role in faciliaing he economic growh. In his paper, we compare wo differen ypes of heavy-ailed disribuion, he Suden s disribuion and he normal reciprocal inverse Gaussian disribuion, wihin he generalized auoregressive condiional heeroskedasiciy (GARCH) framework for he daily sock marke reurns of Souh Korea (KOSPI). Our resuls show wo imporan findings: i) he daily KOSPI reurns exhibi condiional heavy ails even afer volailiy clusering effec has been accouned for; and ii) he NRIG disribuion has a beer in-sample performance han he Suden s disribuion. Keywords: sock marke, GARCH model, heavy-ailed disribuion, KOSPI