GARCH Model With Fat-Tailed Distributions and Bitcoin Exchange Rate Returns

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1 Journal of Accouning, Business and Finance Research ISSN: Vol., No., pp DOI: / GARCH Model Wih Fa-Tailed Disribuions and Bicoin Exchange Rae Reurns Ruiping Liu Zhichao Shao Guodong Wei 3 Wei Wang 4 School of Economics, Guizhou Universiy, Guiyang, Guizhou, China Business School, Guangxi Universiy, Nanning, Guangxi, China 3 Deparmen of Economics, Universiy of Uah, Sal Lake Ciy, Uah, Unied Saes 4 School of Economics, Guizhou Universiy, Guiyang, Guizhou, China. Absrac In he era of diminishing power from US dollar and increasing compeiion among world currencies, Bicoin, as a compleely new concep as a medium of exchange, has received increasing aenions over he world. Nowadays, Bicoin also becomes an invesmen vehicle, which carries aracive opporuniies bu also significan risks for he invesmen communiy. In his paper, we have compared he empirical performance of a newly-developed heavy-ailed disribuion, he normal reciprocal inverse Gaussian (NRIG), wih he mos popular heavy-ailed disribuion, he Suden s disribuion, under he GARCH framework in fiing he daily Bicoin exchange rae reurns. Our resuls indicae he heavy-ailed disribuion has beer performance in capure he daily Bicoin exchange rae reurns dynamics han he sandard normal disribuion. Our resuls also show he older fashioned Suden s disribuion sill performs beer han he new heavy-ailed disribuion. Keywords: Suden s disribuion GARCH model Bicoin. JEL Classificaions C; C5; G7. Licensed: This work is licensed under a Creaive Commons Aribuion 4.0 License. Publisher: Scienific Publishing Insiue. Inroducion Since he creaion of Bicoin in 009, his crypocurrency has become more and more popular over he world. People from developed counries o developing counries and from big ciies o small village are araced o inves his new invesmen vehicle. Using Bicoin, people have heir ransacions aking place beween each oher direcly, wihou an inermediary. The exchange rae of Bicoin o USD is currenly above $700, saring from almos nohing in 00. The marke capializaion has been over $46 billions and is considered o coninue growing. The number of acive Bicoin users has increased significanly since is inroducion o he marke. There are a lo of economiss who are quie ineresed in he economics of Bicoin. Alhough Bicoin was invened as a medium of exchange, i is now more as an invesmen ool since many speculaive invesors buy Bicoin for is invesmen value raher han as a currency. Because Bicoin lacks of guaraneed value and digial naure, i means ha he purchase and use of Bicoin has significan financial risks. Moreover, he regulaors do no have efficien ools o regulae he marke as a lack of sufficien experience. This means invesmen of Bicoin carries serious regulaory risk. Finally, since he Bicoin o dollar exchange rae has been so volaile, he invesmen also faces severe marke risk. In he las several decades, he generalized auoregressive condiional heeroscedasiciy (GARCH) model has emerged as a sandard ool for quaniaive risk managemen. In his paper, we follow he rend and invesigae he GARCH model in risk managemen of Bicoin exchange rae reurns. Our ineress are on one of he special case of he generalized hyperbolic disribuion, he normal reciprocal inverse Gaussian (NRIG) disribuion, and we compare is empirical performance wih he Suden s disribuion. Thus, our research is essenially similar as he work in Guo (07a), bu we focus on a very new asse class, Bicoin, which has been rarely sudied in he financial economeric lieraure. 7

2 Journal of Accouning, Business and Finance Research, 07, Vol., No., pp Lieraure Review GARCH models wih heavy-ailed disribuion have been very frequenly invesigaed in he lieraure, which covers a variey of asse classes. For insance, Bollerslev (987) firsly developed a GARCH model wih he Suden s disribuion and showed he model could explain sylized facs and dynamics of a variey of foreign exchange raes and sock price indices reurns. Also invesigaing he sock marke bu focusing on individual securiies, Tavares, Curo and Tavare (008) inroduce he heavy ails and asymmeric effec on socks reurns volailiy ino he GARCH framework, and showed ha he heavy-ailed disribuions clearly ouperform he normal disribuion in fiing a series of he US sock reurns. Guo (07a) inroduced several differen ypes of heavy-ailed disribuion ino he GARCH models o compare heir empirical performance in fiing a variey of individual socks and sock indexes reurns. Hung, Lee and Liu (008) and Guo (07b) inroduce he GARCH models wih heavy-ailed disribuion ino he energy commodiy marke. Hung, Lee and Liu showed he Value a Risk esimaes generaed by he GARCH models have good accuracy a boh low and high confidence levels. Guo showed ha he GARCH models wih heavy-ailed disribuions could provide saisfacory margin calculaions. In his paper, we are ineresed in he Bicoin exchange marke and inroduce he GARCH model wih heavy-ailed disribuions ino his new special asse class. Alhough he lieraure on Bicoin is limied overall, here are sill several relaed sudies. Dyhrberg (06a) invesigaed he hedging capabiliies of Bicoin by applying he asymmeric GARCH mehodology and showed ha Bicoin can clearly be used as a hedge agains socks in he Financial Times Sock Exchange Index and agains he American dollar in he shorerm. Dyhrberg (06b) explored he financial asse capabiliies of Bicoin using GARCH models and showed several similariies o gold and he dollar indicaing hedging capabiliies and advanages as a medium of exchange. Kasiampa (07) explored he opimal condiional heeroscedasiciy model wih regards o goodness-of-fi o Bicoin price daa, and found ha he bes model is he AR-CGARCH model, indicaing significance of including boh a shor-run and a long-run componen of he condiional variance. Urquhar (07) examined he volailiy of Bicoin as well as shedding ligh on he forecasing abiliy of GARCH models and heerogeneous auo-regressive (HAR) models in he Bicoin marke. Urquhar found no evidence of he leverage effec in Bicoin and ha he HAR models are superior in modeling Bicoin volailiy o radiional GARCH models. None of he above lieraure has discussed he choice of heavy-ailed disribuions wihin he GARCH framework in fiing he Bicoin exchange rae reurns. In his paper, we uilize he model framework in Guo (07a) and compare he wo ypes of heavy-ailed disribuions, he Suden s disribuion and he NRIG disribuion. 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. Finally, we conclude in Secion The Models A simple GARCH(,) process given by: 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 insead of dynamics of reurns. The variable e is idenically and independenly disribued (i.i.d.). Firsly, we consider a benchmark case ha e follows a normal disribuion: e f( e ) exp( ), 4. (.3) Then, we invesigae wo ypes of heavy-ailed disribuions: 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. (.4) / ( )[( ) ] ( ) denoes he -field generaed by all he available informaion up hrough ime. where The NRIG is a special class of he widely-used generalized hyperbolic disribuion. The generalized hyperbolic disribuion is specified as in Prause (999): (.) (.) 7

3 Journal of Accouning, Business and Finance Research, 07, Vol., No., pp ( / ) K /( ( e ) ) f ( e,,, ) exp( ( e )) where () When f / ( ( e ) / ) K ( ), (.5) K is he modified Bessel funcion of he hird kind and index and: 0, 0., we have he normalized NRIG disribuion as: K0( ( ) ) ( ) exp( ). (.6) 4. Daa and Summary Saisics We invesigae empirical performance of GARCH models wih heavy-ailed disribuion by using he daily Bicoin exchange rae reurns series. The daily Bicoin exchange raes daa are colleced from Coindesk ( for he period from July 9, 00 o July 3, 07, covering all he available daa in Coindesk. There are in oal 56 observaions. Figure illusraes ha he exchange rae of Bicoin o USD is currenly above $700, saring from almos nohing in 00. Figure-. Daily Bicoin exchange rae prices Figure illusraes he dynamics of daily Bicoin exchange rae reurns. The daily exchange rae reurns are very volaile and a lo of spikes are observed. The figure also exhibis significan volailiy clusering. Figure-. Daily Bicoin exchange rae 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 73

4 Journal of Accouning, Business and Finance Research, 07, Vol., No., pp 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 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 4/T, where T is he number of observaions. Almos all series exhibi saisically significan saisic is lepokurosis, suggesing ha accouning for heavy-ailedness is more pressing han skewness in modelling asse prices dynamics. Table-. Summary saisics. 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. Series Obs. Mean Sd. Skewness Kurosis BJ Q(5) Q ARCH (5) Q (5) Daily Bicoin % 5.95% 0.86*.73** 0.** 9.39* ** reurns * and ** denoe a skewness, kurosis, BJ or Q saisically significan a he 5% and % level respecively. 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. The evidence of linear dependence in he squared demeaned reurns, which is an indicaion of volailiy clusering effecs, is significan. 5. Esimaion Resuls To simplify he analysis, only he GARCH (,) model wih he Suden s and he NRIG disribuions is invesigaed. The model is esimaed by maximizing he following log-likelihood funcion of equaion: T ˆ arg max log( f (,, )). (4.) Table repors esimaion resuls of he GARCH(,) model wih he wo ypes of heavy-ailed disribuion for all he daily Bicoin exchange rae reurn series. All he parameers are significanly differen from zero. There resuls show he Suden s disribuion has beer in-sample performance. The Akaike informaion crierion (AIC) and he Bayesian informaion crierion (BIC) also indicae ha he heavy-ailed disribuions have beer performance han he sandard normal disribuion and he Suden s disribuion has he bes in-sample performance. Table-. Esimaion of he GARCH model wih heavy-ailed innovaions alpha bea /nu (/alpha) log-likelihood AIC BIC Normal 0.094** 0.837** Suden's 0.0** 0.80** 0.43** NRIG 0.4** 0.8** 0.87** * and ** denoe saisical significance a he 5% and % level respecively. 6. Conclusion Since is firs appearance, Bicoin has gained more and more aenion from eiher indusry praciioners or academia. People are araced by his new myserious concep. Following he rend, we have compared empirical performance of a newly-developed heavy-ailed disribuion, he normal reciprocal inverse Gaussian, wih he mos popular heavy-ailed disribuion, he Suden s disribuion, under he GARCH framework in fiing he daily Bicoin exchange rae reurns. Our resuls indicae he heavy-ailed disribuion has beer performance in capure he daily Bicoin exchange rae reurns dynamics han he sandard normal disribuion. We furher show he Suden s disribuion has he bes in-sample performance. In his paper, we only consider he single Bicoin marke. I would be ineresing o invesigae he iner-connecions beween he Bicoin marke wih some oher markes using he GARCH framework as in Guo (07c). We hope ha will be anoher conribuion o he lieraure on Bicoin. References Bollerslev, T. (987) A condiional heeroskedasic ime series model for securiy prices and raes of reurn daa. Review of Economics and Saisics, 69: 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, 3: Dyhrberg, A. (06a) Hedging capabiliies of Bicoin. Is i he virual gold? Finance Research Leers, 6: Dyhrberg, A. (06b) Bicoin, gold and he dollar A GARCH volailiy analysis. Finance Research Leers, 6:

5 Journal of Accouning, Business and Finance Research, 07, Vol., No., pp Guo, Z. (07a) Empirical Performance of GARCH Models wih Heavy-ailed Innovaions, Working paper. Guo, Z. (07b) Models wih shor-erm variaions and long-erm dynamics in risk managemen of commodiy derivaives, Working paper. Guo, Z. (07c) How informaion is ransmied across he naions? An empirical invesigaion of he US and Chinese commodiy markes. Global Journal of Managemen and Business Research, 7: -. Huang, J., M. Lee and H. Liu (008) Esimaion of value-a-risk for energy commodiies via fa-ailed GARCH models. Energy Economics, 8: Kasiampa, P. (07) Volailiy esimaion for Bicoin: A comparison of GARCH models. Economics Leers, 58: 3-6. Prause, K. (999) The generalized hyperbolic model: esimaion, financial derivaives, and risk measures. Ph.D. Disseraion. Tavares, A., J. Curo and G. Tavares (008) Modelling heavy ails and asymmery using ARCH-ype models wih sable Pareian disribuions. Nonlinear Dynamics, 5(): Urquhar, A. (07) The volailiy of Bicoin, Working paper. 75