Heavy-tailed distribution, GARCH models and the silver returns

Transcription

1 In. J Laes Trends Fin. Eco. Sc. Vol-XX No. X Monh, 0 Heavy-ailed disribuion, GARCH models and he silver reurns Andrew Maree #, Peer Card, Paul Kidman #3 # Macro Financial Policy Deparmen, Reserve Bank of New Zealand, Wellingon, New Zealand amaree056@gmail.com Deparmen of Economics, Universiy of Auckland, New Zealand peer.j.card@gmail.com 35 Absrac - Afer serving as a medium of exchange for he human sociey, silver is sill widely used in our daily life. From he jewellery, elecronic and elecrical indusries as well as medicine, opics, he power indusry, auomoive indusry and many oher indusries, silver is sill playing a very acive role. In addiion o he indusrial usage, silver also serves as an invesmen ool for many financial insiuions. Thus, i is crucial o develop effecive quaniaive risk managemen ool for hose financial insiuions. In his paper, we invesigae he condiional heavy ails of daily silver spo reurns under he GARCH framework. Our resuls indicae ha ha i is imporan o inroduce heavy-ailed disribuions o he GARCH framework and he normal reciprocal inverse Gaussian (NRIG) disribuion, a newly-developed disribuion, has he bes empirical performance in capure he daily silver spo reurns dynamics. Keywords - normal reciprocal inverse Gaussian; GARCH model; silver spo reurns; specious meals; daily daa. Inroducion Silver has been regarded as a form of money and sore of value for more han 4000 years. I has been considered as he mos widely-used precious meal. Silver is he whies, mos malleable and mos conducive meal available. I has enjoyed a variey of uses hroughou hisory, mos noably as a form of money and jewelry. According o he sudy by he Silver Insiue, he main demand for silver was for indusrial applicaions (40%), jewelry, bullion coins, and exchange-raded producs in 00. While silver is less rare han gold, i has played a significan role and been used as a currency for more han 700 years. The name of Briish pound derives originaed from he fac ha a Briish pound was once considered o be worh one pound of serling silver. Over 4 languages use synonymous erms for silver and money. In fac, he U.S. dollar prior o he Civil War was also backed by silver insead of gold, which served as a back of dollar unil 970s. Inernaional Journal of Laes Trends in Finance & Economic Sciences IJLTFES, E-ISSN: Copyrigh ExcelingTech, Pub, UK (hp://excelingech.co.uk/) As a class of invesmen asses, he price of silver is driven by speculaion and supply and demand, like mos commodiies. The price of silver is nooriously volaile compared o ha of oher commodiies because of lower marke liquidiy and demand flucuaions beween indusrial and sore of value uses. Thus, risk managemen of silver price flucuaions becomes even more urgen han ha of oher commodiies. As well-known in he finance lieraure, he majoriy of he asse reurns exhibi wo sylized facs: heavy ails and volailiy clusering. In his paper, we reconsider he wo sylized facs, bu focus on he reurns of daily silver spo prices. As in Guo (07a), we inroduce wo ypes of heavy-ailed disribuion ino he generalized auoregressive condiional heeroscedasiciy (GARCH) framework as in Bollerslev (987), including he Suden s disribuion and he normal reciprocal inverse Gaussian (NRIG) disribuion. We are ineresed in he case if he newly-developed NRIG disribuion ouperforms he Suden s disribuion in fiing he reurns of daily silver spo prices.. Lieraure Review There are quie a few sudies on GARCH models and reurns of he silver marke. However, he majoriy of he sudies focused on eiher he fuures marke or iner-linkage beween he silver marke and oher markes. Papadamou and Markopoulos (04) sudied he inerrelaionship beween major exchange rae reurns (EUR/USD, GBP/USD, and JPY/USD) and precious meal reurns (gold and silver) using a vecor auoregressive model in a mulivariae asymmeric GARCH framework on he inraday frequency. The auhors found a unidirecional volailiy ransmission from he majoriy of our currencies (EUR/USD, GBP/USD) o precious meals, and he sluggish response of silver volailiy o currency volailiy shocks permied implemenaion of inraday profiable sraegies. Khalifa, Miao and Ramchander (00) esimaed four measures of inegraed volailiy - daily absolue

2 In. J Laes Trends Fin. Eco. Sc. Vol-XX No. X Monh, 0 35 reurns, realized volailiy, realized bipower volailiy, and inegraed volailiy by Fourier ransformaion (IVFT) - for gold, silver, and copper fuures markes by using high-frequency daa for he period 999 hrough 008. Khalifa, e al. found ha IVFT could help he realized volailiy proxies produce he smalles forecasing errors of fuures prices, and increasing he ime frequency of esimaing inegraed volailiy does no necessarily improve forecas accuracy. Akgiray, e al. (99) invesigaed he imeseries properies of gold and silver spo prices, and found boh precious meal price series exhibied ime dependence and pronounced generalized auoregressive condiional heeroscedasic (GARCH) effecs. Abidin, e al. (03) examined he join relaionship beween he percenage price change and he rading volume of silver and plainum fuures conracs raded on Commodiy Exchange, Inc. (COMEX) using he daily ime series which covering a period of en years, and found ha lagged causaliy in mean running from he price change o rading volume bu no for opposie direcion under he original AR-GARCH model. Some oher sudies on markes inerdependence and silver fuures include Lucey and Tully (006), Auer (05), Benes (06), Kruse, Tischer and Wiig (07), Shen, Meng and Meng (07), and so on. Here, we wan o invesigae he condiional heavy ails under he GARCH framework for he silver reurns. The heavy-ailed disribuions have been quie frequenly invesigaed under ino he GARCH framework o accoun for condiional heavy ails (lepokurosis). For insance, Bollerslev (987) firsly considered he Suden s disribuion and he GARCH model so ha he Suden s disribuion could capure condiional heavy ails of a variey of foreign exchange raes and sock price indices reurns. Tavares, e al. (008) incorporaed he heavy ails and asymmeric effec on socks reurns volailiy ino he GARCH framework, and showed he Suden s and he sable Pareian disribuion clearly ouperform he Gaussian disribuion in fiing S&P 500 reurns and FTSE reurns. Su and Hung (0) considered 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) and are paricularly ineresed in he NRIG disribuion, a newly-developed heavy-ailed 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. Secion 5 concludes. 3. 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 insead of dynamics of reurns. The variable e is idenically and independenly disribued (i.i.d.). Two ypes of fa-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) where denoes he -field generaed by all he available informaion up 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 )) / ( ( e ) / ) K ( ) (.4) where K () is he modified Bessel funcion of he hird kind and index and: 0, 0. When, we have he normalized NRIG disribuion as:

3 In. J Laes Trends Fin. Eco. Sc. Vol-XX No. X Monh, f K0( ( ) ) ( ) exp( ). (.5) 4. Daa and Summary Saisics Figure illusraes he daily silver spo prices in he Chicago Mercanile Exchange. Currenly, CME is he larges commodiy exchange in he world. I merged wih he Chicago Board of Trade in July 007 o become he larges commodiy derivaive exchange. The daa covers he period from June, 99 o June 30, 07. There are in oal 7508 observaions. Figure illusraes he dynamics of he silver spo reurns. There are significan volailiy clusering phenomenon and high volailiies are observed in he Grea Recession period. 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 saisic is 4/T, where T is he number of observaions. The daa exhibis saisically significan heavy ails. Series Obs Mea n Silver spo reurn s % Sd..66 % Skew ness Kuros is BJ Q(5) Q ARC 0. 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, QARCH(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. and denoe a skewness, kurosis, BJ or Q saisically significan a he 5% and % level respecively H (5) Q (5) Figure : Daily silver spo prices Figure : Daily silver spo reurns Table presens summary saisics of he daa. 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 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 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. 5. Esimaion Resuls The GARCH(,) model wih he Suden s and he NRIG disribuions 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 silver spo reurn series. All he parameers are significanly differen from zero. There resuls show ha i is crucial o inroduce heavy-ailed disribuions ino he GARCH framework and he NRIG disribuion has he bes insample performance. Since he wo disribuions has he same number of parameers, he Akaike

4 In. J Laes Trends Fin. Eco. Sc. Vol-XX No. X Monh, informaion crierion (AIC) and he Bayesian informaion crierion (BIC) also indicae he NRIG disribuion has bes empirical performance. alpha bea /nu (/alpha) loglikelihood Normal Suden's NRIG Table : Esimaion of he GARCH model wih heavy-ailed innovaions and denoe saisical significance a he 5% and % level respecively 6. Conclusion To explain condiional heavy ails of daily silver spo reurns, we inroduce a newly-developed heavyailed disribuion, he normal reciprocal inverse Gaussian, under he GARCH framework in fiing he reurns series. We compare is empirical performance wih ha of he mos widely-used heavy-ailed disribuion, he Suden s disribuion, and he benchmark Gaussian disribuion. Our resuls indicae ha i is crucial o inroduce heavy-ailed disribuions o he GARCH framework and moreover he NRIG disribuion has he bes performance in capure he daily silver spo reurns dynamics. In his paper, we do no decompose he silver spo reurns ino several componens. Guo (07b, 07c) showed he spo reurns could be decomposed ino a long-run facor and a shor-run facor for a variey of energy commodiies. I would be ineresing o invesigae if he silver spo reurns could be decomposed ino several sochasic facors and if he sochasic facors model could be useful for risk managemen of he silver commodiy. In addiion, Glosen, Jagannahan and Runkle (993) inroduce asymmeric responses of condiional volailiy o negaive and posiive shocks under he GARCH framework, and i would be ineresing o consider such a leverage effec exiss in he daily silver spo reurns process References AIC [] Abidin, S., A. Banchi, R. Lou and Q. Niu (03). Informaion flow and causaliy beween price change and rading volume in silver and plainum fuures conracs. BIC Inernaional Journal of Economics, Finance and Managemen, vol., pp [] Akgiray, V., G. Booh, J. Haem and C. Musafa (99). Condiional dependence in precious meal prices. The Financial Review, vol. 6, pp [3] Auer, B. (05), Supersiious seasonaliy in precious meals markes? Evidence from GARCH models wih ime-varying skewness and kurosis. Applied Economics, vol. 47, pp [4] Benes, S. (06), Long memory volailiy of gold price reurns: How srong is he evidence from disinc economic cycles? Physica A: Saisical Mechanics and is Applicaions, vol. 443, pp [5] 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 [6] 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 [7] Glosen, L., R. Jagannahan and D. Runkle (993), "On he relaion beween he expeced value and he volailiy of nominal excess reurn on socks." Journal of Finance, vol. 5, pp [8] Guo, Z. (07a), Empirical Performance of GARCH Models wih Heavy-ailed Innovaions. Working paper. [9] Guo, Z. (07b), A Sochasic Facor Model for Risk Managemen of Commodiy Derivaives, Proceedings of he 7h Economic and Finance Conference, pp. 6-4; [0] Guo, Z. (07c), Models wih Shor-Term Variaions and Long-Term Dynamics in Risk Managemen of Commodiy Derivaives, Working paper. [] Khalifa, A., H. Miao and S. Ramchander (00). Reurn disribuions and volailiy forecasing in meal fuures markes: Evidence from gold, silver, and copper. Journal of Fuures Markes, vol. 3, pp

5 In. J Laes Trends Fin. Eco. Sc. Vol-XX No. X Monh, [] Kruse, S., T. Tischer and T. Wiig (07), A new empirical invesigaion of he plainum spo reurns. Journal of Smar Economic Growh, vol., no., pp [3] Lucey, B. and E. Tully (006). Seasonaliy, risk and reurn in daily COMEX gold and silver daa Applied Financial Economics, vol. 6, pp [4] Papadamou, S. and T. Markopoulos (04), Invesigaing inraday inerdependence beween gold, silver and hree major currencies: he Euro, Briish Pound and Japanese Yen. Inernaional Advances in Economic Research, vol. 0, pp [5] Prause, K. (999), "The generalized hyperbolic model: esimaion, financial derivaives, and risk measures." Ph.D. Disseraion. [6] Shen, H., X. Meng and X. Meng (07), Heavy-ailed disribuion and risk managemen of gold reurns. Inernaional Journal of Academic Research in Economics and Managemen Sciences, vol. 6, no. 3, pp. 5-4; [7] Su, J. and J. Hung (0), Empirical analysis of jump dynamics, heavy-ails and skewness on value-a-risk esimaion. Economic Modelling, vol. 8, no. 3, pp [8] Tavares, A., J. Curo and G. Tavares (008), Modelling heavy ails and asymmery using ARCH-ype models wih sable Pareian disribuions. Nonlinear Dynamics, vol. 5, no., pp [9] The Silver Insiue (00), Demand and supply in 00.