Modelling and forecasing he volailiy of peroleum fuures prices Sang Hoon Kang a, Seong-Min Yoon b, * a Deparmen of Business Adminisraion, Pusan Naional Universiy, Busan 609-735, Korea b Deparmen of Economics, Pusan Naional Universiy, Busan 609-735, Korea Absrac We invesigae volailiy models and heir forecasing abiliies for hree ypes of peroleum fuures conracs raded on he New York Mercanile Exchange (Wes Texas Inermediae crude oil, heaing oil #, and unleaded gasoline) and sugges some sylized facs abou he volailiy of hese fuures markes, paricularly in regard o volailiy persisence (or longmemory properies). In his conex, we examine he persisence of marke reurns and volailiy simulaneously using he following - GARCH-class models: -GARCH, -IGARCH, and -FIGARCH. Alhough he -FIGARCH model beer capures long-memory properies of reurns and volailiy, he ou-of-sample analysis indicaes no unique model for all hree ypes of peroleum fuures conracs, suggesing ha invesors should be careful when measuring and forecasing he volailiy (risk) of peroleum fuures markes. JEL Classificaion: C3; C5; G17; Q40 Keywords: DM es; forecasing abiliy; long memory; persisence; peroleum fuures * Corresponding auhor. Tel.: +8-51-510-557; fax: +8-51-581-3143. E-mail address: smyoon@pusan.ac.kr (S.-M. Yoon). 1
1. Inroducion Modelling and forecasing peroleum fuures prices and heir volailiy are of grea ineres because accuraely measuring he volailiy of peroleum fuures prices is an imporan componen of he price linkage beween spo and fuures markes (Silvapulle and Moosa, 1999; Lin and Tamvakis, 001; Hammoudeh, Li and Jeon, 003; Hammoudeh and Li, 004; Huang, Yang and Hwang, 009), risk managemen, such as value-a-risk (Sadorsky, 006; Aloui and Mabrouk, 010), jump, or regime swiching in energy fuures markes (Fong and Kim, 00; Lee, Hu and Chiou, 010), and opion pricing formulas for fuures conracs (Wang, Wu and Yang, 008). Thus, a beer undersanding of he dynamics of peroleum fuures prices and heir volailiy should be useful o energy researchers, marke paricipans, and policymakers. Previous empirical sudies have examined sochasic properies of peroleum fuures prices by considering various economeric echniques and daa frequencies. In paricular, some have invesigaed wheher imes series of peroleum prices demonsrae long-memory properies of reurns or volailiy (Brunei and Gilber, 000; Serleis and Andreadis, 004; Elder and Serleis, 008; Tabak and Cajueiro, 007; Aloui and Mabrouk, 010). Long memory is a paricularly ineresing feaure in ha is presence direcly conflics wih he validiy of he weak-form efficiency of he peroleum marke. Thus, he presence of long memory provides evidence of nonlinear dependence and of a predicable componen of reurns and volailiy. Some empirical sudies have addressed he modelling and forecasing of longmemory volailiy in crude oil or peroleum markes using GARCH-ype models (Sadorsky, 006; Agnolucci, 009; Kang, Kang and Yoon, 009; Mohammadi and Su, 010), bu hey have considered long memories in reurns and volailiy o be
irrelevanly appearing phenomena. I is well-known ha marke shocks have considerable influence on reurns and volailiy a he same ime, and hus hey have dual long-memory properies. On he basis of his idea, some empirical sudies have considered he relaionship beween reurns and volailiy for various economic and financial ime series using a join -FIGARCH model (Conrad and Karanasos, 005a, 005b; Kang and Yoon, 007; Kasman, Kasman and Torun, 009). The -FIGARCH model can faciliae he analysis of a relaionship beween reurns and volailiy for a process exhibiing dual long-memory properies. The primary objecive of his sudy was o model and forecas price volailiy for hree ypes of peroleum fuures conracs raded on he New York Mercanile Exchange (NYMEX): Wes Texas Inermediae (WTI) crude oil, heaing oil #, and unleaded gasoline. This sudy exends he work of Kang, Kang and Yoon (009) using -FIGARCH models ha can capure long-memory properies of reurns and price volailiy for peroleum fuures simulaneously. Addiionally, his sudy demonsraes he superior predicabiliy of -FIGARCH models using wo forecas error saisics wih muliple forecas horizons (e.g., 1-, 5-, and 0-day-ahead horizons). The res of his paper is organized as follows. Secion presens he saisical characerisics of he daa. Secion 3 discusses he -GARCH-class models and forecas error saisics. Secion 4 presens he volailiy model esimaion and ouof-sample forecasing resuls, and Secion 5 provides conclusions. 3
. Daa We invesigaed he dynamics of fuures prices of WTI crude oil, heaing oil #, and unleaded gasoline. In his repor, fuures conracs refer o hose conracs wih he earlies delivery dae (Conrac 1). Such fuures conracs are raded on NYMEX, and daa regarding hese conracs are available from he U.S. Energy Informaion Adminisraion (EIA). Table 1 Descripive saisics and uni roo es resuls for peroleum fuures reurns Panel A: Descripive saisics WTI crude oil Heaing oil # Unleaded gasoline Mean 0.04 0.038 0.036 Sd. dev..30.415.578 Maximum 14.3 10.40 19.49 Minimum -16.54-0.97-5.45 Skewness -0.85-0.673-0.305 Kurosis 6.335 8.681 9.349 Jarque-Bera 1431*** 460*** 5085*** Q (4) 33.1 44.88*** 6.41 Q s(4) 18.3*** 151.48*** 76.97*** Panel B: Uni roo ess ADF -40.8*** -55.81*** -53.1*** PP -53.35*** -56.6*** -53.18*** KPSS 0.070 0.075 0.039 Noes: The Jarque-Bera es corresponds o he es saisic for he null hypohesis of normaliy in he disribuion of sample reurns. The Ljung-Box saisics, Qn ( ) and Qs ( n ), check for serial correlaion of he reurn series and he squared reurns up o he n h order, respecively. MacKinnon s (1991) 1% criical value is -3.435 for he ADF and PP ess. The criical value for he KPSS es is 0.739 a he 1% significance level. *** indicaes rejecion of he null hypohesis a he 1% significance level. 4
The daa used were of daily frequency for he period 3 January 1995 o 9 December 006; daa for he las one year were used o evaluae he accuracy of ouof-sample volailiy forecass. 1 The price series were convered ino logarihmic percenage reurn series; ha is, y 100 ln( P P 1) for 1,,, T, where y indicaes reurns for each price a ime, P is he curren price, and P 1 is he price on he previous day. Following Sadorsky (006), he acual daily volailiy (variance) is measured by daily squared reurns ( r ). Fig. 1 shows he dynamics of reurns and price volailiy for he hree ypes of peroleum fuures conracs. Table 1 shows he descripive saisics and he resuls of he uni roo es for boh sample reurns. As shown in Panel A of Table 1, he mean of hese reurn series is quie small, whereas he corresponding sandard deviaion of he reurns is subsanially higher. As indicaed by he skewness, kurosis, and Jarque-Bera resuls, he reurns are no normally disribued. We also examined he null hypohesis of a whie-noise process for sample reurns using he Box-Pierce es for reurns Q (4) and squared reurns Q s (4). The reurn series provides suppor for he null hypohesis of no serial correlaion (excep for heaing oil fuures), whereas he squared reurn series provides evidence of serial correlaion a he 1% significance level. Panel B of Table 1 presens he resuls of hree ypes of uni roo ess for each of he sample reurns: augmened Dickey-Fuller (ADF), Phillips-Perron (PP), and Kwiakowski, Phillips, Schmid, and Shin (KPSS). We considered he null hypohesis of a uni roo for he ADF and PP ess and he null hypohesis of saionariy for he KPSS es. The null hypohesis of a uni roo is rejeced (large negaive values), 1 The long-memory propery is ofen confused wih srucural breaks in a ime series (Lamoreaux and Lasrapes, 1990; Diebold and Inoue, 001). Srucural breaks disor he long-memory propery in reurns and volailiy. To avoid possible srucural breaks, his paper excludes recen volaile price daa. 5
whereas he null hypohesis of saionariy is no rejeced a he 1% significance level. Thus, we concluded ha he reurn series is a saionary process. Fig. 1. Dynamics of daily reurns and price volailiy for hree ypes of peroleum fuures conracs. 6
Fig.. Auocorrelaion of daily reurns and price volailiy for hree ypes of peroleum fuures conracs. Fig. displays he auocorrelaion funcion (ACF) of daily reurns and volailiy up o 10 ime inervals wih wo-sided 5% criical values 1.96 1/ T 7. For he reurns, mos auocorrelaions are small, and some significan auocorrelaions die ou
quickly. There seems o be no sysemic paern in he reurn series of hese peroleum fuures conracs. However, he auocorrelaions for he volailiy series are significanly posiive and persisence lass for a subsanial number of lags. This indicaes ha he volailiy of peroleum fuures conracs exhibis a long-memory process. 3. Model framework 3.1. -FIGARCH model The model, a well-known parameric mehod for esing long-memory properies in financial ime series, considers he fracionally inegraed process Id ( ) in he condiional mean. The ( n,, s) model can be expressed as a generalizaion of he ARIMA model, as follows: z, z ~ N(0,1 ), (1) L L ( y L 1 ), () where is independenly disribued wih variance, L denoes he lag operaor, L 1 1L L nl and n and s ( L) 11L L are, sl respecively, he auoregressive (AR) and moving-average (MA) polynomials for which all roos lie ouside he uni circle. The parameers of he model are,, and i. i, 8
According o Hosking (1981), if 0.5 0. 5, hen he y process is saionary and inverible. For such processes, effecs of shocks o on y decay slowly o zero. If 0, hen he process is saionary (or shor memory), and he effecs of shocks o on y decay geomerically. For 1, he process follows a uni roo process. If 0 0.5, hen he process exhibis posiive dependence beween disan observaions, indicaing long memory. If 0.5 0, hen he process exhibis negaive dependence beween disan observaions: ha is, ani-persisence. Similar research on volailiy has exended he represenaion of, leading o he FIGARCH model of Baillie, Bollerslev and Mikkelsen (1996). The FIGARCH p, d, q can be expressed as follows: d L1 L 1 L, (3) q p where L 1 L L, ql L 1 L L pl, and. The } process can be inerpreed as innovaions for he condiional { variance and is serially uncorrelaed wih a mean equal o zero. All he roos of L and L 1 lie ouside he uni roo circle. The FIGARCH model provides greaer flexibiliy for modelling he condiional variance in ha i can accommodae he covariance saionary GARCH model for d 0 and he nonsaionary IGARCH model for d 1. Thus, he araciveness of he FIGARCH model is ha for 0 d 1, i is sufficienly flexible o allow for an inermediae range of persisence. The parameers of he -FIGARCH model can be esimaed using nonlinear opimizaion procedures o maximize he logarihm of he Gaussian 9
likelihood funcion. Under he assumpion ha he random variable z ~ N0,1, he log-likelihood of he Gaussian or normal disribuion ( L Norm ) can be expressed as 1 L Norm z, (4) T ln ln 1 where T is he number of observaions. The esimaion procedure for - FIGARCH-class models requires a minimum number of observaions. This minimum number is relaed o he runcaion order of fracional differencing operaors 1 L and 1 L d. Following he sandard procedure used in previous research, we se he runcaion order of infinie L 1 and d 1 L o 1,000 lags as follows: 1 L 1000 k L k k k0 1. (5) 3.. Evaluaion of forecass To measure forecasing accuracy, we calculaed he mean square error ( MSE ) and mean absolue error ( MAE ) of volailiy forecass as follows: 1 MSE T 1 MAE T T i1 T i1 ( f, a, ), (6), (7) f, a, 10
where T denoes he number of forecas daa poins, is he volailiy forecas for day, and a, is he acual volailiy on day. A smaller forecas error saisic indicaes he superior forecasing abiliy of a given model. Alhough he above forecas error saisics are useful for comparing esimaed models, hey do no allow for saisical analyses of differences in forecas accuracy beween wo forecasing models. Thus, i is imporan o deermine wheher any reducion in forecas errors is saisically significan insead of comparing forecas error saisics beween forecasing models. For his reason, Diebold and Mariano (1995) developed a es of forecas accuracy for wo ses of forecass. Having generaed n, h -sep-ahead forecass from wo differen forecasing models, he f, forecaser has wo ses of forecas errors e, e, 1 and, where 1,,, n. Wih g ) as a funcion of forecas errors, he hypohesis of equal forecas accuracy can ( e 1, be represened as Ed 0, where d g( e1, ) g( e, ) and E is he expecaion operaor. The mean of differences beween forecas errors d _ n 1 d has he approximae asympoic variance of n 1 V d n h 0 k, (8) k 1 1 1 where k is he k h auocovariance of d, which can be esimaed as ^ n 1 n d d d k d. (9) k1 Diebold and Mariano s (1995) es saisic for he null hypohesis of equal forecas accuracy is 11
^ DM Vd 1/ d, (10) where DM has an asympoic sandard normal disribuion under he null hypohesis. In his sudy, he DM es was conduced using he loss differenial based on he MSE and MAE of he differen forecasing models. 4. Empirical resuls 4.1. Tesing long memory in peroleum fuures conracs To examine he long-memory propery, we used several long-memory ess: Lo s (1991) modified R/S analysis and wo semi-parameric esimaors of he long-memory parameer: he log-periodogram regression (GPH) of Geweke and Porer-Hudak (1983) and he Gaussian semi-parameric (GSP) of Robinson and Henry (1999). Panel A of Table provides he resuls of Lo s R/S es for daily reurns and volailiy. For reurns, he value of he modified R/S saisic suppors he null hypohesis of shor memory, while he volailiy displays srong evidence of persisence. However, Panels B and C of Table show ha boh he semi-parameric es (GPH and GSP ess) resuls rejec he null hypohesis of shor memory in reurns and volailiy of sample prices. 3 As a resul, he evidence of long memory in reurns is inconclusive by The choice of hese alernaive ess is jusified by he fac ha several auhors have quesioned he relevance of Lo s (1991) modified R/S. Pracically, Lo s modified R/S analysis has a srong preference for acceping he null hypohesis of no long-range dependence, regardless of wheher long memory is presen in a ime series (Hiemsra and Jones, 1997; Teverovsky, Taqqu and Willinger, 1999). 3 0.5 0.6 0.8 The GPH es was implemened wih differen bandwidhs: m T, m T, m T. The GSP es saisic was also esimaed wih diverse bandwidhs: m T / 4, m T / 16, m T / 64. 1
hese differen long-memory ess, while he volailiies of crude oil, heaing oil, and gasoline seem o be well fied by a fracionally inegraed process. From his poin, our research evolved wih he -FIGARCH model o idenify he longmemory propery in reurns and volailiy in he hree energy markes. Table Resuls of long-memory ess: Lo s R/S es, he GPH es, and he GSP es Panel A: Lo s R/S es WTI crude oil Heaing Oil Gasoline Reurns 0.937 1.039 1.054 Volailiy 3.45***.948***.988*** Panel B: GPH es Reurns 0.5 m T 0.004 0.07-0.046 0.6 m T -0.085-0.059-0.077 0.8 m T -0.045*** -0.053** -0.068*** Volailiy 0.5 m T 0.447*** 0.433*** 0.334*** 0.6 m T 0.316*** 0.64*** 0.64*** 0.8 m T 0.04*** 0.153*** 0.13*** Panel C: GSP es Reurns m T / 4-0.054*** -0.036** -0.045*** m T /16-0.011-0.036-0.004 m T /64-0.071-0.055-0.14* Volailiy m T / 4 0.19*** 0.150*** 0.133*** m T /16 0.89*** 0.301*** 0.45*** m T /64 0.539*** 0.435*** 0.331*** Noes: The criical value of Lo s modified R/S analysis is.098 a he 1% significance level. m denoes he bandwidh for he GHP and he GSP ess. *, **, and *** indicae significance levels of 10%, 5%, and 1%, respecively. 13
4.. Esimaion resuls using he -FIGARCH model We esimaed he -FIGARCH model described by Equaions () and (3) o capure possible long-memory properies in boh he mean and condiional variance. We also evaluaed he performance of he -GARCH, -IGARCH, and -FIGARCH models in erms of heir abiliy o capure long-memory properies of reurns and volailiy simulaneously. We esimaed he - FIGARCH model using he quasi-maximum likelihood esimaion mehod of Bollerslev and Wooldridge (199). Table 3 shows he esimaion resuls obained using hese models. 4 In he mean equaion, an 0,,1 model is he bes represenaion of boh WTI and heaing oil reurns for a long-memory process, whereas an 1,,0 is he bes represenaion of he long-memory process for he gasoline reurns. This suggess ha gasoline prices have more pronounced shor-run dynamics relaive o hose of WTI and heaing oil prices, which are affeced by marke shocks. Generally, he resuls confirm he abiliy of he -FIGARCH model o capure he dynamics of reurns for he hree ypes of peroleum fuures conracs. For example, he esimaed values of he parameer are negaive and saisically differen from zero, providing evidence of negaive dependence (or ani-persisence) among he reurns. 5 This resul is consisen wih he findings of Elder and Serleis (008), who examined prices of crude oil fuures conracs using GPH and wavele OLS esimaors. 4 For he (n, d, s) specificaion in Equaion (), an MA (1) specificaion was reained for WTI and heaing oil fuures reurns, whereas an AR (1) specificaion was chosen for unleaded gasoline fuures reurns. 5 Ani-persisence is a form of long memory characerized by negaive auocorrelaion ha decays very slowly. Peers (1994, p. 61) argued ha ani-persisence ime series reverses iself more ofen han a random one would. An ani-persisence process refers o a mean-revering process. 14
The esimaes of he long-memory parameer d are posiive and significan a he 1% level, indicaing rejecion of d 0 (GARCH model) and d 1 (IGARCH model). This suggess ha he volailiy of peroleum fuures reurns has long-memory properies. Previous sudies have repored similar findings (Brunei and Gilber, 000; Sadorsky, 006; Kang, Kang and Yoon, 009; Aloui and Mabrouk, 010). In Table 4, we presen he accuracy of model specificaions using several diagnosic ess: hree residual ess and hree model selecion crieria. To check he residual es, we applied he Box-Pierce es, Q (4) for up 4 h -order serial correlaion in he residuals, Engle s (198) LM ARCH (10) es for he presence of ARCH effecs in residuals up o lag 10, and he RBD (10) es for condiional heeroscedasiciy in residuals up o lags 10. 6 Addiionally, he Akaike informaion crierion (AIC), he Shibaa crierion (SC), and he Hannan-Quinn crierion (HQ) were used o choose he bes specificaion model among he given models in Table 3. As presened in Table 4, he resuls of Q (4) and he ARCH (10) show no serial correlaion and no remaining ARCH effec. The insignificance of RBD (10) saisics indicaes ha he -FIGARCH model is suiable for depicing heeroscedasiciy exhibied in he peroleum fuures markes, indicaing ha here is no saisically significan evidence of misspecificaion in he -FIGARCH model. Addiionally, he lowes values of hree model selecion crieria (AIC, SB, and HQ) indicae ha he -FIGARCH model bes capures he long-memory dynamics of boh reurns and price volailiy simulaneously for peroleum fuures conracs. 6 Tse (00) developed residual-based diagnosics (RBD) for condiional heeroscedasiciy o es he n ull hypohesis of a correc model specificaion. 15
Table 3 Esimaion resuls of volailiy models Series WTI crude oil Heaing oil # Unleaded gasoline Model -GARCH -IGARCH -FIGARCH -GARCH -IGARCH -FIGARCH -GARCH -IGARCH -FIGARCH Mean equaion 0.05 (0.05)** 0.050 (0.04)** 0.054 (0.06)** 0.045 (0.030) 1 - - - - - 1 Variance equaion 1 1-0.088 (0.06)** 0.103 (0.031)** 0.08 (0.054) 0.047 (0.017)** 0.938 (0.03)** - - d - - -0.091 (0.06)** 0.106 (0.030)** 0.07 (0.017) 0.051 (0.015)** 1-0.051-0.078 (0.06)** 0.093 (0.03)** 0.513 (0.00)** - 0.415 (0.097)** 0.191 (0.090)** 0.85 (0.065)** -0.057 (0.08)** 0.050 (0.034) 0.110 (0.056)* 0.074 (0.01)** 0.910 (0.04)** - - - - 0.047 (0.030) -0.058 (0.07)** 0.051 (0.033) 0.051 (0.05)** 0.079 (0.00)** 1-0.079 0.04 (0.031) -0.055 (0.08)** 0.049 (0.034) 0.75 (0.177) 0.040 (0.031) 0.101 (0.040)** -0.073 (0.09)** 0.08 (0.030) 0.107 (0.040)** -0.077 (0.09)** - - - 0.197 (0.174) 0.059 (0.041) 0.913 (0.060)** 0.047 (0.093) 0.057 (0.074) 0.043 (0.031) 0.097 (0.041)** -0.07 (0.030)** 0.645 (0.80)** Noes: Sandard errors are in parenheses below he corresponding parameer esimaes. ***, **, and * indicae rejecion of he null hypohesis a he 1%, 5%, and 10% significance levels, respecively. - 0.550 (0.185)** 0.5 (0.13)* 0.409 (0.130)** - - - - 1-0.057-0.538 (0.138)** 0.36 (0.169)** 0.60 (0.081)** 16
Table 4 Diagnosic ess of volailiy models Series WTI crude oil Heaing oil # Unleaded gasoline Model Q(4) Q s (4) ARCH (10) RBD (10) -GARCH 17.46 [0.785] 6.43 [0.33] 1.179 [0.99] 9.336 [0.500] -IGARCH 16.84 [0.816] 7.30 [0.199] 1.43 [0.57] 13.38 [0.0] -FIGARCH 17.90 [0.76] 5.71 [0.64] 0.91 [0.50] 9.873 [0.451] -GARCH 3.38 [0.438] 0.17 [0.57] 0.670 [0.753] 6.665 [0.756] -IGARCH.03 [0.518] 3.90 [0.35] 0.84 [0.604] 9.645 [0.47] -FIGARCH.07 [0.515] 17.59 [0.79] 0.599 [0.815] 5.511 [0.854] -GARCH 15.61 [0.871].10 [0.453] 1.45 [0.56] 10.85 [0.369] -IGARCH 16.7 [0.8] 0.84 [0.530] 1.188 [0.93] 11.77 [0.300] -FIGARCH 16.17 [0.847].09 [0.454] 1.186 [0.94] 9.583 [0.477] AIC 4.434757 4.43747 4.433171 4.499449 4.501740 4.499199 4.688999 4.69466 4.680864 SC 4.434749 4.437467 4.433160 4.49944 4.501734 4.499188 4.688991 4.69461 4.680853 HQ 4.439078 4.441073 4.4381 4.503770 4.505340 4.50440 4.69330 4.6987 4.685905 ln(l) -6151.81-6157.01-6148.79-616.46-619.97-615.16-643.54-6440.9-640.67 Noes: Q (4) and Q s(4) are Box-Pierce saisics for reurn series and squared reurn series, respecively, for up o 4 h -order serial correlaion. ARCH (10) is Engle s (198) ARCH LM es o check he presence of ARCH effecs in residuals up o lag 10. RBD (10) is he residual-based diagnosic for condiional heeroscedasiciy, using 10 lags. ln(l) is he maximized Gaussian log-likelihood value. Numbers in brackes are p-values. 17
Table 5 Accuracy of ou-of-sample forecass for peroleum fuures volailiy Mean square error ( MSE) Mean absolue error ( MAE) Series Models 1-day horizon 5-day horizon 0-day horizon 1-day horizon 5-day horizon 0-day horizon WTI crude oil Heaing oil # Unleaded gasoline - FIGARCH - IGARCH - GARCH - FIGARCH - IGARCH - GARCH - FIGARCH - IGARCH - GARCH MSE DM MSE DM MSE DM MAE DM MAE DM MAE DM 19.74-19.80-0.37-3.77-3.77-3.8-33.1-7.19** 34.0-3.93** 35.64-3.13** 5.10-10.77** 5.0-5.31** 5.36 -.77** 1.78-8.40** 1.99 4.6**.75 -.69** 4.04-10.19** 4.07-4.47** 4.15-3.35** 37.04-9.69** 37.0-8.99** 37.4-9.61** 5.8-14.41** 5.7-13.97** 5.7-1.56** 117.0-13.8** 11.8-6.4** 17.5-3.38** 9.83-17.6** 10.08-8.00** 10.37-4.16** 35.3-35.1-35.66-5.09-5.08-5.09-115.7-1.90* 118.7 -.01** 14.6 -.49** 7.6-9.69** 7.8-10.70** 7.47-14.8** 19.9-6.1** 00.4-4.63** 11.5-3.88** 11.89-14.01** 1.17-8.07** 1.66-4.65** 11.6-116.1-1.0-6.6-6.66-6.84 - Noes: Values in bold ype refer o he lowes value for boh MSE and MAE saisics. The DM es saisic was used o evaluae he null hypohesis of no difference in forecas accuracy beween he FIGARCH model and he GARCH or IGARCH model. ** indicaes rejecion of he null hypohesis for he DM es a he 5% significance level. 18
4.3. Ou-of-sample forecas resuls Alhough he -FIGARCH model capures he dynamics of he hree peroleum fuures ime series well, an imporan quesion remains as o which -GARCH-class model bes forecass volailiy. To address his, we evaluaed 49 ou-of-sample volailiy forecass beween 3 January 006 and 9 December 006 and assessed he accuracy of hese forecass. We obained ou-of-sample forecass using parameer esimaes for he volailiy models in Table 5. Addiionally, we esed he null hypohesis of no difference in forecas accuracy beween he models using he DM es saisic in Equaion (10). The ou-of-sample forecas analysis considered 1, 5, and 0 forecas horizons, corresponding o 1-day, 1-week, and 1- monh rading periods, respecively. Table 5 presens he calculaed values of he ou-of-sample volailiy forecas error saisics and he resuls of he DM es. In he case of WTI crude oil fuures, he - FIGARCH model provides he lowes MSE and MAE values and shows a superior abiliy o forecas volailiy for all hree forecas horizons. I is noeworhy ha in he cases of boh heaing oil # and unleaded gasoline fuures conracs, he -GARCH model is more suiable han he oher models (i.e., he -IGARCH and -FIGARCH models). Addiionally, he values of he DM es saisic are negaive and rejec he null hypohesis of no difference a he 5% significance level, indicaing beer performance by he -GARCH model han he oher models in hese cases. Thus, he resuls of he ouof-sample analysis indicae ha none of he models assessed provides he bes fi for all of he hree series considered. In conras o Kang, Kang, and Yoon (009), who suggesed ha he fracionally inegraed model provided he bes fi for he volailiy of crude oil spo prices, he resuls of 19
he presen sudy indicae ha shocks o he volailiy of heaing oil and unleaded gasoline fuures reurns dissipae exponenially, poining o he GARCH model. These findings have imporan implicaions for measuring value-a-risk esimaions, deermining opimal hedging raios, and pricing derivaives in peroleum fuures markes. For example, (1) an appropriae volailiy model provides accuracy for capial reserve requiremens in quanifying value-arisk esimaions (Fan e al., 008; Aloui and Mabrouk, 010), () accurae condiional variance from he volailiy model is used for calculaing hedging raios and enhancing hedging effeciveness in he price change regression (Wilson, Aggarwal and Inclan, 1996; Zanoi, Gabbi and Geranio, 010), and (3) an accurae long-memory volailiy model is an imporan inpu in measuring opion pricing in he Black-Scholes model (Bollerslev and Mikkelsen, 1996; Taylor, 000). 5. Conclusions In his sudy, we sough o idenify a good model for forecasing volailiy and examined some sylized facs abou he volailiy (paricularly in regard o long memory or persisence) of hree ypes of peroleum fuures conracs. For his, we calculaed he ou-of-sample forecass of he volailiy and evaluaed he performance of he -GARCH, -IGARCH, and -FIGARCH models in erms of heir abiliy o capure long-memory properies of reurns and volailiy simulaneously. The esimaion resuls sugges ha he -FIGARCH model can beer capure long-memory feaures han can he -GARCH or -IGARCH models, indicaing ha reurns and volailiy for he hree ypes of peroleum fuures conracs have 0
dual long-memory properies. The presence of long-memory properies cass doub on he weak-form efficiency of peroleum fuures markes. However, he ou-of-sample analyses sugges ha none of he volailiy models is adequae for all hree peroleum fuures series. This suggess ha invesors should be careful when measuring volailiy (risk) in peroleum fuures markes. The findings of his sudy should be useful in faciliaing accurae value-a-risk managemen, developing fuures pricing models, and deermining opimal hedge raios wih respec o peroleum markes. A number of avenues could be followed o exend his research. Firs, i would be ineresing o consider high-frequency daa in measuring he long-memory propery in energy markes. Second, our long-memory resul would be sensiive o he presence of srucural breaks in energy markes. Thus, i would be worhwhile o include Markov swiching-ype volailiy models in capuring regime shifs and comparing he forecasing abiliy wih he long-memory volailiy models. Third, i would be ineresing o check he relevance of differen reurn disribuions o enhance he forecasing abiliy of volailiy models. Acknowledgmens This work was suppored by he Naional Research Foundaion of Korea Gran funded by he Korean Governmen (NRF-010-371-B00008). The auhors hank Chae Woo Nam, Jong Hyun Choi, and seminar paricipans a he 6 h Conference of he Asia-Pacific Associaion of Derivaives (APAD 010) for many helpful commens and simulaing discussions. 1
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