The Macrotheme Review A multidisciplinary journal of global macro trends

Transcription

1 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 The Macroheme Review A mulidisciplinary journal of global macro rends Modelling Long Memory in Volailiy of Oil Fuures Reurns Saada Abba Abdullahi*, Zahid Muhammad** and Reza Kouhy*** *Kano Universiy of Science and Technology, Deparmen of Agriculural Economics, Nigeria. **Universiy of Dammam, Deparmen of Finance and Economics, Saudi Arabia. ***Universiy of Aberay Dundee, Dundee Business School, Scoland, Unied Kingdom. Absrac This paper examines long memory in he Wes Texas Inermediae (WTI and Bren crude oil fuures markes using he GARCH-class models. The resuls provide srong evidence of long erm dependence in reurns for boh markes a differen mauriies. Also, he presence of asymmeric leverage effec was deeced in he oil fuures prices for all markes. The findings sugges ha he wo oil fuures markes have similar paern in heir reurns volailiy a differen mauriies which violaes he marke efficien hypohesis. Keywords: Crude Oil Prices, Fuures Markes, GRACH Models, Long Memory 1. Inroducion One of he imporan non-linear dynamics properies of oil prices is long-erm dependence. Long memory (or long-erm dependence is a special form of non-linear dynamics where a ime series has non-linear dependence in is firs and second momens and beween disan observaions, and a predicable componen ha increases is forecas abiliy (Thupayagale, 010. I also means ha a ime series displays slow decay in is auocorrelaion funcions (Belkhouja and Bouahary, 011. Consequenly, he presence of long memory implies ha energy prices and in paricular oil prices (acually reurns end o be highly volaile, wih price changes ha ofen parially cancel ou, alhough he original shock akes a long ime o work hrough he sysem (Arouri e al, 011. The exisence of long memory also invalidaes he weak-form efficiency of he oil markes because he oil price reurns can be predicable (Elder and Serleis, 008. A large body of lieraure has examined long memory in he crude oil markes. However, majoriy of he previous sudies have focused on he volailiy of oil spo prices (see e.g; Alvarez-Ramirez e al, 008; Ayadi e al, 009; Kang e al, 009; Cheong, 009; Gui e al, 010; Wang e al, 010; Power and Turvey, 010; Fernandez, 010; Wei e al, 010; Mohammadi and Su, 010; Wang e al, 011; Hou and Suardi, 011. To our knowledge, here are only a handful of sudies ha have examined long memory using oil fuures prices (see e.g; Tabak and Cajueiro, 007; Cunado e al, 010; Wang e al, 011; Arouri e al, 01; Ozdemir e al, 013. This paper conribues o he lieraure as follows: Firs, he exising lieraure has been devoed on he Wes Texas Inermediae (WTI crude oil marke. As an exension, his paper examines long memory in he WTI and Bren crude oil fuures markes o find wheher hese prices have 174

2 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 similar paern or can serve as alernaive. Second, i invesigaes long memory in hese markes a differen mauriies because i is well known ha fuures conracs for differen mauriies can exhibi dissimilar paerns since hey are raded for delivery on differen periods. To our knowledge no previous sudy has examined wheher oil fuures markes have similar paern of persisence across mauriies. These resuls can help in making invesmen decisions, porfolio diversificaion and risk managemen. Third, he paper invesigaes wheher he response of oil prices o shocks are similar across he markes and mauriies. The res of he paper is organized as follows. Secion discusses he mehodology employed. Secion 3 describes he daa used and is properies for his paper. Secion 4 presens he empirical resuls and Secion 5 concludes and make some recommendaions.. Mehodology This paper examines long memory in oil fuures markes using he generalized auoregressive condiional heeroskedasiciy (GARCH models. The GARCH model was inroduced by Bollerslev (1986 where he curren condiional variance depends on is own lagged values and he variance includes boh auoregressive and moving average elemens. The model can be described as follows: r where, ~ iid N(0, / 1 1 where r represens he dependen variable which is reurn, (.1 (. is he condiional mean, is he condiional variance, is he uncondiional mean value which is consan, 1 is he GARCH erm which capure informaion on he pas forecas error variance, 1 is he ARCH erm which capure informaion on volailiy from he pas period. The parameers and are expeced o be posiive wih he resricions 0, 0, 0 o ensure posiive condiional variance. The sum of he parameers measures he persisence of shock on volailiy; where 1 implies ha shock o volailiy would be unsable. As an exension, Nelson (1991 proposed he exponenial GARCH (EGARCH model ha allows for asymmeric response of he condiional variance o boh posiive and negaive shocks, and non-negaiviy in he parameers of he condiional variance. The EGARCH (1, 1 model can be wrien as: log z 1 z E z log 1 where, is he parameer ha capures he asymmeric effec of shock o condiional variance. The condiion 0 means ha posiive shock leads o less volailiy han negaive shock while 0 is he reverse condiion of high volailiy han negaive shock (Mohammadi and Su, 011. Ding e al (1993 developed he asymmeric power auoregressive condiional heeroskedasiciy (APARCH model o capure asymmeric effec of shock on condiional variance and assumes ha he effec on residuals follow exponenial rae of decay. The APARCH model can be wrien as: 1 (.3 175

3 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall (.4 where is he coefficien ha capures he asymmeric leverage effec on condiional variance and deermines he bes specificaion of he model. The resricions 0 and 1 1are imposed on he parameers and he model akes he form of GARCH (1, 1 when and 0. The condiion 1means ha he condiional sandard deviaion is he bes for modelling shock o volailiy while sugges condiional variance. The models discussed above deals wih shor memory because hey assume exponenial decay in heir condiional variance. Baillie e al (1996 developed he fracionally inegraed GARCH (FIGARCH model which capures he long memory in condiional variance and allows he auocorrelaion in volailiy o die a slow hyperbolic rae. The FIGARCH ( 1, d,1 can be wrien as: 1 1 ( L 1 1 ( L d 1 ( L(1 L where d is he fracional inegraed parameer ha capures long memory and L is he lag operaor. The parameers mus ake he form 0 d 1 and 0, 1, 1 o ensure posiive condiional variance. The superioriy of he FIGARCH model is ha i permis hree differen condiions: he inermediae range of persisence (long memory when 0 d 1, infinie persisence when d 1 and geomeric decay when d 1. Bollerslev and Mikkelsen (1996 exend he EGARCH model o capure boh asymmeric response and long memory in he condiional variance. However, he fracional inegraed EGARCH (FIEGARCH model assumes non-negaiviy in he parameers of he condiional variance no as in he case of FIGARCH model. The FIEGARCH model can be wrien as: 1 ( L g ( z 1 d ln ( L (1 L 1 where d 1 means ha shock on condiional variance decay a slow hyperbolic rae. Tse (1998 furher proposed he fracional inegraed APARCH (FIAPARCH model o capure long memory and asymmeric effec of shock on condiional variance. The FIAPARCH ( 1, d,1 model can be wrien as: 1 1 d 1 ( L 1 1 ( L ( L(1 L where 0 d 1, 0, 0,, 1 and 1 1. The model reduces o APARCH model when d 0 and FIGARCH when and 0. Davidson (004 developed he hyperbolic GARCH (HYGARCH model which is more powerful han he FIGARCH in accouning for long memory in condiional variance. The HYGARCH ( 1, d,1 model can be wrien as: 1 1 d 1 ( L 1 1 ( L ( L(1 k (1 L 1 where 0 d 1, 0, k 0,, 1. The model reduces o GARCH when d 0, FIGARCH when k 1, IGARCH when d 1and k 1, nonsaionary k 1 and saionary k 1. (.5 (.6 (.7 (.8 176

4 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall Daa The paper used daily closing fuures prices for Wes Texas Inermediae and Bren a one and hree-monh conrac o mauriies from 3, January 000 o 5, Ocober 013. Daa for he sudy were obained from he Daa Sream Inernaional. All he price series were convered ino log reurns series calculaed as r log( / 1 p p 100, where r is he fuures reurn, p is he curren fuures price and p 1is lagged fuures price for one period. The following noaions: WTI-C1, WTI-C3, Bren-C1, and Bren-C3 are used o denoe fuures price reurns a one and hree-monh mauriies. 4. Empirical Resuls Table 1 and Table repors he resuls of he condiional mean and variance equaions esimaed from he GARCH models along wih heir diagnosic ess for he oil markes. Firs, he resuls of he GARCH (1, 1 model show ha he parameer esimaes for he condiional variance equaion and are posiive and significan in all markes a he differen mauriies. The esimaes of he measure of persisence parameer are abou he same and very close o uniy in he wo oil fuures markes wihin he mauriies. The resuls show ha he values are beween and across he markes, suggesing ha he oil fuures markes have high degree of persisence consisen wih hose of Arouri e al (01 and Wang e al (010 who sudied he WTI fuures marke. However, he resuls of he EGARCH model show ha he esimaed value of he measure of persisence is greaer han uniy in all markes excep Bren a one monh mauriy. The values repored are beween 1.1 and across he markes and mauriies which sugges permanen persisence in he reurns series, supporing Wang e al (010. Secondly, he resuls repored for he asymmeric parameer, in he EGARCH, APARCH, FIEGARCH and FIAPARCH models are posiive and significanly differen from zero in each case excep a Bren one-monh mauriy. The resuls indicae ha he esimaes range across he models beween and 0.44 wihin he markes and mauriies. This implies ha here is srong evidence of a leverage effec in he oil markes excep Bren one-monh mauriy. Thirdly, he resuls indicae ha he esimaes of he power parameer which selec he bes specificaion for modelling oil fuures reurns across he APARCH and FIAPARCH models are beween and across he markes. The resuls canno rejec he null hypohesis of 1 a he 5% significance level in all he markes, suggesing ha heir reurns are beer invesigaed wih condiional sandard deviaion which suppors he presence of long memory. Fourhly, he resuls of he esimaed values of he long memory parameers d show ha he fracional inegraed coefficien are all significan and differen from zero in he FIGARCH, FIEGARCH, FIAPARCH and HYGARCH models. For he WTI marke, he values of he esimaes range from 0.39 o across he models wihin he mauriies, resuls consisen wih Arouri e al (01 and Wang e al (010. In he Bren marke, he values repored ranges from o across he models wihin he mauriies. These sugges ha he oil fuures reurns for boh markes have long memory in heir condiional variance. The resuls also fail o suppor he hypohesis of d 0 and d 1, implying ha he FIGARCH model does no reduce o eiher he IGARCH or GARCH models in each marke. Boh oil markes herefore have similar paern of long erm dependence in heir reurns a he differen mauriies. The resuls suppor Tabak and Cajueiro (007 and Cunado e al (010 ha repored long memory in he oil fuures marke using a differen approach. Finally, he Box-piers es and ARCH es for serial 177

5 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 correlaion are conduced o confirm he finess of he models. Boh ess canno rejec heir null hypohesis of no serial correlaion a he 5% level. In sum, he analysis using he differen GARCH models show evidence of long memory in he WTI and Bren crude oil fuures markes a he differen mauriies. Table 1: Esimaed Resuls for WTI Marke Panel A: Resuls for WTI 1-monh fuures conrac GARCH EGARCH APARCH FIGARCH FIEGARCH FIAPARCH HYGARCH 0.079**( ( ( **( ( ( **( ( *( **( ( *( *( ( *( ( *( *( *( *( *( **( *( *( *( *( *( *( *( **(0.395 d 0.410*( *( *( *( *( ( *( *(0.078 Q( [0.717] 16.08[0.71] 14.61[0.798] 16.81[0.665] 16.08[0.71] 16.49[0.686] [0.656] ARCH( [0.850] 0.583[0.89] 1.99[0.5] 0.34[0.993] 0.583[0.89] 0.430[0.933] 0.043[0.999] Log(L Noe: Figures in bracke are he sandard errors in parenhesis beside he parameers. Q (0 is he Box-piers es Q-saisics of order 0 for he sandardised residuals. ARCH (10 is he -saisics of he homoscedasiciy es wih 10 lags. P-values are repored in he square bracke. Significan a 1% and 5% level are represened by * and **, respecively. Log (L represens he logarihm maximum likelihood funcion. 178

6 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 Panel B: Resuls for WTI 3-monh fuures conrac GARCH EGARCH APARCH FIGARCH FIEGARCH FIAPARCH HYGARCH 0.077**( **( *( *( ( *( *( *( *( ( **( *( *( *( *( *( **( *( ( *( ( *( ( *( *( *( **( *( **( *(0.074 d Q(0 ARCH( [0.741] 0.919[0.514] 16.85[0.663 ] 0.855[0.575 ] 0.39*( *( [0.756] 16.13[0.709] 1.419[0.165] 0.464[0.914] 0.670*( *( [0.700 ] 0.833[0.597 ] 0.413*( *( *( *( [0.665] 16.17[0.706] 0.704[0.71] 0.50[0.890] Log(L Noe: Figures in bracke are he sandard errors in parenhesis beside he parameers. Q (0 is he Box-piers es Q-saisics of order 0 for he sandardised residuals. ARCH (10 is he -saisics of he homoscedasiciy es wih 10 lags. P-values are repored in he square bracke. Significan a 1% and 5% level are represened by * and **, respecively. Log (L represens he logarihm maximum likelihood funcion. 179

7 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 Table 3: Esimaed Resuls for Bren Marke Panel A: Resuls for Bren 1-monh fuures conrac GARCH EGARCH APARCH FIGARCH FIEGARCH FIAPARCH HYGARCH 0.066**( **( ( *( *( ( **( ( **( **( ( **( ( ( ( ( *( *( *( *( *( ( *( **( ( ( ( *( *( *(0.00 d 0.683*( *( *( *( ( *( *( *( [0.517] 18.1 [0.573] Q( [0.514] [0.589] [0.443] 18.09[0.58] [0.591] ARCH(10.050[0.05] 1.989[0.031].157[0.018] 1.360[0.193] 1.757[0.063] 1.369[0.188] 1.360[0.193] Log(L Noe: Figures in bracke are he sandard errors in parenhesis beside he parameers. Q (0 is he Box-piers es Q-saisics of order 0 for he sandardised residuals. ARCH (10 is he - saisics of he homoscedasiciy es wih 10 lags. P-values are repored in he square bracke. Significan a 1% and 5% level are represened by * and **, respecively. Log (L represens he logarihm maximum likelihood funcion. 180

8 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 Panel B: Resuls for Bren 3-monhs fuures conrac GARCH EGARCH APARCH FIGARCH FIEGARCH FIAPARCH HYGARCH 0.067**( **( ( *( **( **( *( ( *( *( *( ( *( *( *( *( ( *( *( *( *( *( ( *( *( **( *( ( *( *(0.334 d 0.379*( *( *( *( *( **( *( *( Q(0 1.7[0.889] 1.1[0.908] 13. [0.868] 1.8[0.906 ] 13.69[0.846] 13.7[0.865] 1.31[0.905 ] ARCH( [ [ [0.069] 0.796[0.63] 3.779[0.000] ] 0.7[0.705] 0.781[0.648] ] Log(L Noe: Figures in bracke are he sandard errors in parenhesis beside he parameers. Q (0 is he Box-piers es Q-saisics of order 0 for he sandardised residuals. ARCH (10 is he - saisics of he homoscedasiciy es wih 10 lags. P-values are repored in he square bracke. Significan a 1% and 5% level are represened by * and **, respecively. Log (L represens he logarihm maximum likelihood funcion. 5. Conclusion In his paper, we examine long memory propery of he WTI and Bren oil fuures prices using GARCH-class models. Empirical resuls provide srong evidence of long memory in he oil prices for boh markes a differen mauriies suggesing ha shock on heir condiional volailiy disappear slowly a hyperbolic rae. The presence of asymmeric leverage effec is deeced in he oil fuures reurns series. The implicaion of hese findings is ha he markes rejec he weak form efficien hypohesis because heir reurns are predicable. Second, he presence of long memory suggess ha hese oil markes will have low reurns in he long erm because heir fuure prices are predicable. Lasly, hedging aciviies will no be effecive while speculaive aciviies will be profiable because pas informaion can be used o help exploi arbirage opporuniies in hese markes. 181

9 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 References Alvarez-Ramirez, J., Alvarez, J., Rodriguez, E., 008. Shor-erm predicabiliy of crude oil markes: A derended flucuaion analysis approach. Energy Econ, 30, Arouri, M. E. H., Lahiani, A., Levy, A., Nguyen, D. K., 011. Forecasing he condiional volailiy of oil spo and fuures prices wih srucural breaks and long memory models, Energy Econ, 34, Arouri, M. E. H., Jawadi, F., Nguyen, D. K. 01. Nonlineariies in carbon spo-fuures relaionships during Phase II of he EU ETS, Econ Model, Vol. 1. Ayadi, O. F., Williams, J., Hyman, L. M., 009. Fracional dynamic behaviour in Forcados oil price series: An applicaion of derended flucuaion analysis, Energy Sus Dev 13, Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., Fracionally inegraed generalized auoregressive condiional heeroskedasiciy, J Economerics, 74, Belkhouja, M., Bouahary, M., 011. Modelling volailiy wih ime-varying FIGARCH models, Econ Model. Bollerslev, T., Generalized auoregressive condiional heeroskedasiciy, J Economerics, 3, Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., Fracionally inegraed generalized auoregressive condiional heeroskedasiciy, J Economerics. 74, Cheong, C. W., 009. Modelling and forecasing crude oil markes using ARCH-ype models, Energy Policy, 37, Davidson, J.,004. Momen and memory properies of linear condiional heeroscedasiciy models, and a new model, J Bus Econ Sa,, Dickey, D. A., Fuller, W. A., 1981, Likelihood raio saisics for auoregressive ime series wih a uni roo, Economerica, 49, Elder, J., Jin, H. J., 007, Long memory in commodiy fuures volailiy: A wavele perspecive, J Fuures Markes, 7, Elder, J., Serleis, A., 008. Long memory in energy fuures prices. Rev Financ Econ, 17, Fernandez, V., 010. Commodiy fuures and marke efficiency: A fracional inegraed approach, Resources Policy, 35, Gui, R., Chen, H.,Wang, Y.,010. Mulifracal analysis on inernaional crude oil markes based on he mulifracal derended flucuaion analysis, Physica A: Saisical Mechanics and is Applicaions, 389, Hou, A. and Suardi, S., 01. A nonparameric GARCH model of crude oil price reurn volailiy. Energy Econ, 34, Kang, S. H., Kang, S.-M., Yoon, S.-M., 009. Forecasing volailiy of crude oil markes, Energy Econ, 31,

10 Saada Abba Abdullahi, Zahid Muhammad and Reza Kouhy, The Macroheme Review 3(8, Fall 014 Mohammadi, H., Su, L., 010. Inernaional evidence on crude oil price dynamics: Applicaions of ARIMA-GARCH models, Energy Econ, 3, Nelson, D. B.,1991. Condiional heeroskedasiciy in asse reurns: A new approach. Economerica, 59, Ozdemir. Z. A, Gokmenoglu. K., Ekinci. C., 01, Persisence in crude oil spo and fuures prices. Energy, 59, Power, G. J., Turvey, C. G., 010. Long-range dependence in he volailiy of commodiy fuures prices: Wavele-based evidence, Physica A: Sa Mech Appl, 389, Tabak, B. M., Cajueiro, D. O., 007. Are he crude oil markes becoming weakly efficien over ime? A es for ime-varying long-range dependence in prices and volailiy, Energy Econ, 9, Thupayagale, P., 010. Essays in long memory: evidence from African sock markes. PhD hesis Universiy of S. Andrews. Tse, Y. K., The condiional heeroscedasiciy of he yen dollar exchange rae, J Appl Econom, 13, Wang, T., Yang, J., 010. Nonlineariy and inraday efficiency ess on energy fuures markes, Energy Econ, 3, Wang, Y., Wu, C., Wei, Y., 011. Can GARCH-class models capure long memory in WTI crude oil markes, Econ Model, 8, Wei, Y., Wang, Y.,Huang, D., 010. Forecasing crude oil marke volailiy: Furher evidence using GARCH-class models, Energy Econ, 3,