Forecasting Volatility and Spillovers in Crude Oil Spot, Forward and Futures Markets

Transcription

1 CIRJE-F-641 Forecasing Volailiy and Spillovers in Crude Oil Spo, Forward and Fuures Markes Chia-Lin Chang Naional Chung Hsing Universiy Michael McAleer Erasmus Universiy Roerdam and Tinbergen Insiue and CIRJE, Faculy of Economics, Universiy of Tokyo Roengchai Tansucha Maejo Universiy andchiangmaiuniversiy Augus 2009 CIRJE Discussion Papers can be downloaded wihou charge from: hp:// Discussion Papers are a series of manuscrips in heir draf form. They are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Discussion Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 Forecasing Volailiy and Spillovers in Crude Oil Spo, Forward and Fuures Markes Chia-Lin Chang Deparmen of Applied Economics Naional Chung Hsing Universiy Taichung, Taiwan Michael McAleer Economeric Insiue Erasmus School of Economics Erasmus Universiy Roerdam and Tinbergen Insiue The Neherlands and Cener for Inernaional Research on he Japanese Economy (CIRJE) Faculy of Economics Universiy of Tokyo Roengchai Tansucha Faculy of Economics Maejo Universiy Thailand and Faculy of Economics Chiang Mai Universiy Thailand Augus 2009

3 Absrac Crude oil price volailiy has been analyzed exensively for organized spo, forward and fuures markes for well over a decade, and is crucial for forecasing volailiy and Value-a- Risk (VaR). There are four major benchmarks in he inernaional oil marke, namely Wes Texas Inermediae (USA), Bren (Norh Sea), Dubai/Oman (Middle Eas), and Tapis (Asia- Pacific), which are likely o be highly correlaed. This paper analyses he volailiy spillover effecs across and wihin he four markes, using hree mulivariae GARCH models, namely he CCC, and models. A rolling window approach is used o forecas he 1-day ahead condiional correlaions. The paper presens evidence of volailiy spillovers and asymmeric effecs on he condiional variances for mos pairs of series. In addiion, he forecased condiional correlaions beween pairs of crude oil reurns have boh posiive and negaive rends. Keywords: Volailiy spillovers, mulivariae GARCH, condiional correlaions, crude oil spo prices, spo reurns, forward reurns, fuures reurns. JEL Classificaions: C22, C32, G17, G32 2

4 1. Inroducion Over he pas years, oil has become he bigges raded commodiy in he world. In he crude oil marke, oil is sold under a variey of conrac arrangemens and in spo ransacions, and is also raded in fuures markes which se he spo, forward and fuures prices. Crude oil is usually sold close o he poin of producion, and is ransferred as he oil flows from he loading erminal o he ship FOB (free on board). Thus, spo prices are quoed for immediae delivery of crude oil as FOB prices. Forward prices are he agreed upon price of crude oil in forward conracs. Fuures prices are prices quoed for delivering a specified quaniy of crude oil a a specific ime and place in he fuure in a paricular rading cener. The four major benchmarks in he world of inernaional rading oday are: (1) Wes Texas Inermediae (WTI), he reference crude for USA, (2) Bren, he reference crude oil for he Norh Sea, (3) Dubai, he benchmark crude oil for he Middle Eas and Far Eas, and (4) Tapis, he benchmark crude oil for he Asia-Pacific region. Volailiy (or risk) is imporan in finance and is ypically unobservable, and volailiy spillovers appear o be widespread in financial markes (Milunovich and Thorp, 2006), including energy fuures markes (Lin and Tamvakis, 2001). Consequenly, a volailiy spillover occurs when changes in volailiy in one marke produce a lagged impac on volailiy in oher markes, over and above local effecs. Accurae modelling of volailiy is crucial in finance and for commodiies. Shocks o reurns can be divided ino predicable and unpredicable componens. The mos frequenly analyzed predicable componen in shocks o reurns is he volailiy in he ime-varying condiional variance. The success of he Generalized Auoregressive Condiional Heeroskecasiciy (GARCH) model of Engle (1982) and Bollerslev (1986) has subsequenly led o a family of univariae and mulivariae GARCH models which can capure differen behaviour in financial reurns, including ime-varying volailiy, persisence and clusering of volailiy, and asymmeric effecs of posiive and negaive shocks of equal magniude. In modelling mulivariae reurns, such as spo, forward and fuures reurns, shocks o reurns no only have dynamic inerdependence in risks, bu also in he condiional correlaions which are key elemens in porfolio consrucion and he esing of unbiasedness and he efficien marke hypohesis. The hypohesis of efficien markes is essenial for undersanding opimal decision making, especially for hedging and speculaion. Subsanial research has been conduced on spillover effecs in energy fuures markes. Lin and Tamvakis (2001) invesigaed volailiy spillover effecs beween NYMEX 3

5 and IPE crude oil conracs in boh non-overlapping and simulaneous rading hours. They found ha subsanial spillover effecs exis when boh markes are rading simulaneously, alhough IPE morning prices seem o be affeced considerably by he close of he previous day on NYMEX. Ewing e al. (2002) examined he ransmission of volailiy beween he oil and naural gas markes using daily reurns daa, and found ha changes in volailiy in one marke have spillovers o he oher marke. Sola e al. (2002) analyzed volailiy links beween differen markes based on a bivariae Markov swiching model, and discovered ha i enables idenificaion of he probabilisic srucure, iming and he duraion of he volailiy ransmission mechanism from one counry o anoher. Hammoudeh e al. (2003) examined he ime series properies of daily spo and fuures prices for hree peroleum ypes raded a five commodiy cenres wihin and ouside he USA by using mulivariae vecor error-correcion models, causaliy models and he GARCH models. They found ha WTI crude oil NYMEX 1-monh fuures prices involves causaliy and volailiy spillovers, NYMEX gasoline has bi-direcional causaliy relaionships among all he gasoline spo and fuures prices, spo prices produce he greaes spillovers, and NYMEX heaing oil for 1- and 3-monh fuures are paricularly srong and significan. Hammoudeh e al. (2009) examined he dynamic volailiy and volailiy ransmission in a mulivariae seing for four Gulf Cooperaion Council economies, and analysed he opimal weighs and hedge raios for secoral porfolio holdings. Of four major crude oil markes, only he mos well known oil markes, namely WTI and Bren, have spo, forward and fuures prices, while he Dubai and Tapis markes have only spo and forward prices. I would seem ha no research has ye esed he spillover effecs for each of he spo, forward and fuures crude oil prices in and across all markes. Several mulivariae GARCH models specify risk of one asse as depending dynamically on is own pas and on he pas of oher asses (see McAleer, 2005). da Veiga, Chan and McAleer (2008) analysed he mulivariae model of Ling and McAleer (2003) and model of McAleer, Hoi and Chan (2009), and found ha hey were o superior o he GARCH model of Bollerslev (1986) and GJR model of Glosen, Jagannahan and Runkle (1992). In his paper we invesigae he imporance of volailiy spillover effecs and asymmeric effecs of negaive and posiive shocks on he condiional variance when modelling crude oil volailiy in he reurns of spo, forward and fuures prices in he Bren, WTI, Dubai and Tapis markes, and across hese markes, using mulivariae condiional volailiy models. The spillover effecs beween reurns in he markes and across markes are 4

6 also esimaed. A rolling window is used o forecas 1-day ahead condiional correlaions and o explain he condiional correlaions movemens, which are imporan for porfolio consrucion and hedging. The plan of he paper is as follows. Secion 2 discusses he univariae and mulivariae GARCH models o be esimaed. Secion 3 explains he daa, descripive saisics and uni roo ess. Secion 4 describes he empirical esimaes and some diagnosic ess of he univariae and mulivariae models, and forecass 1-day ahead condiional correlaions. Secion 5 provides some concluding remarks. 2. Economeric models This secion presens he CCC model of Bollerslev (1990), model of Ling and McAleer (2003), and model of McAleer, Hoi and Chan (2009). These models assume consan condiional correlaions, and do no suffer from he curse of dimensionaliy, as compared wih he VECH and BEKK models (see McAleer e al. (2008) and Caporin and McAleer (2009)). The model of Ling and McAleer (2003) assumes symmery in he effecs of posiive and negaive shocks of equal magniude on he condiional volailiy, and is given by 12 where D diag hi, 1 Y E Y F (1) LY L (2) D (3) H W A B H r s l l l i, j l1 l1,,, H,..., h1 h m W,..., 1 m 2 2 of independenly and idenically (iid) random vecors, i,..., m (4),..., 1 m is a sequence, A and B l are m m marices wih ypical elemens ij and ij, respecively, for i, j 1,..., m, I diag I i is an m m 1... marix, L I 1 L... q L Im L ql are polynomials in L, he lag operaor, m and L p p F is he pas informaion available o ime, GARCH effec. l represens he ARCH effec, and l represens he 5

7 Spillover effecs, or he dependence of condiional variances across crude oil reurns are given in he condiional volailiy for each asse in he porfolio. Based on equaion (3), he model also assumes ha he marix of condiional correlaions is given by E( ). If m 1, equaion (4) reduces o he univariae GARCH model of Bollerslev (1986): h p q 2 2 ii ih i i1 i1 (5) The model assumes ha negaive and posiive shocks of equal magniude have idenical impacs on he condiional variance. An exension of he VARMA- GARCH model o accommodae asymmeric impacs of posiive and negaive shocks is he model of McAleer, Hoi and Chan (2009), which capures asymmeric spillover effecs from oher crude oil reurns. asymmeries wih respec o i is given by H W r A l l l1 l1 in which i hi for all i and, Cl are m m r C I l An exension of (4) o accommodae l l s l1 B H l l marices, and i (6) I is an indicaor variable disinguishing beween he effecs of posiive and negaive shocks of equal magniude on condiional volailiy, such ha I i 0, i 0 1, i 0 When m 1, equaion (4) reduces o he asymmeric univariae GARCH, or GJR model of Glosen e al. (1992): r s 2 (8) h I h j j j j j j j1 j1 For he underlying asympoic heory, see McAleer e al. (2007) and, for an alernaive asymmeric GARCH model, namely EGARCH, see Nelson (1991). reduces o: If Cl 0, wih A l and B l being diagonal marices for all l, hen h r h i i l i, l l i, l l1 l1 s (7) (9) which is he consan condiional correlaion (CCC) model of Bollerslev (1990). As given in equaion (7), he CCC model does no have volailiy spillover effecs across differen 6

8 financial asses, and hence is inrinsically univariae in naure. In addiion, CCC does no capure he asymmeric effecs of posiive and negaive shocks on condiional volailiy. The parameers in model (1), (4), (6) and (9) can be obained by maximum likelihood esimaion (MLE) using a join normal densiy, namely ˆ 1 arg min log 2 n 1 Q Q (10) 1 where denoes he vecor of parameers o be esimaed on he condiional log-likelihood funcion, and Q denoes he deerminan of Q, he condiional covariance marix. When does no follow a join mulivariae normal disribuion, he appropriae esimaors are defined as he Quasi-MLE (QMLE). In order o forecas 1-day ahead condiional correlaion, we use he rolling windows echnique and examine he ime-varying naure of he condiional correlaions using and. Rolling windows are a recursive esimaion procedure whereby he model is esimaed for a resriced sample, and is hen re-esimaed by adding one observaion a he end of he sample and deleing one observaion from he beginning of he sample. The process is repeaed unil he end of he sample. In order o srike a balance beween efficiency in esimaion and a viable number of rolling regressions, he rolling window size is se a 2008 for all daa ses. 3. Daa The univariae and mulivariae GARCH models are esimaed using 3,007 observaions of daily daa for crude oil spo, forward and fuures prices in he Bren, WTI, Dubai and Tapis markes for he period 30 April 1997 o 10 November All prices are expressed in US dollars. In he WTI marke, prices are crude oil-wti spo cushing price ($/BBL), crude oil-wti one-monh forward price ($/BBL) and NYMEX one-monh fuures prices, while he prices in he Bren marke are crude oil-bren spo price FOB ($/BBL), crude oil-bren one-monh forward price ($/BBL), and one-monh fuures prices. In he Dubai marke, he prices are crude oil-arab Gulf Dubai spo price FOB ($/BBL) and crude oil-dubai one-monh forward price ($/BBL), whereas in he Tapis marke, he prices are crude oil-malaysia Tapis spo price FOB ($/BBL) and crude oil-tapis one-monh forward price ($/BBL). Three series are obained from DaaSream daabase service, while he series for Tapis are colleced from Reuers. 7

9 The synchronous price reurns i for each marke j are compued on a coninuous compounding basis as he logarihm of he closing price a he end of he period minus he logarihm of he closing price a he beginning of he period, which is defined as rij, log Pij, Pij, 1. [Inser Figure 1 here] [Inser Tables 1-2 here] Table 1 presens he descripive saisics for he reurns series of crude oil prices. The average reurn of spo, forward and fuures in Bren, WTI and Dubai are similar, while Tapis has he lowes average reurns. The normal disribuion has a skewness saisic equal o zero and a kurosis saisic of 3, bu hese crude oil reurns series have high kurosis, suggesing he presence of fa ails, and negaive skewness saisics signifying he series has a longer lef ail (exreme losses) han righ ail (exreme gain). The Jarque-Bera Lagrange muliplier saisics of he crude oil reurns in each marke are saisically significan, hereby signifying ha he disribuions of hese prices are no normal, which may be due o he presence of exreme observaions. Figure 1 presens he plo of synchronous crude oil price reurns. These indicae volailiy clusering, or periods of high volailiy followed by periods of ranquiliy, such ha crude oil reurns oscillae in a range smaller han he normal disribuion. However, here are some circumsances where crude oil reurns flucuae in a much wider scale han is permied under normaliy. The uni roo ess for all crude oil reurns in each marke are summarized in Table 2. The Augmened Dickey-Fuller (ADF) and Phillips-Perron (PP) ess were used o es he null hypohesis of a uni roo agains he alernaive hypohesis of saionariy. The ess yield large negaive values in all cases for levels such ha he individual reurns series rejec he null hypohesis a he 1% significance level, so ha all reurns series are saionary. Since he univariae ARMA-GARCH model is nesed in he model, and ARMA-GJR is nesed in wih condiional variance specified in (5) and (8), he univariae ARMA-GARCH and ARMA-GJR models are esimaed. I is sensible o exend univariae models o heir mulivariae counerpars if he properies of univariae 8

10 models are saisfied. All esimaion is conduced using he EViews 6 economeric sofware package. 4. Empirical resuls From Tables 3 and 4, he univariae ARMA(1,1)-GARCH(1,1) and ARMA(1,1)- GJR(1,1) models are esimaed o check wheher he condiional variance follows he GARCH process. In Table 3, no all he coefficiens in he mean equaions of ARMA(1,1)- GARCH(1,1) are significan, whereas all he coefficiens in he condiional variance equaion are saisically significan. Table 4 shows ha he long-run coefficiens are all saisically significan in he variance equaion, bu rbrefu (bren fuures reurn), rwisp (WTI spo reurn), rwifor (WTI forward reurn), rapsp (Tapis spo reurn), and rapfor (Tapis forward reurn) are only significan in he shor run. In addiion, he asymmeric effec of negaive and posiive shocks on he condiional variance are generally saisically significan. [Inser Tables 3-5 here] In order o check he sufficien condiion for consisency and asympoic normaliy of he QMLE for GARCH and GJR, he second momen condiions are 11 1 and 1 ( 2) 1 1, respecively. Table 5 shows ha all of he esimaed second momen condiion are less han one. In order o derive he saisical properies of he QMLE, Lee and Hausen (1997) derived he log-momen condiion for GARCH(1,1) as log I ( ) 0 E log 0, while McAleer e al. (2007) esablished he log-momen condiion for E GJR(1,1) as. Table 5 shows ha he esimaed log-momen condiion for boh models is saisfied for all reurns. For he spo, forward and fuures reurns of four crude oil markes, here are en series of reurns o be analyzed. Consequenly, 45 bivariae models need o be esimaed. The calculaed consan condiional correlaions beween he volailiies of wo reurns wihin markes and across markes using he CCC model and he Bollerslev and Wooldridge (1992) robus -raios are presened in Table 6. The highes esimaed consan condiional correlaion is 0.935, namely beween he sandardized shocks in Bren spo reurns (rbresp) and Bren forward reurns (rbrefor). 9

11 [Inser Table 6 here] Corresponding mulivariae esimaes of he condiional variance from he VARMA(1,1)-GARCH(1,1) and VARMA(1,1)-AGARCH(1,1) models are also esimaed. The esimaes of volailiy and asymmeric spillovers are presened in Table 7, which shows ha volailiy spillovers for and are eviden in 32 and 31 of 45 cases, respecively. The significan inerdependences in he condiional volailiies among reurns are boh 3 of 45 cases for and. In addiion, asymmeric effecs are eviden in 27 of 45 cases. Consequenly, he evidence of volailiy spillovers and asymmeric effecs of negaive and posiive shocks on condiional variance sugges ha is superior o he and CCC models. [Inser Table 7 here] The esimas of he condiional variances based on he and models repored in Table 7 sugges he presence of volailiy spillovers beween Bren and WTI reurns, namely volailiy spillovers from Bren fuures reurns o spo and Bren forward reurns, from Bren spo reurns o WTI spo reurns, from WTI fuures reurns o Bren spo reurns, and from WTI fuures reurns o Bren spo reurns. In addiion, he resuls show ha mos of he Dubai and Tapis reurns have volailiy spillover effecs from Bren and WTI reurns. This evidence is in agreemen wih he knowledge ha he Bren and WTI markes are wo marker crudes ha se he crude oil prices and influence he oher crude oil markes. [Inser Figure 2 here] The condiional correlaion forecass are obained from a rolling window echnique. Figures 2 plos he dynamic pahs of he condiional correlaions from and. All he condiional correlaions display significan variabiliy, which suggess ha he assumpion of consan condiional correlaion is no valid. I is ineresing o noe ha he correlaions are posiive for all pairs of crude oil reurns, and rapsp_rapfor has he highes correlaion, a In addiion, he condiional correlaion forecass of some 10

12 pairs of crude oil reurns exhibi an upward rend in 22 of 45 cases and a downward rend in 20 of 45 cases. This evidence should also be considered in diversifying a porfolio conaining hese asses. 5. Conclusion The empirical analysis in he paper examined he spillover effecs in he reurns on spo, forward and fuures prices of four major benchmarks in he inernaional oil marke, namely Wes Texas Inermediae (USA), Bren (Norh Sea), Dubai/Oman (Middle Eas) and Tapis (Asia-Pacific) for he period 30 April 1997 o 10 November Alernaive mulivariae condiional volailiy models were used, namely he CCC model of Bollerslev (1990), model of Ling and McAleer (2003), and model of McAleer e al. (2009). Boh he ARCH and GARCH esimaes were significan for all reurns in he ARMA(1,1)-GARCH(1,1) models. However, in he case of he ARMA(1,1)- GJR(1,1) models, only he GARCH esimaes were saisically significan, and mos of he esimaes of he asymmeric effecs were significan. Based on he asympoic sandard errors, he and models showed evidence of volailiy spillovers and asymmeric effecs of negaive and posiive shocks on he condiional variances, which suggesed ha was superior o boh and CCC. The paper also presened some volailiy spillover effecs from Bren and WTI reurns, and from he Bren and WTI crude oil markes o he Dubai and Tapis markes, which confirms ha he Bren and WTI crude oil markes are he world references for crude oil. The paper also compared 1-day ahead condiional correlaion forecass from he VARMA- GARCH and models using he rolling window approach, and showed ha he condiional correlaion forecass exhibied boh upward and downward rends. Acknowledgemens The auhors wish o hank Felix Chan and Abdul Hakim for providing he compuer programs. For financial suppor, he firs auhor is mos graeful o he Naional Science Council, Taiwan, he second auhor wishes o hank he Ausralian Research Council, and he hird auhor acknowledges he Energy Conservaion Promoion Fund, Minisry of Energy, Faculy of Economics, Maejo Universiy, and Faculy of Economics, Chiang Mai Universiy. 11

13 References Bollerslev, T., 1986, Generalized auoregressive condiional heeroscedasiciy. Journal of Economerices 31, Bollerslev, T., 1990, Modelling he coherence in shor-run nominal exchange rae: A mulivariae generalized ARCH approach. Review of Economics and Saisics 72, Bollerslev, T. and J. Wooldridge, 1992, Quasi-maximum likelihood esimaion and inference in dynamic models wih ime-varying covariances. Economeric Reviews 11, Caporin, M. and M. McAleer, 2009, Do we really need boh BEKK and DCC? A ale of wo covariance models. Available a SSRN: hp://ssrn.com/absrac= Engle, R.F., 1982, Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 50, Ewing, B., F. Malik and O. Ozfiden, 2002, Volailiy ransmission in he oil and naural gas markes. Energy Economics 24, Glosen, L., R. Jagannahan and D. Runkle, 1992, On he relaion beween he expeced value and volailiy and of he nominal excess reurns on socks. Journal of Finance 46, Hammoudeh, S., H. Li and B. Jeon, 2003, Causaliy and volailiy spillovers among peroleum prices of WTI, gasoline and heaing oil. Norh American Journal of Economics and Finance 14, Hammoudeh, S., Y. Yuan and M. McAleer, 2009, Shock and volailiy spillovers among equiy secors of he Gulf Arab sock markes. To appear in Quarerly Review of Economics and Finance. Lee, S.W. and B.E. Hansen, 1994, Asympoic heory for he GARCH(1,1) quasi-maximum likelihood esimaor. Economeric Theory 10, Lin, S. and M. Tamvakis, 2001, Spillover effecs in energy fuures markes. Energy Economics 23, Ling, S. and M. McAleer, 2003, Asympoic heory for a vecor ARMA-GARCH model. Economeric Theory 19, McAleer, M. 2005, Auomaed inference and learning in modelling financial volailiy. Economeric Theory 21,

14 McAleer, M., S. Hoi and F. Chan, 2009, Srucure and asympoic heory for mulivariae asymmeric condiional volailiy. Economeric Reviews 28, McAleer, M., F. Chan, S. Hoi and O. Lieberman, 2008, Generalized auoregressive condiional correlaion. Economeric Theory 24, McAleer, M., F. Chan and D. Marinova, 2007, An economeric analysis of asymmeric volailiy: Theory and applicaion o paens. Journal of Economerics 139, Milunovich, G. and S. Thorp, 2006, Valuing volailiy spillover. Global Finance Journal 17, Nelson, D., 1991, Condiional heeroscedasiciy in asse reurns: A new approach. Economerica 59, Sola, M., F. Spagnolo, and N. Spagnolo, 2002, A es for volailiy spillovers. Economics Leers 76, da Veiga, B., F. Chan and M. McAleer, 2008, Modelling he volailiy ransmission and condiional correlaions beween A and B shares in forecasing value-a-risk. Mahemaics and Compuers in Simulaion 78,

15 Table 1 Descripive Saisics for Crude Oil Price Reurns Reurns Mean Max Min S.D. Skewness Kurosis Jarque-Bera rbresp rbrefor rbrefu rwisp rwifor rwifu rdubsp rdubfor rapsp rapfor Table 2 Uni Roo Tes for Reurns ADF es Phillips-Perron es Reurns Consan Consan None Consan None Consan and Trend and Trend rbresp * * * * * * rbrefor * * * * * * rbrefu * * * * * * rwisp * * * * * * rwifor * * * * * * rwifu * * * * * * rdubsp * * * * * * rdubfor * * * * * * rapsp * * * * * * rapfor * * * * * * Noe: * significan a he 1% level. 14

16 Table 3 Univariae ARMA(1,1)-GARCH(1,1) Mean equaion Condiional Variance equaion Reurns C AR(1) MA(1) ˆ ˆ rbresp * * * * * * rbrefor * * * * rbrefu * * * * rwisp * * * * * rwifor * * * rwifu * * * * * * rdubsp * * * * rdubfor * * * * rapsp * * * * rapfor * * * Noes: (1) The wo enries for each parameer are heir respecive esimae and he Bollerslev and Wooldridge (1992) robus - raios. (2) * significan a he 5% level. 15

17 Reurns Table 4 Univariae ARMA(1,1)-GJR (1,1) Mean equaion Condiional variance equaion C AR(1) MA(1) ˆ ˆ rbresp * * * * * rbrefor * * * rbrefu * * * rwisp * * * * * rwifor * * * rwifu * * * * * rdubsp * * * * rdubfor * * * * rapsp * * * * rapfor * * * * Noes: (1) The wo enries for each parameer are heir respecive esimae and he Bollerslev and Wooldridge (1992) robus - raios. (2) * significan a he 5% level. ˆ 16

18 Table 5 Log-momen and second momen condiions for ARMA(1,1)-GARCH(1,1) and ARMA(1,1)-GJR(1,1) Reurn ARMA-GARCH ARMA-GJR Log-Momen Second momen Log-Momen Second momen rbresp rbrefor rbrefu rwisp rwifor rwifu rdubsp rdubfor rapsp rapfor

19 Table 6 Consan condiional correlaions for CCC-GARCH(1-1) Reurns rbresp rbrefor rbrefu rwisp rwifor rwifu rdubsp rdubfor rapsp rapfor rbresp ( ) (74.699) (57.939) (87.222) (61.139) (45.118) (57.787) 09 (13.994) 04 (14.047) rbrefor (75.679) (66.055) (99.892) (64.702) (64.702) (44.895) (16.679) 03 (14.199) rbrefu ( ) (90.429) ( ) (37.236) (22.395) (11.102) (10.188) rwisp ( ) ( ) (22.564) (18.390) (9.418) (8.286) rwifor ( ) (20.303) (24.507) (6.294) (6.329) rwifu (19.881) (21.240) (10.239) (9.031) rdubsp ( ) (19.442) (20.383) rdubfor (22.445) (16.468) rapsp ( ) rapfor Noes: (1) The wo enries for each variable are heir condiional correlaions and he Bollerslev and Wooldridge (1992) robus - raios. (2) Bold denoes significan a he 5% level. 18

20 Table 7 Summary of volailiy spillovers and asymmeric effecs of negaive and posiive shocks No. Reurns Number of volailiy spillovers Number of VARMA-GJR Asymmeric effecs 1 rbresp_rbrefor rbresp_rbrefu 1( ) 1( ) 0 3 rbrefor_rbrefu 1( ) 1( ) 0 4 rbresp_rwisp 1( ) 1( ) 1 5 rbrefor_rwisp rbrefu_rwisp rbresp_rwifor rbrefor_rwifor rbrefu_rwifor rwisp_rwifor rbresp_rwifu 1( ) 1( ) 1 12 rbrefor_rwifu rbrefu_rwifu rwisp_rwifu rwifor_rwifu 1( ) rbresp_rdubsp rbrefor_rdubsp 1( ) 1( ) 1 18 rbrefu_rdubsp 0 1( ) 0 19 rwisp_rdubsp 2 ( ) 2( ) 1 20 rwifor_rdubsp 1( ) 1( ) 1 21 rwifu_rdubsp 1( ) 1( ) 1 22 rbresp_rdubfor 1( ) 1( ) 0 23 rbrefor_rdubfor 1( ) 1( ) 0 24 rbrefu_rdubfor 1( ) 1( ) 0 25 rwisp_rdubfor 1( ) 1( ) 1 26 rwifor_rdubfor 1( ) 1( ) 0 27 rwifu_rdubfor 1( ) 1( ) 0 28 rdubsp_rdubfor 1( ) rbresp_rapsp 1( ) 1( ) 2 30 rbrefor_rapsp 1( ) 1( ) 2 31 rbrefu_rapsp 1( ) 1( ) 1 32 rwisp_rapsp 2 ( ) 2 ( ) 1 33 rwifor_rapsp 1( ) 1( ) 1 34 rwifu_rapsp 1( ) 1( ) 1 35 rdubsp_rapsp 1( ) 1( ) 2 36 rdubfor_rapsp 1( ) 1( ) 2 37 rbresp_rapfor 1( ) 1( ) 1 38 rbrefor_rapfor 1( ) 1( ) 1 39 rbrefu_rapfor 1( ) 1( ) 0 40 rwisp_rapfor 2 ( ) 2 ( ) 0 41 rwifor_rapfor rwifu_rapfor 1( ) 1( ) 0 43 rdubsp_rapfor 1( ) 1( ) 1 44 rdubfor_rapfor 1( ) 1( ) 1 45 rapsp_rapfor 1( ) 1( ) 1 Noes: The symbols ( ) indicae he direcion of volailiy spillovers from A reurns o B reurns (B reurns o A reurns), means hey are inerdependen, and 0 means here are no volailiy spillovers beween pairs of reurns. 19

21 Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Figure 1 Reurns of daily spo, forward and fuures prices for Bren, WTI, Dubai and Tapis RBRESP RWTISP Observaions Observaions Observaions RBREFOR Observaions RWTIFOR RBREFU Observaions Observaions RWTIFU RDUBSP RTAPSP Observaions Observaions RDUBFOR RTAPFOR Observaions Observaions 20

22 Figure 2 Forecass of he condiional correlaions beween pairs of reurns from and rbresp_rbrefor rbresp_rbrefu rbrefor_rbrefu rbresp_rwisp rbrefor_rwisp rbrefu_rwisp rbresp_rwifor rbrefor_rwifor rbrefu_rwifor rwisp_rwifor rbresp_rwifu rbrefor_rwifu rbrefu_rwifu rwifor_rwifu rwisp_rwifu

23 Figure 2 (coninued) rbresp_rdubsp rbrefor_rdubsp rbrefu_rdubsp rwisp_rdubsp rwifor_rdubsp rwifu_rdubsp rbresp_rdubfor rbrefor_rdubfor rbrefu_rdubfor rwisp_rdubfor rwifor_rdubfor rwifu_rdubfor rdubsp_rdubfor rbresp_rapsp.44 rbrefor_rapsp

24 Figure 2 (coninued) rbrefu_rapsp rwisp_rapsp rwifor_rapsp rwifu_rapsp rdubsp_rapsp rdubfor_rapsp rbresp_rapfor rbrefor_rapfor rbrefu_rapfor rwisp_rapfor rwifor_rapfor rwifu_rapfor rdubsp_rapfor rdubfor_rapfor rapsp_rapfor