CARF Working Paper CARF-F-162. Modelling Conditional Correlations for Risk Diversification in Crude Oil Markets

Transcription

1 CARF Working Paper CARF-F-162 Modelling Condiional Correlaions for Risk Diversificaion in Crude Oil Markes Chia-Lin Chang Naional Chung Hsing Universiy Michael McAleer Erasmus Universiy Roerdam Tinbergen Insiue The Universiy of Tokyo Roengchai Tansucha Maejo Universiy Chiang Mai Universiy Augus 2009 CARF is presenly suppored by Bank of Tokyo-Misubishi UFJ, Ld., Ciigroup, Dai-ichi Muual Life Insurance Company, Meiji Yasuda Life Insurance Company, Nippon Life Insurance Company, Nomura Holdings, Inc. and Sumiomo Misui Banking Corporaion (in alphabeical order). This financial suppor enables us o issue CARF Working Papers. CARF Working Papers can be downloaded wihou charge from: hp:// Working 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 Working Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 Modelling Condiional Correlaions for Risk Diversificaion in Crude Oil 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

3 Absrac This paper esimaes univariae and mulivariae condiional volailiy and condiional correlaion models of spo, forward and fuures reurns from hree major benchmarks of inernaional crude oil markes, namely Bren, WTI and Dubai, o aid in risk diversificaion. Condiional correlaions are esimaed using he CCC model of Bollerslev (1990), VARMA- GARCH model of Ling and McAleer (2003), VARMA-AGARCH model of McAleer e al. (2009), and DCC model of Engle (2002). The paper also presens he ARCH and GARCH effecs for reurns and shows he presence of significan inerdependences in he condiional volailiies across reurns for each marke. The esimaes of volailiy spillovers and asymmeric effecs for negaive and posiive shocks on condiional variance sugges ha VARMA-GARCH is superior o he VARMA-AGARCH model. In addiion, he DCC model gives saisically significan esimaes for he reurns in each marke, which shows ha consan condiional correlaions do no hold in pracice. Keywords: Condiional correlaions, crude oil spo prices, forward prices, fuures prices, risk diversificaion. JEL Classificaions: C22, C32, G17, G32 2

4 1. Inroducion Crude oil is arguably he world s mos influenial physical commodiy as i provides energy for all kinds of human aciviies in he form of refined energy producs, such as liquefied peroleum gases (LPGs), gasoline and diesel. Consequenly, crude oil is a dynamically raded commodiy ha affecs many economies. For insance, Sadorsky (1999) found ha oil price volailiy shocks have asymmeric effecs on he economy, namely changes in oil prices affec economic aciviy, bu changes in economic aciviy have lile impac on oil prices, so ha oil price flucuaions have large macroeconomic impacs. Guo and Kliesen (2005) argued ha changes in oil prices affec aggregae economic aciviy hrough changes in he dollar price of crude oil (relaive price change), and increases in uncerainy regarding fuure price. Subsanial research has been conduced on he volailiy of spo, forward and fuures prices. Models of crude oil price volailiy can be univariae or mulivariae. In he former case, Fong and See (2002) examined he emporal behaviour for daily reurns for crude oil fuures using a Markov swiching model of condiional volailiy. Lanza e al. (2006) used he AR(1)-GARCH(1,1) and AR(1)-GJR(1,1) models o esimae condiional volailiy based on forward and fuures reurns. Manera e al. (2006) used univariae ARCH and GARCH models o esimae spo and forward reurns. Sandard diagnosic ess also showed ha he AR(1)-GARCH(1,1) and AR(1)-GJR(1,1) specificaions were saisically adequae for boh he condiional mean and condiional variance. Sadorsky (2006) invesigaed he forecas performance of a large number of models. The fied model for heaing oil and naural gas volailiy was TGARCH, whereas GARCH was used for crude oil and unleaded gasoline volailiy. Lee and Zyren (2007) calculaed hisorical volailiy and GARCH models o compare he hisorical price volailiy behaviour of crude oil, moor gasoline and heaing oil in U.S. markes since They combined he shifing variable in GARCH and TARCH models o capure he response from changes in OPEC s pricing behaviour. Narayan and Narayan (2007) modelled crude oil price volailiy using daily daa by using he EGARCH model o gauge wo feaures of crude oil price volailiy, namely asymmery and he persisence of shocks. For he mulivariae condiional volailiy model, Lanza e al. (2006) modelled condiional correlaions in he WTI oil forward and fuure reurns using he CCC model of Bollerslev (1990) and DCC model of Engle (2002). They found ha DCC could vary dramaically, being negaive in four of en cases and close o zero in anoher five cases. Only 3

5 in he case of dynamic volailiies of he hree-monh and six-monh fuure reurns was he range of variaion relaively narrow. Manera e al. (2006) esimaed DCC in he reurns for Tapis oil spo and one-monh forward prices using CCC, VARMA-GARCH model of Ling and McAleer (2003), VARMA-AGARCH model of McAleer e al. (2009), and DCC, and also esed and compared volailiy specificaions. Trojani and Audrino (2005) proposed a mulivariae ree-srucured DCC model by incorporaing mulivariae hresholds in condiional volailiies and correlaions. They found in some Mone Carlo simulaions ha he model was able o capure GARCH-ype dynamics and a complex hreshold srucure in condiional volailiies and correlaions. In he empirical daa for inernaional equiy markes, he esimaed condiional volailiies were srongly influenced by GARCH and mulivariae hreshold effecs. They concluded ha condiional correlaions were deermined by simple hreshold srucures, whereas no GARCH-ype effecs could be idenified. The purpose of his paper is o esimae univariae and mulivariae condiional volailiy models for he reurns on spo, forward and fuures prices for Bren, WTI and Dubai o aid in risk diversificaion in crude oil markes. The remainder of he paper is organized 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. Secion 5 provides some concluding remarks. 2. Economeric models 2.1 Univariae condiional volailiy models Following Engle (1982), consider he ime series y 1 ( y, where 1( y ) E ) E is he condiional expecaion of y a 1 ime and is he associaed error. The generalized auoregressive condiional heeroskedasiy (GARCH) model of Bollerslev (1986) is given as follows: h, ~ N(0,1 ) (1) h p q 2 j j jh j j1 j1 (2) 4

6 where 0, 0 and 0 are sufficien condiions o ensure ha he condiional j j variance h 0. The parameer j represens he ARCH effec, or he shor run persisence of shocks o reurns, and j represens he GARCH effec, where j j measures he persisence of he conribuion of shocks o reurn i o long run persisence. Equaion (2) assumes ha he condiional variance is a funcion of he magniudes of he lagged residuals and no heir signs, such ha a posiive shock ( 0 ) has he same impac on condiional variance as a negaive shock ( 0 ) of equal magniude. In order o accommodae differenial impacs on he condiional variance of posiive and negaive shocks, Glosen e al. (1992) proposed he asymmeric GARCH, or GJR model, which is given by where r s 2 (3) h I h j j j j j j j1 j1 I i 0, i 0 1, i 0 is an indicaor funcion o differeniae beween posiive and negaive shocks. When rs 1, sufficien condiions o ensure he condiional variance, h 0, are 0, 1 0, 11 0 and 1 0. The shor run persisence of posiive and negaive shocks are given by 1 and 1 1, respecively. When he condiional shocks,, follow a symmeric disribuion, he shor run persisence is 2 1 1, and he conribuion of shocks o expeced long-run persisence is In order o esimae he parameers of model (1)-(3), maximum likelihood esimaion is used wih a join normal disribuion of. However, when does no follow a normal disribuion, or he condiional disribuion is no known, quasi-mle (QMLE) is used o maximize he likelihood funcion. Bollerslev (1986) showed he necessary and sufficien condiion for he second-order saionariy of GARCH is r s i 1. For he GARCH(1,1) model, Nelson (1991) i i1 i1 obained he log-momen condiion for sric saionary and ergodiciy as E log 0, which is imporan in deriving he saisical properies of he QMLE. For GJR(1,1), Ling and McAleer (2002a, 2002b) presened he necessary and sufficien 5

7 2 condiion for E as McAleer e al. (2007) esablished he logmomen condiion for GJR(1,1) as E I sufficien for consisency and asympoic normaliy of he QMLE. 2 log , and showed ha i is 2.2 Mulivariae condiional volailiy models The ypical specificaion underlying he mulivariae condiional mean and condiional variance in reurns is given as follows: where, y,..., y1 y m disribued (i.i.d.) random vecors, 1 y E y F (4) D,..., 1 m is a sequence of independenly and idenically F is he pas informaion available o ime, D diag h1,..., hm, m is he number of reurns, and 1,..., n, (see Li, Ling and McAleer (2002), McAleer (2005), and Bauwens e al. (2006)). The consan condiional correlaion (CCC) model of Bollerslev (1990) assumes ha he condiional variance for each reurn, h i, i 1,.., m, follows a univariae GARCH process, ha is where h r s 2 i i iji, j ijhi, j j1 j1 (5) ij represens he ARCH effec, or shor run persisence of shocks o reurn i, and represens he GARCH effec, or he conribuion of shocks o reurn i o long run persisence, namely r j1 ij s ij. j1 The condiional correlaion marix of CCC is E F E for i, j 1,..., m. From (4), D D Q D D, where marix is defined as 1 ij, where, D diag Q 12 i, and E F 1 Q is he condiional covariance marix. The condiional correlaion D Q D 1 1, and each condiional correlaion coefficien is esimaed from he sandardized residuals in (4) and (5). Therefore, here is no mulivariae esimaion involved for CCC, which involves m univariae GARCH models, excep in he calculaion of he condiional correlaions. 6

8 Alhough he CCC specificaion in (5) is a compuaionally sraighforward mulivariae GARCH model, i assumes independence of he condiional variances across reurns and does no accommodae asymmeric behaviour. In order o incorporae inerdependencies, Ling and McAleer (2003) proposed a vecor auoregressive moving average (VARMA) specificaion of he condiional mean in (4), and he following specificaion for he condiional variance: H,..., h1 h m 2 2,,... 1 m where H W A B H r s i i j j i1 j1 (6), and W, A i for i 1,.., r and B j for j 1,.., s are m m marices. As in he univariae GARCH model, VARMA-GARCH assumes ha negaive and posiive shocks have idenical impacs on he condiional variance. In order o separae he asymmeric impacs of he posiive and negaive shocks, McAleer, Hoi and Chan (2009) proposed he VARMA-AGARCH specificaion for he condiional variance, namely where C i are m m marices for i 1,.., r r r s H W A C I B H i i i i i j j i1 i1 j1, and I diag I,..., I I i 0, 1, 1 m 0 i 0 i, where If m 1, (6) collapses o he asymmeric GARCH, or GJR, model. Moreover, VARMA- AGARCH reduces o VARMA-GARCH when Ci 0 for all i. If Ci 0 and A i and (7) B j are diagonal marices for all i and j, hen VARMA-AGARCH reduces o he CCCmodel. The parameers of model (4)-(7) are obained by maximum likelihood esimaion (MLE) using a join normal densiy. When does no follow a join mulivariae normal disribuion, he appropriae esimaor is QMLE. Unless is a sequence of iid random vecors, or alernaively a maringale difference process, he assumpion ha he condiional correlaions are consan may seen unrealisic. In order o make he condiional correlaion marix ime dependen, Engle (2002) proposed a dynamic condiional correlaion (DCC) model, which is defined as y ~ (0, Q ), 1,2,..., n (8) 1 Q D D, (9) 7

9 where D diag h,..., h is a diagonal marix of condiional variances, and is he 1 k informaion se available o ime. The condiional variance, univariae GARCH model as follows: h i, can be defined as a h p h i i ik i, k il i, l k1 l1 q. (10) If is a vecor of i.i.d. random variables, wih zero mean and uni variance, Q in (9) is he condiional covariance marix (afer sandardizaion, i yi hi ). The i are used o esimae he dynamic condiional correlaions, as follows: 1/2 1/2 ( diag( Q ) Q ( diag( Q ) (11) where he k k symmeric posiive definie marix Q is given by Q (1 ) Q Q (12) in which 1 and 2 are scalar parameers o capure he effecs of previous shocks and previous dynamic condiional correlaions on he curren dynamic condiional correlaion, and 1 and 2 are non-negaive scalar parameers. As Q is condiional on he vecor of sandardized residuals, (12) is a condiional covariance marix, and Q is he k k uncondiional variance marix of. For furher deails, and criique of he DCC model, see Caporin and McAleer (2009). 3. Daa The daa used in his paper are daily synchronous closing price of spo, forward and fuures crude oil prices from hree major crude oil markes, namely Bren, WTI and Dubai. The 4,659 price observaions from 2 January 1991 o 10 November 2008 are obained from he DaaSream daabase service. The reurns of crude oil prices i of marke j a ime in a coninuous compound basis are calculaed as rij, log Pij, Pij, 1, where P ij, and Pij, 1 are he closing prices of crude oil price i of marke j for days and 1, respecively. The univariae and mulivariae condiional volailiy models are esimaed using he EViews 6 economeric sofware package. The descripive saisics for he crude oil reurns series are summarized in Table 1. The sample mean is quie small, bu he corresponding variance of reurns is much higher. Boh negaive skewness and high kurosis sugges ha reurns are no disribued normally. 8

10 Similarly, he null hypohesis of normaliy is also rejeced for he sample reurn series by he Jarque-Bera(J-B) es lagrange muliplier saisics. The logarihms of crude oil prices are ploed in Figure 1. I is clear ha here is subsanial clusering of volailiies, such ha a urbulen rading day ends o be followed by anoher urbulen day, while a ranquil period ends o be followed by anoher ranquil period. [Inser Tables 1-2 here] [Inser Figure 1 here] The empirical resuls of he uni roo ess for he sample reurns in each marke are summarized in Table 2. The Augmened Dickey-Fuller (ADF) and Phillips-Perron (PP) ess are used o es for uni roos in he individual series. The large negaive values in all cases indicae rejecion of he null hypohesis a he 1% level, such ha all reurns are saionary. 4. Empirical Resuls Univariae esimaes of he condiional volailiies, GARCH(1,1) and GJR(1,1), wih differen condiional mean equaion models based on spo, forward and fuures reurns in each marke, are given in Tables 3-5, which repor he respecive QMLE and he Bollerslev- Woodridge (1992) robus -raios. The log-momen and second momen condiions are also presened o confirm he saisical properies of he esimaes. The second momens of GARCH(1,1) and GJR(1,1), namely 1 1 and , are less han 1, and he 2 esimaed log-momens of GARCH(1,1) and GJR(1,1), which are given as E log 1 1 and (log( 1 1I )) 1 E 2, respecively, are less han 0, so he QMLE are consisen and asympoically normal (see McAleer (2005) and McAleer e al. (2007)). The univariae GARCH esimaes for Bren are given in Table 3. The coefficiens in he mean equaions in Panel 3a are no all saisically significan. The mean equaion of AR(1)-GARCH(1,1) is significan only for forward reurns, while ARMA(1,1)-GARCH(1,1) is significan in all reurns series. In addiion, he coefficien in he condiional variance equaions for boh AR(1)-GARCH(1,1) and ARMA(1,1)-GARCH(1,1) are all significan. Consequenly, ARMA(1,1)-GARCH(1,1) is preferred o AR(1)-GARCH(1,1). 9

11 In he case of he asymmeric GARCH(1,1) model in Panel 3b, only he coefficiens in he mean equaion for ARMA(1,1) are significan. The esimaes of he asymmeric effec for he univariae model are no saisically significan, excep for spo reurns. The resuls for univariae esimaion of he WTI marke are repored in Table 4. The robus -raios show ha he ARMA(1,1)-GARCH(1,1) specificaion for all reurns is saisically adequae in boh he condiional mean and condiional variance equaions, bu he coefficiens in he condiional mean equaion of AR(1)-GARCH(1,1) are insignifican. The univariae GJR models are presened in Panel 4b in Table 4, where only he forward reurns for ARMA-GARCH model are significan. However, asymmery beween negaive and posiive shocks on he condiional variance is no observed. For he Dubai marke in Table 5, he coefficiens in he mean equaion for spo and forward reurns in Panels 5a and 5b are significan only for AR(1)-GARCH(1,1) and AR(1)- GJR(1,1). Panel 5a shows ha he coefficiens in he condiional variance equaion for AR(1)-GARCH(1,1) are all saisically significan, whereas in Panel 5b, he condiional variance coefficiens are significan only in spo reurns. These resuls show ha here is an asymmeric effec beween negaive and posiive shocks on he condiional variance. [Inser Tables 3-5 here] Table 6 presens he consan condiional correlaions for he spo, forward and fuures reurns in each marke using he CCC model based on univariae GARCH(1,1) esimaes. Three reurns in he Bren and WTI markes in Panels 6a and 6b provide six condiional correlaions, while wo reurns in he Dubai marke in Panel 6c give one condiional correlaion. The highes esimaed condiional correlaion in he Bren marke is 0.940, namely beween he sandardized shocks o he volailiy of he spo and forward reurns. In he case of he WTI marke, he highes esimaed condiional correlaion for Bren is 0.883, namely beween he sandardized shocks o he volailiy of spo and fuures reurns, and fuures and forward reurns. The condiional correlaion beween he shocks o spo and forward reurns for he Dubai marke is [Inser Table 6 here] [Inser Figure 6 here] 10

12 The esimaes of he dynamic condiional correlaions and he descripive saisics for DCC across he shocks o reurns in each marke are presened in Table7, Panels 7a and 7b, respecively. Based on he Bollerslev and Wooldridge (1992) robus -raios, he esimaes of he DCC parameers, ˆ 1 and ˆ 2, in each marke are always saisically significan. This indicaes ha he assumpion of consan condiional correlaion for all shocks o reurns is no suppored empirically. In addiion, he mean of he dynamic condiional correlaions of each pair is idenical o he consan condiional correlaion esimaes repored in Table 6. The shor run persisence of shocks on he dynamic condiional correlaions is greaes for WTI a 0.264, while he larges long run persisence of shocks o he condiional correlaions is for Bren, namely = [Inser Tables 7-10 here] The corresponding mulivariae esimaes for he VARMA(1,1)-GARCH and VARMA(1,1)-AGARCH models for each marke are given in Tables I is clear from Table 8, Panel a, ha he forward reurns are significan only for ARCH and GARCH, while he spo and fuures reurns are only significan for ARCH. Moreover, here are significan inerdependences in he condiional volailiy beween spo and forward reurns, and beween spo and fuures reurns. The resuls in Panel b show ha he ARCH and GARCH effecs are significan in he condiional volailiy model for spo, forward and fuures reurns. There are also significan inerdependences in he condiional volailiy beween spo and fuures reurns. In addiion, as he asymmeric effecs for each reurn in Panel 8a are insignifican, if follows ha VARMA-GARCH model dominaes is asymmeric counerpar, VARMA- AGARCH. Table 9, Panel a, for Bren presens he VARMA-GARCH model, in which he ARCH and GARCH effecs are significan in he condiional volailiy model for spo, forward and fuures reurns. Also presen are he spillover effecs across he spo, forward and fuures reurns. In conras, Panel 9b shows ha he ARCH and GARCH effecs are insignifican, excep for he GARCH effec for forward reurns. In addiion, he asymmeric spillover effecs for each of he reurns is no saisically significan, such ha VARMA-AGARCH is dominaed by VARMA-GARCH. Table 10 presens he VARMA-GARCH and VARMA-AGARCH esimaes for Dubai. I is clear ha he ARCH and GARCH effecs for spo and forward reurns are 11

13 significan, and here is a significan display of inerdependences in he condiional volailiies beween he spo and forward reurns. In Panel 10b, he ARCH and GARCH effecs are saisically significan only for forward reurns, bu he ARCH effec is significan for spo reurns. There is also he presence of inerdependences beween spo and forward reurns, while he asymmeric spillover effecs for each of he reurns is insignifican. Consequenly, VARMA-GARCH is preferred o VARMA-AGARCH. 5. Conclusion This paper esimaed four mulivariae volailiy models, namely CCC, DCC, VARMA-GARCH and VARMA-AGARCH, for he spo, forward and fuures reurns for hree major benchmark inernaional crude oil markes, namely Bren, WTI and Dubai. The reurns for he period 2 January 1991 o 10 November 2008 were esimaed using mulivariae condiional volailiy and condiional correlaion models. Boh he univariae ARCH and GARCH componens of he GARCH(1,1) and GJR(1,1) models were significan for all reurns, whereas mos of he esimaed asymmeric effecs for GJR(1,1) were no significan. The calculaed consan condiional correlaions across he condiional volailiies of reurns using he CCC model were high. The paper also presened he ARCH and GARCH effecs for reurns, and significan inerdependences in he condiional volailiies across reurns in each marke. The esimaes of volailiy spillovers and asymmeric effecs for negaive and posiive shocks on he condiional variances suggesed ha he VARMA- GARCH model was superior o he asymmeric VARMA-AGARCH. In addiion, he esimaes of he DCC model for reurns in each marke were saisically significan. In shor, consan condiional correlaions were no suppored in he empirical examples. Such esimaes of he dynamic condiional correlaions of shocks o reurns associaed wih spo, forward and fuures prices can be used as an aid o risk diversificaion in crude oil markes. 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. 12

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16 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

17 Table 2. Uni roo ess 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 1%. 16

18 Table 3. Univariae volailiy models of crude oil reurns for Bren Panel 3a. AR(1)-GARCH(1,1) and ARMA(1,1)-GARCH(1,1) esimaes Mean equaion Variance equaion Log- Reurns c AR(1) MA(1) ˆ ˆ ˆ Momen Spo Forward Fuures Second momen AIC SIC Panel 3b. AR(1)-GJR(1,1) and ARMA(1,1)-GJR(1,1) esimaes Mean equaion Variance equaion LogzReurns c AR(1) MA(1) ˆ ˆ ˆ ˆ Momen Second momen Spo Forward Fuures Noes: (1) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (2) Enries in bold are significan a 5%. AIC SIC 17

19 Table 4. Univariae volailiy models of crude oil reurns for WTI Panel 4a. AR(1)-GARCH(1,1) and ARMA(1,1)-GARCH(1,1) esimaes Mean equaion Variance equaion Log- Reurns c AR(1) MA(1) ˆ ˆ ˆ Momen Spo Forward Fuures E Second momen AIC SIC Panel 4b. AR(1)-GJR(1,1) and ARMA(1,1)-GJR(1,1) esimaes Mean equaion Variance equaion Log- Second Reurns Momen momen AIC SIC c AR(1) MA(1) ˆ ˆ ˆ ˆ Spo Forward Fuures Noes: (1) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (2) Enries in bold are significan a he 95% level 18

20 Table 5. Univariae volailiy models of crude oil reurns for Dubai Panel 5a. AR(1)-GARCH(1,1) and ARMA(1,1)-GARCH(1,1) esimaes Mean equaion Variance equaion Log- Reurns c AR(1) MA(1) ˆ ˆ ˆ Momen Spo Forward Second momen AIC SIC Panel 5b. AR(1)-GJR(1,1) and ARMA(1,1)-GJR(1,1) esimaes Mean equaion Variance equaion Log- Second Reurns Momen momen AIC SIC c AR(1) MA(1) ˆ ˆ ˆ ˆ Spo Forward Noes: (1) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (2) Enries in bold are significan a 5%. 19

21 Table 6. Consan condiional correlaions (CCC) based on GARCH(1,1) Panel 6a: Bren Reurns rbresp rbrefor rbrefu rbresp rbrefor rbrefu Panel 6b: WTI Reurns rwisp rwifor rwifu rwisp rwifor rwifu Panel 6c: Dubai Reurns rdubsp rdubfor rdubsp rdubfor

22 Table 7. Dynamic condiional correlaions (DCC) based on GARCH(1,1) Panel 7a. Esimaes of Q Reurns ˆ ˆ 1 2 rbresp_rbrefor_rbrefu rwisp_rwifor_rwifu rdubsp_rdubfor Noe: Two enries for each parameers are heir respecive esimae and robus -raio Panel 7b. Descripive saisics Reurns Mean Max Min S.D. Skewness Kurosis rbresp_rbrefor rbresp_rbrefu rbrefor_rbrefu rwisp_rwifor rwisp_rwifu rwifor_rwifu rdubsp_rdubfor

23 Table 8. VARMA-GARCH and VARMA-AGARCH models for Bren Panel a. VARMA(1,1)-GARCH(1,1) Reurns bresp rbresp (4.085) (1.735) rbrefor (1.390) (-0.690) rbrefu (-0.079) (-9.420) brefor brefu bresp (-0.509) (-1.377) (3.656) (3.163) (4.703) (2.465) (79.990) (3.179) (3.252) brefor brefu (0.231) (-0.882) (-0.472) (-2.140) (2.777) (16.441) Panel b.varma(1,1)-agarch(1,1) Reurns bresp brefor rbresp rbrefor rbrefu Noes: Enries in bold are significan a 5%. brefu bresp brefor brefu

24 Table 9. VARMA-GARCH and VARMA-AGARCH models for WTI Panel a. VARMA(1,1)-GARCH(1,1) Reurns rwisp rwisp (0.062) (0.818) rwifor (5.365) (-1.311) rwifu (-0.179) (-1.851) wifor wifu wisp wifor wifu (2.331) (1.976) (1.829) (-0.279) (2.669) (1.075) (4.001) (1.452) (1.534) (1.184) ( ) (0.876) (0.643) (-4.697) (4.583) Panel b.varma(1,1)-agarch(1,1) wifor wifu Reurns wisp rwisp (-0.078) (0.314) (2.395) rwifor (5.641) (-0.960) (1.277) rwifu (-0.146) (-1.760) (1.757) Noes: Enries in bold are significan a 5% (-0.195) (2.448) (0.676) (0.843) (0.743) (0.658) wisp wifor wifu (3.805) (1.178) (1.380) (1.349) ( ) (0.978) (0.596) (-4.502) (4.948) 23

25 Table 10. VARMA-GARCH and VARMA-AGARCH models for Dubai Panel a. VARMA(1,1)-GARCH(1,1) Reurns dubsp rdubsp (6.403) (0.524) rdubfor (1.070) (1.069) dubfor dubsp dubfor (4.672) (0.260) ( ) (0.598) (-4.757) (1.526) Panel b.varma(1,1)-agarch(1,1) dubfor Reurns dubsp rdubsp (5.510) (-1.123) rdubfor (1.653) (0.884) Noes: Enries in bold are significan a 5% (5.409) (0.052) (2.421) (1.164) dubsp dubfor ( ) (1.016) (-3.650) (4.639) 24

26 Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Reurns (%) Figure 1 Reurns of daily spo, forward and fuures reurns for Bren, WTI and Dubai RBRESP RWTISP Observaions Observaions RWTIFOR Observaions Observaions RBREFOR RWTIFU RBREFU Observaions Observaions RDUBSP RDUBFOR Observaions 25 Observaions

27 Figure 2 Dynamic condiional correlaions RBRESP_RBREFOR RWTISP_RWTIFOR RBRESP_RBREFU RWTISP_RWTIFU RBREFOR_RBREFU RWTIFOR_RWTIFU RDUBSP_RDUBFOR 26