DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND

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1 DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Analyzing and Forecasing Volailiy Spillovers, Asymmeries and Hedging in Major Oil Markes* Chia-Lin Chang, Michael McAleer, and Roengchai Tansucha WORKING PAPER No. 19/2010 Deparmen of Economics and Finance College of Business and Economics Universiy of Canerbury Privae Bag 4800, Chrischurch New Zealand

2 WORKING PAPER No. 19/2010 Analyzing and Forecasing Volailiy Spillovers, Asymmeries and Hedging in Major Oil Markes Chia-Lin Chang 1, Michael McAleer 2 and Roengchai Tansucha 3 April 2010 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 and asymmeric effecs across and wihin he four markes, using hree mulivariae GARCH models, namely he consan condiional correlaion (CCC), vecor ARMA-GARCH () and vecor ARMA-asymmeric GARCH (VARMA- AGARCH) 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 forecas condiional correlaions beween pairs of crude oil reurns have boh posiive and negaive rends. Moreover, he opimal hedge raios and opimal porfolio weighs of crude oil across differen asses and marke porfolios are evaluaed in order o provide imporan policy implicaions for risk managemen in crude oil markes. Keywords: Volailiy spillovers, mulivariae GARCH, condiional correlaion, asymmeries, hedging. JEL Classificaions: C22, C32, G17, G32. Acknowledgemens: The auhors wish o hank wo referees for helpful commens and suggesions, and 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, Naional Science Council, Taiwan, and a Visiing Erskine Fellowship a he Universiy of Canerbury, New Zealand, and he hird auhor acknowledges he Faculy of Economics, Maejo Universiy, Thailand and he Energy Conservaion Promoion Fund, Minisry of Energy, Thailand. 1 Deparmen of Applied Economics, Naional Chung Hsing Universiy, Taichung, Taiwan 2 Economeric Insiue, Erasmus School of Economics, Erasmus Universiy Roerdam and Tinbergen Insiue, The Neherlands 3 Faculy of Economics, Maejo Universiy, Thailand Corresponding Auhor: Michael McAleer, michael.mcaleer@gmail.com 1

3 WORKING PAPER No. 19/2010 Analyzing and Forecasing Volailiy Spillovers, Asymmeries and Hedging in Major Oil Markes 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 price are prices quoed for delivering in a specified quaniy of crude oil a a specified ime and place in he fuure in a paricular rading cenre. 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). These resuls hold even when markes do no necessarily rade a he same ime. 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. Volailiy spillovers and asymmeries among hose four major benchmarks are likely o be imporan for consrucing hedge raios and opimal porfolios. As research has ypically focused on oil spo and fuures prices o he neglec of forward prices, his paper analyses all hree oil prices. Accurae modelling of volailiy is crucial in finance and for commodiy. 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 Heeroskedasiciy (GARCH) model of Engle (1982) and Bollerslev (1986) has subsequenly led o a family of univariae and mulivariae GARCH models which can capure differen 2

4 behavior in financial reurns, including ime-varying volailiy, persisence and clusering of volailiy, and he 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 New York Mercanile Exchange (NYMEX) and Inernaional Peroleum Exchange (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 may 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 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. Chang e al. (2009) examined mulivariae condiional volailiy and condiional correlaion models of spo, forward, and fuures reurns from hree crude oil markes, namely Bren, WTI and Dubai, and provided evidence of significan volailiy spillovers and asymmeric effecs in he condiional volailiies across reurns for each marke. Of he four major crude oil markes, only he mos well known oil markes, namely WTI and Bren, he ligh swee grade caegory, have spo, forward and fuures prices, while he Dubai and Tapis markes, he heavier and less swee grade caegory, have only spo and forward prices. I would seem ha no research has ye esed he spillover effecs for each of 3

5 he spo, forward and fuures crude oil prices in and across all markes, or esimaed he opimal porfolio weighs and opimal hedge raios for purposes of risk diversificaion. Spo, fuures and forward oil markes have differen fundamenals and conrac radabiliy and liquidiy, and hus differen volailiy behaviour. Forward markes are usually less volaile han spo and fuures markes. I would herefore be ineresing o deermine if his sylized characerisic holds across he major oil benchmarks. Several mulivariae GARCH models specify risk for 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) analyzed he mulivariae vecor ARMA-GARCH (VARMA- GARCH) model of Ling and McAleer (2003) and vecor ARMA-asymmeric GARCH () model of McAleer, Hoi and Chan (2009), and found ha hey were superior o he GARCH model of Bollerslev (1986) and he GJR model of Glosen, Jagannahan and Runkle (1992). This paper has wo main objecives, as follows: (1) We invesigae he imporance of volailiy spillovers and asymmeric effecs of negaive and posiive shocks of equal magniude on he condiional variance for modelling crude oil volailiy in he reurns of spo, forward and fuures prices wihin and across he Bren, WTI, Dubai and Tapis markes, using mulivariae condiional volailiy models. The spillover effecs beween reurns in and across markes are 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. (2) We apply he esimaed resuls o compue he opimal hedge raios and opimal porfolio weighs of he crude oil porfolio, which provides imporan policy implicaions for risk managemen in crude oil markes. 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 of 1-day ahead condiional correlaions. Secion 5 presens he economic implicaions for opimal hedge raios and opimal porfolio weighs. Secion 6 provides some concluding remarks. 2. Economeric Models This secion presens he consan condiional correlaion (CCC) model of Bollerslev (1990), he model of Ling and McAleer (2003) and 4

6 model of McAleer, Hoi and Chan (2009). These models assume consan condiional correlaions, and do no suffer from he problem of dimensionaliy, as compared wih he VECH and BEKK models, and also possess regulariy and saisical properies, unlike he DCC model (see McAleer e al. (2008) and Carporin and McAleer (2009, 2010) for deailed explanaions of hese issues). In explaining a vecor of oil prices, Y, he 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 1 Y E Y F (1) LY L (2) D (3) r s l l l i, j l1 l1 (4) H W A B H where (1) denoes he decomposiion of Y ino is predicable (condiional mean) and random 12 componens, D diag hi,,,, H,..., h1 h m W,..., 1 m,..., 1 m is a, A and 2 2 sequence of independenly and idenically (iid) random vecors, i,..., m B l are m m marices wih ypical elemens ij and I diag I i is an m m 1..., respecively, for i, j 1,..., m, marix. L I 1 L... q L Im L ql are polynomials in L, he lag operaor, and informaion available o ime. GARCH effec. l represens he ARCH effec, and ij m and L p p F is he pas l represens he 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 E. If m 1, equaion (4) reduces o he univariae GARCH model of Bollerslev by (1986): 5

7 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. An exension of (4) o accommodae asymmeries wih respec o i is given by r r s (6) H W A C I B H l l l l l l l l1 l1 l1 in which i i hi for all i and, l C are m m marices, and I is an indicaor i 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 (7) 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 (9) 6

8 which is he CCC model of Bollerslev (1990). As given in equaion (7), he CCC model does no have volailiy spillover effecs across differen financial asses, and hence is inrinsically univariae in naure. In addiion, CCC also 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 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, 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 3,009 observaions of daily daa on 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. The 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, 7

9 he prices are crude oil-arab Gulf Dubai spo price FOB ($/BBL) and crude oil-dubai onemonh forward price ($/BBL). 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. The synchronous price reurns i for each marke j are compued on a coninuous compounding basis as he logarihm of 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. Bren and WTI reurns are more volaile han hose of Dubai/Oman and Tapis, as shown by he esimaes of heir respecive sandard errors. This may be explained by he fac ha ligh swee crude oil is less pleniful and in greaer demand han he more sour and heavier grades, or due o he presence of differen regulaory resricions in hese markes. I also seems ha he forward reurns are less volaile han hose of spo and fuures (if hey exis) prices, wih he excepion of Tapis. This has o do wih he naure and characerisics of he forward conracs relaive o hose of he spo and fuures conracs Figure 1 presens he plo of synchronous crude oil price reurns. These indicae volailiy clusering or period 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 8

10 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 are 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 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 regulariy condiions of univariae models are saisfied, so ha he QMLE will be consisen and asympoically normal. 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 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 effecs 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 model, he second momen condiions are 11 1 and 2 1, respecively. Table 5 shows ha all of he esimaed second momen 1 1 condiions 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 9

11 2 1 1 E log 0, while McAleer e al. (2007) esablished he log-momen condiion for GJR(1,1) as E I log 0. Table 5 shows ha he esimaed log-momen condiion for boh models is saisfied for all reurns. The high persisence of volailiy shown in Table 5 can be explained by he reinforcing mechanism beween oil invenories and he oil basis = (fuures spo). For he spo, forward and fuures reurns in he 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 volailiy of wo reurns wihin 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). [Inser Tables 6 here] Corresponding mulivariae esimaes of he condiional variances 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 volailiy among reurns hold for 3 of 45 cases for boh 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 he condiional variance sugges ha is superior o he and CCC models. [Inser Tables 7 here] The esimaes 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 Bren spo and forward reurns, from Bren spo reurns o WTI spo reurns, and from WTI fuures reurns o Bren spo reurns. In addiion, he resuls show ha mos of he Dubai and 10

12 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 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. Figure 2 plos he dynamic pahs of he condiional correlaions derived from VARMA- GARCH 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 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. Implicaions for Porfolio Design and Hedging Sraegies This secion presens opimal hedge raios and opimal porfolio weighs among crude oil reurns and across markes. Theoreically, hedging involves he deerminaion of he opimal hedge raio. One of he mos widely used hedging sraegies is based on he minimizaion of he variance of he porfolio, he so-called minimum variance hedge raio (see, for example, Kroner and Sulan (1993), Lien and Tse (2002), and Chen e al. (2003)). In order o minimize risk, he dynamic hedge raio, based on condiional informaion available a, is given by: h 12, 12, (11) h22, where 12, is he risk-minimizing hedge raio for wo crude oil asses, h 12, is he condiional covariance beween crude oil asses 1 and 2, and h 22, is he condiional variance of crude oil asse 2. In order o minimize risk, a long posiion of one dollar aken in one crude oil asse 11

13 should be hedged by a shor posiion of $ in anoher crude oil asse a ime (Hammoudeh e al. (2009)). The average values of he opimal hedge raio ( ) using esimaes from he model are presened in he firs column of Table 8. By following he esimaed hedge sraegy, he highes average opimal hedge raio is (rwisp/rwifu), meaning one dollar long in WTI spo should be shored by 95 cens in WTI fuures. The lowes average opimal hedge raio is (rapfor/rwifor), meaning one dollar long in Tapis forward should be shored by 12 cens in WTI forward. Ineresingly, we find ha he average opimal hedge raio across markes, namely Dubai and WTI, Tapis and Bren, and Tapis and WTI, are very low, signifying one dollar long in he firs marke should be shored by only a few cens in he second marke. In he case of opimal porfolio weighs, he esimaed covariance marices from he model are used o compue he opimal porfolio holdings ha minimize porfolio risk, assuming he expeced reurns are zero. Applying he mehods of Kroner and Ng (1998), he opimal porfolio weigh of crude oil asse 1/2 holding ( w 12, ) is given by: w 12, h22, h12, h 2h h 11, 12, 22, (12) and 0, if w12, < 0 w12, w12,, if 0 < w12, 0 1, if w12, > 0 (13) where w 12,, is he porfolio weigh of he firs asse relaive o he second asse a ime. The average of he weighs w 12, means he opimal porfolio holdings for he firs asse should be w 12, cens o a dollar. Obviously, he opimal porfolio holding for he second asse would be (1- w 12, ) o a dollar. The average values of w 12, based on he esimaes are presened in he second column of Table 8. For insance, he highes average opimal hedge raio is (rbrefor/rbresp), suggesing ha he opimal holding of Bren forward in one dollar of 12

14 forward/spo for Bren marke is 97 cens, compared wih 3 cens for Bren spo. These opimal porfolio weighs sugges ha invesors should have much more Bren forward han Bren spo in heir porfolio. Surprisingly, he average opimal porfolio weighs across markes, namely Dubai and Bren, Dubai and WTI, Tapis and Bren, and Tapis and WTI, sugges ha invesors should own WTI and Bren (he ligh swee grade caegory) in greaer proporions han Dubai and Tapis (he heavier and less swee grade caegory). 6. 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 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 confirmed 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 rend and downward rends. In order o design opimal porfolio holdings across wo crude oil grade caegories, he opimal porfolio weighs sugges holding he ligh swee grade caegory (WTI and Bren) in a greaer proporion han he heavier and less swee grade caegory (Dubai and Tapis). In he case of minimizing risk by using a hedge, a long posiion of one dollar in he ligh swee 13

15 grade caegory (WTI) should be shored by only a few cens in he heavier and less swee grade caegory (Dubai and Tapis). 14

16 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= Caporin, M. and M. McAleer, 2010, Do we really need boh BEKK and DCC? A ale of wo mulivariae GARCH models. Available a SSRN: hp://ssrn.com/absrac= Chen, S.-S., C.-F. Lee and K. Shresha, 2003, Fuures hedge raios: A review. Quarerly Review of Economics and Finance 43, Chang, C.-L., M. McAleer and R. Tansucha, 2009, Modeling condiional correlaions for risk diversificaion in crude oil markes. Journal of Energy Markes 2, 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. Quarerly Review of Economics and Finance 49, Hassan, S. and F. Malik, 2007, Mulivariae GARCH modeling of secor volailiy 15

17 ransmission. Quarerly Review of Economics and Finance 47, Kroner, K.F. and J. Sulan, Time-varying disribuions and dynamic hedging wih foreign currency fuures. Journal of Financial and Quaniaive Analysis 28, Kroner, K.F. and V.K. Ng, Modeling asymmeric movemen of asse prices. Review of Financial Sudies 11, Lee, S.W. and B.E. Hansen, 1994, Asympoic heory for he GARCH(1,1) quasi-maximum likelihood esimaor. Economeric Theory 10, Lien, D. and Y.K. Tse, 2002, Some recen developmens in fuures hedging. Journal of Economic Surveys 16(3), 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, 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,

18 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 Tess 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: * denoes significance a he 1% level. 17

19 Reurns Table 3 Univariae ARMA(1,1)-GARCH(1,1) Mean equaion Variance equaion C AR(1) MA(1) ˆ rbresp * * * * * * rbrefor * * * * rbrefu * * * * rwisp * * * * * rwifor * * * rwifu * * * * * * rdubsp * * * * rdubfor * * * * rapsp * * * * rapfor * * * Noes: (1) The wo enries for each parameer are heir respecive parameer esimae and he Bollerslev and Wooldridge (1992) robus - raios. (2) * denoes significance a he 1% level. ˆ 18

20 Reurns Table 4 Univariae ARMA(1,1)-GJR (1,1) Mean equaion C AR(1) MA(1) ˆ Variance equaion rbresp * * * * * rbrefor * * * rbrefu * * * rwisp * * * * * rwifor * * * rwifu * * * * * rdubsp * * * * rdubfor * * * * rapsp * * * * rapfor * * * * Noes: (1) The wo enries for each parameer are heir respecive parameer esimaes and Bollerslev and Wooldridge (1992) robus - raios. (2) * denoes significance a he 1% level. ˆ ˆ 19

21 Table 5 Log-momen and Second Momen Condiions for he ARMA(1,1)-GARCH(1,1) and ARMA(1,1)-GJR(1,1) models Reurns ARMA-GARCH ARMA-GJR Log-Momen Second momen Log-Momen Second momen rbresp rbrefor rbrefu rwisp rwifor rwifu rdubsp rdubfor rapsp rapfor

22 Table 6 Consan Condiional Correlaion for CCC-GARCH(1-1) Model Reurns rbresp rbrefor rbrefu rwisp rwifor rwifu rdubsp rdubfor rapsp rapfor rbresp ( ) (74.699) (57.939) (87.222) (61.139) (45.118) (57.787) (13.994) (14.047) rbrefor (75.679) (66.055) (99.892) (64.702) (64.702) (44.895) (16.679) (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) ubfor (22.445) (16.468) rapsp ( ) rapfor Noes: (1) The wo enries for each parameer are heir respecive esimaed condiional correlaion and Bollerslev and Wooldridge (1992) robus - raios. (2) Bold denoes significance a he 5% level.

23 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. 22

24 Table 8 Summary of Volailiy Spillovers and Asymmeric Effecs of Negaive and Posiive Shocks No. Porfolio Average Opimal Hedge Raio ( ) Opimal Porfolio Weighs (w 12, ) of firs crude oil reurn in 1$ porfolio 1 rbrefor/rbresp rbresp/rbrefu rbrefor/rbrefu rbresp/rwisp rwisp/rbrefor rwisp/rbrefu rbresp/rwifor rbrefor/rwifor rbrefu/rwifor rwisp/rwifor rbresp/rwifu rbrefor/rwifu rbrefu/rwifu rwisp/rwifu rwifor/rwifu rdubsp/rbresp rdubsp/rbrefor rdubsp/rbrefu rdubsp/rwisp rdubsp/rwifor rdubsp/rwifu rdubfor/rbresp rdubfor/rbrefor rdubfor/rbrefu rdubfor/rwisp rdubfor/rwifor rdubfor/rwifu rdubsp/rdubfor rapsp/rbresp rapsp/rbrefor rapsp/rbrefu rapsp/rwisp rapsp/rwifor rapsp/rwifu rapsp/rdubsp rapsp/rdubfor rapfor/rbresp rapfor/ rbrefor rapfor/rbrefu rapfor/rwisp rapfor/rwifor rapfor/rwifu rdubsp/apfor rapfor/rdubfor rapsp/rapfor Noes: Average ( ) is he risk-minimizing hedge raio for wo crude oil asses. (w 12, ) is he porfolio weigh of wo asses a ime. 23

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

26 rbresp_rbrefor Figure 2 Forecass of he condiional correlaions beween pair of reurns from he and 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

27 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.35 rbresp_rapsp.44 rbrefor_rapsp

28 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