Modelling Dynamic Conditional Correlations in the Volatility of Spot and Forward Oil Price Returns

Transcription

1 Modelling Dynamic Condiional Coelaions in he Volailiy of Spo and Fowad Oil Pice Reuns Maeo Manea a, Michael McAlee b and Magheia Gasso c a Depamen of Saisics, Univesiy of Milan-Bicocca and FEEM, Milan, Ialy b School of Economics and Commece, Univesiy of Wesen Ausalia c FEEM, Milan, Ialy Absac: This pape esimaes he dynamic condiional coelaions in he euns on Tapis oil spo and onemonh fowad pices fo he peiod June 99 o 6 Januay 004, using ecenly developed mulivaiae condiional volailiy models, namely he Consan Condiional Coelaion Mulivaiae GARCH (CCC- MGARCH) model of Bolleslev [990], Veco Auoegessive Moving Aveage GARCH (VARMA- GARCH) model of Ling and McAlee [003], VARMA Asymmeic GARCH (VARMA-AGARCH) model of Chan e al. [00], and he Dynamic Condiional Coelaion (DCC) model of Engle [00]. The mulivaiae esimaes show ha he ARCH and GARCH effecs fo spo (fowad) euns ae significan in he condiional volailiy model fo spo (fowad) euns. Moeove, hee ae significan inedependences in he condiional volailiies beween he spo and fowad makes. The mulivaiae asymmeic effecs ae significan fo boh spo and fowad euns. The calculaed consan condiional coelaions beween he condiional volailiies of spo and fowad euns using CCC-GARCH(,), VAR()-GARCH(,) and VAR()-AGARCH(,) ae viually idenical. Finally, he esimaes of he wo DCC paamees ae saisically significan, which makes i clea ha he assumpion of consan condiional coelaion is no suppoed empiically. Keywods: Asymmeic effecs; Dynamic condiional coelaions; Mulivaiae GARCH models; Fowad pices and euns; Spo pices and euns. Pesening auho: Maeo Manea, Depamen of Saisics, Univesiy of Milano-Bicocca, Via Bicocca degli Acimboldi 8, 06 Milano, Ialy. Maeo.Manea@unimib.i. INTRODUCTION Spo and fowad pices of physical commodiies, including oil, have been invesigaed ove an exended peiod. Subsanial eseach has been undeaken o analyze he elaionship beween spo and fowad pices, and hei associaed euns. The efficien make hypohesis is cucial fo undesanding opimal decision making wih egad o hedging and speculaion, and also fo making financial decisions abou he opimal allocaion of pofolios of asses wih egad o hei mulivaiae euns and associaed isks. To dae, hee has been lile eseach egading an analysis of he volailiies (o isks) associaed wih pofolios of euns fo physical asses a he mulivaiae level. Such shocks o euns can be decomposed ino pedicable and unpedicable componens. The mos fequenly analysed pedicable componen in shocks o euns is he volailiy in he condiional vaiance. When hese condiional volailiies vay ove ime, GARCH models (see Engle [98] and Bolleslev [986]) may be used o capue dynamic cluseing behaviou. In he las wo decades, univaiae and mulivaiae GARCH models have become widely esablished in heoeical and empiical financial economics and economeics. The sucual and saisical popeies of hese models have been fully developed, and he compuaional equiemens ae geneally saighfowad. In modelling mulivaiae euns, such as on he spo and fowad pices of oil, he shocks o euns no only have dynamic inedependence in isks, bu also in he condiional coelaions. This is an exension of he consan (o saic) condiional coelaion appoach o analyzing mulivaiae isks associaed wih pofolios of asses. Thee ae seveal widely used oil makes, he mos well known of which ae Ben and WTI. Howeve WTI spo pices ae no available, so

2 ha i is no possible o es he unbiasedness o efficien make hypohesis fo his physical commodiy. I follows ha i is also no possible o deemine opimal hedging saegies based on whehe shocks o spo and fowad euns ae high and posiively o negaively coelaed. One epesenaive oil make fo ligh swee cudes in he Asia and Pacific egion, namely Tapis, has boh spo and fowad pices. Consequenly, i is possible o deemine whehe o hedge o no, based on deemining if he shocks o spo and fowad euns ae, in fac, high and eihe posiively o negaively coelaed. The pupose of his pape is o esimae he dynamic condiional coelaions in he euns on Tapis oil spo and one-monh fowad pices, using ecenly developed mulivaiae condiional volailiy models. The dynamic coelaions will enable a deeminaion of whehe he spo and fowad euns ae subsiues o complemens, which can be used o hedge agains coningencies. The plan of he pape is as follows. Secion discusses biefly he univaiae and mulivaiae GARCH models o be esimaed. Secion 3 descibes he daa and he empiical esimaes of he univaiae models, he mulivaiae models wih consan condiional coelaions, and he mulivaiae models wih dynamic condiional coelaions. Secion 4 povides some concluding commens.. ECONOMETRIC MODELS This secion pesens models of he volailiy in Tapis oil spo and fowad pices euns, namely he Consan Condiional Coelaion Mulivaiae GARCH (CCC-MGARCH) model of Bolleslev [990], Veco Auoegessive Moving Aveage GARCH (VARMA-GARCH) model of Ling and McAlee [003], VARMA Asymmeic GARCH (VARMA-AGARCH) model of Chan e al. [00], and he Dynamic Condiional Coelaion (DCC) model of Engle [00]. The specificaion, and sucual and saisical popeies, of hese models ae discussed biefly in his secion. Conside he following specificaion: ( ) y = E y F + ε ε = Dη, () whee = ( ), = ( ) y,..., y y m η,..., η η m is a sequence of independenly and idenically disibued (iid) andom vecos, F is infomaion / / D = diag h h, m available o ime, (,..., m ) is he numbe of euns, and =,,n. Bolleslev [990] assumed ha he condiional vaiance fo each eun, h, i =,,m, follows a univaiae GARCH pocess, ha is, h i s i = ωi + αεi, j + βhi, j j= j= () whee α epesens he ARCH effecs (o he sho-un pesisence of shocks o eun i) and β epesens he GARCH effecs (o he conibuion of shocks o eun i o long-un pesisence, namely α + β s j= j= ). CCC- MGARCH assumes independence of he condiional vaiances acoss euns and does no accommodae asymmeic behaviou. In ode o accommodae inedependence in he condiional vaiance, Ling and McAlee [003] poposed and esablished he sucual and saisical popeies fo: H W A B H s = + iε i + j j i= j= H,..., h h m whee ( ) (3) =, ε = ( ε,..., ε m ) and W, A i (i =,,) and B j (i =,,s) ae m m maices. VARMA-GARCH assumes ha negaive and posiive shocks have idenical impacs on he condiional vaiance. In ode o accommodae asymmeic effecs, Chan e al. [00] poposed and esablished he sucual and saisical popeies fo he VARMA-AGARCH specificaion: s H = W+ Aε + CI ε + B H i i i i i j j i= i= j= (4) whee C i ae m m maices fo i =,,, and I = diag I I, whee I i = 0 when ε i > ( ),..., m 0 and I i = when ε i < 0. If m =, equaion (4) educes o he asymmeic univaiae GARCH, o GJR, model of Glosen e al. [99]. Moeove, VARMA-AGARCH educes o VARMA- GARCH when C i = 0 fo all i. If C i = 0, wih A i and B j being diagonal maices fo all i, j, hen VARMA-AGARCH educes o CCC-MGARCH. The paamees of models ()-(4) ae obained by maximum likelihood esimaion (MLE) using a join nomal densiy. When η does no follow a, )

3 join mulivaiae nomal disibuion, he appopiae esimao is defined as he Quasi- MLE (QMLE). The condiional coelaion is assumed o be consan fo all hee models discussed above. Fom equaion (), i follows ha εε = D ηη D, E F = Ω = D Γ D. The so ha ( ε ε ) condiional coelaion maix is defined as Γ = D Ω D, whee Γ has ypical consan ρ ρ elemen = ji fo i, j =,,m and =,,n. When m = = s =, he necessay and sufficien condiion fo he exisence of he second momen of β ε, ha is ( ) E ε <, is α + <. This condiion is also sufficien fo ( ) he QMLE o be consisen and asympoically nomal. Jeanheau [998] showed ha he log- momen condiion, E ( α η ) β log + < 0, is sufficien fo he QMLE o be consisen fo GARCH(,), while Boussama [000] showed ha he QMLE is asympoically nomal fo GARCH(,) unde he same condiion. McAlee e al. [00] esablished he log-momen condiion ( (( α γ η ) )) η β fo GJR(,), namely, E I( ) log + + < 0 and showed ha i is sufficien fo consisency and asympoic nomaliy of he QMLE. Hence, he second momen condiion α + γ + β < is also sufficien fo consisency and asympoic nomaliy of he QMLE fo GJR(,) (see Ling and McAlee [00]). In empiical examples, he paamees ae eplaced by hei especive QMLE, η is eplaced by he esimaed sandadized esiduals fo =,,n, and expeced values ae eplaced by hei especive sample means. Unless η is a sequence of iid andom vecos, he assumpion of consan condiional coelaion is no valid. In ode o capue he dynamics of ime-vaying condiional coelaion, Γ, Engle [00] and Tse and Tsui [00] poposed he closely elaed Dynamic Condiional Coelaion (DCC) and he Vaiable Condiional Coelaion Mulivaiae GARCH models, especively. The DCC model is given as ( ) Γ = θ θ Γ + θ η η + θ Γ, (5) in which θ and θ ae scala paamees o capue he effecs of pevious sandadized shocks and dynamic condiional coelaions on cuen dynamic condiional coelaions, especively. Chan e al. [003] poposed he Genealized Auoegessive Condiional Coelaion (GARCC) model, which conains boh DCC and VCC-MGARCH as special cases, and esablished he sucual and saisical popeies of GARCC. They showed ha, if η follows an auoegessive pocess wih sochasic coefficiens ahe han being a sequence of iid andom vecos, model ()-() is equivalen o Engle s [00] DCC model in (5). 3. DATA AND EMPIRICAL ESTIMATES The univaiae and mulivaiae GARCH models ae esimaed using daa on spo and fowad euns fo he peiod June 99 o 6 Januay 004. Figue shows he euns o he spo and fowad pices, fo which he coelaion coefficien is I is clea fom Figue ha hee is subsanial cluseing of euns, and hence also in he volailiies. The univaiae esimaes of he condiional volailiies based on he spo and fowad euns ae given in Tables and. The hee enies fo each paamee ae hei especive esimaes, asympoic -aios and Bolleslev and Wooldidge [99] obus -aios. The esuls in Table ae used o esimae he CCC model of Bolleslev [990] and he DCC model of Engle [00]. Boh he ARCH and GARCH esimaes ae significan fo spo and fowad euns. Alhough he second momen condiion is no saisfied, he log-momen condiion is saisfied, so ha he QMLE ae consisen and asympoically nomal. The univaiae GJR esimaes in Table ae easonably simila o he coesponding esimaes in Table. The esimaes of he asymmeic effec a he univaiae level ae no saisically significan fo eihe spo o fowad euns. Moeove, he obus -aios exceed he asympoic counepa in 6 of 8 cases. As in Table, he second momen condiion is no saisfied fo eihe spo o fowad euns, bu he log-momen condiion is saisfied, so ha he QMLE ae consisen and asympoically nomal. Coesponding mulivaiae esimaes fo he VAR()-GARCH(,) and VAR()-AGARCH(,) models ae given in Tables 3 and 4, especively. The ARCH and GARCH effecs fo spo (fowad) euns ae significan in he condiional volailiy model fo spo (fowad) euns. I is also clea fom Table 3 ha hee ae significan inedependences in he condiional volailiies beween he spo and fowad makes, specifically he fowad GARCH effec is 3

4 significan fo spo euns, while boh he ARCH and GARCH spo effecs ae significan fo fowad euns. The esuls in Table 4 mio hose in Table 3, bu moe significanly. In paicula, he ARCH and GARCH effecs fo spo (fowad) euns ae significan in he condiional volailiy model fo spo (fowad) euns. Thee ae also significan inedependences in he condiional volailiies beween he spo and fowad makes, specifically he fowad (spo) ARCH and GARCH effecs ae significan fo spo (fowad) euns. As compaed wih he insignifican asymmeic effec of he univaiae esimaes in Table, he mulivaiae asymmeic effecs in Table 4 ae significan fo boh spo and fowad euns. Oveall he mulivaiae VAR()- AGARCH(,) esuls in Table 4 dominae hose in Tables -3. Consan condiional coelaions beween he condiional volailiies of spo and fowad euns using hee mulivaiae GARCH models, namely CCC-GARCH(,), VAR()-GARCH(,) and VAR()-AGARCH(,), ae given in Table 5. The wo enies fo each paamee ae hei especive esimaes and asympoic -aios. In spie of he esimaes in Tables, 3 and 4 having diffeen saisical implicaions, he consan condiional coelaions fo he hee models in Table 5 ae viually idenical a Finally, he DCC-GARCH(,) esimaes ae given in Table 6. As he hee models in Table 5 yield vey simila esimaes of he consan condiional coelaion, he DCC esimaes in Table 6 ae based only on he CCC model. The esimaes of he wo DCC paamees ae saisically significan, which makes i clea ha he assumpion of consan condiional coelaion is no suppoed empiically. This is highlighed by he dynamic condiional coelaions beween spo and fowad euns in Figue, fo which he mean, a 0.933, is viually idenical o he consan condiional coelaion epoed in Table 5. The dynamic condiional coelaions ae in he ange (0.47, 0.993), signifying medium o exeme inedependence. Moeove, he skewness and kuosis of he dynamic condiional coelaion indicae a song negaively skewed disibuion. In summay, he dynamic volailiies in he euns in Tapis oil spo and fowad makes ae geneally inedependen ove ime, some imes vey songly. 4. CONCLUSION The pupose of his pape was o esimae he dynamic condiional coelaions in he euns on Tapis oil spo and one-monh fowad pices fo he peiod June 99 o 6 Januay 004, using ecenly developed mulivaiae condiional volailiy models. The mulivaiae esimaes showed ha he ARCH and GARCH effecs fo spo (fowad) euns wee significan in he condiional volailiy model fo spo (fowad) euns. Moeove, hee wee significan inedependences in he condiional volailiies beween he spo and fowad makes. As compaed wih he insignifican asymmeic effec of he univaiae esimaes, he mulivaiae asymmeic effecs wee significan fo boh spo and fowad euns. The calculaed consan condiional coelaions beween he condiional volailiies of spo and fowad euns using CCC-GARCH(,), VAR()-GARCH(,) and VAR()-AGARCH(,) ae viually idenical. Finally, he esimaes of he wo DCC paamees wee saisically significan, which makes i clea ha he assumpion of consan condiional coelaion was no suppoed empiically. The dynamic volailiies in he euns in Tapis oil spo and fowad makes wee geneally inedependen ove ime. These findings sugges ha a sensible hedging saegy would conside spo and fowad makes as being chaaceized by diffeen degees of subsiuabiliy. 5. ACKNOWLEDGEMENTS The auhos wish o hank Felix Chan, Alessando Lanza, and semina paicipans a he Fondazione Eni Enico Maei (FEEM) fo helpful commens. Tom Doan kindly povided a bea es vesion of RATS 6 fo he mulivaiae GARCH models. The second auho is mos gaeful fo he hospialiy of FEEM and he financial suppo of he Ausalian Reseach Council. 6 REFERENCES Bolleslev, T., Genealised auoegessive condiional heeoscedasiciy, Jounal of Economeics, 3, , 986. Bolleslev, T., Modelling he coheence in shoun nominal exchange aes: a mulivaiae genealized ARCH appoach, Review of Economics and Saisics, 7, , 990. Bolleslev, T. and J.M. Wooldidge, Quasimaximum likelihood esimaion and infeence in dynamic models wih ime-vaying covaiances, Economeic Reviews,, 43-7, 99. Boussama, F., Asympoic nomaliy fo he quasi-maximum likelihood esimao of a GARCH model, Compes Rendus de 4

5 l Academie des Sciences, Seie I, 33, 8-84 (in Fench), 000. Chan, F., S. Hoi and M. McAlee, Sucue and asympoic heoy fo mulivaiae asymmeic volailiy: empiical evidence fo couny isk aings, pape pesened o he 00 Ausalasian Meeing of he Economeic Sociey, Bisbane, Ausalia, July 00. Chan, F., S. Hoi and M. McAlee, Genealized auoegessive condiional coelaion, unpublished pape, School of Economics and Commece, Univesiy of Wesen Ausalia, 003. Engle, R.F., Auoegessive condiional heeoscedasiciy wih esimaes of he vaiance of Unied Kingdom inflaion, Economeica, 50, , 98. Engle, R.F., Dynamic condiional coelaion: a new simple class of mulivaiae GARCH models, Jounal of Business and Economic Saisics, 0, , 00. Glosen L., R. Jagannahan and D. Runkle, On he elaion beween he expeced value and volailiy of nominal excess eun on sock, Jounal of Finance, 46, , 99. Jeanheau, T., Song consisency of esimaos fo mulivaiae ARCH models, Economeic Theoy, 4, 70-86, 998. Ling, S. and M. McAlee, Saionaiy and he exisence of momens of a family of GARCH pocesses, Jounal of Economeics, 06, 09-7, 00. Ling, S. and M. McAlee, Asympoic heoy fo a veco ARMA-GARCH model, Economeic Theoy, 9, , 003. McAlee, M., F. Chan and D. Mainova, An economeic analysis of asymmeic volailiy: Theoy and applicaion o paens, pape pesened o he Ausalasian meeing of he Economeic Sociey, Bisbane, Ausalia, July 00, o appea in Jounal of Economeics. Tse, Y.K. and A.K.C. Tsui, A mulivaiae genealized auoegessive condiional heeoscedasiciy model wih ime-vaying coelaions, Jounal of Business and Economic Saisics, 0, 35-36, SPOT FORWARD Coelaion Coefficien beween Spo and Fowad Reuns = Figue. Reuns o Spo and Fowad Pices fo Tapis, June 99 6 Januay Mean = Min. = 0.47 Max. = S.D. = Skew. = -.49 Ku. =.474 S_F Figue. Dynamic Condiional Coelaions Beween Spo and Fowad Reuns 5

6 Table. Univaiae AR()-GARCH(,) Esimaes Reuns ω α β Log-momen Second momen Spo 3.5E Fowad 3.7E Noe: The hee enies fo each paamee ae hei especive esimaes, asympoic -aios and Bolleslev- Wooldidge (99) obus -aios. Table. Univaiae AR()-GJR(,) Esimaes Reuns ω α γ β α+γ/ Log momen Second momen Spo.95E Fowad 3.47E Noe: The hee enies fo each paamee ae hei especive esimaes, asympoic -aios and Bolleslev- Wooldidge (99) obus -aios. Table 3. VAR() GARCH(,) Esimaes Reuns ω α s β s α f β f Spo 4.67E Fowad 5.64E Noe: The wo enies fo each paamee ae hei especive esimaes and asympoic -aios. Table 4. VAR() AGARCH(,) Esimaes Reuns ω α s γ s β s α f γ f β f Spo 3.37E Fowad 3.55E Noe: The wo enies fo each paamee ae hei especive esimaes and asympoic -aios. Table 5. Consan Condiional Coelaions beween Table 6. DCC-GARCH(,) Esimaes Spo and Fowad Reuns Model ρ Model θ θ ' CCC-GARCH(,) Γ = ( θ θ ) Γ + θη η + θγ VAR()-GARCH(,) Noe: The wo enies fo each paamee ae hei especive esimaes and asympoic -aios. VAR()-AGARCH(,) Noe: The wo enies fo each paamee ae hei especive esimaes and asympoic -aios. 6