A Markov Regime Switching Approach for Hedging Energy Commodities

Transcription

1 A Markov Regime Swiching Approach for Hedging Energy Commodiies Amir H. Alizadeh, Nikos K. Nomikos and Panos K. Pouliasis Faculy of Finance Cass Business School London ECY 8TZ Unied Kingdom and ABSTRACT In his paper we employ a Markov Regime Swiching (MRS) approach for deermining ime-varying minimum variance hedge raio in energy fuures markes. The hedging effeciveness of New York Mercanile Exchange (NYMEX) peroleum fuures conracs is examined using univariae MRS and bivariae MRS VAR wih GARCH error srucure. The raionale behind he use of MRS models sems from he fac ha he dynamic relaionship beween spo and fuures reurns may be characerized by regime shifs, which, in urn, suggess ha by allowing he hedge raio o be sae dependen upon he sae of he marke, one can obain more efficien hedge raios and hence, superior hedging performance compared o oher mehods in he lieraure. Regime swiching in GARCH processes reduces volailiy persisence and improves he forecas abiliy. The performance of he MRS hedge raios is compared o ha of alernaive models such as GARCH, Error Correcion and OLS in he Wes Texas Inermediae Crude oil, Unleaded Gasoline and Heaing oil markes. In and ou-of-sample ess indicae ha MRS hedge raios ouperform he oher mehods in reducing porfolio risk in he peroleum producs markes. In he crude oil marke, he MRS models ouperform he oher hedging sraegies only wihin sample. Overall, he resuls indicae ha by using MRS models marke agens may be able o increase he performance of heir hedges, measured in erms of variance reducion and increase in uiliy.

2 INTRODUCTION Marke paricipans in all financial and commodiy markes operae in an environmen subjec o some degree of variabiliy. The energy commodiies group, prone o large price flucuaions and uncerainy in boh he physical and he financial marke, aracs he ineres of his paper. I was no unil afer he second 970 s oil price crisis ha oil derivaive conracs were inroduced and hereafer developed widely. Since hen, oil price risk managemen became an ineviable challenging ask because of he global naure of oil and is implicaions in he inernaional poliical arena. The primary crude oil disillaes, gasoline, heaing oil, aviaion fuel and fuel oil are indispensable for ransporaion, indusrial and residenial uses. As a resul, crude oil is he world s mos acively raded commodiy. Physical oil rade movemens in he year 004 reached 48. million barrels per day beween expor and impor regions compared o 3.44 million barrels per day in he year 990. Oil producs accouned for 0.96 million barrels per day of he rade figure whereas he laer increased by.38 million barrels per day in 004 for iner-area movemens. Global economic and poliical aciviy has proven o play a crucial role in he sabiliy of oil prices, driving he marke o relaively high levels of volailiy. Even hough recen echnological advances enhanced he developmen of alernaive energy sources, oil sill delivers superior efficiency of use and hus, indusries experience large amoun of risk. Nowadays here are wo major exchanges providing oil derivaive conracs, he New York Mercanile Exchange (NYMEX) and he Inerconinenal Exchange (ICE), formerly Inernaional Peroleum Exchange (IPE). Also, since 999, anoher markeplace for rading oil-relaed conracs (crude oil, gas oil, gasoline and kerosene fuures), is Tokyo Commodiy Exchange (TOCOM). In 004 more han 78 million fuures conracs were raded on NYMEX, of which 5.88,.78 and.88 million reflec he individual raded volumes of crude oil, gasoline and heaing oil, respecively. Derivaive markes allow marke agens o minimize heir exposure o risk by reducing he variance of heir porfolio; hence risk managemen ools and heir effeciveness in erms of hedging are of umos imporance. In his line, i is essenial o evaluae Briish Peroleum Saisical Review 004

3 differen approaches ha ha are employed o consruc efficien and reliable hedging sraegies. The hedge raio i.e. he raio of fuures conracs o buy or sell for each uni of he underlying asse on which he hedger bears risk, is one of he ools ha are used o manage poenial adverse effecs of price changes in he physical marke. Earlier sudies in he lieraure (Johnson, 960; Sein, 96; Ederingon, 979) derive hedge raios ha minimize he variance of he hedged porfolio, based on porfolio heory. Le S and F represen he price changes in spo and fuures prices beween period and -. Then, he minimum-variance hedge raio is he raio of he uncondiional covariance beween cash and fuures price changes over he variance of fuures price changes; his is equivalen o he slope coefficien, γ, in he following regression: S = γ + γ F u u ~ iid(0, ) () 0 + σ Wihin his specificaion, he esimaed R of Equaion () represens he hedging effeciveness of he minimum variance hedge. Empirical sudies ha have employed he above mehodology o esimae hedge raios and measures of hedging effeciveness include: for T-Bill fuures Ederingon (979) and Franckle (980); for oil fuures, Chen e al. (987); for sock indices, Figlewski (984) and Lindahl (99); for currencies Grammaikos and Saunders (983). However, he implici assumpion of Equaion () ha he risk in spo and fuures markes and hus he opimal hedge raio are consan over ime, does no ake ino accoun he fac ha since many asse prices follow imevarying disribuions, he minimum variance hedge raio should be ime-varying (Myers and Thompson, 989; Kroner and Sulan, 993). This, in urn raises concerns regarding he risk reducion properies of hedge raios generaed from Equaion (). To address his issue, a number of sudies apply mulivariae GARCH (Generalised Auoregressive Condiional Heeroscedasiciy) (Engle & Kroner, 995) models and derive ime-varying hedge raios direcly from he second momens (variances of fuures and spo price changes and heir covariance). Examples in he lieraure include: for currency fuures Kroner and Sulan (993); for sock index fuures Park and Swizer (995); for ineres rae fuures Gagnon and Lypny (995), for freigh fuures Kavussanos and Nomikos (000); for corn fuures Moschini and Myers (00); for 3

4 elecriciy fuures Bysrom (003); for peroleum fuures Alizadeh e al. (004). The consensus from hese sudies is ha GARCH-based hedge raios change as new informaion arrives o he marke and on average end o ouperform, in erms of risk reducion, consan hedge raios derived from Equaion (). However, hese gains are marke specific and vary across differen conracs while, occasionally, he benefis in erms of risk reducion seem o be minimal (Lien and Tse, 00). The raionale behind he use of he GARCH models lies in he fac ha asse reurns end o exhibi volailiy clusering; in oher words, large (small) price changes end o be followed by large (small) price changes (Mandelbro, 963). This paern of volailiy behaviour suggess ha alhough acual price changes migh be uncorrelaed, he condiional second momens could be ime dependen. The mos widely used approaches for modelling ime-varying volailiy are he GARCH family models. Empirically, a common feaure of GARCH models is ha hey end o impue a high degree of persisence o he condiional volailiy. This means ha shocks o he condiional variance ha occurred in he disan pas coninue o have a nonrivial impac in he curren esimae of volailiy. Lamoureux and Lasrapes (990) associae hese high levels of volailiy persisence wih srucural breaks or regime shifs in he volailiy process. They demonsrae his by inroducing deerminisic shifs in he condiional variance equaion and find ha his leads o a marked reducion in he degree of volailiy persisence, compared o ha implied by he GARCH models. As an alernaive o GARCH models, Wilson e al. (996) employ an ieraive cumulaive sums-of-squares (ICSS) mehodology and show evidence of sudden changes in he uncondiional volailiy of oil fuures conracs. Wih daa covering he period 984 o 99, hree major volailiy shifs are deeced and he reasons are aribued o he naure and magniude of exogenous shocks (OPEC policy, Iran-Iraq conflic, Gulf War and exreme weaher condiions). By accouning hese shifs in he ARCH framework hey find similar conclusions wih Lamoureux and Lasrapes (990). Fong and See (00; 003) also repor significan regime shifs in he condiional volailiy of crude oil fuures conracs, which dominae he GARCH effecs. In addiion, hey find ha in a high variance regime a negaive basis is more likely o increase he regime persisence han a posiive basis and associae volailiy regimes wih specific marke evens. The exisence of regime shifs in he relaionship beween spo and fuures reurns is also demonsraed by Sarno and Valene (000) who show ha regime swiching models 4

5 explain he relaionship beween spo and fuures prices beer han simple linear models in he FTSE-00 and S&P 500 sock index fuures markes. The evidence presened above suggess ha by allowing he volailiy o swich sochasically beween differen processes under differen marke condiions, one may obain more robus esimaes of he condiional second momens and, as a resul, more efficien hedge raios compared o he mehods which are currenly being employed, such as GARCH models or OLS. Wheher his is he case is an issue ha is examined empirically in his paper for he crude oil, gasoline and heaing oil fuures conracs, raded on NYMEX. Alizadeh and Nomikos (004) examined he hedging effeciveness of FTSE-00 and S&P 500 sock indices, aking ino accoun regime shifs in he sae of he marke, by inroducing Markov Swiching Models for he esimaion of dynamic hedge raios. Allowing Equaion () o swich beween wo sae processes, hey provided evidence in favour of hose models in erms of variance reducion and increase in uiliy, boh inand ou-of-sample. Following Gray (996), Lee and Yoder (006) exend he mulivariae MRS model, o a sae dependen bivariae GARCH model. They apply heir model in he corn and nickel fuures markes and hey repor higher, ye insignifican, variance reducion compared o OLS and he single-regime GARCH hedging sraegy. Therefore by invesigaing he hedging effeciveness of Markov regime swiching models we conribue o he exising lieraure in a number of ways. Firs, NYMEX oil fuures are used o generae hedge raios ha are regime dependen and change as marke condiions change. Second, along wih simple MRS univariae models, we exend his approach o a bivariae Regime Swiching VAR model wih GARCH error srucure. The regimes of he models are reaed as laen variables since hey are esimaed along wih he oher parameers of he model using maximum likelihood echniques. Third, we evaluae he hedging effeciveness of hese models in he Unied Saes energy marke, using boh in- and ou-of-sample ess. The ou-of-sample ess, in paricular, are performed by forecasing he regime probabiliies using he esimaed ransiion probabiliy marices, and calculaing hedge raios based on hese forecass. Finally, he performance of he regime swiching hedge raios is compared o ha of 5

6 alernaive hedge raios generaed from a variey of models ha have been proposed in he lieraure such as GARCH and error-correcion models. This way we provide robus evidence on he performance of he proposed hedging sraegy. Our paper is differen from he Lee and Yoder (006) Regime Swiching - BEKK sudy in he sense ha for more parsimonious represenaion we employ he diagonal BEKK parameerizaion of variance covariance marix reducing he compuaional burden and number of parameers o be esimaed. Furhermore we allow for lagged cross erms in he mean equaion in order o capure he ineracions of spo- fuures prices, since exclusion of any relevan explanaory variables from he mean equaion could increase he variance of he error erm, hreaening he GARCH resuls. A las, we employ a baery of alernaive models, o provide evidence for he robusness of MRS hedge raios. The srucure of his paper is as follows. We nex presen he minimum-variance hedge raio mehodology and illusrae he univariae MRS models used in his sudy. We hen presen he MRS-BEKK model esimaion procedure. Daa and heir properies wih our empirical resuls are repored and discussed. This is followed by an evaluaion of he hedging effeciveness of he proposed sraegies; conclusions are given in he las secion. MARKOV REGIME SWITCHING MODELS AND HEDGING Marke paricipans in fuures markes choose a hedging sraegy ha reflecs heir individual goals and aiudes owards risk. The degree of hedging effeciveness in fuures markes depends on he relaive variaion of spo and fuures price changes as well as he hedge raio, ha is he raio of fuures conracs o buy or sell for each uni of he underlying asse. The hedge raio ha minimises he variance of he hedge porfolio is derived as he slope coefficien of spo price changes on fuures price changes, as in Equaion (). This can also be expressed as: 6

7 γ = Cov( S, F ) Var( F ) () Therefore, he minimum variance hedge raio of Equaion () is he raio of he uncondiional covariance beween cash and fuures price changes over he variance of he fuures price changes. Equaion () can also be exended o accommodae he condiional minimum-variance hedge raio, γ,, which is he ime varying equivalen of he convenional hedge raio γ, in Equaion (). This is believed o be more efficien in reducing he risk of a hedged posiion, compared o he convenional hedge raio, because i is updaed as i responds o he arrival of new informaion in he marke. To esimae such a dynamic hedge raio, he second momens of spo and fuures reurns in Equaion () are condiioned on he informaion se available a ime -, primarily using mulivariae GARCH models. Sarno and Valene (000) provide a furher dimension o he lieraure by showing ha changes in marke condiions may affec he relaionship of spo and fuures prices. Using a mulivariae exension of he Markov Swiching Model (MRS) proposed by Hamilon (989) and Krolzig (999), hey find ha he relaionship beween spo and fuures reurns in he S&P 500 and FTSE-00 marke is regime dependen. This in urn suggess ha shifs in he spo-fuures relaionship may have an impac on he magniude of he hedge raio and consequenly, on he hedging effeciveness of he fuures marke. Alizadeh and Nomikos (004) allow for changes in he marke condiions o affec he hedge raios. They exend Equaion () o a wo-sae MRS model in order o allow for swiches beween wo differen processes, dicaed by he sae of he marke. This can be shown mahemaically as: S = γ + γ F + ε ε ~ iid(0, σ ) (3) 0, s, s, s ε, s I can be shown ha if expeced fuures reurns are zero, i.e. if fuures follow a maringale process E (F + ) = F hen, he minimum variance hedge raio of Eq. () is equivalen o he uiliy-maximizing hedge raio. A proof of his resul is available a Benninga e al. (984) and Kroner and Sulan (993). The maringale assumpion of fuures reurns implies ha he expeced reurns from he hedged porfolio are unaffeced by he number of fuures conracs held, so ha risk minimizaion becomes equivalen o uiliy maximizaion. The assumpion of zero expeced reurns is also in line wih he descripive saisics presened in Table I, which show ha he uncondiional fuures reurns have a mean of zero. 7

8 where, s = {, } indicaes he sae in which he marke is in. The link beween he wo saes of he marke in Equaion (3) is provided hrough a firs-order Markov process wih he following ransiion probabiliies: Pr(s Pr(s = s = s - - = ) = P = ) = P Pr(s Pr(s = s = s - - = ) = P = ) = P = (- P = (- P ) ) (4) where he ransiion probabiliy P gives he probabiliy ha sae will be followed by sae, and he ransiion probabiliy P gives he probabiliy ha sae will be followed by sae. Transiion probabiliies P and P give he probabiliies ha here will be no change in he sae of he marke in he following period. These ransiion probabiliies are assumed o remain consan beween successive periods and can be esimaed along wih he oher parameers of he model. Once he densiy funcions for each sae of he marke and probabiliies of being in respecive saes are defined, he likelihood funcion for he enire sample is formed by eliminaing he unobserved erm s, and summing up he possible values of i. The corresponding log-likelihood is consruced as: T L( θ) = log = π ( S exp γ γ F ) + ( S exp γ, 0,,, 0, πσ σ,, πσ, π σ γ,, F ) (5) where θ = (γ 0,s, γ,s, σ ), s s =, is he vecor of parameers o be esimaed 3 and π,, π, are he probabiliies of he regime being in sae or, respecively. L(θ) can be 3 Some sudies argue ha consan ransiion probabiliies and/or variances are resricive assumpions. Sudies such as Diebold e al. (994), Marsh (000), Fong and See (00;003) and Alizadeh and Nomikos (004) condiion ransiion probabiliies on observable variables ha are par of he informaion se e.g. he basis. Moreover, Perez-Quiros and Timmerman (000) condiion he variances of sock reurns on he lagged levels of reasury bills. Alizadeh and Nomikos (004) use he lagged average basis, o condiion he variances. Oher sudies such as Hamilon and Susmel (994), Gray (996) and Dueker (997) use ARCH and GARCH specificaions o model he dynamics of volailiies. In his sudy, we use consan ransiion probabiliies, alhough he models can easily be exended o allow for a ime-varying ransiion probabiliy marix P. As for he variances, he focus is on he bivariae MRS GARCH case. The univariae MRS is used as benchmark, hus variances are no condiioned on he available informaion se and considered consan wihin each regime. However, wo alernaive models were esimaed as in Alizadeh and Nomikos (004): a) an MRS model in which he ransiion probabiliies are condiioned on he lagged basis and b) an MRS model in which boh he ransiion probabiliies and variances are condiioned on he lagged basis. These resuls are no presened here and are available from he auhors. 8

9 maximised using numerical opimizaion mehods, subjec o he consrains ha π, + π, = and 0 π,, π,. Esimaing Equaion (3), using he MRS specificaion oulined above, yields wo hedge raios, γ, and γ,, which represen he minimum variance hedge raios, given he sae of he marke. In fac, hese wo hedge raios can be considered as he upper and lower bounds of he opimal hedge raio. Since he probabiliy of he marke being in sae or a any poin in ime is given by π, and π, = - π,, where 0 π, and 0 π,, he opimal hedge raio a any poin in ime can be deermined as he weighed average of he wo esimaed hedge raios, weighed according o heir respecive probabiliies. Hence, he opimal hedge raio a ime will be dependen on he probabiliy of he marke being in sae or and can be expressed as: γ * = π γ + ( π ) γ (6),,,, Esimaing he opimal hedge raio using he Markov Regime Swiching model oulined above allows for shifs in he mean and volailiy processes and recognizes any changes in he relaionships beween hem. This ensures a beer esimae of opimal hedge raio compared o OLS or GARCH models as he firs esimaes a consan hedge raio and he second esimaes a hedge raio which is ime-varying bu mainly auoregressive in naure. 3 MARKOV REGIME SWITCHING ARCH MODELS AND HEDGING An alernaive way o esimae he opimum hedge raio would be o use Equaion (). Condiional second momens of spo and fuures reurns are measured by he family of ARCH models, inroduced by Engle (98). For his purpose we employ a VAR model for he condiional means of spo and fuures reurns wih a mulivariae GARCH error 9

10 srucure. Allowing for regime shifs in he inercep erm 4, he condiional means of spo and fuures reurns are specified using he following VAR : ε p S,, s X = µ s + Γ i X + ε,s ; ε,s = Ω i= ε F,, s ~ IN(0, H,s ) (7) where X ( F ) = S is he vecor of spo and fuures reurns, Γ i is a x coefficien marix measuring changes in X and ε,s ( ) = ε S,, s ε F,, s is a vecor of Gaussian whie noise processes wih ime varying sae dependen covariance marix H,s. The unobserved sae variable s follows a wo-sae, firs order Markov process wih consan ransiion probabiliies, as hey are specified in Equaion (4). The condiional second momens of spo and fuures reurns are specified as a GARCH(,) model (Bollerslev,986). However, in he regime-swiching framework, he GARCH model in is basic form would be inracable because boh, condiional variance and condiional covariance would be a funcion of all pas informaion, raher han a funcion of he curren regime alone. This pah-dependency problem would require he inegraion of an exponenially increasing number of regime pahs, a each sep, delivering an infeasible model o esimae. Hamilon and Susmel (994) and Cai (994) solve he pah dependency problem by eliminaing he GARCH erm. The main drawback of heir model is ha many lags of ARCH erms are needed in order o capure he dynamics of volailiies. Gray (996) suggess a possible formulaion for he condiional variance process by using he condiional expecaion of he variance. Lee and Yoder (006) exend Gray s model o he bivariae case and fully solve he pah dependency problem by developing a similar collapsing procedure for he covariance. Following augmened Baba e al. (987) (henceforh BEKK) represenaion (see Engle and Kroner, 995), he GARCH-like formulaion of he variance/covariance marix can be wrien as: 4 In his paper we allow for regime shifs only in he inercep erm of he VAR sysem in Equaion (7). Furher exension of his model o allow swiching in he auoregressive erms is sraighforward. In addiion o his model, we also esimae models allow for swiching in all he parameers of he mean equaion, or even include he error correcion erm as esimaed following Johansen procedure (988). However, he above model seleced as he more parsimonious, overcoming convergence problems, whereas he gains of is exension were minimal and is sill an under research area. 0

11 H, s C scs + A sε ε As + Β sh Β s = (8) for s = {,}, where, C s is a x lower riangular marix of sae dependen coefficiens, A s and B s are x sae dependen coefficien marices resriced o be diagonal 5, wih α ii,s + β ii,s <, i=,, for saionariy wihin each regime. This formulaion, guaranees H,s o be posiive definie for all and, in conras o he consan correlaion model of Bollerslev (986) i allows he condiional covariance of spo and fuures reurns o change signs over ime 6. Moreover, in his diagonal represenaion he sae dependen condiional variances are a funcion of he lagged values of boh he lagged aggregaed variances and aggregaed error erms (afer inegraing he unobserved sae variable). Similarly, he sae dependen condiional covariance is a funcion of lagged aggregaed covariance and lagged cross producs of he aggregaed error erms. Gray s (996) inegraing mehod of he sae dependen variances as applied for boh spo-fuures reurns can be expressed as: h [ X Ω ] Ε[ X Ω ] = Ε π = [ π µ + ( π ) µ ], ( µ, + h, ) + ( π, )( µ, + h, ),,,, (9) where X represens he dependen variable, fuures or spo reurns, h is he aggregae variance, µ s, he sae dependen mean equaion, π s, he regime probabiliies as defined in Equaion (4) and h s, he sae dependen variances, for s = {,}. Similarly, he collapsing procedure for he sae dependen residuals can be wrien as: ε [ X Ω ] = X [ π µ + ( π µ ] = (0) X Ε,,, ), Addiionally, in a bivariae model under he formulaion of Equaion (8) he pah dependency problem mus be solved for he covariance as well. We follow he mehod 5 Coefficiens marices A and B are resriced o be diagonal for a more parsimonious represenaion of he condiional variance (see Bollerslev e al. 994). 6 For a discussion of he properies of his model and alernaive mulivariae represenaions of he condiional variance marix see Bollerslev e al. (994) and Engle and Kroner (995).

12 proposed by Lee and Yoder (006) o inegrae he regime pahs a each sep. The collapsing procedure for he covariance is specified as: h π sf,, = Ε [ S, F Ω ] Ε [ S Ω ] Ε [ F Ω ] = () s f sf s f sf s s f f [ µ µ + h ] + ( π )[ µ µ + h ] [ π µ + ( π ) µ ] [ π µ + ( π ) µ ],,,,,,,,,,,,,,, where h sf, is he aggregae covariance, µ and s s, f µ s, are he mean equaions for spo and fuures reurns, respecively and h, he sae dependen covariances for s = {,}. sf s Under he specificaions of Equaions (8), (9), (0) and () he Markov Regime Swiching BEKK model becomes pah-independen because variance/covariance marix depends on he curren regime alone and no on he enire hisory. Consequenly, he Markov propery for a firs order Markov process is no violaed and we can allow for he condiional means of spo and fuures a mulivariae GARCH error srucure. Finally, assuming ha he sae dependen residuals follow a mulivariae normal disribuion wih mean zero and ime varying sae dependen covariance marix H.s. he densiy funcion for each regime (sae of he marke) can be wrien as follows: s ; θ) = H π exp ε ( f X,s,sH,sε, s ; s = {,} () where θ = α, b, γ, α, b, γ, c, c, c, a, a, β, β, P, ) is he ( S, s S, S, F, s F, F,, s, s, s, s, s, s, s P vecor of parameers o be esimaed, ε,s and H,s are defined in Equaions (7) and (8), respecively. Once he densiy funcions for each sae of he marke and probabiliies of being in respecive saes are defined, he likelihood funcion for he enire sample is formed in he same way as in he univariae case, by a mixure of he probabiliy disribuion of he sae variable and he densiy funcion for each regime as follows: π, π ( X = + ; θ) H, f, exp ε,h,ε, H, exp ε,h,ε, (3) π π

13 The log-likelihood of he above densiy funcion can hen be defined as: T L ( θ) = log f ( X ; θ) (4) = L(θ) can be maximised using numerical opimizaion mehods, subjec o he consrains ha π, + π, = and 0 π,, π,. Esimaing Equaions (7), (8), (9), (0) and () using he MRS specificaions oulined above, he second momens of spo and fuures reurns are condiioned on he informaion se available a ime -. Based on Equaion () he esimaed hedge raio a ime, given all he available informaion up o - can be wrien as: Cov( S, ) * F Ω γ Ω = (5) Var( F Ω ) where Cov( S, F ) and Var( F ) are calculaed from he collapsing Ω Ω procedure, as presened in Equaions () and (9), respecively. Esimaing he opimal hedge raio using he Markov Regime Swiching BEKK model oulined above furher allows for srucural changes in he GARCH processes and overcomes some of he limiaions ha radiional GARCH models exhibi. Firs, by allowing he volailiy equaion o swich across differen saes, we relax he assumpion of consan parameers hroughou he esimaion period improving he fi of he daa. Second, accouning for regime swiching, he high volailiy persisence imposed by single regime models decreases and he forecasing performance is expeced o be beer (see for example Lamoureux and Lasrapes, 990; Cai, 994 and Dueker, 997). Moreover, he esimaed hedge raio is dependen on he sae of he marke and is Markovian formulaion abolishes he auoregressive naure of GARCH hedge raios. Consequenly, one expecs MRS hedge raios esimaed by he variance/covariance marix o ouperform he convenional hedging sraegies. 3

14 4 DESCRIPTION OF THE DATA AND PRELIMINARY ANALYSIS The daa se for his sudy comprises of weekly spo and fuures prices for hree energy commodiies: WTI crude oil, Unleaded Gasoline and Heaing oil, covering he period January 3, 99 o July 8, 004, resuling 706 weekly observaions. Spo and fuures prices are Wednesday prices; when a holiday occurs on Wednesday, Tuesday s observaion is used in is place. The above daase was obained from CRB-Infoech CD and Daasream along wih volume and open ineres daa. Daa for he period January 3, 99 o July 30, 003 (654 observaions) are used for he in-sample analysis; ou-ofsample analysis is carried ou using he remaining daa from he period Augaus 6, 003 o July 8, 004 (5 observaions). All he commodiies under sudy are raded on he New York Mercanile Exchange (NYMEX). WTI conracs are raded for all deliveries wihin he nex 30 consecuive monhs as well as for specific long-daed deliveries such as 36, 48, 7 and 84 monhs from delivery. Each conrac is raded unil he close of business on he hird business day prior o he 5h calendar day of he monh preceding he delivery monh. If he 5h calendar day of he monh is a non-business day, rading shall cease on he hird business day prior o he business day preceding he 5h calendar day. Unleaded Gasoline conracs are raded for all deliveries up o consecuive monhs; NYMEX Heaing oil conracs are raded for all deliveries wihin he nex 8 monhs. Each conrac of eiher Gasoline or Heaing oil is erminaed he las business day of he monh preceding he delivery monh. WTI conrac is quoed in US dollars per barrel (US$/bbl) whereas he wo crude producs are quoed in US dollar per US gallon. The size of each conrac is 4,000 gallons (,000 barrels). While for he wo crude oil producs a reliable cash marke exiss, reformulaed gasoline and heavy fuel oil in New York harbour, respecively, crude oil spo daa lack consisency, so insead we used he global spo index published by he US Deparmen of Energy (DOE) (obained from he CRB Infoech daabase). One problem encounered in he analysis of fuures conracs is ha individual conracs expire. In order o deal wih hin rading and expiraion effecs, i is assumed ha he 4

15 hedger will swich conracs, he nex business day afer rading aciviy has shifed from he neares o he second neares o mauriy conrac. Therefore, we uilize he volume and open ineres daase o discriminae liquidiy beween he firs and second neares o mauriy conracs since he mos effecive hedge is he nearby conrac (Chen e al, 987). Consequenly, in all cases, he neares conrac available is chosen as he appropriae hedge mechanism, and rolling over o he fron monh conrac occurs he business day following he day ha boh rading volume and open ineres exceed ha of he neares o expiry conrac 7. Having consruced a coninuous ime series for he fuures conracs prices, spo and fuures prices are hen ransformed ino naural logarihms. Summary saisics of he levels and reurn series are presened in Table I, Panel A. As expeced, spo prices are more volaile han he fuures prices. Jarque-Bera (980) ess indicae significan deparures from normaliy for all he commodiies and for boh spo and fuures prices, wih he excepion of Heaing oil spo prices a 0% significance level. The disribuions of he reurns appear o be normal. The Ljung-Box (978) Q saisic on he firs six lags of he sample auocorrelaion funcion is significan for all spo and fuures prices revealing ha serial correlaion is presen in boh spo and fuures prices. The same is eviden for he spo reurns ime series. Fuures prices reurns show no significan signs of serial correlaion wih he excepion of Unleaded Gasoline conrac a 0% significance level. Engle s (98) ARCH es, carried ou as he Ljung-Box Q saisic on he squared series, indicaes he exisence of heeroscedasiciy for all he level and reurn series, wih he excepion of WTI fuures reurns. Finally, Phillips and Perron (988) non parameric uni roo ess on he levels and firs differences indicae ha he spo and fuures prices are firs difference saionary. Having idenified ha spo and fuures prices are I () variables, coinegraion echniques are used nex o invesigae he exisence of a long run relaionship beween hese series. Johansen (988) coinegraion ess, presened in Table I, Panel B, indicae ha all physical commodiy prices sand in a long-run relaionship wih he corresponding fuures conracs. The normalized coefficien esimaes of he 7 For insance he November 00 WTI fuures conrac expires on Ocober,. The rollover o he December 00 conrac akes place five business days prior expiry, on he 5 h of he same monh, because open ineres crossover beween he wo nearby conracs occurred on he 0 h, while volume crossover on Ocober, 4. 5

16 coinegraing vecor β = (β β 0 β ) represen his long-run relaionship beween he series. Furhermore he resuls of likelihood raio ess on he hypohesis ha here is a one-o-one relaionship beween spo and fuures prices; ha is, he coinegraing vecor is he lagged basis: H 0 : β = (, 0, -), show ha he hypohesis can be rejeced a all significan levels, wih he excepion of Heaing oil. Therefore, for WTI crude oil and Unleaded Gasoline we use he unresriced coinegraing vecors, in he join esimaion of he condiional mean and he condiional variance (VECM-GARCH), whereas for he Heaing oil marke he coinegraing vecor is resriced o be he lagged basis. 5 EMPIRICAL RESULTS Markov Regime Swiching Models wih differen specificaions are esimaed assuming wo saes. The choice of a wo-sae process is moivaed by he fac ha his model capures he dynamics of he spo and fuures reurns in a more efficien way han a Markov process wih more han wo regimes. For insance, Fong and See (00;003) use a wo-sae process in he crude oil fuures marke. On he oher hand, Sarno and Valene (000) use a hree-sae process o model spo-fuures relaionship in sock indices; noneheless, in heir sudy he hird sae seems o capure only jumps in he fuures prices a he ime of swiching beween conracs of differen mauriies and does no reflec fundamenal changes in marke condiions. Finally, he wo-sae process is inuiively appealing since i allows for periods of low and high volailiy. The MRS model of Equaion (3) is hen esimaed by maximizing he log-likelihood funcion of Equaion (5); we call his he univariae MRS model. Furhermore, we exend he regime swiching model o a bivariae MRS-VAR allowing for a GARCH error srucure, wih consan ransiion probabiliies; we call his model, he MRS-BEKK model. The resuls of boh he univariae and bivariae models are presened in Table II. Several observaions meri aenion. Firs, regarding he univariae MRS model, repored in Table II, Panel A, i can be seen ha for he esimaes of he volailiies (sandard deviaions σ and σ ) under he wo saes, he resricion of equal volailiies is srongly rejeced in each case, according o he Likelihood Raio (LR) ess, presened in he same able. As indicaed by he LR ess, here is also marked asymmery across 6

17 he hedge raios, γ,s, s =,, beween he wo saes. This suggess ha he dynamics of he spo-fuures relaionship are differen under hese wo differen regimes. We can noe an eviden associaion beween he magniude of he hedge raio and he sae of he porfolio volailiy; as expeced, a high variance sae is associaed wih a low hedge raio and vice versa. Tha is because during high volailiy periods he correlaion of he shor run dynamics (reurns) is expeced o fall and hus, spo-fuures prices o diverge. Furhermore, he OLS hedge raio of Equaion (), lies beween hese wo hedge raios as he OLS echnique esimaes he average hedge raio over he sample period, as opposed o he MRS model ha allows he hedge raio o depend on he sae of he marke. These findings sugges markedly differen dynamics of spo-fuures relaionship under he wo regimes and are consisen across all hree models, for all hree markes. Looking a he esimaed MRS-BEKK models, presened in Panel B of Table II, alhough he inclusion of a consan is significan only in he case of Heaing oil, he signs of hese coefficiens are reverse, in each regime. Moreover, he inclusion of lagged cross auoregressive erms, in he mean equaion displays significance only in he spo equaion. This implies ha spo prices respond o changes in he shor run dynamics, whereas fuures remain unresponsive o lagged cross reurns. Spo prices are usually more sensiive o informaion due o he fac ha hey incorporae higher liquidiy and hus, informaion is auomaically absorbed in he spo markes whereas in he fuures markes he speed of adjusmen o he available informaion is a funcion of liquidiy, mauriy ec. This is expeced, since oil fuures prices are deermined by supply and demand in he oil physical marke and suggess ha peroleum fuures are exogenous o spo prices. The degree of persisence in he variance in each regime is measured by he sum of α ii,s + β ii,s coefficiens for s =,. In all cases he sum less han uniy indicaing ha he GARCH sysem is covariance saionary. As indicaed by hese sums, low variance saes are characerized by lower persisency, whereas in he high variance sae persisency increases. This is in line wih oher sudies in he lieraure such as Fong and See (00) in he crude oil fuures marke. The only excepion is he heaing oil equaion, where he low fuures variance sae is associaed wih longer memory. This can be aribued o he fac ha he high variance regime occurs infrequenly a he poin of he upward jumps of he basis. Visual inspecion of he fuures and spo prices 7

18 shows ha hese jumps are caused solely by spo price spikes. Descripive saisics (Table I) show ha heaing oil involves he higher spo price volailiy and he lowes fuures price volailiy of all hree commodiies. As a resul, he low variance sae is dominan hroughou he sample period and occasional spo price jumps are capured by he model as he high variance sae. However, for he high variance sae error and GARCH coefficiens are no significan, probably because he high variance regime is of shor duraion whereas he low variance regime is he prevailing sae. From he esimaed ransiion probabiliies P and P we can calculae he duraion of being in each regime. For insance in he case of WTI crude oil marke he ransiion probabiliies of MRS-BEKK (Panel B, Table II) are esimaed as P = 5.4% and P = 3.5%; hese indicae ha he average expeced duraion of being in regime is abou 4 (=/0.54) weeks and he average expeced duraion of being in regime is abou 7 (=/0.35) weeks 8. Thus, high variance saes are less sable and are characerized by shorer duraion compared o low variance saes. The univariae MRS model (Panel A, Table II) indicaes ha he saes are equally sable wih he excepion of Heaing oil. Consequenly, allowing he condiional variances o be boh ime varying and regime swiching, he persisency of regimes is reduced, in oher words, he ransiion probabiliies are higher, allowing for more frequen swiching. For example, in he WTI crude oil marke he ransiion probabiliies as esimaed from he MRS model are P = 0.% and P = 0.5% indicaing a duraion of abou 5 weeks for each regime, compared o 4 and 7 weeks as esimaed by he MRS-BEKK. The smooh regime probabiliies for he WTI crude oil, unleaded gasoline and heaing oil markes derived from he esimaed MRS-BEKK model are presened in Figures, 4 and 7, respecively 9. These indicae he likelihood of being in sae (low variance sae). The shaded areas in he graphs idenify he periods when he marke is in he high 8 The average expeced duraion of being in sae is calculaed using he formula suggesed by Hamilon (989): i ip ( P ) = ( P ) = ( P i = ) 9 Based upon he esimaed parameer vecorθˆ, esimaed from daa spanning he period = o T, hree esimaes abou he unobserved sae variable s, can be made. The firs is he esimaed probabiliy ha he unobserved sae variable a ime equals given he observaions o < T and is ermed he filered probabiliy abou s. he second is he esimaed probabiliy ha he unobserved sae a ime equals given he enire sample of observaions from o T, ermed he smooh probabiliy. The hird is he esimaed probabiliy ha he unobserved sae variable a ime T+ equals given observaions o T and is ermed he expeced or prediced probabiliy abou s. See Hamilon (994) for furher deails. 8

19 variance sae. In he case of WTI crude oil, sae wo is prevailing over he periods 99 o lae 995, mid 996 o lae 997 and early 00 o mid 00. Figure illusraes ha WTI is characerized mainly by he low variance sae unil 995, aribued o he resoraion of Kuwai s producion afer he Gulf war and overproducion from he OPEC counries in combinaion wih relaively weak demand. The low variance sae is hen disurbed by bad weaher condiions in he US and Europe as well as he ension in he Middle Eas, Asian crisis in 998 ec, which creaes insabiliy in he high/low variance regimes occurrence i.e. shorer duraion regimes. In he case of Unleaded Gasoline, each of he wo regimes is characerized by less persisency han WTI, bu he paern, in Figure 4, is very similar o Figure. As expeced, mainly due o seasonal facors, regimes are less sable, since backwardaions and supply shorages for a ligh disillae are more highly likely and frequen, due o consrained refining capaciy and he fac ha he producion is subjec o he qualiy of he crude. Even in periods of crude oil oversupply (e.g. 993), consrained refining capaciy may disurb he supply/demand dynamics of he refined producs. Concerning heaing oil, he regimes are more disinc wih he low variance sae dominan. Alhough also seasonal, he behaviour of he basis is differen as i seems o be more sable, wih occasional jumps ha persis only for a shor period of ime. So, high variance periods are he early 99, mainly due o he Persian Gulf War and he effecs of he resoluion of he Sovie Union. The nex hree jumps of he basis occur in he lae 993, lae 995 and lae 996, which can be associaed wih cold weaher and regional supply demand imbalances. Afer he year 000, he high variance sae becomes more frequen. This is due o he srong demand and igh producion, followed by he Sepember, 00 erroris aacks and he following recession in he US. Invenory levels in combinaion wih consrained refining capaciy increased he volailiy of he marke. Figures, 5 and 8 plo he basis (defined as he logarihm of spo minus logarihm of fuures) of he WTI crude oil, Unleaded gasoline and Heaing oil, respecively. Figures 3, 6 and 9 show he in-sample OLS, GARCH and MRS-BEKK hedge raios for all hree markes. The shaded areas, which specify he periods ha he marke is in sae, are also projeced o hese graphs o faciliae he visual comparison of he sae of he marke and he magniude of he basis and he hedge raio. The graphs of he hedge raios indicae ha he flucuaions of he MRS hedge raios are similar o hose of he smooh probabiliies, bu smooher; his is expeced since hey are 9

20 consruced based on hese regime probabiliies. Turning nex o he graphs of he basis in Figures, 5 and 8, we can noe ha when he basis is close o zero and he basis is relaively less volaile, he marke is in he low variance sae (sae ). During hese periods he hedge raio is higher and less volaile. Similarly, when he marke is in sae (high variance sae) he basis is furher away from zero. This indicaes ha here is a posiive relaionship beween he volailiy and he magniude of he basis. This is consisen wih he findings of oher sudies such as Lee (994), Chouldry (997) and Kavussanos and Nomikos (000) who found ha when he spread beween spo and fuures (i.e. he basis) increases hen he volailiy in he marke increases as well. 6 TIME VARYING HEDGE RATIOS AND HEDGING EFFECTIVENESS Following esimaion of he univariae MRS models, smooh probabiliy esimaes are used o calculae an in-sample sae-dependen hedge raio for each marke using Equaion (6). Similarly, esimaion of he bivariae MRS-BEKK smooh probabiliy esimaes are used o calculae an in-sample sae-dependen hedge raio for each marke using Equaion (5). To formally assess he performance of hese hedges, porfolios implied by he compued hedge raios each week are consruced and he variance of reurns of hese porfolios over he sample is calculaed. In mahemaical form we evaluae: Var ( S γ F ) (6) * * where γ are he compued hedge raios. To compare he hedging performance of he MRS agains alernaive models ha have been proposed in he lieraure, we also esimae and calculae hedge raios based on he OLS model of Equaion (), on a bivariae Vecor Error Correcion Model (VECM) of spo and fuures reurns (Engle & Granger, 987; Johansen, 988), as well as ime varying hedge raios generaed from a 0

21 bivariae VECM model wih GARCH error srucure. 0 For benchmarking purposes, we also consider he use of a naïve hedge by aking a fuures posiion which exacly offses * he spo posiion (i.e. seing γ = ). The porfolio variances for he hree energy commodiies are presened in Table III. The same able also presens he incremenal variance improvemen of he MRS-BEKK model agains he oher models. I can be seen ha he MRS hedging sraegies ouperform he oher models in erms of in-sample variance reducion. Among he MRS models, he MRS-BEKK is he bes model for boh he WTI crude oil and Heaing oil marke. In he Unleaded Gasoline marke he univariae MRS provides beer variance reducion compared o alernaive sraegies. Finally, anoher feaure of he of he insample resuls is ha MRS models provide greaer variance reducion in boh he crude and heaing oil marke as opposed o he unleaded gasoline marke, where MRS-BEKK fails o deliver beer variance reducion compared o consan hedge raio sraegies. However, dynamic hedging sraegies are more cosly o implemen han saic models since hey require frequen updaing and rebalancing of he hedged porfolio. Consequenly, hedging effeciveness is more appropriaely assessed by considering he economic benefis from hedging as obained from he hedger s uiliy funcion. Consider an invesor wih he following mean variance uiliy funcion as in Kroner and Sulan (993), Gagnon e al. (998), and Lafuene and Novales (003): E U x ) E ( x ) kvar ( x ) (7) ( + = + + where k is he degree of risk aversion (k > 0) of he individual invesor and x + represens he reurns from he lagged porfolio. From Table III, he average weekly variance of reurns from he hedged posiion in he WTI crude oil marke is when he consan hedge raio is used and when he MRS-BEKK model is used. Assuming ha expeced reurns from he hedged porfolio are equal o zero and he degree of risk aversion is 4 hen, on average, one obains a weekly uiliy of U ) = - ( x + 4 (6.785)= if he consan hedge raio is used and U ) = - 4 (6.065) = - ( x + 0 The VECM-GARCH model is specified using he Baba e al (987) (BEKK) represenaion (Engle & Kroner, 995). See Kavussanos and Nomikos (000) for deails on he specificaion of hese models. Esimaion resuls for hese models are available from he auhors upon reques.

22 4.6 when he MRS-BEKK hedge raio is used. Hence, by using he MRS-BEKK model, hedgers in he marke can benefi from an increase in he average weekly uiliy of y, over he consan hedge raio, where y represens he reduced reurns caused by he ransacion coss incurred due o porfolio rebalancing. Therefore, a sraegy based on he MRS-BEKK hedge raio will be preferred over a consan sraegy if y < Assuming ransacion coss in he range of %, even hough he percenage variance reducion of he univariae Markov-based hedging sraegy MRS is no dramaic, MRS hedge would sill resul in an improvemen in uiliy for an invesor wih a mean variance uiliy funcion and k =4, even afer accouning for ransacion coss. Therefore, an invesor wih a mean-variance uiliy funcion would prefer he MRS-based sraegies o he consan sraegy since, on average, he increase in uiliy more han offses he higher ransacion coss due o rebalancing. Finally, i is also worh noing ha all he MRS sraegies ouperform he GARCH model on he basis of uiliy comparisons. The in-sample performance of he alernaive hedging sraegies gives an indicaion of heir hisorical performance. However, invesors are more concerned wih how well hey can do in he fuure using alernaive hedging sraegies. So, ou of sample performance is a more realisic way o evaluae he effeciveness of he condiional hedge raios. For ha reason, we wihhold he las 5 observaions of he sample for each marke, for he period Augus 6, 003 o July 8, 004, and esimae he models using only daa up o his dae. In he case of he univariae Markov Regime Swiching models (MRS), hedge raios a ime + are obained using a wo sep procedure. Firs, esimaes of he ransiion marix a ime, P ), and he esimaed smooh regime probabiliies a ime, Pr( s ) = ˆ π, ˆ = and Pr( s = ) = π, are used o forecas regime probabiliies a ime e e +, ha is, π and π as follows:,+,+ ˆ ˆ π (8) e e ( ) ( ) P, P, =, + π, + πˆ, πˆ, Pˆ, Pˆ, These assumpions are in line wih mos empirical sudies in he lieraure (Kroner & Sulan, 993; Park & Swizer, 995; Lafuene & Novales, 003)

23 Second, he one-sep ahead forecass of regime probabiliies are used o deermine he opimal hedge raio a ime +, γ * +. This is done by muliplying he one-sep ahead regime probabiliies by he respecive mean hedge raio for each regime; i.e. by he mean of he hedge raio under each marke condiion, and adding hem up (see Hamilon, 994 for more deails). γ * γ + = + (9) e e, ( ) π, + π, γ, In he case of Markov Regime Swiching BEKK model, hedge raios a ime + are obained using a four sep procedure. Firs, we esimae he forecas of he regime e e probabiliies a ime +, π and π as described above.,+,+ Second, we perform one sep ahead forecass of he sae-dependen covariance and variance as follows: Ε Ε [ h Ω ] SF, s, + [ h Ω ] = c + c + a ε + β h ; for s = {, } FF, s, + = c, s, s c, s + a, s, s a, s, s ε S, F, ε F, + β, s, s FF, β, s h SF, (0) Third we perform one sep ahead forecass of he mean equaion for spo and fuures reurns, respecively, as follows: [ S Ω ] Ε + [ F Ω ] = a + b S + γ F ; for s = {, } Ε + = a S, s F, s + b S, F, S + γ S, F, F () The above forecass of Equaion () are used o inegrae he sae variable s a each sep of he recursive esimaion, in he collapsing procedure of he variance-covariance marix. Afer eliminaing he sae variable s using Equaions (9) and (), he one sep ahead forecas of he hedge raio is compued as: 3

24 [ hsf, + Ω ] [ h Ω ] Ε * γ + Ω = () Ε FF, + The following week (Augus 3, 003) his exercise is repeaed wih he new observaion included in he daase. As in he case of he in-sample ess, he hedging performance of MRS-based hedge raios is compared o ha of alernaive compeing models. In he case of he GARCH-based hedges, he model is re-esimaed each week during he ou-of-sample period and ou-of-sample hedge raios are generaed by obaining one-sep ahead forecass of he ime varying variance-covariance marix, similar o Equaions (0) and () of he MRS-BEKK forecasing procedure. Also, in he case of VECM a differen hedge raio is obained each week by re-esimaing he model each week during he ou-of-sample period. The resuls from he ou-of-sample performance from alernaive hedging sraegies are presened in Table IV. Looking a he resuls for boh Unleaded Gasoline and Heaing Oil marke, we can see ha he highes reducion in he ou-of-sample porfolio variance is achieved by he wo regime MRS-BEKK model. Compared o consan (OLS) hedge he gain in variance reducion is 7.5% for Unleaded Gasoline and.9% for Heaing oil. This compares favourably wih findings in oher fuures markes. For ou-of-sample variance reducions, Kroner and Sulan (993) repor percenage variance reducions of he GARCH hedges relaive o he OLS hedge ranging beween 4.94% and 0.96% for five currencies; Gagnon and Lypny (995) find.87% variance reducion for he Canadian ineres rae fuures, Bera e al (997) esimae.74% and 5.70% variance reducions for he corn and soybeans fuures. Kavussanos and Nomikos (000) repor variance reducions for individual freigh roues using freigh fuures conracs (BIFFEX) ranging beween 0.43% and 5.7%. Alizadeh e al. (004) repor a gain of.6% o 6.8%, assessing he effeciveness of peroleum fuures o hedge bunker price flucuaions. In addiion, he univariae MRS models ouperform he alernaive non-markovian sraegies in he Heaing oil marke. However, in he case of WTI crude and Unleaded Gasoline, he univariae MRS fails o deliver beer variance reducion han boh he naïve and GARCH hedges, bu i ouperforms he convenional OLS hedge. Concerning he crude oil marke, he greaes variance reducion is provided by he single regime VECM-GARCH hedging sraegy whereas MRS-BEKK model achieves 4