On he shor- and long-run effcency of energy and precous meal markes Mohamed El Hed Arour, Shawka Hammoudeh, Duc Khuong Nguyen, Amne Lahan To ce hs verson: Mohamed El Hed Arour, Shawka Hammoudeh, Duc Khuong Nguyen, Amne Lahan. On he shorand long-run effcency of energy and precous meal markes. 2013. <hal-00798036> HAL Id: hal-00798036 hps://hal.archves-ouveres.fr/hal-00798036 Submed on 7 Mar 2013 HAL s a mul-dscplnary open access archve for he depos and dssemnaon of scenfc research documens, wheher hey are publshed or no. The documens may come from eachng and research nsuons n France or abroad, or from publc or prvae research ceners. L archve ouvere plurdscplnare HAL, es desnée au dépô e à la dffuson de documens scenfques de nveau recherche, publés ou non, émanan des éablssemens d ensegnemen e de recherche franças ou érangers, des laboraores publcs ou prvés.
On he shor- and long-run effcency of energy and precous meal markes Mohamed El Hed Arour a, Shawka Hammoudeh b, Amne Lahan c, and Duc Khuong Nguyen d a Unversy of Auvergne and EDHEC Busness School, France. mohamed.arour@edhec.edu b LeBow College of Busness, Drexel Unversy, USA. Shawka.M.Hammoudeh@drexel.edu; c LEO, Unversy of Orleans and ESC Rennes Busness School, France. amne.lahan@unv-orleans.fr; d ISC Pars School of Managemen, France. dnguyen@scpars.com Absrac Ths arcle conrbues o he relaed leraure by emprcally nvesgang he effcency of nne energy and precous meal markes over he las decades, employng several pronounced models. We es for boh he shorand he long-run effcency usng, n addon o lnear conegraon models, nonlnear conegraon and errorcorrecon models (ECM) whch allow he effcency nensy o change per regme. Our fndngs can be summarzed as follows: ) fuures prces are found o be conegraed wh spo prces, bu hey do no consue unbased predcors of fuure spo prces; ) he hypohess of rsk neuraly s rejeced and here s some evdence of me-varyng rsk prema; ) he shor-run effcency hypohess s rejeced, suggesng ha usng pas fuures prce reurns mproves he modelng and forecasng of fuure spo prces; and v) he nonlnear modelng suggess he presence of wo dsnc regmes where n he frs regme he effcency hypohess s suppored, whereas n he second s rejeced. The emprcal fndngs have mporan mplcaons for producers, hedgers, speculaors and polcymakers. Keywords: marke effcency, precous meals, energy markes, lnear and nonlnear ECM models JEL Classfcaons: C3, C5, G1 1
1. Inroducon Inernaonal fnancal markes have known a successon of serous crses snce 1987 (e.g., he 1997-1998 Asan crss, he 2001 do com recesson, he 2001 Argenna economc crss and he 2007-2010 global fnancal crss), whch are commonly characerzed by hgh volaly and conagon effecs (Forbes and Rgobon, 2002; Lee e al., 2007; Markwa e al., 2009). Recen sudes also sugges lower dversfcaon benefs from equy nvesmens due o he ncreased correlaons beween equy markes around he world, parcularly durng mes of hgh and exreme volaly (Chan-Lau e al., 2004; Damands, 2009). These sylzed facs have undenably encouraged nvesors o consder alernave nvesmen nsrumens as a hedge agans ncreasng rsk and uncerany n equy markes. Energy producs (manly ol, ol-relaed and naural gas conracs) and precous meals (manly gold, palladum, planum and slver) have emerged as naural desrable asse classes for nernaonal porfolo dversfcaon because of her dfferen volale reurns and low correlaons wh socks (Arour and Nguyen, 2010; Conovor e al., 2010; Daskalak and Skadopoulos, 2011; Hammoudeh and Araújo-Sanos, 2012). The flgh-o-qualy phenomenon equally occurs when fnancal nsably ncreases and deepens n he sock markes or when he prce of ol exhbs long swngs. Indeed, mos nvesors, for fears of losses, allocae her nvesmens o precous meals whch are vewed as safe-haven and refuge asses durng wdespread marke pancs. However, he observed ncreases n prce speculaons and he hgh degree of elasc subsuon among energy producs and beween precous meal conracs n boh consumpon and producon call for careful nvesgaon of her prce dynamcs. All are more lkely o be nfluenced by demand, supply, and expecaons abou fuure busness cycles. Energy and precous meals fuures conracs allow hedgers o secure he prces of her expeced purchase or sale of energy producs and precous meals a a specfc delvery dae n he fuure. The prces of fuures conracs hus convey nformaon abou expecaons of marke parcpans concernng he spo prces a he maury dae. Such nformaon s crucal for agens no fully hedged as well as for agens plannng for fuure producon or use of precous meals and energy producs. The mporance of fuures prces hus arses n parcular wh her ably o forecas spo prces a specfed fuure daes as hey provde economc agens wh means of managng he rsks relaed o radng of energy producs or precous meals n he spo markes. Whle all rsk managemen ools share a common neres,.e., mnmzng he rsk agans an unfavourable evoluon of fuure spo prces, her use s condonal on some marke condons among whch nformaonal effcency s he mos m- 2
poran. Havng s roo n he well-known effcen markes heory of fnancal economcs, nformaonal effcency refers o he degree o whch marke prces reflec accuraely and nsananeously all he relevan nformaon abou he rue underlyng value of fnancal secures. In hs schema of hngs, he nformaonal effcency maers n wo man ways. Frs, f a parcular marke s neffcen, nvesors may buld up varous radng sraeges ha lead o earnng excess reurns. Second, f all relevan nformaon s ncorporaed n fnancal secures prces as soon as hey appear, new nvesmen capal goes o s hghes-valued use. These feaures hus hghlgh he necessy of research on he effcency of asse markes. The effcen marke hypohess (EMH), formally developed by Fama (1965, 1970), has been esed for a varey of asse classes ncludng commodes. As far as he energy and precous meal markes are concerned, hs hypohess mples ha fuures prces consue he bes unbased forecass of fuure spo prces plus or mnus a me-varyng rsk premum, and hus speculaors canno earn abnormal profs. On he oher hand, fuures prces are unbased forecass of fuure spo prces f one or more speculaors are rsk-neural. Therefore, he queson of wheher or no commody prces behave accordng o he marke effcency hypohess maers because effcency enables o know f speculave reurns could be earned. To dae, several emprcal sudes have addressed hs ssue for commody markes (Booh and Kaen, 1979; Sol and Swanson, 1981; Aggarwal and Sundararaghavan, 1987; Tabak and Cajuero, 2007; Alvarez-Ramrez e al., 2010; Arour e al., 2010,2011,2012; Orz-Cruz e al., 2012), bu her focus s manly on he sochasc properes of successve spo and/or fuures prce changes of gold, slver and crude ol. Comparng o prevous sudes, hs arcle ess he hypoheses of nformaonal effcency and rsk neuraly for energy and precous meals markes over he shor- and he long- run, usng boh lnear and nonlnear echnques among whch he exponenal smooh ranson error-correcon model (ESTECM) s of parcular neres. Theorecally, a marke may be nformaonally effcen and unbased n he long run, bu may pass hrough perods of neffcency n he shor run, and vce versa. These dfferen paerns of prce behavor have obvously mporan bu also very dfferen mplcaons for marke operaors. Moreover, because of ransacon coss, nformaon asymmery and nvesors heerogeneous expecaons, markes can be effcen durng a ceran regme, and as a resul he use of nonlnear models s of parcular neres for capurng shor-run changes n he effcency nensy over dfferen regmes. Under he effcen and rsk neuraly hypoheses, he fuures prce wll be an opmal forecas of he fuure spo prce a he conrac ermnaon. 3
In s smples form, he EMH can be reduced o he jon hypohess ha economc agens are, n he aggregae, endowed wh raonal expecaons and are rsk neural so ha he fuures prce s an unbased esmaor of he fuure spo prce (Taylor, 1995). Furhermore, he effcency hypohess also saes ha asse prces fully and nsananeously reflec all avalable nformaon so ha no raders can conssenly earn abnormal profs by speculang n he fuures prces. Thus n hs paper, we conrbue o he leraure by proposng an negraed approach o emprcally es he marke effcency and rsk-neural hypoheses n presence of nonlneary a boh he shor- and long-run levels for peroleum (WTI, gasolne, heang ol, and propane), naural gas and precous meals markes (gold, slver, palladum, and planum). We parcularly examne he dynamc relaonshps beween spo and fuures prces of hese markes, mos of whch have no been researched well n he marke effcency leraure. The remander of hs arcle s organzed as follows. Secon 2 brefly revews he relaed leraure. The emprcal framework s nroduced n Secon 3. Secon 4 descrbes he daa used and repors he obaned resuls. Secon 5 concludes and dscusses he man mplcaons of he emprcal resuls. 2. Leraure revew As noed earler, pas leraure has been mosly concerned by esng he effcency hypohess of spo and/or fuures markes for crude ol, gold, and slver. For ol and olrelaed produc markes, hs leraure begns wh Green and Mork (1991) ha examnes wheher he offcal prces of crude-ol conracs are effcen n he sense of Fama (1970),.e., wheher he prce of a fuures conrac on crude ol s an effcen predcor of he ex-pos spo prce a he me of merchandse delvery, f all he relevan nformaon was avalable a he me when he conrac was se up. Usng he generalzed mehod of momens (GMM) o make nferences abou he predcably of monhly prces on Mdeas Lgh and Afrcan Lgh/Norh Sea crude ols, Green and Mork (1991) rejec he weak-form effcency for he whole sample perod 1978-1985. They however show evdence of effcency mprovemen over me when subsample perods are used. More recenly, Swzer and El-Khoury (2007) es he effcency of NYMEX (New York Mercanle Exchange) lgh swee crude ol fuures conrac marke durng he recen perod of exreme volaly, and hey fnd ha he prces of crude ol fuures conrac are conegraed wh he spo prces. Maslyuk and Smyh (2008) 4
examne he effcency of crude ol markes by analyzng he weekly spo and fuures prces for boh Wes Texas Inermedae (WTI) and Bren crude ol prces over he perod from January 1991 hrough December 2004. They employ Lagrange Mulpler un roo ess allowng for one and wo srucural breaks, and show ha each of he ol prce seres follows a random walk,.e., he crude ol markes under consderaon are weak-form effcen. Dfferenly, Shambora and Rosser (2007) fnd evdence agans he valdy of he EMH for NYMEX crude ol fuures conracs, as her resuls from an arfcal neural nework (ANN) model and several echncal radng rules show sgnfcan predcably n he fuures marke for ol. There s also evdence o suppor he hypohess of evolvng effcency hrough me (Tabak and Cajuero, 2007; Elder and Serles, 2008; Arour e al., 2010; Alvarez-Ramrez e al., 2008; Alvarez-Ramrez e al., 2010; Orz-Cruz e al., 2012). Tabak and Cajuero (2007) nvesgae he me-varyng degrees of long-range dependence n he Bren and WTI crude-ol reurns over he perod 1983-2004 by means of esmang he Lo (1991) s modfed Hurs exponen by rescaled range analyss. They fnd ha crude ol markes have become more effcen over me. The resuls of Alvarez-Ramrez e al. (2008) are conssen wh hose of Tabak and Cajuero (2007), as he crude ol markes hey consder converge owards weakform effcency over me. Usng dfferen approaches (.e., sem-paramerc wavele-based esmaor, me-varyng parameer model wh GARCH effecs, and derended flucuaon analyss), Elder and Serles (2008), Arour e al. (2010) and Alvarez-Ramrez e al. (2010) documen he presence of me-varyng shor-erm predcably n ol prce changes. Orz- Cruz e al. (2012) analyze he evoluon of he nformaonal complexy and effcency of he WTI crude ol marke hrough mulscale enropy analyss. They show ha he crude ol marke s nformaonally effcen over he sudy perod, excep for wo perods ha correspond o he early 1990s and lae 2000s US recessons. As for meal markes, Goss (1981) uses daa from he London Mercanle Exchange (LME) over he perod 1971-1978 o examne he hypohess ha fuures prces are unbased predcors of he subsequen spo prces for he markes for copper, n, lead and znc. Whle he null hypohess for lead and n s rejeced, he auhor repors conrary resuls for he case of copper and znc fuures conracs. Goss (1985) revsed hs 1981 paper by nroducng jon ess for he same meals raded n LME and exendng he sample perod o cover he perod 1966-1984. Hs resuls demonsrae ha he EMH s generally no rejeced. These fndngs are confrmed by hose of Canarella and Pollard (1986) who sudy boh overlappng and non-overlappng daa for fuures conracs of copper, lead, n and znc over he perod 5
1975-1983. Those auhors canno rejec he unbasedness hypohess. Gross (1988) proposes a sem-srong es of effcency of he alumnum and copper markes over he perod 1983-1984. He compares he predcve performance of several compeng models and shows ha he effcency hypohess canno be rejeced. By conras, Sephon and Cochrane (1990, 1991) examne he unbasedness hypohess n he LME for sx meals over he perod 1976-1985, and conclude ha he LME for meals s no an effcen marke. Chowdhury (1991) and Beck (1994) reach he same concluson for copper usng conegraon models. In a relaed sudy, Wakns and McAleer (2006) analyze daa on hree-monh fuures conracs for alumnum, alumnum alloy, copper, lead, nckel, n and znc. In mos of he samples consdered for he seven meals markes, conegraon ess deec he exsence of only one sgnfcan long-run relaonshp among he fuures prce, spo prce, sock level and neres rae. Fguerola- Ferre and Glber (2008) employ a bvarae FIGARCH model accommodang for long memory n volaly process o examne he prce dynamcs of he LME 3-monh alumnum and copper fuures markes. They fnd ha he condonal volaly of spo and fuures alumnum and copper prces exhbs a common degree of fraconal symmerc negraon. Among he meal markes, gold marke has receved he mos aenon from academc researchers snce hs yellow meal s wdely vewed as a hedge asse durng mes of fnancal urbulences and crses. The dynamc properes of gold prces, ncludng gold fuures prces, have been exensvely nvesgaed (e.g., Ball e al., 1985; Berus and Sanhouse, 2001; Cner, 2001; Cho and Hammoudeh, 2010; Hammoudeh e al., 2011). Tschoegl (1980) examnes he effcency of he gold marke wh respec o he nformaon conaned n sequences of successve prce changes and fnds ha alhough some shor-erm dependence exss, nobody can use hese reurn relaonshps o make abnormal profs. In conras, Neal (1989) and Beckers (1984) fnd evdence of marke effcency n he gold fuures marke and n he gold opons marke, respecvely. Conssenly, Marshall and Sengos (1994) es he effcency of spo gold reurns a varous frequences n he conex of Sms nsananeous unpredcably propery. They fnd no evdence of ou-of-sample forecasably n he reurn seres, mplyng ha he gold marke s weak-form effcen. More recenly, Wang e al. (2011) use he mulfracal derended flucuaon analyss o nvesgae he effcency and mulfracaly of he gold prces raded n COMEX durng he perod 1990-2009. Usng he rollng wndow approach, hey fnd ha he gold marke became more and more effcen over me, especally afer 2001. In addon, he gold marke s more effcen durng he upward perods han durng he downward perods. 6
Dfferenly, several sudes have focused on he queson of wheher gold prces can be predced from nformaon relaed o oher fnancal and commody prces. For example, Basu and Clouse (1993) sugges ha he gold marke s neffcen durng he perod from Ocober 1, 1989 o Sepember 30, 1990 because of sgnfcan correlaons beween he gold spo marke prce and oher marke varables (equy, bond, and currency). Narayan e al. (2010) fnd evdence of conegraon beween gold and ol spo and fuures markes,.e., he ol marke can be used o predc he gold marke prces and vce versa, mplyng ha hese wo markes are jonly neffcen. Kumar (2004) focuses on agrculural commodes n Inda and fnds no evdence of a long-run equlbrum relaonshp beween fuures and spo markes. The auhor concludes ha he fuures markes are no effcen as hey fal o dscover prces and provde effcen hedge agans he rsk emergng from prce volaly. Moreover, fuures markes appear o be unable o effecvely ncorporae nformaon. Wang and Ke (2005) use smlar mehods o nvesgae he effcency of he Chnese whea and soybean fuures markes. The auhors show a long-erm equlbrum relaonshp beween fuures prces and spo prces for soybean and a weak shor-erm effcency for he soybean fuures marke. The fuures marke for whea s however neffcen, whch may be caused by over-speculaon and governmen nervenon. Overall, he emprcal resuls of he prevous leraure are que mxed when a sngle commody marke s consdered,.e., he marke under consderaon can be effcen or neffcen dependng on he parcular sudy. They ndcae, however, ha commodes markes are generally neffcen when nformaon from oher markes s used o predc prces n one parcular marke. In hs paper, we conrbue o he above leraure by proposng an negraed approach o emprcally es he marke effcency and rsk-neural hypoheses a boh he shor- and long-run levels for energy and precous meals markes. We parcularly look a he relaonshps beween spo and fuures prces of hese markes. 3. Emprcal mehodology In hs secon, we dscuss four nesed lnear and nonlnear models o examne he effcency and rsk neuraly hypoheses for energy and precous meal markes over boh he shor and long run. These models nclude he lnear conegraon, he lnear error-correcon 7
model (ECM), he lnear ECM wh me-varyng volaly, and he exponenal smooh ranson ECM (ESTECM). Accordng o he prmary speculave effcency hypohess, he forward prces are he bes unbased forecas of fuure spo prces plus or mnus a me varyng-rsk prema (Blson, 1981). In effcen markes whou frcons, he arbrage free or he cos-of-carry model leads o he followng relaonshp beween he fuure prce a me ( F ) and he spo prce a me ( S ): F ( r + c y) ( T ) S e = (1) where r s he neres rae, c he sorage cos, y he convenence yeld and T he expraon dae of he fuures conrac. In he leraure, he non arbrage relaonshp (1) s dffcul o es emprcally because c and y are no drecly observed. Thus, researchers sugges esng he Fama (1970, 1991) weak-form effcency hypohess accordng o whch he fuures prce for a rsk-neural speculaor s an unbased predcor of fuure spo prce: where = α + β + ε (2) S F F refers o he offcal prce of fuures conrac maurng n perod, and ε s he raonal expecaon error and assumed o be serally uncorrelaed. The marke effcency and he rsk neuraly requre ha he resrcons β = 1 and α = 0, respecvely. The rejecon of hese resrcons means ha eher he marke s neffcen or a sgnfcan rsk prema may exs ( α 0), makng marke forecass based bu possbly effcen. If he relaonshp (2) does no hold, a rsk-neural speculaor can have a free lunch and earn money on shor or long fuures posons. Economercally, f he seres S and F are no saonary, he esmaon of Eq. (2) encouners he problem of spurous regresson, unless he wo varables are conegraed. In he case of conegraed varables, Eq. (2) expresses he equlbrum long-run relaonshp beween S and F, and hus an error correcon model (ECM) can be esmaed o nvesgae he shor- and long-run dynamcs. Ths approach allows us o nvesgae he effcency hypohess over me. Indeed, he effcency hypohess becomes more complcaed when he me dmenson s nroduced. A marke may be effcen n he long run bu experences shor 8
run neffcences. Whle usng sandard conegraon echnques enables o es for he longrun effcency and for a consan average rsk prema, he ECM wh GARCH-n-mean effecs allows esng for he shor-run effcency and a me-varyng rsk prema. We should, however, noe ha he conegraon beween S and F s only a necessary bu no a suffcen condon for he marke effcency. Whle under he conegraon hypohess, S and F are governed by he same fundamenals, comove and do no end o drf apar from each oher over me, he valdy of he effcen marke hypohess requres ha secury prces fully reflec all avalable nformaon and ha no prof opporunes be lef unexploed (Fama, 1970, 1991). I s hen clear ha conegraon does no rule ou shorrun neffcences as pas spo and fuures prces may mprove fuure spo prce forecass even f S and F are conegraed. If S and F are conegraed, we can esmae he followng lnear ECM: S p q 1 + b F n + φ F n + ϕ j S j ν (3) = 1 j = 1 = λ ρε + where ρ > 0 because changes n he spo prce respond o devaons from he long-run equlbrum gven by Eq. (2) and ν s a saonary error erm. Idenfyng Eq. (2) and Eq. (3), he effcency hypohess mples ha ρ = 1, λ = ρ α, b = ρ β 0 and φ ϕ = 0. If hese resrcons do no hold, pas spo and fuures prces = j can be used o mprove he forecass of fuure spo prce developmens, whch goes agans he marke effcency hypohess. Accordng o Eq. (3), he shor-run marke effcency can be nvesgaed by esng: ) b = ρ β 0 as any nformaon abou fuures prce change s nsananeously refleced n he = j curren spo prce change; ) φ ϕ = 0 as pas fuure and spo prce changes are already refleced n he curren prces and ) ρ = 1 o have a saonary relaonshp. Noe ha Eq. (3) allows for he exsence of a sgnfcan rsk premum as we do no mpose he resrcon α = 0 as requred n Eq. 2. Moreover, from Eq. (2) we learn ha β s he coeffcen of and ha for he long-run marke effcency o hold hs should be equal o uny ( β = 1). F 9
To sum up, f he ess from Eq. (2) do no lead o rejecon of he long-run effcency hypohess (.e., S and F are conegraed and β = 1), he resrcons mposed o es he shor-run marke effcency hypohess based on Eq. (3) are: (H1): ρ = 1, φ ϕ = 0, b = 1 and λ = 0 under rsk neuraly hypohess; = j (H2): ρ = 1, φ ϕ = 0, b = 1 and λ 0 under consan rsk prema hypohess; = j (H3): ρ = 1, φ ϕ = 0, b = 1 and λ = λ() under me-varyng rsk prema hypohess. = j Fnally, gven he exensve volaly observed n he meal and energy markes over he las decades, GARCH(1,1)-n-mean approach can be used o es for a me-varyng rsk prema. Tha s, o es for he hypohess (H3), he rsk prema s assumed o be a funcon of he condonal sandard devaon of he change n he spo prce. In hs conex, Eq. (3) can be rewren as follows: S h 2 = λ + γ h 2 = r + kν 1 ρε + lh 2 1 1 + b F n + p = 1 φ F n + q j= 1 ϕ S j j + ν (4) Several sudes have recenly shown some evdence of nonlneary and asymmery n energy and meal prce dynamcs (e.g., McMllan and Quroga, 2008; Ah, 2009; Arour and Nguyen, 2010). These characerscs may be explaned by dfferen facors such as ransacons coss, nformaon asymmery, and agen heerogeney. Such marke mperfecons consue mporan barrers o effcency nsofar as hey dscourage arbrage operaons and preven asse prces o converge o her effcen level. To he exen ha hese mperfecons may lead o lmng he exchange of asses, parcularly when he expeced poenal gan s nferor o he nduced coss, her presence mples wo dfferen zones. Frs, here exss a non-exchange zone whn whch arbrage radng s nacve. Prces may connue o flucuae far from her effcen values wh devaons close o a un roo ha naurally amplfes he neffcency of markes. In he second zone, called exchange zone, prce adjusmen s acve and s speed s as hgh as he dsequlbrum beween acual prces and her effcen values ncreases. Therefore, one way o mprove he model n Eq. (4) would conss of nroducng nonlneary n he mean equaon descrbng he spo prce adjusmen and o esmae he followng nonlnear smooh ranson ECM as gven n Eq. (5): S p q 1 1 + b F n + φ F n + ϕ j S j ρ2 ε 1Ψ( z, δ, c) ν (5) = 1 j = 1 = λ ρ ε + 10
{ } 2 where Ψ( z,, c) = 1 exp δ ( ε c) δ denoes he ranson funcon ha depends on he 1 hreshold parameer (c), he ranson varable z = ε 1 and he ranson speed (δ ). The parameers ρ 1 and ρ 2 denoe he adjusmen erms n he frs and n he second regmes, respecvely. They are he mos mporan parameers, specfyng he prce adjusmen dynamcs and defnng s convergence speed oward he equlbrum. When ρ 2 and ρ 1 + ρ2 are sgnfcanly posve, even f ρ 1 s negave, he nonlnear mean reverson n prces s valdaed. Ths mples ha for a mnor dsequlbrum, prce devaons would dverge from he equlbrum and would be characerzed by a un roo or explosve behavor, bu for large devaons, he adjusmen process would be mean-reverng. 4. Daa and emprcal resuls 4.1 Daa and prelmnary analyss The sample daa conss of he daly closng spo and fuures prces for four precous meal prces (gold, slver, palladum, and planum), four peroleum producs (WTI, gasolne, heang ol, and propane) and naural gas. The fuures prces are he closng prces of hreemonh fuures conracs on he respecve commodes. The precous meals are raded n he New York COMEX (Commody Exchange) and her spo and fuures prces are exraced from he Bloomberg daabase. Peroleum and naural gas prces are obaned from Daasream. All prces are expressed n US dollars, and daly reurns are compued by akng he dfferences n he logarhm of wo successve prces. Our sudy perod runs from 01/04/1999 o 03/31/2011 for precous meals and naural gas, from 01/02/1997 o 01/31/2011 for WTI and heang ol, from 10/03/2005 o 03/31/2011 for gasolne, and from 01/04/1999 o 09/18/2009 for propane. Fgure 1 plos spo and fuures of he energy and precous meal prces we sudy. We frs use he Augmened Dckey Fuller (ADF) and Phllps-Perron (PP) un roo ess o check he sably hypohess for all he prce seres. We also perform he Andrews and Zvo (1992) es as he ADF and PP es are no robus o evenual srucural breaks characerzng he commody prce me-seres. The resuls, repored n Table 1, show ha spo and fuures prces of almos all commodes are negraed of order one, I(1). The nonsaonary of commody prces hus gves us he opporuny o nvesgae her jon dynamcs n he long run and o es he effcency hypohess over me. 11
Fgure 1 Spo and fuures prce dynamcs Table 1 Resuls of un roo ess ADF PP Z&A Seres Level Level Level LSWTI -3.619(c) -38.906(a) -3.437(c) -81.209(a) -5.096-81.147 LSGasolne -1.931(b) -39.696(a) -1.887(b) -39.698(a) -6.160-80.110 LSHeang -2.763(c) -42.990(a) -2.859(c) -78.379(a) -5.015-78.359 LSPropane -2.768(c) -62.128(a) -2.856(c) -62.134(a) -3.618-62.417 LSGold -3.123(c) -56.689(b) -3.166(c) -56.694(b) -4.769-56.826 LSSlver -1.840(c) -60.811(c) -1.815(c) -60.934(c) -3.351-60.999 LSPlanum -2.660(c) -57.980(b) -2.534(c) -58.086(b) -5.153-58.152 LSPalladum 0.582(a) -54.020(a) 0.527(a) -54.093(a) -3.229-54.278 LSNaural Gas -2.451(b) -50.765(a) -2.592(b) -59.377(a) -5.827-59.235 LFWTI -3.043(c) -79.789(a) -2.934(c) -79.908(a) -4.478-80.110 LFGasolne -1.889(b) -38.849(a) -1.888(b) -38.851(a) -5.015-80.110 LFHeang -2.227(c) -79.715(a) -2.200(c) -79.768(a) -5.015-79.841 LFPropane -2.328(c) -58.506(a) -2.360(c) -58.571(a) -3.159-58.977 LFGold -3.150(c) -55.605(b) -3.117(c) -55.626(b) -4.849-55.718 LFSlver -1.930(c) -56.964(b) -1.817(c) -57.025(b) -3.271-57.173 LFPlanum -2.608(c) -55.689(b) -2.563(c) -55.689(b) -5.244-55.861 LFPalladum 0.514(a) -51.400(a) 0.547(a) -51.325(c) -3.221-51.683 LFNaural Gas -2.054(b) -63.141(a) -2.009(b) -60.129(a) -4.746-63.220 Noe: LS and LF desgnae respecvely he spo and fuures prces n logarhm. The logarhmc ransformaon ams o reduce he varance of all seres. Level and desgnae respecvely seres n level and hose n he frs dfference. (a): model wh neher rend nor consan; (b): model wh consan bu whou rend and (c): model wh rend and consan. The crcal values for he ADF and PP ess a he 5% level are -1.95 for model (a), -2.89 for model (b) and -3.45 for model (c). The crcal value for he Zvo and Andrews (1992) es, denoed by Z&A, s -5.08 a he 5% level. Numbers n bold face ndcae ha he null hypohess of un roo s rejeced a he 5% level. 12
Table 2 summarzes he summary sascs for he daly spo and fuures reurns of he commodes, as well as her sochasc properes. On average, we fnd ha over our sample perod he precous meals have hgher daly reurns han energy producs. The hghes average reurns are obaned for he spo slver and slver fuures (0.063%), followed closely by he average reurns on spo gold and gold fuures (0.050%), and on spo planum and planum fuures. Ths confrms he sayng on Wall Sree f you wan o buy gold, buy slver (Hammoudeh e al., 2011). The spo reurns of naural gas yeld a negave reurn average (- 0.001%), whle s fuures conracs on naural gas generae he lowes posve average (0.012%). Ths s mos lkely has o do he dscovery of he new exracon echnque hydraulc drllng. 1 Table 2 Descrpve sascs and sochasc properes of reurn seres WTI Gasolne Heang Propane Gold Slver Planum Palladum Naural gas Panel A - Spo Reurns Mean( 100) 0.017 0.016 0.030 0.031 0.050 0.063 0.049 0.026-0.001 Sd. dev. 0.026 0.027 0.026 0.025 0.011 0.019 0.016 0.022 0.046 Skewness -0.769-0.164-1.606-2.454-0.070-0.422-0.466-0.312 0.488 Kuross 17.315 5.882 39.747 67.328 8.315 11.045 15.958 9.146 22.689 JB 55764 a 528 a 359853 a 676889 a 3762 a 8709 a 22463 a 5080 a 59575 a Q(5) 15.725 a 3.582 3.158 4.070 3.021 8.592 3.808 15.076 b 14.507 b Q²(5) 375.217 a 225.870 a 1340.755 a 142.534 a 338.435 a 123.858 a 218.735 a 243.427 a 1814.547 a ARCH (10) 64.601 a 20.630 a 56.800 a 33.923 a 19.322 b 25.887 a 15.119 34.726 a 44.121 a Panel B - Fuures Reurns Mean( 100) 0.019 0.020 0.030 0.033 0.050 0.063 0.049 0.026 0.012 Sd. dev. 0.020 0.024 0.019 0.019 0.011 0.019 0.015 0.022 0.030 Skewness -0.789-0.058-0.744-1.037 0.186-0.779 0.201-0.273 0.053 Kuross 15.497 6.165 13.815 15.112 9.053 10.481 18.735 7.447 9.117 JB 42683 a 629 a 31522 a 24560 a 4895 a 7771 a 32974 a 2672 a 5739 a Q(5) 10.194 c 3.158 4.770 9.106 7.554 0.825 4.236 17.628 a 9.499 c Q²(5) 245.635 a 311.957 a 185.589 a 279.421 a 231.889 a 272.506 a 52.174 a 310.637 a 50.478 a ARCH (10) 42.791 a 16.576 c 14.986 29.900 a 28.779 a 13.250 13.441 42.319 a 46.587 a Noes: JB and ARCH(10) are respecvely he emprcal sascs of Jarque-Bera es for normaly and he LM ARCH es for condonal heeroscedascy. Q(5) and Q²(5) refer o he emprcal sascs of Ljung-Box es for seral correlaon appled o reurn and squared reurn seres, respecvely. a, b, and c ndcae he rejecon of he null hypohess of normaly and no ARCH effecs a he 1%, 5% and 10%, respecvely. The uncondonal volaly of all he daly spo and fuures reurns, as measured by sandard devaons, s subsanal wh daly values rangng from 0.011 (gold spo and gold fuures) o 0.046 (naural gas). Energy producs are more volale han precous meals. Wh respec o he rsk-reurn profle, naural gas spo and fuures reurns experenced he lowes performance as hey have he hghes volaly, bu he lowes reurns. Gold spo and fuures reurns have he hghes rsk-adjused reurn rao followed by slver spo and fuures reurns and planum spo and fuures reurns. These fndngs sugges ha mos precous meals mgh 1 hp://en.wkpeda.org/wk/drllng_rg 13
be a good hedge for porfolos of socks and oher asses, especally when fnancal markes pass hrough perods of urbulences and crses. The descrpve sascs also show ha skewness s negave n mos cases and ha excess kuross s hghly sgnfcan. Clearly, mos of he energy and precous meal reurns have faer als and longer lef als (.e., he probably of observng exreme negave reurns s hgher) han he normal dsrbuon. The Jarque-Bera es (JB) confrms hese fndngs snce he normaly s srongly rejeced for all cases a he 1% level. The Ljung-Box sascs, Q(5) and Q²(5), ndcae srong evdence of auocorrelaon n squared reurns, bu only some evdence of auocorrelaon n spo and fuures reurns (WTI, palladum and naural gas). These resuls ypcally show sgns of hgh degree of perssence n he condonal volaly process of energy and precous meal prce reurns. Resuls from he ARCH ess for condonal heeroscedascy are conssen wh hose from he Ljung-Box es appled o squared reurns as ARCH effecs are sgnfcanly presen n almos all reurn seres. Taken ogeher, hese fndngs sugges he usefulness and he suably of GARCH-ype models for modelng he me-varyng condonal volaly of he consdered commodes. Overall, we observe ha spo and fuures prce reurns of our commodes follow smlar dynamc paerns n general, whch s a pror no conradcory o he speculave effcency. We nex nvesgae he conegraon and effcency hypohess usng boh prce and reurn seres. 4.2 Conegraon ess and long-run analyss 4.2.1 Conegraon resuls We frs esmae Eq. (2) n order o nvesgae he long-run relaonshp beween spo and fuures prces of each commody asse. We hen es he null hypohess of conegraon usng he Engle-Granger framework and he Johansen procedure. The opmal number of lags s 13 as s seleced by boh AIC and BIC nformaon crera. 2 Resuls of conegraon ess are repored n Tables 3-4. From Table 3, we see ha boh ADF and Z&A ess do no rejec he conegraon hypohess, whch suggess ha spo and fuures prces of all commodes we consder converge owards a long-run equlbrum. When he Johansen procedure s used o es for conegraon (Table 4), we reach he same concluson for all commodes, excep for gasolne 2 The resuls for he selecon of opmal lag lengh n lnear conegraon framework can be made enrely avalable on reques addressed o he correspondng auhor. 14
and naural gas where we fnd wo conegraon relaonshps. For hese laer commodes, spo and fuures prces are no conegraed and ha a VAR model s suffcen o model her dynamc neracons. Overall, he resuls from boh ess sugges ha he same underlyng facors derve he spo and fuures prces of all consdered commodes. 3 Table 3 Conegraon ess whn he Engle-Granger framework WTI Gasolne Heang Propane Gold Slver Planum Palladum Naural gas Model s esmaes α 0.061 *** -0.061 *** -0.005 *** 0.0128 *** -0.006 *** -0.007 *** 0.097 *** -0.047 *** -0.021 *** (0.003) (0.005) (0.001) (0.002) (0.002) (0.002) (0.006) (0.005) (0.001) β 0.983 *** 1.051 *** 0.986 *** 0.996 *** 1.001 *** 1.001 *** 0.986 *** 1.007 *** 0.974 *** (0.001) (0.006) (0.002) (0.002) (0.000) (0.000) (0.001) (0.001) (0.004) R ² 0.994 0.947 0.985 0.981 0.999 0.999 0.986 0.998 0.994 AIC -3.238-2.771-2.342-2.404-6.330-6.000-4.671-5.021-1.331 SIC -3.236-2.764-2.340-2.401-6.327-5.996-4.668-5.017-1.328 Resdual dagnoss Q(10) 27985.5 7318.38 41928.9 26648.4 31.724 113.091 8870.05 217.696 24122.1 Q²(10) 12313 3988 19993 14094 517 396 11665 561 11795 ARCH(5) 1447.4 459.75 3547.0 2008.4 50.707 34.275 1667.3 68.469 2042.8 ARCH(10) 772.21 230.64 1827.4 1021.8 27.269 21.928 880.17 35.168 1012.5 ADF (p) -6.533 a (13) -5.671 a (3) -8.277 a (6) -6.353 a (6) -56.458 a (0) -21.152 a (4) -10.137 a (4) -20.698 a (4) -7.957 a (3) Z&A -6.883-7.121-8.775-6.666-53.936-22.217-15.831-22.010-8.305 Noes: Q(10) and Q²(10) refer o he emprcal sascs of Ljung-Box es for seral correlaon appled o resduals and squared resduals, respecvely. ARCH, ADF and Z&A denoe he emprcal sascs of he LM ARCH es for condonal heeroscedascy, and Augmened Dckey-Fuller and Zvo and Andrews (1992) ess for un roo, respecvely. The numbers n parenhess are he sandard devaons of he esmaed coeffcens. p denoes he lag lengh n he ADF es seleced by he SIC nformaon creron. a denoes he ADF es wh neher rend nor consan. *** ndcaes sgnfcance of he coeffcens a he 1% level. The crcal value for he ADF a he 5% level s -1.95. The crcal value for he Zvo and Andrews (1992) es a he 5% level, noed Z&A, s -5.08. Table 4 Resuls of Johansen conegraon es WTISPOT/WTI3M Trace Sasc 5% Crcal Value 1% crcal value r = 0 *** 147.7076 15.41 20.04 r = 1 0.7512 3.76 6.65 GasolneSpo/Gasolne3M r=0 *** 37.5415 15.41 20.04 r=1 *** 4.6201 3.76 6.65 HeangSpo/Heang3M r=0 *** 81.7803 15.41 20.04 r=1 0.6320 3.76 6.65 PropaneSpo/Propane3M r=0 *** 55.9924 15.41 20.04 r=1 1.7286 3.76 6.65 GoldSpo/Gold3M r=0 *** 1136.1140 15.41 20.04 r=1 0.3449 3.76 6.65 SlverSpo/Slver3M r=0 *** 1236.4860 15.41 20.04 r=1 0.4974 3.76 6.65 PlanumSpo/Planum3M 3 Smlar resuls are obaned afer correcon for overlappng observaon problems. 15
r=0 *** 185.3933 15.41 20.04 r=1 1.2373 3.76 6.65 PalladumSpo/Palladum3M r=0 *** 887.9372 15.41 20.04 r=1 1.0307 3.76 6.65 NaGasSpo/NaGas3M r=0 *** 98.0159 15.41 20.04 r=1 *** 3.9412 3.76 6.65 Noes: Resuls show one conegraon relaonshp for he par of spo and fuures prces for all seres are conegraed excep for gasolne and naural gas where wo conegraon relaonshps are found, ndcang ha hese seres are no conegraed. *** ndcaes he rejecon of he null hypohess a he 1% level. 4.2.2 Tess for he speculave effcency hypohess We now use he Wald es o examne he speculave effcency hypohess for he 18 energy and precous meal prces. The resuls are summarzed n Table 5. Accordngly, he rsk neuraly hypohess ( β = 1) and he marke effcency hypohess ( α = 0 ) are ndvdually rejeced a he 1% level for all commodes. Ths fndng hus goes agans he frs necessary condon for long-run marke effcency. The Wald es also rejecs he jon resrcons of marke effcency and rsk neuraly ( α = 0 and β = 1), ndcang ha neher he marke s effcen and/or ha a sgnfcan rsk prema may exs. These condons make marke forecass based bu possbly effcen. Table 5 Resuls of hypohess ess WTI Gasolne Heang Propane Gold Slver Planum Palladum Naural gas α = 0 340.436 151.492 26.787 51.021 8.405 [0.004] 8.053 [0.005] 283.793 103.850 10.154 [0.001] β = 1 300.591 64.165 84.081 3.368 [0.067] 6.645 [0.010] 6.367 [0.011] 272.188 73.754 39.058 α = 0, β = 1 180.287 144.888 45.358 87.923 10.915 10.643 150.944 274.956 473.820 Noes: he able repors he sascs of Wald es ha examnes he ndvdual hypoheses of marke effcency and rsk neuraly as well as he jon hypohess ( α = 0 and β = 1) as gven n Eq. (2). Numbers n brackes are he assocaed p-values. All n all, our fndngs show ha all he energy and precous meal spo and fuures prces are conegraed, bu we are no able o valdae he long-run effcency and he rsk-neuraly hypoheses. To he exen ha Eq. (2) characerzes he long-run equlbrum beween spo and fuures prces, we can es he shor-run effcency hypohess by esmang an ECM. Ths modelng approach enables us no only o nvesgae he effcency hypohess over me bu also o lnk he shor- and long-run effcency hypoheses hrough an error-correcon adjusmen process ha predcs fuure spo prces from nformaon conaned n fuures conracs. 16
4.3 Shor-run analyss 4.3.1 Lnear ECM esmaon The esmaon resuls of he lnear ECM n Eq. (3) are presened n Table 6. The adjusmen erms ρ are sgnfcan and have he expeced posve sgn, hus confrmng he fndngs of our conegraon ess and showng an acve mean-reverson mechansm n he relaonshp beween spo and fuures prces for all consdered commodes. The sgnfcance of he coeffcens φ 1 and ϕ 1 n numerous cases suggess ha he use of pas fuures and spo reurns mproves he forecas of fuure spo reurns. Moreover, he economerc specfcaon of Eq. (3) seems o successfully descrbe he shor-run dynamcs of spo reurns as he F- sascs of all esmaed ECMs are very large and he esmaed resduals do no exhb seral correlaon, excep for naural gas. However, here s srong and sgnfcan evdence of ARCH effecs n he resdual seres, whch needed o be accouned for n order o beer reproduce he dynamcs of spo reurns over me. Table 6 Esmaon resuls of he lnear ECM Coeffcens WTI Gasolne Heang Propane Gold Slver Planum Palladum Naural Gas λ ( 100) 0.016 0.018 0.031 0.022 0.015 8.11E-5 0.036 6.54E-5-8.55E-5 ρ (0.031) (0.067)) (0.033) (0.039) (0.018) (2.10E-4) (0.026) (3.39E-4) (6.7E3-4)) 0.048 *** 0.054 *** 0.030 *** 0.033 *** 0.684 *** 0.785 *** 0.096 *** 0.615 *** 0.052 *** (0.008) (0.012) (0.004) (0.006) (0.044) (0.027) (0.013) (0.033) (0.005) b 0.024 (0.029) 0.147 *** (0.051) -0.022 (0.003) 0.342 *** (0.026) 0.835 *** (0.030 0.976 *** (0.013) 0.496 *** (0.022) 0.882 *** (0.024) 0.724 *** (2.30E-5)) φ 1-0.041 *** (0.015) -0.018 (0.028) -0.016 (0.016) 0.046 ** (0.021) -0.013 (0.016) -0.004 (0.015) 0.051 *** (0.018) -0.017 (0.017) -0.023 (0,025) ϕ 0.018-0.087 ** 0.053 *** -0.133 *** -0.114 *** -0.098 *** -0.267 *** -0.109 *** -0.039 ** 1 (0.235) (0.044) (0.020) (0.020) (0.004) (0.021) (0.023) (0.027) (0.017) 2 R 0.007 0.016 0.007 0.046 0.197 0.639 0.140 0.305 0.230 Q(4) 1.685 [0.793] 2.050 [0.726] 4.122 [0.389] 2.039 [0.728] 1.043 [0.903] 2.886 [0.577] 7.388 [0.116] 3.823 [0.430] 15.407 [0.004] Q 2 (4) 348.021 142.821 1157.882 230.439 254.537 234.828 419.988 400.894 1657.570 ARCH(10) 31.352 18.378 [0.048] 54.731 24.174 23.156 55.208 34.516 41.831 164.909 Noe: Q(4) and Q 2 (4) denoe he emprcal sascs of he Ljung-Box es for seral correlaon appled o resduals and squared resduals. ARCH(10) refers o he emprcal sascs of he LM ARCH es for condonal heeroscedascy of resduals. The sandard errors are presened n parenheses and he p-values are n brackes. *, ** and *** denoe sgnfcance a he 10%, 5% and 1% levels, respecvely. Ths able s based on esmaon of Eq. (3). 4.3.2 Tess of he effcency and rsk-neuraly hypoheses We presen n Table 7 he resuls from he ess for he hypoheses (H1) and (H2) whn he lnear ECM framework,.e., hypohess of effcency and rsk neuraly ( ρ = 1, φ ϕ =0, b = 1 and λ = 0 ), and he hypohess of effcency and consan rsk prema ( ρ = 1, = j φ ϕ =0, b = 1 and λ 0 ). Boh hypoheses are srongly rejeced for all he energy and pre- = j 17
cous meal producs under consderaon. Thus, our fndngs go agans he energy and precous meal marke effcency hypohess alhough a mean-reverson process exss. However, hese fndngs may reflec a msspecfcaon assocaed wh he lnear adjusmen model. To address ha, we consran n Eq. (3) he rsk prema o be null or me-nvaran, whereas nvesors may no only requre a rsk premum bu also expec o be me-varyng, gven he hgh volaly of he energy and precous meal markes. Table 7 Wald es for ECM parameer resrcons Hypohess WTI Gasolne Heang Propane Gold Slver Planum Palladum Naural Gas H1: ρ = 1, φ ϕ = 0, = j b =1 and λ = 0. H2: ρ = 1, φ ϕ = 0, = j 3067.325 3834.154 1156.960 1446.192 9291.917 11614.900 5832.981 7291.224 12.522 15.652 20.362 25.453 948.395 1185.494 36.813 46.017 6095.862 7619.801 b =1 and λ 0 Noes: Ths able shows he resuls of he Wald ess (F-sasc) of he hypoheses H1 and H2 based on he esmaon resuls repored n Table 6. The p-values are presened n brackes. 4.3.3 ECM-GARCH-M and me-varyng rsk premum esmaon We now relax he consrans on he rsk premum by allowng o vary hrough me. I urns ou ha we can esmae an ECM-GARCH-M model, descrbed n Eq. (4), o characerze he dynamc of each commody s spo reurns. The GARCH-n-Mean specfcaon echnque models he rsk prema as a funcon of he condonal volaly of spo prce reurns. Here, nformaon crera lead us o rean a GARCH (1, 1) specfcaon. We repor he obaned resuls n Table 8, whch enable us o es for he hypohess H3 of effcency under me-varyng rsk prema: ρ = 1, φ ϕ = 0, b = 1 and λ = λ(). = j Table 8 Esmaon resuls of ECM-GARCH-M model WTI Gasolne Heang Propane Gold Slver Planum Palladum ECM equaon λ( 100) -3.73E-3 (0.073) 0.418 (0.309) γ 0.016-0.136 ρ (0.035) (0.125) 0.052 *** 0.036 *** (0.006) (0.011) b 0.098 *** 0.053 (0.029) (0,045) φ 1-0.005-0.006 (0.013) (0.029) ϕ -0.036-0.014 1 (0.024) (0.040) Varance equaon r( 100) 6.77E-4 *** 1.28E-3 *** (9.90E-5) (4.11E-4) k 0.093 *** 0.049 *** (0.004) (0.008) l 0.901 *** 0.932 *** (0.005) (0.012) 0.129 (0.092) -0.053 (0.044) 0.024 *** (0.004) -0.010 (0.024) -0.0006 (0.013) 0.033 (0.022) 6.21 E-3 (0.084) -0.045 (0.049) 0.053 *** (0.006) 0.346 *** (0.021) 0.076 *** (0.019) -0.141 *** (0.022) 0.066 (0.052) -0.076 (0.062) 0.645 *** (0.004) 0.804 *** (0.025) -0.001 (0.011) -0.123 *** (0.030) 0.093 ** (0.038) 0.093 ** (0.047) 0.594 *** (0.024) 0.927 *** (0.009) 0.051 *** (0.018) -0.157 *** (-0.019) -0.047 (0.063) 0.047 (0.054) 0.137 *** (0.015) 0.542 *** (0.013) 0.078 *** (0.018) -0.282 *** (0.023) 0.130 (0.083) -0.056 (0.053) 0.500 *** (0.025) 0.836 *** (0.014) -0.001 (0.011) -0.157 *** (0.026) 1.23E-3 *** (1.44E-4) 1.10E-2 *** (0.001) 1.83E-4 *** (1.90E-5) 1.58E-4 *** (1.86E-5) 4.49E-4 *** (4.13E-5) 1.04E-3 *** (1.16E-4) 0.103 *** 0.188 *** 0.062 *** 0.102 *** 0.128 *** 0.162 *** (0.004) (0.007) (0.004) (0.006) (0.008) (0.010) 0.878 *** 0.815 *** 0.918 *** 0.891 *** 0.856 *** 0.823 *** (0.006) (0.007) (0.005) (0.005) (0.007) (0.008) Naural gas 0.005 *** (9.11E-4) -0.136 *** (0.035) 0.093 *** (0.005) 0.745 *** (0.009) 0.019 (0. 020) -0.149 *** (0.019) 1.84E-3 *** (1.69E-4) 0.190 *** (0.009) 0.826 *** (0.006) 18
2 R 0.005 0.015 0.006 0.041 0.196 0.632 0.136 0.302 0.205 Q(4) 5.355 [0.253] 3.266 [0.514] 4.439 [0.350] 14.850 [0.005] 2.220 [0.695] 14.354 [0.006] 15.514 [0.004] 4.008 [0.405] 48.298 Q 2 (4) 3.344 [0.502] 5.127 [0.274] 10.314 [0.035] 1.252 [0.869] 4.053 [0.399] 2.183 [0.702] 16.246 [0.003] 4.756 [0.313] 1.092 [0.895] ARCH(10) 1.396 [0.174] 1.436 [0.158] 1.373 [0.185] 0.249 [0.990] 0.637 [0.782] 0.603 [0.812] 2.442 [0.006] 1.118 [0.343] 0.468 [0.910] Noes: Q(4) and Q 2 (4) denoe he emprcal sascs of he Ljung-Box es for seral correlaon appled o resduals and squared resduals. ARCH(10) refers o he emprcal sascs of he LM ARCH es for condonal heeroscedascy of resduals. K represens he ARCH effec, and l he GARCH effec. The sandard errors are presened n parenheses and he p-values n brackes. ** and *** represen sgnfcance a he 5% and 1%, respecvely. Esmaons are based on Eq. (4). Mos esmaed coeffcens have he expeced sgns, and he hypoheses of seral correlaon and ARCH effecs are rejeced n mos cases. As for he varance equaon, he coeffcens of he GARCH specfcaon have he expeced sgns and are also sascally sgnfcan, suggesng he exsence of he me-varyng paern of he spo reurn volaly and confrmng he presence of he ARCH effecs we found n he reurn seres. However, he coeffcen γ s sgnfcan only for slver and naural gas, and as a resul he hypohess of a mevaryng rsk prema canno be valdaed for he seven oher producs. All n all, our fndngs reveal several neresng facs. Frs, hey underscore a sgnfcan conegraon relaonshp beween he spo and fuures prces, whch favors he frs necessary condon for long-run effcency. Second, he effcency hypohess s raher rejeced n he shor- and long-run. Ths resul mples ha he fuures prces are no good and unbased esmaes of he fuure spo prce and ha he pas prce reurns are relevan o forecas fuure prces. Thrd, we demonsrae some evdence o suppor he assumpon of mevaryng rsk prema. These conclusons can have mporan mplcaons for he energy and precous meal marke parcpans as hey sugges ha here are sll nvesmen opporunes n hese markes hrough speculang on he nformaon ncorporaed on fuures conracs. 4.3.4 Nonlnear ESTECM and effcency per regme I s also possble o mprove he model specfcaon by nroducng nonlneary n he mean-equaon descrbng he spo prce adjusmen. The specfcaon n Eq. (5) corresponds o a wo-regme Exponenal Swchng Transon ECM (ESTECM) ha s ofen useful o characerze he dynamcs of fnancal me seres. In parcular, hs model allows for a dynamc adjusmen beween spo and fuures prce reurns o vary accordng o he prevalng regme. Economercally, hs ESTECM corresponds o he nonlnear form of Eq. (3) whch defnes he dynamcs of spo reurns wh respec o wo dfferen regmes. We esmae an ESTECM for he spo reurn of each commody by he nonlnear leas squares usng 19
he esmaon procedure dealed n Van Djk e al. (2002). The man resuls are summarzed n Table 9. Table 9 Esmaon resuls of ESTECM model Coeffcens WTI Gasolne Heang Propane Gold Slver Planum λ (*10) 0.003 (0.003) ρ 1 0.453 *** (0,052) b 0.006 * (0.026) φ 1-0.039 *** (0.015) ϕ 0.032 1 (0.023) ρ 2 0.416 *** (0,052) δ 13.061 *** (2.617) C 0.156 *** -0.003 (0.008) -0.039 (0.106) 0.145 ** (0.049) -0.019 (0,028) -0.085 * (0.043) 0.098 (0.106) 9.235 (24.496) 0.021 (0.019) 1.921 [0.750] 3.034 [0.552] 0.016 *** (0.005) -0.027 (0.018) -0.042 * (0.025) -0.018 (0.015) 0.056 *** (0.019) 0.071 *** (0.020) 15.611 (13.840) -0.049 *** (0.015) 4.083 [0.395] 54.400 0.003 (0.004) -0.169 ** (0.084) 0.339 *** (0.026) 0.046 ** (0.020) -0.135 *** (0.019) 0.204 ** (0.084) 144.961 ** (71.269) -0.091 *** (0.002) 2.029 [0.730] 14.359 [0.006] 0.000 (0.000) 0.401 *** (0.082) 0.838 *** (0.028) -0.019 (0.017) -0.107 *** (0.033) 0.337 *** (0.075) 0.313 * (0.170) 0.045 *** (0.003 0.527 [0.970] 1.139 [0.887] 0.001 (0.002) 0.137 (0.184) 0.976 *** (0.013) -0.004 (-0.015) -0.098 *** (0.020) 0.656 *** (0.185) 64.721 *** (27.148) -0.033 *** (4.57E-4) 2.755 [0.599] 6.800 [0.146] 0.007 ** (0.003) 0.220 *** (0.044) 0.506 *** (0.022) 0.047 ** (0.018) -0.254 *** (0.020) 0.138 *** (0.045) 0.701 * (0.417) 0.017 (0.008) 7.132 [0.129] 21.101 Palladum 0.002 (0.003) 0.882 *** (0.087) 0.883 *** (0.022) -0.020 (0.017) -0.108 *** (0.026) 0.294 *** (0.085) 27.704 ** (10.864) 0.078 *** (0.005) 3.675 [0.451] 7.507 [0.111] Naural gas -0.003 (0.007) -0.244 *** (0.084) 0.725 *** (0.024) -0.024 (-0.024) -0.038 ** (0.016) 0.299 *** (0.084) 214.039 ** (96.982) 0.188 *** (0.002) 16.076 [0.003] 133.046 (0.002) Q(4) 1.367 [p-value] [0.849] ARCH(10) 3.271 [p-value] [0.513] var(nonlnear) /var(lnear) 0.990 1.003 0.999 0.999 0.994 0.998 0.998 0.997 0.997 Noes: Q(4) denoes he emprcal sascs of he Ljung-Box es for seral correlaon appled o resduals. ARCH(10) refers o he emprcal sascs of he LM ARCH es for condonal heeroscedascy of resduals. The sandard errors are presened n parenheses whle he p-values are n brackes. *, ** and *** represen sgnfcance a he 10%, 5% and 1%, respecvely. Ths able s based on he esmaon of Eq. (5). The resuls repored n Table 9 show ha for mos seres here are no remanng seral correlaons and ARCH effecs n he esmaed resdual seres, whch sugges he appropraeness of he ESTECM. Ineresngly, n all cases excep heang ol, ρ 2 > 0 and ρ 1 + ρ2 > 0. These fndngs hus show srong evdence of nonlnear mean reverson beween spo and fuures prces. Moreover, he rao of resdual nonlnear varance o resdual lnear varance s less han he uny for all seres, excep gasolne, ndcang ha he nroducon of nonlneary enables o mprove he ably of he radonal ECM o forecas he spo prce dynamcs. The lagged values of spo and fuures reurns are also found o sgnfcanly affec he spo reurn dynamcs, and consequenly we can conclude on he rejecon of he weak-form effcency for he energy and precous meal markes under consderaon. The observed nonlneary n he behavor of commody spo reurns s clearly dsplayed va he esmaed ranson funcon. Excep for heang ol and gasolne, he parameers of he ranson funcon are sascally sgnfcan and hus confrm our choce of he exponenal funcon. Two dsnc regmes are denfed for he dynamc adjusmens of spo reurns. Frs, a cenral regme s esablshed and characerzed by small devaons beween 20
spo and fuures prces. Whn hs regme, fuures prces may no conan sgnfcan nformaon regardng fuure spo prces and he prce adjusmen process may no be acve and he arbrage operaons are raher absen. The second regme s acvaed for large devaons of spo prces from fuures prces and when he ranson funcon reaches he uny. In hs case, he arbrage becomes acve and fuures conracs would provde helpful nformaon o forecas fuure spo prces. I s obvous ha he effcency hypohess canno be rejeced f spo reurn dynamcs say n he frs regme. However, when he gap beween he spo and fuures prces ges wder and wder, he adjusmen would become more acve, and hus he marke for a parcular commody would be consdered raher neffcen. Fgure 2 Esmaed ranson funcons Gold Palladum Planum 0.30 0.25 1.0 0.9 0.8 0.7 0.6 Transon Funcon 0.20 0.15 0.10 0.05 Transon Funcon 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Transon Funcon 0.5 0.4 0.3 0.2 0.1-0.06-0.05-0.04-0.03-0.02-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Transon Varable Slver -0.125-0.100-0.075-0.050-0.025 0.000 0.025 0.050 0.075 0.100 Transon Varable WTI -0.125-0.100-0.075-0.050-0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 Transon Varable Gasolne 1.0 1.0 1.0 0.9 0.8 0.8 0.9 0.8 Transon Funcon 0.7 0.6 0.5 0.4 0.3 0.2 Transon Funcon 0.6 0.4 0.2 Transon Funcon 0.7 0.6 0.5 0.4 0.3 0.2 0.1-0.10-0.08-0.06-0.04-0.02 0.00 0.02 0.04 0.06 0.08 0.10 Transon Varable Heang ol -0.35-0.30-0.25-0.20-0.15-0.10-0.05 0.00 0.05 0.10 0.15 0.20 Transon Varable Propane 0.1-0.25-0.20-0.15-0.10-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Transon Varable Naural gas 1.0 1.0 1.0 0.9 0.9 0.9 0.8 0.8 0.8 Transon Funcon 0.7 0.6 0.5 0.4 0.3 Transon Funcon 0.7 0.6 0.5 0.4 0.3 Transon Funcon 0.7 0.6 0.5 0.4 0.3 0.2 0.2 0.2 0.1 0.1 0.1-0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Transon Varable -0.2-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Transon Varable -0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 Transon Varable The esmaon of he ranson speed vares consderably across he energy and precous meal producs. The esmaed ranson funcons, ploed n Fgure 2 agans he ranson varable, llusrae he shfs beween dfferen regmes and show he relaonshp beween spo and fuure prces n each regme as well. For all seres, he ranson funcon 21