Prediction of Oil Price using ARMA Method for Years 2003 to 2011

Transcription

1 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3, No., Jauary 03, pp ISSN: HRMARS Predicio of Oil Price usig ARMA Mehod for Years 003 o 0 Abdalali MONSEF Amir HORTMANI Khadijeh HAMZEH 3 Deparme of Ecoomics, Payame Noor Uiversiy, Ira mosefali@yahoo.com Islamic Azad Uiversiy, Dehagha Brach, Ira Amir_horamai@yahoo.com 3 Isfaha Payame Noor Uiversiy, Ira Pu_alm@yahoo.com Absrac Oe mai purpose of ecoomic aalysis is predicio of ecoomic variables. For his reaso, various mehods have bee developed i his coex. Oe impora challege i predicio of ime series is precise predicio wihou ay compuaioal complexiies. I is usually assumed ha auoregressive movig-average (ARMA models have his abiliy wih a high accuracy. I his sudy, ARMA mehod has bee used o predic ime series of oil price. To deermie he order of his process Akaike ad Schwarz s Bayesia crieria have bee used. The resuls show ha he bes aswers are obaied by ARMA (, 0 model for saic predicios ad ARMA (3, model for dyamic predicios. Key words Predicio, ime series models, ARMA, oil price. Iroducio Predicig or foreseeig fuure codiios ad accides as a key eleme i risk maageme ad corol decisio makigs is very impora for may orgaizaios ad isiuios. I oher words, every orgaizaio eeds o predic is fuure siuaio o make coscious decisios because fial efficiecy of each decisio depeds o he aure of a sequece of accides which occurs subseque o he decisio. The abiliy o guess o-corollable aspecs of hese accides before makig ay decisio allows a beer selecio comparig o he cases i which his abiliy is abse, so maageme sysems ypically have a predicio fucio o pla ad corol operaio of a orgaizaio. O his basis, i seems ha awareess abou he reurs of he ivesme is esseial for ivesig o differe projecs. I oday s world i which uceraiy is icreasig day by day, seppig i he pah of o-reurable ad cosly ivesme wihou ecoomic evaluaio ad esimaio of is beefis ad disadvaages is a idioic ac. Today ivesors ofe ry o simulae fuure ad ake a look a he projec oulook aimig a miimizig risks due o heir ivesmes ad icreasig heir expeced beefis. Lack of awareess abou oucome of he projec ad payig o aeio o ivesmes reurs ca lead o irreparable resuls. I his coex, oil ad gas projecs are especially impora due o heir high coss. O he oher had, uceraiy i his kid of projecs is raher more because prices of hese eergy carriers ad also execuio cos of his kid of projecs have always bee flucuaig because of is variable ad exclusive echology. Therefore, before akig ay acio for ivesme o hese projecs fuure should be simulaed by lookig hrough he pas o reach maximum social beefis by execuio of hese projecs. I oher words, he red i earlier prices of hese producs i global markes should be sudied ad based o hese prices he fuure prices ca be simulaed usig special mehodologies. I his way, he risk due o ivesme o hese projecs ca be reduced.

2 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS The mos impora challege which govermes ad ivesor compaies ecouer is uceraiy abou he price of his valuable aural maerial. This impora is doubled whe oe coury bases he majoriy of is ecoomic o oil ad supplies he mai par of is budge by direc sale of crude oil. O he oher had, refieries, ivesme compaies ad oher acive compaies i operaioal oil projecs are very ieresed o beau are oil prices red ad heir values i he fuure. I his paper, wo hypoheses were iiially cosidered: Shor erm predicio of oil price is feasible by he bes ARMA model ad is red i ascedig. Evaluaio crieria cofirmed he accuracy of predicio by he bes ARMA model ad will predic he oil price well. The mai purpose of his sudy is preseig he bes ARMA model for shor-erm predicio of oil prices.. Crude oil marke As oil marke is compeiive crude oil prices varies severely, which iesely iflueces oil rasacios paers. Before oil revoluios i 970s, oil marke had bee almos maaged by large oil compaies, herefore oil coracs mosly were performed uder he framework of fixed-erm coracs. This paer had bee useful if oil price remaied sable because relaive sabiliy of oil price allowed producers ad cosumers o pla beer for heir producio or cosumpio uder fixed-erm coracs. I he circumsaces ha prices chage severely; hese coracs are o ecoomical because hey resul i heavy profis or losses (Derakhsha, 7: Spo markes Firsly, i should be oed ha cosiderable perceage of produced crude oil is direcly sold o refieries or oher applicas by producers wih posed or official prices. This kid of coracs is usually cocluded for oe moh o oe year ad hey ca be exedable. Noeheless, oil compaies usually have o accurae esimaio of marke demad for crude oil, so fixed-erm coracs cao fully fulfill heir eeds. Usig fixed-erm coracs causes ha oil compaies have someimes oil shorage for sale ad someimes surplus oil. Spo markes which are also called ope marke ca balace surplus ad shorage supply of oil compaies, i.e. oil compaies ca sell heir surplus supply or buy heir oil shorage i spo markes (Derakhsha, 33:004.. Forward markes ad fuures markes Forward markes have bee developed for various kids of crude oil bechmarks such as Bre Bled of Norh Sea i Europe ad WTI i U.S.A ad Dubai crude oil i Persia Gulf. To become familiar wih forward marke, i is eough o oe ha mohly producio of Bre Bled is allocaed bewee he compaies o which owership of Norh Sea oil belog as cargo los of 500,000 barrel. However he value barrels is a compuaioal basis for each cargo lo, his value ca chageup o ±0 perce. I is usually plaed o load 30 o 40 cargo los per moh. Oil producio is mohly allocaed o he compaies ad loadig is performed i hree days (Derakhsha, 35-36:004. I forward rasacios of Bre crude oil, he seller gives a buyer a miimum 5 days oice of he ieded hree daes for loadig crude oil cargo durig his 5 days. The cargos which are loaded durig 5- day period are so called Daed Bre while forward cargos are so called 5-day Bre (Derakhsha, 36:004. Therefore, raders i forward markes for Bre crude oil have a Porefeuille or Porfolio o sell ad buy5-day Bre coracs wih various prices. These raders ca rade o a day basis. Similarly, ew buyer ca receive he crude oil cargo or rasfer is righ o hird buyer. I his way, buyers ca exi he marke wihou receivig crude oil ad jus by rasferrig heir righs o he ohers. Traders profi ad loss is deermied hrough book-ous seleme or briefly seleme (Derakhsha, 36-37:004. Crude oil fuures marke is he compleed form of crude oil forward markes because i hese markes crude oil is raded i sadard uis which are so called corac. The corac rades i uis of 7

3 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS,000 barrels of crude oil whose qualiy mus be sadard. Sadard crude oil allows raders o be iformed abou he properies of raded crude oil i Exchage, specially is uspecific graviy (Derakhsha, 37: Time series aalysis mehods Time series paers which are ofe used for shor-erm predicio ry o explai he behavior of a variable based o is pas values (ad possibly oher cosidered variables. These paers are able o make accurae predicios whe he ecoomic paer has uclear ifrasrucure. I corary o ecoomeric paers which eeds saisical daa ad ecoomic heories, ime series paers jus work wih saisical daa of variables wihou ay eed o ecoomic heories. These paers which relae curre values of a variable o is pas values are uivariae ime series models some of which are auoregressive processes, movig average processes, auoregressive movig average processes ad auoregressive iegraed movig average processes (Nofresi, 7-8:999.. Movig average process: y series have movig average process if: y = α + θ(lε Where: ( θ(l = + θ L + θ L +... ε ~iid(0,σ + ε θ L q q ( Above process is he movig average model of order q which is represeed by MA (q. I ca be easily proved ha all movig average processes are saioary.. Auoregressive movig average process: his process is a combiaio of auoregressive ad movig average processes ad is wrie: φ(ly = α + θ(lε (3 Equaios ad are also valid i he above formula. The same cosrais as hose i auoregressive processes are eeded i order ha he model remais saioary. These process are called auoregressive movig average models of order p ad q ad are represeed by ARMA (p,q. 3. Auoregressive iegraed movig average processes: if a ime series shows evidece of osaioary ad o-saioary ca be removed by differecig i d ime, i ca be wrie as: d φ(l( L y = α + θ(lε (4 Where p ad q represe he order of auoregressive ad movig average processes ad d deoes he order of differecig required i order ha y become saioary. This process is referred by ARMA (p, d, q. Here equaios ad are also valid. I fac, all processes meioed above are a special (simple form of his process. For example if p = q = d = 0, a radom walk process wih a acceleraio erm is obaied ad if p =, d = q = 0, a auoregressive process of order is obaied (Abasi, 48:007. I his paper, below crieria were used o deermie p ad q; a Akaike Iformaio Crierio AIC(p = Tlogσ ^ p + p b Schwarz Bayesia Iformaio Crierio SBIC = Tlogσ ^ p + plogt (5 (6 73

4 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS c Haa-Qui Iformaio Crierio HQIC(p = Tlogσ ^ p + ploglogt The followig crieria were also employed for comparig predicios of various models: a Mea Squared Error(MSE or Roo Mea Squared Error(RMSE MSE = T + = (yˆ y (7 (8 RMSE = T + = (yˆ y b Mea Absolue Deviaio(MAD or Mea Absolue Perceage Error(MAPE MAD = MAPE = = = yˆ y yˆ y y c Their Iequaliy Coefficie TIC = yˆ = = (yˆ y + y = I all above meioed crieria, is he sample size ad is he period i which predicio is made (Abrishami, Mehrara, 8-5:00. Fially i order o deermie wheher performace of he models based o above crieria is saisically sigifica or o, below crieria were used:. F es: if square of firs model error is greaer ha secod model, his crieria ca be wrie as: F MSE H i= = = H MSE e e i i i= ( Uder he ull hypohesis, i.e. equaliy of boh wo square values of predicio error, ad below codiios his value has a f-disribuio wih (H, H degree of freedom. H is he umber of predicio periods. Predicio errors have a ormal disribuio wih a zero mea. Predicio errors have o correlaio. 3 Predicio errors are coemporaeously ucorrelaed. Bu akig io accou ha i mos cases hese codiios are o me, wo followig ess are also recommeded: (9 (0 ( 74

5 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS. Grager-Newbold es: his es has bee suggesed assumig ha wo firs codiios are me. Assume ha: x = e i + e,z = e i e If firs ad secod model errors are equal, correlaio coefficie bewee x ad y (r xz is equal o zero. Therefore, uder ull hypohesis, equaliy of predicio errors, below value has disribuio wih H- degree of freedom: rxz GN = ( rxz H (4 If his correlaio coefficie is posiive, he firs model has a greaer error, oherwise secod model has a greaer error. 3. Diebold-Mariao es: Whe firs ad secod codiios are o me, his mehod is highly applicable. I addiio, oher predicio crieria such as absolue perce predicio error (or oher fucios such sag (e i ca be used i his mehod. We have: d = H H [ g(ei g(ei ] i= Null hypohesis or equaliy of predicive power is valid i boh wo models. Therefore, wo siuaios ca be recogized: a Terms d i ( di = [ g(ei g(ei ] are ucorrelaed: cosequely below phrase has a disribuio wih H- degree of freedom: DM = d γ0 (H (6 Where γ0is sample variace of d i series. b Terms d i are correlaed. I his circumsaces, below saic s has a disribuio wih H- degree of freedom uder ull hypohesis: DM = (γ 0 + δ d γ (H q Where γ i is he i value of auocorrelaio of d series. q Is he iiial value of i 4. A review o previous researches (3 (5 (7 γ.(eders, 8-86:00 Due o imporace of he subjec, may researchers have bee coduced i his area ad may mehods have bee used. There is a wide rage of mehods from simple liear regressio mehod o advace oliear mehods. Some of hese researches are as follows: Posali ad Picchei (006 preseed a quaiaive aalysis for oil price pah. They idiced ha oil price durig 00 years have wo edogeous srucural breaks. Therefore, ull hypohesis of ui roo was rejeced. Their resuls showed ha oil price ca be prediced usig geomeric Browia mehod. I his research, o oher predicio mehod (e.g. ARMA was used o compare he resuls. 75

6 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS Malik ad Nasereddi (006 used differe predicio mehods such as arificial eural ework o forecas Gross domesic produc of U.S.A wih he help of oil prices. They foud ha eural ework mehod lead o he miimum predicio error. I had bee saisfacory if hey had compared predicive power of differe mehods usig appropriae crieria. De Sais (003 ried o explai why oil prices flucuae. For his aim, a geeral equilibrium model was applied o Saudi Arabia which is a big oil producer. The resuls showed ha Saudi Arabia s share i OPEC ad some decisios of OECD couries ca ifluece oil prices. Tag ad Hammoudeh (00 modeled world oil price behavior usig firs-geeraio arge zoe model. They used daa durig he period of ad revealed ha OPEC ried o maiai a weak arge zoe regime for he oil price. I addiio hey foud ha based o his model (firs-geeraio arge zoe oliear models show higher predicive power. Buchaaa, Hodges ad Theis (00 preseed a mehod o predic he direcio of aural gas spo price movemes. They revealed ha raders play a impora role i deermiig he direcio of aural gas price moveme. All above meioed researches preseed o log-erm or shor-erm red for oil price ad mosly cosidered aalysis aspecs of oil relaed issues. I oher words, heses researches mosly cosidered descripive aspecs of he research ad less used mahemaical ad compuaioal models ad validiy ess. Azar ad Rajabzadeh (003 prediced demad for OPEC oil usig Box-Jekis mehod. They used aual daa durig he period of 960 o 00. They also employed mea absolue ad mea squared error crieria o evaluae he models. However his research is more comprehesive ha he oher meioed researches, some deficiecies such as use of oo few crieria for model selecio, lack of sigificace esig of differece bewee validiy crieria ad use of low orders of ARMA ca be observed i his research. 5. Esimaio of he models ad heir resuls Daa durig he period of early 003 o May 0 was used o esimae ime series models. Takig io cosideraio he modelig priciples of ime series, which were described previously, saioary of respecive ime series was examied. If he series is eiher level saioary or red saioary, i ca be easily modeled; oherwise i mus be differeced o become saioary. For his aim, ADF es was used. Prices of wo well-kow bechmark ypes of oil, i.e. Bre ad WTI, always closely rack each oher ad i oher words each oe is a muliple of he oher. Therefore, we used prices of Bre oil which is more wellkow hroughou he world ad based o which mos of crude oils are classified. Curve. Saioary es was performed for cosidered ime series usig augmeed Dickey-Fuller es. The resuls i Table show ha his series is saioery a he 5% level. Ref: he research resuls 76

7 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS Table. Dickey-Fuller es resuls es saisics Value Criical values Sigificace level -4/05439 % -3/ /4563 %5-3/5399 %0 Ref: he research resuls 5.. Dyamic predicios Modelig was performed afer his es wihou ay eed o differecig. Table liss he resuls of his modelig. As i is observed i his Table, ARMA (3, ad ARMA (3, 3 models are largely differe from he oher models. Table. Saisics of roo mea squared error mea absolue error ad Theil iequaliy coefficie i dyamic siuaio RooMea Squared Error(RMSE Mea Absolue Error(MAE Theil Iequaliy Coefficie (TIC model ARMA(, ARMA(, ARMA(0, ARMA(, ARMA(, ARMA(, ARMA(, ARMA(0, ARMA(3, ARMA(3, ARMA(3, ARMA(0, ARMA(, ARMA(, ARMA(3, Ref: he research resuls I should be oed ha ime series models wih auoregressive ad movig average of lag greaer ha 3 were also esed, bu sice o improveme was observed i predicio saisics, accordig o parsimoy priciple hey were o used. Based o hese facs, eiher ARMA (3, or ARMA (3, 3 model mus be seleced for dyamic predicios. Table shows he resuls of compariso bewee wo meioed models. As i is obvious, oly Grager-Newbold crierio differeiaes bewee wo models ad cosiders ARMA (3, predicio as a beer predicio ha ARMA (3, 3, bu o sigifica differeces ca be disiguished by wo oher crieria. Table 3. Predicio evaluaio ess: compariso of ARMA (3, ad ARMA (3, 3 models Tes Saisics P-Value The F Tes The Grager-Newbold Tes The Diebold-Mariao Tes 3- g(e=e g(e=e Ref: he research resuls 77

8 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS I oher words, however absolue values of he error calculaed by various crieria are differe, hese differeces is o saisically sigifica. Based o his fac ad parsimoy priciple, he simples model, i.e. ARMA (3, was seleced. I addiio, he absolue values of predicio error i his model are smaller ha hose i oher models. Therefore, he bes model for dyamic predicios is ARMA (3, model ad he bes order for MA ad AR is ad 3, respecively. 5.. Saic predicios A his sage, saic predicios were made usig he esimaed models ad desirable saisics were calculaed. These saisics are lised i able 4. A problem which arises i his codiio is ha he resuls of he majoriy of models are very close o each oher ad his makes he selecio of wo models ad heir compariso difficul. Oe soluio o his problem is ha models wih iordiae differeces are ake aside (models wih o AR process ad amog oher remaied models, he es is performed o wo models wih maximum differeces. If o sigifica differeces are observed bewee he wo models, i ca be cocluded ha he models whose predicio saisics are locaed i his ierval, show o sigifica differeces ad agai accordig o parsimoy priciple he simples model is seleced. Table 4. Saisics of roo mea squared error mea absolue error ad Thiele iequaliy coefficie i saic siuaio model Roo Mea Squared Error(RMSE Mea Absolue Error(MAE Theil Iequaliy Coefficie(TIC ARMA(, ARMA(, ARMA(0, ARMA(, ARMA(, ARMA(, ARMA(, ARMA(0, ARMA(3, ARMA(3, ARMA(3, ARMA(0, ARMA(, ARMA(, ARMA(3, Ref: he research resuls Based o Table 4, amog models which have AR process ARMA (, 0 ad ARMA (, 0 models have maximum differeces i predicio saisics, herefore he es is performed o hese wo models. Table 5 liss he resuls of his es. Table 5. Predicio evaluaio ess: compariso of ARMA (, 0 ad ARMA (, 0 models Forecas Evaluaio Tes Saisics P-Value The F Tes The Grager-Newbold Tes The Diebold-Mariao Tes 3- g(e=e g(e=e Ref: he research resuls 78

9 Ieraioal Joural of Academic Research i Accouig, Fiace ad Maageme Scieces Vol. 3 (, pp. 7 79, 03 HRMARS Based o he obaied resuls, oe of he ess cofirms he differece bewee he wo models. I oher words, o saisically sigifica differece is observed bewee he wo models which had maximum differece i predicio saisics. Therefore, all models whose predicio saisics are locaed i his ierval, have he same performace. The, accordig o parsimoy priciple he simples model, i.e. ARMA (, 0, is seleced ad he bes order of AR ad MA processes is ad 0, respecively. 6. Coclusios ad recommedaios The ess resuls revealed ha amog he models, ARMA (3, is he bes model for dyamic predicio of crude oil price series akig io cosideraio all aspecs of modelig. For saic predicios, i should be oed ha he majoriy of he models lead o very close resuls ad comparaive es showed ha hese models have o saisically sigifica differeces. Therefore, he simples model, i.e. ARMA (, 0, was seleced for predicio. Also i was foud ha saic predicios are always more accurae ha dyamic predicios. This is aribues o how daa are used i hese wo kids of predicio. I real world, dyamic predicios are more impora ad saic predicios are also possible for jus oe sep forward. Fially he bes seleced models for saic ad dyamic predicios were used o make a fuurisic predicio for fuure ime periods. I is obvious ha i saic codiio oly oe sep forward predicio was possible ad he prediced value was obaied o be 44.5 dollars per oil barrel. I dyamic predicio, wo seps forward predicios were obaied o be equal o 83.8 ad 83.9 dollars per barrel (for ex wo moh. As discussed, i oe sep forward predicios, he prioriy is give o saic predicios. Refereces. Abrishami, Hamid & Mehrara, Mohse, (00. Applied ecoomerics (ew approaches, firs ediio, Uiversiy of Tehra press.. Azar, Adel & Rajabzade, Ali, (003. Evaluaig compoud predicio i ecoomics area hrough classic eural ework approaches, ecoomic joural of Uiversiy of Tehra, fall ad wier, No Derakhsha, Masoud, (004. Risk maageme ad derivaives i oil markes, ieraioal isiue of eergy sudies. 4. Abbasiejad, Hossei, (007. Advace ecoomerics, firs ediio, Baradar press. 5. Nofresi, Mohammad, (999. Ui roo ad Co iegraio ecoomerics, firs ediio, Rasa culural services isiue. 6. Buchaaa W.K., Hodges, P, Theis. J (00. Which way he aural gas price: a aemp o predic he direcio of aural gas spo price movemes usig rader posiios. Joural of Eergy Ecoomics 3, De Sais. R. (003. Crude oil price flucuaios ad Saudi Arabia s behavior. Joural of Eergy Ecoomics 5,