DO WTI OIL PRICE MOVEMENTS FOLLOW A MARKOV...

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DOWTI OIL PRICEMOVEMENTSFOLLOWAMARKOVCHAIN?

ByRichardR.ConnCMA,MBA,CPA,ABV,ERP

Thispaperisthefirstinatwo‐partserieswhichis,ultimately,dedicatedtooutliningamethodologyforpredictingfutureoilpricesusingMarkovproperties.Farfrombeinguniqueorinnovative,therehavebeenmanyexcellentacademicworksonthissubjectalreadywritten(seeReferences).However,thedifficultywithacademicpapersisthattheyaretypicallywrittenfortheacademiccommunityandnottheaveragepractitioner.Thispaperissquarelyaimedatprovidingthetypicalfinancialanalyst(whoalreadyhasarudimentaryunderstandingofmatrixalgebraandthebasicqualitiesofMarkovprobabilities)withanoverviewofcreatingaWTI(WestTexasIntermediate)MonteCarlo‐styledoilpricepredictionmodel.

Theprimarydifferencebetweentheapproachtakenherecomparedwiththeacademicpapersisthatthefocusofthisserieswillbeuponforecastingthefuturestatesoftheoutcomes(i.e.directionallythestepsinvolvedingettingfromthecurrentWTIleveltosomefuturepredictionofWTI)comparedwithestimatingthefuturepricelevelofWTI.Mostoftheacademicpapersareintentuponestimatingwhetherfutureoilpricewillbehigherorlowerthancurrentpricesand,ifso,buthowmuch.We,instead,willbeinterestedinpredictingtherandomwalkthatWTIislikelytotakebetweennowandthelong‐termforeseeablefutureratherthanuponattemptingtoestimatethesizeofeachofthosesteps(i.e.themagnitudeofthepricechangeateachpointintime).Onlyafterwehavedevisedapaththatfutureoilpricemovementsarelikelytofollowwillweturnourattentiontothequestionofestimatinghowlargeeachofthosepricechangesreasonablycanbeexpectedtobe.Thepracticalityofthisapproachwillbecomeevidentasthemethodologyisunveiled.

ANOVERVIEWOFTHEMECHANICS

Whiletheexplanationwillbetedious,thedetailsofwhattheMarkovpropertiesareandwhytheyareimportanttothemovementsofoilpricesisunavoidable.Atthesametime,aslittlespaceaspossiblewillbededicatedtowardstheunderlyingtheory–thisseriesisdedicatedtowardscomingupwithapracticalandworkingmodel.

MarkovChainsareprobabilisticmodels/observationsaboutmovementswithinastatespace.Consideringthemovementofoilprices(andweshallonlyconcernourselveswiththemovementofWTI)thereareonlythreestates.Attheendofanygivenday,forexample,theWTIpricecouldhavemovedup,moveddownorstayedthesame,relativetotheclosingpriceofthepreviousday.Thislatterstatewouldbedescribedasa‘loop’–thepriceloopedbacktowhereitwassuchthatthenew‘state’isthesameastheold.

Suchanobservationshouldleadtheanalyticallycurioustothequestion:‘WithwhatfrequencyhaveWTIpricesincreasedinthepastrelativetohowoftentheyhavedeclinedorremainedthesame?’BeingabletoanswerthisquestionmayleadtobeingabletomakebetterpredictionsaboutthefuturepathofWTIprices.IfweabbreviatesuchmovementsasD,S,UforDown,Same,Uprespectively,thenwehaveashorthandmeansofquicklysymbolizinglongstringsorchainsofpricemovements.If,forexample,wearedescribingasequenceofpricechangesoversixdays(i.e.fivemovements)asDDSUD,then,thatmeanstheSeconddayendingpricewasDownfromthefirst;theThirdwasDownfromthesecond;theFourthwastheSameastheThird;theFifthwasUpfromtheFourth;andfinallytheSixwasDownfromtheFifth.

TheprimarycharacteristicofMarkovianmovementsbetweenthevariousstatesisthatonlythecurrentpositionwithinthestatespaceisinanywayrelevanttothenextmovement.Intheexampleabove,ifweareattemptingtopredictwhereWTIwillbeattheendofDaySeven(eitherD,S,orUrelativetoDaySix),theonlyinformationthatisusefulinmakingthatpredictionisthatDaySixendedasa“D”day.ThefactthatthepreviousfourmovementspriortoDaySixwereDDSUinnowayimprovesourstatisticalchancesofcorrectlyestimatinghowDaySevenwillend.AnotherwaytosaythisisthatXXXXDD(whereDaySevenendsasa‘Down’Day,forexample)hasjustasmuchprobabilityofoccurringasDDSUDDorUUUUDDorUSDSDDoranypermutationwhereXXXXrepresentsanyrandomstringof“D”,”S”,”U”(andthemathematicallyastutewillrecognizethatthereare3^4=81permutationspossiblehere).InotherwordsthehistoryorpaththatWTIpricestookbeforearrivingattheDofDaySixhasnobearinguponthedirectionthatpriceswilltakeonDaySeven.AnotherwayofsayingthisisthatthepricemovementonDaySeveniscompletelyindependentofthedirectionofthepricemovementsatanytimepriortoDaySix.

WhyshouldwecareifWTIpricesexhibitMarkoviancharacteristics?Because,ifWTIpricesareindependentofthepathupontheytooktogettothecurrentstateANDwecanidentifyordetermineaprobabilityforwhichstatetheymightnextmoveto,thenthiswillquiteeasilyallowustobuildasimpleprobabilisticmodelforpredictingwhereWTIpricesmaybegoing.Conversely,if

CurrentWTIPrice

FIGURE1

NewHigherPrice(U)

NewLowerPrice(D)

PriceStaysSame(S)

WTIpricesarepathdependent,thenthelevelofcomplexityinvolvedincalculatingwheretheycannextbeexpectedtomovetoincreasesexponentially.Imagine,forexample,thatitisnowDaySixwhichhasendedasaDDayandtheprobabilityofwherepricemayendtomorrowisdependentuponthepathpricestookinthepreviousfivedays(i.e.thefourpricemovementspriortothe“D”ofDaySix).Thatwouldmeanthateachofthepossible81differentpricepathsthatmayhaveoccurredinthelastfivedayscouldhaveitsownuniqueprobabilityindeterminingwherepricemovestoonDaySeven.Andnowimaginehowcomplexitmaybetopredictpricemovementsiftomorrow’spricewereinsomewaydependentuponwhichpricepathhadbeentakenovertheprior50or100or1,000tradingdays?Thatwouldbe3^49or3^99or3^999possiblepermutationsrespectively–numberstoolargetocomprehend,muchlessassignprobabilitiesto.

SowewantWTIpricemovementstopossessMarkoviancharacteristics,butwecannotsimplyassumethattheydo.SomeformaltestprocedureneedstobeappliedtoempiricalWTIhistoricdatainordertoassessifthosemovementshavefollowedaMarkovchain(i.e.arepathindependent)

ANINTUITIVEAPPROACH

EvenbeforeconsideringamorerigorousapproachthatinvolvestheuseofmatrixalgebraandtheChi‐squaretest,thereisanintuitiveapproachthatshouldallowustosetalower‐boundaryabouttheprice‐pathdependencyofhistoricWTIpricemovements.WhilethisinformalmethodwillnotallowustoabsolutelyconcludethatWTIpricemovementsdofollowaMarkovchain,itwillalertusintheeventthattheyclearlydonot.

IfWTIpricemovementsareMarkovianandprice‐pathindependent,thenitisonlythecurrentstatethathasanybearinguponwherepricewillmovenext.Thismeansthat,selectingasub‐sampleof,sayDD,pricemovementsfromaverylargesampleofWTIpricehistorywewouldexpecttofindthesamerelativefrequencyofDDpricechainsastheoccurrenceofXDDpricechains.Thatis,giventhattheprobabilityforpricestodeclinetwiceinsuccession(whichnecessarilywouldrequirea3dayobservationperiod),wewouldexpecttofindthesameprobabilityforanXDDpricechaintooccuroverafourdayobservationperiod(wherethe“X”couldrepresenteitheraUp,DownorSamemovement).Ifthiswerenotthecase,thenonemightbegintosuspectthatprice‐pathhadsomebearinguponwhatcoursethethirdpricemovementtook.

Expandingthislogic,ifposthocWTIpricemovementsareprice‐pathindependent,wewouldexpectXXDDtooccurwiththesamerelativefrequenciesasDDchains.And,similarly,XXXXXXXDDtendayobservationsshouldoccurwiththesamefrequencyandsoshouldone‐hundred‐daychainsendinginDDoccurinjustthesamefrequency.1

1Thecaveathereisthatwearespeakingaboutrelativefrequency.If,forexample,oursubsampleis1,001dayslong,thereare1,000chainsoflengthtwopossible.Thisisbecause,onthefirstday,thereisnomovement,soittakestwodaysuponwhichtoobservethefirstmovement.Sequentially,then1,000two‐periodchainscouldbeobservedovera1,001daysubsample.Ifnisthenumberofdaysinthesubsample,andkisthelengthofthechain,then,n‐(k‐1)=numberofchainspossible.Therefore,1,001–(2–1)=1,000.Similarly,ifwewanttoobservethefrequencyoflengththreechains,nowtherewouldonlybe1,001–(3–1)=999chainspossible.InthatcasewemaybemeasuringtheratioofXDDchainsthatoccurover999possible

EIAEMPIRICALEVIDENCE:TheU.S.EnergyInformationAdministration(EIA)maintainsalargedatabaseofhistoricalWTIprices(spotprices,measuredatCushing,OK)andmakesthatavailableforeasydownloadineither,daily,weeklyormonth‐endform.ThatdatabasebeginsonJan.2,1986andthereforeincludesapproximately7,100tradingdayobservationsuptopresentday(thispaperwaswritteninearlyMay,2014).Bydrawingourconclusionsfroma28yearsample,weareimplicitlyconditioningourresultstobelong‐termedinnature.Anysingle‐eventWTIpriceshockthathasoccurredsinceJanuary1986will,tosomeextent,bereflectedinthedata–buttemperedbythelong‐termstatusquo.Similarly,anyshortormid‐termcyclesthattheWTIhasexhibitedwillbesomewhatmitigatedinourresults.Hadwewantedtoconstructamid‐termestimatorofWTIprices,wewouldhavebeenfacedwiththedifficultyofdecidingwhichportionofthehistoricdatabestreflectedtheexpectedcharacteristicsoftheforthcomingmid‐term.Wewill,however,examinethedataforshortandlong‐termvolatility,withtheintentofidentifyinganypermanenttransitionsthatmayhaveoccurredovertime.Butwewillnotdosountilafterourobservationsaboutfrequencyhasbeenpresented.

outcomes.Similarly,achainoflength100(i.e.wouldrequire101observationdays)couldonlypossiblyoccur1,001–(100–1)=902inthissubsampleandthereforeitwouldbefrequencyofthenumberlength100chainsendinginDDrelativetothe902possibleoutcomesofachainthislengththatwouldprovidetheimportantstatisticinthiscase.

TABLE1:FREQUENCYOF‘XX’ENDINGCHAINSINEIADATA

Two Day Chains:

Day n Observation

Day (n + 1) Observation

D 22.11% 0.71% 24.63%

S 0.83% 0.07% 0.84%

U 24.50% 0.95% 25.37%

Ten Day Chains:

Day n Observation

D 22.10% 0.71% 24.63%

S 0.83% 0.07% 0.84%

U 24.50% 0.95% 25.37%

One-Hundred Day Chains:

Day n Observation

D 22.03% 0.71% 24.67%

S 0.82% 0.07% 0.84%

U 24.54% 0.95% 25.37%

One-Thousand Day Chains:

Day n Observation

D 22.49% 0.60% 24.67%

S 0.65% 0.08% 0.72%

U 24.61% 0.76% 25.42%

Lookingfirstatthe2DayChaindata,attheintersectionofDD,thistellsusgivenallthepossible2daychainsthatcouldhavebeenobservedintheEIAdata(therewere7,143)22.11%ofthetimeaDownmovementonDaynwasfollowedbyasubsequentDownmovement(i.e.DD)onDayn+1.Similarly,25.37%ofthetimeanUpmovementwasfollowed,onthenextday,byanotherUpmovement(i.e.UU).Movingtothedatarepresentingthe1,000daychainswefindthat,ofallthepossible1,000daychainsthatcouldhavebeenobservedinthatdata(ofwhichtherewere7,144–(1000–1)=6,145uniqueobservations),weseeverysimilarstatisticsasthein2Dayobservations.Thereisroughlya22.5%chanceofobservinga1,000daychainthatendsinDD.Thereisonlyasixth‐tenthsofone‐percentchanceofobservinga1,000daychainthatendsinSD.And,thereisapproximatelya25.4%chanceofa1,000daychainendinginUU.Theseareverysimilarnumberstotheobservationsinthe2daychainswhichare,inturn,veryclosetothosestatisticsobservedwiththe10daychains

andthe100daychains.Hadweobservedadramaticallychangingstatisticasthechainsgotlonger,wemighthavebeguntosuspectthattheWTImovesinaprice‐pathdependentmanner.Butthedailyoutcomes2haveremainedremarkablyconsistentregardlessofthelengthofthechain,thereforeitisappropriatetodosomemoreformaltesting.

MARKOVPROBABILITIESINMATRIXFORM

ItisconvenienttopresentMarkovsingle‐stagedatainMatrixform.Foreaseofexposition,wewillspeakofthedailypricemovementswhenexplainingthemethodology,butlatertheweeklyobservationswillalsobepresentedandcomparedtothedaily.Therearetwopresentationalternativespossible(oneisthetranspositionoftheother)andafewwordsonMatrixprotocolareprobablyinorder.

FIGURE2

Day n Observations

Down Same Up

Day (n +1) Observations

Down DD SD UD

Same DS SS US

Up DU SU UU

100% 100% 100%

Day (n + 1) Observations Down Same Up

Day n Observations

Down DD DS DU 100% Same SD SS SU 100%

Up UD US UU 100% Thedifferencebetweenthetwopresentationsabovedealswithwhetherweareidentifyingthe‘Dayn’observationsasrowsandthe‘Dayn+1’observationsascolumns,ortheotherwayaround.Theconventionusuallyistheformer,‘Dayn’arerows,‘Dayn+1’arecolumnsand,asaresult,thetotalofeachrowsumsto100%.Thedifferencebetweenthetwoisamatterofpersonalpreference,butitisimportanttokeepinmindthatthestatisticweareinterestedinisthatgiventhatweareatcurrentstateXasat‘Dayn’,wewishtoknowtheprobabilityofmovingtostateXon‘Dayn+1’.Forexample,giventhatwehaveobserveda‘DState’on‘Dayn’weareinterestedinlearningthe

2ForreasonsofspacetheweeklystatisticswillnotbepresentedherebyreservedforAppendix1.Similartothedailystats,thereisagooddealofconsistencymovingbetweenthelengthsofchains.OnenotabledistinctionbetweentheDailyandWeeklyobservationswasthatitwasnoticeablymorelikelyforatwo‐weekchaintofinishonaUU(about30%ofthetime),thanitwasforatwo‐daychaintodoso(onlyabout25%ofthetime).

frequencywithwhichthistransitionstoa‘DU’chain,forexample,onDayn+1.Therefore,thesumofallthe‘DX’chainsinthatsubsampletotal100%(which,bydefinition,theymust).Itwouldbeamistaketoobtaintheteststatisticinthewrongorder.Forexample,todeterminethefrequencyof‘XD’.Inwordsthiswouldbeakintoaskingourselves,giventhefactthat‘Dayn’hasendedinthe‘DState’whatistheprobabilityofobservingthe‘XState’on‘Dayn–1’?SuchanobservationwouldnotbeinaccordancewithMarkoviancharacteristics.Forreasonsofpersonalpreference,thispaperwillusetheunconventionalDaynObservationsascolumns(ergo,theColumnswillsumto100%),andapologizesareofferedtothosereadersmoreaccustomedtothemorefrequentformwheretherowssumto100%.

MARKOVPROBABILITIESOBSERVEDINWTIPRICEMOVEMENTSSINCE1986

ThefollowingtableshowstheprobabilityofobservinganXpricemovementonDayn+1giventheoccurrenceofanXmovementontheprecedingday:

TABLE2:MARKOVPROBABILITIESINWTI

2 Day Chain3 Day n Observation

D 46.61% 41.13% 48.44% S 1.74% 4.03% 1.65% U 51.65% 54.84% 49.90%

100.00% 100.00% 100.00%

10 Day Chain Day n Observation

D 46.60% 41.13% 48.44% S 1.74% 4.03% 1.65% U 51.66% 54.84% 49.90%

100.00% 100.00% 100.00%

D 46.48% 40.98% 48.49% S 1.74% 4.10% 1.65% U 51.78% 54.92% 49.86%

100.00% 100.00% 100.00%

D 47.10% 41.57% 48.56% S 1.36% 5.62% 1.41% U 51.53% 52.81% 50.03%

100.00% 100.00% 100.00%

3Somelicenceistakenwiththenamingconventionhere.Recognizethatanobservationoftwoperiods(e.g.SU)requiresthreeseparatedaysbecauseitcanonlybeknownwhatthe‘n’daymovementisattheendofthesecondday.Similarly,then+1dayobservationcanonlybecompletedatendofdaythree.Therefore,whenwespeakofaTwoDayChain,wereallymeanthreedayshaveelapsed.Further,forthe10‐DayChain,Day‘n’istheobservationtakenonDay10(the9thpricemovement)andDay(n+1)istheobservationtakenonDay11.Usingposthocdatalookingbackwards,wearealwaysinterestedintheXXmovementsattheendofthechain.ThiswillnotbethecasewhenweuseMarkovprobabilitiestopredictfuturepricemovements.

Similartoourobservationsregardingthesimplefrequenciestable,wenotethattheMarkovprobabilitiesalsodisplayasurprisingconsistencyregardlessofhowlongthepricemovementchainis.Itisnotasintuitivelyobvious,inthiscasewhetherthisconsistencyisagoodthingornot.However,thestatisticalinterpretationofoutcomesissimilar.Lookingatthe2DayChaindata,forexample,ofalltheoccasionswhereanoccurrenceofDXwasobserved,46.60%ofthosetimesresultedinaDDobservation,1.74%ofthetimeaDSwasobservedand51.65%ofthetimesaDUeventhadoccurred.BeforeproceedingonwithamoreindepthdescriptionofMarkovprobabilitiesandhowtheymightbetested,itwillbeusefultodigresstoacomparisonoftheDailyvs.WeeklyMarkovMatricesasthisinformationwillbecomerelevantlaterinthediscussion.ThefollowingissimplyarepeatofTable2asabove,exceptnowtheweek‐endclosingpricedatahasbeenjuxtaposed(ontheright)alongsidethedailydata:

TABLE3:DAILYVSWEEKLYMARKOVPROBABILITIES

Daily WTI Price Chain Data Weekly (i.e. week-ending) WTI Price Chain Data 2 Day Chain 2 Week Chain

Day n Observation Weekly n Observation

D S U D S U

D 46.61% 41.13% 48.44% Weekly (n +1) Observations

D 51.66% 50.00% 42.51%

S 1.74% 4.03% 1.65% S 0.58% 0.00% 0.26%

U 51.65% 54.84% 49.90% U 47.76% 50.00% 57.23%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

10 Day Chain 10 Week Chain Day n Observation Weekly n Observation

D S U D S U

D 51.17% 50.00% 42.44%

S 1.74% 4.03% 1.65% S 0.59% 0.00% 0.26%

U 51.66% 54.84% 49.90% U 48.25% 50.00% 57.31%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

D S U D S U

D 51.55% 50.00% 42.47%

S 1.74% 4.10% 1.65% S 0.62% 0.00% 0.27%

U 51.78% 54.92% 49.86% U 47.83% 50.00% 57.26%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

D S U D S U

D 51.14% 0.00% 41.54%

S 1.36% 5.62% 1.41% S 0.46% 0.00% 0.00%

U 51.53% 52.81% 50.03% U 48.40% 100.00% 58.46%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

Salientobservationsincomparingthedailyvs.weeklyprobabilitiesare:

1. GiventhataparticularweekhasendedinaUstate,itisabout8%morelikelythatitwillbefollowedbyanotherUweek‐endthanisitthecasethataUUoccurrencewouldbeobservedinthedailydata.

2. Thecorollaryisthat,giventhataweekhasendedinaDstate,itismarginallylesslikelythatitwillbefollowedbyaU(DU’sonlyoccurapproximately48%ofthetimeintheweekly

data)thanwouldbethecaseinthedaily(whereDU’soccurapproximately52%ofthetime).

TheotherreadilyapparentobservationwecanmakeabouttheWeeklyData,isthatthismuchlessstabilityintheoccurrenceoftheSXchains.Thisstandstoreasonbecausethenumberofoccasionsuponwhichthefirstweekwillendatexactlythesamepriceasthestartingobservationwillbequiterare.

THEMARKOVTRANSITIONMATRIX

Ifweacceptthatonlythecurrentstateisanindicatorofwherethenextstateislikelytobe(and,therefore,allthepreviouspricemovementspriortothecurrentstatehavenobearinginthematter),thenthereisasinglematrixthatwillencapsulateallthatcanbeknownaboutprobablepricemovements.ThisiscalledtheTransitionMatrixand,inthecaseofthedailyWTIpricedataisrepresentedby:

TABLE4:MARKOVTRANSITIONMATRIX

D 46.61% 41.13% 48.44% S 1.74% 4.03% 1.65% U 51.65% 54.84% 49.90%

100.00% 100.00% 100.00%

Thistellsusthat,giventhatwehavealreadyobservedaDmovementtoday(onDayn),thereisa46.61%probabilityofanotherDmovementhappeningtomorrow(onDay(n+1)).Butdoesitgiveusanypredictiveinsightintowhatislikelytohappenthedayaftertomorrow(onDay(n+2))?Forexample,giventhattodayaDmovementhasalreadyoccurred,whatistheprobabilitythatanotherDmovementwilloccurtomorrowfollowedbyanotherDmovementthedayafterthat?Well,todayweknowthereisa46.61%probabilitythatasubsequentDwilloccuronDay(n+1)andthentomorrow,giventhataDmovementhadalreadyhappenedonthatday,theprobabilityofanotherD(onDay(n+2))is46.61%again.So,theprobabilityofaDDDis46.61%^2=21.7249%ButwhatifwewereattemptingtopredictthemorerelevantinstancewhereweknowtodayhasendedwithaDmovement(forexample),butwewishtopredictifDay(n+2)wouldendwithaDmovement?Symbolically,thiscanberepresentedbythechainDXD.TheprobabilityofobservingaDonDay(n+2)giventhataDhasoccurredonDaynmustbethesumoftheprobabilities:DDD+DSD+DUD.ThesearetheonlythreepathsbeginningwiththecurrentstateofDthatwillallowustofinishwithaDonDay(n+2).Usingthe2DayprobabilitiesfromtheTransitionMatrix,wecancalculatethisprobabilityas:

DDD=(DD)x(DD) =46.61%x46.61% =21.7249% DSD= (DS)x(SD) =1.74%x41.13% =0.7157% DUD=(DU)x(UD) =51.65%x48.44% =25.0193% PROBABILITYOFDXD =47.46%4Markovchainshavethissomewhatcounter‐intuitiveandironicquality:pastmovementshavenopredictivepower–knowingthepricepathuponwhichtoday’sstatecameintobeingisofnopracticalassistanceinestimatingwhattomorrow’spricemovementwillbe–onlytoday’sstatehasanybearingupontomorrow.Andyet,whenestimatingfurtherforward,saytothedayaftertomorrow,thenthepricepathdoesbecomerelevant.Thisisbecausetomorrow’sstateisdependentuponwhatstateweareattodayandsotoowillbethestateonDay(n+2)becauseweareattemptingtopredictitpriortoknowingwithabsolutecertaintyhowtomorrowwillend.Thisisthe‘chain‐like’featureofMarkovchains.WhilewecouldcontinuetocalculateallthepossiblepathsremainingaswedidaboveforDXD,namely:DXS,DXU,SXD,SXS,SXU,UXD,UXSandUXU,itwillbeeasieratthispointjusttonotethat,becauseofthemarvelsofmatrixalgebra,alltheseoutcomeswillbeprovidedsimplybyraisingtheTransitionMatrixtothepoweroftwo:FIGURE3:TRANSITIONMATRIXRAISEDTO2NDPOWER

Matrix A Matrix A2 D S U D S U

D 46.61% 41.13% 48.44% ^2 47.46% 47.39% 47.43%S 1.74% 4.03% 1.65% = 1.73% 1.78% 1.73%U 51.65% 54.84% 49.90% 50.81% 50.82% 50.83%Therefore,if,onDaynweareinStateS,andweareattemptingtoestimatewhattheprobabilityisthatDay(n+2)willendalsoinStateS,MatrixA2tellsusthereisa1.78%chanceofthatoccurring.And,itshouldbereadilyapparentthatestimatingtheoutcomeonDay(n+3)isrepresentedinMatrixA3andthattheprobableoutcomesonDay(n+100)arecapturedinMatrixA100,etc.Therefore,ifwewishtoknowwhattheprobabilityisthatWTIwillcloseeitherinanD,SorUstate1,000dayshence,weonlyneedaninexpensivematrixcalculatortokeyinMatrixAasaboveandraisethistothepowerof1,000.Couldanyapproachbesimpler?Inthiscase,theanswerisYES,because,aswewillnextdiscuss,theprocessveryquicklyapproachesasteadystate.

STEADYSTATELIMITVALUEOFTRANSITIONMATRIX

Ignoringallthereasoningastowhythisistrue5,itturnsoutthat,iftheTransitionMatrixis‘regular’(meaningallitselementsarepositive–and,ofcourse,thecolumnsallsumto1),thenraisingthe

4ThosewithagreaterfamiliarityofMatrixAlgebrawillrecognizethisasthedotproductofthefirstcolumnandthetranspositionofthefirstrow.

matrixtoincreasinglylargerpowerswilleventuallybringittoa‘steadystate’whereinalltheelementsapproachalimitingvalue(referredtoasanasymptoticquality).Asithappens,theWTItransitionmatrixreachesitslimitveryquickly(inlessthan5days),ascanbeseenbelow:

5Thesteady‐statevectorcanbefoundbyfirstfindingtheeigenvectorresultingfromtheeigenvalue+1.Thiseigenvectorwillbeamultipleofthesteady‐statevectorwhichcanbefoundbyscalingitscomponentssuchthattheysumto1.

TABLE5:TRANISITIONMATRIXRAISEDTOINCREASINGPOWERS

Matrix A

D 46.61% 41.13% 48.44%

S 1.74% 4.03% 1.65%

U 51.65% 54.84% 49.90%

Matrix A2

D 47.46% 47.39% 47.43%

S 1.73% 1.78% 1.73%

U 50.81% 50.82% 50.83%

Matrix A3

D 47.44% 47.44% 47.45%

S 1.74% 1.74% 1.74%

U 50.82% 50.82% 50.82%

Matrix A4

D 47.44% 47.44% 47.44%

S 1.74% 1.74% 1.74%

U 50.82% 50.82% 50.82%

Matrix A10

D 47.44% 47.44% 47.44%

S 1.74% 1.74% 1.74%

U 50.82% 50.82% 50.82%

Matrix A100

D 47.44% 47.44% 47.44%

S 1.74% 1.74% 1.74%

U 50.82% 50.82% 50.82%

Matrix A1000

D 47.44% 47.44% 47.44%

S 1.74% 1.74% 1.74%

U 50.82% 50.82% 50.82%

ItcanbeseenintheabovethattheWTIsteadystateisachievedbyDay(n+4)–theprobabilitiesinMatrixAraisedtothe4thpowerareexactlythesamewhenraisedtothe10th,100thor1,000thpower.Noteaswellthattheprobabilitiesineachofthecolumnsduplicateeachother–thismeansthat,bythefourthday,itdoesnotmatterwhattheinitialstatewas–theoutcomeprobabilitiesarethesameregardlessofwhereonestarted.Therefore,forlong‐termestimation,oneneednottakeintoaccountthestartingstate,theprobabilityofexperiencingaD,SorUonanygivendayinthefutureremainsstatic,namely;D=47.44%,S=1.74%,andU=50.82%6Similarly,fortheWeeklyData,theTransitionMatrixcanbeseeninTable3above(theTwo‐Weekmatrix)andthismatrixisalsoquicklyasymptotic,reachingasteady‐stateafterbeingraisedtothepowerofseven.TABLE6:WEEKLYSTEADYSTATEPROBABILITIES

Weekly Data - Steady State

D 46.827% 46.827% 46.827%

S 0.406% 0.406% 0.406%

U 52.767% 52.767% 52.767%

100.00% 100.00% 100.00%

CHI‐SQUARETESTS

Atlastwehavesufficientbackgroundtoattendtotheinitialquestion:“DoWTIoilpricemovementsfollowaMarkovChain?”TheChi‐Squarestatisticisagoodness‐of‐fittestthatwecanusetoaskthequestion;‘Howlikelyisit,baseduponrandomvariationalone,thattheobservedWTIoutcomeswoulddifferasmuchastheydofromtheexpectedoutcomes–assumingthatWTIpricedoesmoveinaMarkovchain?’TheChi‐squarestatisticis:

² ∑

However,onecannotuse‘scaled’orproportionateinputsforthechi‐squarecalculation–thepercentagedatathathasbeensoconvenienttouseinmatrixformuptothispointwillgive

6Somemaypointoutthatthesesteady‐stateprobabilitiesderivedfromtheTransitionMatrixaresimplythesumoftherowsintheveryfirst2‐dayfrequencystatisticsfromTable1.D=47.45%=(22.11%+0.71%+24.63%);S=1.74%=(0.83%+0.07%+0.84%);U=50.82%=(24.50%+0.95%+25.37%).Thisisoftenthecasewithmoresophisticatedmeasurementfunctions–theyonlyservetoconfirmwhatweintuitivelywouldhavesuspectedtobethetruthbutwithoutstatisticalproof.Withonlytherelativefrequencies,forexample,wedidnotknowthattheMarkovprobabilitieswouldconvergetoasteady‐statelimitorhowquicklythismayhappen.

erroneouschi‐squarestatistics.7Wemustactuallyusethecountdata,which,forexample,forthetenday‐chain,was:TABLE7

10 Day-Chain Observations

D 1,577 51 1,757

S 59 5 60

U 1,748 68 1,810

3,384 124 3,627

Theten‐dayexpectedmatrixdataisafunctionoftheA10matrixinTable5–thatis,iftheWTIpricesdofollowaMarkovchain,wewouldexpectthepossibilityofaDstateonday10tobe47.44%(regardlessofthestartingstate),anSoutcomewouldoccur1.74%ofthetimeand,finallyanUoutcomehastheprobabilityof50.82%.Thisinformation,combinedwiththefactthatweknowtherewere7,13510‐daychainsinthehistoricdata(simplythesumofthecolumntotalsfromTable6)leadsustotheconclusionthatthe10‐DayExpectedMatrixwouldbe:TABLE8

10-Day Expected Outcome

D 1,606 59 1,721

S 59 2 63

U 1,720 63 1,843

3,385 124 3,627

Theresultingchi‐squarestatisticfromthecomparisonofthesetwomatricesis8.44.Applicable‘DegreesofFreedom‘forthistestis8.8Wewillassumea.05or5%alphasignificance.OurnullhypothesisisthattheWTIpricesdofollowaMarkovchainandthereforeanyvariancewefindbetweentheactualobservedcountsandthehypotheticallyexpectedarejusttheresultofrandomvariation(i.e.chance).Thechi‐squarestatisticcalculateshowprobableitwouldbetoencounterthecalculatedvariancejustbaseduponrandomchancealone.Ifwefindthatthereislessthana5%chancethatsuchachi‐squareoutcomecouldbeobservedasaresultofrandomchance,thenwemustrejectthenullhypothesisandconcludethatWTIpricesdonotfollowaMarkovchain.ConsultingaChi‐squaretablequicklyshowsthat,at8dfandχ2=8.44thestatisticiswellabovethe25%probabilitymark(anyreliablestatisticssoftwarepackagewillreportthattheactualprobabilityis39.1%).Therefore,baseduponthe10‐daychainalone,wewouldnotrejecttheconclusionthatWTIpricesfollowaMarkovchain.

7Specifically,alltheexpectedcountsmustbe>thanone–andbydefinitionaprobability%willnotqualify.8ThereisoftenconsiderableconfusionaboutDegreesofFreedom(df)forthechi‐squaretests.Theissuedependsuponwhetherasingle‐sampletestofindependenceisbeingperformed(inwhichcase,therelevantdfwouldbe(rows–1)(columns–1)andwouldamountto4fortheabove.However,agoodness‐of‐fittestisacomparisonbetweentwosamples:theactualobservedandthehypotheticallyexpected,thereforethedfis(k–1)wherekisthenumberofcellsinthetwo‐waytable.

Similarly,theχ2wascalculatedforallthe10‐Day,100‐Dayand1,000‐Dayand10‐Week,100‐Weekand1,000‐Weekchains.Innocasewastherereasontorejectthenullhypothesis.WhilewecanreasonablyconcludethatWTIpricesdofollowaMarkovchain9,wearemoreinterestedintheresultsofthelonger1,000‐periodchainsasweareattemptingtomodellong‐term,ratherthanshorttermWTIprices.

9Tobeprecise,wehavenoevidencetorejecttheassumptionthatWTIfollowsaMarkovchain.ThisisdecidedlydifferentthanhavingproofpositivethatWTIdoesfollowaMarkovchain.

TABLE9:CHI‐SQUARESTATISTICS

Chi-square results (8 df) CHAIN LENGTH χ2 α Prob. Reject Ho?

10-Day Chains 8.45 39.1% No

100-Day Chains 8.89 35.2% No

1000-Day Chains 11.00 20.2% No

10-Week Chains 12.68 12.3% No

TIMEHOMOGENEITY

OneoftherequirementsofMarkovprobabilitiesistheassumptionoftimehomogeneity.Thatis,theprobabilityofmovingfromagivenstatetothenextstateisassumedtoremainconstantovertime.InthecaseofWTIpricemovements,wecanspeculatethatthisisarequirementthatisnotstrictlyadheredto.Weknowforcertain,forexample,thatWTIpricesexhibitsperiodsofgreaterandlesservolatility(seeFigure4).TheseperiodsmayalsobeassociatedwithvaryingprobabilitiesbetweenD,S,andUstates.Wemightalsospeculatethat,duringtimesofeconomicboon,thefrequencyofUmovementsincreasesand,correspondingly,theDfrequencymayincreaseduringperiodsofeconomicrecession.Thepertinentquestionbecomes:‘HowmightthisinfractionofthetimehomogeneityrequirementimpairourabilitytouseaMarkovprobabilitymodeltopredictfuturepricesteps?

Theanswertothisquestionisbeyondthescopeofthispaperandiscertainlydeservingofapaperallonitsown.Sufficeittosaythat,onayear‐over‐yearbasis,thereisconsiderablevariationamongstthestatespaceprobabilities.Completedetailsofall28individualyearsareprovidedinAppendix2.Tohighlightthedailyextremes,however,insomeyearstheDDchainrepresentedasmuchas54.5%ofallDXmovementswhereasinothersitdroppedtoonly36.8%.InpeakyearstheUUchainoccurred61.7%ofallUXmovementsandthisdroppedtoaslowas38.8%inotheryears.

Inspiteofdemonstratingloweroverallpricevolatility,theweeklychaindatademonstratedanevengreatervariance.InsomeyearstheDDchainrepresented71%ofallpossibleDXoutcomeswhereasinothersitdroppedto23.5%.Attheotherextreme,UUchainsrangedbetweenahighof71%ofallUXoutcomestoalowof36.8%.

Iftherelevantperioduponwhichtomeasuretimehomogeneityisone‐yearinlength,thenwecandefinitelyconcludethattheWTIpricemovementsarecompromisedinthischaracteristic.WTIpricemovementsdonotadheretoahomogenousprobabilitymatrixthroughtime.

FIGURE410

CONCLUSIONS

ThispaperhasdiscussedtheprimarycharacteristicsofMarkovchainsandhowtheymayapplytothemovementofWTIpricesbetweenprice“states”(Down,SameandUp).ThemostimportantMarkovpropertyisthatonlythecurrentstatehasanypredictivebearinguponthenextstate–allpreviouspricemovementsbeforethecurrentstateareirrelevantinthepredictionofwhereWTIpricemaygonext.WetestedfortheexistenceofaMarkovchainusingsimplestatisticsaswellasviaaformalChi‐squareapproachandfound,inallcases,thattheWTIpricemovementsdostronglyexhibitthisqualityofaMarkovchain.Further,weidentifiedthetransitionmatrixforboththedailyandweeklyMarkovprobabilitiesandfoundthat,inbothcases,theasymptoticlimitsarequicklyachievedandasteadystateisrealizedonlyafterafewfutureperiods.Aconfoundingcomplexity,however,isthattheposthocWTIhistoricaldatadoesnotdemonstratetimehomogeneity.Infact,thereisconsiderablevariationinthesingleyearMarkovprobabilitiesovertimeanditremainstobeseenjusthowsuchadefectwillimpairourabilitytoreasonablypredictfutureWTIstates.

10Theweeklyvolatilityparallelstheebbsandflowsofdailydata,butisexpectedlylessineachyear.Thisisbecausethedailyfluctuationsadds‘noise’tothevolatilitycalculation.

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

80.00%1986

Annual WTI Price Volatility: Daily vs. Weekly Data

Daily Data: Annualized Single‐Year WTI Volatility

Weekly Data: Annualized Single‐Year WTI Volatility

Wehavenotyetbeguntoconsiderhowlargethesizeofeachstepbetweenthevariousstatesshouldbe(i.e.thedollaramountofaUorDmove).Thistopicislefttobeinvestigatedinthefollowingpaper.

WehavediscussedtheconceptualissuessurroundingMarkovprobabilitiesandmatrixalgebrasufficientlythatweshouldnowbeabletoapplythesteady‐stateprobabilitiesinanactualMonteCarlomodeldesignedtopredictfuturestatesoftheWTIpricemovements.Thiswillbethesubjectofthenextandconcludingpaperinthisseries.

APPENDIX1

WEEKLY DATA: Frequency of 'XX' Ending Chains in EIA Data

Two-Week Chains Week n Observation

Week (n + 1) Observation

D 24.15% 0.20% 22.46%S 0.27% 0.00% 0.14%U 22.33% 0.20% 30.24%

100% *

Ten-Week Chains Week n Observation

D 23.81% 0.20% 22.52%S 0.27% 0.00% 0.14%U 22.45% 0.20% 30.41%

100% *

D 24.06% 0.22% 22.46%S 0.29% 0.00% 0.14%U 22.32% 0.22% 30.29%

100% *

D 23.33% 0.00% 22.50%S 0.21% 0.00% 0.00%U 22.08% 0.21% 31.67%

100% * * The sum of all 9 observations total 100% as expected

APPENDIX2

APPENDIX 2: ANNUAL MARKOV PROBABILITIES 1986 THROUGH 2013 DAILY DATA WEEKLY DATA

YEAR YEAR

1986 D S U 1986 D S U

D 49.57% 33.33% 45.16% D 56.00% 0.00% 42.31%

S 3.42% 0.00% 4.03% S 0.00% 0.00% 0.00%

U 47.01% 66.67% 50.81% U 44.00% 0.00% 57.69%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1987 D S U 1987 D S U

D 50.00% 50.00% 44.26% D 53.85% 0.00% 46.15%

S 5.00% 0.00% 4.92% S 0.00% 0.00% 0.00%

U 45.00% 50.00% 50.82% U 46.15% 0.00% 53.85%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1988 D S U 1988 D S U

D 36.84% 50.00% 50.37% D 45.83% 0.00% 44.83%

S 4.39% 0.00% 0.73% S 0.00% 0.00% 0.00%

U 58.77% 50.00% 48.91% U 54.17% 0.00% 55.17%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1989 D S U 1989 D S U

D 38.26% 25.00% 51.49% D 42.86% 0.00% 38.71%

S 3.48% 0.00% 2.99% S 0.00% 0.00% 0.00%

U 58.26% 75.00% 45.52% U 57.14% 0.00% 61.29%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1990 D S U 1990 D S U

D 50.39% 40.00% 51.22% D 46.15% 0.00% 57.69%

S 2.33% 0.00% 1.63% S 0.00% 0.00% 0.00%

U 47.29% 60.00% 47.15% U 53.85% 0.00% 42.31%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1991 D S U 1991 D S U

D 43.44% 50.00% 50.78% D 53.85% 0.00% 42.31%

S 3.28% 0.00% 1.56% S 0.00% 0.00% 0.00%

U 53.28% 50.00% 47.66% U 46.15% 0.00% 57.69%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1992 D S U 1992 D S U

D 45.24% 54.55% 53.33% D 52.17% 50.00% 40.74%

S 2.38% 0.00% 6.67% S 4.35% 0.00% 3.70%

U 52.38% 45.46% 40.00% U 43.48% 50.00% 55.56%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

1993 D S U 1993 D S U

D 50.00% 14.29% 61.68% D 61.77% 0.00% 63.16%

S 2.94% 14.29% 1.87% S 0.00% 0.00% 0.00%

U 47.06% 71.43% 36.45% U 38.24% 0.00% 36.84%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1994 D S U 1994 D S U

D 51.24% 0.00% 46.51% D 31.58% 0.00% 42.42%

S 0.83% 0.00% 0.78% S 0.00% 0.00% 0.00%

U 47.93% 100.00% 52.71% U 68.42% 0.00% 57.58%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1995 D S U 1995 D S U

D 44.14% 33.33% 44.03% D 45.83% 0.00% 42.86%

S 1.80% 0.00% 2.99% S 0.00% 0.00% 0.00%

U 54.05% 66.67% 52.99% U 54.17% 0.00% 57.14%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1996 D S U 1996 D S U

D 42.59% 33.33% 43.57% D 42.86% 0.00% 38.71%

S 1.85% 16.67% 2.14% S 0.00% 0.00% 0.00%

U 55.56% 50.00% 54.29% U 57.14% 0.00% 61.29%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1997 D S U 1997 D S U

D 51.88% 46.15% 54.72% D 70.97% 50.00% 47.37%

S 3.76% 15.39% 5.66% S 6.45% 0.00% 0.00%

U 44.36% 38.46% 39.62% U 22.58% 50.00% 52.63%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

1998 D S U 1998 D S U

D 54.48% 66.67% 50.88% D 67.65% 0.00% 55.56%

S 1.49% 0.00% 0.88% S 0.00% 0.00% 0.00%

U 44.03% 33.33% 48.25% U 32.35% 0.00% 44.44%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

1999 D S U 1999 D S U

D 47.66% 60.00% 38.85% D 36.84% 0.00% 38.24%

S 0.94% 0.00% 2.88% S 0.00% 0.00% 0.00%

U 51.40% 40.00% 58.27% U 63.16% 0.00% 61.77%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2000 D S U 2000 D S U

D 44.14% 50.00% 43.80% D 45.46% 0.00% 36.67%

S 0.00% 0.00% 1.46% S 0.00% 0.00% 0.00%

U 55.86% 50.00% 54.75% U 54.55% 0.00% 63.33%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2001 D S U 2001 D S U

D 53.49% 0.00% 50.85% D 53.57% 0.00% 58.33%

S 1.55% 0.00% 0.85% S 0.00% 0.00% 0.00%

U 44.96% 100.00% 48.31% U 46.43% 0.00% 41.67%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2002 D S U 2002 D S U

D 44.44% 14.29% 43.70% D 45.46% 0.00% 40.00%

S 2.78% 14.29% 2.22% S 0.00% 0.00% 0.00%

U 52.78% 71.43% 54.07% U 54.55% 0.00% 60.00%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2003 D S U 2003 D S U

D 41.38% 50.00% 50.76% D 47.83% 100.00% 35.71%

S 1.72% 0.00% 0.00% S 0.00% 0.00% 3.57%

U 56.90% 50.00% 49.24% U 52.17% 0.00% 60.71%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

2004 D S U 2004 D S U

D 41.18% 0.00% 55.04% D 59.09% 0.00% 29.03%

S 0.00% 0.00% 0.78% S 0.00% 0.00% 0.00%

U 58.82% 100.00% 44.19% U 40.91% 0.00% 70.97%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2005 D S U 2005 D S U

D 40.74% 75.00% 43.17% D 54.17% 0.00% 39.29%

S 2.78% 0.00% 0.72% S 0.00% 0.00% 0.00%

U 56.48% 25.00% 56.12% U 45.83% 0.00% 60.71%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2006 D S U 2006 D S U

D 46.28% 0.00% 51.56% D 45.83% 0.00% 50.00%

S 0.00% 0.00% 0.00% S 0.00% 0.00% 0.00%

U 53.72% 0.00% 48.44% U 54.17% 0.00% 50.00%

100.00% 0.00% 100.00% 100.00% 0.00% 100.00%

2007 D S U 2007 D S U

D 43.70% 66.67% 49.23% D 33.33% 0.00% 32.35%

S 1.68% 0.00% 0.77% S 0.00% 0.00% 0.00%

U 54.62% 33.33% 50.00% U 66.67% 0.00% 67.65%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2008 D S U 2008 D S U

D 53.49% 0.00% 48.78% D 68.75% 0.00% 50.00%

S 0.00% 0.00% 0.81% S 0.00% 0.00% 0.00%

U 46.51% 100.00% 50.41% U 31.25% 0.00% 50.00%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2009 D S U 2009 D S U

D 50.41% 0.00% 45.80% D 23.53% 0.00% 37.14%

S 0.00% 0.00% 0.00% S 0.00% 0.00% 0.00%

U 49.59% 0.00% 54.20% U 76.47% 0.00% 62.86%

100.00% 0.00% 100.00% 100.00% 0.00% 100.00%

2010 D S U 2010 D S U

D 52.42% 100.00% 45.67% D 47.62% 0.00% 37.50%

S 0.00% 0.00% 0.79% S 0.00% 0.00% 0.00%

U 47.58% 0.00% 53.54% U 52.38% 0.00% 62.50%

100.00% 100.00% 100.00% 100.00% 0.00% 100.00%

2011 D S U 2011 D S U

D 42.02% 0.00% 51.88% D 41.67% 0.00% 46.43%

S 0.00% 0.00% 0.00% S 0.00% 0.00% 0.00%

U 57.98% 0.00% 48.12% U 58.33% 0.00% 53.57%

100.00% 0.00% 100.00% 100.00% 0.00% 100.00%

2012 D S U 2012 D S U

D 47.86% 100.00% 44.78% D 65.52% 0.00% 45.46%

S 0.86% 0.00% 0.00% S 3.45% 0.00% 0.00%

U 51.28% 0.00% 55.22% U 31.03% 100.00% 54.55%

100.00% 100.00% 100.00% 100.00% 100.00% 100.00%

2013 D S U 2013 D S U

D 45.38% 0.00% 49.62% D 60.00% 0.00% 40.74%

S 0.00% 0.00% 0.00% S 0.00% 0.00% 0.00%

U 54.62% 0.00% 50.38% U 40.00% 0.00% 59.26%

100.00% 0.00% 100.00% 100.00% 0.00% 100.00%

REFERENCES

Xu,H.,Zhang,Z.X.,ATrendDeductionModelofFluctuationOilPrices,(revised2011),InstituteofInternationalStudies,FudanUniversity,Shanghai

Mostafaei,H.etal.,ModellingthefluctuationsofBrentoilpricesbyaprobabilisticMarkovchain,(2011),JournalofComputations&Modelling,vol.1,no.2,17–26

Abazi,A.,StochasticVolatilityintheCrudeOilPrices:AMarkovChainMonteCarloApproach,(2003),unpublishedthesis

Sirdari,M.Z.etal.,AsationaritytestonMarkovchainmodelsbasedonmarginaldistribution,SchoolofMathematicalSciences,UniversitiSainsMalaysia

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