第7章-Modelling-long-run-relationship-in-finance(金融计量经济学导论-东北财经大学-陈磊)课件

Chapter7Modellinglong-runrelationshipinfinance©ChrisBrooks2002

陈磊200411StationarityandUnitRootTestingThestationarityorotherwiseofaseriescanstronglyinfluenceitsbehaviourandproperties-e.g.persistenceofshockswillbeinfinitefornonstationaryseriesSpuriousregressions.Iftwovariablesaretrendingovertime,aregressionofoneontheothercouldhaveahighR2evenifthetwoaretotallyunrelatedIfthevariablesintheregressionmodelarenotstationary,thenitcanbeprovedthatthestandardassumptionsforasymptoticanalysiswillnotbevalid.Inotherwords,theusual“t-ratios”willnotfollowat-distribution,sowecannotvalidlyundertakehypothesistestsabouttheregressionparameters.1.1WhydoweneedtotestforNon-Stationarity?©ChrisBrooks2002

陈磊20042ValueofR2for1000SetsofRegressionsofa

Non-stationaryVariableonanotherIndependent

Non-stationaryVariable

©ChrisBrooks2002

陈磊20043

Valueoft-ratioonSlopeCoefficientfor1000SetsofRegressionsofaNon-stationaryVariableonanotherIndependentNon-stationaryVariable

©ChrisBrooks2002

陈磊200441.2TwotypesofNon-StationarityVariousdefinitionsofnon-stationarityexistInthischapter,wearereallyreferringtotheweakformorcovariancestationarity

Therearetwomodelswhichhavebeenfrequentlyusedtocharacterisenon-stationarity:therandomwalkmodelwithdrift:

yt=+yt-1+ut (1) andthedeterministictrendprocess:

yt=+t+ut

(2)

whereutisiidinbothcases.©ChrisBrooks2002

陈磊20045StochasticNon-StationarityNotethatthemodel(1)couldbegeneralisedtothecasewhereytisanexplosiveprocess:

yt=+yt-1+ut where>1.Typically,theexplosivecaseisignoredandweuse=1tocharacterisethenon-stationaritybecause>1doesnotdescribemanydataseriesineconomicsandfinance.>1hasanintuitivelyunappealingproperty:shockstothesystemarenotonlypersistentthroughtime,theyarepropagatedsothatagivenshockwillhaveanincreasinglylargeinfluence.

©ChrisBrooks2002

陈磊20046StochasticNon-stationarity:TheImpactofShocksToseethis,considerthegeneralcaseofanAR(1)withnodrift:

yt=yt-1+ut (3) Lettakeanyvaluefornow.Wecanwrite: yt-1=yt-2+ut-1 yt-2=yt-3

+ut-2Substitutinginto(3)yields: yt=(yt-2

+ut-1)+ut =2yt-2+ut-1+utSubstitutingagainforyt-2: yt=2(yt-3+ut-2)+ut-1+ut =3yt-3+2ut-2+ut-1+utSuccessivesubstitutionsofthistypeleadto:*

yt=Ty0+ut-1

+2ut-2+3ut-3+...+T-1u1+ut©ChrisBrooks2002

陈磊20047TheImpactofShocksfor

StationaryandNon-stationarySeries

Wehave3cases:1.||<1

T0asT Sotheshockstothesystemgraduallydieaway. 2.=1

T=1

T Soshockspersistinthesystemandneverdieaway.Weobtain: asT

Sojustaninfinitesumofpastshocksplussomestartingvalueofy0. 3.>1.Nowgivenshocksbecomemoreinfluentialastimegoeson,sinceif>1,3>2>etc.©ChrisBrooks2002

陈磊20048DetrendingaNon-stationarySeries

Goingbacktoour2characterisationsofnon-stationarity,ther.w.withdrift:

yt=+yt-1+ut (1) andthetrend-stationaryprocess

yt=+t+ut

(2)Thetwowillrequiredifferenttreatmentstoinducestationarity.Thesecondcaseisknownasdeterministicnon-stationarityandwhatisrequiredisdetrending去势.Thefirstcaseisknownasstochasticnon-stationarity.Ifwelet

yt=yt-yt-1=yt-Lyt

=(1-L)

yt

then yt

=(1-L)

yt=+ut wehaveinducedstationarityby“differencingonce”.

单位根过程©ChrisBrooks2002

陈磊20049DetrendingaSeries:

UsingtheRightMethod

Althoughtrend-stationaryanddifference-stationaryseriesareboth“trending”overtime,thecorrectapproachneedstobeusedineachcase.Ifwefirstdifferencethetrend-stationaryseries,itwould“remove”thenon-stationarity,butattheexpenseonintroducinganMA(1)structureintotheerrors.p372Converselyifwetrytodetrendaserieswhichhasstochastictrend,thenwewillnotremovethenon-stationarity.Wewillnowconcentrateonthestochasticnon-stationaritymodelsincedeterministicnon-stationaritydoesnotadequatelydescribemostseriesineconomicsorfinance.©ChrisBrooks2002

陈磊200410SamplePlotsforvariousStochasticProcesses:AWhiteNoiseProcess©ChrisBrooks2002

陈磊200411RandomWalkand

aRandomWalkwithDrift©ChrisBrooks2002

陈磊200412ADeterministicTrendProcess©ChrisBrooks2002

陈磊200413AutoregressiveProcesseswith

differingvaluesof(0,0.8,1)©ChrisBrooks2002

陈磊2004141.3DefinitionofNon-Stationarity

Consideragainthesimpleststochastictrendmodel:

yt=yt-1+ut

oryt=ut Wecangeneralisethisconcepttoconsiderthecasewheretheseriescontainsmorethanone“unitroot”.Thatis,wewouldneedtoapplythefirstdifferenceoperator,

,morethanoncetoinducestationarity.

Definition Ifanon-stationaryseries,ytmustbedifferenceddtimesbeforeitbecomesstationary,thenitissaidtobeintegratedoforderd.Wewriteyt

I(d).

Soifyt

I(d)thendytI(0). AnI(0)seriesisastationaryseries AnI(1)seriescontainsoneunitroot, e.g.yt=yt-1+ut©ChrisBrooks2002

陈磊200415CharacteristicsofI(0),I(1)andI(2)Series

AnI(2)seriescontainstwounitrootsandsowouldrequiredifferencingtwicetoinducestationarity.p376I(1)andI(2)seriescanwanderalongwayfromtheirmeanvalueandcrossthismeanvaluerarely.I(0)seriesshouldcrossthemeanfrequently.

Themajorityofeconomicandfinancialseriescontainasingleunitroot,althoughsomearestationaryandconsumerpriceshavebeenarguedtohave2unitroots.©ChrisBrooks2002

陈磊2004161.4Testforaunitroot

无法使用acf或pacf确定序列是否是单位根过程。TheearlyandpioneeringworkontestingforaunitrootintimeserieswasdonebyDickeyandFuller(DickeyandFuller1979,Fuller1976).Thebasicobjectiveofthetestistotestthenullhypothesisthat=1in:

yt=yt-1+ut againsttheone-sidedalternative<1.Sowehave H0:seriescontainsaunitroot vs. H1:seriesisstationary.Weusuallyusetheregression:

yt=yt-1+ut

atestof=1isequivalenttoatestof=0(since-1=).©ChrisBrooks2002

陈磊200417DifferentformsfortheDFTestDickeyFullertestsarealsoknownastests:,,.Thenull(H0)andalternative(H1)modelsineachcaseare i) H0:yt

=yt-1+ut H1:yt=yt-1+ut,

<1 Thisisatestforarandomwalkagainstastationaryautoregressiveprocessoforderone(AR(1)) ii) H0:yt=yt-1+ut H1:yt=yt-1++ut,

<1 ThisisatestforarandomwalkagainstastationaryAR(1)withdrift. iii) H0:yt=yt-1+ut H1:

yt=yt-1++t+ut

,

<1 ThisisatestforarandomwalkagainstastationaryAR(1)withdriftandatimetrend.©ChrisBrooks2002

陈磊200418ComputingtheDFTestStatisticWecanwrite yt=ut

whereyt=yt-yt-1,andthealternativesmaybeexpressedas

yt=yt-1++t+ut

with

==0incasei),and=0incaseii)and=-1.Ineachcase,thetestsarebasedonthet-ratioontheyt-1termintheestimatedregressionofytonyt-1,plusaconstantincaseii)andaconstantandtrendincaseiii).Theteststatisticsaredefinedas teststatistic=

Theteststatisticdoesnotfollowtheusualt-distributionunderthenull,sincethenullisoneofnon-stationarity,butratherfollowsanon-standarddistribution.CriticalvaluesarederivedfromMonteCarloexperimentsin,forexample,Fuller(1976).Relevantexamplesofthedistributionareshownintable7.1below©ChrisBrooks2002

陈磊200419CriticalValuesfortheDFTest

Thenullhypothesisofaunitrootisrejectedinfavourofthestationaryalternativeineachcaseiftheteststatisticismorenegativethanthecriticalvalue.临界值还与样本容量有关，参见675页表A2.7。©ChrisBrooks2002

陈磊200420TheAugmentedDickeyFullerTest(ADF)Thetestsaboveareonlyvalidifutiswhitenoise.Inparticular,utwillbeautocorrelatediftherewasautocorrelationinthedependentvariableoftheregression(yt)whichwehavenotmodelled.Thesolutionisto“augment”thetestusingplagsofthedependentvariable.Thealternativemodelincase(i)isnowwritten:

ThesamecriticalvaluesfromtheDFtablesareusedasbefore.Aproblemnowarisesindeterminingtheoptimalnumberoflagsofthedependentvariable. Thereare2ways -usethefrequencyofthedatatodecide -useinformationcriteria©ChrisBrooks2002

陈磊2004211.5TestingforHigherOrdersofIntegrationConsiderthesimpleregression: yt=yt-1+ut WetestH0:=0vs.H1:<0.IfH0isrejectedwesimplyconcludethatytdoesnotcontainaunitroot.ButwhatdoweconcludeifH0isnotrejected?Theseriescontainsaunitroot,butisthatit?No!Whatifyt

I(2)?Wewouldstillnothaverejected.Sowenowneedtotest H0:yt

I(2)vs.H1:yt

I(1) Wenowregress2ytonyt-1(pluslagsof2ytifnecessary).NowwetestH0:ytI(1)whichisequivalenttoH0:ytI(2).Sointhiscase,ifwedonotreject,thenytisatleastI(2).continuetotestforafurtherunitrootuntilwerejectedH0.DickeyandPantula(1987)认为，检验顺序应从高阶至低阶。©ChrisBrooks2002

陈磊2004221.6ThePhillips-PerronTest

PhillipsandPerronhavedevelopedamorecomprehensivetheoryofunitrootnonstationarity.ThetestsaresimilartoADFtests,buttheyincorporateanautomaticcorrectiontotheDFproceduretoallowforautocorrelatedresiduals.ThetestsusuallygivethesameconclusionsastheADFtests,andthecalculationoftheteststatisticsiscomplex.©ChrisBrooks2002

陈磊2004231.7CriticismofDickey-Fullerand

Phillips-Perron-typetests

Maincriticismisthatthepowerofthetestsislowiftheprocessisstationarybutwitharootclosetothenon-stationaryboundary. e.g.thetestsarepooratdecidingif=1or=0.95,especiallywithsmallsamplesizes.

Ifthetruedatageneratingprocess(dgp)is

yt=0.95yt-1+ut thenthenullhypothesisofaunitrootshouldberejected.

Onewaytogetaroundthisistouseastationaritytestaswellastheunitroottestswehavelookedat.

©ChrisBrooks2002

陈磊200424Stationaritytests

Stationaritytestshave H0:ytisstationary versus H1:ytisnon-stationary Sothatbydefaultunderthenullthedatawillappearstationary.

OnesuchstationaritytestistheKPSStest(Kwaitowski,Phillips,SchmidtandShin,1992).

ThuswecancomparetheresultsofthesetestswiththeADF/PPproceduretoseeifweobtainthesameconclusion.

©ChrisBrooks2002

陈磊200425Stationaritytests(cont’d)

AComparison

ADF/PP KPSS H0:yt

I(1) H0:yt

I(0) H1:yt

I(0) H1:yt

I(1)

Therearefourpossibleoutcomes 1.RejectH0 and DonotrejectH0

2.DonotrejectH0 and RejectH0

3.RejectH0 and RejectH0

4.DonotrejectH0 and DonotrejectH0

©ChrisBrooks2002

陈磊200426有关单位根和协整的一些参考书1.董文泉等，《经济周期波动的分析与预测方法》第十二章，吉林大学出版社，1998年。2.李子奈、叶阿忠，《高等计量经济学》第二章第五节，清华大学出版社，2000年。3.张晓峒，《计量经济分析》，经济科学出版社，2000年。4.赵国庆主编，《计量经济学》第八章，中国人民大学出版社，2001年。5.王维国主编，《计量经济学》第十章，东北财经大学出版社，2002年。6.PhilipH.Franses,《商业和经济预测中的时间序列模型》第四章，中国人民大学出版社，2002年。©ChrisBrooks2002

陈磊2004272TestingforunitrootsinEViewsp383-385©ChrisBrooks2002

陈磊200428

3Cointegration:AnIntroduction

Inmostcases,ifwecombinetwovariableswhichareI(1),thenthecombinationwillalsobeI(1).Moregenerally,ifwecombinevariableswithdifferingordersofintegration,thecombinationwillhaveanorderofintegrationequaltothelargest.i.e., ifXi,t

I(di)fori=1,2,3,...,k sowehavekvariableseachintegratedoforderdi.

Let (1)

Thenzt

I(maxdi)

©ChrisBrooks2002

陈磊200429LinearCombinationsofNon-stationaryVariablesRearranging(1),wecanwrite

where

Thisisjustaregressionequation. Butthedisturbanceswouldhavesomeveryundesirableproperties:zt´isnotstationaryandisautocorrelatedifalloftheXiareI(1). WewanttoensurethatthedisturbancesareI(0).Underwhatcircumstanceswillthisbethecase?©ChrisBrooks2002

陈磊200430

DefinitionofCointegration

(Engle&Granger,1987)

Letztbeak1vectorofvariables,thenthecomponentsofztarecointegratedoforder(d,b)if i)AllcomponentsofztareI(d) ii)Thereisatleastonevectorofcoefficientssuchthat

zt

I(d-b)0<b≤dIfvariablesarecointegrated,itmeansthatalinearcombinationofthemwillbestationary.Theremaybeuptorlinearlyindependentcointegratingrelationships(whererk-1),alsoknownascointegratingvectors.risalsoknownasthecointegratingrankofzt.Acointegratingrelationshipmayalsobeseenasalongtermrelationship.Manytimeseriesarenon-stationarybut“movetogether”overtime.©ChrisBrooks2002

陈磊200431CointegrationandEquilibriumExamplesofpossibleCointegratingRelationshipsinfinance:spotandfuturespricesratioofrelativepricesandanexchangerateequitypricesanddividendsMarketforcesarisingfromnoarbitrageconditionsshouldensureanequilibriumrelationship.Nocointegrationimpliesthatseriescouldwanderapartwithoutboundinthelongrun.©ChrisBrooks2002

陈磊200432

4EquilibriumCorrectionorErrorCorrectionModels

Whentheconceptofnon-stationaritywasfirstconsidered,ausualresponsewastoindependentlytakethefirstdifferencesofaseriesofI(1)variables.Theproblemwiththisapproachisthatpurefirstdifferencemodelshavenolongrunsolution. e.g.ConsiderytandxtbothI(1). Themodelwemaywanttoestimateis

yt=xt+ut

Butthiscollapsestonothinginthelongrun. Thedefinitionofthelongrunthatweuseiswhere

yt=yt-1=y;xt=xt-1=x. Henceallthedifferencetermswillbezero,i.e.

yt=0;xt=0.©ChrisBrooks2002

陈磊200433SpecifyinganECM

Onewaytogetaroundthisproblemistousebothfirstdifferenceandlaggedlevelterms,e.g.

yt=1xt+2(yt-1-xt-1)+

ut (2)yt-1-xt-1isknownastheerrorcorrectionterm. Providingthatyt

and

xtarecointegratedwithcointegratingcoefficient,then(yt-1-xt-1)willbeI(0)eventhoughtheconstituentsareI(1).WecanthusvalidlyuseOLSon(2).*TheGrangerrepresentationtheoremshowsthatanycointegratingrelationshipcanbeexpressedasanequilibriumcorrectionmodel.©ChrisBrooks2002

陈磊200434

5TestingforCointegrationinRegression

Themodelfortheequilibriumcorrectiontermcanbegeneralisedtoincludemorethantwovariables:

yt=1+2x2t+3x3t+…+kxkt+ut (3)

utshouldbeI(0)ifthevariablesyt,x2t,...xktarecointegrated.Sowhatwewanttotestistheresidualsofequation(3)toseeiftheyarenon-stationaryorstationary.WecanusetheDF/ADFtestonut. Sowehavetheregression withvt

iid.However,sincethisisatestontheresidualsofanactualmodel,,thenthecriticalvaluesarechanged.因为参数估计的偏差会改变针对残差的检验统计量的分布。©ChrisBrooks2002

陈磊200435TestingforCointegrationinRegression:Conclusions

EngleandGranger(1987)havetabulatedanewsetofcriticalvalues（见A2.8）andhencethetestisknownastheEngleGranger(E-G)test.

WecanalsousetheDurbinWatsonteststatistic（CRDW）orthePhillipsPerronapproachtotestfornon-stationarityof.Whatarethenullandalternativehypothesesforatestontheresidualsofapotentiallycointegratingregression? H0:unitrootincointegratingregression’sresiduals H1:residualsfromcointegratingregressionarestationary

如果拒绝H0，应建立ECM; 否则，只能对差分后变量建立回归模型。©ChrisBrooks2002

陈磊2004366MethodsofmodellingCointegratedSystems:Thereare(atleast)3methodswecoulduse:EngleGranger,EngleandYoo,andJohansen.TheEngle-Granger2StepMethod

Step1: -MakesurethatalltheindividualvariablesareI(1). -ThenestimatethecointegratingregressionusingOLS. -Savetheresidualsofthecointegratingregression,. -TesttheseresidualstoensurethattheyareI(0).

Step2: -Usethestep1residualsasonevariableintheerrorcorrectionmodele.g.

yt=1xt+2()+

ut where=yt-1-xt-1

©ChrisBrooks2002

陈磊200437TheEngle-GrangerApproach:

SomeDrawbacks

1.Unitrootandcointegrationtestshavelowpowerinfinitesamples 2.Weareforcedtotreatthevariablesasymmetricallyandtospecifyoneasthedependentandtheotherasindependentvariables.即使可能不存在理论根据。 3.Cannotperformanyhypothesistestsabouttheactualcointegratingrelationshipestimatedatstage1. -Problem1isasmallsampleproblemthatshoulddisappearasymptotically. -Problem2isaddressedbytheJohansenapproach. -Problem3isaddressedbytheEngleandYooapproachortheJohansenapproach.©ChrisBrooks2002

陈磊200438OneoftheproblemswiththeEG2-stepmethodisthatwecannotmakeanyinferencesabouttheactualcointegratingregression.TheEngle&Yoo(EY,1987)3-stepproceduretakesitsfirsttwostepsfromEG.

EYaddathirdstepgivingupdatedestimatesofthecointegratingvectoranditsstandarderrors.

由于存在针对协整关系及其假设检验的更好估计方法，E-Y方法在实证研究中很少使用。

TheEngle&Yoo3-StepMethod

©ChrisBrooks2002

陈磊200439Inthegeneralcase,wherewehavemorethantwovariableswhichmaybecointegrated,therecouldbemorethanonecointegratingrelationship.

Infacttherecanbeuptorlinearlyindependentcointegratingvectors(whererg-1),wheregisthenumberofvariablesintotal.So,inthecasewherewejusthadyandx,thenrcanonlybeoneorzero.Andifthereareothers,howdoweknowhowmanythereareorwhetherwehavefoundthe“best”?

Theanswertothisistouseasystemsapproachtocointegrationwhichwillallowdeterminationofallr

cointegratingrelationships-Johansen’smethod.多个协整关系©ChrisBrooks2002

陈磊200440多个协整关系两个非平稳变量之间最多只能有一个线性组合是平稳的,即最多只存在一个协整关系。一个包括K个变量的非平稳系统，可以存在最多K-1个线性独立的协整关系。前面叙述的回归方法只能发现一个协整关系。如果存在多个协整关系，如何知道是否存在其他的协整关系？是否已经找到最“好”或最强的协整关系？这需要检验协整关系的系统方法。©ChrisBrooks2002

陈磊2004417AnExample:Lead-LagRelationshipsbetweenSpotandFuturesPrices

BackgroundWeexpectchangesinthespotpriceofafinancialassetanditscorrespondingfuturespricetobeperfectlycontemporaneouslycorrelatedandnottobecross-autocorrelated. i.e.expect Corr(ln(Ft),ln(St))1

Corr(ln(Ft),ln(St-k))0

k

Corr(ln(Ft-j),ln(St))0

jWecantestthisideabymodellingthelead-lagrelationshipbetweenthetwo.WewillconsidertwopapersTse(1995)andBrooksetal(2001).

©ChrisBrooks2002

陈磊200442Futures&SpotData

Tse(1995):1055dailyobservationsonNSA东京证券交易所stockindexandstockindexfuturesvaluesfromDecember1988-April1993.Brooksetal(2001):13,03510-minutelyobservationsontheFTSE100stockindexandstockindexfuturespricesforalltradingdaysintheperiodJune1996–1997.©ChrisBrooks2002

陈磊200443MethodologyThefairfuturespriceisgivenby whereFt*isthefairfuturesprice,Stisthespotprice,risacontinuouslycompoundedrisk-freerateofinterest,disthecontinuouslycompoundedyieldintermsofdividendsderivedfromthestockindexuntilthefuturescontractmatures,and(T-t)isthetimetomaturityofthefuturescontract.TakinglogarithmsofbothsidesofequationabovegivesFirst,testftandstfornonstationarity.©ChrisBrooks2002

陈磊200444Dickey-FullerTestsonLog-PricesandReturnsforHighFrequencyFTSEData©ChrisBrooks2002

陈磊200445CointegrationTestRegressionandTestonResidualsConclusion:logFtandlogStarenotstationary,butlogFtandlogStarestationary.Butamodelcontainingonlyfirstdifferenceshasnolongrunrelationship.Solutionistoseeifthereexistsacointegratingrelationshipbetweenftandstwhichwouldmeanthatwecanvalidlyincludelevelstermsinthisframework.Potentialcointegratingregression: whereztisadisturbanceterm.Estimatetheregression,collecttheresiduals,,andtestwhethertheyarestationary.©ChrisBrooks2002

陈磊200446EstimatedEquationandTestforCointegrationfor

HighFrequencyFTSEData©ChrisBrooks2002

陈磊200447ConclusionsfromUnitRootandCointegrationTestsConclusion:arestationaryandthereforewehaveacointegratingrelationshipbetweenlogFtandlogSt.FinalstageinEngle-Granger2-stepmethodistousethefirststageresiduals,astheequilibriumcorrectionterminthegeneralequation.Theoverallmodelis©ChrisBrooks2002

陈磊200448EstimatedErrorCorrectionModelfor

HighFrequencyFTSEDataLookatthesignsandsignificancesofthecoefficients: ispositiveandhighlysignificant ispositiveandhighlysignificant isnegativeandhighlysignificant©ChrisBrooks2002

陈磊200449ForecastingHighFrequencyFTSEReturnsIsitpossibletousetheerrorcorrectionmodeltoproducesuperiorforecaststoothermodels?ComparisonofOutofSampleForecastingAccuracy©ChrisBrooks2002

陈磊200450CanProfitableTradingRulesbeDerivedfromtheECM-COCForecasts?Thetradingstrategyinvolvesanalysingtheforecastforthespotreturn,andincorporatingthedecisiondictatedbythetradingrulesdescribedbelow.Itisassumedthattheoriginalinvestmentis£1000,andiftheholdinginthestockindexiszero,theinvestmentearnstheriskfreerate.

LiquidTradingStrategy-makingaroundtriptrade(i.e.apurchaseandsaleoftheFTSE100stocks)everytenminutesthatthereturnispredictedtobepositivebythemodel.Buy-&-HoldwhileForecastPositiveStrategy-allowsthetradertocontinueholdingtheindexifthereturnatthenextpredictedinvestmentperiodispositive.FilterStrategy:BetterPredictedReturnThanAverage-involvespurchasingtheindexonlyifthepredictedreturnsaregreaterthantheaveragepositivereturn.FilterStrategy:BetterPredictedReturnThanFirstDecile-onlythereturnspredictedtobeinthetop10%ofallreturnsaretradedonFilterStrategy:HighArbitraryCutOff-Anarbitraryfilterof0.0075%isimposed,©ChrisBrooks2002

陈磊200451SpotTradingStrategyResultsforErrorCorrectionModelIncorporatingtheCostofCarry©ChrisBrooks2002

陈磊200452ConclusionsThefuturesmarket“leads”thespotmarketbecause:thestockindexisnotasingleentity,sosomecomponentsoftheindexareinfrequentlytradeditismoreexpensivetotransactinthespotmarketstockmarketindicesareonlyrecalculatedeveryminuteSpot&futuresmarketsdoindeedhavealongrunrelationship.Sinceitappearsimpossibletoprofitfromlead/lagrelationships,theirexistenceisentirelyconsistentwiththeabsenceofarbitrageopportunitiesandinaccordancewithmoderndefinitionsoftheefficientmarketshypothesis.©ChrisBrooks2002

陈磊200453TouseJohansen’smethod,weneedtoturntheVARoftheform

yt=

1yt-1+2yt-2+...+k

yt-k+ut

g×1g×g

g×1g×g

g×1g×gg×1g×1

intoaVECM,whichcanbewrittenas

yt=

yt-k+1

yt-1+2

yt-2+...+k-1

yt-(k-1)+ut

where=and

isalongruncoefficientmatrixsincealltheyt-i=0.8TestingforandEstimatingCointegratingSystemsUsingtheJohansenTechnique©ChrisBrooks2002

陈磊200454Letdenoteaggsquarematrixandletcdenoteag1non-zerovector,andletdenoteasetofscalars.

iscalledacharacteristicrootofifwecanwrite

c=c

gg

g1g1

Wecanalsowrite

c=

Igc

andhence

(-

Ig)c=0 whereIgisanidentitymatrix.

ReviewofMatrixAlgebra

necessaryfortheJohansenTest

©ChrisBrooks2002

陈磊200455Sincec

0bydefinition,thenforthissystemtohavenon-zerosolution,werequirethematrix(-

Ig)tobesingular(i.e.tohavezerodeterminant).

-

Ig

=0

Forexample,letbethe22matrix

Thenthecharacteristicequationis

-

Ig

Thisgivesthesolutions=6and=3.

ReviewofMatrixAlgebra(cont’d)

©ChrisBrooks2002

陈磊200456ThecharacteristicrootsarealsoknownasEigenvalues.

Therankofamatrixisequaltothenumberoflinearlyindependentrowsorcolumnsinthematrix.

WewriteRank()=r

Therankofamatrixisequaltotheorderofthelargestsquarematrixwecanobtainfromwhichhasanon-zerodeterminant.

Forexample,thedeterminantofabove0,thereforeithasrank2.Somepropertiesoftheeigenvaluesofanysquarematrix:

1.thesumoftheeigenvaluesisthetrace2.theproductoftheeigenvaluesisthedeterminant3.thenumberofnon-zeroeigenvaluesi