时间序列分析报告及VAR模型

1、实用文档Lecture66. Timeseriesanalysis:Multivariatemodels6.1 Learningoutcomes Vectorautoregression(VAR) Cointegration Vectorerrorcorrectionmodel(VECM) Application:pairstrading6.2 Vectorautoregression(VAR)向量自回归Theclassicallinearregressionmodelassumesstrictexogeneity;hence,thereisnoserialcorrelationbetween

2、errortermsandanyrealisationofanyindependentvariable(leadorlag).Aswediscovered,serialcorrelation(orautocorrelation)isverycommoninfinancialtimeseriesandpaneldata.Furthermore,weassumedapre-definedrelationofcausality:explanatoryvariableaffectthedependentvariable.传统的线性回归模型假设严格的外生性，误差项与可实现的独立变量之间没有序列相关性。金

3、融时间序列及面板数据往往都有很强的自相关性，假定解释变量影响因变量。WenowrelaxbothassumptionsusingaVARmodel.VARmodelscanberegardedasageneralisationofAR(p)processesbyaddingadditionaltimeseries.Hence,weenterthefieldofmultivariatetimeseriesanalysis.VAR模型可以当作是在一般的自回归过程中加入时问序列。Let'slookatastandardAR(p)proctortwovariables(ytandxt).The

4、nextstepistoallowthatlaggedvaluesofxtcanaffectytandviceversa.Thismeansthatweobtainasystemofequationsfortwodependentvariables(ytandxt).Bothdependentvariablesareinfluencedbypastrealisationsofytandxt.Bydoingthat,weviolatestrictexogeneity(seeLecture2);however,wecanuseamorerelaxedconcept,namelyweakexogen

5、eityAsweuselaggedvaluesofbothdependentvariables,wecanarguethattheselaggedvaluesareknowntous,asweobservedtheminthepreviousperiod.WecallthesevariablespredeterminedPredetermined(lagged)variablesfulfilweakexogeneityinthesensethattheyhavetobeuncorrelatedwiththecontemporaneouserrortermint.WecanstilluseOLS

6、toestimatethefollowingsystemofequations,whichiscalledaVARinreducedform.标准文案实用文档Thebeautyofthismodelisthatwedon'tneedtopredefinewhetherxoryareendogenous(thedependentvariable).Infact,wecantestwhetherx(y)isendogenousorexogenoususingGrangercausalitytestsTheideaofGrangercausalityisthatpastobservation

7、s(laggeddependentvariables)caninfluencecurrentobservations-butnotviceversa.Sotheideaisrathersimple:thepastaffectsthepresent,andthepresentdoesnotaffectthepast.STATAprovidesGrangercausalitytestsafterconductingaVARanalysis,whichisbasedontestingthejointhypothesisthatpastrealisationsdonotGrangercausethep

8、resentrealisationofthedependentvariable.Inmanyapplications,VARmodelsmakealotofsense,asacleardirectionofcausalitycannotbepredefined.Forinstance,thereisasubstantialliteratureonthebenefitsofinternationalisation(e.g.enteringforeignmarketthroughcross-borderM&A).Thereisevidencethatmultinationalsoutper

9、formlocalpeersduetothebenefitsofoperatinginmanycountries.Atthesametime,weknowthathigh-performingcompaniesaremorelikelytoenterforeignmarketsduetotheirownershipspecificadvantages.ThisargumentisbasedontheResource-basedViewandtheOLSframeworkdevelopedbyDunningandRugman(ReadingSchoolofInternationalBusines

10、s).TheVARmodelallowsyoutoincorporatebotheffects:infactyoucantestwhetherperformancedrivesinternationalisationorinternationalisationdrivesperformance.BeforeyoustartusingaVARmodel,youhavetomakesurethatthetimeseriesarestationary.SothefirststepistocheckwhetherthetimeseriesisstationaryusingDickey-Fullerte

11、stsandKPSStests.Thesecondstepistospecifytheoptimallaglength(p)ofthemodel.Thisisdonebycomparingdifferentmodelspecificationsusinginformationcriteria.ApartfromusingAkaike(AIC)andBayesianSchwarz(BIC),theHannan-Quinn(HQIC)iscommonlyused.MostappliedeconometriciansfavourtheHannan-Quinn(HQIC)criterion.STATA

12、willhelpyoutomakeagoodchoice.Afterspecifyingyourmodel,youneedtocheckstabilityconditions.ThecoefficientmatrixofthereducedformVARhastoensurethattheiterationsequenceconvergestoalong-termvalue.STATAwillhelpyouincheckingstability.Tobeprecise,youneedtoshowthattheeigenvaluesofthecoefficientmatrixliewithint

13、heunitcircle.Thereasonbehinditcanbeonlyunderstoodwhenyouunderstandthemethodofdiagonalizingamatrix.VARmodelsofferanothernicefeature:impulseresponsefunctionsVARmodelscapturethedynamicsoftwo(ormore)stationarytimeseries;hence,wecanassessthedynamicimpactofamarginalchangeofonevariableonanother.Thestandard

14、OLSregressionprovidescoefficients,andcoefficientsrefertothepartialimpactofanexplanatoryvariableonthedependentvariable.InthecaseofVARmodels,therelationshipbecomesdynamic,asachangeofonevariable(sayx)intcanaffectxandyint+1.Theimpactonxandyint+1inturnaffectsxandyint+2andsoonuntiltheimpactdiesout.Impulse

15、responsefunctionsareveryusefulin川ustratingtheshort-termdynamicsinamodel.标准文案实用文档LetslookatanexampletoseehowVARmodellingworks.InLecture,wetriedveryhardtounderstandgoldprices.Weextendourunivariatemodelbyexploringtherelationshipsbetweengoldandsilverprices.Linkingtwo(similar)assetsorsecuritiesisaverycom

16、montradingstrategy,whichiscalledpairs-trading.Beforewedoanysophisticatedmodelling,itisalwaysbeneficialtolookatsomelinecharts.Figure1showstheindexedtimeseriesofnominalgoldandsilverpricesfrom1900to2010.Figure1:Nominalgoldandsilverprices,indexed,1900-2010indexsindexgOA5RwbzOHWU20051nwuolWecanseethatthe

17、reisacertaindegreeofco-movement,whichwemightbeabletoexploitforourtradingstrategy.BeforewecanuseVAR,weneedtoensurethatbothtimeseriesarestationary.ItisobviousfromFigure1thatgoldandsilverpricesarenotstationary.However,aftertakingafirst-differencewecanshowthatpricechangesarestationary.Sobothtimeseriesar

18、eI(1).Thenextstepistodeterminetheoptimallaglengthusinginformationcriteria.Table1showsdifferentspecificationsusingthevarsoccommand.Table 1: Determiningtheoptimallaglengthusinginformationcriteria标准文案实用文档Numberofobs=105Selection-ordercriteriaSample:1906-2010lagllLRdfpFPEAICHQICSBIC234520.1

19、24.130.118551015819826126.00049-1.94511-1.92463-1.8945618.86612.9271.60868.013610.599*444440.0010.0120.8070.0910.031.000442-2.0486-1.98714-1.89694*.000422*-2.09552*-1.9931*-1.84276.000448-2.03465-1.89126-1.68079.000448-2.03478-1.85042-1.57982.000438-2.05954-1.83421-1.50347Endogenous:return_greturn_s

20、Exogenous:_consBasedontheAICandHQIC,twolagsareoptimal;however,the(S)BICprefersonlyonelag.IwouldpreferHQICandtrytwolagsfirst.Ifthesecondlagdoesnotexhibitsignificantcoefficient,wecouldtrytoreducethelaglengthinlinewith(S)BIC.WerunaVARwithtwolagstoexplaincurrentpricechangesingoldandsilver.Table2provides

21、theOLSestimates.Table 2: VARmodelwithtwolagsVectorautoregressionSample:1903-2010Loglikelihood=126.0166FPE=.0004Det(Sigma_ml)=.0003323EquationParmsRMSENo.ofobs=108AIC=-2.148455HQIC=-2.04776SBIC=-1.90011R-sqchi2P>chi2return_g5.1269270.242534.57860.0000returns5.1965690.130616.227630.0027Coef.Std.Err

22、.zP>|z|95%Conf.Intervalreturn_greturn_gL1一.4864107L2.-.0139809.1229856.1228173.96-0.110.0000.909.2453633-.2546979.7274581.2267361return_sL1.-.0068126L2.-.207786.0805903.0807151-0.08-2.570.9330.010-.1647668-.3659847.1511415-.0495874_cons.0277213.01248572.220.026.0032497.0521929return_sreturn_gL1.3

23、143786L2.1085011.1904648.19020381.650.570.0990.568-.0589257-.2642915.6876828.4812937return_sL1.1094293L2.-.3201805.1248083.12500150.88-2.560.3810.010-.1351905-.5651789.354049-.0751821_cons.024511.01933631.270.205-.0133875.0624095Weseethatsilverprices(lag2)affectcurrentgoldprices,andwecanestablishaut

24、ocorrelationinbothtimeseries.TotestwhethergoldGrangercausessilverorviceversa,werunGrangercausalitytestsreportedinTable3.Table 3: Grangercausalitytests标准文案实用文档GrangercausalityWaldtestsEquationExcludedchi2dfProb>chi2return_greturn_s6.76520.034return_gALL6.76520.034return_sreturn_g3.961520.138return

25、_sALL3.961520.138Hence,weconfirmthatpastchangesinsilverpricescanpredictfuturegoldpricechanges.Thisisveryinteresting,asitcanbeusedtodevelopatradingstrategy.Finally,weneedtoshowthattheVARisstable(seeTable4).Table4:StabilityconditionoftheVAREigenvaluestabilityconditionEigenvalueModulus2367286+.362415i.

26、432882367286-.362415i.4328806119136+.3747777i.3797406119136-.3747777i.37974Alltheeigenvalueslieinsidetheunitcircle.VARsatisfiesstabilitycondition.Finally,wecan川ustratetheimpactofsilverpricechangesonfuturegoldpricechangesusinganimpulseresponsefunction.Figure2showstheimpulseresponsefunctionandconfiden

27、ceintervalsderivedfrombootstrapping.Ifsilverpricesincreasetodayby1%,weshouldexpectasignificantdeclineingoldpricesintwoyearsby0.2%.Figure2:Impulseresponsefunction标准文案实用文档step95%CIimpulseresponsefunction(irf)Graphsbyirfname,impulsevariable,andresponsevariable标准文案实用文档6.3 CointegrationWhenweexploreFigur

28、e1abitmorecarefully,wecanseethatsilverandgoldpricesexhibitacertaindegreeofco-movement.Wecouldalmostarguethattheysharecammonstochastictrend.ThelimitationofARIMAandVARmodelsisthattheycanbeonlyusedifthetimeseriesarestationary.Inourcase,wehadtofirst-differenceyourtimeseriestoensurestationarity.First-dif

29、ferencingeliminatesalotofinformationinthetimeseries.Istherenobetterwaytoanalysegoldandsilverprices.Longbeforethedevelopmentofmultivariatetimeserieseconometrics,peoplerealisedthatgoldandsilverseemtohaveacommonmovementaroundalong-termequilibrium(goldsilverpriceratio).Moreover,theideaofequilibriumcondi

30、tionsineconomicsandtheavailabilityofmacroeconomictimeseriesledtothedevelopmentofcointegrationanalysis.Theideaisverysimple.Eveniftwo(ormore)timeseriesarenon-stationaryandhencehavestochastictrends,theymightbestilldrivenbythesameunderlyingfactorsthatleadtotheirstochasticbehaviour.Therefore,weanalysethe

31、timeseriesinlevelsandseewhetherwecanfindalong-termequilibrium-aso-calledcointegratingvector.BeforeweexploretheJohansenprocedurelet'slookatthe-goverratioovertimeshowninFigure3.Figure3:Thegold-silverratio,1900-2010Theratiolookslikeamean-revertingprocessthus,inthelongrunittendstogobacktoitslong-ter

32、mequilibrium(mean).Basedontheratio,wecouldarguethatgoldseemstobeovervaluedcomparedtosilveratthemoment.标准文案实用文档Ofcourse,takingtheratiosuggestsaverysimplecointegratingvector-infactweassumeaone-to-onerelationship.BeforewecanusetheJohansenprocedurewehavetomakesurethatthetimeserieshavethesameorderofinteg

33、rationI(p).WealreadyknowthatgoldandsilverpricesarebothI(1)timeseries.Table5showstheresultsoftheJohansentestforcointegration.InlinewiththeVARmodel,weusetwolags.Table5:Johansentest.johansprice_gprice_s,lags(2)Johansen-JuseliuscointegrationranktestSample:1901to2010Numberofobs=109H1:H0:Max-lambdaTraEige

34、nvaluesrank<=(r)statisticss(lambda)r(rank<=(r+1)(rancetatisticsk<=(p=2).11763126013.6408312.0758622118.599456582.240287.5994565Osterwald-LenumCriticalvalues(95%interval):Table/Case:1*(assumption:interceptinCE)H0:Max-lambdaTrace015.6719.9619.249.24Table/Case:1(assumption:interceptinVAR)H0:Ma

35、x-lambdaTrace014.0715.4113.763.76NormalizedBeta'price_gprice_svec1-1.549e-07.0000176vec24.183e-07-.00001694NormalizedAlphavec1vec2price_g69248.321447701.12price_s-5468.401514662.554Thenullhypothesisthatthereisnocointegration(r=0)canberejectedifweusethetracestatistic.However,thenullhypothesisthat

36、wehaveonecointegratingvector(r=1)cannotberejected.Theproblemisthatthemax-lambdastatisticdoesnotsupportcointegration.Ialsotriedlog-pricesinstead,whichiscommoninanalysinggold-silverratios;however,Idon'tobtainclearresults.Giventheextremeincreaseinvolatilityinprices,itmightbelikelythattherearestruct

37、uralbreaksinanallegedcointegrationvector.Structuralbreaksaredifficulttohandle.Anotherwaytolookatthisproblemistotestwhetherpriceratiosorlog-priceratiosarestationarytimeseries.Iftheyarestationary,thenthetwounderlyingtimeseriesarecointegratedandtheratioindicatesthecointegrationvector.AgainDickey-Fuller

38、testscannotrejectthenullhypothesis;hence,bothratiosdon'tseemtobestationary.标准文案实用文档6.4 Vectorerror-correctionmodel(VECM)TheVECMcombinesVARandcointegrationintooneframework.TheVARisextendedbyincludingdeviationsfromthelong-termequilibriumdefinedbythecointegrationvector.Thecoefficientofthedeviationf

39、romthelong-termequilibriumindicatesthespeedofadjustmentbackintoequilibrium.TheVECMcapturethelong-termrelationshipandtheshort-termdynamicsoftwoormoretimeseries.Let'sseehowitworksinthecaseofgoldandsilverprices.Table6reportstheVECMspecification,whichresemblestheVARwithtwolags.ItalsocontainstheCEcom

40、ponent;theco-callederror-correctioncomponentthatcapturesthedeviationfromthelong-termequilibriuminthepreviousperiod.SotheCEisalaggedandhencepredeterminedvariable,asrequiredbyOLSandtheVARframework.Table6:VECMbasedongoldandsilverpricesNo.ofobs=108=54.15731HQIC=54.28821SBIC=54.48015R-sqchi2P>chi2Vect

41、orerror-correctionmodelSample:1903-2010AICLoglikelihood=-2911.494Det(Sigma_ml)=8.93e+20EquationParmsRMSED_price_g61.5e+060.294942.663730.0000D_price_s649748.20.259835.794240.0000Coef.Std.Err.zP>|z|95%Conf.IntervalD_price_g_ce111.-.0048672.0135324-0.360.719-.0313903.0216559price_gLD.2714343.238468

42、21.140.255-.1959548.7388233L2D.8565381.23517553.640.000.39560261.317474pricesLD.56377846.5867810.090.932-12.3460713.47363L2D.-26.234715.670113-4.630.000-37.34793-15.1215_cons2813.244472046.20.010.995-922380.4928006.9D_price_s_ce111.0005916.00043921.350.178-.0002693.0014524price_gLD.-.007626i3.007739

43、9-0.990.324-.0227962.0075436L2D.0357519.0076334.680.000.0207914.0507123price_sLD.2898047.21378551.360.175-.1292072.7088166L2D.-1.064965.1840335-5.790.000-1.425664-.7042665_cons231462915321.091.510.131-6882.48753175.08Thespeedofadjustmentisnotsignificant,whichunderminestheideaofalong-runequilibriumin

44、goldandsilverpricesassuggestedbytheliterature.Wecanexplorestructuralchanges,whenweplotthepredictedlong-termequilibriumovertime.标准文案实用文档Figure4:Predictedlong-termequilibriumbasedonVECM7n¥eQUZ7n¥eQU40+eQMR-7n¥enwnQ_8A¥eonBo+QnMZ1-195020001900yearItisveryobviousthatthelong-termequil

45、ibriumundergoesstructuralbreaks.Wecouldsplitthetimeperiodintoastableandunstableperiod.Yetthemainissuewithstructuralbreaksisthattheyappeartobeobviousexpos匕butnearlyimpossibletopredictexante.Forinstance,ifwefocuson1900-1980,weobtainaverystrongresultthatunderlinescointegrationandadjustmentsintothelong-

46、termequilibrium.Hence,weconcludethatthegold-silverratioisnolongerareliablephenomenonthatwecouldrelyon.Nevertheless,wecanusetheshort-termdynamicscapturedintheVARtodoshort-termtrading.6.5Pairstrading-APPLICATIONTheideaofpairstradingisthatwetradetwosimilarsharesthataredrivenbysimilarmacroeconomicfactor

47、s.Inourexample,wefocusontheUSsteelindustryandtrytoidentifyatradingstrategybasedonUnitedStatesSteelCorporationandTitanInternational.Firstweneedtomodifythetimedimension,asNASDAQreportsthelatestsharepricesfirst.Weselected5-yearsofdailyclosingpricesforouranalysis.*Timedimensionneedstobemodifiedgent=_nre

48、placet=1264-t标准文案实用文档tssett*Linecharttwoway(lineus_steelt)(linetitant)Let'hsaveaquicklookatalinechartcombiningbothshareprices.Obviouslybothsharepricesarenon-stationary,whichweshouldconfirmfirstusingDickey-Fullertests.US_steelTitanWerunDickey-Fullertestsbasedonsharepricesandfirst-differencedtimes

49、eries.ThetestsconfirmthatbothtimeseriesareI(1).Hence,wecantrytofindacointegrationrelationfollowingtheJohansenprocedure.Beforewedothat,IsuggestthatweexploreaVARmodelanddeterminetheoptimallagstructure.*Returnsgenr_us=ln(us_steel)-ln(l.us_steel)genr_titan=ln(titan)-ln(l.titan)*Dickey-Fullerdfullerus_st

50、eeldfullertitandfullerr_usdfullerr_titan标准文案实用文档.varsocr_usr_titanSelection-ordercriteriaSample:6-1263Numberofobs=1258lagllLRdfpFPEAICHQICSBIC()460,1.882.3e-06*-7.31777*-7.3147*-7.30961*460；5.821.876140.7592.3e-06-7.31291-7.3037-7.2884460；3.846.04640.1962.3e-06-7.31135-7.29601-7.270523461).192.69234

51、0.6112.3e-06-7.30713-7.28565-7.249964I4611.913.438540.4872.3e-06-7.30351-7.27588-7.23Endogenous:r_usr_titanExogenous:_consWecannotestablishanyVARlagstructure,whichshowsthatthereishardlyanyserialcorrelationofreturns.ThisisgreatnewsfortheEfficientMarketHypothesis-butbadnewsforus.WhenyouexploreACandPAC

52、function,youwillalsodiscoverthatunivariateanalysiswon'tgoanywhere,asautocorrelationinreturnsisnotpresent.Luckily,wecanfindcointegrationwithTitanadjustingbacktolong-termequilibriumconditions.BasedontheVECM,wecanpredictthelevels(shareprices)ofbothstocks.Theseareone-stepaheadforecast.Whenshouldyoubuyandsell?Thisalsodependsontransactioncosts.Inpairstrading,yougolonginonestockandshortinthesecondstocktohedgeyourposition.Weneedtoavoidthatwehavetotradetoofrequentlytoreducetransactioncosts.Thefollowingfigurecumulatesthedeviationsfromthelong-termequilibriumandsh