第5章 Univariate time series modelling and forecasting(金融计量经济学导论东北财经大学 陈磊)

Chapter5Univariatetimeseriesmodellingandforecasting11introduction单变量时间序列模型只利用变量的过去信息和可能的误差项的当前和过去值来建模和预测的一类模型(设定)。与结构模型不同；通常不依赖于经济和金融理论用于描述被观测数据的经验性相关特征ARIMA（AutoRegressiveIntegratedMovingAverage）是一类重要的时间序列模型Box-Jenkins1976当结构模型不适用时，时间序列模型却很有用如引起因变量变化的因素中包含不可观测因素，解释变量等观测频率较低。结构模型常常不适用于进行预测本章主要解决两个问题一个给定参数的时间序列模型，其变动特征是什么？给定一组具有确定性特征的数据，描述它们的合适模型是什么？2AStrictlyStationaryProcessAstrictlystationaryprocessisonewhere

Foranyt1,t2,…,tn∈Z,anym∈Z,n=1,2,…AWeaklyStationaryProcessIfaseriessatisfiesthenextthreeequations,itissaidtobeweaklyorcovariance

stationary1.E(yt)=, t=1,2,...,2.3.

t1

,t22SomeNotationandConcepts3Soiftheprocessiscovariancestationary,allthevariancesarethesameandallthecovariancesdependonthedifferencebetweent1

andt2.Themoments ,s=0,1,2,... areknownasthecovariancefunction.Thecovariances,s,areknownasautocovariances.

However,thevalueoftheautocovariancesdependontheunitsofmeasurementofyt.Itisthusmoreconvenienttousetheautocorrelationswhicharetheautocovariancesnormalisedbydividingbythevariance: ,s=0,1,2,...

Ifweplotsagainsts=0,1,2,...thenweobtaintheautocorrelationfunction(acf)or

correlogram.

SomeNotationandConcepts4Awhitenoiseprocessisonewithnodiscerniblestructure.

Thustheautocorrelationfunctionwillbezeroapartfromasinglepeakof1ats=0.如果假设yt服从标准正态分布,则approximatelyN(0,1/T)

Wecanusethistodosignificancetestsfortheautocorrelationcoefficientsbyconstructingaconfidenceinterval.

a95%confidenceintervalwouldbegivenby

.

Ifthesampleautocorrelationcoefficient,,fallsoutsidethisregionforanyvalueofs,thenwerejectthenullhypothesisthatthetruevalueofthecoefficientatlagsiszero.

AWhiteNoiseProcess

5WecanalsotestthejointhypothesisthatallmofthekcorrelationcoefficientsaresimultaneouslyequaltozerousingtheQ-statisticdevelopedbyBoxandPierce:

whereT=samplesize,m=maximumlaglengthTheQ-statisticisasymptoticallydistributedasa.

However,theBoxPiercetesthaspoorsmallsampleproperties,soavarianthasbeendeveloped,calledtheLjung-Boxstatistic:

Thisstatisticisveryusefulasaportmanteau(general)testoflineardependenceintimeseries.JointHypothesisTests6Question: Supposethatwehadestimatedthefirst5autocorrelationcoefficientsusingaseriesoflength100observations,andfoundthemtobe(from1to5):0.207,-0.013,0.086,0.005,-0.022. Testeachoftheindividualcoefficientforsignificance,anduseboththeBox-PierceandLjung-Boxteststoestablishwhethertheyarejointlysignificant.Solution: Acoefficientwouldbesignificantifitliesoutside(-0.196,+0.196)atthe5%level,soonlythefirstautocorrelationcoefficientissignificant.

Q=5.09andQ*=5.26 Comparedwithatabulated2(5)=11.1atthe5%level,sothe5coefficientsarejointlyinsignificant.

AnACFExample(p234)7Letut(t=1,2,3,...)beasequenceofindependentlyandidenticallydistributed(iid)randomvariableswithE(ut)=0andVar(ut)=2,then yt=+ut+1ut-1

+2ut-2+...+qut-q

isaqthordermovingaveragemodelMA(q).Orusingthelagoperatornotation:

Lyt=yt-1

Liyt=yt-i通常，可以将常数项从方程中去掉，而并不失一般性。3MovingAverageProcesses

8移动平均过程的性质Itspropertiesare E(

yt

)=

Var(

yt

)=0=(1+)2 Covariances 自相关函数9ConsiderthefollowingMA(2)process:

whereutisazeromeanwhitenoiseprocesswithvariance. (i)CalculatethemeanandvarianceofXt (ii)Derivetheautocorrelationfunctionforthisprocess(i.e.expresstheautocorrelations,1,2,...asfunctionsoftheparameters1and2). (iii)If1=-0.5and2=0.25,sketchtheacfofXt.ExampleofanMAProcess

10(i)IfE(ut

)=0,thenE(ut-i)=0

i.So E(Xt

)=E(ut+1ut-1+2ut-2

)=E(ut

)+1E(ut-1

)+2E(ut-2

)=0

Var(Xt

) =E[Xt

-E(Xt

)][Xt

-E(Xt

)] Var(Xt) =E[(Xt

)(Xt

)] =E[(ut+1ut-1+2ut-2)(ut+1ut-1+2ut-2)] =E[+cross-products] ButE[cross-products]=0,sinceCov(ut,ut-s)=0fors0.So

Var(Xt

)

=0=E[

]

=

=Solution11(ii)TheacfofXt

1 =E[Xt-E(Xt

)][Xt-1-E(Xt-1

)] =E[Xt

][Xt-1

] =E[(ut+1ut-1+2ut-2

)(ut-1+1ut-2+2ut-3

)] =E[(

)] = =

2 =E[Xt

-E(Xt

)][Xt-2

-E(Xt-2

)] =E[Xt

][Xt-2

] =E[(ut+1ut-1+2ut-2

)(ut-2+1ut-3+2ut-4

)] =E[()] =

Solution(cont’d)12

3 =E[Xt

][Xt-3

] =E[(ut+1ut-1+2ut-2

)(ut-3+1ut-4+2ut-5

)] =0

Sos=0fors>2.

nowcalculatetheautocorrelations:Solution(cont’d)13(iii)For1=-0.5and2=0.25,substitutingtheseintotheformulaeabovegives1

=-

0.476,2=0.190.

Thustheacfplotwillappearasfollows:ACFPlot14Anautoregressivemodeloforderp,AR(p)canbeexpressedas Orusingthelagoperatornotation: Lyt=yt-1

Liyt=yt-i

or or where

4AutoregressiveProcesses

15平稳性使AR模型具有一些很好的性质。如前期误差项对当前值的影响随时间递减。TheconditionforstationarityofageneralAR(p)modelisthattherootsof特征方程

alllieoutsidetheunitcircle.Example1:Isyt=yt-1+utstationary? Thecharacteristicrootis1,soitisaunitrootprocess(sonon-stationary)Example2:p241AstationaryAR(p)modelisrequiredforittohaveanMA()representation.

TheStationaryCondition

foranARModel

16Statesthatanystationaryseriescanbedecomposedintothesumoftwounrelatedprocesses,apurelydeterministicpartandapurelystochasticpart,whichwillbeanMA().

FortheAR(p)model,

,ignoringtheintercept,theWolddecompositionis

where,

可以证明，算子多项式R(L)的集合与代数多项式R(z)的集合是同结构的，因此可以对算子L做加、减、乘和比率运算。Wold’sDecompositionTheorem

17Themomentsofanautoregressiveprocessareasfollows.Themeanisgivenby*TheautocovariancesandautocorrelationfunctionscanbeobtainedbysolvingwhatareknownastheYule-Walkerequations: *IftheARmodelisstationary,theautocorrelationfunctionwilldecayexponentiallytozero.TheMomentsofanAutoregressiveProcess18ConsiderthefollowingsimpleAR(1)model

(i)Calculatethe(unconditional)meanofyt.Fortheremainderofthequestion,set=0forsimplicity.(ii)Calculatethe(unconditional)varianceofyt.(iii)Derivetheautocorrelationfunctionforyt.

SampleARProblem19(i) E(yt)=E(+1yt-1) =+1E(yt-1) Butalso

E(yt)=

+1

(

+1E(yt-2)) =

+1

+12

E(yt-2))

=

+1

+12

(

+1E(yt-3)) =

+1

+12

+13

E(yt-3)

AninfinitenumberofsuchsubstitutionswouldgiveE(yt)=

(1+1+12

+...)+1

y0

Solongasthemodelisstationary,i.e.

,then1=0.

SoE(yt)=

(1+1+12

+...)=Solution

20(ii)Calculatingthevarianceofyt

:*

FromWold’sdecompositiontheorem:

Solongas,thiswillconverge.

Solution(cont’d)

21Var(yt) =E[yt-E(yt)][yt-E(yt)]butE(yt)=0,sincewearesetting=0.Var(yt)=E[(yt)(yt)]

*有简便方法

=E[

]

=E[

=E[

=

=

=Solution(cont’d)

22(iii)Turningnowtocalculatingtheacf,firstcalculatetheautocovariances:

（*用简便方法） 1=Cov(yt,yt-1)=E[yt-E(yt)][yt-1-E(yt-1)] 1=E[ytyt-1]

1=E[

]

=E[

=

=Solution(cont’d)

23Solution(cont’d)Forthesecondautocorrelationcoefficient, 2=Cov(yt,yt-2)=E[yt-E(yt)][yt-2-E(yt-2)]Usingthesamerulesasappliedaboveforthelag1covariance

2=E[yt

yt-2]

=E[

] =E[

==

=24Solution(cont’d)Ifthesestepswererepeatedfor3,thefollowingexpressionwouldbeobtained3=andforanylags,theautocovariancewouldbegivenbys=Theacfcannowbeobtainedbydividingthecovariancesbythevariance:25Solution(cont’d)

0=

1= 2=

3=…

s=26Measuresthecorrelationbetweenanobservationkperiodsagoandthecurrentobservation,aftercontrollingforobservationsatintermediatelags(i.e.alllags<k).yt-k与

yt之间的偏自相关函数kk

是在给定yt-k+1,yt-k+2,…,yt-1

的条件下，yt-k与

yt之间的部分相关。Sokkmeasuresthecorrelationbetweenytandyt-kafterremovingtheeffectsofyt-k+1,yt-k+2,…,yt-1.

或者说，偏自相关函数kk

是对yt-k与

yt之间未被yt-k+1,yt-k+2,…,yt-1所解释的相关的度量。Atlag1,theacf=pacfalwaysAtlag2,22=(2-12)/(1-12)Forlags3+,theformulaearemorecomplex.

5ThePartialAutocorrelationFunction(denotedkk)

27ThepacfisusefulfortellingthedifferencebetweenanARprocessandan

MAprocess.InthecaseofanAR(p),therearedirectconnectionsbetweenytandyt-sonly

fors

p.SoforanAR(p),thetheoreticalpacfwillbezeroafterlagp.InthecaseofanMA(q),thiscanbewrittenasanAR(),sotherearedirectconnectionsbetweenytandallitspreviousvalues.ForanMA(q),thetheoreticalpacfwillbegeometricallydeclining.ThePartialAutocorrelationFunction28TheinvertibilityconditionIfMA(q)processcanbeexpressedasanAR(∞),thenMA(q)isinvertible.MA(q)的可逆性条件：特征方程根的绝对值大于1。 从而有29BycombiningtheAR(p)andMA(q)models,wecanobtainanARMA(p,q)model:

where and or with6ARMAProcesses

30ARMA过程的特征是AR和MA的组合。可逆性条件：Similartothestationaritycondition,wetypicallyrequiretheMA(q)partofthemodeltohaverootsof(z)=0greaterthanoneinabsolutevalue.

ThemeanofanARMAseriesisgivenby

TheautocorrelationfunctionforanARMAprocesswilldisplaycombinationsofbehaviourderivedfromtheARandMAparts,butforlagsbeyondq,theacfwillsimplybeidenticaltotheindividualAR(p)model.

ARMA过程的特征31Anautoregressiveprocesshasageometricallydecayingacf:拖尾numberofspikes尖峰信号ofpacf=ARorder

：截尾

AmovingaverageprocesshasNumberofspikesofacf=MAorder：截尾ageometricallydecayingpacf

：拖尾AARMAprocesshasageometricallydecayingacf:拖尾ageometricallydecayingpacf

：拖尾SummaryoftheBehaviouroftheacfandpacffor

ARandMAProcesses32 Theacfandpacfareestimatedusing100,000simulatedobservationswithdisturbancesdrawnfromanormaldistribution.

ACFandPACFforanMA(1)Model:yt=–0.5ut-1+utSomesampleacfandpacfplots

forstandardprocesses33ACFandPACFforanMA(2)Model:

yt=0.5ut-1-

0.25ut-2+ut

34ACFandPACFforaslowlydecayingAR(1)Model:

yt=0.9

yt-1+ut

35ACFandPACFforamorerapidlydecayingAR(1)Model:yt=0.5

yt-1+ut

36ACFandPACFforaAR(1)ModelwithNegativeCoefficient:yt=-0.5

yt-1+ut

37ACFandPACFforaNon-stationaryModel

(aunitcoefficient):yt=yt-1+ut

38ACFandPACFforanARMA(1,1):

yt=0.5yt-1+0.5ut-1+ut

39BoxandJenkins(1970)werethefirsttoapproachthetaskofestimatinganARMAmodelinasystematicmanner.Thereare3stepstotheirapproach: 1.Identification 2.Estimation 3.Modeldiagnosticchecking

Step1:

-Involvesdeterminingtheorderofthemodel. -Useofgraphicalprocedures:data,acf,pacf -Abetterprocedureisnowavailable

7BuildingARMAModels

-TheBoxJenkinsApproach

40Step2: -Estimationoftheparameters -Canbedoneusingleastsquaresormaximumlikelihooddependingonthemodel.Step3:

-ModelcheckingBoxandJenkinssuggest2methods: -deliberateoverfitting -residualdiagnostics

一般只涉及自相关检验。Box–Jenkins诊断检验方法只能针对参数不足的模型，而不能针对参数过多的模型。BuildingARMAModels

-TheBoxJenkinsApproach(cont’d)

41Wewanttoformaparsimoniousmodelbecause -varianceofestimatorsisinverselyproportionaltothenumberofdegreesoffreedom. -modelswhicharebigmightbeinclinedtofittodataspecificfeatures,whichwouldnotbereplicatedout-of-sample.

Identificationwouldtypicallynotbedoneusingacfandpacf.Thisgivesmotivationforusinginformationcriteria,whichembody2factors -atermwhichisafunctionoftheRSS -somepenaltyforthelossofdegreesoffreedomfromaddingextraparametersTheobjectistochoosethenumberofparameterswhichminimisestheinformationcriterion.SomeMoreRecentDevelopmentsin

ARMAModelling

42Theinformationcriteriavaryaccordingtohowstiffthepenaltytermis.

ThethreemostpopularcriteriaareAkaike’s(1974)informationcriterion(AIC),Schwarz’s(1978)Bayesianinformationcriterion(SBIC),theHannan-Quinncriterion(HQIC).

wherek=p+q+1,T=samplesize.

Sowemin.IC

s.t.

InformationCriteriaforModelSelection

43SBICembodiesastifferpenaltytermthanAIC.WhichICshouldbepreferrediftheysuggestdifferentmodelorders?SBICisstronglyconsistent(butinefficient).AICisnotconsistent,andwilltypicallypick“bigger”models.NocriterionisdefinitelysuperiortoothersInformationCriteriaforModelSelection

44AsdistinctfromARMAmodels.TheIstandsforintegrated.Anintegratedautoregressiveprocessisonewithacharacteristicrootontheunitcircle.TypicallyresearchersdifferencethevariableasnecessaryandthenbuildanARMAmodelonthosedifferencedvariables.

AnARMA(p,q)modelinthevariabledifferenceddtimesisequivalenttoanARIMA(p,d,q)modelontheoriginaldata.ARIMAModels

45Anothermodellingandforecastingtechnique

Howmuchweightdoweattachtopreviousobservations?

Expectrecentobservationstohavethemostpowerinhelpingtoforecastfuturevaluesofaseries.

Theequationforthemodel

St=yt+(1-)St-1 (1) or St=St-1

+

(yt-1-St-1)

isthesmoothingconstant,with01 yt isthecurrentrealisedvalue St isthecurrentsmoothedvalue11ExponentialSmoothing

46Lagging(1)byoneperiodwecanwrite

St-1=yt-1+(1-)St-2 (2)andlaggingagain

St-2=yt-2+(1-)St-3 (3)

Substitutinginto(1)forSt-1from(2) St =yt+(1-)(yt-1+(1-)St-2)

=yt+(1-)yt-1+(1-)2St-2 (4)

Substitutinginto(4)forSt-2from(3) St

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

=yt+(1-)yt-1+(1-)2(yt-2+(1-)St-3) =yt+(1-)yt-1+(1-)2yt-2+(1-)3St-3

ExponentialSmoothing47Tsuccessivesubstitutionsofthiskindwouldleadto

since0,theeffectofeachobservationdeclinesexponentiallyaswemoveanotherobservationforwardintime.

Forecastsaregeneratedby

ft+s=St

forallstepsintothefutures=1,2,...Thistechniqueiscalledsingle(orsimple)exponentialsmoothing.

ExponentialSmoothing48Itdoesn’tworkwellforfinancialdatabecauseitissimplisticandinflexiblethereislittlestructuretosmoothitisanARIMA(0,1,1)withMAcoefficient(1-)itcannotallowforseasonalityforecastsdonotconvergeonlongtermmeanassCanmodifysingleexponentialsmoothingtoallowfortrends(Holt’smethod)toallowforseasonality(Winter’smethod).

AdvantagesofExponentialSmoothingVerysimpletouseEasytoupdatethemodelifanewrealisationbecomesavailable.

ExponentialSmoothing49Forecasting=prediction.尝试确定序列可能的将来值。Animportanttestoftheadequacyofamodel.预测是非常有用的。e.g.-Forecastingtomorrow’sreturnonaparticularshare-Forecastingthepriceofahousegivenitscharacteristics-Forecastingtheriskinessofaportfoliooverthenextyear-ForecastingthevolatilityofbondreturnsWecandistinguishtwoapproaches: -Econometric(structural)forecasting,较适合长期

-TimeseriesforecastingThedistinctionissomewhatblurred(e.g,VARs).Pointforecast点预测：predictasinglevalue对每一时段Intervalforecast区间预测：给定置信区间ForecastinginEconometrics

50Expectthe“forecast”ofthemodeltobegoodin-sample.

Saywehavesomedata-e.g.monthlyFTSEreturnsfor120months:1990M1–1999M12.Wecoulduseallofittobuildthemodel,orkeepsomeobservationsback:

Agoodtestofthemodelsincewehavenotusedtheinformationfrom 1999M1onwardswhenweestimatedthemodelparameters.In-SampleVersusOut-of-Sample

51SometerminologyOne-stepaheadforecastmulti-stepaheadforecastssstepaheadforecastsstepaheadextrapolationforecastForecastinghorizon范围，水平：预测期的时间跨度预测随预测长度的变化而变化最优预测模型也随着预测长度变化而变化Recursivewindows(samples)：样本长度随时间增加Rollingwindows(samples)：样本随时间移动但长度固定e.g.p28152HowtoproduceforecastsTounderstandhowtoconstructforecasts,weneedtheideaofconditionalexpectations: E(yt+1

t)Wecannotforecastawhitenoiseprocess:

E(ut+s

t)=0

s>0.Thetwosimplestforecasting“methods” 1.Assumenochange:f(yt+s)=E(yt+s

t)=

yt

，对随机游动是最优预测 2.Forecastsarethelongtermaveragef(yt+s)=

对平稳序列来说优于1。53ModelsforForecasting对于时间序列预测，时间序列模型通常优于结构模型Structuralmodels e.g. Toforecasty,werequiretheconditionalexpectationofitsfuturevalue:

Butwhatareetc.?Wecoulduseetc.,so

=

!!54ModelsforForecastingTimeSeriesModels Thecurrentvalueofaseries,yt,ismodelledasafunctiononlyofitspreviousvaluesandthecurrentvalueofanerrorterm(andpossiblypreviousvaluesoftheerrorterm).Modelsinclude:simpleunweightedaveragesexponentiallyweightedaveragesARIMAmodelsNon-linearmodels–e.g.thresholdmodels,GARCH,bilinearmodels,etc.55Theforecastingmodeltypicallyusedisoftheform:

whereft,s=yt+s,s0;

ut+s=0,s>0

=ut+s,s0

ForecastingwithARMAModels

56AnMA(q)onlyhasmemoryoflengthq.

e.g.saywehaveestimatedanMA(3)model:

yt=+1ut-1+

2ut-2+

3ut-3+ut yt+1=+

1ut+

2ut-1+

3ut-2+ut+1 yt+2=+

1ut+1+

2ut+

3ut-1+ut+2 yt+3=+

1ut+2+

2ut+1+

3ut+ut+3

Weareattimetandwewanttoforecast1,2,...,sstepsahead.

Weknowyt,yt-1,...,andut,ut-1，...。

ForecastingwithMAModels

57ft,1

=yt+1,

t=E(yt+1

t)=E(+

1ut+

2ut-1+

3ut-2+ut+1)

=+

1ut+

2ut-1+

3ut-2

ft,2

=yt+2

,

t=E(yt+2

t)=E(+

1ut+1+

2ut+

3ut-1+ut+2)

=+

2ut+

3ut-1

ft,3

=yt+3

,

t=E(yt+3

t) =E(+

1ut+2+

2ut+1+

3ut+ut+3)

=+

3ut

ft,4

=yt+4

,

t=E(yt+4

t) =

ft,s

=yt+s

,

t=E(yt+s

t) =

s

4ForecastingwithMAModels58自回归模型有无限记忆。SaywehaveestimatedanAR(2)

yt=+1yt-1+

2yt-2+ut yt+1=+

1yt+

2yt-1+ut+1 yt+2=+

1yt+1+

2yt+ut+2 yt+3=+

1yt+2+

2yt+1+ut+3

ft,1=E(yt+1

t)=E(+

1yt+

2yt-1+ut+1)

=+

1E(yt)+

2E(yt-1)

=+

1yt+

2yt-1

ft,2=E(yt+2

t)=E(+

1yt+1+

2yt+ut+2)

=+

1E(yt+1)+

2E(yt)

=+