lec213經(jīng)典教材《金融時間序列分析》RueyS.Tsay英文第三版高清教材以及最新2013年高清講義

1、 # LectureNotesofBus41202(Spring2013)AnalysisofFinancialTimeSeriesRueyS.TsaySimpleARmodels:(Regressionwithlaggedvariables.)Motivatingexample:ThegrowthrateofU.S.quarterlyrealGNPfrom1947to1991.Recallthatthemodeldiscussedbeforeisrt=0.005+0.35r/_i+0.18心_20.14幾_3+%aa=0.01.ThisiscalledanAR(3)modelbecauset

2、hegrowthrate門dependsonthegrowthratesofthepastthreequarters.Howdowespecifythismodelfromthedata?Isitadequateforthedata?Whataretheimplicationsofthemodel?Thesearethequestionsweshalladdressinthislecture.Anotherexample:U.S.monthlyunemploymentrate.AR(1)model:Form:幾=+01門_1+%where0and機arerealnumbers,whichare

3、referredtoasparameters(tobeestimatedfromthedatainanapplication).Forexample,rt=0.005+0.2n_i+atStationarity:necessaryandsufficientcondition|0i|2.Stochasticbusinesscycle:if+4020,then幾showschar-ncteristicsofbusinesscycleswithaveragelength.27Twherethecosineinverseisstatedinradian.Ifwedenotethesolutionsof

4、thepolynomialasbi,wherei=/T5thenwehave0i=2aand02=(以+以)sothat27TInRorS-Plus,onecanobtain/a2+b2usingthecommandMod.Forecasts:SimilartoAR(1)modelsSimulationinR:Usethecommandarima.simyl=arimasim(model=list(ar=c(13,一.4),1000)y2=arimasim(model=list(ar=c(8,一.7),1000)ChecktheACFandPACFoftheabovetwosimulateds

5、eries.Discussion:(Referenceonly)AnAR(2)modelcanbewrittenasanAR(1)modelifoneexpandsthedimension.Specifically,wehave4=0i(n_i“)+血(幾2“)+心rt-i=rt-i4(anidentity.)Now,puttingthetwoequationstogether,wehave幾一“.n-i一“010210rt-i-“atThisisa2-dimensionalAR(1)model.SeveralpropertiesoftheAR(2)modelcanbeobtainedfrom

6、theexpandedAR(1)model.BuildinganARmodelOrderspecificationPartialACF:(naive,buteffective)UseconsecutivefittingsSeeText(p.40)fordetailsKeyfeature:PACFcutsoffatlagpforanAR(p)model.Illustration:SeethePACFoftheU.S.quarterlygrowthrateofGNP.Akaikeinformationcriterion2/4W)=ln()+foranAR()model,where進istheMLE

7、ofresidualvariance.FindtheARorderwithminimumAICfor60,P.BICcriterion:BIC)=ln&)+轡1Series:dgnpcooCMoLLonames(ml)1rder116Hn.used1111series*11ar11order.max”frequency11Hvar.pred1x-mean11aic11Hpartialacf11resid11methodcallasy.var.coefplot(ml$resid,type=;1;)%Plotresidualsofthefittedmodel(notshown)Box.test(m

8、l$resid,lag=10,type=;Ljungy)%ModelcheckingBox-Ljungtestdata:ml$residX-squared=7.0808,df=10,p-value=0.7178m2=arima(x,order=c(3,0,0)%Anotherapproachwithordergiven.m2Call:Coefficients:arlar2ar3034800.1793-0.1423s.e.0.07450.077800745arima(x=x,order=c(3,0,0)intercept%Fittedmodelis0.0077%y(t)=0.348y(t-1)+

9、0.179y(t-2)0.0012%-0142y(t-3)+a(t),%wherey(t)=x(t)-0.0077sigma*2estimatedas9427e-05:loglikelihood=565.84,aic=-1121.68names(m2)1Hcoef”sigma2var.coef11maskloglik11aic117,armaresiduals11call11series11code11Hn.condH13modelBoxtest(m2$residuals,lag=10,type=Ljung)Box-Ljungtestdata:m2$residualsX-squared=7.0

10、169,df=10,p-value=0.7239plot(m2$residuals,type=J1J)%Residualplottsdiag(m2)%obtain3plotsofmodelchecking(notshowninhandout)pl=c(l,-m2$coef1:3)%Furtheranalysisofthefittedmodelroots=polyroot(pl)roots11.590253+1.063882e+00i-1.920152-3530887e-17i1.590253-1.063882e+00iMod(roots)11.9133081.9201521.913308k=2

11、*pi/acos(1.590253/1.913308)k110.65638predict(m2,8)%Predictionl-stepto8-stepahead$predTimeSeries:Start=177End=184Frequency=110.0012362540.0045555190.0074549060.00795851850.0081814420.0079368450.0078200460.007703826$seTimeSeries:Start=177End=184Frequency=110.0097093220.0102805100.0106863050.0106889945

12、0.0106897330.0106947710.0106955110.010696190Anotherexample:IVIonthlyUSunemploymentTatefromJanuary1948toFebruary2013.Demonstration:inclass,includingtheRscriptsfore,foreplot：andbacktest.require(quantmod)getSymbols(NRATE,src=,FREDu)rate=as.numeric(UNRATE$UNRATE)unrate=ts(rate,frequency=12,start=c(1948,

13、1)plot(unrate)head(UNRATE)UNRATE1948-01-013.41948-02-013.81948-06-013.6acf(rate)acf(diff(rate)par(mfcol=c(2,l)acf(rate)acf(diff(rate)ml=ar(diff(rate),method=Hmleu)varp:redHpartialacfncallx.mean11aic11resid11method11asy.var.coef11names(ml)1,ordernar6Hnused11order.max111series1frequencyml$order112ml=a

14、rima(rate,order=c(12,1,0)mlCall:Coefficients:arlar2ar3ar4ar5ar6ar7ar8ar90.02330.21940.15310.08890.1239-0.0059-0.02490.0166-0.0014s.e.0.03560.03560.03630.03690.03700.0373003730.03710.03700)arima(x=rate,order=c(12,1,arlOarilarl2-0.09550.0350-0.12820.03650.03570.0358sigma*2estimatedas0.03778:loglikelih

15、ood=17075,aic=-3155tsdiag(ml,gof=24)m2=arima(rate,order=c(2,1,1),seasonal=list(order=c(l,0,1),period=12)m2Call:arima(x=rate,order=c(2,1,1),seasonal=list(order=c(l,0,1),period=12)Coefficients:sigma*2estimatedas0.0363:loglikelihood=185.07,aic=-358.13tsdiag(m2,gof=24)#useforecastoriginatt=770.arlar20.5

16、9820.2300s.e.0.06250.0391malsarismal-0.59020.5515-0.81350.05770.07100.0521source(fore.R11)fore(ml,rate,770,12)TimeSeries:Start=771End=782Frequency=118.2155248.1151508.0808518.0106707.9903777.9635257.9560297.95121897.9905477.9899618.0290948.041167TimeSeries:Start=771End=782Frequency=110.19522650.2793

17、9620.37015620.46097550.55196690.64942920.741492780.83073920.91916381.00459921.08094941.1574140rate771:78218.28.18.28.28.28.17.87.97.87.87.97.7p2=fore(m2,rate,770,12)source(foreplot.R)foreplot(p2,rate,770,750)%Noshowninthehandoutsource(backtest.R)backtest(ml,rate,760,1)1HRMSEofout-of-sampleforecasts1

18、110.1288231111Meanabsoluteerrorofout-of-sampleforecasts1110.1029483backtest(m2,rate,760,1)1HRMSEofout-of-sampleforecasts1110.1156268111Meanabsoluteerrorofout-of-sampleforecasts10.09050361Moving-average(MA)modelModelwithfinitememory!SomedailystockreturnshaveminorserialcorrelationsandcanbemodeledasMAo

19、rARmodels.MA(1)modelForm:rt=“+9at_iStationarity:alwaysstationary.Mean(orexpectation):E(心)=“Variance:Var(n)=(1+的代.Autocovariance:Lag1:Cov(幾”_i)=一吠Lag0:Cov(n,=0for1.Thus,rtisnotrelatedto化_2,幾一3：ACF:pi=磊，pg=0for1.Finitememory!MA(1)modelsdonotrememberwhathappentwotimeperiodsago.Forecast(atorigint=n):14

20、1-stepahead:rn,(l)=“一Oan.Why?Becauseattimen,anisknown,butan+iisnot.1-stepaheadforecasterror:en(l)=aniwithvarianceMulti-stepahead:=“for01.Thus,foranMA(1)model,themulti-stepaheadforecastsarejustthemeanoftheseries.Why?Becausethemodelhasmemoryof1timeperiod.Multi-stepaheadforecasterror:Varianceofmulti-st

21、epaheadforecasterror:(1+02)必=varianceofn.Invertibility:Concept:rtisaproperlinearcombinationofatandthepastobservationsn_i?心_2,.Whyisitimportant?Itprovidesasimplewaytoobtaintheshockat，F(xiàn)oraninvertiblemodel,thedependenceofrton化_0convergestozeroas0increases.Condition:62.Forecastsgothethemeanafter2periods

22、.BuildinganMAmodelSpecification:UsesampleACFSampleACFsareallsmallafterlagqforanMA(g)series.(SeetestofACF.)Constantterm?Checkthesamplemean.Estimation:usemaximumlikelihoodmethodConditional:Assumeat=0for0Exact:Treata/with1,butpi=01-況/Var(77)豐0i.ThisisthedifferencebetweenAR(1)andARMA(1,1)models.PACF:doe

23、snotcutoffatfinitelags.BuildinganARMA(1,1)modelSpecification:useEACForAICWhatisEACF?Howtouseit?Seetext.Estimation:cond.orexnctlikelihoodmethodModelchecking:asbeforeForecast:MA(1)affectsthe1-stepaheadforecast.OthersaresimilartothoseofAR(1)models.Threemodelrepresentations:ARMAform:compact.,usefulinest

24、imationandforecastingARrepresentation:(bylongdivision)rt=(/)()+at+7Tin_i+7T2心_2+Ittellshowrtdependsonitspastvalues.MArepresentation:(bylongdivision)rt=/i+at+妙g_i+妙2他_2TIttellshowrtdependsonthepastshocks.Forastationaryseries,血convergestozeroasiToo.Thus,theeffectofanyshockistransitory.TheMArepresentat

25、ionisparticularlyusefulincomputingvariancesofforecasterrors.Fora-st.epaheadforecast,theforecasterrorisen()=an+e+血+仇_1術(shù)+1ThevarianceofforecasterrorisVar%(0)=(1+悄卜詭_1)龍.Unit-rootNonstationarityRandomwalkFormpt=pt-i+atUnitroot?ItisanAR(1)modelwithcoefficient0i=1.Nonstationary:Why?Becausethevarianceof幾d

26、ivergestoinfinityastincreases.Strongmemory:sampleACFapproaches1foranyfinitelag.RepeatedsubstitutionshowsTOC o 1-5 h zooooPt=工=工i=0i=0where冊=1foralli.Thus,血doesnotconvergetozero.Theeffectofanyshockispermanent.RandomwalkwithdriftForm:pt=M+Pt-i+g“豐0.HasaunitrootNonstationarvzStrongmemoryHasatimetrendwi

27、thslopeWhy?differencing1stdifference:rt=PtPt-iIfptisthelogprice,thenthe1stdifferenceissimplythelogreturn.Typically,1stdifferencemeansthechange：or*incre-mentoftheoriginalseries.Seasonaldifference:yt=Pt.Pt-s.wheresistheperiodicity,e.g.s=4forquarterlyseriesands=12formonthlyseries.Ifptdenotesquarterlyea

28、rnings,thenytisthechangeinearningfromthesamequarteroneyearbefore.MeaningoftheconstantterminamodelMAmodel:meanARmodel:relatedtomean1stdifferenced:timeslope,etc.PracticalimplicationinfinancialtimeseriesExample:MonthlylogreturnsofGeneralElectrics(GE)from1926to1999(74years)Samplemean:1.04%,std(/z)=0.26V

29、erysignificant!isabout12.45%ayear$1investmentinthebeginningof1926isworthannualcompoundedpayment:$5907quarterlycompoundedpayment:$8720monthlycompoundedpayment:$9570Continuouslycompounded?Unit-roottestLetptbethelogpriceofanasset.Totestthatptisnotpredictable(i.e.hasaunitroot),twomodelsarecommonlyemploy

30、ed:Pt=01仇1+etPt=00+血仇_1+勺Thehypothesisofinterestis耳：0i=1vsHa:0ida=read.table(rq-gdpc96.txt,header=T)gdp=log(da,4)adfTest(gdp,lag=4,type=c(11cH)#AssumeanAR(4)modelfortheseriesTitle:AugmentedDickey-FullerTestTestResults:PARAMETER:LagOrder:4STATISTIC:Dickey-Fuller:-1.7433PVALUE:0.4076#cannotrejectthenu

31、llhypothesisofaunitroot*Amorecarefulanalysisx=diff(gdp)ord=ar(x)#identifyanARmodelforthedifferencedseriesordCall:ar(x=x)Coefficients:1230.34290.1238-0.1226Orderselected3sigma*2estimatedas8.522e-05#AnAR(3)forthedifferenceddataisconfirmed#OurpreviousanalysisisjustifiedDiscussion:ThecommandarimaonR.Dealingwiththeconstantterm.Ifthereisanydifferencing,noconstantisused.Thesubcommandinclude.mean=Tinthearimacommand.Fixingsomeparameters.Usesubcommandfixedinarima.Useunemploymentrateseriesasanexample.RDemonstration:Handlingoutliersrl=ml