金融时间序列分析l

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1、LectureNotesofBus41202(Spring2007)AnalysisofFinancialTimeSeriesRueyS.TsaySimpleARmodels:(Regressionwithlaggedvariables.)AR(1)model:1.Form:rt=φ0+φ1rt−1+at,whereφ0andφ1arerealnumbers,whicharereferredtoas“parameters”(tobeestimatedfromthedatainanapplication).Forexa

2、mple,rt=0.005+0.2rt−1+at2.Stationarity:necessaryandsufficientcondition

3、φ1

4、<1.Why?φ03.Mean:E(rt)=1−φ1σ24.Variance:Var(r)=a.t1−φ215.Autocorrelations:ρ=φ,ρ=φ2,etc.Ingeneral,ρ=φk1121k1andACFρkdecaysexponentiallyaskincreases,6.Forecast(minimumsquarederror):(a)1-stepah

5、eadforecastattimen,theforecastorigin:rˆn(1)=φ0+φ1rn(b)1-stepaheadforecasterror:en(1)=rn+1−rˆn(1)=an+11Thus,an+1istheun-predictablepartofrn+1.Itistheshockattimen+1!(c)Varianceof1-stepaheadforecasterror:2Var[en(1)]=Var(an+1)=σa.(d)2-stepaheadforecast:rˆn(2)=φ0+φ1

6、rˆn(1)(e)2-stepaheadforecasterror:en(2)=rn+2−rˆn(2)=an+2+φ1an+1(f)Varianceof2-stepaheadforecasterror:22Var[en(2)]=(1+φ1)σawhichisgreaterthanorequaltoVar[en(1)],implyingthatuncertaintyinforecastsincreasesasthenumberofstepsin-creases.(g)Behaviorofmulti-stepaheadf

7、orecasts.7.Acompactform:(1−φ1B)rt=φ0+at.AR(2)model:1.Form:rt=φ0+φ1rt−1+φ2rt−2+at,or2(1−φ1B−φ2B)rt=φ0+at.2.Stationaritycondition:(factorofpolynomial)23.Characteristicequation:(1−φx−φx2)=012φ04.Mean:E(rt)=1−φ1−φ2φ15.ACF:ρ0=1,ρ1=1−φ2,ρ`=φ1ρ`−1+φ2ρ`−1,`≥2.6.Stochas

8、ticbusinesscycle:ifφ2+4φ<0,thenrshowschar-12tacteristicsofbusinesscycleswithaveragelength2πk=√,cos−1[φ1/(2−φ2)]wherethecosineinverseisstatedinradian.Ifwedenotethe√solutionsofthepolynomialasa±bi,wherei=−1,thenwehaveφ=2aandφ=−(a2+b2)sothat122πk=√.cos−1(a/a2+b2)√I

9、nRorS-Plus,onecanobtaina2+b2usingthecommandMod.7.Forecasts:SimilartoAR(1)modelsBuildinganARmodel•Orderspecification1.PartialACF:(naive,buteffective)–Useconsecutivefittings–SeeText(p.40)fordetails3–Keyfeature:PACFcutsoffatlagpforanAR(p)model.–Illustration2.Akaikeinf

10、ormationcriterion2`2AIC(`)=ln(σ˜`)+,TforanAR(`)model,where˜σ2istheMLEofresidualvari-`ance.FindtheARorderwithminimumAICfor`∈[0,···,P].3.BICcriterion:`ln(T)2BIC=ln(σ˜`)+.T•Nee