时间序列homework03-solutions

SOLUTIONSIE409:TimeSeriesAnalysisFall2011Homework318November2011(1)(B&D3.5)LetfYtgbetheARMAplusnoisetimeseriesdenedbyYt=Xt+Wt;wherefWtgWN(0;2w),fXtgistheARMA(p;q)processsatisfying(B)Xt=(B)Zt;fZtgWN(0;2z);andE(WsZt)=0forallsandt.(a)ShowthatfYtgisstationaryandnditsautocovarianceintermsof2wandtheACVFoffXtg.(b)ShowthattheprocessUt:=(B)Ytisr-correlated,wherer=max(p;q)andhence,byProposition2.1.1,isanMA(r)process.ConcludethatfYtgisanARMA(p;r)process.Answer:(a)ThemeanfunctionforfYtgisthesameasthemeanfunctionforthestationaryprocessfXtg.ThevarianceofYtisY(t;t)=E(Yt)(Yt)=E((Xt)+Wt)((Xt)+Wt)=X(t;t)+2w;andsimilarly,theautocovarianceforfYtgatlagh0isY(t;t+h)=E(Yt)(Yt+h)=E((Xt)+Wt)((Xt+h)+Wt+h)=X(t;t+h):Thesequantitiesareindependentoft,sofYtgisstationary.(b)TheprocessfUtgsatisesUt=(B)Yt=(B)(Xt+Wt)=(B)Zt+(B)Wt=Zt+1Zt1+


+qZtq+Wt1Wt1


pWtp;fromwhichitiseasytoseethatitisr-correlated.Thus,thereexistsfVtgWN(0;2v)andapolynomiale(z)ofdegreersuchthat(B)Yt=e(B)Vt;meaningthatfYtgisanARMA(p;r)process.(2)(B&D3.9)(a)Calculatetheautocovariancefunction(
)ofthestationarytimeseriesYt=+Zt+1Zt1+12Zt12;fZtgWN(0;2):1(b)UsetheprogramITSMtocomputethesamplemeanandsampleautocovariances^(h),0h20,offrr12Xtg,wherefXt;t=1;:::;72gistheaccidentaldeathsseriesDEATHS.TSMofExample1.1.3.(c)Byequating^(1),^(11),and^(12)frompart(b)to(1),(11),and(12),respectively,frompart(a),ndamodeloftheformdenedin(a)torepresentfrr12Xtg.Answer:(a)ThemeanfunctionforfYtgisE(Yt)=E(+Zt+1Zt1+12Zt12)=;sotheautocovarianceatlaghisE(YtYt+h)=E(Zt+1Zt1+12Zt12)(Zt+h+1Zt+h1+12Zt+h12)=8:2(1+21+212)ifh=021ifjhj=12112ifjhj=11212ifjhj=120otherwise.(b)ThesampleautocorrelationsarefoundbyITSMtobe^(1)^(0)=0:3588;^(11)^(0)=0:1952;and^(12)^(0)=0:3332:(c)Amodeloftheformdenedin(a)has^=28:831;^1=^(11)^(12)=0:19520:3332=0:5858;^12=^(11)^(1)=0:19520:3588=0:5440;and^2=^(1)^1=92740:(3)(B&D5.3)ConsidertheAR(2)processfXtgsatisfyingXtXt12Xt2=Zt;fZtgWN(0;2):(a)Forwhatvaluesofisthisacausalprocess?(b)ThefollowingsamplemomentswerecomputedafterobservingX1;:::;X200:^(0)=6:06;^(1)=0:687:Findestimatesofand2bysolvingtheYule-Walkerequations.(Ifyoundmorethanonesolution,choosetheonethatiscausal.)Answer:(a)Theautoregressivepolynomialforthisprocessisgivenby(z)=1z2z2;whichhasrootsp24(2)22=1p52:Thus,onecanverifythattheprocessiscausalifandonlyifjjp5120:618:2(b)TheYule-Walkerequationsforthisprocessare^(0)^(1)^(1)^(0)^^2=^(1)^(2)alongwith^2=^(0)^^(1)^2^(2):Therstequationimplies^(0)^+^(1)^2=^(1))0:687^0:687^2=0;fromwhichwend^2f0:509;1:965g:Wepreferthecausalsolution,sobypart(a)wechoose^=0:509.ThesecondYule-Walkerequationthenstates^(2)=^(1)^+^2)^(2)=0:687(0:509)+(0:509)2=0:609;andsothelastYule-Walkerequationyields^2=^(0)(1^^(1)^2^(2))=2:985:(4)(B&D5.4)Twohundredobservationsofatimeseries,X1;:::;X200,gavethefollowingsamplestatistics:samplemean:x200=3:82;samplevariance:^(0)=1:15;sampleACF:^(1)=0:427;^(2)=0:475;^(3)=0:169:(a)Basedonthesesamplestatistics,isitreasonabletosupposethatfXtgisWhiteNoise?(b)AssumingthatfXtgcanbemodeledastheAR(2)processXt1(Xt1)2(Xt2)=Zt;wherefZtgIID(0;2),ndestimatesof,1,2,and2.(c)Wouldyouconcludethat=0?(d)Construct95%condenceintervalsfor1and2.(e)AssumingthatthedataweregeneratedfromanAR(2)model,deriveestimatesofthePACFforalllagsh1.Answer:(a)UnderthehypothesisthatthedataareindependentWhiteNoise,forlargenthesampleautocor-relations^(h)areIIDN(0;1=n)randomvariables.Thus,forn=200,wehavethecontrollevels1:96=pn0:1386.Allof^(1),^(2),and^(3)areoutsidetheselevels,sowerejecttheWhiteNoisehypothesis.(b)Weestimatethemeanby^=x=3:82:TheremainingestimatorscanbefoundbysolvingtheYule-Walkerequations^(0)^(1)^(1)^(0)^1^2=^(1)^(2))10:4270:4271^1^2=0:4270:4753alongwith^2=^(0)(1^1^(1)^2^(2))=1:15(1^1(0:427)^2(0:475));fromwhichweobtain^1=0:274;^2=0:358;and^2=0:820:(c)Forlargen,wehavetheapproximatedistributionXnN0@0;1nXjhj1(h)1A:WecanapproximatethevariancewiththegivensampleautocovariancestocomputeXjhj1(h)Xjhj3^(h)=3:61:(Note:Amuchbetterapproximationcanbefoundbyusingthespectraldensityofthettedmodelinpart(b).Theapproximationgivenhereisagrossunderestimate.)Since3:821:96r3:612000:26;werejectthehypothesisthat=0.(d)ThelargesampledistributionoftheYule-WalkerEstimatorsforanAR(p)processare^N0;2n1p:Plugginginestimatesfor2and1p,wend^N0;^2n^1p=N0;^2n^(0)^(0)^(1)^(1)^(0)1!=N0;0:8202001:1510:4270:42711!=N0;0:00440:00190:00190:0044:Thus,95%condenceboundsfortheestimatesare1=0:2741:96p0:0044=[0:144;0:404]and2=0:3581:96p0:0044=[0:228;0:488]:(e)RecallthatingeneralthesamplePACFisgivenby^(0)=1and^(h)=^hh;h1;where^hhisthelastcomponentof^h=^R1h^h.Thus,forh=1wend^(0)^1=^(1))^(1)=^(1)=0:427;4andforh=2wend(usingtheresultinpart(b))^(0)^(1)^(1)^(0)^1^2=^(1)^(2))^(2)=^2=0:358:Finally,sinceweassumethatthedataaregeneratedfr