时间序列homework02-solutions

SOLUTIONSIE409:TimeSeriesAnalysisFall2011Homework24October2011(1)(B&D1.5)LetfXtgbethemoving-averageprocessoforder2givenbyXt=Zt+Zt2;wherefZtgisWN(0;1).(a)Findtheautocovarianceandautocorrelationfunctionsforthisprocesswhen=0:8.(b)Computethevarianceofthesamplemean(X1+X2+X3+X4)=4when=0:8.(c)Repeat(b)when=0:8andcompareyouranswerwiththeresultobtainedin(b).Answer:(a)TheautocorrelationatlaghisE(XtXt+h)=E((Zt+Zt2)(Zt+h+Zt+h2))=E(ZtZt+h)+(E(ZtZt+h2)+E(Zt2Zt+h))+2E(Zt+hZt+h2)Thus,wendh=8:1+2ifh=0ifh=20otherwise.andh=8:1ifh=01+2ifh=20otherwise.For=0:8,thisimpliesh=8:11625ifh=045ifh=20otherwise.andh=8:1ifh=02041ifh=20otherwise.(b)ThevarianceofthesamplemeanisingeneralVar(X4)=116Var(X1+X2+X3+X4)=1164Xi=14Xj=1Cov(XiXj)=116(4(1+2)+4);sofor=0:8thevarianceis0.61.(c)Usingtheresultinpart(b),thevarianceofthesamplemeanwhen=0:8is0.21.When=0:8,thenegativecorrelationatlagh=2meansthat,forexample,apositivedeviationfromthemeanattimetwilltendtobemetbyanegativedeviationfromthemeanattimet+2.Thiswillreducethevarianceofthesamplemeanastheaverageofconsecutivevalueswillmorelikelybeclosetothepopulationmeanthanifeveryothervaluewaspositivelycorrelated.(2)(B&D1.6)LetfXtgbetheAR(1)processdenedinExample1.4.5.(a)Computethevarianceofthesamplemean(X1+X2+X3+X4)=4when=0:9and2=1.(b)Repeat(a)when=0:9andcompareyouranswerwiththeresultobtainedin(a).1Answer:(a)ThevarianceofthesamplemeanisingeneralVar(X4)=116Var(X1+X2+X3+X4)=1164Xi=14Xj=1Cov(XiXj)=1164212+6212+42212+22312:Thisisbecausethereare4individualvalues,3pairsofvaluesthatareonetimeunitapart,2pairsofvaluesthataretwotimeunitsapart,and1pairofvaluesthatarethreetimeunitsapart.When=0:9and2=1,thisvarianceis4.64.(b)Usingtheresultinpart(a),thevarianceofthesamplemeanwhen=0:9and2=1is0.13.Thenegativecorrelationatoddlagsmeansthatthedatawilltendtoalternatebetweenpositiveandnegativevalues.Sumsofcontiguousvalueswillthustendtobeclosertozerothanifconsecutivedatavalueswerepositivelycorrelated.(3)(B&D1.9)Letfx1;:::;xngbeobservedvaluesofatimeseriesattimes1;:::;n,andlet^(h)bethesampleACFatlaghasinDenition1.4.4.(a)Ifxt=a+bt,whereaandbareconstantsandb6=0,showthatforeachxedh1,^(h)!1asn!1:(b)Ifxt=ccos(!t),wherecand!areconstants(c6=0and!2(;]),showthatforeachxedh,^(h)!cos(!h)asn!1:Answer:(a)Thesampleautocorrelationatlagh1isgivenby^(h)=1nPnht=1(xt+h1nPnj=1xj)(xt1nPnj=1xj)1nPnt=1(xt1nPnj=1xj)2:Pluggingxt=a+btintothedenominatoryields^(0)=1nnXt=10@a+bt1nnXj=1(a+bj)1A2=1nnXt=1btb(n+1)22=1nnXt=1b2t2b2(n+1)t+b2(n+1)24=b2n(n+1)(2n+1)6b2(n+1)22+b2(n+1)24:Notethatthisisapolynomialofnofdegree3withleadingcoecient(i.e.,thecoecientofn3)2equaltob2=3.Similarly,pluggingxt=a+btintothenumeratorof^(h)yields^(h)=1nnhXt=10@a+b(t+h)1nnXj=1(a+bj)1A0@a+bt1nnXj=1(a+bj)1A=1nnhXt=1b(t+h)b(n+1)2btb(n+1)2=1nnhXt=1b2t(t+h)b2(n+1)(2t+h)2+b2(n+1)24=b2(nh)(nh+1)(2(nh)+1)6n+b2h(nh)(nh+1)2n2b2(n+1)(nh)(nh+1)4nb2(n+1)h(nh)2n+b2(n+1)2(nh)4n:Again,thisisapolynomialofnofdegree3withleadingcoecientequaltob2=3.Thus,foranyxedh,limn!1^(h)=limn!1b23n3+


b23n3+


=b23b23=1:(b)RecallthatnXt=0cos(!t)=cos(12!n)sin(12!(n+1))sin(12!)andnXt=0cos2(!t)=3+2n+csc(!)sin(!(1+2n))4:Plugginginxt=ccos(!t),wend^(0)=1nnXt=10@ccos(!t)1nnXj=1ccos(!j)1A2=1nnXt=1ccos(!t)cncos(12!n)sin(12!(n+1))sin(12!)12=c2n3+2n+csc(!)sin(!(1+2n))4c2n2cos(12!n)sin(12!(n+1))sin(12!)12:Noticethat^(0)!c22asn!1.Similarly,plugginginxt=ccos(!t),wend^(h)=1nnhXt=10@ccos(!(t+h))1nnXj=1ccos(!j)1A0@ccos(!t)1nnXj=1ccos(!j)1A:Here,itisprudenttorealizethatthelimitof^(h)willbedictatedonlybythelimitoftheleading3term;i.e.,limn!1^(h)=limn!11nnhXt=1c2cos(!(t+h))cos(!t)=limn!1c22nnhXt=1c2(cos(!(2t+h))+cos(!h))=limn!1c2(nh)cos(!h)2n=c2cos(!h)2:Combiningthisresultwiththelimitof^(0),wendthat^(h)!cos(!h)asn!1.(4)(B&D1.18)(UsingITSMtoanalyzethedeathsdata.)OpentheleDEATHS.TSM,selectTransformClassical,checktheboxmarkedSeasonalFit,andenter12fortheperiod.MakesurethattheboxlabeledPolynomialFitisnotchecked,andclick,OK.Youwillthenseethegraph(Figure1.24)ofthede-seasonalizeddata.Thisgraphsuggeststhepresenceofanadditionalquadratictrendfunction.Totsuchatrendtothedeseasonalizeddata,selectTransformUndoClassicaltoretrievetheoriginaldata.ThenselectTransformClassicalandchecktheboxesmarkedSeasonalFitandPolynomialTrend,entering12fortheperiodandQuadraticforthetrend.ClickOKandyouwillobtainthetrendfunction^mt=995271:82t+0:8260t2;1t72:AtthispointthedatastoredinITSMconsistsoftheestimatednoise^Yt=xt^mt^st;t=1;:::;72;obtainedbysubtractingtheestimatedseasonalandtrendcomponentsfromtheoriginaldata.ThesampleautocorrelationfunctioncanbeplottedbyclickingonthesecondyellowbuttonatthetopoftheITSMwindow.FurthertestsfordependencecanbecarriedoutbyselectingtheoptionsStatisticsResidualAnalysisTestsofRandomness.Itisclearfromthesethatthereissub-stantialdependenceintheseriesfYtg.Toforecastthedatawithoutallowingforthisdependence,selecttheoptionForecastingARMA.Specify24forthenumberofvaluestobeforecast,andthepr