lecture13HW123

Homework1Homework2Homework3TimeSeriesAnalysisLecture13:HomeworkReviewProfessorFrankE.CurtisLehighUniversityFall2011TimeSeriesAnalysisLecture13:HomeworkReview(1of97)Homework1Homework2Homework3OutlineHomework1Homework2Homework3TimeSeriesAnalysisLecture13:HomeworkReview(2of97)Homework1Homework2Homework3OutlineHomework1Homework2Homework3TimeSeriesAnalysisLecture13:HomeworkReview(3of97)Homework1Homework2Homework3Problem1(B&D1.4)LetfZtgbeasequenceofindependentnormalrandomvariables,eachwithmean0andvariance2,andleta,b,andcbeconstants.Which,ifany,ofthefollowingprocessesarestationary?Foreachstationaryprocessspecifythemeanandautocovariancefunction.(a)Xt=a+bZt+cZt2(b)Xt=Z1cos(ct)+Z2sin(ct)(c)Xt=Ztcos(ct)+Zt1sin(ct)(d)Xt=a+bZ0(e)Xt=Z0cos(ct)(f)Xt=ZtZt1TimeSeriesAnalysisLecture13:HomeworkReview(4of97)Homework1Homework2Homework3ThingstolearnICalculateautocovarianceatlaghbycomputingE(Xt)(Xt+h):IProcessisstationaryifandonlyifthemeanfunctionandACVFareindependentoft.IBecarefulaboutwhatappearsstationary!Z1cos(ct))notstationary.Z1cos(ct)+Z2sin(ct))stationary.Ztcos(ct)+Zt1sin(ct))notstationary.TimeSeriesAnalysisLecture13:HomeworkReview(5of97)Homework1Homework2Homework3Problem2(B&D1.7)IffXtgandfYtgareuncorrelatedstationarysequences,i.e.,ifXrandYsareuncorrelatedforeveryrands,showthatfXt+YtgisstationarywithautocovariancefunctionequaltothesumoftheautocovariancefunctionsoffXtgandfYtg.TimeSeriesAnalysisLecture13:HomeworkReview(6of97)Homework1Homework2Homework3Problem3(B&D1.10)Ifmt=Ppk=0cktk,t=0;1;:::,showthatrmtisapolynomialofdegreep1intandhencethatrp+1mt=0.TimeSeriesAnalysisLecture13:HomeworkReview(7of97)Homework1Homework2Homework3ThingstolearnIDierencingatlag1atotalofp+1timeswillannihilatepolynomialsofdegreeuptop.IThatis,ifrp+1Xt=Ytisstationary,thenrp+1Xt+pXk=0cktk!=Yt:(Evenifthecoecientsckarerandomvariablesuncorrelatedw/Xt.)IWecaneliminatepolynomialswithoutestimatingcoecients!INoticealsothatdierencingcanannihilatepolynomialswithdierentcoecients,eveniftheyappearinthesametimeseries.TimeSeriesAnalysisLecture13:HomeworkReview(8of97)Homework1Homework2Homework3SinglequadratictrendTimeSeriesAnalysisLecture13:HomeworkReview(9of97)Homework1Homework2Homework3Singlequadratictrend,dierencedtwiceTimeSeriesAnalysisLecture13:HomeworkReview(10of97)Homework1Homework2Homework3TwoquadratictrendsTimeSeriesAnalysisLecture13:HomeworkReview(11of97)Homework1Homework2Homework3Twoquadratictrends,dierencedtwiceTimeSeriesAnalysisLecture13:HomeworkReview(12of97)Homework1Homework2Homework3Problem4(B&D1.11)Considerthesimplemoving-averagelterwithweightsaj=(2q+1)1,qjq.(a)Ifmt=c0+c1t,showthatPqj=qajmtj=mt.(b)IfZt,t=0;1;2;:::,areindependentrandomvariableswithmean0andvariance2,showthatthemovingaverageAt=Pqj=qajZtjis\smallforlargeqinthesensethatE(At)=0andVar(At)=2=(2q+1).TimeSeriesAnalysisLecture13:HomeworkReview(13of97)Homework1Homework2Homework3ThingstolearnIThistypeoflterhelpstoidentifylineartrend,sinceapplyingthelterleavesthetrendcomponentuntouched!IWhatthelterwilldo,however,isremovenoise.Thisisthepointoftheresultinpart(b).ISupposewehaveXt=mt+Zt;fZtgIID(0;2):LetthelteredsequencebeYt=qXj=qajXtj:Then,E(Yt)=E(Xt)=mtandVar(Yt)=Var(Xt)=22q+1:TimeSeriesAnalysisLecture13:HomeworkReview(14of97)Homework1Homework2Homework3LineartrendplusIIDnoiseTimeSeriesAnalysisLecture13:HomeworkReview(15of97)Homework1Homework2Homework3LineartrendplusIIDnoise,lterw/q=1TimeSeriesAnalysisLecture13:HomeworkReview(16of97)Homework1Homework2Homework3LineartrendplusIIDnoise,lterw/q=9TimeSeriesAnalysisLecture13:HomeworkReview(17of97)Homework1Homework2Homework3Problem5(B&D1.14)Showthatthelterwithcoecients[a2;a1;a0;a1;a2]=19[1;4;3;4;1]passesthird-degreepolynomialsandeliminatesseasonalcomponentswithperiod3.TimeSeriesAnalysisLecture13:HomeworkReview(18of97)Homework1Homework2Homework3ThingstolearnIFilterscanbeatoolforestimationaswellaselimination!ISupposewehaveXt=mt+st+Ztwheremtisathirddegreepolynomialandstisaseasonalcomponentwithperiod3.Then,thelteredsequenceisYt=mt+Wt;whereWtisadierentnoisesequence.(ItmaybethatthelterhasreducedthevarianceofWt,inwhichcasewemayhavealreadyapproximatedmt.Morelikely,however,anotherlterisnecessarytoidentifymtinYt.)TimeSeriesAnalysisLecture13:HomeworkReview(19of97)Homework1Homework2Homework3Problem6(B&D1.16)(UsingITSMtosmooththestrikesdata.)Double-clickontheITSMicon,selectFileProjectOpenUnivariate,clickOK,andopentheleSTRIKES.TSM.Thegraphofthedatawillthenappearonyourscreen.TosmooththedataselectSmoothMovingAve,SmoothExponential,orSmoothFFT.TryusingeachofthesetoreproducetheresultsshowninFigures1.18,1.21,and1.22.TimeSeriesAnalysisLecture13:HomeworkReview(20of97)Homework1Homework2Homework3Problem7(B&D1.19)Useatexteditor,e.g.,WORDPADorNOTEPAD,toconstructandsaveatextlenamedTEST.TSM,whichconsistsofasinglecolumnof30numbers,fx1;:::;x30g,denedbyx1;:::;x10:486;474;434;441;435;401;414;414;386;405;x11;:::;x20:411;389;414;426;410;441;459;449;486;510;x21;:::;x30:506;549;579;581;630;666;674;729;771;785:Thisseriesisinfactthesumofa