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SOLUTIONSIE409:TimeSeriesAnalysisFall2011MidtermExamOctober21,2011(1)(10points)Considerthelinear lterde nedbythecoecients(a 2;a 1;a0;a1;a2;a3)=120(2; 5;20;10; 10;3):Whatisthehighestdegreepolynomialthatthis lterwillpasswithoutdistortion?Answer:Alinear lterfajgpassesarbitrarypolynomialsofdegreekwithoutdistortionifandonlyifXjaj=1andXjjraj=0forr=1;:::;k:Forthegivenlinear lter,we nda 2+a 1+a0+a1+a2+a3=220 520+2020+1020 1020+320=1; 2a 2 a 1+0a0+a1+2a2+3a3= 420+520 020+1020 2020+920=0;4a 2+a 1+0a0+a1+4a2+9a3=820 520+020+1020 4020+2720=0; 8a 2 a 1+0a0+a1+8a2+27a3= 1620+520 020+1020 8020+8120=0;16a 2+a 1+0a0+a1+16a2+81a3=3220 520+020+1020 16020+24320=6;Thus,this lterpassesuptodegree3polynomialswithoutdistortion.(2)(10points)Supposeyouaregivenatimeserieswithdatafx1;:::;xk 1;xk;xk+1;xk+2;:::;xng:Atwo-sidedmovingaveragewithparameterq=2yieldsthesmootheddatafr1;:::;rk 1;rk;rk+1;rk+2;:::;rng:Nowsupposeithasbeendeterminedthattheoriginaldatawasallcorrectexcepttherewereerrorswiththekthand(k+1)thdatapoints.Inparticular,therealdatashouldactuallybefx1;:::;xk 1;xk+1;xk+1 1;xk+2;:::;xng:1Atwo-sidedmovingaveragewithparameterq=2wouldthenyieldthesmootheddatafs1;:::;sk 1;sk;sk+1;sk+2;:::;sng:Whatisthee ectonthesmootheddatawiththischangeinthedata?Thatis,whichvalues(ifany)aredi erentinthesequencesfrtgandfstgandbyhowmuchdotheydi er?(Note:Youmayassumethatkismuchbiggerthan1andmuchsmallerthann.)Answer:Smoothingtheoriginaltimeseriesyieldsrk 2=15(xk 4+xk 3+xk 2+xk 1+xk)rk 1=15(xk 3+xk 2+xk 1+xk+xk+1)rk=15(xk 2+xk 1+xk+xk+1+xk+2)rk+1=15(xk 1+xk+xk+1+xk+2+xk+3)rk+2=15(xk+xk+1+xk+2+xk+3+xk+4)rk+3=15(xk+1+xk+2+xk+3+xk+4+xk+5)However,smoothingthecorrectedtimeseriesyieldssk 2=15(xk 4+xk 3+xk 2+xk 1+xk+1)sk 1=15(xk 3+xk 2+xk 1+xk+xk+1)sk=15(xk 2+xk 1+xk+xk+1+xk+2)sk+1=15(xk 1+xk+xk+1+xk+2+xk+3)sk+2=15(xk+xk+1+xk+2+xk+3+xk+4)sk+3=15(xk+1+xk+2+xk+3+xk+4+xk+5 1)Thus,rt=stforallt2f1;:::;ngnfk 2;k+3gandsk 2=rk 2+15sk+3=rk+3 15:(3)(10points)LetfYtgbeastationaryprocesswithmeanzero,letmt=a+btbealineartrendcomponentwithconstantsaandb,andletstbeaseasonalcomponentwithperiod6.(a)IfXt=a+bt+st+Yt,showthatrr6Xt=(1 L)(1 L6)XtisstationaryandexpressitsautocovariancefunctionintermsofthatoffYtg.(b)IfXt=(a+bt)st+Yt,showthatr26Xt=(1 L6)2XtisstationaryandexpressitsautocovariancefunctionintermsofthatoffYtg.Answer:Thedi erencedprocessinpart(a)satis esWt=(1 L)(1 L6)Xt=Xt Xt 1 Xt 6+Xt 7=(a+bt+st+Yt) (a+b(t 1)+st 1+Yt 1) (a+b(t 6)+st 6+Yt 6)+(a+b(t 7)+st 7+Yt 7)=Yt Yt 1 Yt 6+Yt 7:2TheprocessfWtgisstationarysinceitsmeanfunctionist=E(Wt)=E(Yt Yt 1 Yt 6+Yt 7)=0anditsautocovariancefunction,intermsoftheautocovariancefunctionYhoffYtg,ist;t+h=E(WtWt+h)=E(Yt Yt 1 Yt 6+Yt 7)(Yt+h Yt+h 1 Yt+h 6+Yt+h 7)=4Yh 2Yh 1 2Yh+1+Yh 5+Yh+5 2Yh 6 2Yh+6+Yh 7+Yh+7:whicharebothindependentoft.Similarly,thedi erencedprocessinpart(b)satis esWt=(1 L6)2Xt=Xt 2Xt 6+Xt 12=((a+bt)st+Yt) 2((a+b(t 6))st 6+Yt 6)+((a+b(t 12))st 12+Yt 12)=Yt 2Yt 6+Yt 12:TheprocessfWtgisstationarysinceitsmeanfunctionist=E(Wt)=E(Yt 2Yt 6+Yt 12)=0anditsautocovariancefunction,intermsoftheautocovariancefunctionYhoffYtg,ist;t+h=E(WtWt+h)=E(Yt 2Yt 6+Yt 12)(Yt+h 2Yt+h 6+Yt+h 12)=6Yh 4Yh 6 4Yh+6+Yh 12+Yh+12:whicharebothindependentoft.(4)(15points)Datawasgeneratedtosimulateeachofthetimeseries(a)through(e)belowandthesampleACFandPACFwerecomputedandareshowninplots(i)through(v).MatcheachtimeserieswiththeplotsofitssampleACFandPACF.AssumethatfZtgWN(0;2).ExplainindetailhowyouhavematchedeachtimeserieswiththeplotsofitssampleACFandPACF.(Note:Itisessentialthatyouexplainyouranswerhere.Donotsimplymatchtimeseriestoplotswithoutafullexplanationofhowyoudidit.)(a)Xt=Zt 0:8Zt 1(b)Xt=sin2t12+t60+Zt(c)Xt= 0:8Xt 1+Zt 0:8Zt 1(d)Xt=0:34t++Zt(e)Xt=0:8Xt 1+Zt(i)3(ii)(iii)4(iv)(v)5Answer:Timeseries(a)isaMA(1)processwith1= 0:8.ThetheoreticalACFofsuchaprocesshasanegativevalueatlag1and,forallotherlags,avalueofzero.ThetheoreticalPACFofsuchaprocesshasnegativevaluesatalllagsthatdecaytozeroashincreases.ThebestmatchforthistimeseriesisthesampleACF/PACFinplot(iii).Timeseries(b)isnonstationaryasithasalineartrendcomponentandaseasonalcomponent.Thus,thetheoreticalACFofsuchaseriesdoesnotdecaytozeroandevolvessinusoidally.ThebestmatchforthistimeseriesisthesampleACF/PACFinplot(iv).Timeseries(c)isanARMA(1,1)processwith 1= 0:8and1= 0:8.ThetheoreticalACFofsuchaprocesshasalargenegativevalueatlag1andtheACFatallsubsequentlagsoscillatesbetweenpositiveandnegativevaluesduetothefactthat 10.ThetheoreticalPACFalsohasalargenegativevalueatlag1andwilldecayslowlytozeroduetothefactthat10.ThebestmatchforthistimeseriesisthesampleACF/PACFinplot(v).Timeseries(d)isalineartrendplusnoisesequence.ThetheoreticalACFofsuchaprocesshasallpositivevaluesthatdonotdecaytozero.ThebestmatchforthistimeseriesisthesampleACF/PACFinplot(i).Timeseries(e)isanAR(1)processwith 1=0:8.ThetheoreticalACFofsuchaprocesshasapositivevalueatlag1andvaluesatallsubsequentlagsthatare 1timesthepreviousvalue.ThetheoreticalPACFhasanonzerovalueatlag1andazerovalueatallsubsequentlags.Thebestmatchforthistimeseriesist