andmeanstheratioofleftandrightsidestendsto1issaidtohalongmemoryif0 shortmemoryif0andnegativememoryif beenbelonghthat nitenberofdi erencesproducesanARMAshortmemoryprocessbeingrootmodelscandi eren ID: 217135
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troductionExactandapproximateGaussianmaximumlikelihood("Whittle")estimatesofparametricsta-timeseriesmodelshabeenshownto rst-orderasymptoticpropertiesunderlongmemoryasearliershownundershortmemory(e.g.xandTaqqu,1986,Dahlhaus,1989,Solo,1989,GiraitisandSurgailis,1990,Hosoa,1996).Acoariancestationaryserieswithspectraldensity(sd))satisfying and""meanstheratioofleftandrightsidestendsto1,issaidtohalongmemoryif0 ,shortmemoryif=0andnegativememoryif beenbelonghthat nitenberofdi erencesproducesanARMAshortmemoryprocess,beingrootmodelscandi erencingisassumedtoproduceaserieswithspectrumsatisfying(1),with=0notassumed.,anonstationaryfractionalARIMAseriesissuchthat(1isastationaryandinertibleARMA,where isarealnberandisthelagoperator.withotherparametersinnonstationaryfractionalARIMAmodels.LingandLi(1997)extendedhisapproachtoallowforconditionalheteroscedasticit,whileBeran,BhansaliandOcer(1998)discussedmodelselectionintheautoregressivecase.ediscussBeran'sasymptoticjusti cationbelow,inviewofwhiceanalyzeanalternative,discrete-frequencydomainversionofWhittle. iscowithmeanandsdbeha=0wede nethepseudosd(psd)ofof ;decompositionofthe(in nite)varianceofthenonstationarytimeseries.assuggestedySolo(1992)andHurvichandRay(1995),thepsd)canbeinterpretedasthelimitoftheexpectedsampleperiodogram,asinthestationaryframewThispropertasusedinVstationarytimeseries(seeGeweandPorter-Hudak,1983)haethesamedesirableasymptoticpropertiesfor asshownbyRobinson(1995a,b)for withtaperingneededforlargeenoughortoeliminatepolynomialtrends. 0100200300400500 0.51.01.5 Figure1:Arizonatree-ringwidths(548-1047).possiblyationsofaseriesofannualtree-ringwidthsinArizonafrom548A.D.onardsobtainedbD.A.Graybillin1984andmaintainedbyR.Hyndmanatwww-personal.buseco.monash.edu.au/~behadependenlocalaboutpoinsemiparametricestimateforbandwidths=25and50withtheoriginalanddi erencedseries,adjustingthevalueofinthelattercase.able1:ryoftree-ringwidths.rametricestimates=25=50 t.586(.100)Xt.599(.100) Thoughallestimatesgivcon denceinalsbasedontheasymptoticnormaldistributionandbothypothesesexponential(FEXP(modelsBloom eldexponenmodelingofbehaviour(seeSection4).Theydonotullhypothesiscomparingone-sidedstatisticstoastandardnormalforanyloworderFEXP()model.2thewingt-ratiosindicatethepresenceofsomehigherlevelofmemory:modeling,proceduremodelmodelGaussianlikelihood.ifwseriesisnonstationaryandtak able2:estsoffractionalhypothesisfortree-ringwidths q-5.50-3.97 istheorderoftheFEXP()modelmaintainedunderthenwithsimilar tsfortheAR(3)andAR(2)parameters.paper andasymptoticstandarderrorsforthefrequencydomain(possiblytapered)Whittleestimatesofnonstationaryserieswithnopriorassumptionsonorthestationaryincrementsofthetree-ringwidthsthebest tofWhittleestimatesisgivenbyanAR(2)modelwithal,thesamecriterion ndsthat556withalmostthesameautoregressiveparameters.Thisindicatestheclaimedrobustnesstononstationarityoffrequencydomainestimation,whictaperingbehabelltaperbelow)wcon rmthesmalldegreeFEXP(2)models,obtainingsimilarvaluesof,inagreementwiththescoretests.able3:ryoftree-ringwidths.rametricestimates NotaperCostaperNotaperCostaper .556(.057).536(.082).617(.071).613(.099).563(.057).501(.084).609(.071).574(.099) TherestofthepaperisorganizedasfolloThefollowingsectionpresentsthemodelanddiscussesthepropertiesofthetaperedFouriertransformofnonstationarytimeseries.theirasymptoticproperties.samplepropertiesoftheestimatesareexaminedinaMonteCarloexperimentinSection4,whileSection5appliesthemethodsdiscussedtotoempiricalseries.ThetechnicalassumptionsandresultsaresummarizedinAppendixA,withproofsinAppendixB. eassumethatthepsdofsatis es(2)andbelongstotheparametricclassde nedb ;:::;parametervandscalareassumethat independenbestprocess;extracomplexity(seeRobinson,1978,HosoaandThi,1982).er(3)coersstandardparameterizationsoffractionalARIMAandFEXPmodels.De nethetapereddiscreteFouriertransform(DFT)of;:::;n,andj=nteger,andatapersequence):=taperedperiodogrambothtralpartofthedata.processestaperedperiodogrampoinindependendependenprocessestoorespectAppendixdependenceoftheperiodogramordinatesarea ectedbysharppeaksinthepsd.aperingwsuggestedbey(1967)tocontrolleakageproblemsinspectralestimationwhennonstation-yissuspected,aswascedindi erentframeworksbyZhurbenko(1979),Robinson(1986)andDahlhaus(1988)amongothers.urbenk;:::;assumedtobeinteger,proportionaltothecoecien)givenb+1)=(1+ 1 z!p:4 taperssmoothAlekseev,1996),andwhen=1givethenontaperedDFTw1,when=3theyaresimilartothefullcosinebell 21 whilewhen=4theyareveryclosetoParzen'sw n 2 2t n n 3;NN;2n1 2t n TheasymptoticpropertiesofthetapersequencesdependcruciallyonthekhistheDirichletkernelwhen=1,andweusethemtocharacterizeanextendedclassoftapers.ythatanon-negativsymmetric(around andnormalized(sup=1)sequenceofdatatapersisofifthefollowingtoconditionsaresatis ed: np 1sin[p] =2]p;(5)wherea()isacomplexfunction,whosemodulusisboundedandboundedayfromzero,1derives,allboundedforor ;orsome1,limbellpropertiestapersAppendixtaperingtaperedperiodogramwithataperoforderwillbedenotedasAssuggestedbyareferee,summationbypartsyields,foradi erentiabletaperwhicattheboundaries,withderiv 1 eiwt(Xtt (mod2explainingwhtlysmoothtapertapertlyhigh)candealwitharbitrarylevelsofmemory,justifyingalsode nition(2).Infact,handRaandVecanobtainSolo's(1992)ersioncalculation)inthenonstationarycase, taperedperiodogramnonstationaryseriesatFourierfrequencies=0(mod),nottooclosetotheorigin,thoughbetAppendixurthermore,using(5)foradatataperoforder;:::;psotapersalsoremoepolynomialtrendsintheobservedsequenceaswhene.g.themeanifweconcentrateonthesamesetoffrequencies=0(modoestimateeuseapossiblytaperedversionofHannan's(1973)discretefrequency-domainWhittleobjectivefunction Xj(p)Ip(j) isasumop;:::;nassumingforsimplicitythatisanineomitzerofrequency,formean-correctionpurposesinthestationarycase,whiletheexclusionoffrequenciesbet;:::;isfor(polynomial)trendcorrectionandtoguaranboundednessofperiodogramexpectationundernonstationaritreplacedbyanintegral,correspondingtotheconuousWhittleobjectivefunction,butinancasethediscreteformiscomputationallymorecontandmakesmoredirectuseofthefastouriertransformandfunctionalformforTheomissionoffrequencieswhen1couldbeaoidedtoacegreatereciency,forexampleifitisknownthat =0.AsinHannan(1973)edonotrequireGaussianiteestimate=argminandestimate upperdependsonthetaperingapplied.InourstatementsofTheorems1and2werefertoassumptionslistedinAppendixA. 2cp�dowhen=0o1 2],b!poandb2!p2oasn6 taperedationsandan1,butonlyifuscoeringnonstationarybut"mean-revdata( 1)withoutdrift.=0weneedanincreaseddegreeoftaperingtoeliminatedeterministictrend taperingobtainaperiodogramwithboundedexpectationin(6)whenthepsddivInanycasemoretaperingisneededtoobtainasymptoticallynormallydistributedDependingonthede nitionof,intheproofwehaetoconsiderseparatelythecaseswhereitispossiblethat andthosewhere ,becauseofthenon-uniformbehaappearedGaussiansemiparametricestimationforstationaryandinertiblelongmemoryserieswith 2;1 Thisquestionalsoa ectsBeran's(1995)treatmentofnonstationaryfractionalARIMAmodels.consideredtimedomainapproumlikelihood(ML)tapereddata,whensta-iscorrectlyhisestimatespropertieselihoodbeingprocessesdi ersfromours:forthecaseofasimplefractionallydi erenced(0;d;model,heconsidersiswhitenoise,whereaswetak;:::; HeconsideredtheobjectivefunctionUnder(7)thisisandforconsistencyonehastoconsideruniformconyargumentswithrespecttothewholeparameterspacesofadmissible,andexistenceofanasymptoticglobalminimumatThisinesconsiderationoftheprocesses(1harestationaryfor�dd andnonstationaryotherwise.Infactitistheneighbourhoodof hcausesmostdicultybecause(1isatthestationary/nonstationaryborder.processes,tapers,TheTtencyseemstobecircularsincethe(1)errorintheexpansionfor)isonlyjusti edbourhoodpresupposesprooftapering able4:BiasofractionalARIMA(2,,0)Models=512=200 .6.91.11.4-.4.41.11.4 Notaper, dG-SEM-.037-.041-.036-.023-.078-.350-.206-.105-.267-.006.002.002-.025-.137-.395-.023W-2S.002-.003-.014-.033-.002.004.004-.004-.028-.030-.012 aper, dG-SEM-.068-.064-.059-.048-.059-.261-.253-.037-.026-.023-.018.017-.076-.071W-2S-.009-.013-.016-.007-.031-.033-.006.007.005.030-.010-.011 beloulationssupporthisinsighestimatesoffractionalmodelshaetheclassical ,asymptoticnormalityande-ciencyproperties.Indeed,itisconsistentwithRobinson's(1994b) ndingsthatscoretestsforaunitrootandmanotherstationaryandnonstationarynullhypotheses,whendirectedagainstdependsononlythenulldi erenceddata.Bycontrastunitroottestsagainstautoregressivealternativenonstandardasymptotics(seee.g.Solo,1984).enowdiscusstheasymptoticnormalityof 2o3 (ii)Assumption 2c, Theasymptoticvarianceformulaisthesameasforstationaryseries,with k(;o) anditmaybeshownthat nn 1Xj@ (j;b) isaconsistentestimateofisthetapervariancein ationfactor,=lim)with,whichtakesthevalues1.05000,1.00354and1.00086 able5:rddeviationofractionalARIMA(2,,0)Models=512=200 -.4.91.11.4.6.91.4 Notaper, dG-SEM(.079).088.097.102.086.087.124.178.174.172.081W-p(.046).049.053.056.119.065.093.141.135.132.191.048.056.051.217.049.090.126.089.089.237ML-2S(.046).049.054.055.120.045.088.129.097.099.196 aper, dG-SEM(.112).165.151.152.143.165.201.203.200.199.199W-p(.069).078.081.080.074.065.146.147.146.145.146.049.052.051.049.045.089.091.088.088.094ML-2S(.046).058.058.059.059.055.099.100.103.106.110 urbenkodatatapersrespectivimplyingmodestarianceof5%,.35%and.09%foreachofthedatatapers(apartfromtheextrafactordueto;:::;nbetheusualtaperingvariancecorrection(seee.g.Dahlhaus,1985)and=1byorthogonalitofthesineandcosinefunctions.Thesameresultholdsforthecosinebelltaper(4)when =0(or foran)areknown,whereinthiscaseitispossibletoincludeallfrequencies2,inifactually=1,obtaining Thisaccounforareductionof33%fromsettingas3,inolvingthenonlyasymptoticallyuncorrelatedperiodogramordinates(seeTheorem4inAppendixA).models,becausemodelingbehad;corresponding,forexample,toanARMAorBloom eld(1973)exponentialmodel(see(9)belotheasymptoticoftheisfreeyoftheobservedtimeseries(apartfromthee ectsoftaperingifused),andagainisconsistentwiththenatureofthescoretestsofRobinson(1994b).Initialdi erencingimproasymptoticeciencyonlyifaloerordertaperisused(withsmaller,sincethecontributionofisoflesssigni cance),butthismakesallestimatesmoresensitivetopeaksornonstationaritatotherfrequencies(seetheconclusionsofHauser(1999)forstationaryfractionalARIMAmodelsmethodsincludingtapered-Whittleycase,steeper)at=0,i.e.theorsethe able6:BiasofractionalARIMA(2,,0)Models=512=200 -.4.4.61.4-.41.4 Notaper, 1W-p-.004-.004-.005-.081.011.047-.003.001-.004.003.015.014-.007-.003-.001.006.015.004.009 2W-p.003.006.001.390.585.008.003.005-.004.002.002.007.007.000.002-.008.000-.002.002.002 aper, 1W-p.010.014.013.002.048.042.038.003.005.002.015.017.013-.005-.005-.004-.005.005.003 2W-p-.001-.006-.005-.005.009.000.001.000-.001-.001.027.007.007.000-.003-.004-.004.002.001 canbeexpectedfor nitesamples.performanceandnonstationarydata.egeneratedindependentsamplesoftoGaussiantimeseriesmodelsrespectivcomponeneparameters6and=4,andBloom eld's(1973)exponenmodelasproposedinRobinson(1994a,p.73),withparametershcorresponds1intheusualparameterization),=5and3,leadingtotheFEXP(2)model cos2modelsshape,peakbeha4toanalyzeseriesclosetonon-in4,tocomparewiththestationarycase,nonstationarybutstillmean-revertingseries,anddi erencestationary=0). able7:rddeviationofractionalARIMA(2,,0)Models=512=200 -.4.91.4.9 Notaper, 1W-p(.044).043.047.043.187.088.101.107.106.043.048.041.043.046.077.077.077(.044).043.048.043.042.044.079.080.081 2W-p(.036).036.039.037.188.065.075.081.081.036.038.036.036.031.059.059.059(.036).035.038.036.035.030.058.058.058 aper, 1W-p(.066).063.071.070.063.215.115.115.115.043.046.046.043.197.077.077.076(.044).045.049.049.046.190.083.083.084 2W-p(.054).054.050.050.054.061.082.082.082.036.031.031.036.128.059.059.059(.036).035.031.030.035.058.058.058.058 andthenberofexponentialmodels(9),modeltheappropriatevalueofortheshortmemorysimulationoftheexponentialmodelsweusedautocoariancesobtainedbumericalintegrationofthesd.Non-tapered(=1)andtapered(withZhurbenkotaperoforder=2)datawereconsidered.Alsothecosinebelltaper(4)wasusedbuttheresultsweresimilartothetaperwith=2andarenotreportedhere.Theestimatescomparedarethefollo1999b)withbandwidthnbers=30(20fortheFEXPmodel)and40foreachsampleWhittleestimatestaperingisapplied,butonlytforasymptoticallynormalfor iftheraisused.UsingtheWhittlememoryestimatesfromW-p,itispossibletotoaceapproximateshortmemorystationarityandthenusestandarduntapered able8:BiasofrBloom eldFEXP(2)Models=512=200 1.4 Notaper, dG-SEM-.072.169.258-.144-.480-.149-.009.093-.397-.623W-2S-.121-.380-.639 aper, dG-SEM-.130-.117-.205-.179-.074-.064-.208-.239W-2S-.054-.144-.176 methodspossiblytaperednonstationaryinputs.eproposetoalternativo-step(asymptoticallyequivprocedures,wherethesecondstep'sinputisinbothcasestheuntaperedW-2S:Two-StepWhittleestimates,wherethesecondstepisWhittle(stationary)estima-=1.wherethesecondstepisimplementedbytheS-PLUSfunctionHaslettandRaftery,1989).Allparametricestimatesthrough4.)wn)truemodel,sinceotherwisetheestimatesofarenotguaranteedconsistentforthesecondstep(evenifenoughtaperingwapplied),andtaperingisonlyusedforthe rststepestimates,sinceitishopedthatdi erencingesstationarityofthesecondstepinputs.usingthereportbiasandstandarderrortthetaperingappliedandassumingthattheto-stepestimateshaetheMLasymptoticmodelsables6and7fortheestimatesof able9:rddeviationofrBloom eldFEXP(2)Models=512=200 -.4.4.61.4-.4.91.4 Notaper, dG-SEM(.079).192.020.024.171.162.158.198W-p(.070).167.027.018.193.190.176.216.133.106.187.200.184.204.368 aper, dG-SEM(.119).141.150.158.292.290.278.285W-p(.102).113.131.149.267.266.251.277.147.177.105.263.203.216.193 Whennotaperingisapplied(=1)theestimationofbreaksdownifbutevthesevaluesofthememoryparametertheto-stepproceduresgiveconsistentestimatesbecausetheWhittleproceduretendstoreport1,sothedi erencedserieswiththeseinitialestimatesarestationaryThebiasinTable4decreasesforallestimatesinsamplesize,andthelargebandwidthcgoodtapereddeviationsaretlylargerexpected.ThisincrementinvyoftaperedWhittleleadstoanincreaseinthevarianceofthetstepMLestimates,butnotoftheto-stepWhittleestimates,sotimedomainestimationseemsmoresensitivetopreviousfractionaldi erencing.InTable6taperedWhittleestimationprovidesbetterresultsfortheshortmemoryARMAparametersthanforthememoryparameter,withveryclosebehaviourtothatofthetproceduresintermsofbias,whileinTable7thestandarddeviation,thoughlarger,isverywtaperedproducemethodstaperedproceduresyieldtestimates).sizetheasymptotictheorygivesagoodapproximationtothe nitesamplebehaviourofWhittlebeha able10:BiasofrBloom eldFEXP(2)Models=512=200 -.4.41.1.4.6 Notaper, 1W-p-.725-2.119-3.605-4.703-4.858-4.944-.752-1.074-1.365-2.364-1.813-3.001W-2S-.753-.823-.717-3.648-.909-1.127-1.356-1.589-1.235-2.813b2W-p.9051.6532.4682.9492.9893.0181.1501.5271.7572.3451.7081.793W-2S.961.924.9091.1071.5912.3871.3501.5721.7761.8681.6062.167 aper, 1W-p.132.113.099-3.580-1.031-1.155-1.302-1.691-1.479-1.435-2.024-2.4572.008b2W-p.040.032.029-1.142W-2S2.088.9571.0251.2311.5441.1782.3881.5511.5501.8442.1591.583 ExponenmodelsInTables8to11wereporttheresultsforexponentialmodelswiththesamevaluesofasusedbefore.Theconclusionsfor=200andforalluntaperedestimatesofarerathernegative,withlargebiases(Table8)andvy(Table9)relatingtotheasymptoticvalue,probablyduetoadicultdistinctionbeteentheshortmemoryandlongmemorycomponentsofthisparticularmodel.ertheless,taperedWhittleestimationfor=512producesforallreasonablebiasesandstandarddeviations,thesmallestacrossallmethodsandquiteclosetotheasymptoticones,whilebothtestimationsbreakdowninmanycases.ThesuperiorityoftaperedWhittle"W-p"estimatesforthememoryparameteroffractionalexponenBloom eldmodelstaperedproceduresmodelyparametervaluecombinations(Tables10and11).HereZhurbenktswith=2fortaperedestimatesappearsuperiorthanthecosinebellintermsofbiasformostvaluesofOursimulationsagreewithDahlhaus'(1988) ndingthattaperingisdesirableinestimatingshortmemoryparameterswhenthesdhaspeaksduetoARrootsclosetotheunitcircle,whicaresimilartothezerofrequencysingularityofthepsdoffractionallyintegratedprocesses.Processemperaturereadings(SeriesC)andChemicalProcessConcentrationreadings(SeriesA)fromxandJenkins(1976),whicharealsoamongtheseriestowhichRobinson(1994b)appliedhis able11:rddeviationofrBloom eldFEXP(2)Models=512=200 -.4.61.4.4.91.4 Notaper, W-p(.166)W-2S(.166)W-p(.113)W-2S(.113) aper, W-p(.249)W-2S(.166)W-p(.170)W-2S(.113) able12:CHEMICALSERIES-C.ARFIMA(;d; G-SEMW-pW-2S d.99281.0400.9788.9930(.100)(.091)-(.137)- b 1-.1157.8237-.8389.7973-.8263.8128-(.116)---(.143)- b2-.3171.0186-.0162.0189-.0150.0187 scoretestsagainstfractionalalternativeusethesameestimatesasinthesimulations(urbenko's(=2)andcosinetapers.taperswithnonstationaryserieswith=0,butonlytaperingoforder=2allowsserieswithlineardrift. ndingofabout=1inseriesC,thesevaluesobtainedbyconsideringonlyintegerdegreesofdi erencing.allproceduresinmodelfoundvaluesofdistinguishablefrombothtaperedabouthmayexplainwhythetaperedsemiparametricestimateslargerestimatesof(clearlyaboe1)thanthecorrespondingparametricmethods.Whittleandsemipara-tapering able13:CHEMICALSERIES-C.Robinson's(1994)TestsofNonstationa q3.075129.426026.212920.010511.7572q14.426112.67052.8689-2.8529q8.06807.42091.2513-.6660-2.0282-2.9065q4.81374.4391.3854-.9920-2.0581-2.7742 originaldata).modelsbestproducedaboutmodels,ealsousedRobinson's(1994b)scoretestinTable13(usingthesameBloom eldexponentialmodelstoexplainhighfrequencybehaviour),completinghisresultsforanextendedullvreportedwithstandardasymptoticdistribution.Thescoretestsalwysrejectthehypothesis=2,against2,andypothesis=1,wasimilarpatterntoFEXPWhittleestimates,whichcontrastswiththeARIMAmodellinginTable12,butagreeswiththesemiparametrictaperedestimates.able14:CHEMICALSERIES-C.Memoryestimates W-2SW-2S 1.75221.7510(.052)(.052)(.073)(.052) q1.52311.6183(.083)(.083)(.116)(.083) q1.37031.4125(.106)(.106)(.147)(.106) q1.21261.3286(.125)(.125)(.174)(.125) smaller,about0.45asestimatedforanARFIMA(0,,1)model(Table15)andnowallestimatesareexpectedtobeconsistent(Beran,1995,reportedThetaperedWhittle(=2)andthesemiparametric(cosinebell)estimatesreportedslightlylargervaluesthanotherprocedures.HeretheMA(1)parameterseemsinsigni cant(exceptperhapsfortheWhittleestimatewith=2,whichisthemethodwithahighestestimateofandlargesttrade-o betandthe able15:CHEMICALSERIES-A.ARFIMA(;d; W-pW-2SG-SEMW-pW-2S bd.4237.4408.4572.4674.5502.4592.5178.4515.4578(.100)(.067)-(.140)(.096)- b 1-.0183.0570-.1839.0500-.1116.0577-(.061)---(.086)- b2-.0994.0972-.0819.0974-.0868.0971 able16:CHEMICALSERIES-A.Robinson's(1994)TestsofNonstationa q4.5696-1.5296-4.0191-5.8752-6.6391-6.8677q2.9196-.5299-2.5009-4.0131-4.0675-4.3663q1.7333-.9173-2.6523-4.1686-4.7653-4.7765q1.1621-.6467-2.0962-3.5145-3.9798-3.7932 shortmemorypartofthemodel).IfweeliminatetheparameterinareducedARIMA(0,modelaboutable17:CHEMICALSERIES-A.ARFIMA(;d; W-2S d.4286.4179.3692.4207(.056)-(.078)- b2.0994.0983.0872.0973 Robinson's(1994b)testsalwysreject=1and75and ndsomeevidenceinsupportresultsforWhittleestimatesmodelbetAppendix)withrespecttoshouldbeunderstoodelemenTheyaresimilartothoseinCondi-tionsAofFxandTaqqu(1986)orinDahlhaus(1989)forparametricestimatesorinRobinson able18:CHEMICALSERIES-A.Memoryestimates W-pW-2SW-2S .4217.4171.4179.3692.4207(.056)(.056)(.078)(.056) q.4571.5645.4554.4623.4582(.089)(.089)(.124)(.089) q.4323.4198.4208.3206.4255(.113)(.113)(.158)(.113) q.4470.4397.4469.2704.4301(.134)(.134)(.187)(.134) modelstionalARIMA's,fractionalGaussiannoiseorfractionalexponentialmodels(see1994a,andBeran,1994). (;; @@)andsoonforanallesguemeall;all;all;Thedi erenywithrespecttoisrequiredtoapproximatediscretesumsbyin)hasasingularityattheorigin.describethestationarydi erencedserieswtroducethefollowinglinearprocessassumptionwhichistakenfromRobinson(1995b),andiseinthelinearityitimposes,butnototherwise.otherwise.tjFt 1]=002tjFt 1]=1,a.s.,t=0;1;:::,andFtisthe- eldofeventsgenerdby,andeexistsar,suchthatall periodogramsmoothnessWhennotaperisappliedwewillimpose:all; Assumption5impliesthedi erenyof)asstatedinAssumption2.Thisconditionimposedmodelstaperingpropertiesweassumestrongersmoothnessconditions.all;Inparticular,with1,Assumption6impliesthat,for2,0= 0,whichisthebasisforthetaperingbiasreduction.ortheasymptoticdistributionoftheestimatesandrelatedquadraticformsweneedalsoanextraconditionaboutthefourthmomenthelinearinnoations,againtakRobinson(1995b),andtadditionalconditionstoximatetheasymptoticcoariancematrixofof3tjFt 1]=3,E[4tjFt 1]=4;a.s.,t=0;1;:::Assumption8k(;)hastwocontinuousderivativesinatall(;,with,andthesederivativesarall;)+loganbontinuouslydi erdtwice(withr)undersignand(tapered)possiblynonstationarywingconditionsholdunderAssumptions1and2,wereassumedinthesepapers. 2,s=bd+1 2c,fU(s)()=Gjj 2(d s)+o(jj s))!0:19 all 2,s=bd+1 2c,d ]as 2;=01 log(+1) )]=1+log(+1)++v(j)v(j)]=log(+1)+ v(j)v(k)]=orv1theperiodogramisnotasymptoticallyunbiasedforaperingwsareductionintheorderofmagnitudeofboundsinTheorempossibletheximationofpsdwithlargerus,withthecosinebelltapersimilarresultsgothroughforan Othertapersreduceevenmorethebiasandallowconsiderationofv taperssmoothnessconditionsonallentiableand satis esaLipschitzconditionofdeforall,andfor;E 2,s=bd+1 ,asThisconditionholdsunderAssumption6for1(seealso(10)).econsidernowthefullcosinebelltaper(4)andde nethenormalizedcosine-taperedDFTDFTCosinebell 2;3 2)=0ifd1 log(+1) )]=1++vcos(j)vcos(j)]= vcos(j)vcos(k)]=alltruewith )]= 3+O j 1+ j;j,(c")E[vcos(j) )]= enowanalyzethecoariancematrixofthe(normalized)taperedDFTwithtapersoforderperiodogram teger,buttaperingdestroystheorthogonalityofthesineandcosinetransformsatclose ps+1if6=0],foranysequencesofpkjlog(+1) )]=1++vp(jp)vp(jp)]=log(+1)+ vp(jp)vp(kp)]=In(c)thetermlogonlyappearsif=2.Theorem4'sboundsaresimilartoTheorem5's belltaperpropertieswithtapersoforder=3,thoughitcannot lteroutpolynomialtrends.discreteaeragesofperiodogramordinatesofpossiblynonstationary(andtapered)observhcanbeseenasspeci cquadraticformsof;:::;nAllfunctionsarebeperiodicofperiod2ProofsarecollectedinAppendixB.Lemma6all dp=n 2c[onlydo1 Lemma7all;allp=nThen,underAssumptions1,2and4, 2c[onlyifo1 Theconditionon)inLemma2restrictsthepermittedvaluesofinthecompactsetThenextlemmaanalyzestheasymptoticdistributionoftheperiodogramaLemma8small, Lemma9 p=n Lemma10;:::;p=nLemma11 )cos)cosLemma12 2c�doifo1 +(2p=np=nIp(j) f(j)]+)=Lemma13=Lemma14isaoforn,as AppendixProofsProofefollowtheproofintostepsofTheorem1inRobinson(1995b).Firststepand ,orotherwise forsome0 =argmin)andnonstochthatforall0thereexists0suchthatinfandalsoSincethereisauniqueminimumof)atfromtheidenwsbecausesup0usingLemmas2[)]and5[)]toapproximateuniformlyintegralswithsums,andusingthislastlemmawgetthatsup0asRecallthat 2+r1o 1 2ootherwise.o1 de nerg,or setting"=do 1 )withbecin(0 22=fd:r1o 1 if=andthetheoremisproConsidernowthesituationwhereisnotemptttoshowthatoran ;underAssumption1.thatthe rstprobabilityin(11)isnegligible,notethat Xj(p)Ip(j)k 1(j;)2 Thelastsumconinprobability(seeLemmas4and5again)to 2Xj(p)f(j)jjj2(do+" 1=2)=2 22 2Xj(p)k(j)jjj2" 12 2Z k()jj2" 12 2c1 c2Z jj2" 1d=2 2c1 c22" )canbemadeaslargeasdesiredforan0andthen0suchthat,de nesothe rsttermin(11)tozeroasandthTheconsistencyofwsfromthatofandLemma2.Proofand7andthatProofximatingtheperiodogramof(possi-blytapered)observedseriesbythatofthe(possiblytapered)linearinnoroot;:::;p=nsousingTheorems3and5inAppendixA,andevennessofallfunc- Ip(jp) 2f(jp)Ip;(jp)]+ (n=p) 1]=2c.Wenowdistinguishthecaseswithandwithouttapering.tapering proofproof toshowthattj 1=2(logj)1=2+jdo 1(logj)1=2]);we ndfrom(12)that,using),0his(1)ifTheexpectationofiswithLemma5,usingtheconyof)andandtheinyoffHn]=(2=nd,asw,bysummationbyparts,forapositiveconstan nn?X1f(j) (j)f2(j) 1g C nn?Xrf(r) (r) f(r (r+1)j rXj2(j) 1g +C wingthediscussioninRobinson(1995b,pp.1637-8),eobtainthat,for1andusingthepropertiesof)andandthemeanvaluetheorem,(13)aperingeringp�1].Weobtainedintheproofconditionsofthislemma,lemma,j 1=2+jdo p(logj)1=2]);0�pdexpectationof1iscalculatedasforecanwrite Ph2tnXt2t(2t 1)(15)+1 cos(=lim,therighthandsideof(15) Ph2t(1 nnXth2t2t hh Ph2t n);24 his(1)because fromTheorem1ofHeydeandSeneta(1972)proofyconditionsasbecause1and=lim h2t2tjFt 1]�0a.s.Next,weconsiderthecontributionof(16).or0=)andj;kcos(cos(j;kThen,changingvariablesandusingtrigonometricidentities(seealsoLemma7ofVelasco,1999b),j;k +cos+cosUsingLemmas8and9,thisis so(17)isTherefore(14)holdsfor1andtheLemmafollowaswhen=1using(13).ProofofLemma2.wsfromthepoinwiseconergenceinLemma1andanequicontusingthecompactnessofandthedi erenyof;)withrespecttoHannan,1973).Proofconsideronlythescalarcase,theargumentforthevcasebeingstocwthesameprocedureasintheproofofLemma1.taperingringp=1].Usingthesecondmomentsoftheperiodogrampp.1648-51procedure,inLemma1inVelasco(1999b), e ndthat f(j) 2(j))=Opr1=3(logn)2=3+r1=2n 1=4+r1=(5 4do)(logr)2=4do)25 w,usingthesameargumentsandhis)ifapering[InVelasco(1999b)itisobtainedthat,1,1rn= +logsoadaptingtheproof,of,n+n1 =2+ndo p+1(logn)1=2]=op(n 1=2)ifultaneouslythestressingthetaperingsituation,theuntaperedcasebeingsimplersincemanyboundsareexactlyzeroduetotheexactp=n),fromLemma5,and nXj=1g(f2( 1g:ThenE[H?n]=0 n=pH )cos(methodproofasinRobinson(1995b),wtheyofFirstweneedtoshowthatThelefthandsideisThe rsttermin(19)is(1),sinceithaszeromeanandv,andfromRobinson(topofp.1646),weobtainusingsummationbyparts,foran,and for1ByLemma6Thesecondtermin(19)canbeshowntobeusingthesameargument(seealsoV1999b,Lemma6),becauseithaszeromeanandvbecausethewaresymmetricaroundUsingtheboundsforsincesupthe rsttermis(1)andthesecondhasabsolutevalueboundedandthisisus(18)isproFinallyweneedtowthatthatz2tI(jztj�)]!0forforwhicecancthesuciensucienz4t]!0:FollowingRobinson(1995b),(1995b),z4t]Cn Pn1c2s2=O(n2 1)=o(1)andthecentrallimittheoremfolloProoftionsofLemma1,with1,0ProofLemma5.ThisfollowsfromthediscussioninRobinson(1994a),p.75.ProofLemma6.First,usingtrigonometricidentitieswehaethat nnXj=1nXk=1g(g( nX1h2t! 21 +cosandusingLemma9andthisis nnXjXk(g(nh2((j+k)p2((j k)p)o;(20)27 Lemma8,andwith)foranthetermin)of(20)is nnXj=1g2(Xk:jj kh2((j k)p)+O0@n2 1nXj Xk:jj kj�jj kj 2p1A+n 1nXjkj kjjg(jO jsupjj kjg( g(jjj kj 2p!byLemmais,usingjj kjjg( g(j=O jg(jjkj 1jk jj+jg(jjjj 1jk jj,)2p �kalltheerrortermsbeing(1)onchoosing,e.g., ,withProofofLemma7.Bythede nitionof=(2p=nItfollowsbythemeanvaluetheoremthat),where=(2p=nnIp(j) 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AbstractWhittle pseudo-maximum likelihood estimates of parameters for stationarytime series have been found to be consistent and asumptotically normal in thepresence of long-range dependence. Generalizing the definition of thememory parameter , we extend these results to include possiblynonstationary (0.5 1) or antipersistent (-0.5 0) observations. Usingadequate data tapers we can apply this estimation technique to any degree ofnonstationarity We analyse the performance of the estimates on simulated and real data.Long-range dependence; nonstationary long memory time series;nonstationary fractional models; frequency domain estimation; tapering.JEL No.: C22by the authors. All rights reserved. Short sections of text, not to exceed twoparagraphs, may be quoted without explicit permission provided that fullcredit, including © notice, is given to the source. WHITTLE PSEUDO-MAXIMUM LIKELIHOOD ESTIMATIONFOR NONSTATIONARY TIME SERIESCarlos VelascoUniversidad Carlos III, MadridPeter M RobinsonLondon School of Economics and Political ScienceContents: 1. Introduction2. The Model and the DiscreteFourier Transform3. Whittle Estimates4. Simulation Results5. Illustrative Examples6. Appendix A: Technical Assumptionsand Results7. Appendix B: ProofsReferencesThe Suntory CentreSuntory and Toyota International Centresfor Economics and Related DisciplinesLondon School of Economics and PoliticalScienceDiscussion PaperHoughton StreetNo. EM/00/391London WC2A 2AEMay 2000Tel.: 020-7405 7686 The research of the first author was supported by Spanish Direccion General deEnsenanza Superior, Ref. No. PB98-0025. The second author’s research wassupported by ESRC Grant No. R000235892, and a Leverhulme Trust PersonalProfessorship. The authors thank the editor, the associate editor and the referees forvery helpful comments.