thedifferenceddataMoregenerallythereisthequestionofhowmanytimesweneedtodifferencethedataInthemodelthe ID: 338209
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DIFFERENCINGANDUNITROOTTESTSIntheBox-Jenkinsapproachtoanalyzingtimeseries,akeyquestioniswhethertodifferencethata,i.e.,toreplacetherawdata{}bythedifferencedseries{}.Experienceindicatesthattttosteconomictimeseriestendtowanderandarenotstationary,butthatdifferencingoftenyieldsa thedifferenceddata.Moregenerally,thereisthequestionofhowmanytimesweneedtodifferencethedata.Inthe)model,thethdifferenceisastationary).Theseriesisintegratedodenotedby),whereisanintegerwith1,iftheseriesandallitsdifferencesupto1starenonstationary,butthethdifferenceisstationary.Aseriesissaidtobeintegratedorderzero,denotedby(0),iftheseriesisbothstationaryandinvertible.(Theimportanceofinverti-bilitywillbediscussedlater).Iftheseries{}is)with1,thenthedifferencedseries{tttForanexampleofan(2)process,considerthe(2)series.Thisprocessisotstationary.Equation(1)becomes1,thatis,0.Factoringthisgives1)(0,sotheequationhasunitroots.Sincethelargestroot(i.e.,one)doesnothaveabsolutevalue"lessthanone,theprocessisnotstationary.Itcanbeshownthatthe rstdifferenceisnotstationaryeither.Theseconddifferenceisisxx]x2xx,wttt1t1t2tt1t2hichisequaltobythede nitionofour(2)process.Sincetheseconddifferenceiswhitenoise,}isan(0,2,0).Sincetheseconddifferenceisstationary,{}is(2).Ingeneral,foranyprocesswhichisintegratedoforder,Equation(1)willhaveexactlyunitroots.Inpracticeowever,theonlyintegervaluesofwhichseemtooccurfrequentlyare0and1.Sohere,wewilllimitourdiscussiontothequestionofwhetherornottodifferencethedataonetime.Ifwefailtotakeadifferencewhentheprocessisnonstationary,regressionsontimewilloftenyieldaspuriouslysigni cantlineartrend,andourforecastintervalswillbemuchtoonarrow(optimisic)atlongleadtimes.Foranexampleofthe rstphenomenon,recallthatfortheDe atedDowJonesseries,wegota-statisticfortheslopeof5.27(creatingtheillusionofaverystrongindicationorend),butthemeanofthe rstdifferenceswasnotsigni cantlydifferentfromzero.Foranexampleofthesecondphenomenon,letscomparearandomwalkwithastationary(1)model.Forarandomalk,thevarianceofthe-stepforecasterroris xx]var[hnnn...]hvarvar],-whichgoestoincreases.Thewidthoftheforecastintervalswillbeproportionalto,indicatingthatouruncertaintyaboutthefuturevalueoftheseriesgrowswithoutboundastheleadtimeincreased.Ontheotherhand,forthestationary(1)process1,thebest-stepforecastis,whichgoestozeroasincreases.Thevarianceoftheforecasterrorisisxx],whichtendstotox],a niteconstant.Soastheleadtimeisincreasedthewidthofthe-steppredictionintervalsgrowswithoutboundforarandomwalk,butremainsboundedforastationary(1).Clearly,then,ifourserieswerereallyarandomwalk,butwefailedtoifferenceitandmodeleditinsteadasastationary(1),thenourpredictionintervalswouldgiveusmuchmorefaithinourabilitytopredictatlongleadtimesthanisactuallywarranted.Itisalsoundesirabletotakeadifferencewhentheprocessisstationary.Problemsariseherebecausethedifferenceofastationaryseriesisnotinvertible,i.e.,cannotberepresentedasanorexample,if,sothat{}isreallyastationary(1),thenthe rstdifference{}isthenon-invertible(1,1)process,whichhasmoreparametersthantheorigialprocess.(Recallthatan)isinvertibleifthelargestrootoftheequation0satis es1,whereb,...,baretheparameters.)Becauseofthon-invertibilityof{},itsparameterswillbedif culttoestimate,anditwillbedif culttoconstructaforecastof.Consequently,takinganunnecessarydifference(i.e.,overdifferencing)willtendtodegradethequalityofforecasts.Ideally,then,whatwewouldlikeisawaytodecidewhethertheseriesisstationary,orintegratedoforder1.Amethodinwidespreadusetodayistodeclaretheseriesnonstationaryifthesampleautoorrelationsdecayslowly.Ifthispatternisobserved,thentheseriesisdifferencedandtheautocorrela-tionsofthedifferencedseriesareexaminedtomakesurethattheydecayrapidly,therebyindicatinghatthedifferencedseriesisstationary.Thismethodissomewhatadhoc,however.Whatisreallyneededisamoreobjectivewayofdecidingbetweenthetwohypotheses,(0)and(1),withoutmakin anyfurtherassumptions.Unfortunately,eachofthesehypothesescoversavastrangeofpossibilitiesndanyclassicalapproachtodiscriminatebetweenthemseemsdoomedtofailureunlesswelimitthescopeofthehypotheses.TheDickey-FullerTestofRandomWalkVs.StationaryAR(1)Atestinvolvingmuchmorenarrowly-speci ednullandalternativehypotheseswasproposedbickeyandFullerin1979.Initsmostbasicform,theDickey-Fullertestcomparesthenullhypothesisi.e.,thattheseriesisarandomwalkwithoutdrift,againstthealternativehypothesiareconstantswith1.Accordingto,theprocessisastationary(1)wit).Toseethis,notethat,under,wecanwritesothat otethatbymakingtherandomwalkthehypothesis,DickeyandFullerareexpressingaprefer-encefordifferencingthedataunlessastrongcasecanbemadethattherawseriesisstationary.Thisisonsistentwiththeconventionalwisdomthat,mostofthetime,thedatadorequiredifferencing.ATypeIerrorcorrespondstodecidingtheprocessisstationarywhenitisactuallyarandomwalk.Inthiase,wewillfailtorecognizethatthedatashouldbedifferenced,andwillbuildastationarymodelforournonstationaryseries.ATypeIIerrorcorrespondstodecidingtheprocessisarandomwalkwhenisactuallystationary.Here,wewillbeinclinedtodifferencethedata,eventhoughdifferencingisnotWeshouldmentiontwoadditionalimportantdifferencesbetweenthe(1)andtherandomwalk.Whereastheinnovationhasatemporary(exponentiallydecaying)effectonthe(1),ithaspermanenteffectontherandomwalk.Whereastheexpectedlengthoftimebetweencrossingsof niteforthe(1)(sothe(1) uctuatesarounditsmeanof),theexpectedlengthoftimebetweencrossingsofanyparticularlevelisinfinitefortherandomwalk(sotherandomwalkhasatendencytanderinanon-systematicfashionfromanygivenstartingpoint).TheDickey-Fullertestiseasytoperform.Givendatax,...,x,werunanordinarylineaegressionoftheobservations(x,...,x)ofthe"dependentvariable"{},againsttheobservationsx,...,x)ofthe"independentvariable"{},togetherwithaconstantterm.Underboth,thedataobeythelinearregressionmodelcorrespondstoDenotethe-statisticfortheleastsquaresestimateistheestimatedstandarderrorfor.Notethatiseasytocalculate,sinceandcanbbtaineddirectlyfromtheoutputofthestandardcomputerregressionpackages.FortheDe atedDowdata,regressingx,...,xx,...,x,weobtainthefollowin25471546regressionoutput:ResidualStandardError=0.6933,MultipleR-Square=0.9907N=546,F-statistic=57991.38on1and544df,p-value=0coefstd.errt.statp.valueIntercept0.10950.08321.31650.1886X0.99630.0041240.81400.0000TheR-Squarestatisticis.9907,indicatingaverystronglinearrelationshipbetween{}and}.Theestimatedslopeis9963,and0041.WecalculateNotethatwedoNOTusethetstatistic(240.8140)fromtheoutput,sincethiswascomputedrelativetoanullvalueofzero,insteadof1. Thestatisticcanbeusedtotest.Thepercentilesofaregivenintheattachedtable.Thenullhypothesisisrejectedifislessthanthetabledvalue.Thetabulationsfowerebasedonsimulation,assumingGaussian.Thetabledvaluesfortheasymptoticistribution()arevalidaslongasthewith nitevariance.(NoGaussianassumptionisneededhere.)Itshouldbenotedthatdoesnothaveadistributionin nitesamples,anddoesnohaveastandardnormaldistributionasymptotically.Infact,theasymptoticdistributionislonger-tailedthanthestandardnormal.Forexample,theasymptotic01percentagepointofisat3.43,insteado2.326forastandardnormal.Thus,useofthestandardnormaltablewouldresultinanexcessofspuriusdeclarationsofstationarity.FortheDe atedDowdata,weobtained .9024,whichisnotsigni cantaccordingtotheable.Sowearenotabletorejecttherandomwalkhypothesis.Asusualinstatisticalhypothesistest-ing,thisdoesnotmeanthatweshouldconcludethattheseriesisarandomwalk.Infact,fromourearieranalysiswehavestrongstatisticalevidencethattheseriesisarandomwalk,sincethelag-1autocorrelationforthe rstdifferencesishighlysigni cant.AllwecanconcludefromtheDickey-FulleestisthatthereisnostrongevidencetosupportthehypothesisthattheseriesisastationaryThisisthetypeofalternativethatthetestwasdesignedtodetect.Thequestionofwhetherthe rsifferencehasanyautocorrelationisanotherissuealtogether,andthetestwasnotdesignedtodetectthistypeoffailureoftherandomwalkhypothesis.Inanycase,theresultsofthetestindicatethatiouldbeagoodideatodifferencethedata.Wecouldhavecometothissameconclusionbyexamin-ingtheACFoftherawdata,buttheDickey-FullertestprovidesamoreobjectivebasisformakingthisAsanillustrationofthelongtailsinthedistribution,considertherandomwalkdata(hichwasusedinthelasthandoutforcomparisonwiththeDowandDe atedDowseries.Forthisrandomwalkdataset,weobtainthefollowingregressionoutput. ResidualStandardError=0.9917,MultipleR-Square=0.982=546,F-statistic=30111.83on1and544df,p-value=0coefstd.errt.statp.valuentercept0.07710.05501.40280.1612X0.99130.0057173.52760.000ethereforeget1.53.Ifhadastandardnormaldistribution,wewouldbtaina-valueof.063(one-sided),indicatingsomeevidenceinfavorofthealternativehypothesis(thattheseriesisastationary(1)).Ofcourse,weknowthatthisserieswasinfactarandomwalkndsoitissomewhatdistressingthatwearealmostbeingledtocommitaTypeIerror.Butwhenweusethetruedistributionofunderthenullhypothesis(seetable)we ndthattheactualsigni canclevelissubstantiallygreaterthan.10,althoughthetableisnotpreciseenoughtoallowusto ndtheOfcourse,thenullandalternativehypothesesdescribedabovearetoonarrowtobeveryusefulinawidevarietyofsituations.Often,wewillwanttoconsiderdifferencingthedatecausewehopethedifferencemaybestationary,butwedonotwanttocommitourselvestotheassumptionthattheseriesiseitherarandomwalkorastationary(1).Fortunately,althoughwewilotdescribethedetailshere,thereisasimilartestknownastheAugmentedDickey-Fullertest,whichallowsustotestan,1,0)nullhypothesisversusan1,0,0)alternative,wher0isknown.If1,forexample,thenullhypothesiswouldbethattheseriesisnonstationary,butits rstdifferenceisastationary(1);thealternativehypothesiswouldbethattheseriesisastation-(2).Inretrospect,itseemsthattheDe atedDowseriesisbetterdescribedbytheabovenullhypothesisthanbytheonewhichwasactuallytested,i.e.,therandomwalk.Butitisneveragoodideaochangeastatisticalhypothesisafterlookingatthedata;itcandestroythevalidityofthetest.Furth-ermore,theuseoftherandomwalkasanullhypothesisfor nancialtimeseriesseemswiseasagenralrule.DifferenceStationarityVs.TrendStationarityIntheordinaryDickey-Fuller()test,theseriesisassumedtobefreeofdeterministictrendunderboththenullandalternativehypotheses.Manyactualseriesdohavetrend,however,anditisof nteresttostudythenatureofthistrend.Perhapsthemostimportantissueisthewayinwhichthetrendiscombinedwiththerandomaspectsoftheseries.Inthecaseofarandomwalkwithdrifwhere{}iszeromeanwhitenoise,thereisamixtureofdeterministicandstochasticrend,theprocesshasaunitroot,andtheforecastintervalsgrowwithoutboundastheleadtimeincreases.Differencing{}yieldsastationaryseries,so{}issaidtobedifferencestationary.isthesameasAnotherwaytocombinetrendandrandomnessistostartwithadeterministiclineartrendandburyitinwhitenoise:.Thisisastandardlinearregression(trend-line)model,whiccanbeanalyzedwithoutusingtimeseriesmethods.Iftheparameters((])areknown,theheforecastofissimply).Ifthearenormallydistributed,aforecastintervaldddddisgivenforlargezvar.Thewidthofthisforecastintervaldoesnottoin nityastheleadtimeincreases.Moregenerally,anyseriesformedbyaddingadeterministiclineartrendtoastationary,invertible,zeromean"noise"series{aidtobetrendstationary.Trendstationaryseriesdonotcontainaunitroot.Thewidthoftheirfore-castintervalsforlargeis2zvary,whichdoesnottendtoin nity.Trendstationaryseriesaredddddnotdifferencestationary,sinceitcanbeshownthatthedifferenceof{}isnotinvertible.Sincethrendstationaryseriesobeysaregressionmodelwithautocorrelatederrors,wecanusegeneralizedleastsquares(apopularlinearregressiontechnique)toestimatethetrendandassessitsstatisticaHere,weshowhowtotestaspeci cformofdifferencestationarityagainstaspeci cformoftrendstationarity,usingavariant()oftheDickey-Fullertest.Thenullhypothesisisarandomwalkwithdrift(whichisdifferencestationary),versuttt }istrendstationary,andthe"noise"termis(1).Ifweput0,thenwegetthtrend-linemodel.Itcanbeshownthatunder}canbeexpresseda,(2)100111areconstants.(Speci cally, ).)IfweputhenEquation(2)reducesto.e.,arandomwalkwithdrift.Thus,wewanttotestthenullhypothesisthat1versusthealternative1inEquation(2).Toperformthetest,werunanordinarylinearregressionofthe"dependentvariable"{}againstheexplanatoryvariablestime()and{},togetherwithaconstantterm.Theobservationson{are(x,...,x),theobservationsonare(2),andtheobservationson{}arx,...,x).TheteststatisticisthestandardizedestimateofinEquation(2),AlthoughthismayappeartobethesameastheordinaryDickey-Fullerstatistic,itisactuallifferentbecauseofthepresenceoftimeasanexplanatoryvariable.Thepercentilesofunderthenullhypothesis(1)aregivenintheattachedtable.Thenullhypothesisisrejectedifislessthanthabledvalue.Thepercentilesofareconsiderablylessthanthecorrespondingpercentilesof,indi-atingtheeffectsofincludingtimeasanexplanatoryvariable.Forexample,theasymptotic01percen-tagepointofisat3.96for,comparedwith3.43for Thelog10Dowdataseemstocontainatrend,butwhatisthenatureofthistrend?Woulditboreappropriatetomodelthisdataasarandomwalkwithdrift,orasatrendlineplusstationary(1)errors?Inouroriginalanalysisofthisdata,we rsttriedanordinarytrend-linemodel,andfounhighlysigni canttrend.Wethenquestionedthevalidityofthis nding,sincetheDurbin-Watsonstatisticshowedstrongerrorautocorrelation.Wecouldhavepursuedtheuseofatrendstationarymodei.e.,lineartrendplusautocorrelatederrors)forthisseries,byre-estimatingthetrendlineusinggeneral- -10-zedleastsquares.Thisstillwouldnothaveansweredthequestionastowhethersuchamodelismoreappropriatethanarandomwalkwithdrift,however.Toaddressthisquestion,wenowruntheTheregressiondescribedaboveyieldedResidualStandardError=0.0147,MultipleR-Square=0.9977N=546,F-statistic=119357.4on2and543df,p-value=0coefstd.errt.statp.valueIntercept0.02070.01301.58850.112ime0.00000.00001.35080.1773x.lag0.99230.0054182.19970.000hereTimedenotes(2547),x.lagis(x,...,x)andthedependentvariableis15462547x,...,x).Theestimatedcoef cientofx.lagis9923,and0054.WecalculateSincethisisnotlessthanthetabledvalueof3.42,wedonotrejectthenullhypothesisofrandomwalkwithdriftatlevel05.Infact,examinationofthetablerevealsthatourobservedisnotsmallaall,witha-valuearound.9,indicatingthatthereisvirtuallynoevidenceinfavoroftrendstationaritorthisseries.Thisdoesnotmeanthatthelog10Dowdataisactuallyarandomwalkwithdrift.(Indeed,wepreviouslyfoundstrongevidencethatthedifferencesofthisdataarenotuncorrelated,evenhoughtheyseemtohaveanonzeroexpectation.)Itjustmeansthatwecannotrejecttherandomwalkwithdrifthypothesisinfavoroftrendstationarity.