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1、economicslettersELSEVIEREconomicsLetters48(1995)221-228Anoteonthedistributionoftheleastsquaresestimatorofarandomwalkwithdrift:SomeanalyticalevidenceNielsHaldrup*,SvendHyllebergDepartmentofEconomics,UniversityofAarhus,Universitetsparken,Building350,AarhusC,DK-8000,DenmarkReceived18July1994;accep
2、ted25July1994AbstractInthispaperwedefinethenotionofalocaldriftinaunitrootprocess.Thetheoryprovidesabridgebetweentheapparentdivergingasymptotictheoriesthatapplywhenadriftiseitherpresentorabsentinanintegratedtime-series.OurasymptoticresultshelptoexplaintheMonteCarloresultsofHyllebergandMizon(Econ
3、omicsLetters,1989,29,225-230)andSchmidt(AdvancesinEconometrics,1988,8,161-200).Keywords:Unitroots;Localdrift;BrownianmotionJELclassification:C12;C221.IntroductionWithintheDickey-Fullerclassoftestsforunitrootsitiswellknownthatestimatorsandteststatisticsofvarioushypotheseswillhavenon-standardlimi
4、tingdistributions.However,iftheunderlyingprocessisstationarytheusualGaussiantheoryofinferenceappliesasymp-totically,evenwhentherootisveryclosetoone.Thelimitingdistributionswillthusexhibitadiscontinuityatunity,see,forexample,DickeyandFuller(1979)andPhillips(1987a).However,sincetheexactfinitesamp
5、ledistributionswillbecontinuousforallpossiblevaluesoftheautoregressiveroot,thisseemstosuggestthatlimitingdistributionsinadequatelyapproximatethefinitesampledistributionsiftherootislyingclosetothepointofdiscontinuity.ChanandWei(1987b),Phillips(1987)andHaldrupandHylleberg(1993),amongothers,haveus
6、edthenotionofnearlyintegratedtime-seriestodescribeintermediatedistributionsbetweentheGaussianandthenon-standardunitrootdistributions,whichcanbeexpressedintermsofBrownianmotionfunctionals.Evenforfinitesamplesizesthis*Correspondingauthor.0165-1765/95/$09.50I~)1995ElsevierScienceB.V.Allrightsreser
7、vedSSDI0165-1765(94)00637-7222N.Haldrup,S.Hylleberg/EconomicsLetters48(1995)221-228asymptotictheoryseemstodescribetheobservedfinitesampledistributionsratherwell,see,forexample,Chan(1988)andHaldrupandHylleberg(1993).Inthispaperadet
