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1、EconometricTheory,19,2003,32-48.PrintedintheUnitedStatesofAmerica.DOI:10.1017/S0266466603191025ANOTEONTHEPOWEROFBOOTSTRAPUNITROOTTESTSANDERSRYGHSWENSENUniversityofOsloInthisnoteweconsidertheasymptoticpowerfunctionsofsomebootstrapunitroottestsunderlocalalternativesandshowthatthe
2、yareinfactthesameasforordinaryunitroottests.Thisisregardlessofwhetherthedifferencesoftheobser-vations,i.e.,theso-calledrestrictedresiduals,ortheordinaryleastsquaresresid-ualsareusedtoconstructtheresampledobservations.WealsoconsidermodelscontainingaconstantandalineartrendandtheD
3、F-GLStestsproposedbyEl-liott,Rothenberg,andStock(1996,Econometrica64,813-836).AsmallMonteCarloexperimentisincluded.1.INTRODUCTIONTherecentsurveysonbootstrapmethodsintimeseriesbyLiandMaddala(1996)andBerkowitzandKilian(2000)bothcontainspecialsectionsonhowtodealwithnonstationaryda
4、taandadiscussionofbootstrapunitroottests.Bootstrapproceduresofferanopportunitytotakeintoaccountsuchfactorsassamplesize,variousspecificationsoftheinitialcondition,andthedistributionoftheerrors.Theymaythereforehavemoreaccuratefinite-samplepropertiesthanproceduresmakinguseofasympt
5、oticapproximations,wheresuchele-mentstypicallydonotenter.TabulationsbasedonMonteCarlosimulationswillnecessarilyhavetospecifythesefactors,andtheassumptionsmaynotalwaysbewellsuitedinparticularapplications.Thebootstrapprocedurespresentaninterestingpossibilitymakingmoreextensiveuse
6、ofdataandmaythere-forerepresentanalternativeandsupplementtotabulationsandasymptoticapproximations.MoststudiesofbootstrapmethodsforunitroottestshavebeenconcernedwithtypeIerrorsand,inparticular,thequestionofwhetherthesizesofthetestscorrespondtothenominallevels.FerrettiandRomo(199
7、6)consideraunitrootbootstraptestforAR(l)modelsanddemonstratetheasymptoticva-lidityforindependentandautoregressiveerrors.Forageneralizationwheretheorderoftheautoregressiveerrorseriesisallowedtoincrease,seeSwensenIthanktheassociateeditor,BruceE.Hansen,andthreeanonymousrefereesfor
8、veryconstructivecommentsonthepreviousversionsofthemanu
