blockwise bootstrap testing for stationarity

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1、Statistics&ProbabilityLetters50(2000)389{395BootstraptestsforunitrootsinseasonalautoregressivemodelsZachariasPsaradakisSchoolofEconomics,MathematicsandStatistics,BirkbeckCollege,UniversityofLondon,7-15GresseStreet,LondonW1P2LL,UKReceivedOctober1999;receivedinrevisedformApril2000AbstractThis

2、paperproposesbootstraptestsforthepresenceofunitrootsinaseasonalautoregressivemodel.Theasymp-toticvalidityoftheproposedbootstrapschemeisestablished,andMonteCarloexperimentsareusedtoinvestigatethesmall-sampleperformanceofthetests.c2000ElsevierScienceB.V.AllrightsreservedKeywords:Bootstrap;Hypo

3、thesistesting;Least-squaresestimator;Seasonalautoregressivemodel;Unitroots1.IntroductionLetthereal-valuedtimeseriesfXt:t2NgsatisfytheautoregressivemodelXt=Xt?m+t(t2N);(1)whereft:t2Ngisasequenceofindependent,identicallydistributed(i.i.d.)randomvariableswithE[1]=0andE[2]=22(0;1),and2(?1

4、;1].Model(1)withm>2isasimpleseasonalmodelinwhichtime1seriesobservedsemi-annually,quarterly,ormonthlyarerepresentedbym=2,m=4,orm=12,respectively.Forthesakeofconvenience,itwillbeassumedthroughoutthepaperthatX?m+1=


=X0=0.WeareinterestedintestingthenullhypothesisH0:=1,underwhichtherootsofthec

5、haracteristicequation1?zm=0of(1)areallofunitmodulusandfX:t2Ngisaseasonalrandomwalkwithperiodmt(oranARIMA(0;1;0)mprocess).Thisisaproblemofpracticalimportancesinceitisoftenofinteresttotestwhetheraseriesshouldbeseasonallydierencedinordertorenderitstationary.Unstablemodelslike(1)with=1candesc

6、ribewellatimeserieswithchangingseasonalpatterns,andhavebeenusedextensivelyintheliteraturefollowingthein uentialworkofBoxandJenkins(1970).WhenanitesegmentfXt:t=1;:::;ngfromarealizationoftheprocessdenedby(1)isavailable,atestingmethodologyforthehypothesisH0:=1waspresentedinDickeyetal.(1984).

7、TheirtestsarebasedTel.:+44-207-631-6415;fax:+44-207-631-6416.E-mailaddress:zpsaradakis@econ.bbk.ac.uk(Z.Psaradakis).0167-7152/00/$-seefrontmatterc2000ElsevierScienceB.V.AllrightsreservedPII:S0167-7152(00)00128-0390Z.Psaradakis/Statistics&Probabili