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1、6.3ThresholdEffectsinNon-dynamicPanelsAreregressionfunctionsidenticalacrossallobservationsinasample,ordotheyfallintodiscreteclasses?Thisquestionmaybeaddressedusingthresholdregressiontechniques.Thresholdregressionmodelsspecifythatindividualobservationscanbedivide
2、dintoclassesbasedonthevalueofanobservedvariable.Hansen(1999)introduceseconometrictechniquesappropriateforthresholdregressionwithpaneldata.6.3.1ModelSetupTheobserveddataarefromabalancepanelyqx,,:1iN,1tT.Theitititdependentvariableyisscalar,thethresholdvariab
3、leqisscalar,andtheregressorititxisakvector.ThestructuralequationofinterestisityxIqxIqe(6.52)iti12itititititwhereIistheindicatorfunction.Analternativeintuitivewayofwriting(6.52)isi1xiteit,qity.itxe,qi2itititxIqititAnot
4、hercompactrepresentationof(6.52)istosetxanditxIqititsothat(6.52)equals12yxe(6.53)itiititTheobservationsaredividedintotwo‘regimes’dependingonwhetherthethresholdvariableqissmallerorlargerthanthethreshold.Theregimesaredistinguishedbyi
5、tdifferingregressionslopes,and.Fortheidentificationofand,itis1212requiredthattheelementsofxarenottimeinvariant.Wealsoassumethattheitthresholdvariableqisnottimeinvariant.Theerroreisassumedtobeindependentitit282andidenticallydistributed(iid)withmeanzeroandfini
6、tevariance.Theiidassumptionexcludeslaggeddependentvariablesfromx.TheanalysisisasymptoticwithfixedTasitN.6.3.2ModelEstiamtionOnetraditionalmethodtoeliminatetheindividualeffectistoremoveiindividual-specificmeans.Notethattakingaveragesof(6.52)overthetimeindexta
7、ndtakingthedifferenceof(6.52)andtheaveragesproduces***yxe(6.54)itititT***1whereyityityi,xitxitxi,eiteiteiandyiTyit,t1TT11xiTxit,eiTeit.Usingthematrixnotation,(6.54)isequivalenttot1t1***YXe(6.55)Foranygiven,theslopec
8、oefficientcanbeestimatedbyordinaryleastsquares(OLS).Thatis,1ˆX*X*X*Y*(6.56)Thevectorofregressionresidualsiseˆ*Y*X*ˆ(6.57)andthesumofsq