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6.3ThresholdEffectsinNon-dynamicPanelsAreregressionfunctionsidenticalacrossallobservationsinasample,ordotheyfallintodiscreteclasses?Thisquestionmaybeaddressedusingthresholdregressiontechniques.Thresholdregressionmodelsspecifythatindividualobservationscanbedividedintoclassesbasedonthevalueofanobservedvariable.Hansen(1999)introduceseconometrictechniquesappropriateforthresholdregressionwithpaneldata.6.3.1ModelSetupTheobserveddataarefromabalancepanelyqx,,:1iN,1tT.Theitititdependentvariableyisscalar,thethresholdvariableqisscalar,andtheregressorititxisakvector.ThestructuralequationofinterestisityxIqxIqe(6.52)iti12itititititwhereIistheindicatorfunction.Analternativeintuitivewayofwriting(6.52)isi1xiteit,qity.itxe,qi2itititxIqititAnothercompactrepresentationof(6.52)istosetxanditxIqititsothat(6.52)equals12yxe(6.53)itiititTheobservationsaredividedintotwo‘regimes’dependingonwhetherthethresholdvariableqissmallerorlargerthanthethreshold.Theregimesaredistinguishedbyitdifferingregressionslopes,and.Fortheidentificationofand,itis1212requiredthattheelementsofxarenottimeinvariant.Wealsoassumethattheitthresholdvariableqisnottimeinvariant.Theerroreisassumedtobeindependentitit28 2andidenticallydistributed(iid)withmeanzeroandfinitevariance.Theiidassumptionexcludeslaggeddependentvariablesfromx.TheanalysisisasymptoticwithfixedTasitN.6.3.2ModelEstiamtionOnetraditionalmethodtoeliminatetheindividualeffectistoremoveiindividual-specificmeans.Notethattakingaveragesof(6.52)overthetimeindextandtakingthedifferenceof(6.52)andtheaveragesproduces***yxe(6.54)itititT***1whereyityityi,xitxitxi,eiteiteiandyiTyit,t1TT11xiTxit,eiTeit.Usingthematrixnotation,(6.54)isequivalenttot1t1***YXe(6.55)Foranygiven,theslopecoefficientcanbeestimatedbyordinaryleastsquares(OLS).Thatis,1ˆX*X*X*Y*(6.56)Thevectorofregressionresidualsiseˆ*Y*X*ˆ(6.57)andthesumofsquarederrorsisSeˆ**eˆ11(6.58)Y*IX*X*X*X*Y*Chan(1993)andHansen(2000)recommendestimationofbyleastsquares.Thisiseasiesttoachievebyminimizationoftheconcentratedsumofsquarederrors(6.58).Hencetheleastsquaresestimatorsofis29 ˆargminS(6.59)1Onceˆisobtained,theslopecoefficientestimateisˆˆˆ.Theresidual**vectoriseeˆˆˆandresidualvariance211**ˆeeˆˆSˆ(6.60)1NT(1)NT(1)6.3.3Inference(1)TestingforathresholdItisimportanttodeterminewhetherthethresholdeffectisstatisticallysignificant.Thehypothesisofnothresholdeffectin(6.52)canberepresentedbythelinearconstraintH:012Underthenullhypothesisofnothreshold,themodelisyxe(6.61)iti1ititAfterthefixed-effecttransformationismade,wehave***yxe(6.62)it1itit***ByOLS,wehave、eandthesumofsquarederrorsSee.Thelikelihood1it0ratiotestofH0isbasedon2FSSˆˆ(6.63)101TheasymptoticdistributionofFisnon-standard.Hansen(1996)showsthata1bootstrapprocedureattainsthefirst-orderasymptoticdistribution,sop-valuesconstructedfromthebootstrapareasymptoticallyvalid.(2)AsymptoticdistributionofthresholdestimateWhenthereisathresholdeffectChan(1993)andHansen(2000)have12shownthatˆisconsistentfor(thetruevalueof)andthattheasymptotic0distributionishighlynon-standard.Hansen(1999)arguesthatthebestwaytoformconfidenceintervalsforistoformthe‘no-rejectionregion'usingthelikelihoodratio30 statisticfortestson.TotestthehypothesisH:,thelikelihoodratiotestisto00rejectforlargevaluesofLRwhere102LRSSˆˆ(6.64)111dUndercertainassumptionsandH:,HansenshowsthatLRas001N,whereisarandomvariablewithdistributionfunction2Px1expx2(6.65)Sincetheasymptoticdistributionispivotal,i.e.,thislimitingdistributiondoesnotdependonnuisanceparameters,itmaybeusedtoformvalidasymptoticconfidenceintervals.Furthermore,thedistributionfunction(6.65)hastheinversec2log11(6.66)fromwhichitiseasytocalculatecriticalvalues.Forexample,the5%criticalvalueis7.35.AtestofH:rejectsattheasymptoticlevelifLRexceedsc.0010Toformanasymptoticconfidenceintervalfor,the‘no-rejectionregion'ofconfidencelevel1isthesetofvaluesofsuchthatLRc.Thisis1easiesttofindbyplottingLRagainstanddrawingaflatlineatc.1(3)AsymptoticdistributionofslopecoefficientsTheestimatorˆˆˆdependsonthethresholdestimateˆ,however,Chan(1993)andHansen(2000)showthatthedependenceonthethresholdestimateisnotoffirst-orderasymptoticimportance,soinferenceoncanproceedasifthethresholdestimateˆwerethetruevalue.Henceˆisasymptoticallynormalwithacovariancematrixwhichcanbe1NTVˆx*ˆx*ˆˆ2ititit11Iftheerrorsareallowedtobeconditionallyheteroskedastic,thenaturalcovariancematrixestimatorforˆis31 11NTNTNTˆ*ˆ*ˆ*ˆ*ˆˆ*2*ˆ*ˆVhxitxitxitxiteitxitxiti1t1i1t1i1t16.3.4MultiplethresholdsModel(6.52)hasasinglethreshold.Insomeapplicationstheremaybemultiplethresholds.Forexample,thedoublethresholdmodeltakestheformyxIqxIqxIqe(6.67)iti1itit12it1it23it2ititwherethethresholdsareorderedsothat.Wewillfocusonthisdoublethreshold12modelsincethemethodsextendinastraightforwardmannertohigher-orderthresholdmodels.(1)EstimationForgiven,,(6.67)islinearintheslopes,,soOLSestimationis12123appropriate.Thusforgiven,theconcentratedsumofsquarederrorsS,1212isstraightforwardtocalculate(asinthesinglethresholdmodel).ThejointLSestimatesof12,arebydefinitionthevalueswhichjointlyminimizeS12,).Whiletheseestimatesmightseemdesirable,theymaybequitecumbersometoimplementinpractice.22Agridsearchover,requiresapproximatelynNTregressionswhichmay12beprohibitivelyexpensive.Aremarkableinsightallowsustoescapethiscomputationalburden.Ithasbeenfound(Chong,1994;Bai,1997;BaiandPerron,1998)inthemultiplechangepointmodelthatsequentialestimationisconsistent.Thesamelogicappearstoapplytothemultiplethresholdmodel.Themethodworksasfollows.Step1:LetSbethesinglethresholdsumofsquarederrorsasdefinedin(6.58)1andletˆbethethresholdestimatewhichminimizesS.TheanalysisofChongand11Baisuggeststhatˆwillbeconsistentforeitheror.112Step2:Fixingthefirst-stageestimateˆ,thesecond-stagecriterionis132 rSˆ1,,2ifˆ12S(6.68)22S2,,ˆ1if2ˆ1andsecond-stagethresholdestimateisrrˆargminS(6.69)2222rBai(1997)hasshownthatˆisasymptoticallyefficient,butˆisnot.Thisis21becausetheestimateˆwasobtainedfromasumofsquarederrorsfunctionwhichwas1contaminatedbythepresenceofaneglectedregime.Theasymptoticefficiencyofrˆsuggeststhatˆcanbeimprovedbyathird-stageestimation.Bai(1997)suggests21thefollowingrefinementestimator.rStep3:Fixingˆ,definetherefinementcriterion2rrS1,,ˆ2if1ˆ2rS(6.70)11rrSˆ2,,1ifˆ21andtherefinementestimaterrˆargminS(6.71)1111rBai(1997)showsthattherefinementestimatorˆisasymptoticallyefficientin1changepointestimation,andweexpectsimilarresultstoholdinthresholdregression.(2)DeterminingnumberofthresholdsInthecontextofmodel(6.67),thereareeithernothresholds,onethreshold,ortwothresholds.InSection6.3.3weintroducedFasatestofnothresholdsagainstone1threshold,andsuggestedabootstraptoapproximatetheasymptoticp-value.IfFrejects1thenullofnothreshold,inthecontextofmodel(6.67)weneedafurthertesttodiscriminatebetweenoneandtwothresholds.Theminimizingsumofsquarederrorsfromthesecond-stagethresholdestimateisrr21rrS22ˆwithvarianceestimateˆS22ˆ.ThusanapproximatelikelihoodNT(1)ratiotestofoneversustwothresholdscanbebasedonthe33 rr2F2S1ˆ1S2ˆ2ˆ(6.72)ThehypothesisofonethresholdisrejectedinfavoroftwothresholdsifFislarge.2(3)ConfidenceregionconstructionWefinallyconsidertheconstructionofconfidenceintervalsforthetwothresholdparameters,.Bai(1997)showed(fortheanalogouscaseofchange-pointmodels)12thattherefinementestimatorshavethesameasymptoticdistributionsasthethresholdestimateinasinglethresholdmodel.ThissuggeststhatwecanconstructconfidenceintervalsinthesamewayasinSection6.3.3.Letrrrr2LR2S2S2ˆ2ˆandrrrr2LR1S1S1ˆ1ˆourasymptotic1%confidenceintervalsforandarethesetofvaluesof21rrsuchthatLRcandLRc,respectively.2134