Stata面板门槛回归-南开大学王群勇.pdf

Stata面板门槛回归-南开大学王群勇.pdf

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TheStataJournal（2015）15,Number1,pp.121134Fixed-effectpanelthresholdmodelusingStataQunyongWangInstituteofStatisticsandEconometricsNankaiUniversityTianjin,ChinaQunyongWAbstract.Thresholdmodelsarewidelyusedinmacroeconomicsandfinancialanalysisfortheirsimpleandobviouseconomicimplications.Withthesemodels,however,estimationandinferenceiscomplicatedbytheexistenceofnuisanceparameters.Tocombatthisissue,Hansen（1999,JournalofEconometrics93:

345368）proposedthefixed-effectpanelthresholdmodel.Inthisarticle,Iintroduceanewcommand（xthreg）forimplementingthismodel.IalsouseMonteCarlosimulationstoshowthat,althoughthesizedistortionofthethreshold-effecttestissmall,thecoveragerateoftheconfidenceintervalestimatorisunsatisfactory.Iincludeanexampleonfinancialconstraints（originallyfromHansen1999,JournalofEconometrics93:

345368）tofurtherdemonstratetheuseofxthreg.Keywords:

st0373,xthreg,panelthreshold,fixedeffect1IntroductionHeterogeneityisacommonproblemofpaneldata.Thatistosay,eachindividualinastudyisdifferent,andstructuralrelationshipsmayvaryacrossindividuals.Theclassicalfixedeffectorrandomeffectreflectsonlytheheterogeneityinintercepts.Hsiao（2003）considersmanyvaryingslopemodelsforthisproblem.Amongthesemodels,Hansens（1999）panelthresholdmodelhasasimplespecificationbutobviousimplicationsforeconomicpolicy.Thoughthresholdmodelsarefamiliarintime-seriesanalysis,theirusewithpaneldatahasbeenlimited.Thethresholdmodeldescribesthejumpingcharacterorstructuralbreakinthere-lationshipbetweenvariables.Thismodeltypeispopularinnonlineartimeseries,oneexamplebeingthethresholdautoregressive（TAR）model（Tong1983）.Thismodelcancapturemanyeconomicphenomena.Forexample,usingfive-yearintervalaveragesofstandardmeasuresoffinancialdevelopment,inflation,andgrowthfor84countriesfrom1960to1995,RousseauandWachtel（2009）showedthatthereisaninflationthresh-oldforthefinanceandgrowthrelationshipthatliesbetween1325%.Wheninflationexceedsthethreshold,financeceasestoincreaseeconomicgrowth.Inflationseffectoneconomicgrowthdependsontheinflationlevel.Highlevelsofinflationareharmfultoeconomicgrowth,whilelowlevelsofinflationarebeneficialtoeconomicgrowth.Asan-otherexample,thetechnicalspilloverofforeigndirectinvestment（FDI）hasbeenwidelystudied.Girma（2005）foundthattheproductivitybenefitfromFDIincreaseswithab-sorptivecapacityuntilsomethresholdlevel,atwhichpointitbecomeslesspronounced.ThereisalsoaminimumabsorptivecapacitythresholdlevelbelowwhichproductivityspilloversfromFDIarenegligibleorevennegative.c?

2015StataCorpLPst0373122Fixed-effectpanelthresholdmodelusingStataThisarticleisarrangedasfollows.Insection2,Ireviewsomebasictheoriesaboutfixed-effectpanelthresholdmodels.Ithendescribethenewxthregcommandinsec-tion3.Insection4,IperformMonteCarlosimulationstostudytest-powerdistortionandthecoveragerateofconfidenceintervalestimatorsinfinitesamples.IillustrateuseofthecommandwithanexamplefromHansen（1999）insection5.Insection6,Iconcludethearticle.2Fixed-effectpanelthresholdmodels2.1Single-thresholdmodelConsiderthefollowingsingle-thresholdmodel:

yit=+Xit（qit）1+Xit（qit）2+ui+eit

（1）Thevariableqitisthethresholdvariable,andisthethresholdparameterthatdividestheequationintotworegimeswithcoefficients1and2.Theparameteruiistheindividualeffect,whileeitisthedisturbance.Wecanalsowrite

（1）asyit=+Xit（qit,）+ui+eitwhereXit（qit,）=?

XitI（qit）XitI（qit）Given,theordinaryleast-squaresestimatorofis?

=X（）?

X（）1X（）?

ywhereyandXarewithin-groupdeviations.Theresidualsumofsquares（RSS）isequalto?

e?

e.Toestimate,onecansearchoverasubsetofthethresholdvariableqit.Insteadofsearchingoverthewholesample,werestricttherangewithintheinterval（,）,whicharequantilesofqit.sestimatoristhevaluethatminimizestheRSS,thatis,?

=argminS1（）Ifisknown,themodelisnodifferentfromtheordinarylinearmodel.Butifisunknown,thereisanuisanceparameterproblem,whichmakestheestimatorsdistributionnonstandard.Hansen（1999）provedthat?

isaconsistentestimatorfor,andhearguedthatthebestwaytotest=0istoformtheconfidenceintervalusingthe“no-rejectionregion”methodwithalikelihood-ratio（LR）statistic,asfollows:

LR1（）=LR1（）LR1（?

）?

2PrPr（xF1）,namely,theproportionofFF1inbootstrapnumberB.2.2Multiple-thresholdsmodelIftherearemultiplethresholds（thatis,multipleregimes）,wefitthemodelsequentially.Weuseadouble-thresholdmodelasanexample.yit=+Xit（qit1）1+Xit（1qit2）2+Xit（qit2）3+ui+eitHere,1and2arethethresholdsthatdividetheequationintothreeregimeswithcoefficients1,2,and3.Weneedtocomputethis（NT）2timesusingthegridsearchmethod,whichisinfeasibleinpractice.AccordingtoBai（19