,Vol.78,No.1(January,2010),169–211INFERENCEFORTHEIDENTIFIEDSETINP - PDF document

,Vol.78,No.1(January,2010),169–211INFERENCEFORTHEIDENTIFIEDSETINPARTIALLYIDENTIFIEDECONOMETRICMODELSP.ROMANOANDM.SThispaperprovidescomputationallyintensive,yetfeasiblemethodsforinferenceinaverygeneralclassofpartiallyidentiedeconometricmodels.Letdenotethedistributionoftheobserveddata.Theclassofmodelsweconsiderisdenedbyapop-ulationobjectivefunctionQ(P).ThepointofdeparturefromtheclassicalextremumestimationframeworkisthatitisnotassumedthatQ(P)hasauniqueminimizerintheparameterspace.Thegoalmaybeeithertodrawinferencesaboutsomeunknownpointinthesetofminimizersofthepopulationobjectivefunctionortodrawinferencesaboutthesetofminimizersitself.Inthispaper,theobjectofinterest(P)argminQ(P),andsoweseekrandomsetsthatcontainthissetwithatleastsomeprespeciedprobabilityasymptotically.Wealsoconsidersituationswheretheobjectofinterestistheimageof(P)underaknownfunction.Randomsetsthatsatisfythedesiredcoveragepropertyareconstructedunderweakassumptions.Condi-tionsareprovidedunderwhichthecondenceregionsareasymptoticallyvalidnotonlypointwisein,butalsouniformlyin.Weillustratetheuseofourmethodswithanempiricalstudyoftheimpactoftop-codingoutcomesoninferencesabouttheparame-tersofalinearregression.Finally,amodestsimulationstudyshedssomelightonthenite-samplebehaviorofourprocedure.:Partiallyidentiedmodel,incompletemodel,identiedset,identiableparameter,subsampling,uniformcoverage,condenceregion,momentinequalities.PARTIALLYIDENTIFIEDMODELisanymodelinwhichtheparameterofinter-estisnotuniquelydenedbythedistributionoftheobserveddata.Thispaperprovidescomputationallyintensiveyetfeasiblemethodsforinferenceforonelargeclassofsuchmodels.Letdenotethedistributionoftheobserveddata.TheclassofmodelsweconsiderisdenedbyapopulationobjectivefunctionQ(P).Thepointofdeparturefromtheclassicalextremumestima-tionframeworkisthatitisnotassumedthatQ(P)hasauniqueminimizerintheparameterspace.Thegoalmaybeeithertodrawinferencesaboutsomeunknownpointinthesetofminimizersofthepopulationobjectivefunctionortodrawinferencesaboutthesetofminimizersitself.Inthispaperweconsiderthesecondofthesetwogoals.Theobjectofinterestis(P)argminQ(P)Wehenceforthreferto(P)astheidentiÞedset.Inthisinstance,givenin-dependentandidenticallydistributed(i.i.d.)datan,generatedWewouldliketothankMichaelWolfforacarefulreadingofthepaperandusefulsugges-tions.WealsothankNeseYildizforpointingouttheneedforthenonzerovarianceconditionin©2010TheEconometricSociety10.3982/ECTA6706 J.P.ROMANOANDA.M.SHAIKH,weseekrandomsetsXthatcontaintheidentiedsetwithatleastsomeprespeciedprobabilityasymptotically.Thatis,were-liminf(P)WerefertosuchsetsasconÞdenceregionsfortheidentiÞedsetthatarepointwiseconsistentinlevel.Thisterminologyreectsthefactthatthecondenceregionsarevalidonlyforaprobabilitydistributionandhelpsdistinguishthiscoveragerequirementfromothersdiscussedlaterinwhichwewilldemandthatthecondenceregionsarevaliduniformlyin.Weshowthattheproblemofconstructingthatsatisfy()isequivalenttoamultiplehypothesistestingprobleminwhichonewantstotestthefamilyofnullhypotheses(P)indexedbywhilecontrollingthefamilywiseerrorrate,theprobabilityofevenonefalserejectionunder.Usingthisduality,wegoontoconstructthatsatisfy()underweakassumptionsonIntherstgoal,theobjectofinterestissomeunknownpoint(P)Werefertoany(P)asanidentiÞableparameter.Inthiscase,giveni.i.d.datan,generatedfrom,weseekrandomsetsthatcontaineachidentiableparameterwithatleastsomepre-speciedprobabilityasymptotically.Theproblemofconstructingsuchsetsistreatedinacompanionpaper(RomanoandShaikh(2008)OurresultsoncondenceregionsfortheidentiedsetbuildupontheearlierworkofChernozhukov,Hong,andTamer(2007),whowerethersttocon-siderinferenceforthesameclassofpartiallyidentiedmodels.Animportantfeatureofourprocedureforconstructingcondenceregionsfortheidentiedsetisthatitavoidstheneedforaninitialestimateof(P).Ingeneral,ourprocedureisrst-orderasymptoticallyequivalentwiththeprocedureproposedChernozhukov,Hong,andTamer(2007).Ontheotherhand,whenthesetofminimizersofdoesnotprovideaconsistentestimateof(P),ourresultsprovideajusticationforiteratingtheirprocedureuntilastoppingcri-terionismettoproducecondenceregionsthataretypicallystrictlysmallerwhilestillmaintainingthecoveragerequirement.Inthispaper,wealsowishtoconstructcondenceregionswhosecoverageprobabilityisclosetothenominallevelnotjustforaxedprobabilitydistrib-,butratheruniformlyoverallinsomelargeclassofdistributionsCondenceregionsthatfailtosatisfythisrequirementhavethefeaturethatforeverysamplesize,howeverlarge,thereissomeprobabilitydistributionforwhichthecoverageprobabilityofthecondenceregionunderisnotclosetotheprescribedlevel.Researchersmay,therefore,feelthatinferencesmadeonthebasisofasymptoticapproximationsaremorereliableifthecondenceregionsexhibitgooduniformbehavior.Ofcourse,sucharequirementwilltypi-callyrequirerestrictionsonbeyondthoserequiredforpointwiseconsistencyinlevel.BahadurandSavage(1956),forexample,showedthatifissuitablylarge,thenthereexistsnocondenceintervalforthemeanwithnitelength INFERENCEINPARTIALLYIDENTIFIEDMODELSRomanoandWolf(2005),whoonlyconsideredanitenumberofhypothe-1.Let.Ifsup,thenacceptallhypothesesandstop;otherwise,setandcontinue..Ifsup,thenacceptallhypothesesandstop;otherwise,setandcon-WenowprovethatthisalgorithmprovidesasymptoticcontroloftheFWERunderthemonotonicityassumption()and(denotethetruedistributiongeneratingthedatawithcriticalvaluesthatsatisfy(P)()�(P)ifthecriticalvaluesalsosatisfylimsup:Toestablish(),denotebythesmallestrandomindexforwhichthereisafalserejection;thatis,thereexists(P)suchthat.Bydenitionof,wemusthavethat(P).Thus,by(wehavethat(P).Hence,itmustbethecasethat(P)(P)Thesecondconclusionfollowsimmediately.Q.E.D.ASubsamplingConstructionItfollowsfromTheoremthatunderthetworestrictions()and(),thesetofvaluescorrespondingtotheacceptedhypothesesfromAlgorithm,satises().Wenowprovideaconcreteconstructionofcriticalvaluesthatsatisfythesetwopropertiesunderaweakassumptionontheasymptoticbehav-ioroftheteststatistics J.P.ROMANOANDA.M.SHAIKHendandnumerousapplications,seevanderVaartandWellner(1996).TheconditionsofTheoremareveriedfor undertheseassumptionsinoftheXAMPLE2.2—RegressionWithIntervalOutcomes:Thefollowingexam-pleallowsforinferenceinalinearregressionmodelinwhichthedependentvariableisinterval-censored.Let)i,beani.i.d.sequenceofrandomvariableswithdistribution.Thepa-rameterofinterest,,isknowntosatisfysatisfyYi|Xi]=X\ti0,butisunob-served,whichprecludesconventionalestimationof.Letdenotethedistri-butionoftheobservedrandomvariables.Therandomvariablesareknowntosatisfywithprobability1under(P))Y1i|Xi]X\tiEP[Y2i|Xi]P-a.s.}.ThissetmaybecharacterizedasthesetofminimizersofQ(P))Y1i|Xi]ŠX\ti)2++(X\tiŠEP[Y2i|Xi])2+ManskiandTamer(2002)characterizedtheidentiedsetinthissettingandalsoconsideredthecasewhereisobserved,butisinterval-censored.Letandsupposeissuchthat(i)suppand(ii)thevariancesof(P)(P),exist.Toruleoutdegeneratesituations,assumefurtherthat(iii)thereexist,andmsuchthatthatY\ni|Xi=xj]=x\tjandVararY\ni|Xi=xj]�0.Form,letP) li=|{.Let WenowverifytheconditionsofTheoremundertheseassumptionsfor Tothisend,notethat n n(xj)1 \f n×\t1in(Š1)\nŠ1(Y\niŠx\tj)I{Xi=xj}\f2+ INFERENCEINPARTIALLYIDENTIFIEDMODELSForm(KP)(P))Y\ni|Xi=xj]=x\tjforall(j\n)Hence,exceptforthemultiplicativefactors n/n(x,whichareasymptoti-callyconstantanyway,thestructurehereisthesameasthestructureofExam-.Asaresult,wemayuseargumentsnearlyidenticaltothosegivenforaboveinSectionofthetoshowthatthelimitingbehaviorof(P)isequaltothelimitingbehaviorof(KP)(j\n)  p(xj)1 \f (KP)(j\n)  p(xj)1 \f Y\ni|Xi=xj])I{Xi=xj}\f2+wherep(xj)=P{Xi=xj},themaximumoverisunderstoodtobeoverallsubsetsofm,andthesupremumovertheemptysetisunder-stoodtobezero.Thevectorwhose(j\n)componentisgivenby \f Y\ni|Xi=xj])I{Xi=xj}tendsindistributiontoamultivariatenormalrandomvariable.Letj\n(P)notethe(j\n)componentofthislimitingmultivariatenormalrandomvariable.Itfollowsbythecontinuousmappingtheoremthat()tendsindistributionto(KP)(j\n)  j\n(P)Todetermineforwhich)iscontinuousatits1quantile,rstnotethat)isaconvexfunctionofZ(P).ByTheorem11.1ofDavydov,Lifshits,and J.P.ROMANOANDA.M.SHAIKHSmorodina(1998),thedistributionof()iscontinuouseverywhereexceptpossiblyatzero,but(KP)(j\n)  j\n(P)  (P) ,andareasinassumption(iii)above.Hence,()iscontinuousatits1quantilefor 2.3:ATobit-likemodelisaspecialcaseoftheabovesetupifwesupposefurtherthat0,and0and(orsomelargenegativenumberifthereisaplausiblelowerboundon2.4:Ourconstructionofcriticalvalueshasusedsubsampling.Fol-Andrews(2000),itispossibletoshowthatanaivebootstrapconstruc-tionfailstoapproximatethedistributionof()when(P).Itmaystillbethecasethat()issatised,butinsimulationsitseemstobetooconserv-ativeinpractice.Bugni(2007)showedthatasuitablymodiedversionofthebootstrapcanbeusedtoestimatethedistributionof()when(P)HisapproximationdependscruciallyonthestructureofExampleanddoesnotextendeasilytomoregeneralmodels,butitisworthwhiletonotethatitcanbeusedasaningredientinAlgorithm.Specically,wemayreplace(Kwiththe1quantileofhisbootstrapapproximationtothedis-tributionof().ItfollowsfromtheanalysisofBugni(2007)thatthesecriticalvalueswillsatisfy()underweakassumptions.Sincetheyalsosatisfy(),theconclusionsofTheorem2.5:Whenthesetofminimizersofprovidesaconsistentes-timateof(P)Chernozhukov,Hong,andTamer(2007)proposedconstruct-ingcondenceregionsthatsatisfy()usingasingle-stepmethodinwhichargminandcriticalvaluesaregivenby();thatis,Suchanapproachcanbeshownbyexampletofailtoleadtocondencere-gionsthatsatisfy()whenthesetofminimizersofdoesnotprovideaconsistentestimateof(P).SeeExample2.7ofRomanoandShaikh(2006)fordetails. INFERENCEINPARTIALLYIDENTIFIEDMODELS2.6:Whenthesetofminimizersofdoesnotprovideacon-sistentestimateof(P)Chernozhukov,Hong,andTamer(2007)constructingcondenceregionsthatsatisfy()usingasingle-stepmethodin()\fisapositivesequenceofconstantstendingtozeroslowly.Becauseofthisrestrictionontherateatwhichtendstozero,theywereabletoshow(P)ofTheoremrequiresthattheinitialsetbesuchthat(P),butonecanallowfortoberandomprovidedthatitsatises()withoutaffectingtheargumentinanyway.Hence,usingourresults,itfollowsthatthisconstructionsatises().Unfortunately,thespecicchoiceofinnitesam-plesisarbitraryandthecondenceregionresultingfromapplicationoftheirmethodmaythusbeverylargeorverysmalldependingonthechoiceof.Ourresultsprovideajusticationofiteratingtheirprocedureuntilastoppingcrite-rionismet,therebyremovingthisarbitrariness,andproducetypicallysmallercondenceregionswhilestillmaintainingthecoveragerequirement.2.7:ItfollowsfromthediscussioninRemarkthattherearenorst-orderdifferencesbetweenthecondenceregionsfromourstep-downprocedurewithgivenby()andthoseofChernozhukov,Hong,andTamer.Evenwithsuchadelicatechoiceof,weexpecttheiterativeapproachtoperformbetterinnitesamples.Tothisend,itisworthwhiletoexaminesecond-orderdifferences.InSectionofthe,weshowinthecon-textofasimpleexamplethatourcondenceregionissmallertosecondorderthantheoneproposedbyChernozhukov,Hong,andTamer(2007).Intheex-ampleweconsider,itisimportanttonotethatthesetofminimizersofprovidesaconsistentestimateof(P),soonecouldinsteaduseasingle-stepprocedurewithgivenby().Comparedwiththisprocedure,ourcondenceregionisnotsmallertosecondorder.Wesimplyusetheexampletoillustrateaphenomenonthatweexpecttopersistevenwhenthesetofminimizersofdoesnotprovideaconsistentestimateof(P)UniformConsistencyinLevelWenowprovideconditionsunderwhichthesetofvaluescorrespondingtotheacceptedhypothesesfromAlgorithm,satises( INFERENCEINPARTIALLYIDENTIFIEDMODELSuniformlyinand(ii)iscompactwithrespecttothemetric\b(Pj(Pj((gj(X)(X.Toruleoutdegeneratesitu-ations,assumefurtherthat(iii)thereexists0suchthatforeachthereexist1(P)suchthatthatgj(Xi)]=0andVarrgj(Xi)]\f.Assumption(i)isagainknowntoholdprovidedthattheclassoffunctionsisnottoolarge;forgeneralresultstothisendandnumerousapplications,seevanderVaartandWellner(1996).Inthe,weverifythattherequiredcondition()holdsundertheseassumptions.XAMPLE2.4—RegressionWithIntervalOutcomes:RecallthesetupofEx-.Asarguedthere,thestructureofthisexampleissimilartothatof.SinceweprovidedetailsinthecaseofExampleabove,wedonotdosohere.ConÞdenceRegionsforFunctionsoftheIdentiÞedSetInthissection,weconsidertheproblemofconstructingsetsthatsatisfy(and().Letbegiven.Ourconstructionagainreliesonequivalencewithanappropriatemultipletestingproblem,butinthiscasethefamilyofnullhypothesesisgivenby(P)(P)isdenedby().Thealternativehypothesesareunderstoodto(P)Asbefore,itsufcestoconsidertheproblemoftestingthisfamilyofnullhy-pothesesinawaythatcontrolstheFWERatlevelFor,letf().Notethat(P)\rQ(P)Q(P)ThissuggestsanaturalteststatisticforeachofthesenullhypothesesistheteststatisticusedearliertotestthenullhypothesisthatQ(P) J.P.ROMANOANDA.M.SHAIKHWemaynowproceedasbefore,butwiththisteststatisticinplaceofourearlierteststatistic.For,let(Kbeanestimatorofthe1quantileofdistributionofandconsiderthefollowingmodicationofAlgorithm1.Let.Ifsup,thenacceptallandstop;otherwise,set:inf.Ifsup,thenacceptallandstop;otherwise,set:infWenowprovideconditionsunderwhichthesetofvaluescorrespondingtoacceptedhypothesesfromAlgorithmleadstocondenceregionsthatsatisfy()and().For,let(K nbeanisequenceofrandomvariableswithdistributionandletbeasequenceofpositiveintegerstend-ingtoinÞnitybutsatisfyingb/nP)denotethedistributionof(P)denotethesetofvaluescorre-spondingtoacceptedhypothesesfromAlgorithm(KisgivenP)convergesindistributiontoP)andthatP)iscon-tinuousatitssmallestlimsup(xP)(xP) J.P.ROMANOANDA.M.SHAIKHunderthefunctionofinterest.Suchaconstructionwilltypicallybecon-servativeinthesensethatthecoverageprobabilitywillexceedthenominalEMPIRICALILLUSTRATIONInthissection,weusethetechniquesdevelopedabovetoexaminetheim-pactoftop-codingoutcomesontheinferencesthatcanbemadeabouttheparametersofalinearregression.Bytop-codingarandomvariable,wemeanthepracticeofrecordingtherealizationoftherandomvariableifandonlyifitisbelowacertainthreshhold.ThismodelisaspecialcaseofourExampleandsothetheorydevelopedaboveapplieshereundertheappropriateassump-tions.AsimilarempiricalexamplecanbefoundinChernozhukov,Hong,andTamer(2004)Themotivationforourexercisestemsfromthefollowingobservation.Tostudychangesinthewagestructureandearningsinequality,researchersoftenregressthelogarithmofhourlywagesonvariousdemographiccharacteristics.Datasetsusedforthispurposeinvariablytop-codewagesforreasonsofcon-dentiality.Oneapproachtodealwiththetop-codingofwagesistoreplaceallofthetop-codedoutcomeswithacommonvalue.Inpractice,thiscommonvalueisoftentakentobeascalarmultipleofthethreshhold.Thisapproachisjustiedtheoreticallyundertheassumptionthatthedistributionofwagesconditionalontop-codingisdistributedasaParetorandomvariable.See,forKatzandAutor(1999),whereinthescalarusedforthispurposeistakentobe1.5.Ofcourse,wedonotwishtoimposeanyparametricassump-Toexaminethisissue,webeginwithasampleofobservationsfromtheAn-nualDemographicSupplementoftheCurrentPopulationSurveyfortheyear2000.Foreachindividualinthesurvey,thesurveyrecordsavarietyofde-mographicvariablesaswellasinformationonwagesandsalaries.Weselectobservationswiththefollowingdemographiccharacteristics:(1)raceiswhite;(2)ageisbetween20and24years;(3)atleastcollegegraduates;(4)primarysourceofincomeiswagesandsalaries;(5)workedatleast2hoursperweekonaverage.Thereare305suchobservations,noneofwhichsuffersfromtop-codingofwagesandsalaries.Wetreatthissampleofindividualsasthedistrib-utionoftheobserveddataanddrawani.i.d.sampleof1000observationsfromthis.Wewillanalyzethesedatabothforthebenchmarkcaseofnotop-codingandforcasesinwhichsomeamountoftop-codinghasbeenarticiallyimposedonthedata.RecallthesetupofExample.Toallowforgraphicalillustrationofthecondenceregions,weconsideronlyamodelinwhich2;specically,we,whereis1ifthesexisfemaleand0otherwise.Thelatentoutcomevariable,wherewageistotalwagesandsalaries,whichispossiblyunobservedinthepresenceoftop-coding,andistotal INFERENCEINPARTIALLYIDENTIFIEDMODELShoursworked.Weassumethatwageisboundedaboveby .Inthebenchmarkcaseinwhichthereisnotop-coding,wewillletInthecasesinwhichthereissometop-coding,letwage bethethreshholdabovewhichwagesarenotobserved.Deneifwage otherwise,let i=log(wage /Hi)andY2i= Yi=log( Belowwewillconstructcondenceregionsoflevel195fortheiden-tiedsetforeachofthreedifferentscenarios.Forthesakeofcompleteness,wewillalsoconstructcondenceregionsforidentiableparameters,asde-scribedinRomanoandShaikh(2008).Morespecically,followingRomanoandShaikh(2008),weconsiderisgivenby().Wewillcomparetheinferencesthatcanbedrawnfromthesecondenceregionswiththosethatcanbedrawnfrom,where,inthebenchmarkcaseofnotop-coding,and,incaseswithtop-coding,ifwage andYai=15×Y Beforeproceeding,wediscusssomecomputationaldetails.First,considerthechoiceof.Inpractice,onewouldliketouseadata-dependentsubsamplesize;seePolitis,Romano,andWolf(1999)forareviewofseveralalgorithmsforchoosingthesubsamplesizeinthisway.Forthepurposesofthisexercise,however,weusethesamesubsamplesize,30,ineachoftheconstructions.Asaresult,differencesamongthecondenceregionsbelowarenotdrivenbyvariationinthechoiceofsubsamplesize.Notethattheresultsbelowremainsimilarforsubsamplesizesbetween20and40,therangeofsubsamplesizesforwhichthesimulationresultsinthefollowingsectionsuggestthattheprocedurebehaveswellinnitesamples.Second,whencomputingcriticalvalues,wealsousedanapproximationasdescribedinRemark200becausetoolargetocomputecriticalvaluesexactly.Finally,followingthediscussioninRemark,intherststepofAlgorithm,weletTheresultsbelowremainsimilarformuchlargerchoicesofWerstconsiderthecaseinwhichthereisnotop-coding.Algorithmconvergedafter11stepsandthecondenceregionfortheidentiedsetisgivenbyWealsoregressandobtainaWald-stylecondenceregionofthe J.P.ROMANOANDA.M.SHAIKHusethenalspecicationoftheempiricalillustrationinwhich Asintheempiricalillustration,thesamplesizeis1000and05.Fol-lowingthediscussioninRemark,intherststepofAlgorithm,we.Toassessthesensitivityofourproceduretothechoiceofsubsamplesize,weconsidervaluesofFinally,asdescribedinRemark,weapproximatethecriticalvalueswith200.Foreachof100simulations,wecomputethevariables(i),theiterationatwhichAlgorithmconverged,(ii),thecriticalvaluethat,(iii),(iv)sup(P),and(v)(P)InTable,wepresent,foreachvalueof,(i)theaveragenumberofiter-ationsneededforAlgorithmtoconverge,thatis,theaveragevalueofand(ii)thesimulatedprobabilitythattheidentiedsetiscoveredby.Thesimulationresultsshowthattheaveragenumberofiterationsincreaseswiththesubsamplesize,butitistypicallybetween7and10.Thesimulationresultsalsoshowthatthecoverageprobabilitiesareclosetothenominallevel,1forvaluesofrangingfrom20to40.InTable,wepresent,foreachvalueof,(i)themeanof(ii)themeanof,and(iii)thesimulated1quantileof(P).Welabelthethirdcolumn“ideal”becauseitrepresentsthebestpossiblecriticalvalue.Ofcourse,itisinfeasible,sinceitdependson,whichistypicallyunknown.Fortunately,thesimulationresultsshowthatforvaluesofbetween20and40,isclosetothisidealvalue.Thesimulationre-sultsalsoallowforacomparisonwithsingle-stepprocedures.Toseethis,recallTABLEIIMULATIONESULTS CoverageProbabilityAverageNumberofIterations 150.986.99200.947.02250.977.64300.917.98350.988.04400.968.14450.978.87501.008.98550.989.08600.999.20650.999.77700.989.95750.9910.02801.0010.08851.0010.24 J.P.ROMANOANDA.M.SHAIKHisboundedaboveby \f 2 kn4 2log kn\f1\f(29)+32kvn \f 2 kn4 \f kn\fŠ1 \f kn 2log \f isthestandardnormaldistributionWealsohavethatforany1,(isboundedaboveby \f+1 \f8kvnexp\bŠkn2 :For(VPX (P)(VP)Denotebythesymmetricgroupwithelements.Notethatusingthisnota-tion,wemayrewrite (P))(VP)(VP VPX(n)Notefurtherthat(VP VP(n)whichisasumofidenticallydistributedrandomvariables.Let0begiven.Itfollowsthat(VPX VP(n) J.P.ROMANOANDA.M.SHAIKHTechnicalDetailsforExampleNotethatwemaywrite(P)(P)(P) gj(Xi)(P) \f ))gj(Xi)Let00,butsoslowlythat Šand,for,dene(KP)(P))gj(Xi)]\r(KP)(P))gj(Xi)forall(KP)(P))gj(Xi)0forallNotethat(P)(KP)(KP)mandadopttheconventionthatthesumovertheemptysetandthesupremumovertheemptysetarezero.Hence,()canbeboundedfrombelowasm(KP)(P) gj(Xi)m(KP)(P) gj(Xi)m(KP)(P))Ontheotherhand,()canbeboundedfromaboveasm(KP)(P) gj(Xi)m(KP)(P))m(KP)(P) gj(Xi) INFERENCEINPARTIALLYIDENTIFIEDMODELSm(KP)(P))m(KP)(P) gj(Xi)m(KP)(P))Toseetheequality(),notethat(KP)(P) gj(Xi)(P)(P) Byassumption(i),(P)(P)Itthusfollowsfromtheassumptionthat Šthat()tendsinprob-abilitytozero,whichinturnimpliestheequality(Next,wearguethat(KP)(P))(KP)(P))(KP)(KP),theleft-handsideof()isboundedfrombelowbyzero.Itthereforesufcestoshowthat()isboundedfromabovebyzeroinprobability.Tothisend,notethattheleft-handsideof()isboundedfromabovebyP))(KP)(P))where00and(KP)aresuchthat(KP)(P))P)) INFERENCEINPARTIALLYIDENTIFIEDMODELSZ(P).ItthereforefollowsfromTheorem11.1ofDavydov,Lifshits,andSmorodina(1998)thatthedistributionof()iscontinuouseverywhereexceptpossiblyatzero.Next,notethatm(KP)(P))P)) areasinassumption(iii).Hence,()iscontinuousatits1quantileforall TechnicalDetailsforRemarkLet)in,beani.i.d.sequenceofrandomvariableswith.Let(P)denotethemeanoftherstcomponentofthedistributionandlet(P)denotethemeanofthesecondcompo-nentofthedistribution.Theparameterofinterest,,isknowntosatisfy(P)(P).Theidentiedsetisthereforegivenby(P)(P)(P).Thissetmaybecharacterizedasthesetofminimiz-ersofQ(P)(P)(P))ThesampleanalogofQ(P)isgivenbyLetandsupposeissuchthatthat(XiYi)|4]andthatCramer’sconditionholds,thatis,limsup(s)(s)denotesthecharacteristicfunctionof.Assumefurtherthatwithprobability1under.Thisassumptionisnotessential,butitsimpliestheanalysis,whilestillallowingustomakeourcomparison.SeeRemarkbelowforfurtherdiscussion.Let0isanincreasingsequencetendingtoinnity,butsoslowlythatChernozhukov,Hong,andTamer(2007)suggested,forexample,.ConsiderthecondenceregiongivenbyToobtainasecond-orderaccurateexpressionfor,rstnotethat \rn n¯Yn+ \rn n\n(46) J.P.ROMANOANDA.M.SHAIKHm(KP(P m(KP(P.Toseethis,itsufcestoshowthatforanym(KP(P(KP(P Sincetheleft-handsideof()isboundedfrombelowby(KP(P(P itsufcestoshowthat()isboundedfromabovebyzeroinprobability.Tothisend,let(KPbesuchthat(KP(Pfor00.Thus,()isboundedfromaboveby Sincethefunctiondenedbyiscontinuous,itisuniformlycontinuousonacompactset.Hence,forany0andthereexists(M)�0suchthat b\rn)2�\n(52)Pnmax1jm|Zbj(nPn)|�M+Pn{\f (P b\rnŠ} INFERENCEINPARTIALLYIDENTIFIEDMODELSByassumption(i),(PItthereforefollowsfromtheassumptionthat 0that()tendstozeroandthen.Thedesiredclaimthusfollows.Similarly,wehaveforanysequencem(KP(P gj(Xi)m(KP(Pm(KP(P gj(Xi)m(KP(Pm(KP(P gj(Xi).Toseethis,notethat(KP(P gj(Xi)(P Byassumption(i),(PThedesiredclaimthusfollowsfromtheassumptionthat Š.Infact,wehavethatisidenticallyequaltozerowithprobabilitytendingto1.Wenowusethesefactstoarguebycontradictionthattherequiredcondi-tion()holds.Iftheresultwerefalse,thentherewouldexistasubsequenceandacorrespondingsequencesuchthatxPxP INFERENCEINPARTIALLYIDENTIFIEDMODELSAssumption(i)alsoimpliesthatforany0andall1limsupPj((P)P)Itthereforefollowsfromtheassumptionthat0that()tendstozeroasandthen.Byinterchangingtherolesof(KP(K),weseethattheleft-handsideof()isalsoboundedbelowbyzeroinprobability.Thedesiredresultfollows.bothconvergeinprobabilitytozero,wehavebythecontinuousmappingtheoremthat(xP(xPbothconvergeindistributionto.Sinceiscontinuouson,wehaveforevery0thatxPxPToapplyLemma,wethereforeneedonlyshowthattheseconvergencesholdatConsiderrst(xP.Since0,wehavethatm(KPPm(KPPConversely,since,wehaveforany0thatlimsuplimsup(KPPlimsupm(KPPlimsupm(KPP \f