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Forschungsinstitut zur Zukunft der ArbeitInstitute for the Study of Labor Inference on Causal Effects in aGeneralized Regression Kink DesignIZA DP No. 8757David CardDavid S. LeeZhuan PeiAndrea Weber Inference on Causal Effects in aGeneralized Regression Kink DesignDavid CardUC Berkeley, NBER and IZADavid S. LeePrinceton University and NBER IZA Discussion Paper No. January 2015BSTRACTInference on Causal Effects in aGeneralized Regression Kink DesignWe consider nonparametric identification and estimation in a nonseparable model where a continuousregressor of interest is a known, deterministic, but kinked function of an observed assignment variable.This design arises in many institutional settings where a policy variable (such as weekly unemploymentbenefits) is determined by an observed but potentially endogenous assignment variable (like previousearnings). We provide new results on identification and estimation for these settings, and apply our resultsto obtain estimates of the elasticity of joblessness with respect to UI benefit rates. We characterize abroad class of models in which a sharp “Regression Kink Design” (RKD, or RK Design) identifies a readilyinterpretable treatmentthetreated parameter (Florens et al. (2008)). We also introduce a “fuzzyregression kink design” generalization that allows for omitted variables in the assignment rule, noncompliance,and certain types of measurement errors in the observed values of the assignment variable andthe policy variable. Our identifying assumptions give rise to testable restrictions on the distributions ofthe assignment variable and predetermined covariates around the kink point, similar to the restrictionsdelivered by Lee (2008) for the regression discontinuity design. We then use a fuzzy RKD approach tostudy the effect of unemployment insurance benefits on the duration of joblessness in Austria, where thebenefit schedule has kinks at the minimum and maximum benefit level. Our preferred estimates uggestthat changes in UI benefit generosity exert a relatively large effect on the duration of joblessness of bothlowwage and highwage UI recipients in Austria.JEL Classification:C13, C14, C31Keywordsregression discontinuity designregression kink designtreatment effectsnonseparablemodelsnonparametric estimationCorresponding author:Andrea WeberUniversity of MannheimL7, 35, Room 42068131 MannheimGermanymail:a.weber@unimannheim.d We thank Diane Alexander, Mingyu Chen, Kwabena Donkor, Martina Fink, Samsun Knight, Andrew Langan, Carl Lieberman, Michelle Liu, Steve Mello, Rosa Weber and Pauline Leung for excellent research assistance. We have benefited from the comments and suggestions of Sebastian Calonico, Matias Cattaneo, Andrew Chesher, Nathan Grawe, Bo Honoré, Guido Imbens, Pat Kline and seminar participants at Brandeis, BYU, Brookings, Cornell, Georgetown, GWU, IZA, LSE, Michigan, NAESM, NBER, Princeton, Rutgers, SOLE, Upjohn, UC Berkeley, UCL, Uppsala, Western Michigan, Wharton and Zürich. Andrea Weber gratefully acknowledges research funding from the Austrian Science Fund (NRN Labor Economics and the Welfare State). 1IntroductionAgrowingbodyofresearchconsiderstheidenticationandestimationofnonseparablemodelswithcon-tinuousendogenousregressorsinsemiparametric(e.g.,Lewbel(1998);Lewbel(2000))andnon-parametricsettings(e.g.,BlundellandPowell(2003);Chesher(2003);Florensetal.(2008);ImbensandNewey(2009)).Themethodsproposedintheliteraturesofarrelyoninstrumentalvariablesthatareindependentoftheun-observabletermsinthemodel.Unfortunately,independentinstrumentsareoftenhardtond,particularlywhentheregressorofinterestisadeterministicfunctionofanendogenousassignmentvariable.Unemploy-mentbenets,forexample,aresetasfunctionofpreviousearningsinmostcountries.Anyvariablethatiscorrelatedwithbenetsislikelytobecorrelatedwiththeunobserveddeterminantsofpreviouswagesandisthereforeunlikelytosatisfythenecessaryindependenceassumptionsforavalidinstrument.Nevertheless,manytaxandbenetformulasarepiece-wiselinearfunctionswithkinksintherelation-shipbetweentheassignmentvariableandthepolicyvariablecausedbyminimums,maximums,anddiscreteshiftsinthemarginaltaxorbenetrate.AsnotedbyClassen(1977a),Welch(1977),Guryan(2001),Dahlbergetal.(2008),Nielsenetal.(2010)andSimonsenetal.(Forthcoming),akinkedassignmentruleholdsoutthepossibilityforidenticationofthepolicyvariableeffectevenintheabsenceoftraditionalinstruments.Theideaistolookforaninducedkinkinthemappingbetweentheassignmentvariableandtheoutcomevariablethatcoincideswiththekinkinthepolicyrule,andcomparetherelativemagnitudesofthetwokinks.Thispaperestablishesconditionsunderwhichthebehavioralresponsetoaformulaicpolicyvariablelikeunemploymentbenetscanbeidentiedwithinageneralclassofnonparametricandnonseparableregressionmodels.Specically,weestablishconditionsfortheRKDtoidentifythe“localaverageresponse”denedbyAltonjiandMatzkin(2005)orthe“treatment-on-the-treated”parameterdenedbyFlorensetal.(2008).Thekeyassumptionisthatconditionalontheunobservabledeterminantsoftheoutcomevariable,thedensityoftheassignmentvariableissmooth(i.e.,continuouslydifferentiable)atthekinkpointinthepolicyrule.Weshowthatthissmoothdensityconditionrulesoutdeterministicsortingwhileallowinglessextremeformsofendogeneity–including,forexample,situationswhereagentsendogenouslysortbutmakesmalloptimizationerrors(e.g.,Chetty(2012)).Wealsoshowthatthesmoothdensityconditiongeneratestestablepredictionsforthedistributionofpredeterminedcovariatesamongthepopulationofagentslocatednearthekinkpoint.Thus,asinaregressiondiscontinuity(RD)design(LeeandLemieux(2010);DiNardo1 andLee(2011)),thevalidityoftheregressionkinkdesigncanbeevaluatedempirically.Inmanyrealisticsettings,thepolicyruleofinterestdependsonunobservedindividualcharacteristics,orisimplementedwitherror.Inaddition,boththeassignmentvariableandthepolicyvariablemaybeobservedwitherror.WepresentageneralizationoftheRKD–whichwecalla“fuzzyregressionkinkdesign”–thatallowsforthesefeatures.ThefuzzyRKDestimandreplacestheknownchangeinslopeoftheassignmentruleatthekinkwithanestimatebasedontheobserveddata.Underaseriesofadditionalassumptions,includingamonotonicityconditionanalogoustotheoneintroducedbyImbensandAngrist(1994)(andimplicitinlatentindexmodels(Vytlacil,2002)),weshowthatthefuzzyRKDidentiesaweightedaverageofmarginaleffects,wheretheweightsareproportionaltothemagnitudeoftheindividual-specickinks.1WethenreviewandextendexistingmethodsforthenonparametricestimationofRKDusinglocalpoly-nomialestimation,includingFanandGijbels(1996)–hereafter,FG;ImbensandKalyanaraman(2012)–hereafterIK;andCalonicoetal.(Forthcoming)–hereafter,CCT.Andnally,weuseafuzzyRKDapproachtoanalyzetheeffectofunemploymentinsurance(UI)benetsonthedurationofjoblessnessinAustria.AsintheU.S.,theAustrianUIsystemspeciesabenetlevelthatisproportionaltoearningsinabaseperiodpriortojobloss,subjecttoaminimumandmaximum.Westudytheeffectsofthekinksattheminimumandmaximumbenetlevels,usingdataonalargesampleofjoblessspellsfromtheAustrianSocialSecurityDatabase(seeZweimülleretal.(2009)).Simpleplotsofthedatashowrelativelystrongvisualevidenceofkinksintherelationshipbetweenbaseperiodearningsandthedurationsofjoblessnessatbothkinkpoints.Wealsoexaminetherelationshipbetweenbaseperiodearningsandvariouspredeterminedcovariates(suchasgender,age,andoccupation)aroundthekinkpoints,checkingwhethertheconditionaldistributionsofthecovariatesevolvesmoothlyaroundthekinkpoints.Wepresentarangeofalternativeestimatesofthebehavioraleffectofhigherbenetsonthedurationofjoblessnessderivedfromlocallinearandlocalquadraticpolynomialmodelsusingvariousbandwidthselec-tionalgorithms(includingFG,IK,andCCT,andextensionsofIKandCCTforthefuzzyRKDcase).Foreachofthealternativechoicesofpolynomialorderandbandwidthselectorweshowconventionalkinkesti-matesandcorresponding“bias-corrected”estimatesthatincorporatethecorrectionsuggestedbyCalonicoetal.(Forthcoming).Wealsoinvestigatetheempiricalperformanceofthealternativeestimatorsusingsimula-tionstudiesofdatageneratingprocesses(DGP's)thatarecloselybasedonouractualdata.Inourempirical 1Themarginaleffectsofinterestinthispaperrefertoderivativesofanoutcomevariablewithrespecttoacontinuousendogenousregressor,andshouldnotbeconfusedwiththemarginaltreatmenteffectsdenedinHeckmanandVytlacil(2005),wherethetreatmentisbinary.2 setting,wendthatlocalquadraticestimatorshavesubstantiallylarger(asymptotic)meansquarederrorsthanlocallinearestimatorsandthatCCT'sbiascorrectionprocedureleadstoalossinprecisionwithonlyamodestoffsettingreductioninbias.Ourpreferredestimates–derivedfromuncorrectedlocallinearmodelsusingtheFGbandwidthselectionprocedure–implythatchangesinUIbenetgenerosityexertarelativelylargeeffectonthedurationofjoblessnessofUIrecipientsinAustria.2NonparametricRegressionandtheRegressionKinkDesign2.1BackgroundConsiderthegeneralizednonseparablemodelY=y(B;V;U)(1)whereYisanoutcome,Bisacontinuousregressorofinterest,Visanotherobservedcovariate,andUisapotentiallymulti-dimensionalerrortermthatentersthefunctionyinanon-additiveway.ThisisaparticularcaseofthemodelconsideredbyImbensandNewey(2009);therearetwoobservablecovariatesandinterestcentersontheeffectofBonY.AsnotedbyImbensandNewey(2009)thissetupisgeneralenoughtoencompassavarietyoftreatmenteffectmodels.WhenBisbinary,thetreatmenteffectforaparticularindividualisgivenbyY1�Y0=y(1;V;U)�y(0;V;U);whenBiscontinuous,thetreatmenteffectis¶ ¶bY=¶ ¶by(b;V;U).Insettingswithdiscreteoutcomes,Ycouldbedenedasanindividual-specicprobabilityofaparticularoutcome(asinabinaryresponsemodel)orasanindividual-specicexpectedvalue(e.g.anexpectedduration)thatdependsonB,V,andU,wherethestructuralfunctionofinterestistherelationbetweenBandtheprobabilityorexpectedvalue.2ForthecontinuousregressorcaseFlorensetal.(2008)denethe“treatmentonthetreated”(TT)as:TTbjv(b;v)=Z¶y(b;v;u) ¶bdFUjB=b;V=v(u)whereFUjB=b;V=v(u)isthec.d.f.ofUconditionalonB=b;V=v.AsnotedbyFlorensetal.(2008),thisisequivalenttothe“localaverageresponse”(LAR)parameterofAltonjiandMatzkin(2005).TheTT(or 2Inthesecases,onewouldusetheobservedoutcomeYO(adiscreteoutcome,oranobservedduration),andusethefactthattheexpectationofYOandYareequivalentgiventhesameconditioningstatement,inapplyingalloftheidenticationresultsbelow.3 equivalentlytheLAR)givestheaverageeffectofamarginalincreaseinbatsomespecicvalueofthepair(b;v),holdingxedthedistributionoftheunobservables,FUjB=b;V=v(
).Recentstudies,includingFlorensetal.(2008)andImbensandNewey(2009),haveproposedmethodsthatuseaninstrumentalvariableZtoidentifycausalparameterssuchasTTorLAR.Anappropriateinstru-mentZisassumedtoinuenceB,butisalsoassumedtobeindependentofthenon-additiveerrorsinthemodel.Chesher(2003)observesthatsuchindependenceassumptionsmaybe“strongandunpalatable”,andhenceproposestheuseoflocalindependenceofZtoidentifylocaleffects.Asnotedintheintroduction,therearesomeimportantcontextswherenoinstrumentscanplausiblysatisfytheindependenceassumption,eithergloballyorlocally.Forexample,considerthecasewhereYrep-resentstheexpecteddurationofunemploymentforajob-loser,Brepresentsthelevelofunemploymentben-ets,andVrepresentspre-job-lossearnings.Assume(asinmanyinstitutionalsettings)thatunemploymentbenetsarealinearfunctionofpre-job-lossearningsuptosomemaximum:i.e.,B=b(V)=rmin(V;T).ConditionalonVthereisnovariationinthebenetlevel,somodel(1)isnotnonparametricallyidentied.Onecouldtrytogetaroundthisfundamentalnon-identicationbytreatingVasanerrorcomponentcorre-latedwithB.Butinthiscase,anyvariablethatisindependentofVwill,byconstruction,beindependentoftheregressorofinterestB,soitwillnotbepossibletondinstrumentsforB,holdingconstantthepolicyregime.Nevertheless,itmaybepossibletoexploitthekinkinthebenetruletoidentifythecausaleffectofBonY.TheideaisthatifBexertsacausaleffectonY,andthereisakinkinthedeterministicrelationbetweenBandVatv=TthenweshouldexpecttoseeaninducedkinkintherelationshipbetweenYandVatv=T.3UsingthekinkforidenticationisinasimilarspirittotheregressiondiscontinuitydesignofThistleth-waiteandCampbell(1960),buttheRDapproachcannotbedirectlyappliedwhenthebenetformulab()iscontinuous.Thiskink-basedidenticationstrategyhasbeenemployedinafewempiricalstudies.Guryan(2001),forexample,useskinksinstateeducationaidformulasaspartofaninstrumentalvariablesstrategytostudytheeffectofpublicschoolspending.4Dahlbergetal.(2008)usethesameapproachtoestimatetheimpactofintergovernmentalgrantsonlocalspendingandtaxes.Morerecently,Simonsenetal.(Forthcom- 3Withoutlossofgenerality,wenormalizethekinkthresholdTto0intheremainderofourtheoreticalpresentation.4Guryan(2001)describestheidenticationstrategyasfollows:“InthecaseoftheOverburdenAidformula,theregressionincludescontrolsforthevaluationratio,1989per-capitaincome,andthedifferencebetweenthegrossstandardand1993educationexpenditures(thestandardofeffortgap).BecausethesearetheonlyvariablesonwhichOverburdenAidisbased,theexclusionrestrictiononlyrequiresthatthefunctionalformofthedirectrelationshipbetweentestscoresandanyofthesevariablesisnotthesameasthefunctionalformintheOverburdenAidformula.”4 ing)useakinkedrelationshipbetweentotalexpenditureonprescriptiondrugsandtheirmarginalpricetostudythepricesensitivityofdemandforprescriptiondrugs.Nielsenetal.(2010),whointroducetheterm“RegressionKinkDesign”forthisapproach,useakinkedstudentaidschemetoidentifytheeffectofdirectcostsoncollegeenrollment.Nielsenetal.(2010)makeprecisetheassumptionsneededtoidentifythecausaleffectsintheconstant-effect,additivemodelY=tB+g(V)+e;(2)whereB=b(V)isassumedtobeadeterministic(andcontinuous)functionofVwithakinkatV=0.Theyshowthatifg(
)andE[ejV=v]havederivativesthatarecontinuousinvatv=0,thent=limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+b0(v0)�limv0!0�b0(v0):Theexpressionontherighthandsideofthisequation–theRKDestimand–issimplythechangeinslopeoftheconditionalexpectationfunctionE[YjV=v]atthekinkpoint(v=0),dividedbythechangeintheslopeofthedeterministicassignmentfunctionb(
)at0.5AlsorelatedarepapersbyDongandLewbel(2014)andDong(2013),whichderiveidenticationresultsusingkinksinaregressiondiscontinuitysetting.DongandLewbel(2014)showthatthederivativeoftheRDtreatmenteffectwithrespecttotherunningvariable,whichtheauthorscallTED,isnonparametricallyidentied.Underalocalpolicyinvarianceassumption,TEDcanbeinterpretedasthechangeinthetreatmenteffectthatwouldresultfromamarginalchangeintheRDthreshold.MorecloselyrelatedtoourstudyisDong(2013),whichshowsthatidenticationinanRDdesigncanbeachievedintheabsenceofarststagediscontinuity,providedthereisakinkinthetreatmentprobabilityattheRDcutoff.InRemark6below,weprovideanexamplewheresuchakinkcouldbeexpected.Dong(2013)alsoshowsthataslopeandlevelchangeinthetreatmentprobabilitycanbothbeusedtoidentifytheRDtreatmenteffectwithalocalconstanttreatmenteffectrestriction:wediscussananalogouspointintheRKdesigninRemark3.Belowweprovidethefollowingnewidenticationresults.First,weestablishidenticationconditionsfortheRKdesigninthecontextofthegeneralnonseparablemodel(1).Byallowingtheerrortermtoenternonseparably,weareallowingforunrestrictedheterogeneityinthestructuralrelationbetweenthe 5Inanearlierworkingpaperversion,Nielsenetal.(2010)providesimilarconditionsforidenticationforalessrestrictive,additivemodel,Y=g(B;V)+e.5 endogenousregressorandtheoutcome.Asanexampleoftherelevanceofthisgeneralization,considerthecaseofmodelingtheimpactofUIbenetsonunemploymentdurationswithaproportionalhazardsmodel.EvenifUIbenetsenterthehazardfunctionwithaconstantcoefcient,theshapeofthebaselinehazardwillingeneralcausethetruemodelforexpecteddurationstobeincompatiblewiththeconstant-effects,additivespecicationin(2).Theadditionofmultiplicativeunobservableheterogeneity(asinMeyer(1990))tothebaselinehazardposesanevengreaterchallengetothejusticationofparametricspecicationssuchas(2).Thenonseparablemodel(1),however,containstheimpliedmodelfordurationsinMeyer(1990)asaspecialcase,andgoesfurtherbyallowing(amongotherthings)theunobservedheterogeneitytobecorrelatedwithVandB.Havingintroducedunobservedheterogeneityinthestructuralrelation,weshowthattheRKDestimandtidentiesaneffectthatcanbeviewedastheTT(orLAR)parameter.Giventhattheidentiedeffectisanaverageofmarginaleffectsacrossaheterogeneouspopulation,wealsomakeprecisehowtheRKDestimandimplicitlyweightstheseheterogeneousmarginaleffects.Theweightsareintuitiveandcorrespondtotheweightsthatwoulddeterminetheslopeoftheexperimentalresponsefunctioninarandomizedexperiment.Second,wegeneralizetheRKdesigntoallowforthepresenceofunobserveddeterminantsofBandmeasurementerrorsinBandV.Thatis,whilemaintainingthemodelin(1),weallowforthepossibilitythattheobservedvalueforBdeviatesfromtheamountpredictedbytheformulausingV,eitherbecauseofunobservedinputsintheformula,noncompliancebehaviorormeasurementerrorsinVorB.This“fuzzyRKD”generalizationmayhavebroaderapplicabilitythanthe“SharpRKD”.6Finally,weprovidetestableimplicationsforavalidRKdesign.Aswediscussbelow,akeyconditionforidenticationintheRKDisthatthedistributionofVforeachindividualissufcientlysmooth.ThissmoothdensityconditionrulesoutthecasewhereanindividualcanpreciselymanipulateV,butallowsindividualstoexertsomeinuenceoverV.7Weprovidetwoteststhatcanbeusefulinassessingwhetherthiskeyidentifyingassumptionholdsinpractice. 6Thesharp/fuzzydistinctionintheRKDisanalogoustothatfortheRDDesign(seeHahnetal.(2001)).7Lee(2008)requiresasimilaridentifyingconditioninaregressiondiscontinuitydesign.EventhoughthesmoothdensityconditionisnotnecessaryforanRDD,itleadstomanyintuitivetestableimplications,whichtheminimalcontinuityassumptionsinHahnetal.(2001)donot.6 2.2IdenticationofRegressionKinkDesigns2.2.1SharpRKDWebeginbystatingtheidentifyingassumptionsfortheRKDandmakingprecisetheinterpretationoftheresultingcausaleffect.Inparticular,weprovideconditionsunderwhichtheRKDidentiestheTTbjvparameterdenedabove.SharpRKDesign:Let(V;U)beapairofrandomvariables(withVobservableandUunobservable).WhiletherunningvariableVisone-dimensional,theerrortermUneednotbe,andthisunrestricteddi-mensionalityofheterogeneitymakesthenonseparablemodel(1)equivalenttotreatmenteffectsmodelsasmentionedinsubsection2.1.Denotethec.d.f.andp.d.f.ofVconditionalonU=ubyFVjU=u(v)andfVjU=u(v).DeneBb(V),Yy(B;V;U),y1(b;v;u)¶y(b;v;u) ¶bandy2(b;v;u)¶y(b;v;u) ¶v.LetIVbeanarbitrarilysmallclosedintervalaroundthecutoff0andIb(V)fbjb=b(v)forsomev2IVgbetheimageofIVunderthemappingb.Intheremainderofthissection,weusethenotationIS1;:::;SktodenotetheproductspaceIS1:::ISkwheretheSj'sarerandomvariables.Assumption1.(Regularity)(i)ThesupportofUisbounded:itisasubsetofthearbitrarilylargecompactsetIURm.(ii)y(
;
;
)isacontinuousfunctionandispartiallydifferentiablew.r.t.itsrstandsecondarguments.Inaddition,y1(b;v;u)iscontinuousonIb(V);V;U.Assumption2.(SmootheffectofV)y2(b;v;u)iscontinuousonIb(V);V;U.Assumption3.(Firststageandnon-negligiblepopulationatthekink)(i)b(
)isaknownfunction,everywherecontinuousandcontinuouslydifferentiableonIVnf0g,butlimv!0+b0(v)6=limv!0�b0(v).(ii)ThesetAU=fu:fVjU=u(v)�08v2IVghasapositivemeasureunderU:RAUdFU(u)�0.Assumption4.(Smoothdensity)TheconditionaldensityfVjU=u(v)anditspartialderivativew.r.t.v,¶fVjU=u(v) ¶v,arecontinuousonIV;U.Assumption1(i)canberelaxed,butotherregularityconditions,suchasthedominanceofy1byanintegrablefunctionwithrespecttoFU,willbeneededinsteadtoallowfortheinterchangeofdifferentiationandintegrationinprovingProposition1below.Assumption1(ii)statesthatthemarginaleffectofBmustbeacontinuousfunctionoftheobservablesandtheunobservederrorU.Assumption2isconsiderablyweakerthananexclusionrestrictionthatdictatesVnotenterasanargument,becausehereVisallowedtoaffectY,aslongasitsmarginaleffectiscontinuous.InthecontextofUI,forexample,pre-job-lossearningsmayindependentlyaffectunemploymentduration,butAssumption2issatisedaslongastherelationship7 betweenpre-job-lossearningsissmoothacrossthethreshold.Assumption3(i)statesthattheresearcherknowsthefunctionb(v),andthatthereisakinkintherelationshipbetweenBandVatthethresholdV=0.Thecontinuityofb(v)mayappearrestrictiveasitrulesoutthecasewherethelevelofb(v)alsochangesatv=0,butitsnecessitystemsfromtheexibilityofourmodel,whichwediscussinmoredetailinRemark3.Assumption3(ii)statesthatthedensityofVmustbepositivearoundthethresholdforanon-trivialsubpopulation.Assumption4isthekeyidentifyingassumptionforavalidRKdesign.ButwhereascontinuityoffVjU=u(v)invissufcientforidenticationintheRDdesign,itisinsufcientintheRKdesign.Instead,thesufcientconditionisthecontinuityofthepartialderivativeoffVjU=u(v)withrespecttov.Insubsection4.1belowwediscussasimpleequilibriumsearchmodelwhereAssumption4mayormaynothold.Theimportanceofthisassumptionunderscorestheneedtobeabletoempiricallytestitsimplications.Proposition1.InavalidSharpRKD,thatis,whenAssumptions1-4hold:(a)Pr(U6ujV=v)iscontinuouslydifferentiableinvatv=08u2IU.(b)limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+db(v) dvv=v0�limv0!0�db(v) dvv=v0=E[y1(b0;0;U)jV=0]=Ruy1(b0;0;u)fVjU=u(0) fV(0)dFU(u)=TTb0j0whereb0=b(0).Proof:Forpart(a),weapplyBayes'RuleandwritePr(U6ujV=v)=ZAfVjU=u0(v) fV(v)dFU(u0):whereA=fu0:u06ug.ThecontinuousdifferentiabilityofPr(U6ujV=v)invfollowsfromLemma1andLemma2insubsectionA.1oftheSupplementalAppendix.Forpart(b),inthenumeratorlimv0!0+dE[YjV=v] dvv=v0=limv0!0+d dvZy(b(v);v;u)fVjU=u(v) fV(v)dFU(u)v=v0=limv0!0+Z¶ ¶vy(b(v);v;u)fVjU=u(v) fV(v)dFU(u)v=v0=limv0!0+b0(v0)Zy1(b(v0);v0;u)fVjU=u(v0) fV(v0)dFU(u)+limv0!0+Zfy2(b(v0);v0;u)fVjU=u(v0) fV(v0)+y(b(v0);v0;u)¶ ¶vfVjU=u(v0) fV(v0)gdFU(u):(3)Asimilarexpressionisobtainedforlimv0!0�dE[YjV=v] dvv=v0.TheboundedsupportandcontinuityinAssump-tions1-4allowdifferentiatingundertheintegralsignperRoussas(2004)(p.97).Wealsoinvokethe8 dominatedconvergencetheoremallowedbythecontinuityconditionsoveracompactsetinordertoex-changethelimitoperatorandtheintegral.Itimpliesthatthedifferenceinslopesaboveandbelowthekinkthresholdcanbesimpliedto:limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0=(limv!0+b0(v0)�limv!0�b0(v0))Zy1(b(0);0;u)fVjU=u(0) fV(0)dFU(u):Assumption3(i)statesthatthedenominatorlimv0!0+b0(v0)�limv0!0�b0(v0)isnonzero,andhencewehavelimv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+b0(v0)�limv0!0�b0(v0)=E[y1(b(0);0;U)jV=0]=Zy1(b(0);0;u)fVjU=u(0) fV(0)dFU(u);whichcompletestheproof.Part(a)statesthattherateofchangeintheprobabilitydistributionofindividualtypeswithrespecttotheassignmentvariableViscontinuousatV=0.8Thisleadsdirectlytopart(b):asaconsequenceofthesmoothnessintheunderlyingdistributionoftypesaroundthekink,thediscontinuouschangeintheslopeofE[YjV=v]atv=0dividedbythediscontinuouschangeinslopeinb(V)atthekinkpointidentiesTTb0j0.9Remark1.ItistemptingtointerpretTTb0j0asthe“averagemarginaleffectofBforindividualswithV=0”,whichmayseemveryrestrictivebecausethesmoothdensityconditionimpliesthatV=0isameasure-zeroevent.However,part(b)impliesthatTTb0j0isaweightedaverageofmarginaleffectsacrosstheentirepopulation,wheretheweightassignedtoanindividualoftypeUreectstherelativelikelihoodthatheorshehasV=0.InsettingswhereUishighlycorrelatedwithV,TTb0j0isonlyrepresentativeofthetreatmenteffectforagentswithrealizationsofUthatareassociatedwithvaluesofVcloseto0.InsettingswhereVandUareindependent,theweightsfordifferentindividualsareequal,andRKDidentiestheaveragemarginaleffectevaluatedatB=b0andV=0. 8NotealsothatProposition1(a)impliesProposition2(a)inLee(2008),i.e.,thecontinuityofPr(U6ujV=v)atv=0forallu.ThisisaconsequenceofthestrongersmoothnessassumptionwehaveimposedontheconditionaldistributionofVonU.9Technically,theTTandLARparametersdonotconditiononasecondvariableV.Butinthecasewherethereisaone-to-onerelationshipbetweenBandV,thetrivialintegrationoverthe(degenerate)distributionofVconditionalonB=b0willimplythatTTb0j0=TTb0E[y1(b0;V;U)jB=b0],whichisliterallytheTTparameterdiscussedinFlorensetal.(2008)andtheLARdiscussedinAltonjiandMatzkin(2005).Inourapplicationtounemploymentbenets,BandVarenotone-to-one,sincebeyondV=0,Bisatthemaximumbenetlevel.Inthiscase,TTbwillingeneralbediscontinuouswithrespecttobatb0:TTb=(TTbjvbb0RTTb0jvfVjB(vjb0)dvb=b0;andtheRKDestimandidentieslimb"b0TTb.9 Remark2.TheweightsinProposition1arethesameonesthatwouldbeobtainedfromusingarandomizedexperimenttoidentifytheaveragemarginaleffectofB,evaluatedatB=b0;V=0.Thatis,supposethatBwasassignedrandomlysothatfBjV;U(b)=f(b).Insuchanexperiment,theidenticationofanaveragemarginaleffectofbatV=0wouldinvolvetakingthederivativeoftheexperimentalresponsesurfaceE[YjB=b;V=v]withrespecttobforunitswithV=0.Thiswouldyield¶E[YjB=b;V=0] ¶bb=b0=¶Ry(b;0;u)dFUjV=0;B=b(u) ¶bb=b0=¶Ruy(b;0;u)fBjV=0;U=u(b) fBjV=0(b)fVjU=u(0) fV(0)dFU(u) ¶bb=b0=¶Ry(b;0;u)fVjU=u(0) fV(0)dFU(u) ¶bb=b0=Zy1(b0;0;u)fVjU=u(0) fV(0)dFU(u):EventhoughBisrandomizedinthishypotheticalexperiment,Visnot.Intuitively,althoughrandomizationallowsonetoidentifymarginaleffectsofB,itcannotresolvethefactthatunitswithV=0willingeneralhaveaparticulardistributionofU.Ofcourse,theadvantageofthishypotheticalrandomizedexperimentisthatonecouldpotentiallyidentifytheaveragemarginaleffectofBatallvaluesofBandV,andnotjustatB=b0andV=0.Remark3.IntheproofofProposition1,weneedthecontinuityofb(v)toensurethattheleftandrightlimitsofy1(b(v0);v0;u),y2(b(v0);v0;u)andy(b(v0);v0;u)arethesameasv0approaches0.Inthecasewhereboththeslopeandthelevelofb(v)changeatv=0,theRKestimanddoesnotpointidentifyaninterpretabletreatmenteffectinthenonseparablemodel(1).TheRDestimand,however,stillidentiesanaveragetreatmenteffect.InsubsectionA.2oftheSupplementalAppendix,weshow:limv0!0+E[YjV=v0]�limv0!0�E[YjV=v0] limv0!0+b(v0)�limv0!0�b(v0)=E[y1(˜b;0;U)jV=0]where˜bisavaluebetweenlimv0!0�b(v0)andlimv0!0+b(v0).Inthespecialcaseofaconstanttreatmenteffectmodellike(2),theRDandRKdesignbothidentifythesamecausaleffectparameter.Intheabsenceofstrongaprioriknowledgeabouttreatmenteffecthomogeneity,however,itseemsadvisabletouseanRDdesign.10 10Turner(2013)studiestheeffectofthePellGrantprogramintheU.S.Theformulaforthesegrantshasbothadiscontinuityandaslopechangeatthegranteligibilitythreshold.ShearguesthatthestatusofbeingaPellGrantrecipient,D,mayimpactYindependentlyfromthemarginalnancialeffectofBonY(i.e.,Y=y(B;D;V;U)),andshestudiestheidenticationofthetwo10 2.2.2FuzzyRegressionKinkDesignAlthoughmanyimportantpolicyvariablesaresetaccordingtoadeterministicformula,inpracticethereisoftensomeslippagebetweenthetheoreticalvalueofthevariableascomputedbythestatedruleanditsobservedvalue.Thiscanarisewhentheformula–whiledeterministic–dependsonother(unknown)variablesinadditiontotheprimaryassignmentvariable,whenthereisnon-compliancewiththepolicyformula,orwhenmeasurementerrorsarepresentintheavailabledataset.ThismotivatestheextensiontoafuzzyRKD.11Specically,assumenowthatB=b(V;e),wherethepresenceofeintheformulaforBallowsforunobserveddeterminantsofthepolicyformulaandnon-compliantbehavior.ThevectoreispotentiallycorrelatedwithUandthereforealsowiththeoutcomevariableY.Asanillustration,considerthesimplecasewheretheUIbenetformuladependsonwhetherornotaclaimanthasdependents.LetDbeaclaimantwithdependentsandletNbeaclaimantwithnodependents,andletb1(v)andb0(v)bethebenetformulasforDandN,respectively.SupposeDandNbothhavebaseperiodearningsofv0andthattheonlynon-compliantbehaviorallowedisforDtoclaimb0(v0)orforNtoclaimb1(v0).Inthiscase,wehavetwopotentiallyunobservedvariablesthatdeterminetreatment:whetheraclaimanthasdependentsornot,andwhetheraclaimant“correctly”claimsherbenets.Wecanrepresentthesetwovariableswithatwo-dimensionalvectore=(e1;e2).Thebinaryindicatore1isequalto1ifaclaimanttrulyhasdependents,wherease2takesfourvaluesdenotingwhetheraclaimantwithbaseperiodearningsvisan“alwaystaker”(alwaysclaimingb1(v)),a“nevertaker”(alwaysreceivingb0(v)),a“complier”(claimingbe1(v)),ora“deer”(claimingb1�e1(v)).TherepresentationB=b(V;e1;e2)effectivelycapturesthetreatmentassignmentmechanismdescribedinthissimpleexample.Withsuitabledenitionofeitcanalsobeusedtoallowformanyothertypesofdeviationsfromadeterministicrule.ExceptforaboundedsupportassumptionsimilartothatforU,wedonotneedtoimposeanyotherrestrictionsonthedistributionofe.WewilluseFU;etodenotethemeasureinducedbythejointdistributionofUande.WealsoassumethattheobservedvaluesofBandV,BandVrespectively,differfromtheirtruevalues treatmenteffectsinaspecialcasethatrestrictstreatmenteffectheterogeneity.11SeeHahnetal.(2001)foradenitionofthefuzzyregressiondiscontinuitydesign.11 asfollows:VV+UV;BB+UBUVGV
UV0;UBGB
UB0;whereUV0andUB0arecontinuouslydistributed,andthattheirjointdensityconditionalonUandeiscontin-uousandsupportedonanarbitrarilylargecompactrectangleIUV0;UB0R2;GVandGBarebinaryindicatorswhosejointconditionaldistributionisgivenbythefourprobabilitiespij(V;U;e;UV0;UB0)Pr(GV=i;GB=jjV;U;e;UV0;UB0).NotethattheerrorsintheobservedvaluesofVandBareassumedtobemixturesofconventional(continuously-distributed)measurementerrorandapointmassat0.Therandomvariables(V;U;e;UV0;UB0;GV;GB)determine(B;B;V;Y)andweobserve(B;V;Y).Assumption1a.(Regularity)InadditiontotheconditionsinAssumption1,thesupportofeisbounded:itisasubsetofthearbitrarilylargecompactsetIeRk.Assumption3a.(Firststageandnon-negligiblepopulationatthekink)b(v;e)iscontinuousonIV;eandb1(v;e)iscontinuouson(IVnf0g)Ie.Letb+1(e)limv!0+b1(v;e),b�1(e)limv!0�b1(v;e)andAe=fe:fVje=e(0)�0g,thenRAePr[UV=0jV=0;e=e]jb+1(e)�b�1(e)jfVje=e(0)dFe(e)�0.Assumption4a.(Smoothdensity)LetV;UV0;UB0haveawell-denedjointprobabilitydensityfunctionconditionaloneachU=uande=e,fV;UV0;UB0jU=u;e=e(v;uB;uV0).ThedensityfunctionfV;UV0;UB0jU=u;e=e(v;uB;uV0)anditspartialderivativew.r.t.varecontinuousonIV;UV0;UB0;U;e.Assumption5.(Smoothprobabilityofnomeasurementerror)pij(v;u;e;uV0;uB0)anditspartialderiva-tivew.r.t.varecontinuousonIV;U;e;UV0;UB0foralli;j=0;1.Assumption6.(Monotonicity)Eitherb+1(e)b�1(e)foralleorb+1(e)b�1(e)foralle.ExtendingAssumption1,Assumption1aimposestheboundedsupportassumptionforeinordertoallowtheinterchangeofdifferentiationandintegration.Assumption3amodiesAssumption3andforbidsadiscontinuityinb(
;e)atthethreshold.AnalogouslytothesharpcasediscussedinRemark3,intheabsenceofcontinuityinb(
;e)theRKestimanddoesnotidentifyaweightedaverageofthecausaleffectofinterest,y1,buttheRDestimanddoes–seesubsectionA.2oftheSupplementalAppendixfordetails.Assumption3aalsorequiresanon-negligiblesubsetofindividualswhosimultaneouslyhaveanon-trivialrststage,haveUV=0,andhavepositiveprobabilitythatVisinaneighborhoodof0.ItiscriticalthatthereisamasspointinthedistributionofthemeasurementerrorUVat0.Intheabsenceofsuchamasspoint,wewillnotobserveakinkintherst-stagerelationship,andfurtherassumptionsmustbemadeabout12 themeasurementerrortoachieveidentication(asinthecasewiththeRDdesign).Incontrast,thereisnoneedforamasspointinthedistributionofUBat0,butwesimplyallowthepossibilityhere.AsshowninFigure1forourapplicationbelow,themajorityofthedatapoints(B;V)appeartoliepreciselyonthebenetschedule,afeaturethatweinterpretasevidenceofamasspointatzerointhejointdistributionof(UV;UB).Assumption3acanbeformallytestedbytheexistenceofarst-stagekinkinE[BjV=v]asstatedinRemark4below.Assumption4amodiesAssumption4:foreachU=uande=e,thereisajointdensityofVandthemeasurementerrorcomponentsthatiscontinuouslydifferentiableinv.NotethatthisallowsarelativelygeneralmeasurementerrorstructureinthesensethatV;UV0;UB0canbearbitrarilycorrelated.Assumption5statesthatthemasspointprobabilities,whilepotentiallydependentonallothervariables,aresmoothwithrespecttoV.Assumption6statesthatthedirectionofthekinkiseithernon-negativeornon-positivefortheentirepopulation,anditisanalogoustothemonotonicityconditionofImbensandAngrist(1994).Inparticular,Assumption6rulesoutsituationswheresomeindividualsexperienceapositivekinkatV=0,butothersexperienceanegativekinkatV=0.Inourapplicationbelow,whereactualUIbenetsdependonthe(un-observed)numberofdependents,thisconditionissatisedsincethebenetschedulesfordifferentnumbersofdependentsareallparallel.Proposition2.InavalidFuzzyRKDesign,thatis,whenAssumptions1a,2,3a,4a,5and6hold:(a)Pr(U6u;e6ejV=v)iscontinuouslydifferentiableinvatv=08(u;e)2IU;e.(b)limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+dE[BjV=v] dvv=v0�limv0!0�dE[BjV=v] dvv=v0=Ry1(b(0;e);0;u)j(u;e)dFU;e(u;e)wherej(u;e)=Pr[UV=0jV=0;U=u;e=e](b+1(e)�b�1(e))fVjU=u;e=e(0) fV(0) RPr[UV=0jV=0;e=w](b+1(w)�b�1(w))fVje=w(0) fV(0)dFe(w).TheproofisinsubsectionA.1oftheSupplementalAppendix.Remark4.ThefuzzyRKDcontinuestoestimateaweightedaverageofmarginaleffectsofBonY,buttheweightisnowgivenbyj(u;e).Assumption3aand6ensurethatthedenominatorofj(u;e)isnonzero.Theyalsoensureakinkatv=0intherst-stagerelationshipbetweenBandV,asseenfromtheproofofProposition2.Itfollowsthattheexistenceofarst-stagekinkservesasatestofAssumption3aand6.Remark5.Theweightj(u;e)hasthreecomponents.Therstcomponent,fVjU=u;e=e(0) fV(0),isanalogoustotheweightinasharpRKDandreectstherelativelikelihoodthatanindividualoftypeU=u;e=eis13 situatedatthekink(i.e.,hasV=0).Thesecondcomponent,b+1(e)�b�0(e),reectsthesizeofthekinkinthebenetscheduleatV=0foranindividualoftypee.AnalogouslytotheLATEinterpretationofastandardinstrumentalvariablessetting,thefuzzyRKDestimandupweightstypeswithalargerkinkatthethresholdV=0.IndividualswhosebenetscheduleisnotkinkedatV=0donotcontributetotheestimand.AnimportantpotentialdifferencefromastandardLATEsettingisthatnon-compliersmaystillreceivepositiveweightsifthescheduletheyfollowasnon-compliershasakinkatV=0.Finally,thethirdcomponentPr[UV=0jV=0;U=u;e=e]representstheprobabilitythattheassignmentvariableiscorrectlymeasuredatV=0.Again,thishastheintuitiveimplicationthatobservationswithamismeasuredvalueoftheassignmentvariabledonotcontributetothefuzzyRKDestimand.Notethatifpijisconstantacrossindividualsthenthiscomponentoftheweightisjustaconstant.Remark6.SofarwehavefocusedonacontinuoustreatmentvariableB,buttheRKDframeworkmaybeappliedtoestimatethetreatmenteffectofabinaryvariableaswell.Asmentionedabove,Dong(2013)discussestheidenticationofthetreatmenteffectwithinanRDframeworkwherethetreatmentprobabilityconditionalontherunningvariableiscontinuousbutkinked.Undercertainregularityconditions,Dong(2013)showsthattheRKestimandidentiesthetreatmenteffectattheRDcutoffforthegroupofcompliers.Inpracticeitmaybedifculttondpolicieswheretheprobabilityofabinarytreatmentisstatutorilymandatedtohaveakinkinanobservedrunningvariable.Onepossibility,suggestedbyareferee,isthatthekinkedrelationshipbetweentwocontinuousvariablesBandVmayinduceakinkedrelationshipbetweenTandVwhereTisabinarytreatmentvariableofinterest.Inthiscase,wemayapplytheRKdesigntomeasurethetreatmenteffectofT.Tobemorespecic,letY=y(T;V;U)T=1[T�0]whereT=t(B;V;h)B=b(V)iscontinuousinVwithakinkatV=0:Asanexample,Bistheamountofnancialaidavailable,whichisakinkedfunctionofparentalincomeV.TisalatentindexfunctionofB,V,andaone-dimensionalerrortermh.Astudentwillchoosetoattendcollege(T=1)ifT�0.Weareinterestedinestimatingtheaveragereturnstocollegeeducation,anexpectationofy(1;V;U)�y(0;V;U).Assumingthattismonotonicallyincreasinginitsthirdargumentandthatforevery(b;v)2Ib(V)IVthereexistsannsuchthatt(b(v);v;n)=0,wecandeneacontinuously14 differentiablefunction˜h:Ib(V)IV!Rsuchthatt(b;v;˜h(b;v))=0bytheimplicitfunctiontheorem.WeshowinsubsectionA.3oftheSupplementalAppendixthatunderadditionalregularityconditions,wehavethefollowingidenticationresultforthefuzzyRKestimand:limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+dE[TjV=v] dvv=v0�limv0!0�dE[TjV=v] dvv=v0=Zu[y(1;0;u)�y(0;0;u)]fV;hjU=u(0;n0) fV;h(0;n0)dFU(u)(4)wheren0˜h(b0;0)isthethresholdvalueofhwhenV=0suchthatn�n0,T(b0;0;n)=1.Therighthandsideofequation(4)issimilartothatinpart(b)ofProposition1,andtheweightsreecttherelativelikelihoodofV=0andh=n0forastudentoftypeU.CrucialtothepointidenticationresultaboveistheexclusionrestrictionthatBdoesnotenterthefunctionyasanargument,i.e.thattheamountofnancialaiddoesnothaveanindependenteffectonfutureearningsconditionalonparentalincomeandcollegeattendance.Whenthisrestrictionisnotmet,theRKestimandcanbeusedtoboundtheeffectofTonYiftheorycanshedlightonthesignoftheindependenteffectofBonY.ThedetailsareinsubsectionA.3oftheSupplementalAppendix.WecanalsoallowtherelationshipbetweenBandVtobefuzzybywritingB=b(V;e)andintroducingmeasurementerrorinVasabove.SimilartoProposition2,weshowthatthefuzzyRKestimandstillidentiesaweightedaverageoftreatmenteffectundercertainregularityassumptions.TheweightsaresimilartothoseinProposition2,andtheexactexpressionisintheSupplementalAppendix.2.3TestableImplicationsoftheRKDInthissectionweformalizethetestableimplicationsofavalidRKdesign.Specically,weshowthatthekeysmoothnessconditionsgivenbyAssumptions4and4aleadtotwostrongtestablepredictions.TherstpredictionisgivenbythefollowingcorollaryofPropositions1and2:Corollary1.InavalidSharpRKD,fV(v)iscontinuouslydifferentiableinv.InavalidFuzzyRKD,fV(v)iscontinuouslydifferentiableinv.ThekeyidentifyingassumptionofthesharpRKDisthatthedensityofVissufcientlysmoothforeveryindividual.ThissmoothnessconditioncannotbetrueifweobserveeitherakinkoradiscontinuityinthedensityofV.Thatis,evidencethatthereis“deterministicsorting”inVatthekinkpointimpliesaviolationofthekeyidentifyingsharpRKDassumption.ThisisanalogoustothetestofmanipulationoftheassignmentvariableforRDdesigns,discussedinMcCrary(2008).InafuzzyRKD,bothAssumption4a,the15 smooth-densitycondition,andAssumption5,thesmooth-probability-of-no-measurement-errorcondition,areneededtoensurethesmoothnessoffV(seetheproofofLemma5),andakinkoradiscontinuityinfVindicatesthateitherorbothoftheassumptionsareviolated.Thesecondpredictionpresumestheexistenceofdataon“baselinecharacteristics”–analogoustochar-acteristicsmeasuredpriortotreatmentassignmentinanidealizedrandomizedcontrolledtrial–thataredeterminedpriortoV.Assumption8.Thereexistsanobservablerandomvector,X=x(U)inthesharpdesignandX=x(U;e)inthefuzzydesign,thatisdeterminedpriortoV.XdoesnotincludeVorB,sinceitisdeterminedpriortothosevariables.Inconjunctionwithourbasicidentifyingassumptions,thisleadstothefollowingprediction:Corollary2.InavalidSharpRKD,ifAssumption8holds,thendPr[XxjV=v] dviscontinuousinvatv=0forallx.InavalidFuzzyRKD,ifAssumption8holds,thendPr[XxjV=v] dviscontinuousinvatv=0forallx.ThesmoothnessconditionsrequiredforavalidRKDimplythattheconditionaldistributionfunctionofanypredeterminedcovariatesX(givenVorV)cannotexhibitakinkatV=0orV=0.Therefore,Corollary2canbeusedtotestAssumption4inasharpdesignandAssumption4aand5jointlyinafuzzydesign.Thistestisanalogoustothesimple“testforrandomassignment”thatisoftenconductedinarandomizedtrial,basedoncomparisonsofthebaselinecovariatesinthetreatmentandcontrolgroups.ItalsoparallelsthetestforcontinuityofPr[XxjV=v]emphasizedbyLee(2008)foraregressiondiscontinuitydesign.Importantly,however,theassumptionsforavalidRKDimplythatthederivativesoftheconditionalexpectationfunctions(ortheconditionalquantiles)ofXwithrespecttoV(orV)arecontinuousatthekinkpoint–astrongerimplicationthanthecontinuityimpliedbythesufcientconditionsforavalidRDD.3NonparametricEstimationandInferenceinaRegressionKinkDesignInthissection,wereviewthetheoryofestimationandinferenceinaregressionkinkdesign.Weassumethatestimationiscarriedoutvialocalpolynomialregressions.ForasharpRKdesign,therststagerelationshipb(
)isaknownfunction,andweonlyneedtosolvethefollowingleastsquaresproblemsminf˜b�jgn�Ã¥i=1fY�i�pÃ¥j=0˜b�j(V�i)jg2K(V�i h)(5)16 minf˜b+jgn+Ã¥i=1fY+i�pÃ¥j=0˜b+j(V+i)jg2K(V+i h)(6)wherethe�and+superscriptsdenotequantitiesintheregressionontheleftandrightsideofthekinkpointrespectively,pistheorderofthepolynomial,Kthekernel,andhthebandwidth.Sincek+1=limv!0+b0(v)andk�1=limv!0�b0(v)areknownquantitiesinasharpdesign,thesharpRKDestimatorisdenedasˆtSRKD=ˆb+1�ˆb�1 k+1�k�1:InafuzzyRKD,therststagerelationshipisnolongerdeterministic.Weneedtoestimatetherst-stageslopesontwosidesofthethresholdbysolving12min˜k�jn�Ã¥i=1fB�i�pÃ¥j=0˜k�j(V�i)jg2K(V�i h)(7)min˜k+jn+Ã¥i=1fB+i�pÃ¥j=0˜k+j(V+i)jg2K(V+i h):(8)ThefuzzyRKDestimatorˆtFRKDcanthenbedenedasˆtFRKD=ˆb+1�ˆb�1 ˆk+1�ˆk�1:(9)LemmaA1andA2ofCalonicoetal.(Forthcoming)establishtheasymptoticdistributionsofthesharpandfuzzyRKDestimators,respectively.Itisshownthatundercertainregularityconditionstheestimatorsobtainedfromlocalpolynomialregressionsoforderpareasymptoticallynormal:p nh3(ˆtSRKD;p�tSRKD�hprSRKD;p))N(0;WSRKD;p)p nh3(ˆtFRKD;p�tFRKD�hprFRKD;p))N(0;WFRKD;p)whererandWdenotetheasymptoticbiasandvariancerespectively.13Giventheidenticationassumptionsabove,oneexpectstheconditionalexpectationofYgivenVtobecontinuousatthethreshold.Anaturalquestioniswhetherimposingcontinuityinestimation(asopposedtoestimatingseparatelocalpolynomials 12WeomittheasteriskinBandVnotationsinthefuzzydesigntoeaseexposition.13Incategorizingtheasymptoticbehavioroffuzzyestimators,bothCardetal.(2012)andCalonicoetal.(Forthcoming)assumethattheresearcherobservesthejointdistribution(Y;B;V).Inpractice,theremaybeapplicationswhere(B;V)isobservedinonedatasourcewhereas(Y;V)isobservedinanother,andthethreevariablesdonotappearinthesamedataset.Weinvestigatethetwo-sampleestimationprobleminsubsectionB.1oftheSupplementalAppendix.17 oneithersideofthethreshold)mayaffecttheasymptoticbiasandvarianceofthekinkestimator.Cardetal.(2012)showsthatwhenKisuniformtheasymptoticvariancesarenotaffectedbyimposingcontinuity.Asimilarcalculationrevealsthattheasymptoticbiasesarenotaffectedeither.WhenimplementingtheRKDestimatorinpractice,onemustmakechoicesforthepolynomialorderp,kernelKandbandwidthh.IntheRDcontextwherethequantitiesofinterestaretheintercepttermsontwosidesofthethreshold,Hahnetal.(2001)proposelocallinear(p=1)overlocalconstant(p=0)regressionbecausetheformerleadstoasmallerorderofbias(Op(h2))thanthelatter(Op(h)).Consequently,thelocallinearmodelaffordstheeconometricianasequenceofbandwidthsthatshrinksataslowerrate,whichinturndeliversasmallerorderoftheasymptoticmean-squarederror(MSE).Thesamelogicwouldimplythatalocalquadratic(p=2)shouldbepreferredtolocallinear(p=1)inestimatingboundaryderivativesintheRKdesign.AsweargueinCardetal.(2014),however,argumentsbasedsolelyonasymptoticratescannotjustifyp=1astheuniversallypreferredchoiceforRDDorp=2astheuniversallypreferredchoiceforRKD.Rather,thebestchoiceofpinthemeansquarederrorsensedependsonthesamplesizeandthederivativesoftheconditionalexpectationfunctions,E[YjV=v]andE[BjV=v],intheparticulardatasetofinterest.InCardetal.(2014),weproposetwomethodsforpickingthepolynomialorderforinterestedempiricists:1.evaluatetheempiricalperformanceofthealternativeestimatorsusingsimulationstudiesofDGP'scloselybasedontheactualdata;2.estimatetheasymptoticmeansquarederror(AMSE)andcompareitacrossalternativeestimators.Usingthesemethodsweargueinsection4belowthatthelocallinearestimatorisamoresensiblechoicethanthelocalquadraticfortheAustrianUIdatawestudy.ForthechoiceofKweadoptauniformkernelfollowingImbensandLemieux(2008)andthecommonpracticeintheRDliterature.Theresultsaresimilarwhentheboundaryoptimaltriangularkernel(c.f.Chengetal.(1997))isused.Forthebandwidthchoiceh,weuseandextendexistingselectorsintheliterature.ImbensandKalya-naraman(2012)proposeanalgorithmtocomputetheMSE-optimalRDbandwidth.BuildingonImbensandKalyanaraman(2012),Calonicoetal.(Forthcoming)developanoptimalbandwidthalgorithmfortheestimationofthediscontinuityinthen-thderivative,whichcontainsRKD(n=1)asaspecialcase.14WeexaminealternativestothedirectanalogsofthedefaultIKandtheCCTbandwidthsforRKD,addressingtwospecicissuesthatarerelevantforoursetting.First,bothbandwidthselectorsinvolvea 14TheoptimalbandwidthinCalonicoetal.(Forthcoming)isdevelopedfortheunconstrainedRKDestimator,i.e.withoutimposingcontinuityintheconditionalexpectationofY,butthebandwidthisalsooptimalfortheconstrainedRKDestimatorbecauseithasthesameasymptoticdistributionasstatedabove.18 regularizationtermwhichreectsthevarianceinthebiasestimationandguardsagainstlargebandwidths.WhileIKandCCTarguethattheregularizedbandwidthselectorperformswellforseveralwell-knownregressiondiscontinuitydesigns,wendthattheRKcounterpartsoftheseregularizedselectorsyieldband-widthsthattendtobetoosmallinourempiricalsetting.Sinceomittingtheregularizationtermdoesnotaffecttheasymptoticpropertiesofthebandwidthselector,wealsoinvestigatetheperformanceofIKandCCTbandwidthselectorswithouttheregularizationterm.Second,theCCTbandwidthisasymptoticallyMSE-optimalforthereduced-formkinkinafuzzydesign,eventhoughthefuzzyestimatorˆtFRKDdenedin(9)isthemainobjectofinterest.BasedontheasymptotictheoryinCalonicoetal.(Forthcoming),weproposefuzzyanalogsoftheIKandCCTbandwidthsthatareoptimalforˆtFRKDandderivetheirasymptoticproperties–seethesubsectionB.2ofSupplementalAppendixfordetails.Acomplicationofusingtheoptimalbandwidthisthattheasymptoticbiasisingeneralnonzero.Asaresult,conventionalcondenceintervalsthatignorethebiasmaynothavecorrectcoveragerates.Calonicoetal.(Forthcoming)offerasolutionbyderivingrobustcondenceintervalsfortheRDandRKestimandsthataccountforthisasymptoticbias.ForanRKdesign,theyrstestimatetheasymptoticbiasrpofap-thorderlocalpolynomialestimatorˆtpbyusingaq-thorderlocalpolynomialregression(q�p+1)withpilotbandwidthhq,thenestimatethevariancevarbcpofthebias-correctedestimatorˆtbcpˆtp�hpˆrp,byaccountingforthesamplingvariationinbothˆtpandhpˆrp.15Finally,theyconstructarobust95%condenceintervalas:ˆtbcp1:96q varbcp.UsingMonteCarlosimulations,Calonicoetal.(Forthcoming)demonstratethatthecondenceintervalsconstructedusingtheirbias-correctedprocedureperformwellinRDD's,andthattheassociatedcoverageratesarerobusttodifferentchoicesofh.16Inthefollowingsection,wepresentavarietyofalternativeestimatesofthebehavioraleffectofhigherbenetsonthedurationofjoblessness.Weinvestigatetheperformanceofthesealternativeestimatorsus-ingsimulatedDGP'sthatarecloselybasedonouractualdata.Thecandidatesincludelocallinearandlocalquadraticestimatorswithseveralbandwidthselectors–defaultCCT,CCTwithoutregularization,FuzzyCCT,FuzzyIKandtheFGbandwidth.17WereportuncorrectedRKDestimatesandtheassociated 15Acrucialassumptioninestimatingvarbcpisthatthepilotbandwidthhqandtheoptimalbandwidthhhavethesameshrinkagerate,i.e.hq h!r2(0;Â¥)asn!Â¥.16Inarelatedstudy,GanongandJäger(2014)raiseconcernsaboutthesensitivityoftheRKDestimateswhentherelationshipbetweentherunningvariableandtheoutcomeishighlynonlinear.Theyproposeapermutationtesttoaccountfortheestimationbias.Weperformthetestonourdata,anddiscussthedetailsinsubsectionB.3oftheSupplementalAppendix.17SeeCardetal.(2012)forthedenitionoftheFGbandwidth.WeapplythesamelogictoderivethepilotFGbandwidthforbiasestimation.19 (conventional)samplingerrorsassociatedwitheachpolynomialorderandbandwidthchoice,aswellasbias-correctedestimatesandtheassociatedrobustcondenceintervalssuggestedbyCalonicoetal.(Forth-coming).184TheEffectofUIBenetsontheDurationofJoblessnessInthissection,weuseafuzzyRKDapproachtoestimatetheeffectofhigherunemploymentbenetsonthedurationofjoblessnessamongUIclaimantsinAustria.TheprecisemagnitudeofthedisincentiveeffectofUIbenetsisofsubstantialpolicyinterest.AsshownbyBaily(1978),forexample,anoptimalunemploy-mentinsurancesystemtradesoffthemoralhazardcostsofreducedsearcheffortagainsttherisk-sharingbenetsofmoregenerouspaymentstotheunemployed.19Obtainingcredibleestimatesofthiseffectisdif-cult,however,becauseUIbenetsaredeterminedbypreviousearnings,andarelikelytobecorrelatedwithunobservedcharacteristicsofworkersthataffectbothwagesandtheexpecteddurationofunemployment.SincetheUIbenetformulainAustriahasbothaminimumandmaximum,aregressionkinkapproachcanprovidenewevidenceontheimpactofhigherUIbenetsattwodifferentpointsinthebenetschedule.Webeginwithabriefdiscussionofajobsearchmodelthatweusetoframeouranalysis.WethendescribethebenetsysteminAustria,ourdatasources,andourmainresults.4.1TheoreticalbackgroundInastandardsearchmodel,higherUIbenetsreducetheincentivesforsearchandraisethereservationwage,leadingtoincreasesintheexpecteddurationofjoblessness.Higherbenetscanalsoaffecttheequilibriumdistributionofwages.Christensenetal.(2005),forexample,derivetheequilibriumdistributionofwages,givenaxedUIbenetandalatentdistributionofwageoffers.Intheirmodel,atwicecontinuouslydifferentiabledistributionfunctionforwageoffersensuresthatdistributionofwagesamongnewlylaid-offworkersistwicecontinuouslydifferentiable.InsectionCoftheSupplementalAppendixweextendthismodeltoincorporateaUIbenetschedulethatislinearinthepreviouswageuptosomemaximumTmax.InthiscasethevaluefunctionofunemployedworkersisincreasingintheirpreviouswagewithaakinkatTmax.Likewise,thevaluefunctionassociatedwithajobpayingawagewhasakinkatw=Tmax;reecting 18EventhoughˆtbcpisnotconsistentundertheCCTasymptoticassumptionsregardingtheshrinkagerateofhqandh,itmaystillbeinformativetoreportitsvalueandshedlightonthedirectionandmagnitudeoftheestimatedbias.19TheoriginalanalysisinBaily(1978)hasbeengeneralizedtoallowforliquidityconstraints(Chetty(2010))andvariabletakeup(Kroft(2008)).20 thekinkintheoptionvalueofUIbenetswhenthejobends.Thiskinkcausesakinkintherelationshipbetweenwagesandon-the-jobsearcheffortwhichleadstoakinkinthedensityofwagesatTmax(seetheSupplementalAppendixfordetails).Assumingaconstantrateofjobdestructionthereisasimilarkinkinthedensityofpreviouswagesamongjob-losers.Suchakink–atpreciselythethresholdforthemaximumbenetrate–violatesthesmoothdensitycondition(i.e.,Assumptions4/4a)necessaryforavalidregressionkinkdesignbasedonthechangeintheslopeofthebenetfunctionatTmax.20Nevertheless,wealsoshowthatifworkershavesomeuncertaintyaboutthelocationofthekinkinfutureUIbenets,theequilibriumdensityofpreviouswagesamongjobloserswillbesmoothatTmax.Giventhesetheoreticalpossibilities,itisimportanttoexaminetheactualdistributionofpre-displacementwagesamongjobseekersandtestforthepresenceofkinksaroundtheminimumandmaximumbenetthresholds,aswellasforkinksintheconditionaldistributionsofpredeterminedcovariates.Whilewedonotnecessarilyexpecttondkinksinoursetting(giventhedifcultyofforecastingfuturebenetschedulesinAustria),akinkcouldexistinothersettingswhereminimumormaximumUIbenetsareoftenxedforseveralyearsatthesamenominalvalue.4.2TheUnemploymentInsuranceSysteminAustriaJob-losersinAustriawhohaveworkedatleast52weeksinthepast24monthsareeligibleforUIbenets,witharatethatdependsontheiraveragedailyearningsinthe“baseyear”fortheirbenetclaim,whichiseitherthepreviouscalendaryear,orthesecondmostrecentyear.ThedailyUIbenetiscalculatedas55%ofnetdailyearnings,subjecttoamaximumbenetlevelthatisadjustedeachyear.Claimantswithdependentfamilymembersareeligibleforsupplementalbenetsbasedonthenumberofdependents.Thereisalsoaminimumbenetlevelforlower-wageclaimants,subjecttotheprovisothattotalbenetscannotexceed60%(forasingleindividual)or80%(foraclaimantwithdependents)ofbaseyearnetearnings.TheserulescreateapiecewiselinearrelationshipbetweenbaseyearearningsandUIbenetsthatde-pendsontheSocialSecurityandincometaxratesaswellasthereplacementrateandtheminimumandmaximumbenetamounts.Toillustrate,Figure1plotsactualdailyUIbenetsagainstannualbaseyearearningsforasampleofUIclaimantsin2004.ThehighfractionofclaimantswhoseobservedUIbenets 20Inthemodelaswrittenallworkersareidentical:henceakinkinthedensityofwagesatTmaxwillnotactuallyinvalidateanRKdesign.Morerealistically,however,workersdifferintheircostofsearch(andinotherdimensions)andthekinkatTmaxislargerforsometypesthanothers,causingadiscontinuityintheconditionaldistributionofunobservedheterogeneityatTmaxthatleadstobiasinanRKD.21 areexactlyequaltotheamountpredictedbytheformulaleadstoaseriesofclearlydiscerniblelinesinthegure,thoughtherearealsomanyobservationsscatteredaboveandbelowtheselines.21Specically,inthemiddleofthegurethereare5distinctupward-slopinglinearsegments,correspondingtoclaimantswith0,1,2,3,or4dependents.Theseschedulesallreachanupperkinkpointatthemaximumbenetthreshold(whichisshowninthegraphbyasolidverticalline).Atthelowerend,thesituationismorecomplicated:eachoftheupward-slopingsegmentsreachestheminimumdailybenetatadifferentlevelofearnings,reectingthefactthatthebasicbenetincludesfamilyallowances,buttheminimumdoesnot.Finally,amongthelowest-paidclaimantsthebenetschedulebecomesupward-slopingagain,withtwomajorlinesrepresentingsingleclaimants(whosebenetis60%oftheirbaseearnings,netoftaxes)andthosewithdependents(whosebenetsare80%oftheirnetbaseyearearnings).22OurRKDanalysisexploitsthekinksinducedbytheminimumandmaximumbenetlevels.Sincewedonotobservethenumberofdependentsclaimedbyajobloser,weadoptafuzzyRKDapproachinwhichthenumberofdependentsistreatedasanunobserveddeterminantofbenets.Thisdoesnotaffectthelocationofthe“topkink”associatedwiththemaximumbenet,sinceclaimantswithdifferentnumbersofdependentsallhavethesamethresholdearningslevelTmaxforreachingthemaximum.Forthe“bottomkink”associatedwiththeminimumbenet,wedeneTminasthekinkpointforasingleclaimant:thisisthelevelofannualearningsshowninthegurebyasolidverticalline.TotherightofTminthebenetschedulesforallclaimantgroupsareupward-sloping.Totheleft,benetsforclaimantswithnodependentsareconstant,whereasbenetsforclaimantswithdependentscontinuetofall.ThusweexpecttomeasureakinkintheaveragebenetfunctionatTminthatisproportionaltothefractionofclaimantswithnodependents.Welimitouranalysistoclaimantswhoseearningsarehighenoughtoavoidthe“subminimum”portionofthebenetschedule:thiscutoffisshownbythedashedlineontheleftsideofFigure1.WealsofocusonclaimantswhoseannualearningsarebelowtheSocialSecuritycontributioncap,sinceearningsabovethislevelarecensored.ThiscutoffisshownbythedashedlineontherightsideofFigure1. 21Theseareattributabletosomecombinationoferrorsinthecalculationofbaseyearearnings(duetoerrorsinthecalculationoftheclaimstartdate,forexample),errorsintheSocialSecurityearningsrecordsthatareover-riddenbybenetadministrators,andmis-reportedUIbenets.Similarerrorshavebeenfoundinmanyothersettings–e.g.,KapteynandYpma(2007).22Thelineforlow-earningsingleclaimantsactuallybends,reectingtheearningsthresholdatwhichasingleclaimantbeginspayingincometaxes.22 4.3DataandAnalysisSampleOurdataaredrawnfromtheAustrianSocialSecurityDatabase(ASSD),whichrecordsemploymentandunemploymentspellsonadailybasisforallindividualsemployedintheAustrianprivatesector(seeZweimülleretal.(2009)).TheASSDcontainsinformationonstartingandendingdatesofspellsandearnings(uptotheSocialSecuritycontributioncap)receivedbyeachindividualfromeachemployerinacalendaryear.WemergetheASSDwithUIclaimsrecordsthatincludetheclaimdate,thedailyUIbenetactuallyreceivedbyeachclaimant,andthedurationofthebenetspell.WeusetheUIclaimdatestoassignthebasecalendaryearforeachclaim,andthencalculatebaseyearearningsforeachclaim,whichistheobservedassignmentvariableforourRKDanalysis(i.e.,Vinthenotationofsection2).Inaddition,weobservetheclaimant'sage,gender,education,maritalstatus,jobtenure,andindustry.Ourmainoutcomevariableisthetimebetweentheendoftheoldjobandthestartofanynewjob(whichwecensorat1year).Ouranalysissampleincludesclaimantsfrom2001-2012withatleastoneyearoftenureontheirpreviousjobwhoinitiatedtheirclaimwithinfourweeksofthejobendingdate(eliminatingjob-quitters,whofaceafour-weekwaitingperiod).Wedroppeoplewithzeroearningsinthebaseyear,claimantsolderthan50,andthosewhoseearningsareabovetheSocialSecurityearningscaporsolowthattheyfallonthe“subminimum”portionofthebenetschedule.Wepoolobservationsfromdifferentyearsasfollows.First,wedividetheclaimantsineachyearintotwo(roughly)equalgroupsbasedontheirgrossbaseyearearnings:thosebelowthe50thpercentileareassignedtothe“bottomkink”sample,whilethoseabovethisthresholdareassignedtothe“topkink”sample.Sinceearningshavearight-skeweddistribution,thecutoffthresholdisclosertoTminthanTmax,implyinganarrowersupportforourobservedassignmentvariableV(observedannualbaseyearearnings)aroundthebottomkinkthanthetopkink.Nextwere-centerbaseyearearningsforobservationsinthebottomkinksubsamplearoundTmin,andbaseyearearningsforthoseinthetopkinksubsamplearoundTmax,sobothkinksoccuratV=0.Finally,wepooltheyearlyre-centeredsubsamplesintobottomandtopkinksamples,yieldingabout275,000observationsineachsample.Table1reportsbasicsummarystatisticsforthebottomandtopkinksamples.Meanbaseyearearningsforthebottomkinkgroupareabout C22,000,witharelativelynarrowrangeofvariation(standarddeviation= C2,800),whilemeanearningsinthetopkinkgrouparehigher(mean= C34,000)andmoredispersed(standarddeviation= C6,700).MeandailyUIbenetsare C25.2forthebottomkinkgroup(implyinganannualizedbenetof C9,200,about44%ofTmin),whilemeanbenetsforthetopkinksampleare C33.523 (implyinganannualizedbenetof C12,300,about28%ofTmax).Claimantsinthebottomkinksamplearemorelikelytobefemale,arealittleyounger,lesslikelytobemarried,morelikelytohavehadablue-collaroccupation,andarelesslikelytohavepost-secondaryeducation.Despitethedifferencesindemographiccharacteristicsandmeanpay,themeansofthemainoutcomevariablearequitesimilarinthetwosamples:theaveragedurationofjoblessnessisaround150days.Onlyabout10percentofclaimantsexhausttheirregularUIbenets.AkeyassumptionforvalidinferenceinanRKdesignisthatthedensityoftheassignmentvariable(inourcase,baseyearearnings)issmoothatthekinkpoint.Figures2aand2bshowthefrequencydistributionsofbaseyearearningsinourtwosubsamples,using100-Eurobinsforthebottomkinksampleand300-Eurobinsforthetopkinksample(eachwithabout4,200observationsperbin).Whilethehistogramslookquitesmooth,wetestedthismoreformallybyttingaseriesofpolynomialmodelsthatallowtherstandhigher-orderderivativesofthebinneddensityfunctiontojumpatthekinkpoint.23Wetestforakinkbytestingforajumpinthelineartermofthepolynomialatthekink.AppendixTable1showsthegoodnessoftandAkaikemodelselectionstatisticsforpolynomialmodelsoforder2,3,4,or5,aswellastheestimatedkinksatTminandTmax:WeshowthettedvaluesfromthemodelswiththelowestAkaikecriterion–a3rdordermodelforthebottomkinksampleanda4thordermodelforthetopkinksample–inFigures2aand2b.Inbothcasesthetteddensitiesappeartobequitesmooth.4.4GraphicalOverviewoftheEffectofKinksintheUIBenetScheduleAsastartingpointforourRKDanalysis,Figures3and4showtherelationshipsbetweenbaseyearearningsandactualUIbenetsaroundthebottomandtopkinks.WeplotthedatausingthesamebinsizesasinFigures2aand2b.24TheFiguresshowclearkinksintheempiricalrelationshipbetweenaveragebenetsandbaseyearearnings,withasharpincreaseinslopeasearningspassthroughthelowerthresholdTminandasharpdecreaseastheypassthroughtheupperthresholdTmax.25Figures5and6presentparallelguresforthemeanlogtimetothenextjob.Theseguresalsoshowdiscerniblekinks,thoughthereisclearlymorevariabilityintherelationshipwithbaseyearearnings. 23Weuseaminimumchi-squaredobjective,whichLindsayandQu(2003)showcanbeinterpretedasaoptimallyweightedminimumdistanceobjectiveforthemultinomialdistributionofhistogramfrequencies.24SeeCalonicoetal.(2014a)fornonparametricproceduresforpickingthebinsizeinRD-typeplots.25TheslopesinthemeanbenetfunctionstotheleftofTminandtotherightofTmaxaremainlyattributabletofamilyallowances.MovingleftfromTmintheaveragenumberofdependentallowancesisfalling,asclaimantswithsuccessivelyhighernumbersofdependentshittheminimumbenetlevel(seeFigure1).Likewise,movingrightfromTmaxtheaveragenumberofallowancesisrising,reectingapositivecorrelationbetweenearningsandfamilysize.24 GiventherelativelyshortdurationofUIbenetsinoursample(20-39weeks),itisalsointerestingtolookattheprobabilityaclaimantexhaustsbenets.AppendixFigure1ashowsthisprobabilityaroundthebottomkink.ThediscreteincreaseintheslopewithrespecttobaseyearearningssuggeststhathigherUIbenetsincreasetheprobabilityofexhaustion.AppendixFigure1bpresentsaparallelgrapharoundthetopkink.Theexhaustionprobabilityexhibitsakinkintheexpecteddirection,thoughaswithourmainoutcomevariable,theprobabilitiesarerelativelynoisyintherangeofearningsjustaboveTmax.Finally,weexaminethepatternsofthepredeterminedcovariatesaroundTminandTmax:AppendixFig-ures2and3showtheconditionalmeansoffourmaincovariatesaroundthetwokinkpoints:age,gender,blue-collaroccupation,andanindicatorforwhethertheclaimanthadbeenrecalledtothepreviousjob.26Thegraphsshowsomeevidenceofnon-smoothnessintheconditionalmeansofthecovariatesinthebottomkinksample,particularlyforclaimantage.Toincreasethepowerofthisanalysisweconstructeda“covari-ateindex”–thepredicteddurationofjoblessnessfromasimplelinearregressionmodelrelatingthelogoftimetonextjobtoatotalof59predeterminedcovariates,includinggender,occupation,age,previousjobtenure,quintileofthepreviousdailywage,industry,region,yearoftheclaim,previousrmsize,andtherecallratesofthepreviousemployer.27Thisestimatedcovariateindexfunctioncanbeinterpretedasthebestlinearpredictionofmeanlogtimetonextjobgiventhevectorofpredeterminedvariables.Figures7and8plotthemeanvaluesoftheestimatedcovariateindicesaroundthetopandbottomkinks.Visually,thepredictedtimetonextjobappearstoevolverelativelysmoothlythroughboththetopandbottomkinks.Inthenextsubsection,weprovideamoreformalcomparisonoftheestimatedslopesoftheconditionalmeanfunctionsforthecovariateindices.4.5RKDEstimationResults4.5.1ReducedFormKinksinAssignmentandOutcomeVariablesTable2apresentsreducedformestimatesofthekinksinourendogenouspolicyvariable(logdailybenets)andourmainoutcomevariable(logoftimetonextjob)aroundTminandTmax.Foreachvariableweshowresultsusingthreedifferentbandwidthselectionprocedures:thedefaultCCTprocedure;theCCTbandwidthselectionprocedurewithoutregularization;andtheFGbandwidth.Weshowtheestimatedkinkarisingfrom 26ManyseasonaljobsinAustrialayoffworkersattheendoftheseasonandre-hirethemagainatthestartofthenextseason.Havingbeenrecalledfromunemploymenttotherecentlylostjobisagoodindicatorthatthepresentspellmayendwithrecalltothatjobagain–seeDelBonoandWeber(2008).27Wetasinglepredictionmodelusingthepooledbottomkinkandtopkinksamples.25 eachselectedbandwidth,aswellasthecorrespondingbias-correctedestimateandtheassociatedrobust95%condenceinterval.28Wepresentestimatesfromlocallinearmodelsincolumns1and3,andfromlocalquadraticmodelsincolumns2and4.DespitethestrongvisualevidenceofkinksinthebenetformulainFigures3and4,anexaminationoftheestimated“rststage”kinksinPanelAofTable2suggeststhatnotalltheproceduresyieldstatisticallysignicantkinkestimates.Inparticular,thedefaultCCTbandwidthselectorchoosesrelativelysmallband-widthsforthelocallinearmodelandyieldsaninsignicantestimateofthebottomkink(t=1:7)andonlyamarginallysignicantestimateofthetopkink(t=2:1).Thecorrespondingbias-correctedkinkestimatesaresubstantiallylessprecise,withsamplingerrorsabout40%largerthantheuncorrectedestimates.Al-thoughthedefaultCCTprocedurechoosessomewhatlargerbandwidthsforthelocalquadraticmodels,thisisoffsetbythedifcultyofpreciselyestimatingtheslopesoneithersideofthekinkpointoncethequadratictermsareincluded,andneithertheestimatedbottomkinkortheestimatedtopkinkinthequadraticmod-elsisclosetosignicant.Aswiththelocallinearmodels,thecorrespondingbias-correctedquadratickinkestimatesareevenlessprecise,withverywidecondenceintervals.RelativetothedefaultCCTbandwidthselector,theCCTselectorwithoutregularizationyieldssubstan-tiallylargerbandwidths–over2timeslargerforthelocallinearmodels,and30-50%largerforthelocalquadraticmodels.Theselargerbandwidthsyielduncorrectedrststagekinkestimatesfromthelocallinearmodelsthatarerelativelyprecise(t�10forthebottomkinksample,t=6forthetopkinksample).Thees-timatedkinksfromthelocalquadraticestimates,however,arestillrelativelynoisy,asarethebias-correctedestimatesfromeitherthelocallinearorlocalquadraticmodels.BycomparisonthebandwidthsselectedbytheFGprocedurearerelativelylarge,anddeliversevensignicantrststageestimatesineightcases.Interestingly,acomparisonoftheuncorrectedandbias-correctedestimatesfromtheFGselectionproceduresuggeststhatthebiasesassociatedwiththenaiveFGbandwidthchoicearerelativelysmall,exceptinthecaseofthelocalquadraticestimateforthetopkink.TurningtothereducedformoutcomemodelsinPanelB,theestimatedkinksinthedurationofjobless-nessarelesspreciselyestimatedthanthekinksinlogbenets.Again,thedefaultCCTbandwidthselectorchoosesrelativelysmallbandwidthsandyieldsverynoisyestimatesofthekink.ThebandwidthsundertheCCTprocedurewithouttheregularizationtermaresubstantiallylarger,andyieldmarginallysignicantes- 28RobustcondenceintervalsandtheCCTbandwidthsareobtainedbasedonavariantoftheStatapackagedescribedinCalonicoetal.(inpress)withthenearest-neighborvarianceestimator,whichwealsouseinthesimulationsbelow.UsingtheCCTStatapackagegeneratesverysimilarempiricalestimates.26 timatedkinksintheoutcomevariablefromthelocallinearmodels.TheFGbandwidthsareevenlarger,andyieldestimatedkinksthataresignicantormarginallysignicantinboththelinearandquadraticmodels.Thebias-correctedestimatesareinallbutonecaseinsignicant,however,reectingtheadditionaluncer-taintyassociatedwiththebiascorrectionterm.29Interestingly,thebias-correctedlocallinearestimatesforthebottomandtopkinkarebothlargerinabsolutevaluethanthecorrespondinguncorrectedestimates,suggestingthattheuncorrectedestimatesmaybeconservative.Finally,aswefoundwiththerststagekinkestimates,thelocalquadraticestimatesinthereducedformmodelarequiteimprecise,andthebias-correctedlocalquadraticmodelsareessentiallyuninformative.Innitesamples,theuseofhigherorderpolynomialmodelsandbiascorrectionmaycomeatthecostofanincreaseinvariancerelativetolower-orderuncorrectedmodels.IntheremainderofthissectionweexaminethetradeoffbetweenbiasandvarianceassociatedwiththeCCTbiascorrectionprocedure.Wedeferadiscussionofthepolynomialorderchoicetosubsection4.5.3.Asmentionedinsection3,theintentofCCT'sbiascorrectionistoeliminatethebiasinthep-thorderpolynomialestimatorˆtpbysubtractingofftheestimatedasymptoticbias,hpˆrp.Thecostofbiascorrection,however,isthatthebiastermisimpreciselyestimated,leadingtoapotentialincreaseintheoverallvarianceofthecorrectedestimatorˆtbcprelativetotheuncorrectedestimatorˆtp.30Theusualmetricfortradingoffbiasandvarianceisthe(asymptotic)meansquarederroroftheestimator,whichisthesumofitssquaredbiasanditsvariance.ByLemmaA1andTheoremA1ofCalonicoetal.(Forthcoming),theasymptoticmeansquarederrorsofˆtpandˆtbcp+1areAMSE(ˆtp)=(hprp)2+op(h2p)+var(ˆtp)andAMSE(ˆtbcp)=op(h2p)+varbcp.ItfollowsthatthechangeintheAMSEassociatedwithbiascorrectionisasymptotically�(hprp)2+varbcp�var(ˆtp).InTable2b,wereporttheestimatedbiashpˆrp,itssquare,andthechangeinestimatedvariance,cvarbcp�cvar(ˆtp),fortherststageandreducedformestimatorspresentedinTable2a.TheincreaseinvarianceislargerthantheestimatedsquaredbiasforboththelocallinearandlocalquadraticestimatesusingeitherthedefaultCCTbandwidthselector,orthealternativeversionthatignorestheregularizationterm.ThisisalsothecasefortheFGbandwidthinthebottomkinksample.Inthetopkinksample,however,theestimatedbiasfortheFGbandwidthisquitelarge,andbiascorrectionappearstodecreasetheAMSE.ThissuggeststhatbiascorrectioncouldbeimportantforestimatorsbasedontheFGbandwidthinthetopkinksample.At 29Theoneexceptionisthebias-correctedlocallinearestimateforthetopkinksample.Inthiscase,therobustcondenceintervalisrelativelywidebutthebias-correctedpointestimateisalsorelativelylargeinmagnitude.30Remark5ofCalonicoetal.(Forthcoming)statesthatthevarianceofˆtbcpissmallerthanthatofˆtpforalargen,butthisasymptoticadvantagemaynotmaterializeinagivennitesample,anddoesnotappeartoholdinoursamples.27 thesametime,sincethebiastermisestimated,itmaydeviatefromtheactualbias.Soinsection4.5.3,weevaluatetheperformanceofthevariousestimatorsinMonteCarlosimulationsusingDGP'sapproximatingourdata,wherewecandirectlyobtainthemeansquarederrorswithouthavingtoestimatethebias.4.5.2KinksinConditionalMeansofPredeterminedCovariatesAsdiscussedinsubsection2.3,akeyimplicationofthesmoothdensityassumptionunderlyingavalidRKdesignisthattheconditionaldistributionsofanypre-determinedcovariatesshouldevolvesmoothlyaroundthekinkpointsintheobservedrunningvariable.InthecontextofourUIexample,thismeansthattheconditionalmeansofallpre-determinedclaimantcharacteristicsshouldvarysmoothlywithbase-periodearningsaroundthebottomandtopkinkpoints.Table3presentstestsofthissmoothnesspredictionforthecovariateindexintroducedinFigure7and8,aswellasforthefourmainsub-componentsofthisindex.WeshowestimatedkinksfromlocallinearandlocalquadraticmodelsusingtheFGbandwidthselector,aswellasthecorrespondingbias-correctedkinkestimates(androbust95%condenceintervals).Forthebottomkinksample(PanelA),theestimatespointtosomereasonforconcern.Inparticular,theconditionalmeanofthepredictedmeanofjoblessnessexhibitsapositivekinkatTminabout30%aslargeasthekinkinactuallogtimetonextjob(comparingtheestimatedkinkof0.9forpredictedlogtimetotheestimateof3.0forlogactualtimeinTable2afromalocallinearmodelusingtheFGbandwidth).Lookingattheindividualcovariates,therearerelativelylargekinksintheconditionalmeansofageandtheindicatorforbluecollarstatus.ThesekinksarevisuallyevidentinAppendixFigures2aand2c,andsuggestthattheconditionaldistributionoftheobservedcharacteristicsofclaimantswithearningsaroundTminisnotsmooth.Giventhissituation,wehavetointerprettheestimatedparametersderivedfromthebottomkinksamplecarefully,acknowledgingthatthereisalikelyupwardbiasintheRKDestimateoftheelasticityofthedurationofjoblessnesswithrespecttoUIbenets,drivenbythenon-smoothnessintheobserveddeterminantsofjoblessnessdurations.Forthetopkinksample(PanelB)thereislessevidenceofnon-smoothness,thoughwenotethatthebias-correctedlocallinearmodelandthelocalquadraticmodelspointtoapossiblenegativekinkinthepredictedlogtimetonextjob.TakentogetherwiththerelativelysmoothpatternsinFigure8andAppendixFigures3a-3d,however,webelievethattheassumptionsforavalidregressionkinkareplausiblysatisedaroundthetopkink.28 4.5.3FuzzyRKDEstimatesandComparisonofAlternativeEstimatorsAsanalstepinourempiricalanalysiswepresentfuzzyRKDestimatesoftheelasticityofthedurationofjoblessnesswithrespecttothelevelofUIbenetsandevaluatetheperformanceofalternativeestimators.Asnotedinsubsections2.2.2and4.2,thefuzzyRKestimandinthebottomkinksampleidentiesthebehavioralresponseforclaimantswhosebaselineearningsareclosetoTminandwhofollowthebenetscheduleintendedforsingleclaimants.Inthetopkinksample,thefuzzyRKestimandidentiestheelasticityforclaimantsclosetoTmaxwhofollowanyofthebenetschedulesseeninFigure1.Therefore,applyingaregressionkinkapproachtothetwosamplesallowsustoestimatetheelasticityofjoblessnesswithrespecttoUIbenetgenerosityfortwoverydifferentsubpopulations.31Table4presentsestimatedelasticitiesforthelocallinearandquadraticestimatorsundertheFGbandwidth.Weshowboththeconventionalestimates(columns2and5)andthebias-correctedestimates(columns3and6)withrobustcondenceintervalsthattakeaccountofthesamplingvariabilityofthebiascorrections.Withoutbiascorrection,thelocallinearmodelsyieldestimatedelasticitiesof1.4and2.0forthebottomkinkandtopkinksamples,respectively.Inbothcases,thecorrespondingbias-correctedestimatesarelargerinmagnitude,asaretheestimatesfromthelocalquadraticmodels,suggestingthatifanything,theuncorrectedlocallinearestimatesare“conservative”.InTable5,wepresentelasticitiesalongwithrst-stageestimatesforfouralternativebandwidthselec-tionprocedures.Thealternativesweconsiderare:defaultCCT,CCTwithnoregularization,FuzzyCCTandFuzzyIK.(ExpressionsforthelattertwobandwidthsaregiveninsubsectionB.2oftheSupplementalAppendix).Foreachbandwidthselectorwepresentthevalueofthemainandpilotbandwidth(thepilotbandwidthisforbiasestimationinconstructingtheCCTcondenceinterval),theuncorrectedandbias-correctedrststagekinkestimates,andtheuncorrectedandbias-correctedstructuralelasticities.TherststagekinkestimatesaregenerallysimilartotheestimatespresentedinPanelAofTable2a,buttheyarenotidenticalbecausethebandwidthsinTable5areselectedtobeoptimalforthenumeratoroftheRKDestimand,orfortheestimanditself(inthecaseofthefuzzyCCTandfuzzyIKprocedures)ratherthanfortherststageequation.ThepatternofestimatesinTable5pointtothreemainconclusions.First,aswenotedinthediscussionofTable2a,manyofthebandwidthselectorschooserelativelysmallbandwidthsthatleadtorelativelyim- 31ItmaybetemptingtoapplyasharpRKdesigntoonlytheobservationsthatlieontheUIschedule.However,thisapproachdoesnotingeneralidentifyaninterpretabletreatmenteffect,justasananalysisofthesubsetof“compliers”inarandomizedexperimentislikelytobehighlyproblematic.29 preciserst-stageandstructuralcoefcientestimates.Asecondobservationisthat,asinTable2a,thelocalquadraticestimatorsaregenerallyquitenoisy.Third,thebias-correctedestimatesfromthelocallinearmod-elsaretypicallynottoodifferentfromtheuncorrectedestimates,buttheaddedimprecisionassociatedwithuncertaintyaboutthemagnitudeofthebiascorrectionfactorislarge,leadingtorelativelywidecondenceintervalsforthebiascorrectedestimates.GiventhewiderangeofestimatesinTables4and5,whichspecicationshouldwepick?ThelineofargumentadvancedinCalonicoetal.(Forthcoming)suggeststhatonemightpreferthebias-correctedesti-matesfromlocalquadraticmodelsusingtheCCTbandwidthselectorwithregularization.Inourapplication,theseestimatesareimprecise.ForthebottomkinksamplethischoicesuggeststheelasticityofjoblessnesswithrespecttoUIbenetsis19.0whereasforthetopkinksampletheestimateiswrong-signedandequalto-6.9.However,giventhewidecondenceintervalsbothestimatesareessentiallyuninformative.Attheoppositeextreme,theuncorrectedestimatesfromlocallinearspecicationsusingtheFGbandwidthselector(inTable4)arerelativelypreciselydeterminedandpointtobehavioralelasticitiesintherangeof1to2.Togainadditionalinsight,wedecidedtoconductaseriesofMonteCarlosimulationsbasedonDGP'sthatcloselyresembleouractualsamples.Becauseweareinterestedinthepowerofthecandidateestimators,weimposetherst-stagekinkparameterandtheelasticityparameterinconstructingtheDGP's.Forthebottomkinksample,weimposetherst-stagekinkastB=2:310�5andtheelasticitytFRKD=1:3,whichimpliesareduced-formkinkoftY=tB
tFRKD=3:010�5.Forthetopkinksample,wesettB=�1:410�5andtFRKD=2:0withanimpliedtY=�2:810�5.WethenspecifytheDGP'sforE[BjV]andE[YjV]asseparatequinticsoneachsideofthethreshold,wheretheparametersofthequinticsareestimatedbyregressing,respectively,B�tB
D
VandY�tY
D
VonthepolynomialtermsVjandD
VkwhereD=1[V�0],j=0;1:::;5andk=2;:::;5.Foroursimulation,wesampleVfromitsempiricaldistributionandtheerrors(eB,eY)jointlyfromtheresidualsfromthequinticregressions,andconstructB=E[BjV]+eBandY=E[YjV]+eY.Wedraw1,000repeatedsamplesinourMonteCarloexercise.Tables6aand6bsummarizetheperformanceofthealternativeestimatorsforthesimulatedbottomkinkandtopkinksamples,respectively.Thetwopolynomialorders(linearandquadratic),twobias-correctionchoices(uncorrectedandbias-corrected)andsixbandwidthselectionprocedures(defaultCCT,CCTwithnoregularization,FuzzyCCT,FuzzyIK,FG,andGlobal)giveriseto24candidateestimatorsintotal.Foreachestimator,wereporttheassociatedbandwidth(s)anditsperformanceinestimatingtherst-stageandtheelasticityparameters.30 Thetwomaincriteriaforevaluatingestimatorperformanceare(1)therootmeansquarederror(RMSE)oftheelasticityestimator;and(2)howoftenitscondenceintervalcoversthetrueparametertFRKD.Incolumn(6)and(7)wereporttwomeasuresofRMSE.Column(6)showstherawRMSEasaratiooftheimposedtFRKD.Becauseweworrythatoutliersmaydrivethenumbersincolumn(6),wereportincolumn(7)a“trimmed”RMSEratiobyrstdiscarding5%ofthesimulationsamplewiththegreatestdeviationbetweenˆtFRKDandtFRKD.Incolumn(8),wereportthecoverageratesofthecondenceinterval.AsseenfromTable6,noneoftheestimatorsconsistentlyachievesthelowestRMSEanddeliversthecorrectcoveragerate,buttheconventionallocallinearestimatorwiththeFGbandwidth(LLFGforshort)appearstobeareasonablechoice.Inthebottomkinksample,thisestimatorandtheglobalconventionallinearestimatoroutperformallothercandidates,andthetwoareverysimilarintermsoftheirRMSE'sandcoveragerates(96%and94%respectively).Inthetopkinksample,theLLFGestimatorhasthesmallestRMSE,althoughthecoveragerateofthecorrespondingcondenceintervalisalower82%.ThelowercoveragerateisindicativeofthebiasoftheLLFGestimator,butthebiasisquitesmallinmagnitude(0.11)ascomparedtotheimposedelasticity(2.0).AviablealternativeforthetopkinksampleisthelocallinearestimatorwiththefuzzyIKbandwidth(LLIKforshort):ithasamoderatelyhigherRMSE,butthecorrespondingcondenceintervalhasabettercoveragerateof94%thanLLFG.AsshowninTable5,usingLLIKalsopointstoastatisticallysignicanthighelasticity,whichisconsistentwiththeLLFGresult.AnalogouslytoTable2b,wealsobreakdownthe(trimmed)MSEandreportthebias,squaredbiasandvarianceofeachoftheestimatorsincolumns(9)-(11)ofTable6.Biascorrectionappearstoincreaseboththesquaredbiasandthevarianceforallestimatorsinthebottomkinksimulationsandfor8outof12estimatorsinthetopkinksimulations.In23outofthe24cases,biascorrectionincreasesthe(trimmed)RMSE,withtheonlyexceptionbeingtheglobalquadraticestimatorforthetopkink.Inaparallelsimulationstudy,weassesstheperformanceoftheestimatorsinDGP'swhereweimposethetrueelasticitytFRKDtobezero:theresultsarepresentedinAppendixTable2.IntermsofRMSE,theFGbandwidthstilloutperformsthefourlocalalternatives,andwithineachbandwidthchoice,theconventionallinearestimatordominatestheotherthreecandidates.Intermsofthecoveragerateofthecorresponding95%condenceinterval,however,LLFGnolongerperformswell:thecoveragerateis2%forthebottomkinksimulationand20%forthetop.Moreover,theredoesnotappeartobeanattractivealternativeestimator:thosewithacoveragerateabove90%haveavarianceatleast30timeshigherthanLLFGinthetopkinksampleandatleast80timeshigherinthebottomkinksample.31 ThepoorcoverageratesoftheLLFGcondenceintervalinAppendixTable2raisetheconcernthatthestructuralelasticitiesarisingfromapplyingthisproceduretoouractualdatamaybemisleading.However,thevalueofthebias-correctedLLFGestimatorishigherthantheconventionalLLFGestimatorinouractualdata(1.95versus1.37forthebottomkink;3.67versus2.04forthetopkink),andthisempiricalregularityismatchedbytheDGP'sunderlyingTable6butnotbytheDGP'sunderlyingAppendixTable2.Therefore,webelievethatthesimulationDGP'sunderlyingTable6,asopposedtothoseunderlyingAppendixTable2,arebetterapproximationstotheactualdata.Inanotherexercise,weprovidefurtherevidencethatthelocallinearestimatorshouldbepreferredtothelocalquadraticbydirectlyestimatingtheAMSE'sforthelocallinearandquadraticestimators,perCardetal.(2014).AsshowninAppendixTable3,usingthedefaultCCTbandwidthselectionproceduretheAMSEforthelocalquadraticmodelisatleastanorderofmagnitudelargerthantheAMSEforthelocallinearmodel.32Fortheotherbandwidthchoices,thelinearAMSEisalsomuchsmaller(atleast68%smaller)thanthequadraticAMSEandweomitthemfromAppendixTable3foreaseofexposition.Asanalrobustnesscheck,weinvestigatehowsensitivetheelasticityestimatesarewithrespecttothechoiceofbandwidth.Figures9and10plottheelasticityestimatesforthebottomandtopkinksamplesassociatedwitharangeofpotentialbandwidths.Ruppert(1997)arguesthatonecanusetherelationshipbetweenthepointestimatesandthebandwidthchoiceasanindicatorofpotentialbias,withstabilityintheestimateindicatingtheabsenceofsignicantbias.Figure9showsthattheestimatedelasticityoftimetonextjobwithrespecttoUIbenetsaroundthebottomkinkisrelativelystableatcloseto1.4foraverywiderangeofbandwidths.Figure10showsthattheestimatedelasticityaroundthetopkinkisalittlemoresensitivetobandwidthchoice,withalargerestimate(between2and3)forlowerbandwidths,butanelasticityof2orlessforbandwidthsabove C5,000.Overall,weconcludethattheconventionallocallinearestimatorwiththeFGbandwidthdoesreasonablywellforourempiricalapplication.Thecorrespondingestimatesleadustotwomainndings.First,forjob-losersfromtheupperpartoftheearningsdistribution(aroundTmax),theelasticityofthetimetonextjobwithrespecttoUIbenetsisaround2.OurcondenceinthisestimateisstrengthenedbythefactthattestsforthevalidityofanRKdesignaroundthetopkinkshowlittleevidencethatthedesigniscompromisedbysorting. 32Notethattheestimatedbiascomponentforthelocalquadraticestimatorissubstantiallylargerthanitslocallinearcounterpart,eventhoughtheformershrinkstozeroatafasterrate.32 Asecond,moretentativeconclusionisthatthecorrespondingelasticityforjob-losersinthelowerpartoftheearningsdistribution(aroundTmin)isofasmallermagnitude–perhapscloserto1.OurcautiousassessmentstemsfromthefactthatthetestsforthevalidityofanRKDapproachshowthattheconditionaldistributionsofobservedworkercharacteristicschangeslopesaroundTmin.Thesechangesarenotassoci-atedwithanydiscernible“bunching”atthekinkpointinthebenetschedule,buttheyarelargeenoughtocausea30%upwardbiasintheestimatedjoblesselasticity.Howdoourestimatedbenetelasticitiescomparetothoseintheexistingliterature?AppendixTable4containsabriefsummaryoftheexistingliterature,drawingonthesurveybyKruegerandMeyer(2002)fortheearlierU.S.-basedliterature,allofwhichuseadministrativerecordsonunemploymentinsuranceclaimsandestimatetheeffectofUIbenetsonthedurationoftheinitialspellofinsuredunemployment.Thesestudiespointtoabenetelasticityintherangeof0.3to0.8.33ArecentstudybyLandais(Forthcoming)appliesaregressionkinkdesigntosomeofthesamedatausedintheseearlierstudiesandobtainsestimatesoftheelasticityoftheinitialUIbenetspellthatrangefrom0.20to0.70.AnotherrecentstudybyChetty(2010)usesretrospectiveinterviewdatafromtheSurveyofIncomeandProgramParticipationandobtainsanaveragebenetelasticityofabout0.5.TakentogethertheseU.S.studiessuggestabenchmarkofaround0.5fortheelasticityofinitialUIclaimdurationontheUIbenet.MostoftheEuropeanstudiesincludedinAppendixTable4estimatetheeffectofbenetsonthetimetorstexitfromtheUIsystem,andobtainbenetelasticitiesthataresimilartotheU.S.studies.AnexceptionisCarlingetal.(2011),whostudytheeffectofareductioninthebenetreplacementrateinSwedenin1996ontheexitratefromunemploymentrecipiencytoemployment.Theirestimateoftheelasticityoftimetonextjobwithrespecttothebenetlevelis1.6,whichisnotfarfromourpointestimateof2.0forthetopkinksample.InCardetal.(2012),wearguedthatthehighelasticitywendmaybeaconsequenceofusingtimetonextjobasanoutcomemeasureasopposedtotheinsuredunemploymentdurationtypicallyusedintheliterature.OurndingsunderscorethepotentialvalueinbeingabletomeasuretimetonextjobinassessingtheincentiveeffectsoftheUIsystem. 33KruegerandMeyerattributeanestimatedelasticityof1.0toSolon(1985)whostudiestheeffectofmakingUIbenetstaxableontheunemploymentdurationofhigh-earningclaimants.Hendsthattheintroductionoftaxationcauseda22%reductionintheaveragedurationofinitialUIclaimsbyhigher-earningclaimants(withnoeffectonlow-earners).Assuminganaveragetaxrateof30%,thisimpliesanelasticityof0.73,whichisourpreferredinterpretationofSolon(1985)'sresults.33 5ConclusionInmanyinstitutionalsettingsakeypolicyvariable(likeunemploymentbenetsorpublicpensions)issetbyadeterministicformulathatdependsonanendogenousassignmentvariable(likepreviousearnings).Conventionalapproachestocausalinference,whichrelyontheexistenceofaninstrumentalvariablethatiscorrelatedwiththecovariateofinterestbutindependentofunderlyingerrorsintheoutcome,willnotworkinthesesettings.Whenthepolicyfunctioniscontinuousbutkinked(i.e.,non-differentiable)ataknownthreshold,aregressionkinkdesignprovidesapotentialwayforward(Guryan(2001);Nielsenetal.(2010);Simonsenetal.(Forthcoming)).ThesharpRKDestimandissimplytheratiooftheestimatedkinkintherelationshipbetweentheassignmentvariableandtheoutcomeofinterestatthethresholdpoint,dividedbythecorrespondingkinkinthepolicyfunction.Insettingswherethereisincompletecompliancewiththepolicyrule(ormeasurementerrorintheactualassignmentvariable),a“fuzzyRKD”replacesthedenominatoroftheRKDestimandwiththeestimatedkinkintherelationshipbetweentheassignmentvariableandthepolicyvariable.InthispaperweprovidesufcientconditionsforasharpandfuzzyRKDtoidentifyinterpretablecausaleffectsinageneralnonseparablemodel(e.g.,BlundellandPowell(2003)).Thekeyassumptionisthattheconditionaldensityoftheassignmentvariable,giventheunobservederrorintheoutcome,iscontinuouslydifferentiableatthekinkpoint.Thissmoothdensityconditionrulesoutsituationswherethevalueoftheassignmentvariablecanbepreciselymanipulated,whileallowingtheassignmentvariabletobecorrelatedwiththelatenterrorsintheoutcome.Thus,extremeformsof“bunching”predictedbycertainbehavioralmodels(e.g.,Saez(2010))violatethesmoothdensitycondition,whereassimilarmodelswitherrorsinoptimization(e.g.,Chetty(2010))arepotentiallyconsistentwithanRKDapproach.Inadditiontoyieldingatestablesmoothnesspredictionfortheobserveddistributionoftheassignmentvariable,weshowthatthesmoothdensityconditionalsoimpliesthattheconditionaldistributionsofanypredeterminedcovariateswillbesmoothfunctionsoftheassignmentvariableatthekinkpoint.Thesetwopredictionsareverysimilarinspirittothepredictionsforthedensityoftheassignmentvariableandthedistributionofpredeterminedcovariatesinaregressiondiscontinuitydesign(Lee(2008)).WealsoprovideaprecisecharacterizationofthetreatmenteffectsidentiedbyasharporfuzzyRKD.ThesharpRKDidentiesaweightedaverageofmarginaleffects,wheretheweightforagivenunitreectstherelativeprobabilityofhavingavalueoftheassignmentvariableclosetothekinkpoint.Underanaddi-34 tionalmonotonicityassumptionweshowthatthefuzzyRKDidentiesaslightlymorecomplexweightedaverageofmarginaleffects,wheretheweightalsoincorporatestherelativesizeofthekinkinducedintheactualvalueofthepolicyvariableforthatunit.WeillustratetheuseofafuzzyRKDapproachbystudyingtheeffectofunemploymentbenetsonthedurationofjoblessnessinAustria,wherethebenetschedulehaskinksattheminimumandmaximumbenetlevel.Wepresentavarietyofsimplegraphicalevidenceshowingthatthesekinksinducekinksinthedurationoftotaljoblessnessbetweentheendofthepreviousjobandthestartofthenextjob.Wealsopresentavarietyoftestsofthesmoothdensityassumptionaroundthethresholdsfortheminimumandmaximumbenetamounts.WepresentalternativeestimatesofthebehavioraleffectofhigherbenetsonthedurationofjoblessnessandevaluatetheempiricalperformanceofthealternativeestimatorsusingMonteCarlosimulation.OurpreferredestimatespointtoelasticitiesthatarehigherthanmostofthepreviousstudiesintheU.S.andEurope.35 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10 20 30 40 50Daily UI BeneÞt in Euro 0 10000 20000 30000 40000 Base Year Earnings in Euro Figure 1: UI BeneÞts in 2004 504509501450195024502950345039504450Frequency at= 4501050255040505550Frequency at 23 24 25 26 Average Daily UI Bene -2000 -1000 0 1000 2000 3000 4000 Base Year Earnings Relative to T-min 35 Average Daily UI Bene -14000 -9000 -4000 1000 Base Year Earnings Relative to T-maxTop Kink Sample 4.4 -2000 -1000 0 1000 2000 3000 4000 Base Year Earnings Relative to T-minFigure 5: Log Time to Next Job 4.5 -14000 -9000 -4000 1000 Base Year Earnings Relative to T-maxTop Kink SampleFigure 6: Log Time to Next Job 4.4 -2000 -1000 0 1000 2000 3000 4000 Base Year Earnings Relative to T-minFigure 7: Predicted Time to Next Job 4.5 -14000 -9000 -4000 1000 Base Year Earnings Relative to T-maxTop Kink SampleFigure 8: Predicted Time to Next Job -4 -2 0 2 4Elasticity 1000 2000 3000 4000Notes: local linear estimation, estimated coefÞcients (blue) with conÞdence bounds (dash) Figure : Fuzzy RKD estimation with varying bandwidthlog time to next job bottom kink sample -2 0 2 4 6 8Elasticity 2000 4000 6000 8000 10000 12000Notes: local linear estimation, estimated coefÞcients (blue) with conÞdence bounds (dash) Figure 1: Fuzzy RKD estimation with varying bandwidthlog time to next job top kink sample Table 1: Summary Statistics for Bottom and Top Kink Samples of UI Claimants Mean Std. Dev.MeanStd. Dev. (1) (2)(3)(4) Baseline earnings (euros)22,142 2,805 33,847 6,730 Daily UI benefit (euros) 25.23.033.55.7 Time to next job (days) 148.2129.1147.4131.7 Duration of initial UI spell (days)** 77.667.079.870.3 Total days of UI received 1944.41707.02679.72480.1 Fraction exhausted benefits*** 0.090.290.090.29 Fraction with time to next job censored 0.190.390.200.40 Fraction eligible for extended benefits 0.780.410.900.29 Fraction female 0.380.480.210.41 Mean Age 33.068.4836.047.45 Fraction Austrian nationals 0.800.400.860.35 Fraction married 0.340.470.400.49 Fraction bluecollar occupation 0.650.480.560.50 Fraction with higher education 0.110.310.180.38 Fraction in Vienna 0.190.390.210.41 Tenure in most recent job (Years) 3.393.274.234.20 Recalled to last job 0.200.400.250.44 Industry: Construction Manufacturing 0.20 0.400.230.42 Trade 0.20 0.400.150.35 Services Number observations ime to next job is censored at 365 days.* Claim duration is censored at 39 weeks (maximum entitlement). *** Indicator equals 1 if claim duration = maximum entitlement. Notes: sample contains UI claimants under the age of 50 with claims in 2001-2012, who had at least 1 year of tenure on their previous job, began their claim within 4 weeks of losing their past job, and had a valid UI claim record and non-missing earnings in the base period prior to the claim. Observations in the bottom kink sample have base period earnings in a range around the bottom kink in the UI benefit schedule; observations in the top kink sample have base period earnings in a range around top kink. See text. 275,293 275,665 Bottom Kink Sample Top Kink Sample LinearLocalQuad.LocalLinearLocal(1)(2)(3)(4) Bandwidth448(929)813(1,343)1,374(2,760)1,568Kink1.72.1error)(1.0)(1.6)(0.7)(2.2)Corrected1.21.21.2 confidence interval][1.6,3.9][2.7,5.1][3.6,0.4][Bandwidth1,257(1,677)1,220(1,646)2,913(3,813)2,128Kink2.31.8error)(0.2)(0.9)(0.2)(1.4)Corrected2.11.11.1 confidence interval][1.0,3.1][1.6,3.7][2.0,0.2][Bandwidth2,466(2,380)2,628(4,146)11,603(6,343)7,003Kink2.22.2error)(0.1)(0.3)(0.4)(0.2)Corrected2.41.91.9 confidence interval][1.7,3.1][0.4,3.4][0.7][ Bandwidth537(1,062)598(986)1,593(3,189)2,546Kink2.227.02.59.2error)(6.7)(23.0)(3.4)(6.7)Corrected1.338.53.711.011.0 confidence interval][19.5,22.0][20,97][6.5,13.9][Bandwidth1,330(1,574)4,736(1,971)2,825(3,053)4,225Kink3.15.5error)(1.7)(2.3)(1.4)(3.1)Corrected0.91.31.41.4 confidence interval][9.7,11.5][90,41][9.5,7.0][Bandwidth2,501(3,858)4,259(4,918)4,465(7,255)8,001Kink3.04.8error)(0.8)(2.4)(0.7)(1.5)Corrected4.29.95.20.00.0 confidence interval][1.4,9.8][3.6,23.4][1.9][KinkTop LinearLocalQuad.LocalLinearLocal(1)(2)(3)(4) Bias0.60.90.21.3Squared0.30.80.01.61.21.70.63.0Bias0.30.80.21.2Squared0.10.60.01.40.21.20.32.20.10.3Squared0.00.10.00.30.10.50.10.2 Bias1.0Squared0.9132.31.74.467.2361.815.538.5Bias2.229.4Squared4.8864.43.17.626.41108.915.828.85.52.2Squared1.730.34.618.97.539.62.310.0KinkTop FG Bandwidth Estimated Kink Bias Corrected Kink FG Bandwidth Estimated Kink Bias Corrected Kink (1) (2) (3) (4) (5) (6) A.BottomKink: 0.90.84,5641.24.0 tonextjob (0.2) [1 .0 ,2. 5] (0.8) [0 . 4,8 . 3] Female1,0941.13,440     ‐ (1.0) [4 .0 ,1 .9] (1.2) [8. 1, 4.0 ] Age1,62729.546.83,311 (9.8) [0. 4,94] (20.9) [12 2,89] occup.2,530     ‐1.84,564     ‐ (0.3) [4 . 1,0.6 ] (1.0) [12 .1 ,0. 4 ] job3,199     ‐0.80.53,194 (0.3) [ 1. 9,2 . 9] (1.1) [8 . 1,2 .4 ] B.TopKink: 1.58,095     ‐ tonextjob (0.1) [3.0,0.01] (0.5) [4 . 3,0.6] Female5,364     ‐0.913,908   ‐ (0.2) [1 . 9,0.2] (0.4) [ 4 ,1 . 1] Age6,160     ‐5.39,776     ‐ (2.6) [23 . 1,12 . 6] (7.8) [55.0,27.0 ] occup.2,0671.51.85,4012.64.0 (1.0) [0.7,4 . 3] (0.9) [0.7,7 . 2] job5,0130.61.113,9082.71.1 (0.2) [0.0 2 ,2 . 2] (0.4) [1 . 6,3 . 7] TableEstimatesofKinksinConditionalMeansofCovariates LocalQuadraticModels Notes:standarderrorsinparentheses,robustconfidenceintervalsinsquarebrackets.Seenotesto Table2.Allestimatesaremultipliedby105.Predictedlogtimetonextjobistheestimatedcovariateindexforthisoutcome,fitonthepooledbottomandtopkinksamples.Thevectorof59covariatesincludesdummiesforgender,bluecollaroccupation,andbeingrecalledtothepreviousjob,decileofage(9dummies),decileofpreviousjobtenure(9dummies),quintileofpreviousdailywage(4dummies),majorindustry(6dummies),region(3dummies),yearofclaim(7dummies),decileofpreviousfirmsize(9dummies),anddecileofpreviousfirm'srecallrate(9dummies). LocalLinearModels FGBandwidth Main(Pilot) EstimatedElasticity(std. error) BiasCorrectedEstimate [RobustCI] FGBandwidth Main(Pilot) EstimatedElasticity(std. error) BiasCorrectedEstimate [RobustCI] (1) (2) (3) (4) (5) (6) A.BottomKink: 2,501(3,858) 1.37 1.95 4,259(4,918) 2.35 5.04 (0.37) [0.59,4. 4 9] (1.20) [1.8 7 ,11.9 4 ] B.TopKink: 4,465(7,255) 2.04 3.67 8,011(12,383) 2.71 1.20 (0.52) [1.2 4 ,6.09] (1.07) [4.51,6.90] LocalLinearModels LocalQuadraticModels TableEstimatedElasticitiesofJoblessnessDurationfromFuzzyRegressionKinkDesign,FGBandwidthNotes:Standarderrorsinparentheses.FGbandwidthisderivedbasedontheoutcomevariable.Pointestimatesandstandarderrorsareobtainedfrom2SLSregressionsdescribedinCardetal(2012).RobustCI'sareobtainedbyavariantoftheStatapackagedescribedinCalonicoetal(inpress). StageFirstStageFirstStageFirst 5 )Struct.Model(Coeff×10 5 )Struct.Model(Coeff×10 5 )Struct.Model(Coeff×10 5 (1)(2)(3)(4)(5)(6)(7)(8)2.01.11.222.5error)(0.7)(3.4)(2.4)(51.4)(0.5)(1.8)(1.0)(5.5)1.65.21.919.019.0 conf. interval][0.6,3.7][9.6,11.4][4.3,8.1][110,148][3.2,0.1][7.7,3.3][4.4,1.2][2.11.52.02.71.12.70.81.6error)(0.2)(0.8)(0.3)(1.2)(0.2)(1.4)(0.5)(3.8)1.90.60.91.81.8 conf. interval][0.7,3.0][4.5,5.7][6.9,6.4][43,24][2.1,0.4][6.6,10.2][2.9,0.8][2.31.11.80.11.42.00.81.9error)(0.3)(1.0)(0.6)(3.0)(0.1)(0.5)(0.5)(3.8)2.12.41.74.80.83.00.51.31.3 conf. interval][0.7,3.4][3.1,7.8][1.1,4.4][9.5,19][2.1,0.5][3.4,9.3][1.8,0.9][2.31.32.00.61.32.71.33.5error)(0.3)(1.1)(0.4)(2.0)(0.2)(1.1)(0.3)(1.7)2.21.22.03.40.83.21.10.10.1 conf. interval][1.1,3.2][3.1,5.4][1.0,5.0][9.6,16.4][0.0][1.2,7.6][3.2,1.1][(1412)1558(1595)4567(3781)4208(1679)1887(1725)3125(4909)5796(1062)598(985)1593(3188)2546(1574)4736(1971)2825(3503)4225KinkTopLinearLocalQuadraticLocalLinearLocal Table 6a: Summary of Monte Carlo Studies, DGP Design Based on Bottom Kink Sample (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 1. Local Linear, No Bias Correction Default CCT 464 - 0.34 0.44 0.94 27.1 4.55 1.00 2.02 4.06 31.0 CCT, no regularization 871 - 0.07 0.24 0.89 12.1 2.15 0.97 1.28 1.63 6.18 Fuzzy CCT 871 - 0.06 0.23 0.89 6.94 1.97 0.95 1.22 1.50 5.09 Fuzzy IK 1,407 - 0.00 0.11 0.83 1.24 0.80 0.91 0.31 0.10 0.98 FG 2,437 - 0.00 0.07 0.66 0.28 0.24 0.96 0.10 0.01 0.09 Global (all data) 4,564 - 0.00 0.14 0.00 0.24 0.20 0.94 -0.01 0.00 0.07 2. Local Linear, Bias-Corrected Default CCT 464 958 0.61 0.64 0.93 3315 7.88 1.00 3.77 14.2 91.0 CCT, no regularization 871 1,256 0.37 0.49 0.90 31.8 4.31 0.98 2.57 6.59 24.8 Fuzzy CCT 871 1,270 0.36 0.45 0.92 74.5 4.11 0.95 2.57 6.61 22.0 Fuzzy IK 1,407 1,397 0.33 0.44 0.91 4.36 3.49 0.89 2.66 7.10 13.5 FG 2,437 3,813 0.00 0.16 0.89 1.19 1.04 0.91 0.42 0.17 1.67 Global (all data) 4,564 4,564 0.00 0.18 0.83 1.27 1.12 0.88 0.77 0.59 1.54 3. Quadratic, No Bias Correction Default CCT 680 - 0.77 0.97 0.94 170 11.0 1.00 1.34 1.81 204 CCT, no regularization 1,210 - 0.38 0.49 0.92 76.3 6.00 0.98 2.72 7.41 53.6 Fuzzy CCT 1,292 - 0.33 0.46 0.91 80.0 5.54 0.99 2.67 7.13 44.8 Fuzzy IK 1,555 - 0.14 0.31 0.89 24.1 3.40 0.98 2.70 7.28 12.3 FG 4,210 - 0.00 0.15 0.84 1.49 1.33 0.79 1.11 1.23 1.77 Global (all data) 4,564 - 0.00 0.17 0.76 1.63 1.48 0.74 1.58 2.50 1.20 4. Local Quadratic, Bias-Corrected Default CCT 680 1,078 0.84 1.28 0.94 2226 17.9 1.00 2.99 8.97 534 CCT, no regularization 1,210 1,374 0.76 1.17 0.93 7172 11.2 0.99 3.05 9.33 204 Fuzzy CCT 1,292 1,442 0.75 1.05 0.93 530498 12.5 0.98 3.79 14.3 248 Fuzzy IK 1,555 1,627 0.72 1.09 0.92 2247 9.36 0.95 3.96 15.6 133 FG 4,210 4,893 0.24 0.37 0.90 4.43 3.90 0.78 3.88 15.1 10.6 Global (all data) 4,564 4,564 0.24 0.37 0.89 4.50 3.95 0.79 3.93 15.5 10.9 Notes: based on 1,000 simulations. DGP is based on 5th order polynomial approximation of bottom kink sample. True kink in first stage is: 2.3 -5 . True elasticity is: 1.3. The trimmed statistic are obtained by first trimming the 5% sample in which the estimates deviate the most from the true parameter value. RMSE/true value RMSE/true value (trimmed) C.I. Coverage Rate Elasticity Estimation Summary Bias (trimmed)Bias(trimmed)Variance (trimmed) Median Main b.w. Median Pilot b.w. First Stage Model Estimation Summary Fraction of Replications: C.I. includes 0 RMSE/true value (trimmed) C.I. Coverage Rate Table 6b: Summary of Monte Carlo Studies, DGP Design Based on Top Kink Sample (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 1. Local Linear, No Bias Correction Default CCT 1,371 - 0.62 0.59 0.81 78.8 3.66 0.99 0.50 0.25 53.5 CCT, no regularization 2,683 - 0.15 0.34 0.72 7.37 1.25 0.91 -0.02 0.00 6.30 Fuzzy CCT 2,301 - 0.25 0.43 0.72 81.0 1.97 0.94 0.04 0.00 15.5 Fuzzy IK 3,254 - 0.00 0.20 0.74 0.76 0.52 0.94 0.39 0.15 0.94 FG 4,465 - 0.00 0.15 0.77 0.48 0.36 0.82 0.11 0.01 0.52 Global (all data) 13,908 - 0.00 0.55 0.00 0.81 0.80 0.00 -1.60 2.56 0.01 2. Local Linear, Bias-Corrected Default CCT 1,371 2,842 0.88 0.90 0.79 12323 10.1 0.99 2.32 5.40 400 CCT, no regularization 2,683 3,673 0.68 0.76 0.69 473 3.12 0.93 0.01 0.00 39.1 Fuzzy CCT 2,301 3,570 0.68 0.79 0.69 18102 5.12 0.95 0.61 0.38 104 Fuzzy IK 3,254 4,604 0.44 0.59 0.64 2.36 1.67 0.90 0.77 0.59 10.5 FG 4,465 7,300 0.00 0.19 0.88 1.17 0.95 0.81 1.50 2.26 1.36 Global (all data) 13,908 13,908 0.00 0.22 0.72 0.92 0.87 0.44 1.57 2.45 0.55 3. Quadratic, No Bias Correction Default CCT 2,033 - 0.90 1.10 0.89 93.6 7.95 1.00 -2.04 4.18 249 CCT, no regularization 3,459 - 0.66 0.77 0.72 52.6 6.70 0.96 -0.39 0.15 180 Fuzzy CCT 3,613 - 0.64 0.72 0.69 97.2 6.00 0.97 -0.59 0.35 144 Fuzzy IK 4,964 - 0.34 0.49 0.71 43.4 3.03 0.92 0.26 0.07 36.7 FG 8,060 - 0.00 0.15 0.91 1.06 0.93 0.84 1.59 2.53 0.94 Global (all data) 13,908 - 0.00 0.20 0.58 2.07 1.97 0.03 3.79 14.3 1.16 4. Local Quadratic, Bias-Corrected Default CCT 2,033 3,232 0.91 1.40 0.90 1013 15.0 1.00 -1.40 1.96 904 CCT, no regularization 3,459 4,088 0.91 1.42 0.78 21785 26.5 0.99 1.25 1.55 2803 Fuzzy CCT 3,613 4,264 0.92 2.17 0.79 16114 25.9 0.98 3.77 14.2 2677 Fuzzy IK 4,964 5,033 0.93 1.44 0.68 65325 14.3 0.93 1.03 1.05 821 FG 8,060 12,471 0.71 0.63 0.62 1.89 1.53 0.93 0.36 0.13 9.2 Global (all data) 13,908 13,908 0.79 0.71 0.49 2.36 1.89 0.93 0.94 0.88 13.5 Bias(trimmed)Variance (trimmed) Notes: based on 1,000 simulations. DGP is based on 5th order polynomial approximation of top kink sample. True kink in first stage is: -1.4 -5 . True elasticity is: 2.0. The trimmed statistic are obtained by first trimming the 5% sample in which the estimates deviate the most from the true parameter value. Elasticity Estimation Summary RMSE/true value (trimmed) C.I. Coverage Rate First Stage Model Estimation Summary Median Main b.w. Median Pilot b.w. Fraction of Replications: C.I. includes 0 RMSE/true value (trimmed) C.I. Coverage Rate RMSE/true value Bias (trimmed) SupplementalAppendixAIdenticationA.1ProofsofProposition1and2InordertoproveProposition1,werstpresentandprovethefollowingLemmas.Lemma1.Letj(x;t):I[c;d]!RwhereIisacompactsubsetofRm.Supposej(x;t)anditspartialderivative,j2(x;t),arecontinuousandthatjisintegrablewithrespecttotheprobabilitymeasureaforeacht.Thenf(t)=Rj(x;t)da(x)iscontinuouslydifferentiableon[c;d].Proof:ByTheorem5(p.97)ofRoussas(2004),f0(t)=Rj2(x;t)da(x)forallt2[c;d].Lets1;s22[c;d]jf0(s1)�f0(s2)j=jZj2(x;s1)da(x)�Zj2(x;s2)da(x)j6Zjj2(x;s1)�j2(x;s2)jda(x)Thecontinuityofj2(x;t)onthecompactsetI[c;d]impliesuniformcontinuity,andthereforewecanchooseadsuchthatjs1�s2jdjimpliesjj2(x;s1)�j2(x;s2)je a(I)forallx2I,whichinturnimpliesthatjf0(s1)�f0(s2)je.QEDLemma2.UnderAssumptions1(i),3(ii)and4,f(v)iscontinuouslydifferentiableandstrictlypositiveonIV.Proof:Thecontinuousdifferentiabilityoff(v)followsfromAssumption1(i),Assumption4andLemma1.f(V)&#x]TJ/;ྖ ;.9;‘ ;&#xTf 1;�.9 ;� Td;&#x [00;RAUfVjU=u(v)dFU(u)&#x]TJ/;ྖ ;.9;‘ ;&#xTf 1;�.9 ;� Td;&#x [00;0followsdirectlyfromAssumption3(ii).InordertoproveProposition2,wepresentandprovethefollowinglemmas.LetSbeasub-vectoroftherandomvector(U;e;UB0;UV0)thatatleastincludesUande,andletSdenotethevectoroftherandomvariablesin(U;e;UB0;UV0)butnotinS.LetQbeasub-vectoroftherandomvector(e;UB0;UV0)thatatleastincludese,andletQdenotethevectoroftherandomvariablesin(U;e;UB0;UV0)butnotinQ.LetIV;IS,IS,IQandIQbethesmallestclosedrectanglethatcontainsthesupportofV,S,S,QandQ,respectively.Intheproofs,weonlyconsiderthecasewherepij(V;U;e;UV0;UB0)&#x]TJ/;ྖ ;.9;‘ ;&#xTf 1;�.9 ;� Td;&#x [00;0foralli;j=0;1sincetheproofsforthecaseswheresomeofthepij=0aresimilarinspiritandsimpler.Also,weabstractawayinthefollowing lemmasfromthepotentialissuesinsituationswheretheconditioningset,e.g.S=s,isofmeasure0(cfBorel-Kolmogorovparadox)becausetheintegralsinPropositionis2areoverthedistributionofS.Lemma3.a)fSjS=s(s)iscontinuousonIS;S;b)fQjQ=q(q)iscontinuousonIQ;Q.Proof:Weproveparta),andtheproofforpartb)issimilar.Therearethreecases1)S=UV0,2)S=UB0and3)S=(UV0;UB0).Forcase1),fUV0jUB0=uB0;U=u;e=e(uV0)=fUV0;UB0jU=u;e=e(uV0;uB0) fUB0jU=u;e=e(uB0)(10)=RfV;UV0;UB0jU=u;e=e(v;uV0;uB0)dv RRfV;UV0;UB0jU=u;e=e(v;uV0;uB0)dvduV0Notethatthenumeratorof(10)isexactlyfSjS=s(s)incase3).BoththenumeratorandthedenominatorarecontinuousasguaranteedbyAssumption4aandProposition1in17.5ofZorich(2004).Sincethedenominatorisstrictlypositive,fUV0jUB0=uB0;U=u;e=e(uV0)iscontinuous.Theproofforcase2)isanalogouswiththerolesofUV0andUB0exchanged.Lemma4.a)¶ ¶vfVjS=s(v)iscontinuousonIV;S.b)¶ ¶vfVjQ=q(v)iscontinuousonIV;Q.Proof:Weonlyproveparta),andtheproofofpartb)issimilar.NotethatfVjS=s;S=s(v)=fV;UV0;UB0jU=u;e=e(v;uV0;uB0) fUV0;UB0jU=u;e=e(uV0;uB0)fVjS=s(v)=RfV;UV0;UB0jU=u;e=e(v;uV0;uB0) fUV0;UB0jU=u;e=e(uV0;uB0)fSjS=s(s)dswheretherstlinefollowsbyBayes'rule,andweintegratebothsidesoverfSjS=s(s)toarriveatthesecondline.Takingderivativeswithrespecttovonbothsides,interchangingdifferentiationandintegration(permittedbyAssumption4a,Lemma3andRoussas(2004)),weobtaintheresultfollowingLemma1.Lemma5.a)¶ ¶vfVjS=s(v)iscontinuousonIV;S.b)¶ ¶vfVjQ=q(v)iscontinuousonIV;Q.Proof:Weonlyproveparta),andtheproofofpartb)issimilar.NotethatafterapplyingBayes'Rule andre-arranging,weobtainfVjS=s;S=s(v)=Pr[UV=0jS=s;S=s]fVjS=s;S=s;UV=0(v)+Pr[UV6=0jS=s;S=s]fVjS=s;S=s;UV6=0(v)=Pr[UV=0jS=s;S=s]Pr[UV=0jV=v;S=s;S=s]fVjS=s;S=s(v) Pr[UV=0jS=s;S=s]+Pr[UV6=0jS=s;S=s]Pr[UV6=0jV=v�uV0;S=s;S=s]fVjS=s;S=s(v�uV0) Pr[UV6=0jS=s;S=s]=Pr[UV=0jV=v;S=s;S=s]fVjS=s;S=s(v)+Pr[UV6=0jV=v�uV0;S=s;S=s]fVjS=s;S=s(v�uV0)MultiplyingbothsidesofthelastlinebyfSjS=s(s)andintegratingovers,takingthepartialderivativewithrespecttov,andapplyingAssumptions4aand5andLemmas3and4,wehavethedesiredresult.Lemma6.a)¶ ¶vPr[GV=i;GB=jjV=v;S=s]and¶ ¶vPr[GV=ijV=v;S=s]arecontinuousonthesetf(v;s):fVjS=s(v)�0gfori;j=0;1.b)¶ ¶vPr[GV=i;GB=jjV=v;Q=q]and¶ ¶vPr[GV=ijV=v;Q=q]arecontinuousonthesetf(v;q):fVjQ=q(v)�0gfori;j=0;1.Proof:Againweonlyproveparta).First,notethatthecontinuousdifferentiabilityofPr[GV=i;GB=jjV=v;S=s]andPr[GV=ijV=v;S=s]isonlyneededonthesetf(v;s):fVjS=s(v)�0gforthepurposeofprovingProposition2becausethesequantitiesarealwaysmultipliedbyfVjS=s(v)whentheyappearinsubsequentproofs.Weconsiderthetwocasesofi=0;1separately.Forcase1wherei=0,Pr[GV=0;GB=jjV=v;S=s]=fVjS=s;GV=0;GB=j(v)Pr[GV=0;GB=jjS=s] fVjS=s(v)=fVjS=s;GV=0;GB=j(v)Pr[GV=0;GB=jjS=s] fVjS=s(v)=fVjS=s;GV=0;GB=j(v)Pr[GV=0;GB=jjS=s] fVjS=s(v)fVjS=s(v) fVjS=s(v)=Pr[GV=0;GB=jjV=v;S=s]fVjS=s(v) fVjS=s(v)=Zp0j(v;u;e;uV0;uB0)fSjV=v;S=s(s)dsfVjS=s(v) fVjS=s(v)=Zp0j�v;u;e;uV0;uB0
fVjS=s;S=s(v)fSjS=s(s) fVjS=s(v)dsfVjS=s(v) fVjS=s(v):Thepartialderivativeoftherighthandsidew.r.t.vinthelastlineiscontinuousonIV;SbyAssumption5andLemmas3,4and5.Forcase2wherei=1, Pr[GV=1;GB=jjV=v;S=s]=ZPr[GV=1;GB=jjV=v;S=s;S=s]fSjV=v;S=s(s)ds=ZPr[GV=1;GB=jjV=v�uV0;S=s;S=s]fVjS=s;S=s(v)fSjS=s(s) fVjS=s(v)ds=Zp1j(v�uV0;u;e;uV0;uB0)fVjS=s;S=s(v)fSjS=s(s) fVjS=s(v)ds:Itspartialderivativew.r.t.viscontinuousonIV;Sforthesamereasonasincase1.SincePr[GV=ijV=v;S=s]=Ã¥jPr[GV=i;GB=jjV=v;S=s],thecontinuousdifferentiabilitywithrespecttovofPr[GV=i;GB=jjV=v;S=s]impliesthatofPr[GV=ijV=v;S=s].ProofofProposition2Forpart(a),theproofisthesameasforpart(a)inProposition1,replacingVwithV,lettingthepair(U;e)servetheroleofUandusingLemma5.Forpart(b),wecanwriteE[YjV=v]=ZE[YjV=v;U=u;e=e]dFU;ejV=v(u;e)=Z(E[YjUV=0;V=v;U=u;e=e]Pr[UV=0jV=v;U=u;e=e]+E[YjUV6=0;V=v;U=u;e=e]Pr[UV6=0jV=v;U=u;e=e])dFU;ejV=v(u;e)=Zz1z2+Zz3z4duV0
[1�z2]z5dFU;e(u;e):(11)wherethesecondlinefollowsfromthelawofiteratedexpectations,andtoeaseexpositionbelow,weusethenotation:z1y(b(v;e);v;u)z2Pr[V=VjV=v;U=u;e=e]z3y(b(v�uV0;e);v�uV0;u)z4fUV0jUV6=0;V=v;U=u;e=e(uV0)z5fVjU=u;e=e(v) fV(v):ThederivativeofE[YjV=v]inequation(11)withrespecttovisdE[YjV=v] dv=Zz01z2z5dFU;e(u;e)+Zz1¶(z2z5) ¶vdFU;e(u;e)+Z¶[(Rz3z4duV0)[1�z2]z5] ¶vdFU;e(u;e)(12)wherez0jdenotesthepartialderivativeofzjwithrespecttov,providedthattheintegrandsarecontinuous. Inaparallelfashion,wecanwriteE[BjV=v]=Zf[z6+z8(1�z7)]z13+[(Zz9z10duV0)z11+(Z(z9+uB0)z12duV0duB0)(1�z11)](1�z13)gz14dFe(e)withz6b(v;e)z7Pr[UB=0jUV=0;V=v;e=e]z8ZuB0fUB0jUV=0;UB6=0;V=v;e=e(uB0)duB0z9b(v�uV0;e)z10fUV0jUB=0;UV6=0;V=v;e=e(uV0)z11Pr[UB=0jUV6=0;V=v;e=e]z12fUV0;UB0jUV6=0;UB6=0;V=v;e=e(uV0;uB0)z13Pr[V=VjV=v;e=e]z14fVje=e(v) fV(v):AndtheanalogousderivativewithrespecttovisdE[BjV=v] dv=Zz06z13z14dFe(e)+Zz6¶ ¶v(z13z14)dFe(e)+¶ ¶vZ[z8(1�z7)z13z14]dFe(e)+¶ ¶vZfZz9z10duV0
z11+ZZ(z9+uB0)z12duV0duB0
(1�z11)g(1�z13z14)gdFe(e);(13)providedthattheintegrandsarecontinuous.Theproofofpart(b)followsfromshowingthatthepartialderivativesofz2,Rz3z4duV0,z5,z7,z8,Rz9z10duV0,;z11,RR(z9+uB0)z12duV0duB0andz13z14,withrespecttovarecontinuous,andnotingthatz1andz6arecontinuousbyAssumptions1a,2,and3a.Fromthisitfollowsthatthereisnodiscontinuityinallbutthersttermontherighthandsideof(12)and(13)atv=0andthattheRKDestimandistheratioofthediscontinuitiesinthersttermsofthosetwoequations.AsshownbyLemma6,z2iscontinuouslydifferentiableinv. z4iscontinuouslydifferentiableinvbecausefUV0jUV6=0;V=v;U=u;e=e(uV0)=Pr[UV6=0jUV0=uV0;V=v;U=u;e=e]fUV0jV=v;U=u;e=e(uV0) Pr[UV6=0jV=v;U=u;e=e]=(1�Pr[UV=0jUV0=uV0;V=v;U=u;e=e])fVjUV0=uV0;U=u;e=e(v)fUV0jU=u;e=e(uV0) fVjU=u;e=e(v) 1�z2;andthederivativeofthelastlineiscontinuousbyLemmas3,5and6.WebreakuptheintegralRz3z4duV0intotwopiecesZz3z4duV0=ZvcUV0z3z4duV0+ZdUV0vz3z4duV0=ZvcUV0y�b+(v�uV0;e);v�uV0;u
z4duV0+ZdUV0vy�b�(v�uV0;e);v�uV0;u
z4duV0wherecUV0anddUV0arethelowerandupperendpointofthesupportofUV0,b+(v;e)=b(v;e)forv�0andalleandb�(v;e)=b(v;e)forv60andalle.Denotey(b(v�uV0;e);v�uV0;u)byz3.Notethatz+3andz�3arecontinuouslydifferentiableinvon[cUV0;v]and[v;dUV0]respectivelybyAssumptions1a,2and3a,wherecUV0anddUV0arethelowerandupperendpointsofthesupportIU0V.Sincez4isalsocontinuouslydifferentiableasshownabove,wecanapplytheNewton-Leibnizformula,whichyields¶ ¶v(ZvcUV0z+3z4duV0+ZdUV0vz�3z4duV0)=ZvcUV0¶ ¶v(z+3z4)duV0+ZdUV0v¶ ¶v(z�3z4)duV0+z+3z4juV0=v�z�3z4juV0=v:ByAssumptions1aand3a,z3iscontinuous,anditfollowsthatz+3z4juV0=v�z�3z4juV0=v=0.SinceRvcUV0¶ ¶v(z+3z4)duV0andRdUV0v¶ ¶v(z�3z4)duV0arecontinuous,Rz3z4duV0iscontinuouslydifferentiableinv.z5iscontinuouslydifferentiableinvbyLemma5andAssumption3a–notethatthecontinuousdiffer-entiabilityoffV(v)=RfVjU=u;e=edFU;e(u;e)isapartofCorollary1,anditfollowsdirectlyfromLemma5;z6iscontinuouslydifferentiablebyAssumption3a,andz7iscontinuouslydifferentiableinvbecausePr[UB=0jUV=0;V=v;e=e]=Pr[UV=0;UB=0jV=v;e=e] Pr[UV=0jV=v;e=e]wherethederivativeoftherighthandsideiscontinuousinvbyLemma6. z8iscontinuouslydifferentiableinvbecauseZuB0fUB0jUV=0;UB6=0;V=v;e=e(uB0)duB0=ZuB0Pr[UV=0jUB0=uB0;V=v;e=e]fUB0jV=v;e=e(uB0) Pr[UV=0jV=v;e=e]duB0=ZuB0Pr[UV=0jUB0=uB0;V=v;e=e]fVjUB0=uB0;e=e(v)fUB0je=e(uB0) fVje=e(v)Pr[UV=0jV=v;e=e]duB0wherethecontinuousdifferentiabilityofthelastlineinvisimpliedbyLemmas3,5and6.ByasimilarapplicationoftheBayes'Rule,wecanshowthatz10iscontinuouslydifferentiableinv.Consequently,Rz9z10duV0iscontinuouslydifferentiableinvbyapplyingthesameargumentusedforRz3z4duV0.Thequantityz11iscontinuouslydifferentiableinvbecauseofLemma6andthatz11=Pr[UB=0;UV6=0jV=v;e=e] Pr[UV6=0jV=v;e=e]z12canbeexpressedasfUV0;UB0jUV6=0;UB6=0;V=v;e=e(uV0;uB0)=Pr[UV6=0;UB6=0jUV0=uV0;UB0=uB0;V=v;e=e] Pr[UV6=0;UB6=0jV=v;e=e]fUV0;UB0jV=v;e=e(uV0;uB0)=Pr[UV6=0;UB6=0jUV0=uV0;UB0=uB0;V=v;e=e] Pr[UV6=0;UB6=0jV=v;e=e]
fVjUV0=uV0;UB0=uB0;e=e(v) fVje=e(v)fUV0;UB0je=e(uV0;uB0);andz12iscontinuouslydifferentiablebyLemmas3,5and6.ItfollowsthatRR(z9+uB0)z12duV0duB0iscontinuouslydifferentiablebythesameargumentasthatforRz3z4duV0.Finally,z13iscontinuouslydifferen-tiableinvbyLemma6andz14byLemma5andAssumption3a.AsaresultofthesmoothnessoftheabovetermsandTheorem5onp.97ofRoussas(2004),wecanwritelimv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0=limv0!0+Zz01z2z5dFU;e(u;e)�limv0!0�Zz01z2z5dFU;e(u;e)=Zlimv0!0+z01�limv0!0�z01z2z5jv=v0dFU;e(u;e)=Zy1(b(0;e);0;u)�b+1(e)�b�1(e)
z2z5jv=v0dFU;e(u;e):(14)Theinterchangeoflimitofintegrationisallowedbythedominatedconvergencetheoremsincez01z2z5is continuousoveracompactrectangle.ThelastlinefollowsfromAssumptions1aand3a.Similarly,wecanwritelimv0!0+dE[BjV=v] dvv=v0�limv0!0�dE[BjV=v] dvv=v0=limv0!0+Zz06z13z14dFe(e)�limv0!0�Zz06z13z14dFe(e)=Zlimv0!0+z06�limv0!0�z06z13z14jv=v0dFe(e)=Z�b+1(e)�b�1(e)
z13z14jv=v0dFe(e):(15)Finally,considerthetermz2z5jv=v0.First,asimilarargumentasin6leadstoz2=Pr[V=VjV=v;U=u;e=e]fVjU=u;e=e(v) fVjU=u;e=e(v):AfterapplyingBayes'Ruleandre-arranging,wehavez2z5jv=v0=Pr[V=VjV=0;U=u;e=e]fVjU=u;e=e(0) fVjU=u;e=e(0)fVjU=u;e=e(0) fV(0)=Pr[V=VjV=0;U=u;e=e]fVjU=u;e=e(0) fV(0)fV(0) fV(0):Similarly,wecanderivez13z14jv=v0=Pr[V=VjV=0;e=e]fVje=e(0) fV(0)fV(0) fV(0):BecausefV(0) fV(0)canbepulledoutoftheintegralinboth(14)and(15),wehavetheresultlimv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+dE[BjV=v] dvv=v0�limv0!0�dE[BjV=v] dvv=v0=Zy1(b(0;e);0;u)j(u;e)dFU;e(u;e)wherej(u;e)=Pr[UV=0jV=0;U=u;e=e](b+1(e)�b�1(e))fVjU=u;e=e(0) fV(0) RPr[UV=0jV=0;e=w](b+1(w)�b�1(w))fVje=w(0) fV(0)dFe(w).NotethatAssumptions3aand6guaranteenon-negative,niteweightsandthatRj(u;e)dFU;e(u;e)=1.A.2IdenticationinthePresenceofBothSlopeandLevelChanges–Remark3InRemark3,weconsidertheidenticationofthetreatmenteffectwhenthereisbothalevelchangeandaslopechangeatthethresholdV=0.Toeaseexposition,denelimv0!0+b0(v0)=b0(0+),limv0!0�b0(v0)=b0(0�),limv0!0+b(v0)=b(0+)andlimv0!0�b(v0)=b(0�).Westudythecasewhereb0(0+)6=b0(0�)andb(0+)6=b(0�),butb()isstillasmoothfunctiononIV=f0g.SimilartothederivationintheproofofProposition1,wecanshowthattheRKestimandidentiesthefollowingparameter limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+db(v) dvv=v0�limv0!0�db(v) dvv=v0=b0(0+)Ry1(b(0+);0;u)fVjU=u(0) fV(0)dFU(u)�b0(0�)Ry1(b(0�);0;u)fVjU=u(0) fV(0)dFU(u) b0(0+)�b0(0�)+Rf[y2(b(0+);0;u)�y2(b(0�);0;u)]fVjU=u(0) fV(0)+[y(b(0+);0;u)�y(b(0�);0;u)]¶ ¶vfVjU=u(0) fV(0)gdFU(u) b0(0+)�b0(0�);whichisingeneralnotreadilyinterpretableasaweightedaverageofthecausaleffect.Ontheotherhand,wecanshowthattheRDestimandidentiesaweightedaverageofthecausaleffectofinterestlimv0!0+E[YjV=v0]�limv0!0�E[YjV=v0] limv0!0+b(v0)�limv0!0�b(v0)=limv0!0+E[y(b(v0);v0;U)jV=v0]�limv0!0�E[y(b(v0);v0;U)jV=v0] limv0!0+b(v0)�limv0!0�b(v0)=E[y(b(0+);0;U)�y(b(0�);0;U) b(0+)�b(0�)jV=0]=E[y1(˜b;0;U)jV=0]where˜bisbetweenb(0+)andb(0�)andthelastlinefollowsfromthemeanvaluetheorem.Similarly,inthefuzzyframeworkofsection2.2.2,itcanbeshownthattheRKestimandnolongeridentiesthecausaleffectofinterestifweallowadiscontinuityinb(
;e)forsomeeatthethreshold.However,theRDestimandstillidentiesaweightedaverageofthecausaleffecty1.Toseethis,letlimv0!0+b(v0;e)b(0;+e),limv0!0�b(v0;e)b(0�;e)andmodifyAssumption3aandAssumption6byreplacingb1(e)withb(0;e);usingnotationsfromtheproofofProposition2,wehavelimv0!0+E[YjV=v0]�limv0!0�E[YjV=v0] limv0!0+E[BjV=v0]�limv0!0�E[BjV=v0]=limv0!0+Rz1z2z5dFU;e(u;e)�limv0!0�Rz1z2z5dFU;e(u;e) limv0!0+Rz6z13z14dFU;e(u;e)�limv0!0�Rz6z13z14dFU;e(u;e)=R[y(b(0+;e);0;u)�y(b(0�;e);0;u)]Pr[UV=0jV=0;U=u;e=e]fVjU=u;e=e(0) fV(0)dFU;e(u;e) R[b(0+;e)�b(0�;e)]Pr[UV=0jV=0;e=e]fVje=e(0) fV(0)dFe(e)=Ry(b(0+;e);0;u)�y(b(0�;e);0;u) b(0+;e)�b(0�;e)[b(0+;e)�b(0�;e)]Pr[UV=0jV=0;U=u;e=e]fVjU=u;e=e(0) fV(0)dFU;e(u;e) R[b(0+;e)�b(0�;e)]Pr[UV=0jV=0;e=e]fVje=e(0) fV(0)dFe(e)=Zy(˜b(e);0;u)y(e;u)dFU;e(u;e)where˜b(e)isavaluebetweenb(0+;e)andb(0�;e)foreacheandy(e;u)=[b(0+;e)�b(0�;e)]Pr[UV=0jV=0;U=u;e=e]fVjU=u;e=e(0) fV(0) R[b(0+;e)�b(0�;e)]Pr[UV=0jV=0;e=e]fVje=e(0) fV(0)dFe(e). A.3ApplyingRKDWhentheTreatmentVariableisBinary–Remark6WeprovidedetailsontheRKidenticationresultstatedinRemark6.Theidentifyingassumptionsare:Assumption1c.(Regularity)(i)ThesupportofUandharebounded:theyaresubsetsofthearbitrarilylargecompactsetIURmandIh=[ch;dh]Rrespectively.(ii)y(t;v;u)iscontinuousonIV;Ufort=0;1.(iii)t(b;v;n)iscontinuouslydifferentiableonIb(V);V;handisstrictlyincreasinginnforallb;v2Ib(V);V.ByAssumption1candtheimplicitfunctiontheorem,wecandenethecontinuouslydifferentiablefunction˜h:Ib(V)IV!Rsuchthatt(b;v;˜h(b;v))=0.Let˜h(b(V);V)betheimageofIb(V);Vunderthemapping˜h.Assumption2c.(SmootheffectofV)y2(t;v;u)iscontinuousonIV;Uforeacht=0;1.Assumption3c.(Firststageandnon-negligiblepopulationatthekink)(i)b(
)isaknownfunction,everywherecontinuousandcontinuouslydifferentiableonIVnf0g,butlimv!0+b0(v)6=limv!0�b0(v).(ii)ThesetAUfu:fV;hjU=u(v;n)�08(v;n)2IV;˜h(b(V);V)ghasapositivemeasureunderU:RAUdFU(u)�0.(iii)t1(b0;0;n0)6=0.Assumption4c.(Smoothdensity)TheconditionaldensityfV;hjU=u(v;n)anditspartialderivativew.r.t.v,¶fV;hjU=u(v;n) ¶v,arecontinuousonIV;h;U.Proposition3.UnderAssumptions1c-4c:(a)Pr(U6ujV=v)iscontinuouslydifferentiableinvatv=08u2IU.(b)limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+dE[TjV=v] dvv=v0�limv0!0�dE[TjV=v] dvv=v0=Z[y(1;0;u)�y(0;0;u)]fV;hjU=u(0;n0) fV;h(0;n0)dFU(u)Proof:Theproofof(a)isanalogoustothatofProposition1(a).Forpart(b),notethatd dvE[TjV=v]=d dvE[1[T�0]jV=v]=d dvZdh˜h(b(v);v)fhjV=v(n)dn=Zdh˜h(b(v);v)¶ ¶v[fhjV=v(n)]dn�[˜h1(b(v);v)b0(v)+˜h2(b(v);v)]fhjV=v(˜h(b(v);v)):where˜hkdenotesthethepartialderivativeof˜hwithrespecttoitsk-thargument.ThesecondlinefollowsfromAssumption1c,andtheinterchangeofdifferentiationandintegrationinthethirdlineispermittedby Assumption4c.Itfollowsthatthedenominatorcanbeexpressedas:limv0!0+dE[TjV=v] dvv=v0�limv0!0�dE[TjV=v] dvv=v0=�[limv0!0+b0(v0)�limv0!0�b0(v0)]˜h1(b0;0)fhjV=0(˜h(b0;0))=�[limv0!0+b0(v0)�limv0!0�b0(v0)]˜h1(b0;0)fV;h(0;n0) fV(0):Similarly,byAssumptions1c,2cand4cd dvE[YjV=v]=d dvE[y(T;V;U)jV=v]=d dvZfZdh˜h(b(v);v)y(1;v;u)fhjV=v;U=u(n)dn+Z˜h(b(v);v)chy(0;v;u)fhjV=v;U=u(n)dngdFUjV=v(u)=ZfZdh˜h(b(v);v)¶ ¶v[y(1;v;u)fhjV=v;U=u(n)]dn+Z˜h(b(v);v)ch¶ ¶v[y(0;v;u)fhjV=v;U=u(n)]dngdFUjV=v(u)�Z[y(1;v;u)�y(0;v;u)]fhjV=v;U=u(˜h(b(v);v))[˜h1(b(v);v)b0(v)+˜h2(b(v);v)]dFUjV=v(u);anditfollowsthatthenumeratoris:limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0=�[limv0!0+b0(v0)�limv0!0�b0(v0)]˜h1(b0;0)Z[y(1;0;u)�y(0;0;u)]fhjV=0;U=u(n0)dFUjV=0(u)=�[limv0!0+b0(v0)�limv0!0�b0(v0)]˜h1(b0;0)Z[y(1;0;u)�y(0;0;u)]fV;hjU=u(0;n0) fV(0)dFU(u):Assumption3c(iii)andtheimplicitfunctiontheoremimplythat˜h1(b0;0)6=0,andtherefore,limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+dE[TjV=v] dvv=v0�limv0!0�dE[TjV=v] dvv=v0=Z[y(1;0;u)�y(0;0;u)]fV;hjU=u(0;n0) fV;h(0;n0)dFU(u)(16)byAssumption3c.Whenthebenetvariablebdirectlyaffectstheoutcome,i.e.Y=y(T;B;V;U),thefuzzyRKDestimandnolongeridentiesthecausaleffectofTonY;rather,theeffectofTonYisconfoundedbythedirecteffectofBonY.IfAssumptions1c-4caremodiedaccordingly,itcanbeshownthattheRKestimandidentiestheparameter Z[y(1;0;u)�y(0;0;u)]fV;hjU=u(0;n0) fV;h(0;n0)dFU(u)| {z }(i)�E[y2(T;b(V);V;U)jV=0] ˜h1(b0;0)fhjV=0(n0)| {z }(ii)whereterm(i)isthesameastheRHSofequation(16)andterm(ii)isthecomponentthatdependsonthedirectimpactofBonY.Totheextentthattheresearchercandeterminethesignof(ii),whichinvolvessigningE[y2(T;b(V);V;U)jV=0]and˜h1(b0;0),shecanboundthetreatmenteffect(i)withtheRKDesti-mand.Forexample,whenhrepresentsastudent'sabilityintheempiricalexampleinRemark6,wemayassertthat˜h1(b0;0)0becausetheexpectedreturnfromcollegeattendanceincreaseswiththeamountofnancialaid.TheconditionalexpectationofthedirectimpactofBonY,E[y2(T;b(V);V;U)jV=0],maybepositivebecauseamoregenerousaidpackageallowsastudentmoretimetofocusonherstudy.Iftheseargumentsweretrue,thentheRKDestimandwouldserveasanupperboundontheeconomicreturnstocollegeattendance.AsstatedinRemark6,wecanalsoallowtherelationshipbetweenBandVtobefuzzyasinsection2.2.2.:B=b(V;e).Inaddition,weallowmeasurementerrorinV, UV,whichhasapointmassat0,andweonlyobserveV=V+UV.WedonotneedtoconsiderthemeasurementerrorinBsincetheobservedvalueofBdoesnotappearintheRKestimand.WeabstractawayfrompotentialmeasurementerrorinT,andleaveitforfutureresearch.Themodiedsetofidentifyingassumptionsare:Assumption1d.(Regularity)InadditiontotheconditionsinAssumption1c,thesupportofeisbounded:itisasubsetofthearbitrarilylargecompactsetIeRk.Assumption3d.(Firststageandnon-negligiblepopulationatthekink)(i)b(v;e)iscontinuousonIV;eandb1(v;e)iscontinuouson(IVnf0g)Ie.Letb+1(e)limv!0+b1(v;e),b�1(e)limv!0�b1(v;e),Aefe:fVje=e(0)�0gandn0(e)˜h(b(0;e);0),thenRfjb+1(e)�b�1(e)jj˜h1(b(0;e);0)jPr[UV=0jV=0;e=e;h=n0(e)]fV;hje=e(0;n0(e))dFe(e)�0.Assumption4d.(Smoothdensity)TheconditionaldensityfV;h;UV0jU=u;e=e(v;n;uV0)anditspartialderivativew.r.t.v,¶fV;h;UV0jU=u;e=e(v;n;uV0) ¶v,arecontinuousonIV;h;UV0;U;e.Assumption5d.(SmoothprobabilityofnoerrorinVandB)AsafunctionoftherealizedvaluesofV,U,e,handUV0,theconditionalprobabilityofUV=0,denotedbyp(v;u;e;uV0;uB0),anditspartialderivativew.r.t.varecontinuousonIV;U;e;h;UV0.Assumption6d.(Monotonicity)(i)Eitherb+1(e)b�1(e)foralleorb+1(e)b�1(e)foralle.(ii)t1(b(0;e);0;n0(e))�0foralleort1(b(0;e);0;n0(e))60foralle. Proposition4.UnderAssumptions1d,2,3d-6d:(a)Pr(U6u;e=e;h=njV=v)iscontinuouslydifferentiableinvatv=08(u;e;n)2IU;e;h.(b)limv0!0+dE[YjV=v] dvv=v0�limv0!0�dE[YjV=v] dvv=v0 limv0!0+dE[TjV=v] dvv=v0�limv0!0�dE[TjV=v] dvv=v0=Z[y(1;0;u)�y(0;0;u)]˜j(u;e)dFU;e(u;e)where˜j(u;e)[b+1(e)�b�1(e)]˜h1(b(0;e);0)Pr[UV=0jV=0;U=u;e=e;h=n0(e)]fV;hjU=u;e=e(0;n0(e)) Rf[b+1(e)�b�1(e)]˜h1(b(0;e);0)Pr[UV=0jV=0;e=e;h=n0(e)]fV;hje=e(0;n0(e))dFe(e):Proof:TheproofissimilartothatofProposition2andisomitted.BEstimationB.1Two-sampleRKDAssuggestedbyareferee,thetriplet(Y;B;V)maynotbejointlyobservedfromasingledatasource.Instead,thevectors(Yi;Vi)fori=1;:::;n1areobservedindataset1and(Bj;Vj)forj=1;:::;n2areobservedindataset2.BecauseoftherequirementofazeropointmassintheUVdistributioninAssumption3a,anRKDtypicallycallsforadministrativedataasopposedtosurveysbasedonacomplexsamplingdesign.Therefore,weassumethat(Yi;Vi)and(Bj;Vj)areindependenti.i.d.samplesasperInoueandSolon(2010).Thevariancesoftherst-stageandreduced-formkinkestimators,ˆtB=ˆk+1�ˆk�1andˆtY=ˆb+1�ˆb�1canbecalculatedbyusingthesharpRKDvarianceestimator,andthecovariancebetweenˆtBandˆtYiszerobytheindependenceassumption.ItfollowsthatthevarianceofthefuzzyRKDestimatorˆtY ˆtBcanbecalculatedbyanapplicationofthedelta-method.TherobustcondenceintervalsinCalonicoetal.(Forthcoming)canbeconstructedanalogouslybysettingthecovariancesbetweentherst-stageandreduced-formestimatorstozero.B.2OptimalBandwidthinFuzzyRKDInthissection,weproposebandwidthselectorsthatminimizetheasymptoticMSEofthefuzzyRD/RKDestimators,buildingonthatinImbensandKalyanaraman(2012)(henceforthIKbandwidth)andCalonicoetal.(Forthcoming)(henceforthCCTbandwidth).FirstweintroducenotationsimilartoCalonicoetal.(Forthcoming).Denem(n)
+andm(n)
�asthen-thrightandleftderivativesoftheconditionalexpectationofarandomvariable(YorB)withrespecttoVatV=0;lettY;nm(n)Y+�m(n)Y�andtB;nm(n)B+�m(n)B�.Inaddition,lets2Y+,s2Y�,s2B+,s2B�,sYB+andsYB�betheconditionalvariancesofYandBandtheir conditionalcovarianceontwosidesofthethreshold.Finally,let˜Vn;p;s(h)=1 tB;n[(ˆm(n)Y+(h)�(�1)sˆm(n)Y�(h))�(m(n)Y+�(�1)sm(n)Y�)]+tY;n t2B;n[(ˆm(n)B+(h)�(�1)sˆm(n)B�(h))�(m(n)B+�(�1)sm(n)B�)];whereˆm(n)
+andˆm(n)
�arethep-thorderlocalpolynomialestimatorofm(n)
+andm(n)
�respectively.NextweproposethelemmathatgeneralizesLemma2ofCalonicoetal.(Forthcoming)andservesasthefuzzyanalogofitsLemma1:Lemma7.AssumethatAssumptions1-3inCalonicoetal.(Forthcoming)aresatisedwithS�p+1andn6p.Ifh!0andnh!Â¥,thenMSEn;p;s=E[(˜Vn;p;s(h))2jfVigni=1]=h2(p+1�n)[B2F;n;p;p+1;s+op(1)]+1 nh1+2nn[VF;n;p+op(1)]whereBF;n;p;r;s=(1 tB;nm(r)Y+�(�1)n+r+sm(r)Y� r!�tY;n t2B;nm(r)B+�(�1)n+r+sm(r)B� r!)n!e0nG�1pJp;rVF;n;p=(1 t2B;ns2Y�+s2Y+ f�2tY;n t3B;nsYB�+sYB+ f+t2Y;n t4B;ns2B�+s2B+ f)n!2e0nG�1pYpG�1penwithen,Gp,YpandJp;rasdenedinCalonicoetal.(Forthcoming).If,inaddition,BF;n;p;r;s6=0,thentheasymptoticMSE-optimalbandwidthishMSE;F;n;p=C1 2p+3F;n;p;sn�1 2p+3whereCF;n;p;s=(2n+1)VF;n;p 2(p+1�n)B2F;n;p;p+1;s.Proof.TheproofofLemma7isanalogoustothatofLemmaA2ofCalonicoetal.(Forthcoming). NotethatLemma2ofCalonicoetal.(Forthcoming)isaspecialcaseofLemma7abovewiths=0.Asinthesharpcase,thebiasofthefuzzyRDestimatordependsonthedifferenceorsumofthederivativeestimatorfromtherststageandtheoutcomeequation.Whetheritisadifferenceorsumdependstheorderofthederivativeestimatedaswellastheorderoftheestimatingpolynomial.BasedonLemma7,weproposeprocedurestocomputetheCCTandIKbandwidthsadaptedtothefuzzyRD/RKDdesignsinthetwofollowingsubsections. B.2.1FuzzybandwidthbasedontheCCTprocedureDenethelocalvarianceestimatorˆVF;n;p(h)=1 ˜t2B;nˆVYY;n;p(h)�2˜tY;n ˜t3B;nˆVYB;n;p(h)+˜t2Y;n ˜t4B;nˆVBB;n;p(h)whereˆVR1R2;n;p(h)=ˆVR1R2+;n;p(h)+ˆVR1R2�;n;p(h)=n!2e0nG�1+;p(h)ˆYR1R2+;p(h)G�1+;p(h)en=nh2n+n!2e0nG�1�;p(h)ˆYR1R2�;p(h)G�1�;p(h)en=nh2nwithR1andR2servingasplaceholdersforYandB,andthequantitiesen,G+;p(h)andˆYR1R2+;p(h)asdenedinCalonicoetal.(Forthcoming).TheconstantsGp,Jp;q,Bn;pandCn;p(K)alsofollowthesamedenitionsinCalonicoetal.(Forthcoming).Step0:UsetheCCTbandwidth(optimalintheMSEsenseforestimatingtY;n)toobtainpreliminaryestimates˜tY;nand˜tB;n.Step1:uandc1.u=ConstK
minfSV;IQRV=1:349g
n�1=5whereConstK=(8p pRK(u)2du 3(Ru2K(u)du)2)1=5;S2VandIQRVdenotethesamplevarianceandinterquartilerangeofV.Theselectionofun,whichisbasedontheSilverman'sruleofthumb,isthesameasinCalonicoetal.(Forthcoming).UseutocomputethevarianceestimatorˆVF;q+1;q+1(ˆu),ˆVF;p+1;q(ˆu)andˆVF;n;p(ˆu).2.Runglobalpolynomialsoforderq+2separatelyforBandYoneachsideofthethreshold.Obtainestimatorsofthe(q+2)-thderivativesonbothsidesofthethresholde0q+2ˆgY;q+2ande0q+2ˆgB;q+2,andusethemtocalculatethebandwidthc:c=C1=(2q+5)F;q+1;q+1;n+qn�1=(2q+5);CF;q+1;q+1;n+q=(2q+3)nu2q+3nˆVF;q+1;q+1(ˆu) 2B2q+1;q+1f1 ˜tB;n[e0q+2ˆgY+;q+2�(�1)n+qe0q+2ˆgY�;q+2]�˜tY;n ˜t2B;n[e0q+2ˆgB+;q+2�(�1)n+qe0q+2ˆgB�;q+2]g2:Step2:hqPerformlocalregressionswithbandwidthctoestimatethe(q+1)-thderivativesonbothsidesofthethresholdandcalculatebandwidthhq:ˆhq=ˆC1=(2q+3)F;p+1;q;n+q+1n�1=(2q+3);ˆCF;p+1;q;n+q+1=(2p+3)nu2p+3nˆVF;p+1;q(ˆu) 2(q�p)B2p+1;qf1 ˜tB;n[e0q+1ˆbY+;q+1(ˆc)�(�1)n+q+1e0q+1ˆbY�;q+1(ˆc)]�˜tY;n ˜t2B;n[e0q+1ˆbB+;q+1(ˆc)�(�1)n+q+1e0q+1ˆbB�;q+1(ˆc)]g2:Step3:h PerformlocalregressionwithbandwidthhqtoestimatethebiasinthefuzzyRD/RKDestimatorˆtF;n;pandcalculatetheresultingmainbandwidthh:ˆh=ˆC1=(2p+3)F;n;p;n+p+1n�1=(2p+3);ˆCF;n;p;n+p+1=(2n+1)nu2n+1nˆVF;n;p(ˆu) 2(p+1�n)B2n;pf1 ˜tB;n[e0p+1ˆbY+;q(ˆhq)�(�1)n+p+1e0p+1ˆbY�;q(ˆhq)]�˜tY;n ˜t2B;n[e0p+1ˆbB+;q(ˆhq)�(�1)n+p+1e0p+1ˆbB�;q(ˆhq)]g2:SimilartoCalonicoetal.(Forthcoming),wehavethefollowingconsistencyresultforthefuzzyCCTbandwidthselectorsproposedabove.Proposition5.(ConsistencyoftheCCTBandwidthSelectors)Letn6pq.SupposeAssumptions1-3inCalonicoetal.(Forthcoming)holdwithS&#x]TJ/;྆ ;.9;‘ ;&#xTf 1;�.96; 0 ;&#xTd [;q+2andthat1 ˜tB;n[e0q+2ˆgY+;q+2�(�1)n+qe0q+2ˆgY�;q+2]�˜tY;n ˜t2B;n[e0q+2ˆgB+;q+2�(�1)n+qe0q+2ˆgB�;q+2]p!c6=0:Step1.IfBF;p+1;q;q+1;n+p+16=0,thenˆhq hMSE;F;p+1;q;n+p+1p!1andMSEF;p+1;q;n+p+1(ˆhq) MSEp+1;q;n+p+1(hMSE;p+1;q;n+p+1)p!1Step2.IfBF;n;p;p+1;06=0,thenˆh hMSE;F;n;p;0p!1andMSEF;n;p;0(ˆh) MSEF;n;p;0(hMSE;F;n;p;0)p!1Proof:BecausetheCCTbandwidthoptimalforestimatingtY;nshrinksattherateofn�1 2p+3,theprelim-inaryestimators,˜tY;nand˜tB;n,areconsistent.TherestoftheprooffollowstheargumentsintheproofofTheoremA4inCalonicoetal.(Forthcoming).TheoptimalfuzzyRDbandwidthisproposedinImbensandKalyanaraman(2012).WesuggestanextensiontobeusedinthefuzzyRKDcase(n=1)andstatethebandwidthselectorsforagenericn.Calonicoetal.(Forthcoming),Calonicoetal.(inpress)andCalonicoetal.(2014b)adapttheIKbandwidthselectionproceduretohqsothatitcanbeusedtobias-correcttheRDestimator.Buildinguponthesestudies,weproposeafurtherextensionofthebandwidthselectorforhqtoageneralfuzzydesignwithadiscontinuityinthen-thderivative. B.2.2FuzzybandwidthbasedontheIKprocedureStep1:UsethesharpIKbandwidth(optimalintheMSEsenseforestimatingtY;n)toobtainpreliminaryestimates˜tY;nand˜tB;nStep2:u1.ˆu=1:84
SV
n�1=52.Useˆh1toestimateˆs2Y(ˆu),ˆs2B(ˆu),ˆsYB(ˆu)andˆf(ˆu)asspeciedinImbensandKalyanaraman(2012)(notethatImbensandKalyanaraman(2012)useWtodenotethetreatmentvariableanduseh1todenotethispreliminarybandwidth).Step3:hqRunglobalregressions:Y=dY
1[V�0]
Vn+aY0+aY1V+:::+aYq+2Vq+2+eYB=dB
1[V�0]
Vn+aB0+aB1V+:::+aBq+2Vq+2+eBanduseˆaYq+2andˆaBq+2toconstruct•ˆhY�;q+1=(Cq+1;q+1(KU)ˆs2Y�(ˆu)=ˆf(ˆu) n�(ˆaYq+2)2)1 2q+5•ˆhY+;q+1=(Cq+1;q+1(KU)ˆs2Y+(ˆu)=ˆf(ˆu) n+(ˆaYq+2)2)1 2q+5•ˆhB�;q+1=(Cq+1;q+1(KU)ˆs2B�(ˆu)=ˆf(ˆu) n�(ˆaBq+2)2)1 2q+5•ˆhB+;q+1=(Cq+1;q+1(KU)ˆs2B+(ˆu)=ˆf(ˆu) n+(ˆaBq+2)2)1 2q+5:Perform(q+1)-thorderlocalregressionsofYandBoneachsideofthethresholdwiththeuniformkernelKUandbandwidthsˆhY;q+1andˆhB;q+1.UsingintheresultingestimatorsˆbY;q+1(ˆhY;q+1)andˆbB;q+1(ˆhB;q+1),weobtainˆhq=ˆC1=(2q+3)F;p+1;q;n+q+1n�1=(2q+3);ˆCF;p+1;q;n+q+1=Cp+1;q(K)1 ˆf(ˆu)f1 ˜t2B;n[ˆs2Y+(ˆu)+ˆs2Y�(ˆu)]�2˜tY;n ˜t3B;n[ˆs2YB�(ˆu)+ˆs2YB+(ˆu)]+˜t2Y;n ˜t4B;n[ˆs2B�(ˆu)+ˆs2B+(ˆu)]g f1 ˜tB;n[e0q+1ˆbY+;q+1(ˆhY+;q+1)�(�1)n+q+1e0q+1ˆbY�;q+1(ˆhY�;q+1)]�˜tY;n ˜t2B;n[e0q+1ˆbB+;q+1(ˆhB+;q+1)�(�1)n+q+1e0q+1ˆbB�;q+1(ˆhB�;q+1)]g2:Step4:h Runglobalregressions:Y=dY
1[V�0]
Vn+gY0+gY1V+:::+gYq+1Vq+1+eYB=dB
1[V�0]
Vn+gB0+gB1V+:::+gBq+1Vq+1+eBanduseˆgYq+1andˆgBq+1toconstruct•ˆhY�;q=(Cp+1;q(KU)ˆs2Y�(ˆu)=ˆf(ˆu) n�(ˆgYq+1)2)1 2q+3•ˆhY+;q=(Cp+1;q(KU)ˆs2Y+(ˆu)=ˆf(ˆu) n+(ˆgYq+1)2)1 2q+3•ˆhB�;q=(Cp+1;q(KU)ˆs2B�(ˆu)=ˆf(ˆu) n�(ˆgBq+1)2)1 2q+3•ˆhB+;q=(Cp+1;q(KU)ˆs2B+(ˆu)=ˆf(ˆu) n+(ˆgBq+1)2)1 2q+3:Performq-thorderlocalregressionsofYandBoneachsideofthethresholdwithbandwidthsˆhY;qandˆhB;qandobtainlocalregressionestimatorsˆbY;q(ˆhY;q)andˆbB;q(ˆhB;q).Pluggingthemin,wehaveanestimateofthemainbandwidthh:ˆh=ˆC1=(2p+3)F;n;p;n+p+1n�1=(2p+3);ˆCF;n;p;n+p+1=Cn;p(K)1 ˆf(ˆu)f1 ˜t2B;n[ˆs2Y+(ˆu)+ˆs2Y�(ˆu)]�2˜tY;n ˜t3B;n[ˆs2YB�(ˆu)+ˆs2YB+(ˆu)]+˜t2Y;n ˜t4B;n[ˆs2B�(ˆu)+ˆs2B+(ˆu)]g f1 ˜tB;n[e0p+1ˆbY+;q(ˆhY+;q)�(�1)n+p+1e0p+1ˆbY�;q(ˆhY�;q)]�˜tY;n ˜t2B;n[e0p+1ˆbB+;q(ˆhB+;q)�(�1)n+p+1e0p+1ˆbB�;q(ˆhB�;q)]g2:WehaveasimilarconsistencyresultfortheIKbandwidthselectorsbelow.Proposition6.(ConsistencyoftheIKBandwidthSelectors)Letn6pq.SupposeAssumptions1-3inCalonicoetal.(Forthcoming)holdwithS&#x]TJ/;྆ ;.9;‘ ;&#xTf 1;
.28; 0 ;&#xTd [;q+2andthataYq+2,aBq+2,gYq+1andgBq+1arenonzero.Selectorforhq:IfBF;p+1;q;q+1;n+p+16=0,thenˆhq hMSE;F;p+1;q;n+p+1p!1andMSEF;p+1;q;n+p+1(ˆhq) MSEp+1;q;n+p+1(hMSE;p+1;q;n+p+1)p!1Selectorforh.IfBF;n;p;p+1;06=0,thenˆh hMSE;F;n;p;0p!1andMSEF;n;p;0(ˆh) MSEF;n;p;0(hMSE;F;n;p;0)p!1Proof:BecausetheIKbandwidthoptimalforestimatingtY;nshrinksattherateofn�1 2p+3,thepreliminaryestimators,˜tY;nand˜tB;n,areconsistent.Thedensity,varianceandcovarianceestimatorsareconsistentasarguedinImbensandKalyanaraman(2012).Sincethehigherderivativeestimatorsalsoconvergetotheir populationcounterparts,ˆCF;p+1;q;n+q+1p!CF;p+1;q;n+q+1andˆCF;n;p;n+p+1p!CF;n;p;n+p+1,andtheresultsofthepropositionfollow. B.3DiscussionofGanongandJaeger(2014)IfweplottherelationshipbetweenbaselineearningsandthelogtimetonextjobnotjustlocallyaroundthekinkpointsasinFigures5-8,butgloballyforthewholerangeofbaselineearningsavailableinthedata,itturnsoutthatthisrelationshipishighlynonlinear.Theupper-andlower-leftpanelsinAppendixFigure4showtherawdataplots,withobservationscenteredattheannualcutoffvaluesforthebottomandtop-kinkpointsintheunemploymentbenetformula.Thesegraphsclearlyshowtheslopechangeatthekinkpoints.Butoverthefullrangeofbaselineearningstherelationshipappearstofollowau-shape.WeareinterestedinwhetherthisshapeisdrivenbyinstitutionalfeaturesoftheUIsystemthatarenotaccountedforinouranalysis,orwhetheritisduetocompositionalchangesinobservablesalongthedistributionofbaselineearnings.Forexample,thefractionofloweducatedorfemaleindividualsishigheratlowlevelsofbaselineearningsthanforhighlevels.Toassesstheroleofcompositionalchanges,weperformaregressionoflogtimetothenextjobonallobservablecharacteristicsona5%subsampleoverthefullrangeofbaselineearnings.34Thenweplottheresidualsfrompredictionsbasedonthisregressionfortheremaining95%ofobservationintherightcolumnofAppendixFigure4.Toaccountforchangingthresholdsovertime,wecenterthedataaroundtheannualbottomandtopkinkthresholdlevels.AppendixFigure4showsthatmostofthecurvatureintheglobalrelationshipcanbeexplainedbychangesinobservablecharacteristics.Theresidualplotsaremostlyatalongtherangeofbaselineearnings,andthenonlinearityatthekinkpointsbecomesevenmoresalient.GanongandJäger(2014)raiseconcernsaboutthesensitivityoftheRKDestimateswhentherelationshipbetweentherunningvariableandtheoutcomeishighlynonlinear.Toassessthissensitivity,theysuggestapermutationtestwhichshiftsthethresholdvalueforthekinkacrossthesupportoftherunningvariableandestimatesslopechangesintheseplacebosamples.OneexampletodemonstratetheirmethodisbasedontherawdataplotforthebottomkinksampleinAppendixFigure4.GanongandJäger(2014)arguethatbecauseofthecurvatureintheglobalrelationshipsmanyoftheplaceboestimatesresultinlargeandsignicantslopechanges.Wereplicatethisexercisebasedontherawindividualleveldata.FollowingGanongandJäger(2014),werandomlydraw200cutoffvaluesforboththebottomandtopkinksamplesovertherangeofbaseline 34Thelistofcovariatesincludes:indicatorvariablesforgender,maritalstatus,Austriancitizenship,education(6categories),decilesforage,tenure,size,andrecallrateatthermofthelastjob,monthlywagelastjob(quintiles),daysworkedoverthelast5years(quartiles),8industrygroupsinteractedwithbluecollarstatus,region,year,andmonthdummies. earningwherethereisnokinkinthebenetformula.Theempiricalc.d.f.'softhecoefcientestimateandthecorrespondingt-statisticareshownbythesolidlinesinAppendixFigure5,wherethegraphsinthetoprowcorrespondtothebottomkinksampleandthoseinthebottomrowcorrespondtothetopkinksample.35Thec.d.f.oftheplacebocoefcientestimateissimilartoFigure3inGanongandJäger(2014),althoughthemedianvalueissomewhatlowerinoursample.Buttheplaceboestimatesndahighshareofcoefcientestimatesthataresignicantlydifferentfromzero.Thesamepatternappearsforthetopkinksample,althoughinthiscasethesignoftheplaceboestimatesistheoppositeofthatofthereduced-fromRKDestimateinTable2.Sincethecurvatureintheglobalrelationshipcanbeexplainedbyobservablecharacteristics,werepeatthepermutationtestontheresiduals.Thecorrespondingc.d.f.'softheplacebocoefcientestimateandt-statisticareplottedusingthedashedlinesinthegraphsinAppendixFigure5.Theplacebopointestimatesnowhaveahighdensityaroundzeroandtheabsolutevaluesofthet-statisticsarebelow2.OurexercisehighlightstheimportanceoftheroleofcurvatureheterogeneityinthepermutationtestofGanongandJäger(2014).Specically,thetestmaynotbeinformativewhenthecurvatureoftheconditionalexpectationfunctionm(v)E[YjV=v]changes.Supposem(v)islocallypiece-wiselineararoundthethresholdv=0withanon-zerokinkbuthassubstantialcurvaturefarawayfromthethreshold,asappearstobetrueinourdata.Inthiscase,thepermutationtestmayfailtodetectthekinkatthethreshold,asthecurvatureawayfromthethresholdcanleadtomanysignicantlocallinearestimates.Basedonasimilarargument,thepermutationtestmayalsohavepoorsizecontrolwhenthecurvatureofm(v)changes.Hence,ifresearcherswishtoconductthepermutationtest,itwillbeimportanttocontrolforconfoundingnonlinearitiesbytakingthedistributionofobservablesintoaccount,aspointedoutbyAndo(2014).Weconclude,thataftercontrollingforcovariatesourdesignpassestheGanongandJäger(2014)permutationtest.Inadditiontothepermutationtest,GanongandJäger(2014)alsoassesstheperformanceofvariousestimatorsintwoDGP'sbasedonsinefunctions.Relyingupontheirsimulationresults,theyadviseprac-titionerstouseeither1)whatwecallthebias-correctedlocalquadraticestimatorwiththedefaultCCTbandwidthor2)cubicsplinewiththegeneralizedcrossvalidationbandwidth.Inanadditionalsimulationstudy,wehavemodiedtheGanongandJäger(2014)DGP'sslightly,36yieldingaMonteCarloexercise 35Weestimatelocallinearmodelswithaxedbandwidthof2500forthebottomkinksampleandaxedbandwidthof4000forthetopkinksample.36TheDGP'sinGanongandJäger(2014)arem(v)=10x1[x�0]+sin(k(v�0:1))+v2+ewherek=5;15.InourDGP,weset inwhichtherobustcondenceintervalforthelocalquadraticestimatorwiththedefaultCCTbandwidthselectorhasamuchlowercoveragerate(13.9%)thanitslocallinearcounterpart(80.4%).37Despitetheinferiorperformanceofthebias-correctedlocalquadraticestimatorinthisparticularDGP,wedonotarguethatalocalquadraticestimatorshouldneverbeused.Rather,webelievethatcautionisneededinattempt-ingtodrawpracticaladvicefromspecicMonteCarlostudies,andsuggestinsteadthatresearchersuseacombinationofmethods–includingsimulationsstudiesbasedonDGP'sthatcloselyresembletheiractualdata–todetermineapreferredestimator(orsetofestimators)fortheirparticularsetting. k=25.37Ineachrepetition,wecomputethemainandthepilotCCTbandwidthsandtherobustcondenceintervalusingthenearest-neighborvarianceestimator.ThelowercoverageratesofthelocalquadraticcondenceintervalisaresultofthelargerCCTmainbandwidth(0.14)ascomparedtoitslinearcounterpart(0.04). CAJobSearchModelwithWage-DependentUIBenetsThisAppendixdescribesanequilibriumwagepostingmodelwithawage-dependentUIbenet,andamaximumbenetlevel.WeasktowhatextentthemodelisconsistenttheRKDidentifyingassumptions,andreachtwomainconclusions.First,whenthereisakinkintheUIbenetformula,abaselinemodelpredictsakinkinthedensityofwagesamongjob-losersatthelevelofwagescorrespondingtothemaximumbenet.Second,thispredictionreliesoncompleteinformationaboutthelocationofthekinkinthebenetscheduleandisnotrobusttoallowingforsmallerrorsinagentsbeliefsaboutthelocationofthekinkinthebenetschedule.Setup.Consideraninnitehorizon,discrete-time,posted-wagemodelofjobsearchwithanexogenousdistributionofwageoffers,andequallyefcientsearchamongemployedandunemployedagents.Withalevelofsearchintensitysthearrivalrateofjoboffersisl
s;thereisalsoanexogenousjobdestructionrateofd.Thereisastrictlyincreasingandconvexcost-of-searchfunctionc(
)withc(0)=0.Wageofferscomefromastationary,twicecontinuouslydifferentiablec.d.f.F(
).ThesetupisidenticaltothemodelusedbyChristensenetal.(2005),exceptthatwecasttheproblemindiscretetime(withadiscountrateb)andassumeawage-dependentUIbenet.38Specically,weassumethattheUIbenetbisafunctionofthelastwagereceivedbeforebeinglaidoff,w�1,givenbytheformulab(w�1)b +rmin(w�1;Tmax),wherer1andb(w)wforallw.Asinmostactualbenetsystems,agentswithapreviouswageabovethethresholdTmaxreceiveamaximumbenetlevel¯b=b +rTmax.Thedependenceofbenetsonpreviouswagesaddstwonovelconsiderationstothestandardsearchmodel:1)whenchoosingsearchintensityandwhethertoacceptawageoffer,anagentmusttakeintoaccounttheeffectofthewageonfutureUIbenets;2)whentakinganewjob,anunemployedworkerresetstheirbenetlevel.BecauseweassumethatUIbenetslastindenitely,andthatthebenetisresetimmediatelyupontakinganewjob,ourmodelarguablyover-emphasizesboththeseconsiderationsrelativetoamorerealisticsettingwherebenetscanexpire,andUIentitlementisbasedonearningsoverapreviousbaseperiodofseveralquarters'duration.Anagent'schoiceproblemischaracterizedbytwovaluefunctions:Wem(w),thevaluefunctionforbeing 38Totranslatethemodeltoourgeneralizedregressionsetting,notethatwecanallowforunrestrictedheterogeneityandindexallthemodel'selementsbyU,theunobservedtype.ThediscussionbelowisconditionalonthetypeU,andwesuppressanynotationindicatingthevalueofU. employedwithcurrentwagew,andWun(w�1),forbeingunemployedwithpreviouswagew�1:Wem(w)=maxs0w�c(s)+b(1�d)lsZmaxfWem(x);Wem(w)gdF(x)+(1�ls)Wem(w)+dWun(w)(17)Wun(w�1)=maxs0b(w�1)�c(s)+blsZmaxfWem(x);Wun(w�1)gdF(x)+(1�ls)Wun(w�1):(18)NotethatWunisanincreasingfunctionofthepreviouswageforw�1Tmax,sinceahigherpreviouswageentitlestheagenttohigherbenets.OncethepreviouswagereachesthethresholdTmax,how-ever,thereisnofurtherincreaseinWun:thusthevaluefunctionWuniskinkedatw�1=Tmax,withW0un(w�1)&#x]TJ/;྆ ;.9;‘ ;&#xTf 1;
.81; 0 ;&#xTd [;0forw�1TmaxandW0un(w�1)=0forw�1&#x]TJ/;྆ ;.9;‘ ;&#xTf 1;�.90;
0 ;&#xTd [;Tmax.InspectionofthevaluefunctionsshowsthatthisinturninducesakinkinWem(w)atw=Tmax,providedthatd&#x]TJ/;྆ ;.9;‘ ;&#xTf 1;�.90;
0 ;&#xTd [;0.OptimalSearchBehavior.Itcanbeshownthattheoptimalbehaviorischaracterizedbyareservationwagestrategywhileemployed,anotherreservationwagestrategywhileunemployed,andachoiceofopti-malsearchintensitysem(w)whenemployedatwagewandsun(w�1)whenunemployedwithpreviouswagew�1.39Clearly,anemployedworkerwillacceptanywageofferthatexceedshercurrentwage.Anunem-ployedworkerwithpreviouswagew�1willacceptanywageofferwwithWem(w)Wun(w�1),implyingareservationwageR(w�1)suchthatWem(R(w�1))=Wum(w�1).ItiswellknownthatwhentheUIbenetisaxedconstantbtheoptimalstrategyforanunemployedworkeristotakeanyjobwithwb,implyingR(w�1)=b,sincethereisnoextradisutilityofworkversusunemployment,andthearrivalrateandsearchcostsarethesamewhetherworkingornot.Thissimpleruleisnolongertruewhenthebenetsdependonw�1.Consideranunemployedworkerwithanofferw=b(w�1):Takingthejobwillyieldthesameowutilityasremainingonunemployment,butwhenthejobendsshewillreceivealowerfutureUIbenet(assumingthatthebenet-replacementrateislessthan1).Thus,ahigherwageofferisrequiredforindifference,implyingthatR(w�1)&#x]TJ/;྆ ;.9;‘ ;&#xTf 1;�.90;
0 ;&#xTd [;b(w�1)whenw�1Tmax.Giventhevaluefunctionsabove,andastrictlyincreasing,convex,andtwicecontinuouslydifferentiablecostfunctionc(
),wecanimplicitlysolvefortheoptimalsearchfunctionssun(
)andsem(
)viatherst 39ItcanbeshownthatWem(w)isstrictlyincreasinginwandthatWun(w�1)isincreasinginw�1,whichleadstoreservationwagestrategiesineachcase. orderconditionsforinteriorsolutionsfor(17)and(18),c0(sem(w))=b(1�d)lZ¯ww[Wem(x)�Wem(w)]dF(x)(19)c0(sun(w�1))=blZ¯wR(w�1)[Wem(x)�Wun(w�1)]dF(x);where wistheupperboundofthesupportoftheofferdistribution.ConsiderationoftheserstorderconditionsshowsthattheoptimallevelsofsearchintensitybothhaveakinkatthewagethresholdTmax.Forexample,therightderivativeofsem(w)atw=Tmaxis:s0em(Tmax+)=�b(1�d)l(1�F(T)) c00(sem(T))W0em(Tmax+);whilethederivativefromtheleftis:s0em(Tmax�)=�b(1�d)l(1�F(T)) c00(sem(T))W0em(Tmax�):SinceWem(w)hasakinkatw=Tmax,W0(Tmax+)6=W0(Tmax�)andtheleftandrightlimitsofthederivativeofsem(w)aredifferentatw=Tmax.Asimilarargumentappliestothederivativeofsun(w�1)atw�1=Tmax.SteadyStateWageDistribution.AstandardwagepostingmodelyieldsasteadystateunemploymentrateuandasteadystatedistributionofwagesG(w)thatstochasticallydominatesthedistributionofwageoffersF(w),reectingthefactthatemployedworkersarealwayssearchingforhigherwageoffers.Whenthebenetlevelvariesacrossunemployedworkers,andworkerswithdifferentbenetlevelshavedifferentreservationwages,thereisalsoasteadystatedistributionofpreviouswagesinthestockofunemployedworkers,whichwedenotebyH(w).40Inthesteadystate,theinowintothesetofworkersemployedwithawageofworlessmustequaltheoutow:ulZ¯w0sun(x)[maxfF(w)�F(R(x));0g]dH(x)=dG(w)(1�u)+(1�d)l(1�u)Zw0sem(x)dG(x)(1�F(w))(20)Inflow(w)=Layoff(w)+Offer(w)
(1�F(w)):(21) 40Itcanbeshownthat:H(w)=Rw0dG(x) lsun(x)�lsun(x)F(R(x)) R¯w0dG(x) lsun(x)�lsun(x)F(R(x)): ThequantityInflow(w)isthefractionofthestockofunemployedworkerswhoreceiveawageofferthatexceedstheirreservationwage,butislessthanw.Ontherighthandside,theproportionwithawagelessthanwanddisplacedwithprobabilitydisgivenbyLayoff(w)),whiletheproportionofindividualswhowillleavejobsthatpaylessthanwforjobsthatpaymorethanwisgivenby(Offer(w)
(1�F(w))).NowconsiderawwithinaneighborhoodofthethresholdTmax.41Considertheaboveowequationforobservedwagesbetweenw+handw.Somere-arrangementyields:Inflow(w+h)�Inflow(w)+Offer(w)((F(w+h))�F(w))=Layoff(w+h)�Layoff(w)+(Offer(w+h)�Offer(w))(1�F(w+h)):ApplyingameanvaluetheoremforStieltjesintegralsontherighthandside,re-arranging,anddividingbyh,weobtainInflow(w+h)�Inflow(w) h+Offer(w)F(w+h)�F(w) h d(1�u)+(1�d)l(1�u)cO(1�F(w+h))=G(w+h)�G(w) hwhereinfx2[w;w+h]sem(x)cOsupx2[w;w+h]sem(x).ByassumptionthedistributionofwageoffersF(
)isdifferentiable.Moreover,itcanbeshownthatthesearchintensitychoiceofemployedworkersiscontinuous,andthatInflow(w)isdifferentiableinaneighborhoodofT.42Takingthelimitash!0,weobtain:Inflow0(w)+Offer(w)f(w) (d(1�u)+(1�d)l(1�u)sem(w)(1�F(w)))=g(w)(22)whichmeansthatthedensityofwagesg(w)iswell-denedinthisneighborhood.Itcanbeshownthateveryfunctionofwontheleft-handsideofthisequationiscontinuouslydiffer-entiableatw=Tmaxexceptthesearchintensityfunctionsem(
),whichiskinkedatTmax.Asnotedabove,thisarisesbecauseofthekinksinthevaluefunctionsWem(
)andWum(
)atthewagethresholdTmax.Asaconsequence,thedensityofwagesamongemployedworkershasakinkatw=Tmax.Assumingthatthejobdestructionrateisconstantacrossalljobs,thepopulationofnewUIclaimantshasthesamedistributionofpreviouswagesasthepoolofemployedworkers.Asaconsequence,thismodelimpliesthatthedensityof 41WechooseaneighborhoodofTmaxinwhichw�R(Tmax).SuchaneighborhoodalwaysexistsbecauseTmax�R(Tmax)–aworkerwhoacceptsawageTmaxwillbestrictlybetteroffthanremainingunemployedwiththemaximumbenet¯b.42DifferentiabilityofInflow(
)followsbecauseinaneighborhoodofT,w�R(x)forallx.ThusInflow(w)=ulR¯w0sun(x)[F(w)�F(R(x))]dH(x).WecandifferentiateundertheintegralsignbecausethederivativeoftheintegrandwithrespecttowiscontinuousintherectangledenedbytheneighborhoodofTand[0; w],andF(
)isdifferentiablebyassumption. wagesamongnewUIclaimantshasakinkatw=Tmax.ModelwithImperfectInformationAboutBenetSchedules.Wenowconsideravariantoftheprecedingmodelinwhichagentshaveimperfectinformationonthelocationofthekinkpointinthebenetschedules.Weshowthatthepredictionofakinkeddensityisnotrobusttosmallerrors.Toproceed,assumethatthetruekinkinthebenetscheduleoccursatw=Tmax,buttheagentmakeschoicesassumingthekinkisatTmax+e.Thisleadstovaluefunctions,indexedbytheerrore,Weem(w)andWeun(w)paralleltothoseinequations(17)and(18).Inaddition,thereisanothervaluefunctiondenedby:Weun(w�1)=maxs0b(w�1)�c(s)+blsZmaxfWeem(x);Weun(w�1)gdF(x)+(1�ls)Weun(w�1):Weun(w�1)istheperceivedvalueofunemploymentforaworkerusinganincorrectbenetformula,whereasWeun(w�1)istheperceivedvalueofunemploymentofanunemployedworkerwhoisreceivingbenetsb(w�1)basedonthecorrectformula,butisevaluatingthevalueofpotentialfutureemploymentusingWeem.Theresultofthissmalloptimizationerroristhatactualsearchintensityforanindividual(intheem-ployedandunemployedstate)willbegivenbytherst-orderconditionsc0(seem(w))=b(1�d)lZ¯ww[Weem(x)�Weem(w)]dF(x)c0(seun(w�1))=blZ¯wRe(w�1)[Weem(x)�Weun(w�1)]dF(x):Moreover,thereservationwageforemployedagentsisstilltheircurrentwage,whilethereservationwageifunemployedisRe(w�1),implicitlydenedbyWeem(Re(w�1))=Weun(w�1).WithanerrorintheperceivedkinkthesteadystateowequationforthewagedensityG(w)isthesameasinequation(20),afterreplacingsem(
),sun(
)withseem(
),seun.Asaresult,thesteadystatedensityforapopulationofagentsoftype”e”exhibitsakinkatTmax+e.Ifthetruepopulationcontainsamixtureofagentswithdifferentvaluesofe,drawnfromadensityfe(
),thenthesteadystateowequationforthedensityofwagesisthesameasinequation(20),afterreplacingsem(
),sun(
)withE[seem(
)],E[seun(
)]whereexpectationsaretakenwithrespecttofe(
).Itcanbeshownthatifeiscontinuouslydistributed,thenE[seem(x)]willbecontinuouslydifferentiable,leadingtoacontinuouslydifferentiablesteadystatedensityg(
).43Thus,acontinuousdistributionoferrorsinagents'beliefsaboutthelocationofthekinkpointwillsmoothoutthekinkthatariseswithfullinformation. 43Specically,E[seem(w)]iscontinuouslydifferentiablebecausedseem(w) dwiscontinuousatw=Tmaxalmosteverywhere(itisnon-differentiableonlywhene=0,ameasurezeroevent). .07 .08 .09 .11 -2000 -1000 0 1000 2000 3000 4000 Base Year Earnings Relative to T-min .14 -14000 -9000 -4000 1000 Base Year Earnings Relative to T-maxTop Kink Sample -9000-400010006000Base Year Earnings Relative to T-maxAge.18.2.22.24.26age in years-14000-9000-400010006000Base Year Earnings Relative to T-maxFemale.2.3.4.5.6.7age in years-14000-9000-400010006000Base Year Earnings Relative to T-maxBluecollar.1.15.2.25.3age in years-14000-9000 AppendixFigure4:GlobalRelationshipbetweenLogTimetoNextJobandBaselineEarnings AppendixFigure5:DistributionoftheCoefcientEstimateandt-StatisticinthePermutationTest AppendixTable1:TestingforSmoothDensityofPreviousEarningsUsingParametricModelsofFrequencyDistributionsinBottomKinkandTopKinkSample s 1000PearsonError)(Pvalue)Criterion(1)(2)(3)(4) 20.44683.0293.02(0.172)(2.6%)0.08073.8187.81(0.091)(7.9%)0.24072.4390.43(0.181)(6.9%)50.10969.3991.39(0.308)(7.7%) 0.070142.55152.55(0.035)(0.0%)0.220124.15138.15(0.081)(0.0%)40.20075.6293.62(0.124)(5.0%)50.15875.5097.50(0.213)(3.5%) Appendix Table 2a: Summary of Monte Carlo Studies, DGP Design Based on Bottom Kink Sample (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 1. Local Linear, No Bias Correction Default CCT 470 - 0.31 0.43 0.94 20.3 4.81 1.00 0.72 0.52 22.6 CCT, no regularization 998 - 0.04 0.22 0.89 4.7 2.04 0.77 0.68 0.46 3.69 Fuzzy CCT 973 - 0.04 0.22 0.89 22.3 1.90 0.77 0.75 0.56 3.06 Fuzzy IK 1,415 - 0.00 0.11 0.85 1.60 1.30 0.55 0.83 0.69 1.01 FG 2,600 - 0.00 0.08 0.60 1.40 1.35 0.02 1.32 1.75 0.09 Global (all data) 4,564 - 0.00 0.14 0.00 1.29 1.25 0.01 1.22 1.49 0.07 2. Local Linear, Bias-Corrected Default CCT 470 970 0.60 0.62 0.94 7.0E+02 8.5 1.00 0.71 0.51 71.1 CCT, no regularization 998 1,311 0.33 0.47 0.92 10.6 3.96 0.97 0.38 0.14 15.6 Fuzzy CCT 973 1,363 0.29 0.45 0.92 1.2E+03 3.76 0.98 0.39 0.16 14.0 Fuzzy IK 1,415 1,615 0.21 0.36 0.90 3.84 2.96 0.92 0.31 0.10 8.70 FG 2,600 3,367 0.00 0.16 0.89 1.89 1.69 0.82 0.87 0.75 2.10 Global (all data) 4,564 4,564 0.00 0.18 0.83 2.17 1.97 0.75 1.54 2.36 1.54 3. Quadratic, No Bias Correction Default CCT 681 - 0.78 0.93 0.95 540 13.5 1.00 -0.38 0.15 183 CCT, no regularization 1,211 - 0.38 0.50 0.91 84 6.6 0.98 0.41 0.16 43.2 Fuzzy CCT 1,291 - 0.35 0.47 0.91 67 5.6 0.98 0.57 0.32 31.1 Fuzzy IK 1,778 - 0.09 0.27 0.90 135 2.77 0.95 0.39 0.15 7.55 FG 3,718 - 0.00 0.14 0.88 2.12 1.90 0.72 1.27 1.60 2.00 Global (all data) 4,564 - 0.00 0.17 0.76 2.56 2.39 0.54 2.13 4.54 1.15 4. Local Quadratic, Bias-Corrected Default CCT 681 1,079 0.85 1.25 0.95 703252 22.4 1.00 1.03 1.07 499 CCT, no regularization 1,211 1,384 0.76 1.02 0.92 2.2E+04 13.2 1.00 0.48 0.23 175 Fuzzy CCT 1,291 1,462 0.76 1.00 0.94 4058 12.8 0.99 0.38 0.15 165 Fuzzy IK 1,778 1,657 0.73 1.21 0.92 5.2E+05 10.6 0.96 0.55 0.30 113 FG 3,718 4,945 0.24 0.37 0.90 3.54 3.00 0.95 -0.01 0.00 9.03 Global (all data) 4,564 4,564 0.24 0.37 0.89 3.67 3.11 0.95 -0.01 0.00 9.70 Notes: based on 1,000 simulations. DGP is based on 5th order polynomial approximation of top kink sample. True kink in first stage is: -1.4 × 10-5 . True elasticity is: 0. The trimmed statistic are obtained by first trimming the 5% sample in which the estimates deviate the most from the true parameter value. Bias (trimmed)Bias(trimmed)Variance (trimmed) C.I. Coverage Rate RMSE RMSE (trimmed) C.I. Coverage Rate Elasticity Estimation Summary Median Main b.w. Median Pilot b.w. First Stage Model Estimation Summary Fraction of Replications: C.I. includes 0 RMSE/true value (trimmed) Appendix Table 2b: Summary of Monte Carlo Studies, DGP Design Based on Top Kink Sample (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) 1. Local Linear, No Bias Correction Default CCT 1,395 - 0.60 0.58 0.81 61.2 7.25 1.00 0.39 0.15 52.5 CCT, no regularization 2,810 - 0.13 0.32 0.73 29.6 2.39 0.68 0.72 0.52 5.18 Fuzzy CCT 2,302 - 0.24 0.43 0.70 63.3 3.94 0.77 0.84 0.71 14.8 Fuzzy IK 4,396 - 0.00 0.16 0.73 1.54 1.33 0.32 1.07 1.15 0.63 FG 5,681 - 0.00 0.18 0.64 1.31 1.23 0.20 1.11 1.23 0.27 Global (all data) 13,908 - 0.00 0.55 0.00 0.37 0.35 0.17 0.34 0.11 0.01 2. Local Linear, Bias-Corrected Default CCT 1,395 2,860 0.89 0.89 0.78 2.5E+07 20.0 0.99 0.49 0.24 400 CCT, no regularization 2,810 3,883 0.64 0.74 0.68 1219 5.83 0.92 0.32 0.10 33.9 Fuzzy CCT 2,302 3,623 0.68 0.77 0.68 9.5E+05 9.80 0.95 0.53 0.29 95.8 Fuzzy IK 4,396 4,796 0.43 0.61 0.69 3.74 2.80 0.85 0.62 0.38 7.44 FG 5,681 9,470 0.00 0.17 0.89 2.00 1.82 0.63 1.52 2.31 1.02 Global (all data) 13,908 13,908 0.00 0.22 0.72 2.62 2.51 0.12 2.40 5.75 0.54 3. Quadratic, No Bias Correction Default CCT 2,045 - 0.90 1.08 0.89 134 17.6 1.00 0.42 0.18 309 CCT, no regularization 3,528 - 0.63 0.75 0.74 182 12.6 0.97 0.21 0.04 159 Fuzzy CCT 3,624 - 0.65 0.72 0.71 284 11.4 0.96 0.16 0.03 131 Fuzzy IK 5,128 - 0.30 0.47 0.72 81.1 4.90 0.90 0.50 0.25 23.8 FG 10,455 - 0.00 0.16 0.81 3.21 3.04 0.24 2.82 7.93 1.29 Global (all data) 13,908 - 0.00 0.20 0.58 5.13 4.94 0.00 4.83 23.4 1.03 4. Local Quadratic, Bias-Corrected Default CCT 2,045 3,234 0.92 1.37 0.90 1.8E+04 34.2 1.00 0.26 0.07 1170 CCT, no regularization 3,528 4,181 0.94 1.49 0.79 2.1E+07 56.7 0.99 1.64 2.69 3217 Fuzzy CCT 3,624 4,262 0.92 1.37 0.79 1.7E+05 50.7 0.98 -0.21 0.05 2574 Fuzzy IK 5,128 4,998 0.93 1.49 0.70 2552 21.6 0.93 0.37 0.14 466 FG 10,455 13,218 0.72 0.65 0.60 3.65 3.04 0.93 0.68 0.46 8.81 Global (all data) 13,908 13,908 0.79 0.71 0.49 4.47 3.67 0.91 1.52 2.30 11.2 Notes: based on 1,000 simulations. DGP is based on 5th order polynomial approximation of top kink sample. True kink in first stage is: -1.4 × 10-5 . True elasticity is: 0. The trimmed statistic are obtained by first trimming the 5% sample in which the estimates deviate the most from the true parameter value. Bias (trimmed)Bias(trimmed)Variance (trimmed) C.I. Coverage Rate RMSE RMSE (trimmed) C.I. Coverage Rate Elasticity Estimation Summary Median Main b.w. Median Pilot b.w. First Stage Model Estimation Summary Fraction of Replications: C.I. includes 0 RMSE/true value (trimmed) Appendix Table 3: Estimated AMSE for Local Linear and Local Quadratic Models Default CCT bandwidthAMSEVariance Bias AMSE Variance Bias (1)(2)(3)(4)(5)(6)Bottom Kink12.5112.470.04Top Kink3.983.170.8120.4118.981.43Table entries are estimates of AMSE for the local linear and local quadratic estimators under the default CCT bandwidth, with asymptotic variance and asymptotic squared bias components. Local Linear Local Quadratic AppendixTable4:SummaryofEstimatedBenetElasticitiesinExistingLiterature Authors(date)DataDesignElasticityEstimateorRange U.S.StudiesClassen(1977b)CWBH*,ArizonaPre/post,RKD0.6-1.0Moftt(1985)CWBH,13statesCross-sectional0.4Solon(1985)CWBH,GeorgiaDiff-in-diff,taxpolicychange0.7Meyer(1990)CWBH,allstatesState-by-year0.8KatzandMeyer(1990)CWBH,allstatesState-by-year0.8MeyerandMok(2007)NewYorkStateUIRecordsPre/post0.3Chetty(2010)SIPP(retrospectiveinterviews)State-by-year0.5Landais(Forthcoming)CWBH,Louisiana/WashingtonRKD,maximumbenet0.2-0.7EuropeanStudiesCarlingetal.(2011)Sweden,registerdataDiff-in-diff,repl.ratechange1.6(outcome=timetonextjob)RoedandZhang(2003)Norway,registerdata,previousCalendarvs.spelldating0.3(female)job2years0.9(male)Laliveetal.(2006)Austria,register/SocialSecuritydataDiff-in-diff,repl.ratechange0.15Arrazetal.(2008)Spain,registerdataPre/post0.8 *Note:CWBHistheContinuousWorkandBenetHistorydataset,basedonemploymentandunemploymentrecords.