,Vol.77,No.3(May,2009),953–973UNCONDITIONALQUANTILEREGRESSIONSM.F - PDF document

,Vol.77,No.3(May,2009),953–973UNCONDITIONALQUANTILEREGRESSIONSM.FORTINHOMAS S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXworkersonthemeanwage,dµ(p)/dp.Asiswellknown,thesamecoef-canalsobeinterpretedasanimpactontheconditionalmean.fortunately,thecoefcientfromasingleconditionalquantileregression,,isgenerallydifferentfrom(p)/dpp�Yqq�Yq)/f,theeffectofincreasingtheproportionofunionizedworkersonthethquantileoftheunconditionaldis-tributionofAnewapproachisthereforeneededtoprovidepractitionerswithaneasywaytocompute(p)/dp,especiallywhenisnotunivariateandbinaryasintheaboveexample.Ourapproachbuildsupontheconceptoftheinuencefunction(IF),awidelyusedtoolintherobustestimationofstatisticaloreconometricmod-els.Asitsnamesuggests,theinuencefunctionIFFofadistributional(Frepresentstheinuenceofanindividualobservationonthatdis-tributionalstatistic.Addingbackthestatistic(Ftotheinuencefunctionyieldswhatwecalltherecenteredinßuencefunction(RIF).Oneconvenientfea-tureoftheRIFisthatitsexpectationisequalto(FBecauseinuencefunctionscanbecomputedformostdistributionalstatistics,ourmethodeasilyextendstootherchoicesofbeyondquantiles,suchasthevariance,theGinicoefcient,andothercommonlyusedinequalitymeasures.Forthethquantile,theinuencefunctionIFisknowntobeequalto)/f.Asaresult,RIFissimplyequalto.WecalltheconditionalexpectationoftheRIFFmodeledasafunctionoftheexplanatoryvariables,variables,RIF(Y;F(X),theRIFregressionmodelInthecaseofquantiles,quantiles,RIF(Y;qFY)|X]=m(X)canbeviewedasanunconditionalquantileregression.Weshowthattheaveragederivativeoftheunconditionalquantileregression,regression,m(X),correspondstothemarginaleffectontheunconditionalquantileofasmalllocationshiftinthedistributionofcovariates,holdingeverythingelseconstant.Ourproposedapproachcanbeeasilyimplementedasanordinaryleastsquares(OLS)regression.Inthecaseofquantiles,thedependentvariableintheregressionisRIF)/f.ItiseasilyTheconditionalmeaninterpretationisthewagechangethataworkerwouldexpectwhenherunionstatuschangesfromnon-unionizedtounionized,orE(YE(YSincetheunconditionalmeanisµ(p)pE(Yp)E(Y,itfollowsthatdµ(p)/dpE(YE(YTheexpressionfor(p)/dpisobtainedbyimplicitdifferentiationappliedtooYq|X=1]ŠPr[Yq|X=0])+Pr[Yq|X=0].5Suchpropertyisimportantinsomesituations,althoughforthemarginaleffectsinwhichweareinterestedinthispapertherecenteringisnotfundamental.InFirpo,Fortin,andLemieux),therecenteringisusefulbecauseitallowsustoidentifytheinterceptandperformOaxaca-typedecompositionsatvariousquantiles.SeeFirpo,Fortin,andLemieux()forsuchregressionsonthevarianceandGini.Inthecaseofthemean,sincetheRIFissimplytheoutcomevariable,aregressionofisthesameasanOLSregressionof UNCONDITIONALQUANTILEREGRESSIONScomputedbyestimatingthesamplequantile,estimatingthedensityatthatpointusingkernel(orother)methods,andformingadummyvariable,indicatingwhetherthevalueoftheoutcomevariableisbelowThenwecansimplyrunanOLSregressionofthisnewdependentvariableonthecovariates,althoughwesuggestmoresophisticatedestimationmethodsinWeviewourapproachasanimportantcomplementtotheliteraturecon-cernedwiththeestimationofquantilefunctions.However,unlikeandNewey),andFlorens,Heckman,Meghir,andVyt-lacil(),whoconsideredtheidenticationofstructuralfunctionsdenedfromconditionalquantilerestrictionsinthepresenceofendogenousregres-sors,ourapproachisconcernedsolelywithparametersthatcapturechangesinunconditionalquantilesinthepresenceofexogenousregressors.Thestructureofthepaperisasfollows.Inthenextsection,wedenethekeyobjectofinterest,the“unconditionalquantilepartialeffect”(UQPE)andshowhowRIFregressionsforthequantilecanbeusedtoestimatetheUQPE.Wealsolinkthisparametertothestructuralparametersofageneralmodelandtheconditionalquantilepartialeffects(CQPE).Theestimationissuesaread-dressedinSection.Sectionpresentsanempiricalapplicationofourmethodthatillustrateswellthedifferencebetweenourmethodandconditionalquan-tilesregressions.WeconcludeinSectionUNCONDITIONALPARTIALEFFECTSGeneralConceptsWeassumethatisobservedinthepresenceofcovariates,sothathaveajointdistribution,YXX01],andisthesupportof.Byanalogywithastandardregressioncoefcient,ourobjectofinterestistheeffectofasmallincreaseinthelocationofthedistributionoftheexplanatoryvariableonthethquantileoftheunconditionaldistribution.Werepresentthissmalllocationshiftinthedistributionofintermsofthecounterfactualdistribution.Bydenition,theunconditional(mar-ginal)distributionfunctionofcanbewrittenas(y)Undertheassumptionthattheconditionaldistributionisunaffectedbythissmallmanipulationofthedistributionof,acounterfactualdistribution S.FIRPO,N.M.FORTIN,ANDT.LEMIEUX,canbeobtainedbyreplacing(y)Ourregressionmethodbuildsonsomeelementarypropertiesoftheinu-encefunction,ameasureintroducedbyHampel()tostudythein-nitesimalbehaviorofreal-valuedfunctionals(F,where,andisaclassofdistributionfunctionssuchthat(F)Letbeanotherdistributioninthesameclass.LetYtthemixingdistribution,whichisawayfrominthedirectionoftheproba-bilitydistributionYt,where1.Thedirectionalderivativeofinthedirectionofthedistributioncanbewrittenas(FYt(F (FYt Fd(G)(y)whereIFF(FYt,withdenotingtheprobabilitymea-surethatputsmass1atthevalue.ThevonMises)linearapproximationofthefunctional(FYt(FYt(FFd(G)(y)r(tr(tisaremainderterm.Wedenetherecenteredinßuence(RIF)moreformallyastheleadingtermsoftheaboveexpansionfortheparticularcasewhere1.SinceF(y)bydenition,itfollowsthatF(FF(s)(FFFinally,notethatthelastequalityinequation()alsoholdsforRIFFInthepresenceofcovariates,wecanusethelawofiteratedexpecta-tionstoexpress(FintermsoftheconditionalexpectationofRIFFInsteadofassumingaconstantconditionaldistribution,wecouldallowthecondi-tionaldistributionstovaryaslongastheyconvergeasthemarginaldistributionsoftooneanother. UNCONDITIONALQUANTILEREGRESSIONS(FF(y)FFRIF(Y;Fwheretherstequalityfollowsfromthefactthattheinuencefunctioninte-gratestozero,andthesecondequalitycomesfromsubstitutinginequation(Equation()showsthatwhenweareinterestedintheimpactofcovari-atesonaspecicdistributionalstatistic(Fsuchasaquantile,wesimplyneedtointegrateoveroverRIF(Y;F,whichiseasilydoneusingregres-sionmethods.Bycontrast,inequation()weneedtointegrateoverthewholeconditionaldistribution,whichis,ingeneral,moredifculttoWenowstateourmainresultonhowtheimpactofamarginalchangeinthedistributionof(FcanbeobtainedusingtheconditionalexpectationoftheRIFF.Notethatallproofsareprovidedinthe1—MarginalEffectofaChangeintheDistributionofposewecaninduceasmallperturbationinthedistributionofcovariatesinthedirectionofmaintainingtheconditionaldistributionofThemarginaleffectofthisdistributionalchangeonthefunctional(Fisgivenbyintegratinguptheconditionalexpectationofthefunctionwithrespecttothechangesindistributionofthecovariatesd(G(FYt RIF(Y;Fd(GYtWenextconsideraparticularchange,asmalllocationshift,inthedistribu-tionofcovariates.Letbeacontinuouscovariateinthevector,whereMostotherapproaches,suchastheconditionalquantileregressionmethodofMachadoand),haveessentiallyproposedtoestimateandintegratethewholeconditionaldistribu-overanewdistributiontoobtainthecounterfactualunconditionaldistributionof.SeealsoAlbrecht,Björklund,andVroman()and).Bycon-trast,weshowinSectionthatourapproachrequiresestimatingtheconditionaldistributiondistribution�Yyonlyatonepointofthedistribution.Notethattheseap-proachesdonotgenerateamarginaleffectparameter,butinsteadatotaleffectofchangesinthedistributionofonselectedfeatures(e.g.,quantiles)oftheunconditionaldistributionof S.FIRPO,N.M.FORTIN,ANDT.LEMIEUX.Thenewdistributionwillbethedistributionofarandom,wherek,and.Inthisspe-cialcase,let()denotethepartialeffectofasmallchangeinthedistributionofcovariatesfromonthefunctional(F.Collectingallweconstructthe1vector\t())\tj().Wecanwritetheunconditionalpartialeffect\t()asanaveragederivative.OROLLARY1—UnconditionalPartialEffect:AssumethatdtheboundaryofthesupportofXissuchthatifThenthevector\t()ofpartialeffectsofsmalllocationshiftsinthedistributionofacontinuous(Fcanbewrittenusingthevectorofaveragederivatives\t() dF(x)TheCaseofQuantilesTurningtothespeciccaseofquantiles,considerthethquantile(q).Itfollowsfromthedenitionoftheinuencefunctionthat �yq,andisthedensityofevaluatedat.ThussRIF(Y;q)|X=x]=c1·Pr[�YqFromequation(),theunconditionalpartialeffect,thatwedenote\t()inthecaseofthethquantile,simpliesto\t()Yt �Yq wherethelasttermistheaveragemarginaleffectfromtheprobabilityresponsemodelPrr�Yq.Wecalltheparameter\t())dE[RIF(Yq/dxunconditionalquantilepartialeffect(UQPE),byanalogywiththeWool-ridge()unconditionalaveragepartialeffect(UAPE),whichisdenedasasdE[Y|X]/dxTheexpressionxpressionRIF(Y;)|X=x]/dxisthevectorofpartialderivativesderivativesE[RIF(Y;)|X=x]/xj]kj=1.11TheUAPEisaspecialcaseofCorollaryforthemean(),wherewheredE[Y|X]/dxsinceRIF(Yµ) UNCONDITIONALQUANTILEREGRESSIONSOurnextresultprovidesaninterpretationoftheUQPEintermsofagen-eralstructuralmodel,h(X\n),wheretheunknownmappingisin-vertibleonthesecondargument,andisanunobservabledeterminantoftheoutcomevariable.WealsoshowthattheUQPEcanbewrittenasaweightedaverageofafamilyofconditionalquantilepartialeffects(CQPE),whichistheeffectofasmallchangeofontheconditionalquantileof(x))h(X\n) h(xQ Y|X=x]infq{q:FY|X(q|x)\f}istheconditionalquantileoper-ator.ForthesakeofsimplicityandcomparabilitybetweentheCQPEandtheUQPE,weconsiderthecasewhereareindependent.Thus,wecanusetheunconditionalformforor\n]inthelasttermoftheaboveequation.Inalinearmodelh(X\n),boththeUQPEandtheCQPEaretriviallyequaltotheparameterofthestructuralformforanyquantile.Whilethisspecicresultdoesnotgeneralizebeyondthelinearmodel,usefulconnectionscanstillbedrawnbetweentheUQPEandtheunderlyingstruc-turalform,andbetweentheUQPEandtheCQPE.Toestablishthesecon-nections,wedenethreeauxiliaryfunctions.Therstfunction,isaweightingfunctiondenedastheratiobetweentheconditionaldensityandtheunconditionaldensity:.Thesecondfunction,,istheinversefunction,whichshallex-istundertheassumptionthatisstrictlymonotonicin.Thethirdfunction,,isa“matching”functionindicatingwheretheunconditionalfallsintheconditionaldistributionoffY|X=x]=q}=FY|X(q|X=x)PROPOSITION1—UQPEandtheStructuralForm:Assumingthatthestructuralformh(X\n)isstrictlymonotonicinandthatareindependenttheparameter()willbe()(X)h(X\n(X)) Wecanalsorepresent()asaweightedaverageof(x)x)()(X)(X)X)Inthissetting,theidenticationoftheUQPErequirestobeunaffectedbychangesinthedistributionofcovariates.TheidenticationoftheCQPErequiresquantileindependence,thatis,the-conditionalquantileofequalsthequantileof.Independencebetweenguarantees,therefore,thatboththeUQPEandtheCQPEparametersareidentied. S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXResult(i)ofthePropositionshowsformallythatUQPE()isequaltoaweightedaverage(overthedistributionof)ofthepartialderivativesofthestructuralfunction.Inthesimplecaseofthelinearmodelmentionedabove,itfollowsthath(X\n(X))/xandUQPE()forall.Moregener-ally,theUQPEwilltypicallydependoninnonlinearsettings.Forexample,h(X\n),whereisdifferentiableandstrictlymonotonic,simplealgebrayieldsUQPE(),whichdependson.Finally,notethatindependenceplaysacrucialrolehere.If,instead,wehaddroppedtheindependenceassumptionbetween,wewouldnotbeable,eveninalinearmodel,toexpressUQPE()asasimplefunctionofthestructuralResult(ii)showsthatUQPE()isaweightedaverage(overthedistribu-tionof)ofafamilyofCQPE(X)X)(X),theconditionalquan-tilecorrespondingtothethunconditionalquantileofthedistributionof.Butwhileresult(ii)ofPropositionprovidesamorestructuralin-terpretationoftheUQPE,itisnotpracticalfromanestimationpointofviewasitwouldrequireestimating,thedistributionof,us-ingnonparametricmethods.Asshownbelow,weproposeasimplerwaytoestimatetheUQPEbasedontheestimationofaveragemarginalef-ESTIMATIONInthissection,wediscusstheestimationofUQPE()usingRIFregres-sions.Equation()showsthatthreecomponentsareinvolvedintheestima-tionofUQPE():thequantile,thedensityoftheunconditionaldistributionthatappearsintheconstant,andtheaveragemarginalmarginaldPr[�Yq/dX.Wediscusstheestimationofeachcomponentinturnandthenbrieyaddresstheasymptoticpropertiesofrelatedestima-Theestimatorofthethpopulationquantileofthemarginaldistribution,theusualthsamplequantile,whichcanberepresented,asinKoenkerandBassett),asargminq)Theseexamplesareworkedindetailintheworkingpaperversionofthisarticle.SeeFirpo,Fortin,andLemieux( UNCONDITIONALQUANTILEREGRESSIONSWeestimatethedensityof,usingthekerneldensityestimator N·b·N\ni=1KYYiŠ\tq (z)isakernelfunctionandapositivescalarbandwidth.WesuggestthreeestimationmethodsfortheUQPEbasedonthreeways,amongmany,toestimatetheaveragemarginaleffecteffectdPr[�Yq/dXAsdiscussedinFirpo,Fortin,andLemieux),thersttwoestimatorswillbeconsistentifwecorrectlyimposefunctionalformrestrictions.Thethirdestimatorinvolvesafullynonparametricrststageand,therefore,willbecon-sistentquitegenerallyfortheaveragederivativeparameter.TherstmethodestimatestheaveragemarginaleffecteffectdPr[�Yq/dXwithanOLSregression,whichprovidesconsistentestimatesifPrr�Yq|X=x]islinearin.Thismethod,thatwecallRIF-OLS,consistsofregress-.Thesecondmethodusesalogisticregressionoftoestimatetheaveragemarginaleffect,whichisthenmultipliedby.Again,theaveragemarginaleffectfromthislogitmodelwillbeconsistentifPrr�Yq,whereisthecumulativedistributionfunction(c.d.f.)ofalogisticdistributionandisavectorofcoef-cients.WecallthismethodRIF-Logit.Intheempiricalsection,weusethesetwoestimatorsandndthat,inourapplication,theyyieldestimatesveryclosetothefullynonparametricestimator.Thelastestimationmethod,calledRIF-NP,isbasedonanonparametricesti-matorthatdoesnotrequireanyfunctionalformassumptiononPrr�Yqtobeconsistent.Weusethemethoddiscussedby)andestimateestimate�Yqbypolynomialseries.Astheobjectofinterestistheaverageaverage�Yq/dx,oncewehaveapolynomialfunctionthatapproxi-matestheconditionalprobability,wecaneasilytakederivativesofpolynomialsandaveragethem.Asshownby)fortheaveragederivativecaseandlaterformalizedinamoregeneralsettingby),thechoiceofthenonparametricestimatorforthederivativeisnotcrucialinlargesamples.Averaginganyregularnonparametricestimatorwithrespecttoyieldsanes-timatorthatconvergesattheusualparametricrateandhasthesamelimitingdistributionasotherestimatorsbasedondifferentnonparametricmethods.IntheempiricalsectionweproposeusingtheGaussiankernel.TherequirementsforthekernelandthebandwidtharedescribedinFirpo,Fortin,andLemieux).Weproposeusingthekerneldensityestimator,butotherconsistentestimatorsofthedensitycouldbeusedaswell.NonparametricestimationofPrr�Yqcouldalsobeperformedbyseriesapprox-imationofthelog-oddsratio,whichwouldkeeppredictionsbetween0and1(Hirano,Imbens,andRidder()).Note,however,thatwearemainlyinterestedinanotherobject,thederiva-deriva-�Yq/dx,andimposingthattheconditionalprobabilityliesintheunitintervaldoesnotnecessarilyaddmuchstructuretoitsderivative. S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXWestudytheasymptoticpropertiesofourestimatorsindetailinFortin,andLemieux),whereweestablishthelimitingdistributionsoftheseestimators,discusshowtoestimatetheirasymptoticvariances,andshowhowtoconstructteststatistics.ThreeimportantresultsfromFirpo,Fortin,andLemieux)aresummarizedhere.Therstresultisthattheasymp-toticlinearexpressionofeachoneofthethreeestimatorsconsistsofthreecomponents.Therstcomponentisassociatedwithuncertaintyregardingthedensity;thesecondcomponentisassociatedwiththeuncertaintyregardingthepopulationquantile;andthethirdcomponentisassociatedwiththeaver-agederivativetermtermdPr[�Yq/dX.Thesecondresultstatesthatbe-causethedensityisnonparametricallyestimatedbykernelmethods,therateofconvergenceofthethreeestimatorswillbedominatedbythisslowerterm.Firpo,Fortin,andLemieux),weuseahigherorderexpansiontypeofargumenttoallowforthequantileandtheaveragederivativecomponentstobeexplicitlyincluded.Bydoingso,wecanintroducearenementintheexpressionoftheasymptoticvariance.Finally,thethirdresultisthattotestthenullhypothesisthatUQPE0,wedonotneedtoestimatethedensity,asasdPr[�Yq/dX0.Thus,wecanuseteststatisticsthatconvergeattheparametricrate.Inthiscase,theonlycomponentsthatcon-tributetotheasymptoticvariancearethequantileandtheaveragederivative.Aswithstandardaveragemarginaleffects,wecanalsoestimatetheUQPEforadummycovariatebyestimatingestimatingPr[�Yq]]ŠŠPr[�YqinsteadoffdPr[�Yq/dXusinganyofthethreemethodsdiscussedabove.Likeintheexampleofunionstatusmentionedinthe,theUQPEinsuchcasesrepresentstheimpactofasmallchangeintheprobabilityprobabilityX=1],insteadofthesmalllocationshiftforacontinuouscovariateconsideredinSectionEMPIRICALAPPLICATIONInthissection,wepresentanempiricalapplicationtoillustratehowtheun-conditionalquantileregressionsworkinpracticeusingthethreeestimatorsdiscussedabove.Wealsoshowhowtheresultscomparetostandard(con-ditional)quantileregressions.Ourapplicationconsidersthedirecteffectofunionstatusonmalelogwages,whichiswellknowntobedifferentatdifferentpointsofthewagedistribution.Weusealargesampleof266,956observa-Firpo,Fortin,andLemieux)formoredetail.AStataadolethatimplementstheRIF-OLSestimatorisavailableontheauthor’swebsite,See,forexample,)and).Forsimplicity,wemaintaintheassumptionthatunioncoveragestatusisexogenous.Studiesthathaveusedselectionmod-elsorlongitudinalmethodstoallowtheunionstatustobeendogenouslydetermined(e.g.,Lemieux())suggestthattheexogeneityassumptiononlyintroducessmallbiasesintheesti- UNCONDITIONALQUANTILEREGRESSIONStionsonU.S.malesfromthe1983–1985OutgoingRotationGroup(ORG)supplementoftheCurrentPopulationSurvey.Lookingattheimpactofunionstatusonlogwagesillustrateswellthediffer-encebetweenconditionalandunconditionalquantilesregressions.Consider,forexample,theeffectofunionstatusestimatedatthe90thand10thquan-tiles.Findingthattheeffectofunions(forshort)estimatedusingconditionalquantileregressionsissmalleratthe90ththanatthe10thquantilesimplymeansthatunionsreducewithin-groupdispersion,wherethe“group”consistsofworkerswhosharethesamevaluesofthecovariates(otherthanunionstatus).Thisdoesnotmean,however,thatincreasingtherateofunionizationwouldreduceoverallwagedispersionasmeasuredbythedifferencebetweenthe90thandthe10thquantilesoftheunconditionalwagedispersion.Toan-swerthisquestionwehavetoturntounconditionalquantileregressions.Inadditiontothewithin-groupwagecompressioneffectcapturedbycondi-tionalquantileregressions,unconditionalquantileregressionsalsocaptureaninequality-enhancingbetween-groupeffectlinkedtothefactthatunionsin-creasetheconditionalmeanofwagesofunionworkers.Thiscreatesawedgebetweenotherwisecomparableunionandnon-unionworkers.Asaresult,unionstendtoincreasewagesforlowwagequantileswhereboththebetween-andwithin-groupeffectsgointhesamedirection,butcandecreasewagesforhighwagequantileswherethebetween-andwithin-groupeffectsgoinoppo-sitedirections.Asabenchmark,TablereportstheRIF-OLSestimatedcoefcientsofthelogwagesmodelforthe10th,50th,and90thquantiles.Theresults(labeledUQRforunconditionalquantileregressions)arecomparedwithstandardOLS(conditionalmean)estimatesandwithstandard(conditional)quantileregres-sions(CQR)atthecorrespondingquantiles.Forthesakeofcomparability,weusesimplelinearspecicationsforallestimatedmodels.WealsoshowinFig-howtheestimatedUQPEofunionschangeswhenweusetheRIF-LogitandRIF-NPmethodsinstead.Interestingly,theUQPEofunionsrstincreasesfrom0.198atthe10thquantileto0.349atthemedian,beforeturningnegative(0.137)atthe90thquantile.ThesendingsstronglyconrmthepointdiscussedabovethatunionsWestartwith1983becauseitistherstyearinwhichtheORGsupplementaskedaboutunionstatus.Thedependentvariableistherealloghourlywageforallwageandsalaryworkers,andtheexplanatoryvariablesincludesixeducationclasses,married,non-white,andnineexperi-enceclasses.Thehourlywageismeasureddirectlyforworkerspaidbythehourandisobtainedbydividingusualearningsbyusualhoursofworkforotherworkers.OtherdataprocessingdetailscanbefoundinLemieuxInthecaseofthevariance,itiseasytowritedownananalyticalexpressionforthebetween-andwithin-groupeffects(see,forexample,Card,Lemieux,andRiddell))andndtheconditionsunderwhichoneeffectdominatestheother.Itismuchhardertoascertain,however,whetherthebetween-orthewithin-groupeffecttendstodominateatdifferentpointsofthewage S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXTABLEIOMPARINGOLS,UUANTILEUANTILE(CQR);1983–1985CPSDATAFOR 10thCentile50thCentile90thCentile OLSUQRCQRUQRCQRUQRCQR Unionstatus017901950288033701350Married0140019501660156014600430HSdropoutSomecollege0133005800580179013301540College0406019602520464041405820Post-graduate0478013802870522048208440Constant1742097011451735174425112 Robuststandarderrors(OLS)andbootstrappedstandarderrors(200replications)forUQRandCQRaregiveninparentheses.Allregressionsalsoincludeasetofdummiesforlabormarketexperiencecategories.havedifferenteffectsatdifferentpointsofthewagedistribution.Thecon-ditionalquantileregressionestimatesreportedinthecorrespondingcolumnsshow,asin),thatunionsshiftthelocationofthecondi-tionalwagedistribution(i.e.,positiveeffectonthemedian)butalsoreduceconditionalwagedispersion.ThedifferencebetweentheestimatedeffectofunionsforconditionalandunconditionalquantileregressionestimatesisillustratedinmoredetailinpanelAofFigure,whichplotsbothconditionalandunconditionalquan-tileregressionestimatesofunionstatusat19differentquantiles(fromthe5thtothe95th).AsindicatedinTable,theunconditionalunioneffectishighlynonmonotonic,whiletheconditionaleffectdeclinesmonotonically.Morepre-cisely,theunconditionaleffectrstincreasesfromabout0.1atthe5thquan-tiletoabout0.4atthe35thquantile,beforedecliningandeventuallyreachingNotethattheeffectsareverypreciselyestimatedforallspecicationsandthe(closeto0.40)aresizeableforcross-sectionaldata.Bootstrappedstandarderrorsareprovidedforbothestimates.AnalyticalstandarderrorsfortheUQPEarenontrivialandderivedinFirpo,Fortin,andLemieux UNCONDITIONALQUANTILEREGRESSIONS 1.—Unconditionalandconditionalquantileregressionestimatesoftheeffectofunionstatusonlogwages.alargenegativeeffectofover0.2atthe95thquantile.Bycontrast,stan-dard(conditional)quantileregressionestimatesdeclinealmostlinearlyfromabout0.3atthe5thquantiletobarelymorethan0atthe95thquantile.Atrstglance,thefactthattheeffectofunionsisuniformlypositiveforconditionalquantileregressions,butnegativeabovethe80thquantileforun-conditionalquantileregressionsmayseempuzzling.SincePropositionthattheUQPEisaweightedaverageoftheCQPEs,fortheUQPEtobeneg-ativeitmustbethatsomeoftheCQPEsarenegativetoo.UnliketheUQPE,however,theCQPEgenerallydependson.Forthesakeofclarity,inFig-wereporttheconditionalquantileregressionsusingahighlyrestrictedspecicationwheretheeffectofunionsisnotallowedtodependonarichsetofothercovariates(nointeractionterms).Whenwerelaxthisassumption,wendthatconditionalquantileregressionsestimatesareoftennegativeformore“skilled”workers(inhigheducation/highlabormarketexperiencecells).However,thesenegativeeffectsareaveragedawaybypositiveeffectsinthemoreparsimoniousconditionalquantileregressions.Ontheotherhand,be-causethematchingfunctionfromPropositionreassignssomeofthenegativeunioneffectsfromthe-conditionalquantilestothequantilesatthetopofthewagedistributionandbecausetheweightingfunc- S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXputsmoreweightontheseworkers,theUQPEbecomesnegativeforworkersatthetopendofthewagedistribution.PanelBshowsthattheRIF-OLSandRIF-LogitestimatesoftheUQPEareverysimilar,whichconrmsthe“folkwisdom”inempiricalworkthat,inmanyinstances,usingalinearprobabilitymodeloralogitgivesverysimilaraveragemarginaleffects.Moreimportantly,FigureshowsthattheRIF-NPestimatesarealsoverysimilartotheestimatesobtainedusingthesetwosimplermeth-Thissuggeststhat,atleastforthisparticularapplication,usingasimplelinearspecicationfortheunconditionalquantileregressionsprovidesfairlyaccurateestimatesoftheUQPE.ThesmalldifferencebetweenRIF-OLSandRIF-NPestimatesstandsinsharpcontrasttothelargedifferencesbetweentheRIF-OLSestimatesandtheconditionalquantileregressionestimatesinpanelA.Thelargedifferencesbetweentheconditionalandunconditionalquan-tileregressionsresultshaveimportantimplicationsforunderstandingrecentchangesinwageinequality.Thereisalongtraditioninlaboreconomicsofattemptingtoestimatetheeffectofunionizationonthe(unconditional)distri-butionofwages.Theunconditionalquantileregressionsprovideasimpleanddirectwaytoestimatethiseffectatallpointsofthedistribution.TheestimatesreportedinFigureshowthatunionizationprogressivelyincreaseswagesinthethreelowerquintilesofthedistribution,peakingaroundthe35thquantile,andactuallyreduceswagesinthetopquintileofthedistribution.Asaresult,thedeclineinunionizationoverthelastthreedecadesshouldhavecontributedtoareductioninwageinequalityatthebottomendofthedistributionandtoanincreaseinwageinequalityatthetopend.Thispreciselymirrorstheac-tualU-shapedchangesobservedinthedata.Bycontrast,conditionalquan-tileregressionsresultsdescribeapositivebutmonotonicallydecliningeffectofunionizationonwages,whichfailstoaccountfortheobservedpatternofchangesinthewagedistribution.CONCLUSIONInthispaper,weproposeanewregressionmethodtoestimatetheeffectofexplanatoryvariablesontheunconditionalquantilesofanoutcomevari-able.Theproposedunconditionalquantileregressionmethodconsistsofrun-ningaregressionofthe(recentered)inuencefunctionoftheunconditionalquantileoftheoutcomevariableontheexplanatoryvariables.TheinuenceTheRIF-NPisestimatedusingamodelfullysaturatedwithallpossibleinteractions(upto432parameters)ofourcategoricalvariables,omittingforeachestimatedquantiletheinterac-tionsthatwouldresultinperfectpredictions.FortheRIF-OLS,theguregraphstheestimatedcoefcients,whilefortheRIF-LogitandRIF-NP,theaverageunconditionalpartialeffectsareSee,forexample,)andDiNardo,Fortin,andLemieuxSee,forexample,Autor,Katz,andKearney)andLemieux UNCONDITIONALQUANTILEREGRESSIONSfunctionisawidelyusedtoolinrobustestimationthatcaneasilybecomputedforeachquantileofinterest.Weshowhowstandardpartialeffects,thatwecallunconditionalquantilepartialeffects(UQPE),canbeestimatedusingourregressionapproach.AnotherimportantadvantageoftheproposedmethodisthatitcanbeeasilygeneralizedtootherdistributionalstatisticssuchastheGini,thelogvariance,ortheTheilcoefcient.Oncetherecenteredinuencefunctionforthesesta-tisticsiscomputed,allthatisrequiredisrunningaregressionoftheresultingRIFonthecovariates.Wediscussinacompanionpaper(Firpo,Fortin,andLemieux))howourregressionmethodcanbeusedtogeneralizetra-ditionalOaxaca–Blinderdecompositions,devisedformeans,tootherdistribu-tionalstatistics.Finally,ourmethodcanbeusefulevenwhentheindependenceassumptionisrelaxed.However,theinterpretationoftheidentiedparameterintermsofitsrelationtothestructuralfunctionlinkingobservedandunobservedfactorstothedependentvariablewouldchange.Yet,theUQPEparameterwouldstillbedenedbyholdingunobservedvariablesandothercomponentsofwhenevaluatingthemarginaleffectofchangesinthedistributionofagivenquantileoftheunconditionaldistributionof.Suchstructuralaver-agedmarginaleffectscanbeusefulinpractice.Weplantoshowinfutureworkhowourapproachcanbeusedwheninstrumentalvariablesareavailablefortheendogenouscovariatesandhowconsistentestimatesofmarginaleffectscanbeobtainedbyaddingacontrolfunctionintheunconditionalquantilere-ROOFOF:Theeffectonthefunctionalofthedistributionofaninnitesimalchangeinthedistributionofdenedas(FYt.Giventhatequation()alsoappliestoRIFitfollowsthat(FYt d(G)(y)Substitutinginequations()and(),andapplyingthefactthatthatRIF(Y;)|X=x]=yRIF(y;)·dFY|X(y|X=x)yields(FYt d(GGRIF(Y;)|X=x]·d(GQ.E.D. S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXROOFOFOROLLARY:Considerthedistributionoftherandom,wheree0,whichisvectorofzerosexceptatthethentry,whichequals1.ThedensityofThecounterfactualdistributionwillbe (y)wherethesecondlineisobtainedusingarst-orderexpansion,where(x))/dx,anddfX(x)/xl]kl=1istheofpartialderivativesof.Therefore,O(t.Now,dene()dBytheusualdenitionofthecounterfactualdistribution,wehave(y)(y)dxThuswecanwrite(y)(y)(y))Yt()t))(F (FYt(F t))(FYt (FYt (FYt(FYt Thedensityofand,bydenitionofdensities,() UNCONDITIONALQUANTILEREGRESSIONSwherethelasttermvanishes:(FYt(FYt O( UsingTheorem,itfollowsthat()(FYt RIF(Y;)|X=x]·d(GGRIF(Y;)|X=x]·ej·lX(x)·fX(x)·dxApplyingpartialintegrationandusingtheconditionthatiszeroattheboundaryofthesupportyields dx·fX(x)·dx=E[RIF(Y;)|X=x] dx())RIF(Y;F) dxQ.E.D.ROOFOF(i)Startingfromequation(() fY(q)·dPr[Yq|X=x] andassumingthatthestructuralformh(X\n)ismonotonicin,sothat(xq,wecanwriteeYq|X=x]=Pr[\n\n(X)Takingthederivativewithrespectto,weget (xq x S.FIRPO,N.M.FORTIN,ANDT.LEMIEUXH(x (x)qh(x ,itfollowsthat(xq H(x H(x h(x h(x h(x h(x ) (xq H(x H(x h(x )   = Š1Y|X(q;dYq|X=x] (xq (xq h(x ) Substitutingintheseexpressionsyields() h(x h(x ) h(X (X)) =EfY|X(q| h(X (X)) (X)h(X (X)) (ii)LettheCQPEbedenedas(x) UNCONDITIONALQUANTILEREGRESSIONSdenotethequantileoftheconditionaldistribution:distribution:YQ[Y|X=x]|X=x].Sinceh(X\n)ismonotonicininYQ[Y|X=x]|X=x]=Pr\nhŠ1(XQQY|X=x])|X=x\b=F\nhŠ1(xQQY|X=x])Thus,bytheimplicitfunctiontheorem,(x)(xQQY|X=x]))·hŠ1(xQ (xQQY|X=x]))·hŠ1(xq)/q)/qY|X=x]=ŠŠhxh(xQQY|X=x])/x·h(x\n) Y|X=x])\fŠ1h(x\n) Y|X=x])\fŠ1=h(xh(xQ UsingthematchingfunctionfunctionY|X=x]=q},wecanwrite(sx)forthethconditionalquantileataxedxedY|X=x])thatequals(matches)thethunconditionalquantile()as(sx)(x)x)h(xh(xQ h(xh(xq h(X\n(X)) ()(X)h(X\n(X)) (X)(X)X)Q.E.D.,J.,A.BJÖRKLUNDS.V(2003):“IsThereaGlassCeilinginSweden?”JournalofLaborEconomics,21,145–178.145–178.957]AUTOR,D.H.,L.F.KATZM.S.K(2008):“TrendsinU.S.WageInequality:RevisingtheRevisionists,”ReviewofEconomicsandStatistics,90,300–323. 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UNCONDITIONALQUANTILEREGRESSIONS,J.M.(2004):“EstimatingAveragePartialEffectsUnderConditionalMomentIndependenceAssumptions,”UnpublishedManuscript,MichiganStateUniversity.[EscoladeEconomiadeS‹oPaulo,Funda‹oGetœlioVargas,RuaItapeva474,S‹oPaulo,SP01332-000,Brazil;sergio.Þrpo@fgv.brDept.ofEconomics,UniversityofBritishColumbia,997-1873EastMall,Van-couver,BCV6T1Z1,CanadaandCanadianInstituteforAdvancedResearch,Toronto,CanadaDept.ofEconomics,UniversityofBritishColumbia,997-1873EastMall,Van-couver,BCV6T1Z1,Canada;ManuscriptreceivedNovember,2006;ÞnalrevisionreceivedDecember,2008.