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Quantile regression under random censoring

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Quantile regression under random censoring
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  QuantileRegressionUnderRandomCensoring   BoHonore  PrincetonUniversity  ShakeebKhan    UniversityofRochester  JamesL.Powell  UniversityofCaliforniaatBerkeley  FirstVersion:  October1993  ThisVersion:  August1999  Abstract  Censoredregressionmodelshavereceivedagreatdealofattentioninboththetheoreticaland appliedeconometricliterature.Mostoftheexistingestimationproceduresforeithercrosssec- tionalorpaneldatamodelsaredesignedonlyformodelswithxedcensoring.Inthispaper,a newprocedureforadaptingtheseestimatorsdesignedforxedcensoringtomodelswithrandom censorshipisproposed.ThisprocedureisthenappliedtotheCLADandquantileestimatorsof Powell(1984,1986a)toobtainanestimatoroftheregressioncoecientsunderamildconditional quantilerestrictionontheerrortermthatisapplicabletosamplesexhibitingxedorrandom censoring.Theresultingestimatorisshowntohavedesirableasymptoticproperties,andperforms wellinasmallscalesimulationstudy. JELClassication:  C24,C14,C13. KeyWords:  censoredquantileregression,randomcensoring,Kaplan-Meierproductlimit estimator,acceleratedfailuretimemodel.   Correspondingauthor.Dept.Economics,UniversityofRochester,Rochester,NY14627;e-mail:skhan@troi.cc.rochester.edu.WearegratefultoJambeandRodchenkoforhelpfulcommentsonanear- lierdraft.  1Introduction  Overthepastdecade,thecensoredregressionmodel,knowntoeconomistsastheTobit model(afterTobin1958),hasbeentheobjectofmuchattentionintheeconometriclit- eratureonsemiparametricestimation.Relaxingthetraditionalparametricrestrictionson theformofthedistributionoftheunderlyingerrorterms,anumberofconsistentesti- matorshavebeenproposedwhichrequireonlyweakconditionsonthesedistributions,in- cluding:constantconditionalquantiles(Powell(1984,1986a);Nawata(1990);Neweyand Powell(1990),conditionalsymmetry(Powell(1986b),Lee(1993a,b),Newey(1991)),andinde- pendenceoftheerrorsandregressors(Duncan(1986);Fernandez(1986);HonoreandPow- ell(1993);Horowitz(1986,1988);andMoon(1989)).Theseproposedestimatorsallexploit anassumptionthatthecensoringvaluesforthedependentvariableareknownforallobser- vations,eventhosethatarenotcensored;whilethetypicalestimatorisconstructedunder thepresumptionthatthedependentvariableiscensoredtotheleftatzero,itisgenerally straightforwardtomodifyitforeitherrightorleftcensoreddata(orboth)withvariable censoringvalues.Hereafter,inalooseanalogytopaneldatamodelling,werefertosuch modelsas   xed  censoringmodels,sincethecensoringvalues,thoughpossiblyvariable,may notbedistributedindependentlyfromtheregressors.Aparallelliteratureinthestatisticsandbiometricsliteraturehasbeenconcernedwith estimationoftheparametersofarelatedmodel,theregressionmodelwith  random  censoring.Inthismodelthedependentvariabletypicallyrepresentsthelogarithmofasurvivaltime (inwhichcasetheregressionmodelcorrespondstoan  acceleratedfailuretime  duration model),whichisright-censoredatvaryingcensoringpointswhichareobservedonlywhen theobservationiscensored.Inaddition,thecensoringtimesaregenerally(butnotalways) assumedtobeindependentlydistributedoftheregressorsanderrorterms.Studieswhich proposesemiparametricmethodsunderrandom(right)censorshipincludeMiller(1976), Prentice(1978),BuckleyandJames(1979),Koul,Suslara,andVanRyzin(1981),Leurgans (1987),andRitov(1990),amongothers.Theseestimationmethodstypicallyimposean assumptionofindependenceoftheerrortermsandcovariates;thosethatdonotimpose independenceinsteadrequirestrongconditionsonthecensoringdistributionwhichgenerally ruleoutcensoringataconstantvalue,asistypicalineconometrics.Inthispaperwedescribeamethodforadaptingestimatorsproposedforxedcensoring tosamplingwithrandomrightcensorship.Weapplythismethodtothecensoredleastabso- lutedeviationsandquantileestimatorsofPowell(1984  ;  1986a)toobtainanestimatorofthe 1   regressioncoecientswhichwillbeconsistentunderarelatively-weakquantilerestriction ontheerrorterms,andwhichisequallyapplicabletosampleswithconstantorrandom censoring.Thefollowingsectiondescribesthisestimationapproach,andcomparesthemod- iedformofthecensoredregressionquantileestimatortootherquantile-basedestimators forrandomcensoringthathaveappearedinthestatisticsliterature.Section3givessu- cientconditionstoensuretheroot  n  -consistencyandasymptoticnormalityoftheproposed estimator,andsection4analyzesitsperformanceusingasimulationstudyandanempirical example.Thenalsectiondiscussesapplicationofthegeneralestimationmethodtoother censoredregressionestimatorsintheeconometricliterature,andconsiderswhethertheas- sumptionofindependenceofthecensoringtimesandcovariatescouldberelaxed.Proofsof thelarge-sampleresultsofsection3areavailableinamathematicalappendix. 2TheModelandEstimationMethod  Theobjectofestimationisthe   p  -dimensionalvectorofregressioncoecients    0  inalinear latentvariablemodel  y    i  =  x  0 i    0  +  "  i  ;i  =1  ;:::;n;  (2.1) where  y    i  isthe(uncensoredandscalar)dependentvariableofinterest,  x  i  isanobservable   p  -vectorofcovariates,and  "  i  isanunobservederrorterm.Withrightcensorship,thelatent variable  y    i  isobservedonlywhenitislessthansomescalarcensoringvariable  c  i  ;thatis, theobserveddependentvariable  y  i  is  y  i  =min  f  y    i  ;c  i  g  =min  f  x  0 i    0  +  "  i  ;c  i  g  : (2.2) Inarandomsamplewithxedcensoring,  n  independently-distributedobservationsonthe triple(  y  i  ;c  i  ;x  i  )areassumedtobeavailable;withrandomcensoring,theobservationsareof theform(  y  i  ;d  i  ;x  i  ),where  d  i  isabinaryvariableindicatingwhetherthedependentvariable isuncensored: d  i  =1  f  y    i  <c  i  g  =1  f  x  0 i    0  +  "  i  <c  i  g  ;  (2.3) for\1  f  A  g  "theindicatorfunctionforthesetA.2   Forsampleswithxedcensoring,theestimatorsof    0  citedintheprecedingsectionoften aredenedassolutionstominimizationproblemsand/orestimatingequationsconstructed usingsampleaveragesoffunctionsoftheobservabledataandunknownparameters,i.e., ^    =argmin    1  n  n  X      i  =1    (  y  i  ;c  i  ;x  i  ;  )(2.4) or 0    = 1  n  n  X      i  =1    (  y  i  ;c  i  ;x  i  ;  ^    )(2.5) forcertainfunctions    (    )or    (    ).Ofcourse,someestimatorsinvolvemorecomplicatedmin- imands/estimatingequations,denedusinghigher-order  U  -statisticsorinvolvingprelimi- nary(nonparametric)estimatorsofunknownfunctions,buttheanalysisoftheirlargesample behavior,thoughmoredicult,followsthesamelinesasinthissimplecase.Consistencyof ^    isdemonstratedafterimposingappropriateconditionsontheerrorterms,covariates,and censoringvalues;oneimportantstepintheproofistoshowthatthetrueparametervalue    0  isauniquesolutiontothepopulationversionsoftheminimizationproblemorestimating equations,    0  =argmin    E      (  y  i  ;c  i  ;x  i  ;  )](2.6) or 0=  E      (  y  i  ;c  i  ;x  i  ;  )]i    =    0  : (2.7) Givensuchanidenticationcondition,applicationofauniformlawoflargenumberstothe sampleaveragedening ^    ensuresitsconsistency.Underrandomcensorship,itisnolongerpossibletodeneanestimatorof    0  inthe samefashionasabove,sincethecensoringvariables  f  c  i  g  arenotknownforall  i  .However,if thecensoringvariables  f  c  i  g  areassumedtobeindependentof  f  (  y  i  ;x  i  )  g  ,andifthemarginal c.d.f. G  (  t  )    Pr  f  c  i    t  g  ofthecensoringvalueswereknown,asimplemodicationofthe estimationapproachabovewouldreplacethefunctions    (  y  i  ;x  i  ;c  i  ;  )or    (  y  i  ;x  i  ;c  i  ;  )bytheir conditionalexpectationsgiventheobservablevariables(  y  i  ;d  i  ;x  i  ).Thatis,an  M  -estimator of    0  correspondingtotheforegoingminimizationproblemwouldbe 3   ^    =argmin    1  n  n  X    i  =1  E      (  y  i  ;c  i  ;x  i  ;  )  j (  y  i  ;d  i  ;x  i  )](2.8) =argmin    1  n  n  X    i  =1    (1  ?  d  i  )      (  y  i  ;y  i  ;x  i  ;  )+  d  i      S  (  y  i  )] ?  1    Z   1(  y  i  <c  )    (  y  i  ;c;x  i  ;  )  dG  (  c  )    ;  where  S  (  t  )    1  ?  G  (  c  )isthesurvivorfunctionforthecensoringvalue  c  i  .Similarly, ^    might bedenedassolutionstoestimatingequationsoftheform 0    = 1  n  n  X    i  =1    (1  ?  d  i  )      (  y  i  ;y  i  ;x  i  ;  ^    )+  d  i      S  (  y  i  )] ?  1    Z   1(  y  i  <c  )    (  y  i  ;c;x  i  ;  ^    )  dG  (  c  )    : (2.9) Byiteratedexpectations,thepopulationanaloguestothesampleaveragesdening ^    will bethesamemomentfunctions,  E      (  y  i  ;c  i  ;x  i  ;  )]or  E      (  y  i  ;c  i  ;x  i  ;  )],asappearinthexed censorshipcase,sothesameidenticationconditionsimposedforxedcensoringwillapply underrandomcensoring.Unfortunately,whenthecensoringvalues  f  c  i  g  haveanon-degeneratedistributionitis unlikelythatthecensoringdistributionfunction  G  (  t  )willbeknown  apriori  .Nevertheless, becauseoftheassumedindependenceofthecensoringvalue  c  i  andthelatentvariable  y    i  , thisdistributionfunction  G  (  t  )canbeconsistentlyestimatedusingtheKaplan-Meierproduct limitestimator(KaplanandMeier,1958);thisestimator ^  G  (  t  )usesonlythepairs  f  (  y  i  ;d  i  )  g  ofdependentandindicatorvariables,anddoesnotinvolvethecovariates  f  x  i  g  orparam- etervector    .BysubstitutionoftheKaplan-Meierestimator ^  G  (  t  )andsurvivorfunction ^  S  (  t  )=1  ?  ^  G  (  t  )intothepreviousminimizationproblemorestimatingequations,feasible estimatorsof    0  canbeconstructed,andconsistencywillfollowfromademonstrationof uniformconvergenceofthesesamplemomentfunctionstotheirlimitingvalues.TheestimationapproachhereissimilarinspirittothatadoptedbyBuckleyandJames (1979),whichadaptedthe\  EM  algorithm"(Dempster,Laird,andRubin1977)formax- imizationofaparametriccensored-datalikelihoodtothesemiparametricsettingwithun- knownerrordistribution.However,theBuckley-Jamesestimatortreatsthelatentdependent variable  y    i  as\missingdata"whentheobserveddependentvariableisuncensored(usingthe Kaplan-Meierestimatorfortheerrordistribution,appliedtoresiduals  b  "    y  ?  x  0 ^    andtheir censoringpoints  u  ?  x  0 ^    ,toestimatetheconditionaldistributionof  y    i  given  d  i  =0);in contrast,thepresentapproachviewsthecensoringvalue  c  i  as\missing"whenthelatent dependentvariableisuncensored.WhiletheBuckley-Jamesestimatordoesnotrequirethat 4 
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