Wolff,Augustin: - PDF document

Wolff,Augustin: HeapinganditsConsequencesforDurationAnalysis Sonderforschungsbereich386,Paper203(2000) Onlineunter:http://epub.ub.uni-muenchen.de/ Projektpartner ThispaperanalysestheconsequencesofheapingindurationmodisaspecicformresponseerrortretrospeclaborRespondenspellepisodebasedCalendarbasedquestionnairesinsteadmayleadtoabnormalconcentrationsofthestartandorendofspellsatspeciccalendarypeingwhicKrausandSteiner tiedforspelldatafromtheGermanSocioEconomicspecialcaseofanexponentialmodelheapingwithasymmetriczerodoesmodeldependencediscusstheproposalberstudieswkthetheoreticalresultsTheMonteCarlosimalsoshowthatanamountofheapingthatcharacterizestheGSOEPdoeseibullmodelitclearlyleadstospuriousseasonalsomedirectionsoffutureworkareindicated SeminarfurempirischeWirtscungLudwigMaximiliansUnivyofMuhLudwigstrRGD MunichJoacademiestrIDMunichthomas statunim oltroductionRetrospectivrespondentoprovideevthistoriesoflaborforcestatesordurationofsomelaborforcestatesubstantialresponseerrorsmayemergeHeapingorroundingoisaparticularformofsucherrorsTherearetypesofheapingdependingonthedesignofthequestionnaireemergefromcalendarbasedSocioEconomiccalendariumrequiresrespondenpossiblelaborHeapingimpliesthatrespondenroundooruserulesofthreportinglaborrespondenwhobecomeunemploinFebruaryorMarcjustreportthestartoftheirspellheapingpatternforunemplotdurationdataoftheGSOEPWresponsesepisodebasedquestionnairesWhenrespondentsrecalltheentiredurationofspelltspellsfromItalianLaborForceSurveydataaresubjecttothisheapingBoththestudyofTorelliandTellatoaswellasthatofKrausandSteinerproposedmodelstestimatesoftheparametersinthepresenceofheapingKrausandSteinerwhodealtwithheapingincalendarbasedquestionnairesprovidednoinformationaboutthegeneralconsequencesfordurationanalysisofthetypepossibleexponenmodelloglogisticmodelspecic ThereissomeevidencethattheeastGermanunemplotspelldataoftheGSOEParealsocharacterizedbyheapingWolercomparedwiththendingsofKrausandSteinerforthewestGermanunemplotspelldatathisamountofheapingislo olprocessItisawwnresultthatazeromeanmeasurementerrorofthedepenindependendoesmodelsresponsespellsyadvestimationresultsdurationmodelsmodelsheapedaimofthispaperistoulationstudiesreticalconsiderationswhetherandwhenspecicformsofheapinghaThisisparticularlyimportanttoknoifthereisnosucienpaperheapinginlaborforcesurvSectionthreeshowswhetherheapingevenasspellparameterestimatesifthetruespelllengthsaredrawnfromtheexponenulationstohighlighttheconsequencesofheapingforparameterestimatesSectionvesummarizesourverypreliminaryresultsandindicatesextensionsofthisworelliandellato studiedLaborinformationof TheItalianLaborForceSurveyisaquarterlysurveyofrotatingpaneldesignfamilyisinedfortoconsecutivesurveysandthendroppedfortosurveysandedagainfortonaleoplewhoconsiderthemselvesasunemploersareaskedforhowmanymonthstheyhaealreadybeenlookingforajob olresponsesretrospectivlengthofanunemplotspellinprogressatthedateofthesecondsurvrstandsecondof Itconsistsofindividualsagedbeteen and yearswhoareunemploedintherstsurvThissamplevidedevidenceforabnormalconcentrationsofunemplotdurationatpercenreportedspellultiplesof monthsformenandwomenrespectivItisthusclearthatrespondentsusedaruleofthbtoreportthelengthoftheirunemplospellstheirdurationofunemplotomultiplesofsixor durationtotheseabotdurationTheauthorspointedoutthattherearetruebehaioralreasonsthatcouldleadtosuchspikesatsimilarspelllengthsTheymafromthetimeunbenetsforstudiesofunemploTheauthorsdevelopedamodelthatyieldsconsistentestimatesofacontimeparametricdurationmodelinthepresenceofheapingLetTbeaconuousrandomvariableofdurationofunemplotwithprobabilitydentthatcompletedspelldurationsareobservedfornsubjectspelllengthbheapedMarrangedinincreasingorderisaBernoullirandomvariablewhichequalsoneforheapedvaluesandzerootherwiseLetfurther betheythat where isaparametricfunctionknouptodescribesthetruespelllengthTheinferentialproblemistoestimatefromthedatathataresubjecttoheapingandaknosetofheaped olelihoodofarandomsampleiszznonheapedheapedupperheapedspelllength beingconstantheinalselihoodbecomes isdistributionfunctionfactorisingthelatterlikelihoodcomponenmodelolvingtheheapingfunctionemergetheparameterscouldbeconsistentlyestimateddisregardingtheheapingproInordertorevealtheeectsofheapingtheauthorsgenerateddurationdatathatstemfromtheexponentialtheWeibullandtheloglogisticdistributionTheydidnotintroducecoTheresultingspelllengthswerealteredbaheapingpatternthatisclosetotheonethattheyrevealedfromtheItalianLaborexponenelihoodfunctionselihoodofmodelsexponenmodeldoesloglogisticdurationmodelsereconsiderablyapartfromtheirtruevalueswherethebiasappearstobepositivelyrelatedtotheamountofheapingNexttheyconcludedthatacrudeelihoodeparameterestimatesconsiderablyInthecaseofalargesamplesizemodelselihoodestimatesclosetotheirtrue olspellsrespectretrospectivlaborrespondencodehisherlaborforceoccurpeopletooatcertaincalendarmonthsandnotfrequentlyenoughatthemonthsthatareclosetothemThestudyofKrausandSteineridentiedsuchabnormalconanalyzedaninowsampletoregisteredunemplotheGSOEPWestfromJanuary toDecember eoplewhowsampleofuncensoredandrightcensoredunemplotspellsThesepeoplebeenbehafromunemplowsaLaborbothationperiodThestrikingndingsweretoabnormalconcenpopulationperiodspondingpopulationvComparedwiththeirpopulationvaluestheoutowratesoftheGSOEPberboringmonthsOctoberandNobertendtobesomewhatloThesetoheapingpatternsimplythatspelllengthsarereportedastoolongforspellsthatstartintherstquarteroftheyearandorterminateduringitslastquartereconometricdurationmodelsnegativdependencemaybeoerstatedandbiasedcoecientsforseasonaleectsma TheauthorsaccountedforsampleattritionoftheGSOEPWestrelativetotheregisterdatabyappropriatewtingfactorscalculatedonayearlybases olulationstooftheidentiedheapingpatternforstandarddurationmodelsNordidtheydealwiththistopicfromatheoreticalperspectivTheyadjustedthemodelofTorelliandTellatoinordertoincorporatethespecicheapingpatternthattheyspellsofthetoadiscretetimeproportionalhazardsmodelspellsmodelspeciedberspelltheheapingprobabilitiesusingoutsideinformationKrausandSteinerproceededthenbycomparingtheestimatesofseveraldurationmodelsthattakheapingintoaccountorcompletelyignoreitAllmodelsincludedastandardellasabaselinehazardLetussummarizetheirresultsTheauthorsfoundhardlyanydierencebeproportionalhazardmodelwithexiblebaselinehazardwithandwithouttheircorrectionforheapingincorporatedelihoodcoefmodellargethesameasforthoseofthehazardmodelwithaexiblebaselinehazsmootherberproportionalmodelincludingdummiesbercoecienmodelsdummiesandynotbeduetoheaping TheauthorsagainusedinformationaboutthemonthlypopulationinowandoutoaspublishedbytheGermanFederalLaborOceTheseincludedageforeignerdisabilitmaritalstatuseducationhouseholdincomeandtheregionalunemplotrateTheychosealogittransformationoftimenamelyexpexpastheirbaselinehazardfunction ollookmodelInthissectionwetakeasimpliedlookatnoncorrectedmaximumlikelihoodmodeldistributedwithparametersifitsdensitexpdependsdependencyypothesis correspondstomonotonelyincreasinghazardcontainingtheRadistributionwithlinearhazard Thespecialcaseofconstanthazard istheexponentialmodeltroducesoneusuallyparameterizesexpThentheWeibullmodelisaspecialacceleratedfailuretimemodelKalbeiscPren p amodeloftheindependentlyandidenticallydistributedHeretoobtaintheWmodelistakasextremevdistributedandissetAppendixspecialcaseofahomogeneoussamplewithoutindividualcoariatesienerthelessusethereparameterizedform insteadofitselfbecauseitis oltheusualwyoflookingattheWeibullmodelineconomicsitallowstouseforageneralizationofourtocoEstimationoftheunknownparametersistypicallydonerelyingonumlikelihoodprinciplecorrespondingexp expprocedureisneedederifonetreatsasxedtherstlineyieldsanexplicitsolution PniTi elihoodelihoodheapedadditionalconsiderationsareneededheapedelihoodoformallyintroduceaheapingmechanismtypicalforcalendarbasedquestionnairesweassumethateveryspellmaybeheapedwithacertainprobabilywhichisassumedtobeindependentofthecoariatesandthespelllengthDenoteforsomesucientlylargetheprobabilitythataspellisprolongedbqandbtheprobabilitythatthespellofthedurationtimesTT olTTdependonthemonthofenAsshowninAppendixAassumingindepenbehabelowdoesonlydependonthemarginaldistributionoftheThereforeaftercalculatingthemarginalheapingprobabilitiesproceedlossofgeneralitywiththemodeldescribedin plugginginheapedTT lnPniTi  goodqformallymabecomenegativDependingofthequestionnairethismaoftenbeunrealisticonesimplythenwouldnotrecordedthesespellsInthiscasetheheapedornaive maximumlikelihoodestimatew lnPniT  max etacitlyassumethatiswelldenedThisisalwysthecasefororifequalszeroforev olberofspellswithumlikelihoodrespectthisarstexplorationofthebehaviourofsamplesizewingtoinnitywillbeperformedoobtainarstimpressionoftheasymptoticpropertiesofandofrestrictourattentiontotoimportantspecialcasesnamelytheexponendistribution andtheRayleighdistribution TheasymptoticbiascanbegivaclosedProposition welldenefollowing expln exp ByusingtheGeneralBinomialtheoremtheconsiderationsgivenbelowcanbeeasilybeoddproceduretroubleifforsomeThisisthecaseforanyrealisticconstellationBytheirdependenceontheunknotheseadditionalconditionsmaybesometimesButnotethatitisalwyssatisedinthecaseofsymmetricheapingaswellasforthepositivelybiasedonesidedheapingpatternobservedintheGSOEP ol exp qXl ll l exp BBBXl ll l exp qXl ll l expAppendixThebiasgrowsatterbythelogarithmicfunctionNotefurtherthatthebiasproportionalspelllengthsmallerceterisbecomesspellwillshoproposalestimationofBeforediscussingthiswebrieywttolookattheestimateasdenedin andthebehaofbothestimatesinsomespecialcasesProposition proimmediatelyaboundbiasoftheemainvalid ifonerandequalitiesbytherelationsgr olReturningtooextremeybeofspecialinTherstoneistheconstellationwhereintheypeforallulaeaboThesecondoneisthesymmetricsituationwhereforevtheproportionofthespellsprolongedbandtheproportionofthespellsshortenedbypicallywillnotIfinthesituationofProsition theheapingis i e forallq thenisconsistentinthecaseoftheInthesituationofCorollaryoneregularlyhasexponensmallbiasexpectedputtingnegativtozeroTheTorelliandTellato caseisactuallynearlysymmetricSoCoroltionsdiscussedinSectiontInanexponentialmodelthebiasfromignoreibullmodelwithdurationdependenceqestimationpropossible olProposition expln exp qXl l l lvuut Appendixtheresultsgainedbytheseconsiderationsareofcoursequitepresharpeningboundnaturenaturallyprovidingnoconcretestatementonnitesamplebiasdependencetedherenitesamplesizeastronglationbetestimatesofwillbewingsimulationstudy oldoesmodelserrorisofthezeromeansymmetrictypecouldasymptoticallyleadtobiasedestimatesoftheparametersoftheWeibullmodelInthissectionwepresenresultsfromsimulationsinordertostudythenitesamplebehaviourandtheallwypemodelssimilartospellaboutexpectpopulationSecondweattempttoshowwhethertheparameterestimatesarelessprecisetroducesdependenceeusetheestimatingequations and derivedfromtheideallikelihoodheapedulationrunstheestimationproceduretimeseassumethatspelllengthsmeasuredinconuoustimetheunitismonths andthattherenocensoringypetionstudiesconsiderdiereneibulldistributionsasDGPsofthespellspellspecicspellheapedpercenspellbercorrespondingspellpercenspellberbecome oldurationispercenheapedspellsaboutpopulationspellsulationsforbetAlsothedurationdependenceisvdurationdependenceparameterthatresultfromestimationwiththeheapedThecorrespondingaeragestandarderrorsandthosethatareacyMLestimationwiththeoriginaldatapriortoheaping arealsodisplaanela ulationresultswithoutandurationdependenceexponenspellbothdependenceheapedbothcoecienthedurationdependenceparameterthatresultfromtheheapeddatasoforahigheraeragespelllengthThemeancoecientsinthepresenceheapingarevtotheparametervInPanelb theunderlyingDGPisaWeibulldistributionwithpositivedurationdependenceThemeanparameterestimatesinthistablehardlydierfromtheparametersoftheDGPNeitherarethesimulationresultsforheapedoriginaldataecarriedoutthesameanalysisforaWeibullDGPwhereissettoieedurationdependenceTheresultsaredisplaedinpanelc ofTable HereweclearlyseethatthemeanestimateoftheconstantfortheheapeddataexceedssomewhatitstruevAlsotheaerageduration ThistypeofheapingleadstosomenonpositivespelllengthsEgaspellthatlastsforonemonthofwhichthestartisheapedfromDecembertoJanuarywouldneverhabeenobservSowediscardedsuchspellsintheheapedsampleThismeansweanalyzethebehaviouroffromandnotthatoffrom olspellpercentofthespellstartsheapedfromDecemberandFebruarytoJanpercenspellsstartsheapedfromNoberanddependenceexponenspellberofobsheapeddataoriginaldata heapeddataoriginaldataEstimatedmeanEstimatedmeanCoe Coe true                  positivdependence berofobsEstimatedmean Estimatedmeanheapeddataoriginaldata heapeddataoriginaldataCoe Coe true         true       dependence berofobs heapeddataoriginaldata heapeddataoriginaldataEstimatedmeanEstimatedmeanCoe Coe            true        berofsimulations oldependencecoecientishigherthanthetrueNoneofthesedierencestaccordingtotheaeragestandarderrorsNextthereasonforthesymmetricheapingspellsoriginallengthpletelydiscardedduetotheheapingTheproportionofsuchspellsincreasesspellberspellstheislikelytobecomesomewhathigherthanitstruefarissymmetricheapingspellsheapedmodelspellAppendixreectscharacteristicsofarealworldsampleofunemplotspellspoinparameterestimatesspellwingsimSupposespellexponentiallydistributedsothereisnodurationdependenceSomeresponspellsbetspellreportedspellspellexpectedeibulldistributionwiththeheapeddatatheestimatedconstantshouldbeupardbiasedpositivdependencetimewsamplesizeandofheaping Theseresultsareaailableonrequest olspellsstartinJanuarybyonemonthwhileofspellsthatstartinFebruaryareincreasedbomonussomewhatlessthanoftheoriginalspellsareincreasedinlengthAgainwecarriedoutsimulationsestimatingthebothdependenceexpectedsamplesizen InPanelb ofTableweraisedthemeasurementerrorsubstanpercenspellspercenpercenheapedtothestartingmonthJanmorethan ofthespellshadtheirspelllengthscTheresultisthatboththeadependencetheheapeddataincreasedinsizeascomparedtoTablea Wherethetruetisequaltoonetheaerageestimatedconstantis totheestimatedmeanstandarderroritissignicantlydierentfromitstruepositivdependencebecomesmodelspellestimatedparametersremainlargelyunalteredfromthehooseexponenpatternwillbesuchthatonaeragepercentofthespellsthatendinOctoberareprolongedbomonthsandpercentofthosespellsthatendinberwillbecomeonemonthlongerSotheirspellendissettoDecemberexponenofco olableulationresultsforexponentiallydistributedspellspercenspellheapedpercenspellheapedheapeddataoriginaldata heapeddataoriginaldataEstimatedmeanEstimatedmeanCoe Coe   true        percenspellheapedpercenspellsheapedpercenspellsheapedheapeddataoriginaldata heapeddataoriginaldataEstimatedmeanEstimatedmeanCoe Coe        berofobsberofsim olexpospellpercenspellheapedberberpercenspellheapedOctoberberHeapedEstimatedmeanCoe oct berofobservberofsimaryingdummariablesforfromJanOctoberberberableshowsoursimulationresultswhenestimatingtheparameterswiththeheapeddataTheyclearlysuggestthatthereisaspuriousseasonalcoecienOctoberberimpliesthatbeloberbercoecienandsignicaninthismonthistoohightiedeectsofheapingontheparametersestimatesofsomemodelsthatisusualintheGSOEPWestisconsideredwereachedthefollowingcon oldoesdarderrorsthatdierfromthoseestimatedwiththeoriginaldataheapingpatternsofthekindthatKrausandSteinerfoundfortheunemplotdurationdataoftheGSOEPWestarenotsymmetricTheirvstudyfoundthatrespondentstendtoplacethestartoftheirspellstoooftentoJanuaryandnotfrequentlyenoughtoFebruaryandMarcSothespellsmodelypepositivdependencebecomespellypothesesspellsheapedspecicoccurexponenaneectiftheisonlyofaboutofspellsThereareseveralpossibleextensionsofthisworkwhicttoaddresstheoreticallyaswellasbysimFirstitisplausiblethatheapingmadependSupposerespondenbecomeedregularlyataroundthesametimeoftheyearseasonallyunemploOnemayexpectthattheyaremorelikelythanotherstoheapthespellsbecomeunemplofocusailableinmanycounThereforetheuseofoutsideinformationshouldpossibleincorporateindependenelihoodmethods OneotherwytocorrectforheapingcouldbebyincludingasetofdummforthestartingmonthsascoResultsthatareaailableonrequestsuggestthatthisindeedimproestheestimatesoftheconstaneritleadstoalargerpositivbiasofthedurationdependenceparameter olmethodstofheapingasypeofinalcensoringearegratefultoHansforhelpfuldiscussionsandJDandRL  TheStatisticalANewModellingtdurationmodelswithanapplicationtoretrospectivSocioEconomicPModellinginaccuraciesdurationdataEuropeanAppendixProofPropositionproofPropositionberandonthefollowinglemmaWeibullexp  exp  oloccurring du dt  toGammain  exp u   troducestochasticallyindependentofdescribingtheheapingsuchthatAccordingto onehasfor nIE TiE TiHiE TiE Hiexp  Xl l llThereforeplimn   plimnBBBBBBBBnXlTi nCCCCA CCCCAlnBBBBexp  BBBBXll l l  CCCCACCCCA lnBBBBXll l l exp  CCCCA ol   qXl l llqXll llexp  BBBBXll l l exp  p qXll l l exp  CCCCAThereforelimn   plimnBBBB lnBBBBnXi Ti nCCCCA CCCCA  BBBBXll l l exp  p qXll l l exp  CCCCA  BBBXll l l exp  p qXll l l CCCA olAppendixProofPropositionromProposition Parta exp exp exp exp ThereforebySlutzkysTheorem olromProposition Partb BBBBXll l l exp  p qXll l l exp plim Xll l l exp  qXll l l exp exp exp plimUsing theassumption exp madeinPropositionimpliesplimThisguaranquadraticformintheabopossessesawelldenedanduniquesolutionnamelyexp p qXll l lvuu plim ol  qXlp  ll lvuut qXll l lqXll l l A  qXl  ll lvuut plimAppendixDependsdependforeverymonstheheapingprobabilitiesljljdescribingtheentrymonthofunitAssumethatthetruedurationisstochasticallyindependentofandanalogoustoaboethatareconditionallyindependentgivurthertheenistakentobeindependentlyandidenticallydistributedamongallunitsninthesamewItisshownthattheexpectationdependsonlyonthemarginalprobabilitiesljqljq olAppendiypendixAtodiscussbiasandbiascorrectionarealsovalidforthecaseconsideredljljljljljljljljAppendixAppendixberisintroducedtoanexponentialDGPTheheapingisagainsuchthatthereispercenspellberapercenthatthespellstartsberwillbecomeJanTheprobabilitythataspellstartsinaspeciccalendarthisnolonger  Thecalendarstartofthespellaredrawnsuchthat olableBulationresultsforexponentiallydistributedspellsincludpercenspellheapedberpercenspellsheapedberEstimatedmeanCoeagehnicalunivydegreetRatioberofobsberofsimwiththedistributionofthepopulationinoercalendarmonthsintoregisteredunemplotinEastGermanertheperiod to coecienandstandarderrorsofsimulationsofmaximumlikelihoodestimationofelihoodulatedparametertlydierenfromtheirtrue Wolff,Augustin: HeapinganditsConsequencesforDurationAnalysis Sonderforschungsbereich386,Paper203(2000) Onlineunter:http://epub.ub.uni-muenchen.de/ Projektpartner