multinomial and conditional logit models

8/13/2019 Multinomial and Conditional Logit Models

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Multinomial and Conditional Logit Discrete-Choice Models in Demography

Author(s): Saul D. Hoffman and Greg J. DuncanSource: Demography, Vol. 25, No. 3 (Aug., 1988), pp. 415-427Published by: Population Association of AmericaStable URL: http://www.jstor.org/stable/2061541

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Demiiograplhy,ol. 25, No. 3, August 988

Multinomialnd Conditional ogitDiscrete-ChoiceModels n DemographySaul D. HoffmnanDepartmentfEconomics,UniversityfDelaware,Newark, elaware 19716GregJ.DuncanInstituteor ocial Research,UniversityfMiclhigan, nn Arbor,Michigan 8106Althoughdiscrete-choicetatistical eclhniqueshavebeen used with incrcasinigregularityn demographicnialyses, cFaddein's oniditionialogitmodel s less wellknown nd seldomused. Coniditionalogitmodels re appropriate lleil lle choiceamong lterniativess modeled s a functionif thecharacteristicsf the lterniatives,ratherhan or naddition o) the haracteristicsf he ndividLualaking lle hoice.Weargue hat his eaturef oniditionalogitmakes tmore ppropriateor stimatinigbehavioralmodels. In this article, he coniditionalogit model is-presenitednidcomparedwith hemorefamiliarmultinomialogitmodel. The differcniceetweenlthe wo echniquess illustrated ith nl nialysisfthe hoiceofmarital nidwelfarestatus ydivorced rseparated omeni.

Statisticalechniquesfor heanalysisofdiscrete hoices have beeinused with ncreasinlgregularityn demographic analyses. The best known are the binomial logit and probittechniques,oth fwhich re suitable or inaryhoiceproblems. orproblemsnvolvingthechoiceamong hree r more ategories,hemultinomialogit echniquesmost ftenemployed;hecorrespondingrobitmodel s usedrelativelyittle ecause of tscomputa-tionaldifficulty.irtuallynused husfar s a closely elatedechnique alled conditionallogit, model hat swell uited or ehavioralmodelingfpolychotomoushoice ituations.Developed yMcFadden 1973), the onditionalogitmodel swidely sed ntransportationdemand tudiessee Ben-Akiva nd Lerman, 1985)but is seldomused in demographicresearch.Conditional ogit s not simply differentnd arguably referableechniqueforestimatinghekind fmodelsforwhichmultinomialogit s currentlysed.Rather,t sappropriateor differentlassofmodelsnwhich choice mong lternativesstreatedsa function f the characteristicsf the alternatives,ather han or in addition o) thecharacteristicsf the ndividualmakinghechoice.We believe hatmany roblemsf nterestodemographersnd other ocial scientistscan be modeledby using characteristicsf thealternative pproach.Thus they reappropriatelystimated ithconditionialogit. Furthermore,e suggest hat t is oftendifficultoattach behavioralnterpretationotheresultsfmodels hat ocus xclusivelynthe characteristicsfthechooser -that s,those stimated yconventional ultinomiallogit.The next ection f his rticle escribeshebasic tatisticalropertiesf he onditionallogit CLGT) modeland compares t with hebetter nownmultinomialogit MNLGT)model.3 It alsoconsidersheform f heunderlyin-godels f ndividual ehaviorhat eadCopyrightD 1988 opulationissociationifAmicrica

415


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416 Demography,ol. 25, No. 3, August988to MNLGT and CLGT estimation.he third ection resents brief iscussion f somepracticaltatisticalnd estimationssues elatingo theCLGT model.The final ection sesdatafromhePanelStudy f ncomeDynamics o llustratehedifferenceetweenhe wotechniquesnappliedwork.We examine ivorced r eparated omen's hoice mong setofmaritalndwelfaretatus lternativesyfirstsing pureMNLGT model, hen pureCLGT model, ndthen mixed ersion hat ncorporateseaturesf both.

StatisticalndModeling ssuesBoth multinomialogit nd conditionalogit re used to analyze the choice of anindividual mong setofJ lternatives.he central istinctionetween he wo,can e putvery imply:MNLGT focuses n the individual s the unit of analysis nd uses theindividual'sharacteristicss explanatoryariables;ncontrast,LGT focuses n the etofalternativesor ach individual nd theexplanatoryariablesre characteristicsfthosealternatives.LetXistand or he haracteristicsf ndividualandZi,for he haracteristicsf he thalternativeorndividual, with hecorrespondinigarameterectorsenoted y/3 nd a,respectively.et J e thenumber f unordered lternativesfor he moment, ssumedconstantor ll individuals)ndPi, heprobabilityhat ndividualchooses lternative. Thechoiceprobabilitiesn the MNLGT and CLGT models re

JMNLGT: P11 exp(Xi/31) I exp(Xi/3k), (1)Jk=CLGT: Pjj = exp(Zija)/ : exp(Zika). (2)k =j

Ina mixedmodel hat ncludes oth haracteristicsf he lternativesnd the nidividual,hecorrespondingrobabilityanl e writtensJMixed: Pj; = I exp(Xifjl+ Z1jfl)/Cxp(Xi,PkZika). (3)k= I

We discuss hemixedogitmodel 3) furthern thenext ectionndestimateuch a modelin the ast ection f this rticle.Notethe ymmetryn equations 1) and 2). In theMNLGT model,theexplanatoryvariables X), being characteristicsf the individual, re themselvesonstanitcrossthealternatives.onsequently,heonlyway hey an affecthoiceprobabilitiessbyhavingdifferentmpact n thevarious lternatives.hus in practice,MNLGT estimates set ofJ 1 coefficients(31) or ach explanatoryariable. he estimated oefficientshow theeffect ftheX variables nitheprobabilityfchoosing ach alternativeelative o onealternativehat ervess a common enclhmark.here reonlyJ 1 coefficients,ecausethescaling f the coefficientss arbitrary.hus it s necessaryonormalize n one set ofcoefficients,ypicallyy settingt equal to zero. For this lternative,hecorrespondinigprobabilitys 1/E xp(Xi,/31),ince 3= 0 and exp(O) 1.In contrast,ntheCLGT model, he xplanatoryariablesZ) assumedifferentaluesineach alternativenote hepresence f j subscriptnZ butnotX), butthe mpact faunit fZ isusually, lthough otnecessarily,ssumed o be constan-tcross lternatives.nthat ase, only single oefficients estimatedor ach Z variable,o the mpact f variable


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Multinomial nd Conditional ogitModels 417on the choice probabilitieserives rom he differencen its value across alternatives.Consequently,n the standard LGT formulation, Z (or X) variablewithno variationacross lternativesas no impact n choiceprobabi1ities.hen such variables redeemedtobe important,he mixedmodel s required.The basic differenceetween heMNLGT andCLGT formulationss clearerwhenequations 1) and 2) are rewrittenydividing hrough y thenumerator:

JMNLGT: P1j 1 I exp[Xi(3k - Pi)], (4)/k=lCLGT: P1j= 1 E exp[(Zik Zj)aj. (5)

k=iHere,theprobabilityn equation 4) depends n the differencen the coefficientscrossalternatives,hereasnequation5), theprobabilityepends nthedifferencesnthevalueofthe characteristicscross lternatives.The differenceetween heMNLGT andCLGT modelss notmerely ne of tatisticalform. he choice probabilitiesn equations 1) and (2) reflectheunderlying odelsofindividual ehaviorhatnecessarilyeflectypothesesbout hebasison which ndividualsmake hoices mong lternatives.ften his s notmadeexplicit, ndresearchers ove otheirmpiricalstimation ithout irstpecifyingheunderlyingehavioralmodel. n fact,however,t s a crucial tepfor he nterpretationfthe empirical esults.LetVi,standfor he value utility)f alternativeto individual, and assume, s abehavioral ule,that an individual hooseshis or hermost highly alued alternative.Suppose hatViidepends n the ttributesfthe lternativesZ1) hroughomeunspecifiedfunctionalormfi).Thenthechoiceprobleman be representedy pairofequations sfollows:

vil = f(Ziv), (6)Pi, = Pr(Vi,> Vik) all knotequal toj. (7)

With he ddition f n appropriatelyefinedrror erm,6quation 6) leadsto theCLGTmodelrather han he MNLGT model, incethe characteristicsf thealternativesre thedeterminantsfchoice.The estimatedarametersffi providenformationotonly boutthechoiceprobabilitieshroughquation2) but lso about hevalue functionnequation(6). The specific orm f equation 6) will,ofcourse, arywith henature ftheproblemand thediscipline. conomists, or xample, irtuallylways egard tilitys a functionprimarilyfan individual'sevelofconsumptiondefined roadly) r, equivalently,heexogenousncome nd the etofprices e orshefaces.Viewed nthisway, quation6) isa statementbout hefunctionalelationshipetweenhecharacteristicsfthe lternatives(theZi,'s) andtheutilityf eachalternativeo the ndividualtheVq1's)-in hort, utilityfunction. quation 7) representshewell-knownrinciplefutilitymaximizationppliedto a discretehoiceproblem.We estimateversionf quation 6) in the ast ection f hisarticle.Noneconomicmodelsbased on equation 6) could also be formulated,lthoughweknow f no attemptso do so. Forexample, choice modelofbecomingmarried ersusremaininginglemight iew hevalueof hese wo lternativess a functionf he conomicsecurity,ompanionship,ndependence,nd other ttributeshat ach provides, ith he


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418 Demography,ol. 25, No. 3, August 988perceivedxtent f hese ttributesn each alternativebtained hroughurveyuestions. oavoidproblems ith espondents'ationalizingast ecisions, useful esearchesignmightbe a two-wave anel n which he ttitudinalnformationsascertainednthefirst ave ndthe behaviore.g., transitiono marriage)s measured n the second. Thus reportsyunmarriedespondentsbout he ttributesf hevariousmaritaltatesn a first ave ouldbe used to predict heprobabilityf marriagen a follow-upnterview.Another xamplewouldbe a child-care hoice model n which hevalue of a givenchild-caremodel (e.g., day care, sittern own home) is taken to be a function fcharacteristicsuch as its ikelyffectn childdevelopment,ts ost, nd itsreliability.Note that n these two examples he perceived r objective haracteristicsf eachalternative ather han its subjective mportancer satisfactionre used to explain anindividual's hoice. The statisticalmodel would then provide nformationi.e., theestimated oefficients)bout the relative alue that individuals lace on the variouscharacteristics,nferredrom heir ctualbehavior.Whatbehavioralmodel eads to MNLGT estimation?n general,MNLGT willbeappropriate hen quation 6) is replacedwith

Vi = f2(Xi)7 (8)where 2 s someunspecifiedunctionelating i toVii. In equation 8), the value ofanalternatives regardeds a function f the characteristicsf the ndividual. ssuminghatequation 7) stillholds, quation 8) then eads toMNLGT estimation.Multinomial ogitcan also be shown to represent nonbehavioral,educed-formversion f equations 6) and 7). Ifequation 6) holdsbut

zi, = g(Xd, (9)thenVij= h(Xi). (10)

Equation (10) relates he value of the jth alternativeo the characteristicsf the ithindividual, ut withoutncludinghecharacteristicsfthe thalternative.In a sense,the choice betweenMNLGT andCLGT is thechoicebetween modelrepresentedy either quation 8) or equation 10) and one representedy equation 6).Althoughhere s,of ourse, ogeneralule boutwhich spreferable,ethink hat modelbasedon equation 6) andutilizing quation 7) hasmuchto recommendt.The generalnotion hat tructuralodels repreferableo reduced-formodels s oneargumentnfavorofequation 6) and CLGT estimationnstead fequation 10) and MNLGT estimation.Eventhoughmodels uch as equation 8) mayprovide irectnduseful nformationboutwhich ndividualsmakewhich hoices, hey re often otwell suited otesting ypothesesaboutwhy hose hoices re made. ndeed, he nterpretationfmodels uch sequation8)oftenmakereferenceo the untested)haracteristicsfalternativesvailable oparticularindividuals.Models such as equation 6) areespecially ell suited or heanalysis f situationsnwhichgovernmentolicyaffectshe attractivenessf an alternativey changing omerelevantharacteristic.xamples nclude heAidto Females WithDependentChildren(AFDC) program, hichprovidesncome o female-headedamilies ith hildren;chol-arship id forhigher ducation,whichmaymakecollege ttendancemore ttractive;ndsubsidies or aycare,whichmay ncreasehe aborforcectivityfwomen.To assess heeffectfgovernmentolicies ike hese n individualhoices, t snecessary,henpossible,to include hepolicyparametersirectlyn the choiceproblem. ince theseparameters,though,retypicallycharacteristicf he lternativenquestion, conditionalogitmodelsuch as eauation 6) is the DDroDriatemodel.


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420 Demography,ol.25,No. 3, August988record nd alternative. The explanatoryariables resimilarlyonstructedoreflecthevalueof achvariable or ach individualneach alternative.nindividual'shoiceamongthe lternativess ndicated y 1 for he ppropriateecord; heotherlternativesre coded0. Table 1 illustrateshe typical ata structureorCLGT estimation.n this xample,there rethree lternativesor achoffourndividuals, hochoose lternatives, 2, 1, and2, respectively.hereis one X variable nd one Z variable; he inclusion findividualcharacteristics eans that he model s really mixedmodel rather hana pure CLGTmodel.Z11 s the value for ndividual of somecharacteristicn alternative, Z12 s thevalue of that haracteristicor hat ndividualn alternative, Z21 is the value ofthatcharacteristicor ndividual in alternative, andso on. Estimationfthismodelwouldyield single oefficientor .The final wo olumnshowhow nattributehats nvariantcross lternativesan beintroduced o create mixed ogitmodel. Let D2 be a dummy ariable qual to 1 foralternativeand 0 for heother lternatives,nd etD3 be definedimilarlyor lternative3. The variablesnthefinal wo olumns reD2Xand D3X; ust s in MNLGT estimation,theygive the effect f variableX relative o an omitted ategory, ere alternative .Estimation fthismodelwouldyield hree oefficients-oneachfor , XD2, andXD3. Ifdesired,onstanterms or wo lternativesouldbe constructedyusingD2 andD3.There s an additionalndparticularlyseful eaturefCLGT models. nsomechoicesituations, ot every lternatives availableto every ndividual. or example,womenwithout hildren recategoricallyneligibleo receiveAFDC, and onlywomen ivingnstatesfferingheAFDC-UP programanchoose o be bothmarriedndreceiving elfare;onlywidowswith iving hildren an choose to livewith heir hildren; nly ndividualsowningarsor iving earbusroutesan drive r take hebus towork, espectively.akingproper ccount of differencesn thesize and compositionf the choice set availabletospecificndividualss troublesomendermost ircumstancesnd ofteneadstoclumsy,dhoc solutions. he samplemaybepartitionedo that he nalysiss no longer eneral, r0valuesmight e assigned or he ndependentariablesncases ike hat oncerningFDCbenefitsfwomenneligibleo receive FDC.9Amorenaturalolution, owever,ssimplytoeliminaten irrelevantlternativeromhe hoice et or n individual.0 With he hoice

Table1. Typical ataStructureor LGTEstimationDependent

Individuallternative ariable Z XD2 XD31 1 0 Z,1 0 02 0 Z12X1 03 1 Z1 0 Xi2 1 0 Z21 0 02 1 Z22 X2 03 0 Z23 0 X23 1 1 Z31 0 02 0 Z32 X3 03 0 Z33 0 X34 1 0 Z41 0 02 1 Z42 X4 03 0 Z43 0 X4


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MultinomialndConditionalogitModels 423Table3. EstimatesfRemarriagendWelfarehoices fDivorcedndSeparatedWhiteWomen,

PSID, 1969-1982MNLGTmodel CLGTmodel Mixedmodel

Variable Married Single Married Welfare Single Married Welfare SingleConstant -2.918* -4.682* -2.408* - 2.587* - 3.004* - 3.415*(0.873) (0.783) (0.542) (0.499) (1.001) (0.922)No. ofchildren -0.680* -0.818* 0.31 7*(0.093) (0.080) (0.076)Age 0.017 0.094* 0.024 0.081*(0.018) (0.016) (0.020) (0.018)Education 0.363* 0.435* -0.216* -0.249*

(0.065) (0.057) (0.093) (0.093)Urbanresidence -0.731 * -0.613* - 0.578* - 0.548*(0.238) (0.203) (0.270) (0.247)Husband's -0.018 0.047*incomea (0.017) (0.022)AFDC income 0.192* 0.202*(0.051) (0.055)Wageratea 1.102* 1.102* 1.102* 1.492* 1.492* 1.492*(0.107) (0.107) (0.107) (0.162) (0.162) (0.162)Nonlabor - -0.011 0.215* -0.102 0.182*income (0.090) (0.076) (0.092) (0.076)Numberfcases 1,269 1,269 1,269 766 1,269 1,269 766 1,269Log-likelihood -950.4 -828.4 -770.0

a Predictedalue,fterax.*Significantt he percentevel.

the results eflect ifferencesn opportunity,r does behavior iffervengivenlimilaropportunities?he negativeffectfchildren n remarriages illustrative,inceonemightwellhypothesizehatmarriage ouldbe especially ttractiveo womeni ith hildren. heestimatedoefficientsreuseful or etermininighomakeswhich hoice,but hey re essuseful or xplaining hy hedoesso.The pureCLGT model s shown ncolumns -5. We seethere hat he ncome fawoman's potential) ewhusband oesnothave a significantffectn theprobabilityfremarriage;he ffects, nfact, egativeutverymall.4 Incontrast,he mount fAFDCbenefits as a positivend significantffect n theprobabilityhat woman willbe onAFDC. We find hatnonlaborncomemostlyomposed f limonynd/or hild upport)has no effectntheprobabilityfbeingmarriedelativeowelfare,ut hat t ncreases heprobabilityhat womanwillbe single nd notreceiving elfare.5Interpretationf he stimatedffectf woman'swagerate s llustrativef heCLGTapproach.Asshown,hewage oefficients arge, ositive,ndsignificant.ts ocfficientasbeenconstrainedobe equal across lternatives,eflectingheassumptionhat dollarofafter-taxncome sequallyvaluable neachalternative.he positiveoefficient,herefore,indicateshathigherwages ncrease hevalueof an alternative.Although woman'swageratehas thesameeffectn utilityn each alternative,hisdoesnotmean that thasno effectn her hoiceamong he lternatives.he effectf nyvariablen choiceprobabilitieserivesromhedifferencen ts alue cross lternativessee


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424 Demography,ol.25,No. 3,August988eq. (5)]. Thus a woman'swagerate ffectserchoice,dependingn how herwagevariesacrossalternatives. hat variation,n turn,dependson the estimated ncome of herprospectiveusband, he chedule fwelfare enefitsnher tate, nd herownwage rate.Forexample, ecause fter-taxarriedndsinglewages re similar ormostwomen,6thewagerate oes notgreatlyffecthe hoicebetweenmarryingndremainingingle. t does,however,ubstantiallyffecthechoicebetween hose wo lternativesndwelfareecauseafter-tax elfarewagesare sharplyower.Moreover,he differenceetweenwelfarendnonwelfare ages s greatest or wogroups f women:women n states hatprovide owwelfare enefits, practicehat ffectivelymposes veryowmaximumwage rate, ndhigh-wage omen, ince he bsolute ifferenceetween elfarendnonwelfareageratesisgreatestor hem.7Finally, onsiderwhatwouldhappen f herewere $1 increasena woman's retaxwagerate.Utility ouldrise neachalternativey1.102 thecoefficientn thewageratefrom able 3) times heresultingncreasen after-taxages.Thus, for xample,utilitywould ncrease east nthewelfarelternative,gainbecauseof tshigh axrate.Althoughutilityneach alternativesnowhigher,t s,ofcourse,mpossibleor heprobabilityhateach alternatives chosento increase imilarly. ather, heresultinghoiceprobabilitieswouldbe calculated yusing quation2) andsubstitutinghenew et f fter-taxagerates.Inthis ase,theprobabilityf hoosingwelfare ouldfall, ince tsutilityevel s now owerrelativeo the otherwo lternatives.Estimates f the mixed modelare presentedn columns6-8. Coefficientsn theindividualharacteristicslow showthe mpact f these haracteristics,etof a woman'seconomic pportunities;s intheMNLGT model, heyremeasured elativeo the ingle/welfarelternative. anyofthese haracteristicsrenow estimated o havesubstantiallydifferentnd morereadilynterpretableffectsn remarriagend welfare hoices. Forexample, he number f children'8 woman has is nowseento increase heutilityfmarriagend hencetheprobabilityhat hewillremarry,fterontrollingor ermarriageopportunities.his finding icely eparateshenegativempact f children n remarriageopportunitiesromtheirpositive mpact onl remarriage, iventhose opportunities.9Additionalears f ducation owreduce, atherhanncrease,heprobabilityfboth eingmarriednd being single/no elfare, elative o beingon welfare.Urban residence tilllowers herelativerobabilityfbeing ithermarried rsingle/no elfare,ndagesimilarlyincreases heprobabilities. s for he characteristicsfthe alternatives,heincome of awoman'spotential pouse is now estimated o be positive nd statisticallyignificant,althought sstill elativelymall nmagnitude.he effectfAFDC isvirtuallynchangedbythe addition fthe ndividual haracteristics.

Finally,we note thatboththepureCLGT model and the mixedmodel are ideallysuited o simulation fpolicy hangeswhenever,s in this ase,thecharacteristicsf thealternativesredeterminedygovernmentolicy. ne caneasily ssign ewvalues oreflectthepolicy hange f nterestnd then ecalculatehe ppropriaterobabilitiesyusing heestimatedtructuralarameters.0One can also do this or hepureMNLGT model,but heresults f,fornstance,imulatinghe ffectf changenthenumberf hildren womanhas or in her education are less informativend less directly menable to policymanipulation.Summary

This article asprovidedn introductiono and illustrationftheuse of conditionallogit oestimatemultiple-categoryiscrete-choiceroblems. LGT isclosely elatedo thebetter-knownNLGT model,but t derives rom ifferentehavioralssumptionsnd isestimatedn differentorm. he CLOT model s appropriatehenevert s reasonableo


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MultinomialndConditional ogitModels 425assume hat ndividualhoices mong vailable lternativesre a functionf therelevantcharacteristicsfthose lternatives,atherhan he ttributesfthe ndividual.n the attercase, MNLGT estimations appropriate. e argue,however,hat ucha model susuallya reduced-formonbehavioral odel andthus s of somewhatmore imited nterest.Webelieve hatmany ssues f nteresto demographersnd other ocial cientistsallnaturallyinto CLGT model.Statistically,hekey ifferenceetween he womodels nvolves heunit f nalysis:nan MNLGT model, he ndividuals theunitof analysis, hereasn a CLGT model, heset of alternativess the unitofanalysis. he explanatoryariablesfa CLGT modelareprimarilyharacteristicsf the alternatives,ut ndividual-levelariables, uchas personalattributes,an be readily ccommodatedn a CLGT model. Another seful eaturef aCLGT model is its ability o allow fordifferencesn the available alternativesmongindividuals.We illustratedhedifferenceetweenhese pproachesyconsideringhepostdivorcechoicesof womenregarding arital tatus ndwelfare eceipt. stimatesf threemodelswerepresented:1) an MNLGT model thatused ndividualharacteristicss explanatoryvariables; 2) a CLGT model in whichthe after-tax age rate and exogenousncomeavailable oa woman neachofthree lternativesere heexplanatoryariables;nd 3) amixed ogitmodelthat ncluded hevariablesrom he firstwomodels.Inthemixedmodel,we found hatmarriagepportunitiesas measuredy he ncomeof woman's otentialpouse)have modest ositivempactn theprobabilityfremarriageand thatAFDC benefitsave slightlytrongeregativempactnremarriage.nterestingly,we alsofind hatwomenwithmore hildren remore ikelyoremarry,ncewe control ortheir oorermarriagepportunities.

NotesA review f 10 issuesof Demographypublished etween ebruary 984 and May 1986)produced10 examples fdiscrete-choiceesearch sing logitmodel. Sevenofthe 10 involvedtwo-categoryependentariables-MasseyndMullen 1984) nalyzedhepresencefyounghildrenin a household,Landale and Guest 1985) mobility lansand actions,Tienda and Glass (1985)women's aborforceparticipation,ntwisle nd her colleagues Entwisle t al., 1984; Entwisle,Mason,and Hermalin, 986)contraceptionehavior, aVanzo andHabicht 1986) nfantmortality,and Beller nd Graham 1986) thepresence f a child-supportward.Examples fthree-categorychoice models nclude Lehrer nd Kawasaki's 1985) analysis f a child care modal choice andLeibowitz, isen, and Chow's 1986) analysis f teenage regnancyecisionmaking.Robins nd

Dickinson1985) estimated four-categoryodelofwelfarend child upport.2 The threemultiple-categoryodels dentifiedn note1 are all examples fmultinomialogit.Leibowitz,Eisen, and Chow (1986) used conditionalogitto describewhat s morecommonlyconsideredmultinomialogit.3 Although one ofthe tatistical ethods escribed ere s new, nd discussionsf omeoftheissues an be foundnstatisticalnd econometricsextsseeBen-Akivand Lerman, 985; udge t l.,1980;Maddala,1983),we know fno applied iscussion hat ocusesxplicitlyn the ssues iscussedhere.4 For modeling nd estimationurposes,hedistinctionrawn etweenMNLGT andCLGT isuseful nd instructive.he models o, however,hare common ikelihood unction;ee the thirdsection or discussion f this.5As inCLGT, the mixed ogitmodelusesthe lternatives theunit f nalysis.Whatwe callmixedogits sometimeseferredo s multinomialogit,withhepureMNLGT andCLGT models

treateds special ases nwhich nly hecharacteristicsf lternativesr of ndividualsreused seeAmemiya, 985;Ben Akiva nd Lerman, 985).We find his erminologyonfusing,ince t uggeststhat he statistical odel n question s themorefamiliarnd significanitlyifferentureMNLGTmodelofequation 1).


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426 Demography, ol. 25, No. 3, August 9886 Both heCLGT and MNLGT models re basedonl he ssumptionhat heerror erms ollowan extreme alue distributionnd are ndependentcross lternatives.ee the hird ection or etailson the mplicationsfthis ssumption.7 For nstance,f heoriginal robabilitiesor ome ndividualrePI = 0.4, P2 = 0.4, and P3 =0.2, then an increase n PI to 0.52wouldnecessarilyause P2 and P3 to fallto 0.32 and 0.16,respectively.his propertyoldsonlyfor heratio f probabilitiesor n inidividualnd notfor heaggregateroportionf ndividualsmaking particularhoice.8 The discussionhat ollowss based n the equirementsfLIMDEP's Discrete hoice program;Mlogit roceeds omewhat ifferently.9 Assigningvalueswillnot, nfact, roduce he orrectrobabilityor hosendividuals. scanbe seen nequation 2), ifZ = 0 forome lternative,hen xp(Za)= exp(0)= 1. Sincethis lternativedoesnotexist or he ndividualnquestion, heprobabilityf ts electionhouldbe zero; nstead twould urn ut to be 1/(1 , exp(Zikcx),here he ummationstaken ver ll other lternatives.sa result,heother robabilitiesouldbe too ow.10This can be done readily ith IMDEP's Discrete hoiceprogram.he same thing an bedone forMNLGT programs,utwe know fno softwarehat acilitatest.II A similar roceduresing more laboratemixedmodel sdetailednHoffmannd Duncan(1987).12 The otherwelfare ategoryncludesGeneralAssistancend some misreportedFDCincome.Even thoughhe 1thresholds somewhatrbitrary,llwood 1986) howed hat sa practicalmatter,heres little ifferenceetween arious hresholds.13 There s a minor xceptiono this n states hat ermit tlherwiseligiblemarried ouples oreceive enefitsnder heAFDC-UP program.t s sufficientlyare about 150,000 asesnationallyperyear uringheperiodweanalyze) hat ur data etprovideswofew asestopermitnalysis.14Althoughhe ncome f newhusband spresumedobe relevantothedecision f llwomen,

it s observed nly orwomenwho remarry.husweusean estimatedalueofnewhusband'sncomefor ll women n thesample,basedonl regression odel fit n the womenwho remarried. heincome f a woman'snew pouse s estimateds a functionifherownpersonal haracteristicsage,number fchildren, esidence, tc.) and thoseofherformer usband, ncluding is income ndeducation. ince remarriedomenmaynotbe a randomample, venof hosewho re observation-ally dentical,we also corrector ossible election ias, using technique utlined yLee (1983).(Estimatesf the pouse ncome quation re available rom he uthors.)15 The effectfnonlaborncome s measured elativeo being n welfare.We treatedhevariableinthisway likeMNLGT estimation)ecause here s nsufficientariationnnonlaborncome crossthe lternatives.onlabor ncome iffersnly orwomenwitlhlimony-we ssumed hat heywouldlosetheir limonyf hey emarried-and elativelyewwomenreceived ny limony.16 Theydiffern our analysis ecausewe treat er ncomeas marginal o her husband's nd

compute er fter-taxagerate nl hat asis.Averagemarried ages re about90 percentf veragesinglewages, lthough here s somevariation, ependingn the ncomeofa woman's rospectivespouse nd their onlabor ncome.17 These two ffects ay equire urtherlaboration.irst,he bsolute ifferenceetween elfareand nonwelfarefter-taxages s larger orwomenwithhigher onwelfare ages.Second,whenwelfare enefits re relativelyow, the maximum ncomethatcan be earnedwhilemaintainingeligibilitysalso low.Since a woman annot arnmore han his mount, he faceswhat mounts oa zerowagerate n welfare nce shereaches hatevel.Thiseffectsstrongern ow-benefittatesndforhigh-wage omen han ow-wage omen, ince nboth asesthe maximum arningsmount smorereadilyttained.18 Note thatwe have assumed hat henumber f children womanhas affects,ther hingsequal, only hevalue ofthemarriagelternative. e note, npassing,hat he bilityoso constraina coefficients an advantagef theCLGT model.19 We found hat achadditional hildreduced he ncome f potentialpouseby7 percent.20 See Hoffmannd Duncan 1987)for simulationf he ffectsfchangesnwelfare enefitson cumulative emarriageates.


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Multinomial nd Conditional ogitModels 427Acknowledgments

Thisarticlcs based on researclhupportedvNational nstituteor hild -HealthndHumanDevelopmeicnitGrant ROl HD 19339-01. thas beniefitedrom elpful ommcnitsvJohlnounid, orothv uncani,RobertHutchens,WillardRodgers, arvSolon,and Arland hornitoni.e also thanikwo nonivmousefereesnd thedeputv ditor or hcir ommncits. 'lecrder fthe uthors' amtes as determiniedandomlv.

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