chapter 5 flexible model structures for discrete choice ：5章柔性结构的离散选择模型

CHAPTER5FLEXIBLEMODELSTRUCTURESFORDISCRETECHOICEANALYSISCHANDRARBHATTHEUNIVERSITYOFTEXASATAUSTINDEPTOFCIVIL,ARCHITECTURALCHOICEOFTRAVELMODE,DESTINATIONANDCAROWNERSHIPLEVELINTHETRAVELDEMANDFIELDPURCHASEINCIDENCEANDBRANDCHOICEINTHEMARKETINGFIELDANDCHOICEOFMARITALSTATUSANDNUMBEROFCHILDRENINSOCIOLOGYINTHISCHAPTER,WEPROVIDEANOVERVIEWOFTHEMOTIVATIONFOR,ANDSTRUCTUREOF,ADVANCEDDISCRETECHOICEMODELSDERIVEDFROMRANDOMUTILITYMAXIMIZATIONTHEDISCUSSIONISINTENDEDTOFAMILIARIZEREADERSWITHSTRUCTURALALTERNATIVESTOTHEMULTINOMIALLOGITMNLANDTOTHEMODELSDISCUSSEDINCHAPTER13BEFOREPROCEEDINGTOAREVIEWOFADVANCEDDISCRETECHOICEMODELS,THEASSUMPTIONSOFTHEMNLFORMULATIONARESUMMARIZEDTHISISUSEFULSINCEALLOTHERRANDOMUTILITYMAXIMIZINGDISCRETECHOICEMODELSFOCUSONRELAXINGONEORMOREOFTHESEASSUMPTIONSTHEREARETHREEBASICASSUMPTIONSWHICHUNDERLIETHEMNLFORMULATIONTHEFIRSTASSUMPTIONISTHATTHERANDOMCOMPONENTSOFTHEUTILITIESOFTHEDIFFERENTALTERNATIVESAREINDEPENDENTANDIDENTICALLYDISTRIBUTEDIIDWITHATYPEIEXTREMEVALUEORGUMBELDISTRIBUTIONTHEASSUMPTIONOFINDEPENDENCEIMPLIESTHATTHEREARENOCOMMONUNOBSERVEDFACTORSAFFECTINGTHEUTILITIESOFTHEVARIOUSALTERNATIVESTHISASSUMPTIONISVIOLATED,FOREXAMPLE,IFADECISIONMAKERASSIGNSAHIGHERUTILITYTOALLTRANSITMODESBUS,TRAIN,ETCBECAUSEOFTHEOPPORTUNITYTOSOCIALIZEORIFTHEDECISIONMAKERASSIGNSALOWERUTILITYTOALLTHETRANSITMODESBECAUSEOFTHELACKOFPRIVACYINSUCHSITUATIONS,THESAMEUNDERLYINGUNOBSERVEDFACTOROPPORTUNITYTOSOCIALIZEORLACKOFPRIVACYIMPACTSONTHEUTILITIESOFALLTRANSITMODESASINDICATEDINCHAPTER13,PRESENCEOFSUCHCOMMONUNDERLYINGFACTORSACROSSMODALUTILITIESHASIMPLICATIONSFORCOMPETITIVESTRUCTURETHEASSUMPTIONOFIDENTICALLYDISTRIBUTEDACROSSALTERNATIVESRANDOMUTILITYTERMSIMPLIESTHATTHEEXTENTOFVARIATIONINUNOBSERVEDFACTORSAFFECTINGMODALUTILITYISTHESAMEACROSSALLMODESINGENERAL,THEREISNOTHEORETICALREASONTOBELIEVETHATTHISWILLBETHECASEFOREXAMPLE,IFCOMFORTISANUNOBSERVEDVARIABLETHEVALUESOFWHICHVARYCONSIDERABLYFORTHETRAINMODEBASEDON,SAY,THEDEGREEOFCROWDINGONDIFFERENTTRAINROUTESBUTLITTLEFORTHEAUTOMOBILEMODE,THENTHERANDOMCOMPONENTSFORTHEAUTOMOBILEANDTRAINMODESWILLHAVEDIFFERENTVARIANCESUNEQUALERRORVARIANCESHAVESIGNIFICANTIMPLICATIONSFORCOMPETITIVESTRUCTURETHESECONDASSUMPTIONOFTHEMNLMODELISTHATITMAINTAINSHOMOGENEITYINRESPONSIVENESSTOATTRIBUTESOFALTERNATIVESACROSSINDIVIDUALSIE,ANASSUMPTIONOFRESPONSEHOMOGENEITYMORESPECIFICALLY,THEMNLMODELDOESNOTALLOWSENSITIVITYORTASTEVARIATIONSTOANATTRIBUTEEG,TRAVELCOSTORTRAVELTIMEINAMODECHOICEMODELDUETOUNOBSERVEDINDIVIDUALCHARACTERISTICSHOWEVER,UNOBSERVEDINDIVIDUALCHARACTERISTICSCANANDGENERALLYWILLAFFECTRESPONSIVENESSFOREXAMPLE,SOMEINDIVIDUALSBYTHEIRINTRINSICNATUREMAYBEEXTREMELYTIMECONSCIOUSWHILEOTHERINDIVIDUALSMAYBE“LAIDBACK”ANDLESSTIMECONSCIOUSIGNORINGTHEEFFECTOFUNOBSERVEDINDIVIDUALATTRIBUTESCANLEADTOBIASEDANDINCONSISTENTPARAMETERANDCHOICEPROBABILITYESTIMATESSEECHAMBERLAIN,1980THETHIRDASSUMPTIONOFTHEMNLMODELISTHATTHEERRORVARIANCECOVARIANCESTRUCTUREOFTHEALTERNATIVESISIDENTICALACROSSINDIVIDUALSIE,ANASSUMPTIONOFERRORVARIANCECOVARIANCEHOMOGENEITYTHEASSUMPTIONOFIDENTICALVARIANCEACROSSINDIVIDUALSCANBEVIOLATEDIF,FOREXAMPLE,THETRANSITSYSTEMOFFERSDIFFERENTLEVELSOFCOMFORTANUNOBSERVEDVARIABLEONDIFFERENTROUTESIE,SOMEROUTESMAYBESERVEDBYTRANSITVEHICLESWITHMORECOMFORTABLESEATINGANDTEMPERATURECONTROLTHANOTHERSTHEN,THETRANSITERRORVARIANCEACROSSINDIVIDUALSALONGTHETWOROUTESMAYDIFFERTHEASSUMPTIONOFIDENTICALERRORCOVARIANCEOFALTERNATIVESACROSSINDIVIDUALSMAYNOTBEAPPROPRIATEIFTHEEXTENTOFSUBSTITUTABILITYAMONGALTERNATIVESDIFFERSACROSSINDIVIDUALSTOSUMMARIZE,ERRORVARIANCECOVARIANCEHOMOGENEITYIMPLIESTHESAMECOMPETITIVESTRUCTUREAMONGALTERNATIVESFORALLINDIVIDUALS,ANASSUMPTIONWHICHISGENERALLYDIFFICULTTOJUSTIFYTHETHREEASSUMPTIONSDISCUSSEDABOVETOGETHERLEADTOTHESIMPLEANDELEGANTCLOSEDFORMMATHEMATICALSTRUCTUREOFTHEMNLHOWEVER,THESEASSUMPTIONSALSOLEAVETHEMNLMODELSADDLEDWITHTHE“INDEPENDENCEOFIRRELEVANTALTERNATIVES”IIAPROPERTYATTHEINDIVIDUALLEVELLUCEANDSUPPES1965FORADETAILEDDISCUSSIONOFTHISPROPERTYSEEALSOBENAKIVAANDLERMAN1985THUS,RELAXINGTHETHREEASSUMPTIONSMAYBEIMPORTANTINMANYCHOICECONTEXTSINTHISCHAPTERTHEFOCUSISONTHREECLASSESOFDISCRETECHOICEMODELSTHATRELAXONEORMOREOFTHEASSUMPTIONSDISCUSSEDABOVETHEFIRSTCLASSOFMODELSLABELEDAS“HETEROSCEDASTICMODELS”ISRELATIVELYRESTRICTIVETHEYRELAXTHEIDENTICALLYDISTRIBUTEDACROSSALTERNATIVESERRORTERMASSUMPTION,BUTDONOTRELAXTHEINDEPENDENCEASSUMPTIONPARTOFTHEFIRSTASSUMPTIONABOVEORTHEASSUMPTIONOFRESPONSEHOMOGENEITYSECONDASSUMPTIONABOVETHESECONDCLASSOFMODELSLABELEDAS“MIXEDMULTINOMIALLOGITMMNLMODELS”ANDTHETHIRDCLASSOFMODELSLABELEDAS“MIXEDGENERALIZEDEXTREMEVALUEMGEVMODELS”AREVERYGENERALMODELSINTHISCLASSAREFLEXIBLEENOUGHTORELAXTHEINDEPENDENCEANDIDENTICALLYDISTRIBUTEDACROSSALTERNATIVESERRORSTRUCTUREOFTHEMNLASWELLASTORELAXTHEASSUMPTIONOFRESPONSEHOMOGENEITYTHERELAXATIONOFTHETHIRDASSUMPTIONIMPLICITINTHEMULTINOMIALLOGITANDIDENTIFIEDONTHEPREVIOUSPAGEISNOTCONSIDEREDINDETAILINTHISCHAPTER,SINCEITCANBERELAXEDWITHINTHECONTEXTOFANYGIVENDISCRETECHOICEMODELBYPARAMETERIZINGAPPROPRIATEERRORSTRUCTUREVARIANCESANDCOVARIANCESASAFUNCTIONOFINDIVIDUALATTRIBUTESSEEBHAT2007FORADETAILEDDISCUSSIONOFTHESEPROCEDURESTHEREADERWILLNOTETHATTHEGENERALIZEDEXTREMEVALUEGEVMODELSDESCRIBEDINCHAPTER13RELAXTHEIIDASSUMPTIONPARTIALLYBYALLOWINGCORRELATIONINUNOBSERVEDCOMPONENTSOFDIFFERENTALTERNATIVESTHEADVANTAGEOFTHEGEVMODELSISTHATTHEYMAINTAINCLOSEDFORMEXPRESSIONSFORTHECHOICEPROBABILITIESTHELIMITATIONOFTHESEMODELSISTHATTHEYARECONSISTENTWITHUTILITYMAXIMIZATIONONLYUNDERRATHERSTRICTANDOFTENEMPIRICALLYVIOLATEDRESTRICTIONSONTHEDISSIMILARITYANDALLOCATIONPARAMETERSSPECIFICALLY,THEDISSIMILARITYANDALLOCATIONPARAMETERSSHOULDBEBOUNDEDBETWEEN0AND1FORGLOBALCONSISTENCYWITHUTILITYMAXIMIZATION,ANDTHEALLOCATIONPARAMETERSFORANYALTERNATIVESHOULDADDTO1THEORIGINOFTHESERESTRICTIONSCANBETRACEDBACKTOTHEREQUIREMENTTHATTHEVARIANCEOFTHEJOINTALTERNATIVESBEIDENTICALINTHEGEVMODELSALSO,GEVMODELSDONOTRELAXASSUMPTIONSRELATEDTOTASTEHOMOGENEITYINRESPONSETOANATTRIBUTESUCHASTRAVELTIMEORCOSTINAMODECHOICEMODELDUETOUNOBSERVEDDECISIONMAKERCHARACTERISTICS,ANDCANNOTBEAPPLIEDTOPANELDATAWITHTEMPORALCORRELATIONINUNOBSERVEDFACTORSWITHINTHECHOICESOFTHESAMEDECISIONMAKINGAGENTHOWEVER,GEVMODELSDOOFFERCOMPUTATIONALTRACTABILITY,PROVIDEATHEORETICALLYSOUNDMEASUREFORBENEFITVALUATION,ANDCANFORMTHEBASISFORFORMULATINGMIXEDMODELSTHATACCOMMODATERANDOMTASTEVARIATIONSANDTEMPORALCORRELATIONSINPANELDATASEESECTION4THERESTOFTHISCHAPTERISSTRUCTUREDASFOLLOWSTHECLASSOFHETEROSCEDASTICMODELS,MIXEDMULTINOMIALLOGITMODELS,ANDMIXEDGENERALIZEDEXTREMEVALUEMODELSAREDISCUSSEDINSECTIONS2,3,AND4,RESPECTIVELYSECTION5PRESENTSRECENTADVANCESINTHEAREAOFSIMULATIONTECHNIQUESTOESTIMATETHEMIXEDMULTINOMIALANDMIXEDGENERALIZEDEXTREMEVALUECLASSOFMODELSOFSECTION3AND4THEESTIMATIONOFTHEHETEROSCEDASTICMODELSINSECTION2DOESNOTREQUIRETHEUSEOFSIMULATIONANDISDISCUSSEDWITHINSECTION2SECTION6CONCLUDESTHEPAPERWITHASUMMARYOFTHEGROWINGNUMBEROFAPPLICATIONSTHATUSEFLEXIBLEDISCRETECHOICESTRUCTURES2THEHETEROSCEDASTICCLASSOFMODELSTHECONCEPTTHATHETEROSCEDASTICITYINALTERNATIVEERRORTERMSIE,INDEPENDENT,BUTNOTIDENTICALLYDISTRIBUTEDERRORTERMSRELAXESTHEIIAASSUMPTIONHASBEENRECOGNIZEDFORQUITESOMETIMENOWTHREEMODELSHAVEBEENPROPOSEDTHATALLOWNONIDENTICALRANDOMCOMPONENTSTHEFIRSTISTHENEGATIVEEXPONENTIALMODELOFDAGANZO1979,THESECONDISTHEODDBALLALTERNATIVEMODELOFRECKER1995ANDTHETHIRDISTHEHETEROSCEDASTICEXTREMEVALUEHEVMODELOFBHAT1995OFTHESE,DAGANZOSMODELHASNOTSEENMUCHAPPLICATION,SINCEITREQUIRESTHATTHEPERCEIVEDUTILITYOFANYALTERNATIVENOTEXCEEDANUPPERBOUNDTHISARISESBECAUSETHENEGATIVEEXPONENTIALDISTRIBUTIONDOESNOTHAVEAFULLRANGEDAGANZOSMODELALSODOESNOTNESTTHEMNLMODELRECKER1995PROPOSEDTHEODDBALLALTERNATIVEMODELWHICHPERMITSTHERANDOMUTILITYVARIANCEOFONE“ODDBALL”ALTERNATIVETOBELARGERTHANTHERANDOMUTILITYVARIANCESOFOTHERALTERNATIVESTHISSITUATIONMIGHTOCCURBECAUSEOFATTRIBUTESWHICHDEFINETHEUTILITYOFTHEODDBALLALTERNATIVE,BUTAREUNDEFINEDFOROTHERALTERNATIVESRECKERSMODELHASACLOSEDFORMSTRUCTUREFORTHECHOICEPROBABILITIESHOWEVER,ITISRESTRICTIVEINREQUIRINGTHATALLALTERNATIVESEXCEPTONEHAVEIDENTICALVARIANCEBHAT1995FORMULATEDTHEHEVMODEL,WHICHASSUMESTHATTHEALTERNATIVEERRORTERMSAREDISTRIBUTEDWITHATYPEIEXTREMEVALUEDISTRIBUTIONTHEVARIANCESOFTHEALTERNATIVEERRORTERMSAREALLOWEDTOBEDIFFERENTACROSSALLALTERNATIVESWITHTHENORMALIZATIONTHATTHEERRORTERMSOFONEOFTHEALTERNATIVESHAVEASCALEPARAMETEROFONEFORIDENTIFICATIONCONSEQUENTLY,THEHEVMODELCANBEVIEWEDASAGENERALIZATIONOFRECKERSODDBALLALTERNATIVEMODELTHEHEVMODELDOESNOTHAVEACLOSEDFORMSOLUTIONFORTHECHOICEPROBABILITIES,BUTINVOLVESONLYAONEDIMENSIONALINTEGRATIONREGARDLESSOFTHENUMBEROFALTERNATIVESINTHECHOICESETITALSONESTSTHEMNLMODELANDISFLEXIBLEENOUGHTOALLOWDIFFERENTIALCROSSELASTICITIESAMONGALLPAIRSOFALTERNATIVESINTHEREMAINDEROFTHISDISCUSSIONOFHETEROSCEDASTICMODELS,THEFOCUSISONTHEHEVMODEL21HEVMODELSTRUCTURETHERANDOMUTILITYOFALTERNATIVEUIOFALTERNATIVEIFORANINDIVIDUALINRANDOMUTILITYMODELSTAKESTHEFORMWESUPPRESSTHEINDEXFORINDIVIDUALSINTHEFOLLOWINGPRESENTATION,IIIVU1WHEREISTHESYSTEMATICCOMPONENTOFTHEUTILITYOFALTERNATIVEIWHICHISAFUNCTIONOFOBSERVEDIATTRIBUTESOFALTERNATIVEIANDOBSERVEDCHARACTERISTICSOFTHEINDIVIDUAL,ANDISTHERANDOMICOMPONENTOFTHEUTILITYFUNCTIONLETCBETHESETOFALTERNATIVESAVAILABLETOTHEINDIVIDUALLETTHERANDOMCOMPONENTSINTHEUTILITIESOFTHEDIFFERENTALTERNATIVESHAVEATYPEIEXTREMEVALUEDISTRIBUTIONWITHALOCATIONPARAMETEREQUALTOZEROANDASCALEPARAMETEREQUALTOFORTHEITHALTERNATIVETHEIRANDOMCOMPONENTSAREASSUMEDTOBEINDEPENDENT,BUTNONIDENTICALLYDISTRIBUTEDTHUS,THEPROBABILITYDENSITYFUNCTIONANDTHECUMULATIVEDISTRIBUTIONFUNCTIONOFTHERANDOMERRORTERMFORTHEITHALTERNATIVEAREAND1/IZIIIIEIZIEIIDFFF2THERANDOMUTILITYFORMULATIONOFEQUATION1,COMBINEDWITHTHEASSUMEDPROBABILITYDISTRIBUTIONFORTHERANDOMCOMPONENTSINEQUATION2ANDTHEASSUMEDINDEPENDENCEAMONGTHERANDOMCOMPONENTSOFTHEDIFFERENTALTERNATIVES,ENABLESUSTODEVELOPTHEPROBABILITYTHATANINDIVIDUALWILLCHOOSEALTERNATIVEIFROMTHESETCOFAVAILABLEALTERNATIVES3IIIJIIJCIJIJIDVJIUPII1,ALFOR,ROB,WHEREANDARETHEPROBABILITYDENSITYFUNCTIONANDCUMULATIVEDISTRIBUTIONFUNCTIONOFTHESTANDARDTYPEIEXTREMEVALUEDISTRIBUTION,RESPECTIVELY,ANDAREGIVENBYSEEJOHNSONANDKOTZ,19704ANDTTEETTSUBSTITUTINGINEQUATION3,THEPROBABILITYOFCHOOSINGALTERNATIVEICANBEREWRITTENASIW/FOLLOWS5DWVPJIIIJ,CWIIFTHESCALEPARAMETERSOFTHERANDOMCOMPONENTSOFALLALTERNATIVESAREEQUAL,THENTHEPROBABILITYEXPRESSIONINEQUATION5COLLAPSESTOTHATOFTHEMNLNOTETHATTHEVARIANCEOFTHERANDOMERRORTERMOFALTERNATIVEIISEQUALTO,WHEREISTHESCALEPARAMETERI6/2IITHEHEVMODELDISCUSSEDABOVEAVOIDSTHEPITFALLSOFTHEIIAPROPERTYOFTHEMNLMODELBYALLOWINGDIFFERENTSCALEPARAMETERSACROSSALTERNATIVESINTUITIVELY,WECANEXPLAINTHISBYREALIZINGTHATTHEERRORTERMREPRESENTSUNOBSERVEDCHARACTERISTICSOFANALTERNATIVETHATIS,ITREPRESENTSUNCERTAINTYASSOCIATEDWITHTHEEXPECTEDUTILITYORTHESYSTEMATICPARTOFUTILITYOFANALTERNATIVETHESCALEPARAMETEROFTHEERRORTERM,THEREFORE,REPRESENTSTHELEVELOFUNCERTAINTYITSETSTHERELATIVEWEIGHTSOFTHESYSTEMATICANDUNCERTAINCOMPONENTSINESTIMATINGTHECHOICEPROBABILITYWHENTHESYSTEMATICUTILITYOFSOMEALTERNATIVELCHANGES,THISAFFECTSTHESYSTEMATICUTILITYDIFFERENTIALBETWEENANOTHERALTERNATIVEIANDTHEALTERNATIVELHOWEVER,THISCHANGEINTHESYSTEMATICUTILITYDIFFERENTIALISTEMPEREDBYTHEUNOBSERVEDRANDOMCOMPONENTOFALTERNATIVEITHELARGERTHESCALEPARAMETEROREQUIVALENTLY,THEVARIANCEOFTHERANDOMERRORCOMPONENTFORALTERNATIVEI,THEMORETEMPEREDISTHEEFFECTOFTHECHANGEINTHESYSTEMATICUTILITYDIFFERENTIALSEETHENUMERATOROFTHECUMULATIVEDISTRIBUTIONFUNCTIONTERMINEQUATION5ANDSMALLERISTHEELASTICITYEFFECTONTHEPROBABILITYOFCHOOSINGALTERNATIVEIINPARTICULAR,TWOALTERNATIVESWILLHAVETHESAMEELASTICITYEFFECTDUETOACHANGEINTHESYSTEMATICUTILITYOFANOTHERALTERNATIVEONLYIFTHEYHAVETHESAMESCALEPARAMETERONTHERANDOMCOMPONENTSTHISPROPERTYISALOGICALANDINTUITIVEEXTENSIONOFTHECASEOFTHEMNL,INWHICHALLSCALEPARAMETERSARECONSTRAINEDTOBEEQUALAND,THEREFORE,ALLCROSSELASTICITIESAREEQUALASSUMINGALINEARINPARAMETERSFUNCTIONALFORMFORTHESYSTEMATICCOMPONENTOFUTILITYFORALLALTERNATIVES,THERELATIVEMAGNITUDESOFTHECROSSELASTICITIESOFTHECHOICEPROBABILITIESOFANYTWOALTERNATIVESIANDJWITHRESPECTTOACHANGEINTHEKTHLEVELOFSERVICEVARIABLEOFANOTHERALTERNATIVELSAY,ARECHARACTERIZEDBYTHESCALEPARAMETEROFTHERANDOMCOMPONENTSOFALTERNATIVESIANDJKLX6IFIFJPJJJIJIXXKLKLLLJKLIKL22HEVMODELESTIMATIONTHEHEVMODELCANBEESTIMATEDUSINGTHEMAXIMUMLIKELIHOODTECHNIQUEASSUMEALINEARINPARAMETERSSPECIFICATIONFORTHESYSTEMATICUTILITYOFEACHALTERNATIVEGIVENBYFORTHEQTHQIQIXVINDIVIDUALANDITHALTERNATIVETHEINDEXFORINDIVIDUALSISINTRODUCEDINTHEFOLLOWINGPRESENTATIONSINCETHEPURPOSEOFTHEESTIMATIONISTOOBTAINTHEMODELPARAMETERSBYMAXIMIZINGTHELIKELIHOODFUNCTIONOVERALLINDIVIDUALSINTHESAMPLETHEPARAMETERSTOBEESTIMATEDARETHEPARAMETERVECTORANDTHESCALEPARAMETERSOFTHERANDOMCOMPONENTOFEACHOFTHEALTERNATIVESONEOFTHESCALEPARAMETERSISNORMALIZEDTOONEFORIDENTIFIABILITYTHELOGLIKELIHOODFUNCTIONTOBEMAXIMIZEDCANBEWRITTENASQQCIWIJCJIGQIIQDWVYL1,LOG,7WHEREISTHECHOICESETOFALTERNATIVESAVAILABLETOTHEQTHINDIVIDUALANDISDEFINEDASFOLLOWSCQYQI8OTHERWIS0,21,21,EALTRNIVCHOSENDVIULF1IIQYQITHELOGLIKELIHOODFUNCTIONINEQUATION7HASNOCLOSEDFORMEXPRESSION,BUTCANBEESTIMATEDINASTRAIGHTFORWARDMANNERUSINGGAUSSIANQUADRATURETODOSO,DEFINEAVARIABLETHEN,ANDALSODEFINEAFUNCTIONASUEDWULNGQI9JVGIQIIJ,CJQIQEQUATION7CANBEWRITTENASUDEGYLQIUQICIQLOG010THEEXPRESSIONWITHINPARENTHESISINEQUATION7CANBEESTIMATEDUSINGTHELAGUERREGAUSSIANQUADRATUREFORMULA,WHICHREPLACESTHEINTEGRALBYASUMMATIONOFTERMSOVERACERTAINNUMBERSAYKOFSUPPORTPOINTS,EACHTERMCOMPRISINGTHEEVALUATIONOFTHEFUNCTIONGQIATTHESUPPORTPOINTKMULTIPLIEDBYAPROBABILITYMASSORWEIGHTASSOCIATEDWITHTHESUPPORTPOINTTHESUPPORTPOINTSARETHEROOTSOFTHELAGUERREPOLYNOMIALOFORDERK,ANDTHEWEIGHTSARECOMPUTEDBASEDONASETOFTHEOREMSPROVIDEDBYPRESSETAL19923THEMIXEDMULTINOMIALLOGITMMNLCLASSOFMODELSTHEHEVMODELINTHEPREVIOUSSECTIONANDTHEGEVMODELSINCHAPTER13HAVETHEADVANTAGETHATTHEYAREEASYTOESTIMATETHELIKELIHOODFUNCTIONFORTHESEMODELSEITHERINCLUDESAONEDIMENSIONALINTEGRALINTHEHEVMODELORISINCLOSEDFORMINTHEGEVMODELSHOWEVER,THESEMODELSARERESTRICTIVESINCETHEYONLYPARTIALLYRELAXTHEIIDERRORASSUMPTIONACROSSALTERNATIVESINTHISSECTION,WEDISCUSSTHEMMNLCLASSOFMODELSTHATAREFLEXIBLEENOUGHTOCOMPLETELYRELAXTHEINDEPENDENCEANDIDENTICALLYDISTRIBUTEDERRORSTRUCTUREOFTHEMNLASWELLASTORELAXTHEASSUMPTIONOFRESPONSEHOMOGENEITYTHEMIXEDMMNLCLASSOFMODELSINVOLVESTHEINTEGRATIONOFTHEMNLFORMULAOVERTHEDISTRIBUTIONOFUNOBSERVEDRANDOMPARAMETERSITTAKESTHESTRUCTUREWHERE11DFLPQIQI,|QIIXJQIEISTHEPROBABILITYTHATINDIVIDUALQCHOOSESALTERNATIVEI,ISAVECTOROFOBSERVEDVARIABLESSPECIFICQIPQXTOINDIVIDUALQANDALTERNATIVEI,REPRESENTSPARAMETERSWHICHARERANDOMREALIZATIONSFROMADENSITYFUNCTIONF,ANDISAVECTOROFUNDERLYINGMOMENTPARAMETERSCHARACTERIZINGFTHEFIRSTAPPLICATIONSOFTHEMIXEDLOGITSTRUCTUREOFEQUATION11APPEARTOHAVEBEENBYBOYDANDMELLMAN1980ANDCARDELLANDDUNBAR1980HOWEVER,THESEWERENOTINDIVIDUALLEVELMODELSAND,CONSEQUENTLY,THEINTEGRATIONINHERENTINTHEMIXEDLOGITFORMULATIONHADTOBEEVALUATEDONLYONCEFORTHEENTIREMARKETTRAIN1986ANDBENAKIVAETAL1993APPLIEDTHEMIXEDLOGITTOCUSTOMERLEVELDATA,BUTCONSIDEREDONLYONEORTWORANDOMCOEFFICIENTSINTHEIRSPECIFICATIONSTHUS,THEYWEREABLETOUSEQUADRATURETECHNIQUESFORESTIMATIONTHEFIRSTAPPLICATIONSTOREALIZETHEFULLPOTENTIALOFMIXEDLOGITBYALLOWINGSEVERALRANDOMCOEFFICIENTSSIMULTANEOUSLYINCLUDEREVELTANDTRAIN1998ANDBHAT1998A,BOTHOFWHICHWEREORIGINALLYCOMPLETEDINEARLY1996ANDEXPLOITEDTHEADVANCESINSIMULATIONMETHODSTHEMMNLMODELSTRUCTUREOFEQUATION11CANBEMOTIVATEDFROMTWOVERYDIFFERENTBUTFORMALLYEQUIVALENTPERSPECTIVESSPECIFICALLY,AMMNLSTRUCTUREMAYBEGENERATEDFROMANINTRINSICMOTIVATIONTOALLOWFLEXIBLESUBSTITUTIONPATTERNSACROSSALTERNATIVESERRORCOMPONENTSSTRUCTUREORFROMANEEDTOACCOMMODATEUNOBSERVEDHETEROGENEITYACROSSINDIVIDUALSINTHEIRSENSITIVITYTOOBSERVEDEXOGENOUSVARIABLESRANDOMCOEFFICIENTSSTRUCTURE31ERRORCOMPONENTSSTRUCTURETHEERRORCOMPONENTSSTRUCTUREPARTITIONSTHEOVERALLRANDOMTERMASSOCIATEDWITHTHEUTILITYOFEACHALTERNATIVEINTOTWOCOMPONENTSONETHATALLOWSTHEUNOBSERVEDERRORTERMSTOBENONIDENTICALANDNONINDEPENDENTACROSSALTERNATIVES,ANDANOTHERTHATISSPECIFIEDTOBEINDEPENDENTANDIDENTICALLYTYPEIEXTREMEVALUEDISTRIBUTEDACROSSALTERNATIVESSPECIFICALLY,CONSIDERTHEFOLLOWINGUTILITYFUNCTIONFORINDIVIDUALQANDALTERNATIVEI12QIIQIIZYUWHEREANDARETHESYSTEMATICANDRANDOMCOMPONENTSOFUTILITY,ANDISFURTHERPARTITIONEDIIIINTOTWOCOMPONENTS,ANDISAVECTOROFOBSERVEDDATAASSOCIATEDWITHALTERNATIVEI,SOMEQIZQIIZOFTHEELEMENTSOFWHICHMIGHTALSOAPPEARINTHEVECTORISARANDOMVECTORWITHZEROMEANTHEQIYCOMPONENTINDUCESHETEROSCEDASTICITYANDCORRELATIONACROSSUNOBSERVEDUTILITYCOMPONENTSOFTHEQIZALTERNATIVESDEFININGAND,WEOBTAINTHEMMNLMODELSTRUCTUREFORTHE,QIQIZYXCHOICEPROBABILITYOFALTERNATIVEIFORINDIVIDUALQTHEEMPHASISINTHEERRORCOMPONENTSSTRUCTUREISONALLOWINGAFLEXIBLESUBSTITUTIONPATTERNAMONGALTERNATIVESINAPARSIMONIOUSFASHIONTHISISACHIEVEDBYTHE“CLEVER”SPECIFICATIONOFTHEVARIABLEVECTORCOMBINEDWITHUSUALLYTHESPECIFICATIONOFINDEPENDENTNORMALLYDISTRIBUTEDQIZRANDOMELEMENTSINTHEVECTORFOREXAMPLE,MAYBESPECIFIEDTOBEAROWVECTOROFDIMENSIONM,IZWITHEACHROWREPRESENTINGAGROUPMM1,2,MOFALTERNATIVESSHARINGCOMMONUNOBSERVEDCOMPONENTSTHEROWSCORRESPONDINGTOTHEGROUPSOFWHICHIISAMEMBERTAKESAVALUEOFONEANDOTHERROWSTAKEAVALUEOFZEROTHEVECTOROFDIMENSIONMMAYBESPECIFIEDTOHAVEINDEPENDENTELEMENTS,EACHELEMENTHAVINGAVARIANCECOMPONENTTHERESULTOFTHISSPECIFICATIONISACOVARIANCE2MOFAMONGALTERNATIVESINGROUPMANDHETEROSCEDASTICITYACROSSTHEGROUPSOFALTERNATIVESTHIS2MSTRUCTUREISLESSRESTRICTIVETHANTHENESTEDLOGITSTRUCTUREINTHATANALTERNATIVECANBELONGTOMORETHANONEGROUPALSO,BYSTRUCTURE,THEVARIANCEOFTHEALTERNATIVESISDIFFERENTMOREGENERALSTRUCTURESFORINEQUATION12AREPRESENTEDBYBENAKIVAANDBOLDUC1996ANDBROWNSTONEANDTRAIN1999IZEXAMPLESOFTHEERRORCOMPONENTSMOTIVATIONINTHELITERATUREINCLUDEBHAT1998B,JONGETAL2002A,B,WHELANETAL2002,ANDBATLEYETAL2001A,BTHEREADERISALSOREFERREDTOTHEWORKOFWALKERANDHERCOLLEAGUESBENAKIVAETAL,2001WALKER,2002ANDMUNIZAGAANDALVAREZDAZIANO2002FORIMPORTANTIDENTIFICATIONISSUESINTHECONTEXTOFTHEERRORCOMPONENTSMMNLMODEL32RANDOMCOEFFICIENTSSTRUCTURETHERANDOMCOEFFICIENTSSTRUCTUREALLOWSHETEROGENEITYINTHESENSITIVITYOFINDIVIDUALSTOEXOGENOUSATTRIBUTESTHEUTILITYTHATANINDIVIDUALQASSOCIATESWITHALTERNATIVEIISWRITTENAS13QIIQIXUWHEREISAVECTOROFEXOGENOUSATTRIBUTES,ISAVECT