ToBuyanElectricVehicleor

Not?ABayesianAnalysisofConsumerIntentintheUnitedStates

NafisaLohawalaandMohammadArshadRahman

WorkingPaper25-16May2025

AbouttheAuthors

NafisaLohawalaisafellowatResourcesfortheFuture(RFF).SheearnedaPhD

ineconomicsattheUniversityofMichiganafterreceivingaBS-MSdualdegreein

economicswithaminorincomputerscienceandengineering(algorithms)fromtheIndianInstituteofTechnology,Kanpur.Lohawala’sresearchliesattheintersectionofindustrialorganization,energyeconomics,andpublicfinance

MohammadArshadRahmanisanassociateprofessorintheDepartmentof

EconomicSciencesattheIndianInstituteofTechnologyKanpur(IITK),India.His

researchinterestsincludeBayesianEconometrics,QuantileRegression,Discrete

ChoiceModeling,MarkovchainMonteCarloTechniques,MachineLearning,EnergyEconomics,andAppliedEconometrics.

Acknowledgements

WethankBeiaSpiller,DeepMukherjee,andJoshuaLinnforhelpfulcommentsandsuggestions,andtheRFFcommunicationsteamfortheirassistancewithdissemination.Allremainingerrorsareourown.

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ToBuyanElectricVehicleorNot?ABayesianAnalysisof

ConsumerIntentintheUnitedStates

NafisaLohawala

ResourcesfortheFuture,WashingtonDC,USA

MohammadArshadRahman*

DepartmentofEconomicSciences,IndianInstituteofTechnologyKanpur,India.

Abstract

Theadoptionofelectricvehicles(EVs)isconsideredcriticaltoachievingclimategoals,yetithingesonconsumerinterest.ThisstudyexploreshowpublicintenttopurchaseEVsrelatestofourunexaminedfactors(exposuretoEVinformation,perceptionsofEVs’environmentalbenefits,viewsongovernmentclimatepolicy,andconfidenceinfutureEVinfrastructure)whilecontrollingforpriorEVownership,politicalaffiliation,anddemographiccharacteristics(age,gender,education,andgeographiclocation).WeusedatafromthreenationallyrepresentativeopinionpollsbythePewResearchCenter2021—2023andBayesiantechniquestoestimatetheordinalprobitandordinalquantilemodels.ResultsfromordinalprobitshowthatrespondentswhoarewellinformedaboutEVs,perceivethemasenvironmentallybeneficial,orareconfidentindevelopmentofchargingstationsaremorelikelytoexpressstrongpurchaseinterest,withcovariateeffects(CEs)-ametricrarelyreportedinEVresearch-of10.2,15.5,and19.1percentagepoints,respectively.Incontrast,thoseskepticalofgovernmentclimateinitiativesaremorelikelytoexpressnointerest,bymorethan10percentagepoints.PriorEVownershipexhibitsthehighestCE(19.0—23.1percentagepoints),andtheimpactofmostdemographicvariablesisconsistentwiththeliterature.TheordinalquantilemodelsdemonstratesignificantvariationinCEsacrossthedistributionofpurchaseintent,offeringinsightsbeyondtheordinalprobitmodel.WearethefirsttousequantilemodelingtorevealhowCEsdiffersignificantlythroughoutthespectrumofpurchaseintent.

Keywords:Decarbonization,electricvehicle,ordinalprobit,PewResearch,quantileregression,technologyadoption.

Acknowledgements:WethankBeiaSpiller,DeepMukherjee,andJoshuaLinnforhelpfulcommentsandsug-gestions,andtheRFFcommunicationsteamfortheirassistancewithdissemination.Allremainingerrorsareourown.

*Correspondingauthor

Emailaddresses:nlohawala@rff.org(NafisaLohawala),marshad@iitk.ac.in(MohammadArshadRahman)

2

1.Introduction

TransportationisthelargestsourceofgreenhousegasemissionsintheUnitedStates,represent-ing28percentofthetotalemissionsin2022,withlight-dutyvehiclesresponsibleformorethanhalf(

EnvironmentalProtectionAgency

,

2024

).Electricvehicles(EVs)arewidelyregardedasakeysolutionforreducingemissionsandmeetingclimatetargets,giventheirzerotailpipeemissions.ToaccelerateEVadoption,governmentsworldwide,includingtheUnitedStates,haveimplementedarangeofpoliciesandincentives.KeyUSmeasuresincludenationwidegreenhousegasstandardstoreduceemissionsfromnewvehicles;upto$7,500federaltaxcreditsforEVbuyers;investmentsincharginginfrastructure;andstate-levelmandates,suchasCalifornia’sZero-EmissionVehicle(ZEV)program,whichrequiresautomakerstosellanincreasingshareofEVs.

1

Despitetheseefforts,USsalesremainrelativelylow,accountingforonlyabout10percentofglobalnewEVsalesin2023,trailingbehindChina(60percent)andEurope(25percent)(

In-

ternationalEnergyAgency

,

2024

).ThetrajectoryofUSadoptionisalsoincreasinglyuncertain.In2024,severalautomakers,includingGMandFord,scaledbackambitiousEVtargets,citingweakconsumerdemand(

ColiasandOtts

,

2024

).Successiveadministrationshavetakensharplydifferentapproaches,creatinguncertaintyaroundlong-termincentivesandregulations.TheBidenadministrationpromotedEVadoptionthroughsubsidies,infrastructureinvestments,andemis-sionsregulationsaspartofitsbroaderclimateagenda,buttheTrumpadministrationsignaledaderegulatoryshiftwithplanstoweakenbothincentivesandregulations.

2

Thesechangingpoliticalprioritiesmayreflectbroadertensionsbetweenenvironmentalregulations,industryinterests,andfiscalconsiderations.AlthoughEVsreduceemissions—particularlyinregionswithcleanerelectric-itygrids—subsidiesandincentives,alongwithdeclininggastaxrevenue,posebudgetarychallenges(

Bantaetal.

,

2024

).TheTrumpadministrationhasalsocitedconcernsthatstringentemissionsandfueleconomyregulationsincreasecompliancecostsforautomakersandrestrictconsumerchoice.

Ultimately,thefutureoftheEVmarketdependsonconsumerinterest,whichislikelytobeshapedbymultiplefactors,includingcost,convenience,perceptionofenvironmentalbenefits,andawarenessofthesetrade-offs.AlthoughEVsoftenhavehigherup-frontcostsduetothebatterytechnology,theycanofferlong-termsavingsthroughloweroperationalandmaintenanceexpenses,aselectricmotorsrequirelessmaintenance.However,theoverallcostofownershipcanvarysignif-icantly,dependingonfactorssuchaslocalgasolineandelectricityprices,climate,purchaseincen-tives,homechargingaccess,andannualmileage(

Woodyetal.

,

2024

).Accesstoreliablecharginginfrastructurealsoremainsanimportantconsideration.Limitedavailabilityinsomeregionscanexacerbaterangeanxietyandinconvenience,particularlyforlong-distancetravel.Environmental

1Formoredetails,seetheEPA’s

greenhousegasregulations

,theIRS

EVtaxcreditguidelines

,NationalElectricVehicleInfrastructure(

NEVI

)ProgramandCalifornia’s

ZEVProgram

.

2PresidentTrump’sexecutiveorderonJanuary20,2025outlinedplanstohalt$7.5billionincharginginfras-tructurefunding,weakenvehicleemissionsstandardsthatdriveEVproduction,eliminateEVtaxcredits,andre-vokeCalifornia’sEVmandatewaiver,whichwouldlimitstate-levelregulations.See

/

presidential-actions/2025/01/unleashing-american-energy/

3

concernsalsoaffectconsumerattitudes,butperceptionsmayvary,giventheregionalheterogene-ityinshort-termbenefits(

Hollandetal.

,

2016

)andperceptionaboutcriticalmineralsmining(

Lakshman

,

2024

).

Recognizingtheimportanceofpublicinterestinshapingthefutureofthismarket,weexaminehowpublicopiniononpurchasingEVsisassociatedwithfactorssuchasexposuretoEVinfor-mation,perceptionsabouttheenvironmentalimpactofEVs(comparedtogas-poweredvehicles)andgovernmentactiontomitigateclimatechange,confidenceinrelatedinfrastructuredevelop-ment,priorownership,anddemographiccharacteristicssuchasage,gender,education,politicalaffiliation,andgeographicallocationoftherespondents.

Ourpapermakesthreekeycontributions.First,weexploretheroleofnovelfactorsthathavebeenlargelyoverlookedintheEVadoptionliterature.Studieshaveexaminedhowacombinationoffactors,includingrangeanxiety,performanceconcerns,charginginfrastructure,environmentalattitudes,governmentpolicies,peereffects,anddemographiccharacteristics,influencespurchaseintent(

Carleyetal.

,

2013

;

Tiwarietal.

,

2020

;

Zhaoetal.

,

2022

;

Mamkhezri

,

2025

)ortheimpactofspecificfactors,suchasprovidinginformationabouttotalcostofownership(

Dumortieretal.

,

2015

),mediacoverage(

Scherrer

,

2023

),reputation(

BuhmannandCriado

,

2023

),andconsumerincentives(

Xueetal.

,

2023

;

StekelbergandVance

,

2024

).RelatedresearchexaminesfactorsdrivingactualEVsalesratherthanpurchaseintent,includingtaxcreditsandrebates(

DeShazoetal.

,

2017

;

Jennetal.

,

2018

;

ClintonandSteinberg

,

2019

;

Sheldonetal.

,

2023

),charginginfrastructure(

Lietal.

,

2017

),andautomakers’responsestoenvironmentalpolicies(

Aghion,PhilippeandAntoineDechezlepretreand

DavidHemousandRalfMartinandJohnVanReenen

,

2016

;

Gillingham

,

2022

;

Lohawala

,

2023

).WecontributetothisliteraturebyexamininghowEVpurchaseintentisinfluencedbyfactorssuchasconfidenceinfuturecharginginfrastructuredevelopment,perceptionsofgovernmentclimateefforts,exposuretoEV-relatedinformation,andperceptionsofEVs’environmentalbenefits.

Oursecondcontributionisusinglargeandnationallyrepresentativedatasets.ResearchonthetrendsanddeterminantsofpublicopinionandEVpurchaseintenthaspredominantlyreliedonsmall-scalesurveys.Forexample,

Tiwarietal.

(

2020

),

Lasharietal.

(

2021

),

Huetal.

(

2023

),and

Mamkhezri

(

2025

)analyzed1,800responsesfromUKresidents,1,500responsesfromSouthKorea,807responsesfromChina,and1,500responsesfromtheUnitedStates,respectively.

3

Singhetal.

(

2020

)provideameta-analysisof211peer-reviewedresearcharticles2009—2019,identifyingkeyfactorsinfluencingpublicopinionofEVs.Notably,mostsuchstudiesarebasedonasinglesurveyforaparticularyear,notnationallyrepresentative,orconstrainedbysamplesize,whichdiminishestheirgeneralizability.WeovercometheselimitationsbydrawingonthreenationallyrepresentativeopinionpollsconductedbythePewResearchCenterwithasignificantlylargersamplesizethanotherstudies.Specifically,weusedatafrom2021,2022,and2023,with11,052,7,173,and7,201observations,respectively.

3Althoughsomerecentstudies(e.g.,

Panietal.

(

2023

)and

RuanandLv

(

2022

))uselarge-scaleonlinesentimentanalysistotracktheevolutionofpublicperceptionsofEVs,theydonotexamineindividual-levelpredictorsofit.

4

OurthirdcontributionliesinapplyingadvancedmodelingandestimationtechniquestostudyEVpurchaseintent.Researchhasusedvariousmethodstoanalyzeit,suchaslinearregressiontoassesstheeffectsofpersonalbeliefsandpreferences(

Carleyetal.

,

2013

;

Krauseetal.

,

2013

)andpositivemediacoverage(

Scherrer

,

2023

);binaryprobitandlogittoexploremotivationalfactors,suchasreputationinshapingEVpreferences(

BuhmannandCriado

,

2023

);mixedlogitmodelstoexploreattitudestowardvariousEVpolicyincentives,employmentchanges,andelectricitycosts(

Mamkhezri

,

2025

);structuralequationmodelingtoexaminetheroleofadoptionbarriers,suchasrangeanxietyandcharginginfrastructurereliability(

Tiwarietal.

,

2020

),peereffects(

Zhao

etal.

,

2022

),andpolicyincentives(

Huetal.

,

2023

);andtopicmodelingandsentimentanalysistotrackshiftsinonlinesentimenttowardEVsovertime(

RuanandLv

,

2022

).Thesepapersrelyonclassical(frequentist)methods,whichprovidepointestimatesandconfidenceintervalsbutfailtocaptureuncertaintyinmodelparameters.Thiscanleadtobiasedestimatesofcovariateeffects(CEs),particularlyinnonlinearmodels,suchasprobitorlogit,whereparametersdonotdirectlytranslatetochangesinoutcomeprobabilities(

JeliazkovandVossmeyer

,

2018

).

Incontrast,weadoptaBayesianmodelingframeworkthatprovidesaposteriordistributionovertheparameters,systematicallyincorporatesuncertaintyinparameters(andobservations)whencomputingcovariateeffects,andenablesustomakedirectprobabilitystatementsabouttheparameters—ratherthanrelyingonthenotionof“confidence”asintheclassicalframework.Ourempiricalanalysisconsistsoftwostages.First,weestimateanordinalprobitmodelusingMarkovchainMonteCarlo(MCMC)methodstoassesshowdifferentfactorsareassociatedwithconsumerinterestinEVadoption.WepresenttheCEsontheaverageprobabilitiesofoutcomes,anaspectrarelyexploredintheEVliterature.Next,weextendtheanalysisusingordinalquantilemodelsatthe20th,50th,and80thpercentilesofpurchaseintenttoexaminehowdifferentfactorsshapethedecisionsofconsumerswhoaremoreinclinedtoadoptEVscomparedtothosewhoarelessinclined.WearethefirsttoapplyordinalquantileanalysistoEVpurchaseintent.Asnoclassicalestimationmethodsexistforordinalquantilemodels,Bayesianestimationisessential.TheresultsrevealconsiderableheterogeneityinCEsacrossthedistributionofEVpurchaseintent,whichwouldbemissedbyordinalprobitmodels.

Theremainderofthearticleisstructuredasfollows.Section

2

introducestheordinalprobitandordinalquantilemodels,alongwiththeirMCMCalgorithmsforestimation.Section

3

outlinesthedataandprovidesapreliminaryinvestigation.InSections

4

,wepresenttheresultsfromourmodelsanddiscusstheirimplications.Finally,Section

5

offersconcludingremarks.

2.TheModelingFramework

Webrieflydescribetheordinalprobitandordinalquantilemodels.Forboth,weoutlinetheBayesianestimationalgorithmusingMCMCmethodsandexplainthecomputationofCEs.

5

2.1.TheOrdinalProbitModel

Ordinaldatamodels(

JohnsonandAlbert

,

2000

),alsoknownas“orderedchoicemodels,”allowfittingaregressionmodeltoanordinaldependent(outcomeorresponse)variable,typicallydenotedbyy,asafunctionofthecovariates.Thesemodelsaregroundedintherandomutilityframeworkfromeconomicsandexplained,amongothers,in

JeliazkovandRahman

(

2012

)and

Bathametal.

(

2023

).Inatypicalsetting,yhasseveralcategories,andeachoneisassignedascore(valueornumber)thatisinherentlyorderedorranked.However,thesescoresonlycarryordinalmeaningandlackcardinalinterpretation—althoughthecategoriesareranked,thedifferencesbetweenthemcannotbecompareddirectly.Forexample,inourstudy,responsestothequestionaboutpurchasinganEVarecodedasfollows:1fornotatalllikely,2fornottoolikely,3forsomewhatlikely,and4forverylikely.Ascoreof2comparedto1impliesmorepurchaseintent,butwecannotinterpretitastwicetheintent.Similarly,thedifferenceinintentbetween2and1isnotthesameasthatbetween4and3.

Theordinalregressionmodel,following

AlbertandChib

(

1993

),canbewrittenintermsofacontinuouslatentvariableziasfollows:

zi=xβ+εi,?i=1,···,n,(1)

wherexiisak×1covariatevectorincludingacolumnofones,βisak×1parametervector,andnisthenumberofobservations.Weassumetheerrorεisindependentandidenticallydistributed(iid)asastandardnormaldistribution(i.e.,εi～N(0,1)fori=1,2,...,n),whichgivesrisetotheordinalprobitmodel.

Inourstudy,theunobservedvariablezirepresentsintenttopurchaseEVandisassociatedwiththeobserveddiscreteresponseyitothequestionaboutpurchasinganEV,asfollows:

γj-1<zi≤γj←→yi=j,?i=1,···,n;j=1,···,J,(2)

where-∞=γ0<γ1···<γJ-1<γJ=∞arethecutpoints(orthresholds),andJdenotesthenumberofcategoriesoroutcomesofy.Thisrelationshipbetweenthelatentz(intenttopurchaseanEV)andobservedy(responsetothequestionaboutpurchasinganEV)isshowninFigure

1

.P(yi=1)istheareaunderf(zi)totheleftofγ1,P(yi=2)istheareaunderf(zi)betweenγ1andγ2,andsoon.Wesetγ1=0andvar(ε)=1toanchorthelocationandscaleofthedistribution,respectively.Theserestrictionsarerequiredforparameteridentificationandfurtherexplainedin(

Jeliazkovetal.

,

2008

)and

JeliazkovandRahman

(

2012

).

Giventhedatay=(y1,···,yn)′andthecovariates,thelikelihoodfunctionforordinalprobitmodelcanbewrittenasfollows:

whereΦ(·)isthecumulativedistributionfunction(cdf)ofastandardnormaldistributionand

6

f(zi)

P(yi=3)

P(yi=2)

P(yi=1)

P(yi=4)

x'

1=02

i3

Figure1:Thedistributionofthelatentvariablez.ThefourprobabilitiesP(yi=1),P(yi=2),P(yi=3),andP(yi=4)correspondtotheresponsesregardingtheintentiontopurchaseanelectricvehicle(zi)forindividualiwith

meanxβ:notatalllikely,nottoolikely,somewhatlikely,andverylikely.

I(yi=j)isanindicatorfunction,whichequals1iftheconditionwithinparenthesisistrueand0otherwise.Buttheorderingconstraintγ1=0<γ2<...<γJ-1isdifficulttosatisfyduringtheMCMCsampling.So,weapplythetransformationof

AlbertandChib

(

2001

):δj=ln(γj-γj-1),for2≤j≤J-1,andrewritethelikelihoodasafunctionofβandδ=(δ2,...,δJ-1)′,whichallowsustomakeinferencesfromf(y|β,δ).

TocompletetheBayesianframework,weusetheBayestheoremtocombinetheaugmentedlikelihoodf(y,z|β,δ)withthepriordistributions:β～N(β0,B0)andδ～N(d0,D0).Theresultingaugmentedposteriordistributioncanbewrittenas

π(β,δ,z|y)∝f(y|β,δ,z)f(z|β)π(β)π(δ)={if(yi|δ,zi)}f(z|β)π(β)π(δ),(4)

wheref(yi|δ,zi)=1{γj-1<zi≤γj},thecorrespondencebetweenγandδisdeterminedbytheone-to-onemapping,andthecutpointindexjisdeterminedbytheobservedvalueofyi.Additionally,f(z|β)=fN(z|Xβ,In)andthepriorsπ(β)andπ(δ)arepdfsofnormaldistribution.Basedontheaugmentedjointposterior(

4

),theconditionalposteriorscanbederivedandthemodelestimatedusingMCMCsampling,asoutlinedinAlgorithm1.

Afterestimation,animportantconstructistocomputetheCEs(ormarginaleffects)ontheoutcomeprobabilities,as,unlikelinearregression,thecoefficientsdonotgivetheCEofavariable.

Assumethatthel-thcovariate,xi,l,issettotwodistinctvalues,aandb,denotedasx,land

x,l,respectively.Wepartitionthecovariateandparametervectorsasfollows:x=(x,l,xi,-l),

x=(x,l,xi,-l),andβ=(βl,β-l),where-ldenotesallcovariates(andparameters)exceptforthe

l-th.OurgoalistoexaminethedistributionofthedifferencePr(yi=j|x,l)-Pr(yi=j|x,l)for

1≤j≤J,bymarginalizingoverxi,-land(β,δ),whichenablesustoincorporatetheuncertaintyassociatedwithdataandmodelparametersandavoidnontrivialbiasesinestimatingCEs(

Jeliazkov

andVossmeyer

,

2018

).

Algorithm1:MCMCalgorithmforestimatingordinalprobitmodel.

7

1.Sampleδ,z|y,βinoneblockasfollows:

(a)Sampleδ|y,βmarginallyofzbygeneratingδ′fromarandom-walkchainδ′=δ+s,

=argmaxδlnf(y|β,δ).Accepttheproposedvalueδ′withprobability

otherwiserepeatthecurrentvalueδ.

(b)Samplezi|y,β,δ~TN(γj-1,γj)(xβ,1)fori=1,...,n,whereTN(γj-1,γj)denotesa

truncatednormaldistributionconstrainedbetweenγj-1andγj;andγisderivedfromtheone-to-onemappingrelatingγandδ.

2.Sampleβ|z~N(,)where-1=(B0-1+X′X),and=(B0-1β0+X′z).

ToobtaindrawsfromthedistributionPr(yi=j|x,l)-Pr(yi=j|x,l),weusethemethodof

composition(

ChibandJeliazkov

,

2006

).Thisinvolvesselectinganindividual,extractingthecorrespondingcovariatevalues,drawing(β,δ)fromtheirposteriordistributions,andevaluat-

ingthedifferencePr(yi=j|x,l,xi,-l,β,δ)-Pr(yi=j|x,l,xi,-l,β,δ),wherePr(yi=j|x,β,δ)=

Φ(γj-x,lβl-x-lβ-l)-Φ(γj-1-x,lβl-x-lβ-l),forq=b,a,and1≤j≤J.Theprocess

isrepeatedforallindividualsandMCMCdraws.TheaverageCE(ACE)foroutcomejisthencomputedasthemeanofpointwiseprobabilitydifferences:

where(β(m),δ(m))denotestheMCMCdraws,andMisthenumberofafter-burn-inMCMCdraws.

2.2.TheOrdinalQuantileModel

Theparameterestimatesfromtheordinalprobitmodelprovideinformationontheexpectedvalueofthelatentresponsevariableandaverageprobabilitiesofoutcomesconditionalonthecovariates.However,researchersareoftenmoreinterestedinexaminingspecificquantilesofthedependentvariable,particularlythelowerandupperextremes,toanswerpolicy-relevantquestions.Forinstance,confidencethatpubliccharginginfrastructurewillcontinuetoexpand—oneofourkeycovariates—mayhavelesseffectonindividualswithlowinterestinEVs.However,amongthosewhoaremoderatelyinterested,increasedconfidencemaystrengthenintent,reassuringthemthatchargingaccesswillnotbeabarrierandmovingthemfrom“somewhatlikely”to“verylikely”buyers.Tocapturehowtheinfluenceofconfidenceandotherfactorsvariesacrosslevelsof

8

purchaseintent,weuseordinalquantileregression.Theseinsightsprovidesuggestiveevidenceformoretargetedinterventions;forexample,theyshowthatimprovinginfrastructurevisibilitymaybemoreeffectivewhendirectedatconsumersalreadyconsideringEVsratherthanthosewithlittleornointerest.

Theordinalquantileregressionmodel,proposedby

Rahman

(

2016

),canbewrittenintermsofalatentvariableziasfollows:

zi=xβp+εi,

γp,j-1<zi≤γp,j←→yi=j,

?i=1,···?i=1,···

,n,

,n;j=1,···,J,

(6)

wheretheregressioncoefficientβpandcutpointvectorγparequantiledependentasindicatedbythesubscriptp,and,1,p)fori=1,2,...,n;whereALdenotesanasymmetricLaplacedistribution(

YuandZhang

,

2005

).TheALdistributionisessentialtocreateaworkinglikelihoodfunctionandyieldsthefollowingexpression,

(7)

whereFAL(·)denotesthecdfofanALdistributionandI(yi=j)isanindicatorfunction.

Atthisstage,threekeyissuesneedattention.First,theorderingconstraintofcutpointsduringsamplingisaddressedusingthetransformation,δp,j=ln(γp,j-γp,j-1)for2≤j≤J-1,wherenotethatδpisquantiledependent.Second,wesetγp,1=0toanchorthelocationandfixvarianceforanyquantiletomeetthescalerestriction.Third,theALdistributionisnotconvenientforMCMCsampling,asitdoesnotyieldtractableconditionalposteriors.So,following

Kozumiand

Kobayashi

(

2011

),weusethenormal-exponentialmixtureformulation:

wi～E(1),andisindependentofui,whichfollowsanormaldistribution,ui～N(0,1).Basedon

thisformulation,zi|βp,wi～N(xβp+θwi,τ2wi)fori=1,...,n;enablingustoleveragethe

propertiesofnormaldistributionforanefficientMCMCalgorithm.

ByBayestheorem,wecombinetheaugmentedlikelihoodf(y,z|βp,δp,w)withthenormalpriors:

9

βp～N(βp0,Bp0),δp～N(δp0,Dp0),toarriveattheaugmentedjointposteriordistribution,π(z,βp,δp,w|y)∝f(y|z,βp,δp,w)f(z|βp,w)π(w)π(βp)π(δp),

(8)

×N(βp|βp0,Bp0)N(δp|δp0,Dp0).

Inthefirstline,weusethefactorizationf(y,z|βp,δp,w)=f(y|z,βp,δp,w)×f(z|βp,w)andnotethatzisindependentofδp.Inthesecondline,weexploittheindependenceofyifrom(βp,w)given(zi,δp),whichfollowsfromthesecondlineofEquation(

6

),whereyigiven(zi,δp)isdeterminedwithprobability1.Inthethirdandfourthlines,wespecifytheconditionaldistributionofthelatentvariablezandthepriordistributionontheparameters(βp,δp).

Theconditionalposteriordistributionsarederivedfromtheaugmentedjointposteriordistri-bution(Equation

8

),andtheparametersaresampledasperAlgorithm2.Theimplementationofthisalgorithm,availableinthe

bqror

package,isexplainedin

MaheshwariandRahman

(

2023

).

Algorithm2:MCMCalgorithmforestimatingordinalquantilemodel.

1.Sampleβp|z,w～N(p,p),where,

2.Samplewi|βp,zi～GIG(0.5,i,),fori=1,···,n,where,

and.

3.Sampleδp|y,βpmarginallyofw(latentweight)andz(latentdata),bygeneratingδfrom

arandom-walkchainδ=δp+s,wheres～N(0J-2,ι2),ιisatuningparameterand

denotesnegativeinverseHessian,ob