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Bayesian estimation of inefficiency heterogeneity in stochastic frontier models

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30 pages

Estimation of the one sided error component in stochastic frontier models may erroneously attribute firm characteristics to inefficiency if heterogeneity is unaccounted for. However, it is not clear in general in which component of the error distribution the covariates should be included. In the classical context, some studies include covariates in the scale parameter of the inefficiency with the property of preserving the shape of its distribution. We extend this idea to Bayesian inference for stochastic frontier models capturing both observed and unobserved heterogeneity under half normal, truncated and exponential distributed inefficiencies. We use the WinBugs package to implement our approach throughout. Our findings using two real data sets, illustrate the relevant effects on shrinking and separating individual posterior efficiencies when heterogeneity affects the scale of the inefficiency. We also see that the inclusion of unobserved heterogeneity is still relevant when no observable covariates are available.
inancial support from the Spanish Ministry of Education and Science, research projects ECO2009-08100, MTM2010-17323 and SEJ2007-64500 is also gratefully acknowledged.
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 Working Paper 12-10 Statistics and Econometrics Series 07 May 2012   Departamento de Estadística Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624-98-49  BAYESIAN ESTIMATION OF INEFFICIENCY HETEROGENEITY IN STOCHASTIC FRONTIER MODELS  Jorge E. Galán, Helena Veiga and Michael P. Wiper *  Abstract_______________________________________________________________ Estimation of the one sided error component in stochastic frontier models may erroneously attribute firm characteristics to inefficiency if heterogeneity is unaccounted for. However, it is not clear in general in which component of the error distribution the covariates should be included. In the classical context, some studies include covariates in the scale parameter of the inefficiency with the property of preserving the shape of its distribution. We extend this idea to Bayesian inference for stochastic frontier models capturing both observed and unobserved heterogeneity under half normal, truncated and exponential distributed inefficiencies. We use the WinBugs package to implement our approach throughout. Our findings using two real data sets, illustrate the relevant effects on shrinking and separating individual posterior efficiencies when heterogeneity affects the scale of the inefficiency. We also see that the inclusion of unobserved heterogeneity is still relevant when no observable covariates are available. _____________________________________________________________________________________  Keywords: Stochastic Frontier Models; Heterogeneity; Bayesian Inference.      * Galán, Jorge E., Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), e-mail: jegalan@est-econ.uc3m.es. Corresponding author. Veiga, Helena, Departamento de Estadística and Instituto Flores de Lemos, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid). Wiper, Michael P., Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid). Acknowledgments: The authors would like to thank Mark Steel and Jim Griffin for their comments and suggestions as well as the participants of the 33th National Congress on Statistics and Operations Research and the Permanent Seminar on Efficiency and Productivity of Universidad de Oviedo. Financial support from the Spanish Ministry of Education and Science, research projects ECO2009-08100, MTM2010-17323 and SEJ2007-64500 is also gratefully acknowledged.
BayesianEstimationofInefficiencyHeterogeneityinStochasticFrontierModelsJorgeE.Gala´nHelenaVeigaMichaelP.Wiper§ABSTRACTEstimationoftheonesidederrorcomponentinstochasticfrontiermodelsmayerroneouslyattributefirmcharacteristicstoinefficiencyifheterogeneityisunaccountedfor.However,itisnotclearingeneralinwhichcomponentoftheerrordistributionthecovariatesshouldbeincluded.Intheclassicalcontext,somestudiesincludecovariatesinthescaleparameteroftheinefficiencywiththepropertyofpreservingtheshapeofitsdistribution.Weex-tendthisideatoBayesianinferenceforstochasticfrontiermodelscapturingbothobservedandunobservedheterogeneityunderhalfnormal,truncatedandexponentialdistributedinefficiencies.WeusetheWinBugspackagetoimplementourapproachthroughout.Ourfindingsusingtworealdatasets,illustratetherelevanteffectsonshrinkingandseparatingindividualposteriorefficiencieswhenheterogeneityaffectsthescaleoftheinefficiency.Wealsoseethattheinclusionofunobservedheterogeneityisstillrelevantwhennoobservablecovariatesareavailable.JELclassification:C11;C23;C51;D24Keywords:StochasticFrontierModels;Heterogeneity;BayesianInference.I.IntroductionStochasticfrontiermodels,firstintroducedinAigneretal.(1977)andMeeusenandvandenBroeck(1977),areimportanttoolsforefficiencymeasurement.Thesemodelsrequirethespeci-ficationofaneconomic,functionalformbasedonaproductionorcostfunctionwhichincludesTheauthorswouldliketothankMarkSteelandJimGriffinfortheircommentsandsuggestionsaswellastheparticipantsofthe33thNationalCongressonStatisticsandOperationsResearchandthePermanentSeminaronEfficiencyandProductivityofUniversidaddeOviedo.FinancialsupportfromtheSpanishMinistryofEducationandScience,researchprojectsECO2009-08100,MTM2010-17323andSEJ2007-64500isalsogratefullyacknowledged.DepartmentofStatistics,UniversidadCarlosIIIdeMadrid,C/Madrid126,28903Getafe,Spain.Email:jegalan@est-econ.uc3m.es.Correspondingauthor.DepartmentofStatisticsandInstitutoFloresdeLemus,UniversidadCarlosIIIdeMadrid,andBRU/UNIDE,AvenidadasForc¸asArmadas,1600-083Lisboa,Portugal.§DepartmentofStatistics,UniversidadCarlosIIIdeMadrid.
acompositeerrorterm.Thiserrortermcanbedecomposedintotwoparts,firstlyatwo-sided,idiosyncraticerrorandsecondly,anon-negativeinefficiencycomponent.Measuresofefficiencyareobtainedfromthisone-sidederror,whichtypicallyisassumedtofollowsomespecificdis-tribution.Themostcommondistributionsfortheone-sidederrorarethehalf-normal(Aigneretal.,1977),exponential(MeeusenandvandenBroeck,1977),truncatednormal(Stevenson,1980),andgamma(Greene,1990).However,theestimatedinefficiencycomponentoftenincludessomefirmcharacteristicsotherthanoutputs,inputs,orpricesdefinedfromtheproductionorcostfunction,whichshouldnotbeattributedtoinefficiency.Thesefirmcharacteristicsareexogenousvariables(e.g.typeofownership,GDPlevelinthecountryofoperation)thathaveaneffectonthetechnologyusedbythefirmsordirectlyontheirinefficiency.Ifthesevariablesarenottakenintoaccountinthemodelspecification,thismayaffecttheestimationoftheinefficienciesorofthefrontiersignificantly.Thedistinctionbetweenheterogeneityandinefficiencyhasbecomeaveryimportantissueinstochasticfrontiermodels.Firmcharacteristicscanbemodeledinthefrontieriftheyimplyheterogenoustechnologiesorintheone-sidederrorcomponentiftheyaffecttheinefficiency.Intheformercase,covariatesaredirectlyincludedinthefunctionalformandthemaininterestistomodelunobservedhet-erogeneity(seeGreene,2005).Forthecaseofheterogeneityintheinefficiency,covariatesareusuallyincludedintheparametersoftheone-sidederrordistribution(seeHuangandLiu,1994).HeterogeneityinstochasticfrontiermodelshasalsobeenstudiedfromtheBayesiancon-text.TheBayesianapproachtostochasticfrontiersintroducedbyvandenBroecketal.(1994)presentsadvantagesintermsofformallyderivingposteriordensitiesforindividualefficiencies,incorporatingeconomicrestrictions,andintheeasymodelingofrandomparametersthroughhi-erarchicalstructures.Hierarchicalmodelshavebeenusedtocaptureheterogeneoustechnologies(seeTsionas,2002)andheterogeneityintheinefficiencyhasbeenconsideredthroughcovariatesinthedistributionofthenon-negativeerrorcomponent(seeKoopetal.,1997).Modelingob-servedheterogeneityusingnonparametricandflexiblemixturesofinefficiencydistributionsare2