frontier操作演示

STOCHASTICFRONTIERANALYSIS1MOTIVATION•Usualtextbookpresentationstreatproducersassuccessfuloptimizers.Theymaximizeproduction,minimizecost,andmaximizeprofits.•Conventionaleconometrictechniquesbuildonthisparadigmtoestimateproduction/cost/profitfunctionparametersusingregressiontechniqueswheredeviationsofobservedchoicesfromoptimalonesaremodeledasstatisticalnoise.•Howeverthougheveryproducermayattempttooptimize,notallofthemmaysucceedintheirefforts.Forexample,giventhesameinputs,andthesametechnology,somewillproducemoreoutputthanothers,i.e.,someproducerswillbemoreefficientthanothers.•Econometricestimationtechniquesshouldallowforthefactthatdeviationsofobservedchoicesfromoptimalonesareduetotwofactors:failuretooptimizei.e.,inefficiencyduetorandomshocks•StochasticFrontierAnalysisorSFAisonesuchtechniquetomodelproducerbehavior.2USEFULNESSOFSFA•SFAproducesefficiencyestimatesorefficiencyscoresofindividualproducers.Thusonecanidentifythosewhoneedinterventionandcorrectivemeasures.•Sinceefficiencyscoresvaryacrossproducers,theycanberelatedtoproducercharacteristicslikesize,ownership,location,etc.Thusonecanidentifysourceofinefficiency.•SFAprovidesapowerfultoolforexaminingeffectsofintervention.Forexample,hasefficiencyofthebankschangedafterderegulation?Hasthischangevariedacrossownershipgroups?3STRUCTUREOFTHISPRESENTATION•Part1:Theory:IllustratethebasicsofSFAmainlywithanalysisofcostefficiency.ConceptofefficiencyEstimationIdentificationofsourcesofinefficiency•Part2:Empirics:HowtouseFRONTIERprogramtoestimatedifferenttypesofefficiencymodelsAnapplicationofSFAtoIndianBanking(iftimepermits)•References1.Kumbhakar,S.C.andLovell,C.A.K(2000),StochasticFrontierAnalysis,CambridgeUniversityPress,U.K.2.Coelli,T.J.;Rao,D.S.Prasada,andBattese,G.E.(1998),AnIntroductiontoEfficiencyandProductivityAnalysis,KluwerAcademicPublishers,Boston/Dordrecht/London.4TECHNICALEFFICIENCY•ProductionFunction:YM=ƒ(x;β)showsthemaximumoutputYMproduciblefromagivenvectorofinputs(x).Hereβaretheproductionfunctionparameters.ƒ(L,K;β)=2L0.5K0.5ƒ(9,16;β)=2.90.5.160.5=24•Actualoutput,Ycouldbelessthanmaximumoutput.Infact,anyoutputequaltoorlessthanYMcanbeproduced.Y≤ƒ(x;β)=YM•Figure1•TE=Y/YM0≤TE≤1•Y=YM.TE=ƒ(x;β).TE•Characterization:Y=ƒ(x;β)exp(-u)u≥0(1)5STOCHASTICFRONTIER•In(1)thefrontierisdeterministic.Alldeviationsfrommaximumoutputareascribedtoinefficiency.•Howeversometimesmaximumoutputitselfmightbelower(higher)duetoexogenousshocks.Theproductionfrontieritselfmaybeshifting.•Figure2•Y=ƒ(x;β).exp(v).exp(-u)v≤0andu≥0ƒ(x;β)deterministickernelexp(v)effectonoutputofexogenousshocksexp(-u)inefficiencyƒ(x;β).exp(v)stochasticfrontier•TE=Y/ƒ(x;β).exp(v)=ƒ(x;β).exp(v).exp(-u)/ƒ(x;β).exp(v)=exp(-u)•Y=ƒ(x;β).exp(v–u)(v–u)compositeerrorterm6COSTEFFICIENCYORECONOMICEFFICIENCY•Abilitytoproduceobservedoutputatminimumcost,giveninputprices.•Aproducermaybetechnicallyefficient,butyetcostinefficientbecausehefailstochoosecorrectinputcombination.Allocativeinefficiency•Figure3•Ofcourse,aproducermaybebothtechnicallyinefficientaswellasallocativelyinefficient.Cost(oreconomic)inefficiency=technicalinefficiency+allocativeinefficiency•Figure4•Theoreticallythisnotioniswelldefinedbutempiricallyitisinvolvingtosegregatethesetwosourcesofinefficiency.•Similarly,wecandefineprofitefficiency,butitsdecompositionintotechnical,andallocativeinefficiencyisevenmorechallenging.Will,therefore,concentrateoncost-efficiencyintheremainderofthelecture.7COSTEFFICIENCYEi=c(yi;wi;)expfvi+uig(1){c(yi;wi;)isthedeterministickernel{virandomnoise,takespositiveandneg-ativevalues{uicapturesineciency,takesonlyposi-tivevalues{NotethepositivesignbeforeuiUnderthisformulationcosteciencycanbecalculatedasCEi=c(yi;wi;)expfvigEi=expfuig(2)0CEi11EstimationFirstrewriteaslnEi=lnc(yi;wi;)+ui+vi(3)Estimatingequation(3)requires:{specicationofafunctionalformforthedeterministickernelc(yi;wi;),{anassumptionaboutthedistributionoftherandomvariablevi,and{anassumptionaboutthedistributionoftherandomvariableui.Givenaparticularspecicationfortheran-domvariablesuiandvi,theMaximumLike-lihood(ML)techniqueisusedtoestimatetheunknownparameters.2SPECIFICATIONDeterministicKernel{Cobb-Douglas(inlogform){Translog(aexiblefunctionalform)RandomVariablesviandui{viiidN(0;2v){uiiidN+(0;2u){vianduiaredistributedindependentlyofeachother,andoftheregressors.Giventheseassumptions,thelog-likelihoodfunctionforthesampleofsizeIlnL=KIln+Xiln(i)122Xi2i:(4)wherei=ui+vi,2=(2u+2v),=uvand(:)isthestandardnormalcumulativedistributionfunction.WesubstitutelnEilnc(yi;wi;)inplaceofiinthelikelihoodfunction.3DERIVATIONOFLIKELIHOODFUNCTIONThedensityfunctionofviisf(vi)=1p2vexp(v2i22v)Thedensityfunctionofuiisf(ui)=2p2vexp(v2i22v)Giventheindependenceassumption,thejointdensityfunctionofuiandviistheproductoftheirindividualdensityfunction,andso,f(ui;vi)=22uvexp(u2i22uv2i22v)Sincei=vi+ui,thejointdensityfunctionforuiandiis:f(ui;i)=22uvexp(u2i22u(iui)222v)4Themarginaldensityfunctionofiisthenobtainedbyintegratinguioutoff(ui;i)whichyieldsf(i)