搜索
StochasticFrontiers²Inthissectionwetakethemaximumlikelihoodapproachandapplyittoafairlyusefulandpowerfultool-stochasticfrontierestimation.²What'sthebasicidea?Howtoestimateeconomicrelationshipsthatoughttobemodeledasupperorlowerfrontiersratherthanaverages.²Forexample:Considerademandcurve.Fromthetheoryofdemand,thedemandcurveisafrontierwhichtellsthe¯rmthemostitcanchargeforthemarginalunit.²EconometricsviatraditionalOLSwouldgatherPriceandQuantitydataandestimateanaveragedemandcurve.AtanyQ0themodelpredictsP0butthe¯rmcouldactuallychangePA.TheaverageOLSapproachmightoverorunderpredictprice.²Themodeltheunderpredictscoststhe¯rmmoney.Themodeltheoverpredictsmightbecatastrophic.²Frontierestimationtriesto¯xthisproblem.However,notalldataareconducetoSFA.²Uses:Productionfunctions,costfunctions,demandmodels,testofunione®ectiveness,agencycosts,reservationwages,schooloutcomes,pro¯tability,survivorship,mergerandacquisitionanalysis,e®ectofshadowinputssuchascorruption.²Considerthetraditionalproductionfunction:1²Thisisthesingleinputcasewhereq=f(X).Theslopeoftherayfromtheoriginisameasureofproductivityq=X.Notethatq=XatPointAislessthanatPointBandPointC.²Wecanimaginetechnologicalchangeovertimewhichwouldbeashiftintheproductionfrontier²Usingmultipleinputs,thepicturechangesalittle:²PointPisine±cientrelativetoPointM.PointAisbothtechnicallyandallocativelye±ciency.²WecanmeasureTechnicalE±ciencyasTE=OM=OPandTechnicalIne±ciencyas1¡TE=1¡OM=OP.²Allocativee±ciencyismeasuredasON=OM·1.²Overalle±ciencyismeasuredas(OM=OP)(ON=OM)=(ON=OP)2History²Farrell(1957,JournaloftheRoyalStatisticalSociety,SeriesA):Derivesaproductionfunctionapproachandidenti¯estwosourcesof¯rmine±ciency/e±ciency1.TechnicalE±ciency:Producethemostoutputwithagivenlevelofinputs2.AllocativeE±ciency:Produceagivenoutputascheaplyaspossible.²Mostofthetimewefocusontechnicale±ciencyintheexplanation²Todetermineifa¯rmise±cient,wehavetoknowtheproductionfunctionofthefullye±cient¯rm.²However,weneverknowthefullye±cientproductionfunction²Farrellsuggestedestimatingafullye±cientproductionfunction.Therearetwowaystodothis:1.Nonparametrictechniques:Dataenvelopmentanalysis.Thistechniqueassumesthatalldeviationsfromthee±cientfrontierisarealizationofine±ciency2.Parametrictechniques:StochasticFrontierAnalysis.Thistechniqueassumesthatdeviationsfromthee±cientfrontiercanbeeitherarealizationofine±ciencyorarandomshock.²Aigner,LovellandSchmidt(1977)andvandenBroeck(1977)bothintroducedawaytodealwithSFAandproductionfunctions.BasicSetup²Consideraproductionfunctionqi=f(xi;¯)wherexiisavectorofinputs,qiisoutput,and¯isa[k£1]vectorofparameterstobeestimated.3²Wecanthinkofe±ciencybeingmeasuredas³imultipliedbythetheoreticalnormwhere³i2[0;1]suchthatqi=f(xi;¯)³iIf³i=1thenthe¯rmisfullye±cientandproducesthemostitcan.If³i1thenthe¯rmisnotfullye±cient.²Wecanletqi=f(xi;¯)bethelevelofoutputthatshouldhappen.Letq0betheobservedoutputwhereq0qFbecauseofine±ciencyandotherfactors.²Asq0qF=f(xi;¯),AignerandChu(1968)suggestedaddinganon-negativerandomvariabletof(xi¯)whichwouldcapturethetechnicaline±ciencyof¯rmi:qo=f(xi;¯)¡ui²Toestimatethistypeofmodel,onecouldusea¯xede®ectsmodelwhereuiwastreatedasthe¯rm¯xede®ects.²Let'sassume:f(xi;¯)=¯0X¯11X¯22¢¢¢X¯kklnf(xi;¯)=ln¯0+¯1lnX1+¯2lnX2+¢¢¢¯klnXklnqi=¯1lnX1+¯2lnX2+¢¢¢¯klnXk¡uilnqi=¯X¡ui²AignerandChu(1968)suggestedameasureoftechnicale±ciencyofObservedOutputFrontierOutput=qiexp(xi¯)=exp(xi¯¡ui)exp(xi¯)4where0exp(¡ui)·1.²Whilethisisadecentshotattheproblemitdoesleavealottobedesired.²Mainlyuiissupposedtomeasureine±ciencybutitmightalsobecapturingotherrandomshocksthatarebeyondthecontrolofthe¯rm'smanagement.Forexample,wouldwewanttoholdthe¯rm'smanagementfortheimpactsofKatrinaoranearthquakeorsomeotherweatherevent?²Aigner,Lovell,andSchmidt(ALS)in1977suggestedaddingatwo-sidederrortermtotheone-sidederrortermofAC(1968).Itdoesn'tseemlikethatbigofadeal,butitdidtakesomeworktoderivethelikelihoodfunction.²Now,qi=f(xi¯)³iexp(vi)whichyieldslnqi=ln(f(xi¯))+ln(³i)+vi²De¯ningui=¡ln(³i)yieldslnqi=ln(f(xi¯))+vi¡ui²InaCobb-Douglastypeworldlnqi=¯0+kXj=1¯jln(xji)+vi¡uiHowever,wecannotuseOLStogetatthecompositeerrorterm.²Somebasicassumptionsvi»iidN(0;¾2v),ui¸0andcov(vi;ui)=0,wherev=measure-menterror,weather,randomfactors,andu=technicalin±ciency(onesided).²Hence,uirequiresustomakeanassumptionaboutthedistributionofu.5²Populardistributionsinclude8:Half-NormalExponentialTruncatedNormalGamma(rare)9=;STATAcanhandleallofthese²Let'stakeanotherlookatwhatisgoingon²ForFirmi:AisdeterministicoutputlevelA0mightbefrontieroutput:q=exp(xi¯+vi)wherevi0A00mightbeobservedoutput:q=exp(xi¯+vi¡ui)²ForFirmj:BisdeterministicoutputlevelB0mightbefrontieroutput:q=exp(xj¯+vj)wherevj0B00mightbeobservedoutput:q=exp(xj¯+vj¡uj)²Note:Thecompositeerrortermvi¡uidoesn'tcauseaproblemwithOLSaslongasvianduiareindependentoftheinputsx.^¯OLSisunbiased,consistent,ande±cientamongstlinearestimators,excepttheinterceptisnotconsistent.Nevertheless,itisimpossibletoextricate¾2uand¾2v.²Note:MLEyieldsmoree±cient^¯,aconsistentintercept,andaconsistentvar(vi¡