Consistency Conditions for Regulatory Analysis of Financial Institutions: A Comparison of Frontier Efficiency Methods

Consistency Conditions for Regulatory Analysis ofFinancial Institutions: A Comparison ofFrontier Efficiency Methods Pad W.Bauer FederalReserveBankofCleveland Cleveland,OH41101-1387 AllenN. Berger BoardofGovernorsoftheFederalReserveSystem Washington,DC20551 and Whtion FinancialInstitutionsCenter Philadelphia,PA 19104 U.S.A. GaryD.Ferrier UniversityofArkansas Fayetteville,AR 72701-1201 DavidB.Humphrey FloridaStateUniversity Tallahassee,FL32306 Forthcoming,JournalofEconomicsandBusiness,1998 Abstract Weproposea setofconsistencyconditionsthatfrontierefficiencymeasuresshouldmeetto bemost usefd for re@atory analysis or other purposes. The efficiency estimates shotid be consistent in their efficiency levels, rankings, and identification of best and worst fus, consistent over time and with competitiveconditionsinthemarket,andconsistentwithstandardnonfrontiermeasuresofperformance. We provide evidence on these conditions by evaluating and comparing efficiency estimates on U.S. bank efficiencyfromvariantsof allfourofthemajorapproaches--DEA, SFA,TFA, andDFA -- andfmdmixed results. JELClassification:G21,G28,E58 Keywords:FinancialInstitutions,Efficiency,Re@ation TheopinionsexpresseddonotnecessarilyreflectthoseoftheBoardofGovernors,theFederalReserveBank of Cleveland,or their staffs. The authorsthank Bob DeYoungfor helpti suggestions,and Seth Bonirne, MariaFilson,A.J.Matteo,andLindaPittsforvaluableresearchassistance. Please addresscorrespondenceto AllenN. Berger,Mail Stop 153,FederalReserveBoard, 20th and C Sts. NW,Washington,DC20551,call202-452-2903,fax202-452-5295,oremailaberger@fib.gov.

I. Introduction To make informed policy decisionsregarding financialinstitutions,re~ators need to have fairly accurate informationabout the likelyeffects of their decisionson the performanceof the institutionsthey re~ate/supervise.l Specifically,the regulators of commercialbanks, thrifts, credit unions, and insurance companiesshodd havesomeexpertknowledgebased on rigorousempiricalresearchregardingwhetherthe mergersand acquisitionsthey arepetitionedto approvewillrestit inhigheror lowercosts, andwhetherthe “-increases in equity capital ratios they may require will raise costs significantlyand reduce the supply of intermediationservices. Similarly,re@atoW authoritiesshotid be awarewhetherthe observedmanagerial inefficiencytheymay observecouldraisetheprobabilityof financialinstitutionfailuresubstantially,and so codd be used to reallocate scarce supervisory resources to where they are most needed. In addition, re~ators shotid have quantitative evidenceon the performance effects of redatory restrictions on the interest rates and insurance premiums these institutions are allowed to pay and receive, the prudential restrictionson therisksthese fms are allowedto bear, the geographicareasthey are allowedto serve,and the types of financialservicesthey are allowedto offer. Ifregulatoryauthoritiesdonot havethe benefitof qualityinformationbasedupon quantitativeresearchregardingtheperformanceeffectsoftheir actions,then theirdecisionsmayhavetheunintendedconsequencesofraisingthe costsofprovidingfinancialservicesto thepublic,reducingthequantityorqualityoftheseservices,orincreasingsystemicrisk. In recentyears, the academicresearch on the performanceof financialinstitutionshas increasingly focusedonfrontierefficiencyorX-efficiency,whichmeasuresdeviationsinperformancefromthat of“bestpractice”firms onthe efficientfrontier,holdingconstanta numberof exogenousmarketfactors such asthe pricesfacedinlocalmarkets. Thatis.thefrontierefficiencyof aninstitutionmeasureshowwellitperforms relativeto the predictedperformanceof the “best” fms in the industryif these best fms were facing its samemarketconditions. Frontierefficiencyissuperiorformostregulato~ andotherpurposestothestandard financialratiosfromaccountingstatements--suchasreturnonassets(ROA)orthecost/revenueratio --that are commonlyemployed by regulators, financial institution managers, and indus~ constitants to assess performance. This is becausefrontierefficiencymeasuresuse prograrnmingor statisticaltechniquesto try toremovetheeffectsofdifferencesininputpricesandotherexogenousmarketfactorsaffectingthestandard IFor convenience,we simplyuse theterm “regulators”to refer to alllawmakers,superviso~ agencies, antitrustauthorities,etc.thatexerciseanyre~atory orsupervisoryauthorityoverfinancialinstitutions.

2 performanceratiosinordertoobtainbetterestimatesoftheunderlyingperformanceofthemanagers. The financialinstitutionefficiencyliteratureis both large and recent -- a reviewof 130 studiesof financial institution frontier efficiencyacross 21 countriesfound that fully 116were written or published during 1992-1997 (Berger and Humphrey 1997). Frontier inefficiency or X-inefficiency of financial institutionshas generallybeen found to consumea considerableportion of costs on average,to be a much ‘-” greater sourceofperformanceproblemsthan eitherscaleorproductmix inefficiencies,andto havea strong empiricalassociationwith higherprobabilitiesof financialinstitutionfailures over severalyears following theobservationofsubstantialinefficiency. Frontierefficiencyhasbeenusedextensivelyinre~atory analysistomeasuretheeffectsofmergers and acquisitions, capital regulations, dere~ation of deposit rates, removal of geographicrestrictions on branching andholdingcompanyacquisitions,etc.on financialinstitutionperformance. Themain advmtage of fi-ontierefficiencyoverotherindicatorsofperformanceis that it is an objectivelydeterminedquantitative measure that removes the effects of market prices and other exogenous factors that Muence observed performance. This allowsthe researcherto focus on the quantitativeeffects on costs, input use, etc. that changesinre~atory policyarelikelytoengender. Despite intense research effoti, there is no consensuson the best method or set of methods for measuringfrontierefficiency,andthechoiceofmethodmayaffectthepolicyconclusionsthataredrawnfrom the analyses. Inthepast twentyyears, atleastfourmainfrontierapproacheshavebeen developedto assess firmperformancerelativeto someempiricallydefined“best-practice”standard. Thesearethenonparametric linear programmingapproach,often referred to as data envelopmentanalysis(DEA), and threeparametric econometric approaches -- the stochastic frontier approach (SFA), thick frontier approach (TFA), and distribution-freeapproach(DFA). Theseapproachesdifferintheassumptionstheymakeregardingtheshape of the efficientfrontier, the existenceof random error, and (if random error is allowed)the distributional assumptions imposedon theinefficienciesandrandomerrorin orderto disentangleonefromthe other. As discussed below, these approaches also often differ in whether the underlying concept analyzed is technologicalefficiencyversuseconomicefficiency,althoughthisdifferenceneednotoccurinpractice. Inthispaper,wearguethatitisnotnecessarytohaveaconsensusonwhichisthesinglebestfi-ontier approach for measuring efficiencyfor the efficienciesto be usefil for regulatory analysis. Instead, we

3 propose a setof consistencyconditionsthat efficiencymeasuresderivedfromtie variousapproachesshodd meet to be most usefil for re@ators or other decisionmakers. The efficiencyestimatesderivedfrom the differentapproaches shotid be consistentin their efficiencylevels,rankings, and identificationof best and worstfins, consistentovertimeandwithcompetitiveconditionsinthemarket,andconsistentwithstandmd nofiontier measuresofperformance. Specifically,theconsistencyconditionsare: (i) the efficiencyscores generated by the different approaches should have comparable means, standarddeviations,andotherdistributionalproperties; (ii) tie differentapproachesshotid ranktheinstitutionsintheapproximatelythesameorder; (iii) the differentapproachesshodd identi~ mostlythe sameinstitutionsas “best practice” and as “worstpractice;” (iv) all of tie useti approaches shodd demonstratereasonable stability over time, i.e., tend to consistentlyidenti~ the sameinstitutionsasrelativelyefficientor inefficientin differentyears, ratier thanvaryingmmkedlyfromoneyeartothenext; (v) tie efficiencyscoresgeneratedbythedifferentapproachesshotid bereasonablyconsistentwith competitiveconditionsinthemmket;and (vi) themeasuredefficienciesfromalloftheusefulapproachesshodd bereasonablyconsistentwith standard notiontier performancemeasures,such as return on assets or the cost/revenueratio. Consistencyconditions(i), (ii), and (iii)may be thought of as measuringthe degreeto which the Werent approachesaremutuallyconsistent,andconditions(iv),(v),and(vi)maybe thoughtofasmeasuring the degreeto whichthe efficienciesgeneratedby tie differentapproachesare consistentwith realityor are believable. Theformeraremorehelpfulindetermininggwhetherthedifferentapproacheswillgivethe same answerstore@ato~ policyquestionsorotherqueries,andthelatteraremorehelpfi indetermininggwhether theseanswersarelikelytobecorrect. Specificallyfor the mutual-consistencyconditions,if tie approachesgenerate similardistributions of efficiencyasincondition(i),thentie projectedquantitativeeffectsofregulato~ policiesonperformance wodd bemorelikelyto besimilaracrosstheapproaches. Ifthemethodsallranktie institutionsinaboutthe same order as in (ii), then regulatory authoritieswould generallyget the same answer when evaluating whether institutionsthat had undergonemergers and other regulatory-influencedeventshad became more orlessefficientasaresult. As aweakerconditionthan (ii),ifthe approachesatleastfoundmostlythe same institutionstobe inthehighestandlowestefficiencygroups,asincondition(iii),thenre~atory authorities

4 could draw reasonable conclusions about which operating policies and procedures or managerial control structureswere“best-practice”and“worst-practice,”anddesigntheirpoliciesaccordingly. For example,if it weredeterminedthatbranchbankingoruniversalbankingwerebestpracticesthatconsistentlymaximized measured efficiencyacross all the approaches,then re@ators might be less inclinedto put restrictionson branch expansion or circumscribebankingpowers. Importantly,all of the efficiencyapproachescouldbe ‘-mutuallyconsistentas in conditions(i), (ii), and (iii),but stillnot be veryusefil if they arenot realisticor believableasinconditions(iv),(v),and(vi). Fortheconsistent-with-realityorbelievabilityconditions,iftheefficiencyscoresarestableovertime as in condition(iv) instead of the efficienciesbouncingup and down dramaticallyfrom year to year, this would be consistentwith the likelypattern of true managerialefficienciesovertime. Managementusually does notturn overofien,andevenwhenthis occurs,itis difficultto implementnewpoliciesandprocedures quickly. Similarly, some of efficiency differences may arise from differences in technology that are embodiedindurableplantandequipmentthatmaybe difficultandcostlyto replaceintheshortterm. Thus, onlyin exceptionalcaseswotid it be likelythat efficiencieswodd fluctuatemarkedlyover shortperiodsof time. If condition(iv) were met, then authoritiescould also be more confidentthat their policiestargeted towardeitherveryinefficientorveryefficientfirmswouldstillgenerallyidentifythemcorrectlyafternormal policyandimplementationlagsforre@ato~ actions. Inaddition,competitiveconditionsmayhelplimitthe range of believableefficienciesin the market, as in condition(v). For instance,if the entrybarriers to the industry or localmarket are not too steep and the market is reasonablyunconcentrated,then condition(v) suggeststhatmostfms thatremaininbusinessforalongperiodoftimeshodd bereasonablyefficient,since competitionshoulddrive most of the very inefficientfms out of the industry. Finally,if the efficiencies generated by the different approaches all were positivelyrelated to standard financial ratio measures of performanceasincondition(vi),thenauthoritiescouldbemorecotildent thatthemeasuredefficiencieswere accurate indicators of actual accomplishment,and not just artifacts of the assumptions of the efficiency approaches. Itisexpectedthataccurateefficiencymeasureswotid havepositiverank-ordercorrelationswith the standard nonfrontierperformancemeasures, but the correlationsshotid be far from 1.00 because the standard measuresembodynot onlythe efficiencies,but alsothe effects of differencesin input prices md otherexogenousvariablesoverwhichfinancialinstitutionmanagershavenolittleornocontrol.

5 There is some prior evidenceon these points, but there has never been a comprehensivestudy of financialinstitutionsthat has examinedallsix oftheseconsistencyconditionsforre~ato~ use~ess and applied themto allfour of the major frontier approaches. The prior evidencesuggeststhat the efficiency scoreshorn thedifferentapproachesoftenyieldquitedifferentdistributionsofmeasuredefficiencies,contrary to condition(i). Forexample,acomparisonof 118averageannualefficiencyvaluesfrom66 studiesofU.S. “-banksindicatedthatnonparametricmethodssuchasDEAyieldedalowermeanandhigherstandarddeviation than did the parametric methods such as SFA, TFA, and DFA (Berger and Humphrey 1997, Table 2).2 However, these studies differed in their choice of efficiency concept (technological efficiency versus economicefficiency),samplesofbanks,timeperiodschosen,specificationsofinputsandoutputs,functional forms, and employment of different techniques within each approach, and so do not provide a good experimental design for evaluating whether the efficiency approaches are consistent. For evaluating consistency,itis necessaryto holdtheseotherfactorsconstantand applymtitiple efficiencymethodsto the samedataset. There are a few such studiesthat appliedtwo or moremethodsto the samedata set, andthese are reviewedin SectionIIIbelow. As willbe shown,this evidenceisverylimitedatpresent, andtheresdts are quitemixedacrossstudies. Thestudiessometimesfmdtheaverageefficienciesto be similarandsometimes dissimilar across the approaches, and sometimes consistent and sometimes inconsistent with market competitiveconditions,yieldingambiguousevidenceregardingconditions(i) and (v). These studieshave also yieldedmixed evidenceon the issuesof whetherthe differentefficiencyapproachesrank the best and worst institutionssimilarly,as in conditions(ii) and (iii). Thereis also very littleevidenceon the stability of efficiency, and whether measured efficiencies rank firms in the same order as standard notiontier measuresofperformance,asinconditions(iv)and(vi). The mainpurposeofthecurrentstudyisto addto thislimitedinformationsetbyprovidingspecific evidenceon all of the six conditionsforre@atory usefukess by evaluatingand comparingnew efficiency estimatesfromallfourofthemajorapproaches. Tobecomplete,weemploymultipletechniqueswithineach of the four approaches, using single-periodand panel methods, for a total of nine efficiencytechniques ‘Themeanandstandarddeviationofthenonparametricefficiencyscoreswere72Y0and 17V0r,espectively, asopposedto 84°Aand60/.,respectively,fortheparametricestimates.

6 evaluated. To be sure that the applications are comparable, all nine techniquesuse the same efficiency concept(economicefficiency),the same sampleof banks, the sametime period,the same specificationsof inputs and outputs, and (forthe parametricmethods)the samefunctionalform. To be surethat the results do not dependupon any oneparticdar economicenvironmentof the banking industryor any peculiarities of anyonesmallgroupof bds, we estimatethe averageefficiencyovertimeof apanelof 683 banks over ‘-a 12-yearperiod duringwhich there were significantchangesin the banking industry. This experimental designhelps assurethat the observeddifferencesin efficiencyscoresreflectthe effectsofthe differencesin themeasurementtechniques,ratherthananyoftheseotier factors. Our examination of the consistencyconditionsis in the spirit of Charnes, Cooper, and Sueyoshi (1988), who advocatedthe “methodologicalcross-checking”of resdts that have policy importance. Our applicationis also in concordancewith Learnerand Leonard (1983) and Learner(1994), who emphasized assessing the “fragili~” of one’s results by reporting the resdts of diverseor “extreme” modelsto better understandtheimplicationsofone’sanalysis. Webelievethatourestimationofninedifferentmodelsusing 12yearsofdata,andapplyingallsixconsistencyrequirementstotheninemodelsqualifiesas“extreme,”and if thereis “fi-agility”in thefindingsacrossefficiencyapproaches,we are likelyto fmd it. As well,if there are dimensionsof consistencyin the frontier efficiencyapproaches,we shotid also be able to fmd some evidenceoftheseconsistencies. Theremainderof this paper is organizedas follows. SectionII outlinesthe fourfrontierefficiency approaches, and Section III reviews earlier studies that compared two or more approaches. Section IV describesthedata set andmodelspecifications,and SectionV appliesthe consistencyconditionsto thenine efficiencytechniques. SectionVI pulls all of this informationtogetherand draws someconclusionsabout the consistencyof the various frontier efficiencyapproaches for use in re@ato~ analysis, and discusses someavenuesforfiture research. II. TheFourFrontierEfficiencyA~~roachw As noted above,thefourfrontierapproachesdifferinthe assumptionsmade aboutthe shapeofthe frontier, the treatment of random error, and the distributions assumed for inefficiencyand random error. Thesemethodsalsooftendifferinwhethertheunderlyingconceptofefficiencyistechnologicaloreconomic, with the nonparametric DEA studies usually measuring technologicalefficiencyand the parametric SFA,

7 TFA, and DFA studies usually measuring economic efficiency. In this section, we briefly review the methods, focusing on the underlying concepts and assumptions, rather than the technical details of the estimationmethodswhichhavealreadybeenwell-explainedinseveralcomprehensivesurveys.3 We begin by discussing the efficiency concepts, and then assess the four frontier approaches. Technologicalefficiency,ortechnicalefficiencyasit is sometimescalled,focusesonlevelsofinputsrelative ‘-to levelsofoutputs. To betechnologicallyefficient,a firmmust eitherminimizeits inputsgivenoutputsor maximize its outputs giveninputs. Economicefficiencyis a broader conceptthan technologicalefficiency, in that economicefficiencyalso involvesoptimallychoosingthe levelsand mixes of inputs and/oroutputs basedonreactionstomarketprices. Tobeeconomicallyefficient,afirmhastochooseitsinputand/oroutput levels and mixes so as to optimize an economicgoal, usually cost minimizationor profit mtization. Economicefficiencyrequirestechnologicalefficiencyaswellasallocativeefficiency--i.e,theoptimalinputs and/or outputs arechosenbased on both theproductiontechnologyandtherelativepricesinthemarket. It is quite plausible that some firms that are relatively technologicallyefficient are relatively economically inefficient and vice versa, depending upon the relationship between managers’ abilities to use the best technologyandtheir abilitiesto respondto marketsignals. Therefore,theuseofthetwo differentefficiency conceptsmaygivesignificantlydifferentrankingsoffirms,evenforagivenfrontierapproach. Technological efficiency scores will also tend to be higher than economicefficiencyscores on average, all else equal, becauseeconomicefficiencysetsahigherstandardthatincludesallocativeefficiency. Technologicalefficiencyrequiresonlyinput and output data, but economicefficiencyalsorequires pricedata. Mostoftheearlynonparametricfrontiermodels(e.g.,Charnes,Cooper,andRhodes1978)aswell as some of the early parametric frontier models (e.g., Aigner, Lovell, and Schmidt 1977) focused on technologicalefficiency. In fact, DEA was developedspecificallyfor measuringtechnologicalefficiencyin thepublicandnot-for-profitsectors,wherepricesmaynotbeavailableorreliable,andtheassumptionofcost minimizingorprofitmaximizingbehaviormaynotbeappropriate(Charnes,Cooper,andRhodes1978). However, in recent efficiency analyses, there is usually a difference in the efficiency concept 3See,forexample,thesurveysby Banker,Charnes,Cooper,Swarts,andThomas(1989),Bauer(1990), Seiford and Thrall (1990), Ali and Seiford (1993), Greene (1993), Grosskopf (1993), Lovell (1993), or Charnes,Cooper,Lewin,andSeiford(1994).

8 employedbetweenthenonparametricandparametricapproaches. Mostnonparametic DEAstudiescontinue to applytechnologicalefficiencyto inputs andoutputs,althougha fewstudiesdouse cost-breedDEA (e.g., Ferrier and Lovell 1990, Ferrier, Grosskopf, Hayes, and Yaisawarng 1993, Curnrninsand Zi 1998). In contrast, virtuallyallrecentparametric SFA, TFA, and DFA studiesemployprices and examineeconomic efficiency.4 Thismeansthat inmostcases,efficiencyscoresgeneratedby DEA arenot filly comparableto ‘-tiose ofSFA,TFA, andDFA. We arguehere for the appropriatenessof economicefficiencyfor use in the re@ato~ analysisof financial institutions. Price data do exist for financial institutions, and cost minimization and profit maximizationarelikelyimportantbehavioralobjectives. Moreover,theeconomicinefficienciesoffinancial institutionsarebettermeasuresforre@ators touseinevaluatingthecostsandbenefitsto societyofvarious policiesthan are the technologicalinefficiencies,which do not put value weights on the inputs wasted or outputs not produced. Using technological efficiency in place of economic efficiency, thus neglecting allocativeefficiency,would likelyincrease the level of average efficiency(conditioni), affect the overall rankingsof financialinstitutionsandthe identificationof best practiceandworstpracticefms (conditions ii and iii), andmay reducethe consistencyof measuredefficiencywith the state of competitionin banking markets and with standard nonfrontier measures of performance (conditionsv and vi), which generally depend on economicreactions to market prices. Therefore,in all the empiricalapplicationsin this paper (includingDEA),weincorporatepricedataandemploytie conceptofeconomicoptimization. We choosecostminimizationoverprofitmaximizationbecauseitis amorecommonlyspecifiedand acceptedefficiencyconceptin the literature,and becausethere are problemsmeasuringoutput prices from the bank Call Reports during the first part of our sample @riorto 1984). Ideally,both cost and profit specificationswotid be employedand compared,but examiningour six consistencyconditionsover nine differentefficiencytechniquesalreadyseemto straintheverylimitsof spaceandtime. We recommendthat futureinvestigationsfollowthispath. 4 Themoveto economicefficiencyby theparametricstudiesmayhavebeenmotivatedin largepart by anotherconcern--theneedto accountformultipleoutputs. Unlikewith DEA,this cannotbe accomplished in a single parametric production function, but can be handled in cost or profit functions, which would normally include prices as arguments. However, with recent advances in distance fiction estimation, mdtiple outputscannowbehandledinproductionsettings.

9 Data Enve1opment Analvtis (DEA). Nonparametric approaches to measuring efficiency, represented here by DEA (but also includingthe Free Disposal Hull or FDH), use linear programming techniques. Inthe usual radialformsof DEA that arebased on technologicalefficiency,efficientfms are thoseforwhichnootherfm orlinearcombinationoffirmsproducesasmuchormoreofeveryoutput(given inputs)orusesaslittleorlessofeveryinput(givenoutputs). TheDEAefficientfrontieriscomposedofthese “ undominatedfms and the piecewiselinear segmentsthat connecttie set of input/outputcombinationsof these fins, yieldinga convexproductionpossibilitiesset.5 In the version of DEA we applyhere whichis based on economicefficiency,efficientfirms arethosewhichminimizethe costofproducingtheirobserved outputs giventhebest-practicetechnologyand inputprices.GAn obviousbenefitofDEA is that it doesnot requiretheexplicitspecificationof afunctionalformandsoimposesverylittlestructureonthe shapeofthe efficientfrontier. A potentialproblemof“self-identifiers”and“near-self-identifiers”may arisewhenDEAis applied. UndertheusualradialformsofDEA, eachfm canonlybecomparedto fms onthefrontierortheirlinear combinationswiththesameormoreofeveryoutput(giveninputs)orthesameorfewerofeveryinput(given outputs). Inaddition,otherconstraintsareoftenimposedonDEAproblemswhichrequirecomparabilitywith linear combinationsof other fins. Other constraints specified in financial institutionsresearch include qualitycontrols,such as the number of branchesor averagebank accountsize,or environmentalvariables, suchascontrolsforstateregulatoryenvironment. Theseotherconstraintspotentiallyapplyto~ theradial andcost-basedformsofDEA. Havingtomatchotherfirmsinsomanydimensionscanresultinfms being measuredashighlyefficientsolelybecauseno otherfirmsorfewotier fms (andtheirlinearcombinations) have comparablevaluesofinputs,outputs,orotherconstrainedvariables.7 Thatis, somefirmsmaybe selfidentifiedas 100°/0efficientnot becausethey dominateanyotherfins, but simplybecauseno other fms or linear combination of fms are comparable in so many dimensions. Similarly, other fms may be 5DEA presumesthat linearsubstitutionispossiblebetweenobservedinputcombinationson apiecewise linearfrontierwhileFDHpresumesthatnosubstitutionispossible. GIn applyingDEA, we followedproceduresoutlinedin F&e, Grosskopf, and Lovell(1994). Variable returnsto scalewerepermittedthroughuseofasidesummationrestrictioninthelinearprogram. 7Whilenew procedureshave been devisedto test for and limitextraneousspecificationof inputs and outputsorotherconstraints(e.g.,LovellandPastor 1997),theirapplicationisnotyetcommon.

10 measured as 100°/0efficientor nearly 100°/0efficientbecausethere are only a few other observationswith whichthey arecomparable. Theproblemof self-identifiersandnear-self-identifiersmost often ariseswhen there are asmallnumberof observationsrelativeto thenumberof inputs,outputs,andotherconstraints,so that alargeproportionoftheobservationsaredifficultto matchin alldimensions. Someempiricalevidence fromtheliteratureonthispointispresentedbelow. Our DEA applicationtriesto minimizethe self-identifierprobleminthreeways. First,weuseinput prices in a cost-basedDEA methodology. Intheusualradialinput-based(output-based)DEA applications, input mix (outputmix)isheldconstant,sofms withunusualinput(output)mixesmaybefoundtobe selfidentifiers or near-self-identifiers.We can compare any input mix in our applicationby combininginput pricesandquantitiesandcomparingtotalcosts,ratherthanhavingtocomparefms ineveryinputdimension. Second,we donot imposeanyextraconstraintsonourDEAproblem,sofms onlyhaveto minimizecosts relativeto otherfms orlinearcombinationsoffms producingthesameoutputbundle. Byspeci~g costs and by imposingno extra constraints,we can onlyhave self-identifiersor near-self-identifiersto the extent that the output bundles of somebanks cannotbe easilyreplicatedby linearcombinationsof otherbanks.s Third,weusearelativelylargenumberofobservationsrelativetothesmallnumberofconstraintsinourDEA problems, so that most fms willhave quitea few linearcombinationsof otherfms that are comparable. Specifically,we solvecostminimizinglinearprogrammingproblemswithdataon 683 observationsfor each singleyear(DEA-S),specifying4outputsandnootherconstraints. Wealsocombineall12yearsofdatainto a panel(DEA-P),wherethereferencesetisconstantovertheentireperiod,for atotalof 8196observations.9 One potentialproblemwithDEAthatwedonot try to solveisthat DEA usuallydoesnot allowfor random error due to measurementproblems associatedwith using accountingdata, good or bad luck that temporarilyraisesorlowersinputsoroutputs,orspecificationerrorsuchasexcludedinputsandoutputsand imposing the piecewiselinear shape on the frontier. Any random errors that do exist may be counted as differencesin efficiencyby DEA. Presumably,this wouldresult in loweraverageefficiency,as there will ‘The comparabilityproblemsforboth inputs andoutputscouldbe solvedby using aprofit-based DEA approach(e.g.,Ftie andWhittaker1996). 91nourempiricalapplicationofcost-basedDEA, noneofthebanksthatwereidentifiedastechnologically efficientwere found to be cost efficient. This suggeststhat includingprices ad accountingfor allocative inefficiencyhelpsamelioratethepotentialself-identifierproblem.

11 bemoredispersioninthedata,unlessthereissomeunusualstatisticalassociationbetweenrandomerror and “true” efficiency. This effect may be quite large, since the random error in a singleobservation on the efficient frontier will affect the measured efficiencyof all of the firms that are compared to any linear combinationonthefrontierinvolvingthisfm. ’” S~proachi (SFA}. Theparametricmethods-- SFA, TFA, andDFA --have a “-” disadvantagerelativeto the nonparametricmetiods ofhavingto imposemorestructureonthe shapeof the frontierby speci~ing afictional formforit. As notedabove,wechooseacostfunctionspecificationhere. However, an advantageoftheparametricmethodsisthat they allowfor randomerror, sothesemethodsare less likely to misidenti~ measurement error, transito~ differences in cost, or specification error as inefficiency. The primary challengein implementingthe parametric methods is determiningghow best to separaterandomerrorfrominefficiency,sinceneitherof themare observed. TheparametricmethodsSFA, TFA,andDFAdifferinthedistributionalassumptionsimposedto accomplishthisdisentanglement. SFA employsacomposederrormodelinwhichinefficienciesare assumedto followan asymmetric distribution,usually the half-normal,whilerandom errors are assumedto follow a symmetic distribution, usually the standard normal (Aigner, Lovell, and Schmidt 1977). That is, the error term horn the cost functionis givenby ~ = v + u, where v > 0 representsinefficiencyand followsa half-normaldistribution, md u represents random error and behaves according to a normal distribution. The reasoning is that inefficienciescannotsubtractfromcosts,andsomustbedrawnhorn atruncateddistribution,whereasrandom error can both add and subtract costs, and so may be drawn from a symmetricdistribution. Boti the inefficienciesp andtherandomerrorsu areassumedto be orthogonalto the inputprices,output quantities, andmy othercostfunctionregressorsspecified. Theefficiencyofeachfirmisbasedontheconditionalmean (ormode)ofinefficiencytermp,giventheresidualwhichisanestimateofthecomposederror~. Greene (1990) and others have arguedthat alternativedistributionsfor inefficiencymay be more appropriate than the half-normal,and the applicationof differentdistributionssometimesdo matter to the averageefficienciesfoundforfinancialinstitutions(e.g.,Yuengert 1993,Mester 1996,BergerandDeYoung l 10TherearesomeeffortstodealwithrandomerrorinDEAusingbootstrappingtogainstatisticalinference (e.g., SimarandWilson 1995,FerrierandHirschberg 1997)andchance-constrainedprogramminggto reduce the effects of noise (e.g., Land, Lovell. and There 1993). See Grosskopf (1996) for a survey of these approaches.

12 1997). We arguehere fiat ~ distributionalassumptionssimply imposedwitiout basis in fact are quite arbitraryandcotid leadto significanterrorinestimatingindividualfm efficiencies. Forexample,thehalfnormal assumptionon tie inefficienciesp imposesthat most of the firms are clusterednear fl.dlefficiency, but thereisnotheoreticalreasonwhyinefficienciescouldnotbemoreevenlydistributed,ordistributedclose to symmetricallylike the assumed distributionof the random error. In fact, prior studiesusing the DFA “-” approach (describedbelow)--whichimposesno shapeonthedistributionofinefficiencies-- suggested that the inefficienciesbehavedmore likesymmetricnormaldistributionsthan half-normals(Bauer and Hancock 1993,Berger 1993).11 Despite thesepotentialproblemswithmeasuringtie - of efficiency,onepositive aspectoftie SFAapproachisthat itwillalwaysm theefficienciesofthefms inthe sameorderastheircostfunction residuals,no matter whichspecificdistributionalassumptionsareimposed. That is, fms with lowercosts for agivensetofinputprices,outputquantities,andanyothercostfunctionregressorswillalwaysberanked as moreefficient,sincethe conditionalmean or modeof p (giventhe estimateof theresidual ~) is always increasinginthesizeoftheresidual. ThispropertyofSFAhasintuitiveappealforameasureofperformance forregulatorypurposes--afirmismeasuredashighintheefficiencyrankingsifitkeepscostsrelativelylow for its given exogenousconditions. This is likelyto prove helpful in meetingour consistencyconditions, whichareprimarilybasedonrankorderings. In our empirical application of SFA, we use the half-normal - normal assumptions on the inefficienciesand random error, sincethese are the most common assumptionsin the literature, and leave examinationofourconsistencyconditionsforotherdistributionsforfutureresearchefforts. SimilartoDEA- S and DEA-P, we apply SFA to both each singleyear of data separately(SFA-S) andto the 12-yearpanel as awhole(SFA-P). SFA-Sallowsthecoefficientsofthetranslogcostfunctionto varyovertime,whilethe SFA-Pholdsthe slopecoefficientsfixedovertime and allowsthecostfunctioninterceptsto vary overtime withchangesintechnology,re~ato~ environment,andthemacroeconomy. 1lToinvestigatetis issuefurther,wetestedthedistributionsofefficiencyscoresfi-omallninetechniques employedhereforsymmetryusingthenonparametrictestproposedbyD’Agostino,BalangerandD’Agostino (1990). Inallcases,thenullhypothesisofsymmetrywas rejected. ThisisnotsurprisingfortheSFAresults, sincean asymmetricdistributionwasimposedonthesescores,but theothersevenrejectionssuggestthatthe underlyingefficienciesmaynotbedistributedclosetosymmetricallyaswasfoundinthepriorstudies.

13 Thick Frontier ADnroach (TFA]. TFAusesthesamefunctionalformforthefi-ontiercostfiction as SFA,but is based on a regressionthat is estimatedusing onlytheostensiblybestperformers inthe data set --thoseinthelowestaveragecostquartilefortheirsizeclass.12Parameterestimatesfromthisestimation arethenusedtoobtainestimatesofbest-practicecostforallofthefirmsinthedataset(BergerandHumphrey 1991). Banksinthe lowestaveragecostquartileareassumedto haveabove-averageefficiencyandto form ‘-a“thickfrontier.” As itisusuallyimplemented,TFAassumesthatdeviationsfrompredictedpetiorrnancevalueswithin the highest and lowest performance quartiles of fms represent ordy random error, while deviations in predicted performance between the highest and lowest average-costquartilesrepresent only inefficiencies (aspecialcaseofcomposederror)plusexogenousdifferencesintheregressors. Measuredinefficienciesthus are embedded in the difference in predicted costs between the lowest and highest cost quartiles. This differencemayoccurineithertheinterceptsorintheslopeparameters. In most applications,TFA gives an estimate of efficiencydifferencesbetween the best and worst quartileto indicatethe generallevelof overallefficiency,but doesnot providepoint estimatesof efficiency for allindividualfins. Inourapplication,weneedto obtainefficiencyestimatesfor eachbank ineachtime period so that we can compare these estimatesto our other frontier efficiencymethods. This requires an adjustment. Thethickfrontierisestimatedfromdatalimitedtoonlythelowestcostquartileofbanksforeach size class (as is standard). A separate efficiencyterm for every bank (includingbanks not in the thick frontier) is calculatedusing a methodvery similarto tie DFA estimatesdescribedbelow. The estimated residuals for the entire sample are calculated and it is assumed tiat the inefficiency disturbances are ttncorrelatedwithtie regressors,so that a separateinterceptfor eachbank can be recoveredasthe mean of its residuals. The most efficient 1°/0of the sample (7 banks) are assumed to be ftdly efficient and their average residuals aretruncatedto be at the 1°/0point of the sampledistribution,and the efficiencyof each bank is determinedfrom the differencefrom the frontier in these averageresiduals. The TFA efficiency 12Banks are fwststratified into 8 asset size classes and their averagecost overthe entiretime period (measuredhereastotalcostperdollarofassets)iscomputed. Thosebanksineachsizeclasswiththelowest averagecostformthesubsetofthedatausedto estimatethethickfrontierfor eachyearseparatelyorfor all yearstogether. Thisensuresthat anequalnumberofbanksof allsizeclassesareincludedintheestimation.

14 estimates fromthepanel data set (TFA-P) arebased on oneset ofparameterestimatesoverthe entiretime period (correctedfor fwst-orderserialcorrelation),andthe TFA efficiencyestimatesfor eachyear (TFA-S) estimatethecosttiction parametersseparatelyforeachyear. AswasthecaseforSFA,thelevelsofefficiencygeneratedbyTFAarepotentiallysuspect,sincethey are based onrather arbitraryassumptions--thatthelowestaveragecostquartilewithineachsizeclassis an ‘-adequate“thickfrontier”ofefficientfirms,etc. Nevertheless,thereareagainreasonsforoptimismregarding the rank orderinu generated by TFA. Since the efficiencyorderings are determined by cost function residualsafter controllingfor inputprices, output quantities,and possiblyotherfactors,they have intuitive appeal,andarelikelytobeveryconsistentwiththeSFAestimatesandothermeasuresofperformance. Distribution-Free AI)Droach (DFA). DFA specifiesafunctionalformforthecostfunctionasdoes SFA and TFA, but DFA separatesinefficienciesfromrandomerror in a differentway. It doesnot impose a specificshapeonthedistributionofefficiency(asdoesSFA),nordoesitimposethatdeviationswithinone group of firms are all random error and deviationsbetween groups are all inefficiencies(as does TFA). Instead,DFAassumesthatthereisa“core”efficiencyoraverageefficiencyforeachfm thatisconstantover time, whilerandomerror tendsto averageout overtime (Schmidtand Sickles 1984,Berger 1993). Unlike the other approaches,a panel data set is required,andthereforeordypanelestimatesof efficiencyoverthe entiretimeintervalareavailable(DFA-P). Theseestimatesmaybe derivedusingthreedifferenttechniques. Thefist DFAtechnique,DFA-PWITHIN,isafixed-effectsmodelwhichestimatesinefficiencyfrom the value of a fro-specific dummy variable (derived by estimating with all the cost function variables measuredasdeviationsfromfro-specific means). Efficiencyisestimatedusingthedeviationfromthemost efficientfirm’s interceptterm. A singleset of parameters are obtained so inefficiencyis fixed over time. However,sinceinefficiencyis no longera separatelyspecifiedelementin a composederrorterm,we donot needanassumptionthatinefficiencyisuncorrelatedwiththeregressors(asinSFA)andweadjustforpossible fwst-orderserialcorrelation. The secondDFA technique,DFA-P GLS, appliesgeneralizedleast squaresto pmel data, obtainsa single set of parameters, assumes that bank inefficienciesare fixed over time,13and that inefficiencyis 13Thisassumptionisnot strictlynecessary. Cornwell,Schmidt,and Sickles(1990),Kumbhakar(1990), and Battese and Coelli(1992) generalizedthe approachto allowinefficienciesto vary overtime, but in a

15 uncorrelatedwiththeregressors. Inourcostfunction,whichisalsocorrectedforfwst-orderserialcorrelation, a separate interceptforeachfm is recoveredfi-omthepanelestimatesasthe averageresidualforthat fm overthe timeperiod. Thefm withthe smallestaverageresidualis presumedto bethe most efficientfirm andtheinefficiencyofalltheotherfms ismeasuredrelativetothisbenchmark. ThethirdDFAtechnique,DFA-PTRUNCATED,estimatesthecostfunctionseparatelyforeachyear. ‘-- The efficiencyestimatesarebased on the averageresidualsfor eachbank. Sincesomenoisemightalsobe persistentovertime,we followBerger(1993) andtruncatetheresidualsatboth theupper andlower IYoof thedistribution,thuslimitingtheeffectsofextremeaverageresidualsatbothends. As with the other efficiency approaches, there is concern that the levels of the DFA efficiency estimates may be influenced by the somewhat arbitrary assumptions. The measurement of the “core” efficiencymeansthat efficiencyvariationsovertimefor anindividualfm tendto be averagedoutwiththe randomerror. DFA alsoimplicitlyassumesthat inefficiencyistheonlytime-invariantfixedeffect. Iftiere are otherfactors that arepersistentlyaffectinga fnm’s coststhat are not includedin tie regressionmodel, suchasbeingin ahigh-crimelocation,thismaybe countedasinefficiency(althoughthiswotid affectallthe otherfrontierapproachesaswell). Nonetheless, similarto SFA and TFA approaches, DFA is intuitivelyappealing as a measure of economicperformancebecauseit is based on keepingcosts lowfor a givenset of outputs and inputprices over a longperiod of time and overmany changesin economicconditions. We thereforeexpectthe DFA efficiencyranks to behighlycorrelatedwith SFA and TFA ranks andothermeasuresofbank performance. III. ResultsfromEarlierEfficiencvComparisons Althoughthereis a largeliteratureon financialinstitutionefficiency,there is not muchinformation availableon ourconsistencyconditionsbecausemost studiesapplieda singleefficiencyapproachandthese conditionsarebest malyzed bycomparingtheapplicationofmultipleapproachesto asingledataset. Afew studiesdidcomparemultipletechniques,usuallyapplyingtwoefficiencymethodsto the samedata set. The comparisons of bank efficienciesusing more than one approach includeFerrier and Lovell(1990), Bauer, Berger, andHumphrey(1993),Hasan andHunter(1996),BergerandMester(1997),Eisenbeis,Ferner, and structuredmanner.

16 Kwan (1997), Resti (1997), and Berger and Hannan (forthcoming).” We briefly extie some of the evidencefromthesestudieshere.15 The studiesby Bauer,Berger,andHumphrey(1993), Hasan andHunter(1996), BergerandMester (1997), and Berger and Hannan (forthcoming)compared estimates using two or more of the parametric approaches. In most cases, these studiesfound that averageefficiencieswere comparable and reasonably ‘-consistentwith competitiveconditionsin the banking industry-- supportingconsistencyconditions(i) and (v)above--but therewereexceptions. Hasan andHunter(1996)foundSFAaverageefficiencyvaluesto be much higher than TFA, .81 versus .67, respectively, and Berger and Hannan (forthcoming)found SFA averageefficienciesof .92to be quiteabit higherthan the .70 averagefor DFA. Allofthesestudiesfound that the parametric approaches tended to rti the banks similarly and identi~ the same ones as highly efficientand inefficient-- supportingconsistencyconditions(ii)and (iii)--but againthere weredifferences of degree. Forexample,BergerandMester(1997)foundarank-ordercorrelationof .988betweenSFA and DFA efficiencies,but Bauer,Berger,and Humphrey(1993)foundthat thetwo methodsidentifiedthe same banksinthemostandleastefficient25Y0ofthebanks38Y0ofthetimeand46Y0ofthetime,respectively,not all that much higher than the 25°Acorrespondence that wotid be expected by chance alone. When consistencyconditions(iv)and(vi)werelookedatintheseandotherstudies,thelimitedevidencesuggested thattie parametricapproachesappearedtobeyieldefficienciesthatpersistedoverseveralyears(e.g.,Berger and Humphrey 1991,1992,Eisenbeis,Ferner, andKwan 1997,DeYoung 1997),andtheseefficiencieswere related in the expected way (although not always strongly) with standard, nofiontier measures of performancesuchasreturnon assets(e.g.,BergerandHumphrey 1991,BergerandMester 1997,Eisenbeis, Ferrier,andKwm 1997). Perhaps more interesting are the comparisons of bank efficiencies between nonparametric and parametricapproaches,whicharereallymuchmoredissimilarfromeachotherthantheparametricapproaches are from one another. DEA and SFAwere comparedby Ferner and Lovell(1990), Eisenbeis,Ferrier, and 14Afewfrontiermodelcomparisonshavealsobeenmadeusingdataforotherfinancialinstitutions,such as bti branches(Giokas 1991),insurancefu-ms(Fecher,Kessler,Perelman,and Pestieau 1993,Yuengert C 1993, s andZi 1998),mutualfunds(FerrierandPhilpot 1994),andFederalReserveoffices(Bauer andHancock1993). ‘5Someadditionalsummarydetailsmaybe foundinBergerandHumphrey(1997),

17 Kwan(1997),andResti(1997). Thesestudiesreportedfairlycloseaverageefficienciesgeneratedbythetwo approaches. However, this belies the potential problem that the levelsof efficiencyunder DEA may be sensitiveto “self-identifiers”or “near-self-identifiers”when there are too few observationsrelative to the number of constraintsin DEA. Thereis someempiricalevidencethatthis problemmayhaveoccurred. For example, Ferner and Lovell (1990) found that the averageefficiencylevel rose from 54°/0to 83°/0when ‘-constraints on number of branches and average accountsizes were addedto the model,keepingthe same numberofobservations. Sincethe averageefficiencyfor SFAwas 79°/0andthe averageefficiencyforDEA is somewherebetweenvery low (540A)and relativelyhigh (83Yo),the questionof whetherDEA and SFA yieldsimilardistributionsofefficiencythat areconsistentwithcompetitiveconditionsinthebankingindustry -- as in consistencyconditions (i) and (v) -- remains open.]c With regard to consistentrankings -- as in conditions(ii)and(iii)--therestits fromtheliteraturearecontradictory. Resti(1997)fomd veryhighrankordercorrelationsbetweenDEAandSFAof .73to .89,andEisenbeis,Ferrier,andKwan(1997)foundfairly high rank correlations ranging between .44 and .59, but Ferrier and Lovell (1990) found rank-order correlation of only .02, which was not significantly different from zero. With respect to consistency conditions(iv) and (vi),the very smallamountof evidencesuggestedconsistencyovertime, andvery low, butpositivecorrelationwithnonfrontiermeasuresofperformance(Eisenbeis,Ferrier,andKwan 1997).17 ‘cAdditionalempiricalevidenceon this questioncomesfi-omstudiesof bank branches,wherethere are often smallnumbersofobservationsemployedinDEA analyses. Forexample,severalDEA studiesofbank branches used 35 or fewer observations and large numbers of inputs and outputs and usually found most branches to be either 100%efficientor ve~ closeto it (Sherman and Gold 1985,Parkan 1987,Oral and YolahuI1990,VassiglouandGiokas 1990,Giokas 1991,andPastor 1993),whereasaDFAstudyfoundmuch loweraverageefficiencies(Berger,Leusner,andMingo 1997). 17There is also somemixed evidenceregardingthe consistencyof estimateswithinthe sametechnique appliedto the samedata set,wheresomeofthe assumptionsormethodsare altered. Someofthese studies foundstrongconsistency. Forexample,BergerandMester(1997)foundtheDFA efficiencyestimatesto be robust to most changes in specification, Maudos (1996) found very high rank-order correlations of efficienciesgeneratedusingdifferentdistributionalassumptionsontheinefficienciesunderSFA(.86to .99), andFerrier,Kerstens,andVandenEeckaut(1994)obtainedcorrelationsbetween.87 and .99when applying four different radial and nonradialDEA procedures. However,other studiesfound less consistency. For example, Berger and DeYoung (1997) found the average efficiencies to differ significantly when the specificationsof the inefficiencydistributionand thefunctionalform for the cost tiction under SFAwere altered, DeBorger, Ferrier, and Kerstens (forthcoming)comparedradial and nonradialtechnicalefficiency using input-basedandoutput-basedFDH andfoundrank correlationsto vary substantiallybetween .32 and .96, andDeYoung(1998)foundverydifferentidentificationofthebestandworstperformancegroupsunder TFA, dependingupon whetheraveragecostsversussuperviso~ ratingsofmanagement(theM in CAMEL)

18 Thus,theevidenceis quitelimitedandsometimescontradictoryontheextenttowhichtie efficiency approachespass ourconsistencyconditionsforuse inre~atory analysis,orwhichsubsetofthemmaypass and whichmayfail. Wethereforeproceedwith ourempiricalanalysisof thefourmajormethodsusingnine techniquesappliedto alargedata setofbanksoveran extendedperiodoftimeinorderaddto theevidence. IV. Data andSDecificationISSW ConsistentwiththespiritofCharnes,Cooper,andSueyoshi(1988),LearnerandLeonard(1983),and Learner(1994) --who arguedfor usingdiverseor extremeconditionsfor evaluatingmodels--we choosea long and turbulenttime period with many regulato~ changes and many changesin market conditionsfor evaluatingourconsistencyconditions.Ourdatasetiscomposedof 683U.S. banksoverthe 12-period1977- 88. All banks have over $100 millionin assets,comefrom branch-bankingstates, andwere in continuous operation overthe entireperiod. As a group,thebanks accountfor overtwo-thirdsof allassetsintheU.S. banking system. In addition,sinceall statesnow allowbranch banking,our resdts may be taken as fairly representativeofthebankingsystemasawhole.18 This timeperiodis oneofmanychangesintheU.S. bankingindustry. Duringthelate 1970s,rapid inflationand financialmarket innovationin the areas ofcash managementandmoneymarketmutualfunds expandedcompetitioninbank corporateandconsumermarketsfromless-regulatedfinancialintermediaries, who werenot subjectto restrictionson depositrates. In the early 1980s,depositinterestrates and account types at banks and savingsand loanswere substantiallydere~ated and bank chartersbecamemore freely issued,leadingto highercosts and ftier competitionamongfinancialinstitutions. As well, 20 of the 51 states(DistrictofColumbiacountedasastate)eitherrelaxedoreliminatedremaininggeographicrestrictions on branching within the state, and 43 of the 51 removed barriers to interstate banking through holding company acquisitionsduringthis period (Berger, Kashyap, and Scalise 1995, Table B6). In addition,the entry of foreign banks into the market for nonfarm, nofilnancial corporate debt during this period wasusedtodeterminethegroups. 18Iffailedbankshadbeenincludedinthedataset,itislikelythat ourefficiencieswouldhavebeenlower on average. Failingbanks and thrifts typicallyhave lowerthan averageefficiencylevelsthan otherbanks, althoughitisamatterofsomecontroversytheextenttowhichtheinefficienciescausethefailuresversushigh costs createdby dealingwithproblemloansjust beforefailurecausemeasuredefficiencyto be downwardly biased(e.g.,BergerandHumphrey1992,Cebenoyan,Coopeman, andRegister 1993,HermalinandWallace 1994,Barr, Seiford,md Siems1994,BergerandDeYoung1997).

19 dramaticallyreducedthemarket shareofU.S. banks and likelyreducedmarginson theloansthat domestic banks continuedto make. This time interval also witnessed a substantial amount of technological and financial innovation,starting with the substitution of ATMs for human tellers in the late 1970s, and the developmentand refinementof derivativecontracts and other products of financialengineeringduring the 1980s. As arestit of allofthesechanges,manybanksperformedwell,mmy performedpoorly,md atotal ‘-ofover800banksfailedoverthisperiod. Thus, theperiod 1977-88forU.S. banks is almostan idealintervalto determinehowthe different frontier modelsidentifi and measure bank efficiencyover a variety of extreme conditions. As well, it is undersuchextremeconditionsthatitismostimportantforregulatorstobeabletoevaluatetheeffectsoftheir policies Table1showsthemainvariablesemployedinthevariousfrontierefficiencyestimations. Wespeci~ the same four banking outputs md same four inputs in all of our frontiermodels,whetherestimatedfor a seriesof singleyears orpooledwithinapaneldata set. Theoutputs aredemanddeposits,real estateloans, commercial and industrial loans, and installmentloans, all measured in real dollar terms. Production of servicesinthese accountcategoriesis associatedwiththevastmajorityofbankingcosts. Thefourbanking inputs specified are labor, physical capital, small denominationtime and savings deposits, and purchased funds. Theparametricapproachesuseonlytheinputprices,whereasourcost-basedDEAtechniquesspeci~ both input quantitiesandprices.19In allcases,totalcosts --operatingexpensesfromthephysicalinputsof labor and capital, plus interestcosts from the financialinputs of time and savingsdeposits and purchased finds --areincludedtomeasuretotalcostefficiency Theoutputsandinputschosenarefairlybasicandstandard,althoughthereis considerablevariation withintheliterature,andmanystudiesaddotherbankoutputs(e.g.,off-balancesheetactivities),otherinputs (e.g., financial equity capital), other bank characteristics (e.g., nonperforming loans), and environmental factors (e.g., state incomegrowth)to the models. The specificationof the cost functionfor the parametric 19Theinputprices arenot directlyobserved,and so mustbe constructedfrom the availableinformation by dividingflowsof expendituresby stocks. Thepriceof laborequalssalariesandbenefitsdividedby the number of Ml time equivalentworkers. The price of physical capital is expenditureson equipment and premises dividedby thebook valueofphysicalassets, andthe pricesof timedeposits andpurchasedfunds aretheinterestexpensesonthesecategoriesdividedbythedollarsintheseaccounts. Theseprocedurescreate dataerrorsandlikelyaccountsomeofthesubstantialvariationinpricesshowninTable 1.

20 modelsisalsofairlybasicandstandard,atranslogcostmodelwithpartially-restrictedshareequations.20As above,wechoosethemoststandardspecificationsfromtheliterature,ad arepreventedbyspaceandtime constraintsfromtryingallof theinterestingvariationson ournineseparatemodelsinevaluatingoursix separateconsistencyconditions.21Werecommendthatfutureresearchtryto veri~ oroverturnourrestits withrobustnesschecksusingmoreandperhapsbetterspecificationsoftheoutputs,inputs,andfunctional “-” forms. . . V. ~on oftheConsistencvCon- The data presentedin Tables 2-6 and Figures 1and2 providedirectevidenceon our consistency conditionsforthenineefficiencytechniques,arrangedinorderoftheconditions. . . . . . . . Consistencv Condl~ (I) -- Comparisons -encv Dlstrlbutlons with Each Oti . A numberof distributionalchwacteristicsof the efficiencyscoresgeneratedby the nme efficiencytechniques are reported in Table 2. The mean efficiencyfrom the seven SFA, TFA, and DFA parametric models averaged .83 (with a mode of .84), whilemean efficiencyaveragedonly .30 (modeof .21) across the two nonparametric DEA models. The average standard deviationof efficiencyestimates from the parametric models(.06)waslessthanone-halfthatforthenonparametricmodels(.14). The levelandtimepattern ofmean efficiencyfor eachfrontiermethodoverour 12-yearperiod are displayed in Figure 1. By assumption, the three DFA-P panel methods estimate ordy a single “core efficiency”overtime, and soyieldflat linesby construction,but theirlevelscan stillbe comparedwith the othermethods. As shown,theparametricmethodsgenerallyyieldrelativelyhighmeanefficiencies,between about 80°/0and 90°/0,that arereasonablycloseto one anotherin termsof level,and donot varymuch over time evenwhen data for separateyears (S) are used. The ordysignificantexceptionis TFA-S, whichhas 20Thetotalcostfunctionwasjointlyestimatedwithn-1ofthecostshareequationswiththestandardcrossequation Shephard’sLemma restrictions on the slope parameters imposed. The intercepts of the share equationswere allowedto varyto incorporateallocativeinefficiencies. Berger(1993) foundthat efficiency estimates using no share equations, share equations like this with the intercepts free to vary, and Wly restrictedshareequationsgaveverysimilarefficiencyresults. 21Somerecentfrontierefficiencystudiesusemoregloballyflexiblefmctional forms,suchastheFourierflexiblespecification(e.g.,Spong,Sullivan,andDeYoung 1995,Berger,Cummm”s, andWeiss 1997,Berger and DeYoung 1997, Berger, Leusner, and Mingo 1997, Berger and Mester 1997, DeYoung, Hasan, and Kirchhoff1998).

21 mean efficiencyof 67.4°Aandvaries quitea bit overtime. It appearsthatusingonly25°Aofthe data from asingleyeartoestimatethecostfunctionsaddssignificantnoisetothemodel,butthatthisproblemissolved byusinginformationfromtheotheryears,asindicatedbytheTFA-Prestdts, Themoststrikingresuk fromTable2 andFigure 1ishowmuchlowertheefficienciesfromtheDEA ..approachesare. ThemeanefficienciesfromDEA-S andDEA-P aresubstantiallybelowtheefficienciesfrom all the parametric methods. This inconsistencybetween the distributionsof the DEA and the parametric distributionsofefficiencyisftier illustratedinFigure2,whichshowsthecumtiative distributionfunctions for the efficiencies from one panel parametric technique (DFA-P GLS) and one panel nonpararnetric technique(DEA-P).22ThisshowsthattherelativelylowmeanefficiencyfortheDEA methodsismanifested in lowefficienciesfor the greatmajorityof the bds. Thenonpararnetricmethodidentifiesabout 90°/0of the banks as having less than 30Y0efficiency, while the parametric method suggests a much closer correspondenceofefficiencyacrossobservations,withalmostallofthefms near900/0efficiency.23 These data suggesttiat the parametricmethods aregenerallyconsistentwith one anotherin terms ofthe distributionsof theefficienciesgenerated,yieldingrelativelyhighefficienciesforthevast majorityof fms and the nonparametricmethodsare generallymutuallyconsistent,yieldingrelativelylow efficiencies for most fins. The determinationof which set of methodsmaybe more usefi for regtdatoryanalysisor otheruses mustwait for evaluationofthe otherconsistencyconditions,partictiarly whichapproachesyield morerealisticorbelievableefficiencyestimates, -- Rank-OrderCorr&ons oftheEfficiencvDistr*. Although estimates of the levels of cost efficiencyfor the parametric and nonparametricfrontier methods are quite differentacrossbanks,it is stillpossiblethatthesemethodswillgeneratesimilar~ ofbanksbytheir efficiency scores across frontier methods. As discussed above, identi@ingthe rough ordering of which **ThemaximumvalueforDEA-P efficiencyshowninFigure2 is lessthan 1.00becauseitis the average efficiencyovertimeforeachbank,andnobankwasontheDEApanelfrontierineveryperiod. 231tis alsonotablethat the efficiencyscoresare generallynot very stronglyaffectedby the choiceof a single-yearversus panel method or other differencesin techniquewithin each approach,with the possible exceptionofTFA. ThenonpararnetricChi-squareandKolmogorov-Smimovtestscanbeusedtotestwhether a pair of samplesshareacommondistribution. Bothtestproceduresfailedto rejectthentdlhypothesesthat each of the pairs DEA-S and DEA-P, SFA-S and SFA-P, TFA-S and TFA-P, and DFA-P WITHIN and DFA-PGLSbelongtothesamepopulation.

22 financialinstitutionsaremoreefficientthan othersisusuallymore importantforre~ato~ policydecisions than measuringthelevelofefficiency,sothatre~ators candeterminewhetherre~atory-influenced events likemergersresdt in improvedorworsenedfinancialinstitutionfirmefficiency. Ifthemethodsdonotrank institutions similarly, then policy conclusionsmay be “fragile” and depend on which frontier efficiency approachisemployed. Table 3 contains Spearmanrank-order correlationcoefficientsshowinghow closethe rankings of banks are among each of the nine frontiermethodsusing the full sample of banks. The ranking for each method is based on the averageefficiencyvaluefor each bank overthe entire 12-yearperiod. It wouldbe expectedhat the rankingsamongallsevenof tie parametricmethodswodd be relativelyhigh, sinceallof these methodsessentiallyrank the banks by teasing efficienciesfrom random error in the residualsfrom similarlyspecifiedcostfictions. Indeed,the averagerank-ordercorrelationamongthese sevenmethodsis .756, and all of these correlations are statisticallysignificant at the IYolevel. We wodd also expect a relativelyhighrank correlationamongthe two nonparametricmethods,sincethey also generateefficiencies from essentiallythe samemodel. Again,this expectationisjustified, as therank-ordercorrelationbetween themisastatisticallysignificant.895. However, the data suggests that the DEA and the parametric techniques give only very weakly consistent rankings with each other. The average rank-order correlations between the parametric and nonpararnetricmethodsisonly.098. Tenofthefourteencorrelationsarepositiveandstatisticallysignificant, two arenegativeand statisticallysignificant,andtwo arenot statisticallysignificantlydifferentfrom zero.24 Thus, the DEA and tie parametric models cannot be relied upon to generallyrank the banks in the same order,andsomaygiveconflictingresultswhenevaluatingimportantre~atory questions. consistencv condition (iii) -- Identification of Best-Practice and Worst-Pract ice Firms. As discussed above,even if the methodsdo not alwaysrank the financialinstitutionssimilarly,theymay still be usefulfor someregulatorypurposesiftheyareconsistentinidenti~g whicharemostefficientmd least efficient institutions. The upper triangle of the matrix shown in Table 4 reports for each pair of fi-ontier 24ThenonparametricKruskal-Wallistestcanbeusedtotestwhetier mdtiple samplesaredrawnfromthe same population.Giventhe lack of consistencybetweenthe parametric andnonparametrictechniques,it is not surprisingthat aKruskal-Wallistestrejectedthenullhypothesisthattheninesetsofefficiencyscoresall weredrawnfromthesamedistribution.

23 efficiencytechniques,theproportionofbanksthat areidentifiedbyonetechniqueashavingefficiencyscores in thetop 25Y0that are alsoidentifiedinthetop quarterby theothertechnique. For example,ofthe banks identifiedas in the best-practice25°Aby DEA-S, 35.70/0of tiese samebankswere also identifiedas being in the top quarter by SFA-S. This number also describes the proportion of the best qutier of fms as ..identifiedby SFA-S that are also in the top 25°/0by DEA-S, sincethe number of banks in the top 25°/0is alwaysthe same (171 of 683 banks). Random chance alonewould yield an expected value of a 25.0°A correspondence,andthevalueof .357showninthetableisnot statisticallysignificantlydifferentfi-om.250. The sameanalysiswithrespectto the lowestefficiency25Y0ofbanks -- the“worst-practice”-- is shownin thelowertriangleofthetable. Table 4 tells essentiallythe same story as the rank-ordercorrelationsabove -- there is very good consistencyamongthe sevenparametric techniques,very good consistencybetweenthe two nonparametric techniques,but poorconsistencybetweentheparametricandnonparametricmethods. Withintheparametric techniques,the correspondenceof the best practice 25Y0of banks ranges from 49.lYoto 93.OYOi,s always statisticallysignificantlyhigherthan .250 atthe 1°Alevel,md averagesa 69.3°Acorrespondence. Similarly, amongthe SFA, TFA, andDFAmethods,thejoint identificationoftheworst practice25°/0ofbanks ranges from49.IYoto 89.5Y0,isalwaysstatisticallysignificantlydifferentfromrandomchance,andgivesanaverage correspondenceof 69.20A.Thetwo DEA methodsidentifi thebest- andworst-practicequarterofthebanks identically76.0°Aand 74.9°Aof the time. respectively,and both are statisticallysignificantlygreater than .250. In contrast,thecorrespondencesofthebest-practiceandworst-practicebanksbetweenthetwoDEA methodsandthesevenparametricmethodsgoesonlyashighas37.40A,andisbelowtherandomexpectation of 25°/0in severalcases. For best-practice,the averagecorrespondenceis 31.1°/0,and for worst-practice,it is 32.8°/0,and in no cases are the correspondencesstatistically significmtly different from .250. Thus, although the parametric methods tend to identi~ the same firms as efficient and inefficient and the nonparametic methods are also internallyconsistent in this regard, the two types of approaches are not consistentintheiridentificationofthebest-practiceandworst-practicefins. As arestit, re~atory policies targetedateitherefficientorinefficientfirmswouldhitdifferenttargets,dependinguponwhichsetoffrontier efficiencyapproacheswereusedtoframethepolicy.

24 consistencv Condition (iv) -- The Stabilitv of Measured Efficiency Over Time, As discussed above,to be usefil for re~ato~ policypurposes,it is importantthat the efficiencymeasuresdemonstrate reasonablestabilityovertime, and do not vary markedlyfrom oneyear to the next. Althoughsomebanks may marginallyimproveor worsen theirperformanceover short periodsof time, it is unlikelythat a very efficient bank in oneyem wodd becomevery inefficientthe next, ordyto return to high efficiencyin the followingyear. Consequently,measuredefficiencyby acceptableapproachesshotid yieldefficiencieswhich are fairly stable over time, and regulatory policies targeted specifically at either very efficient or very inefficientfirmsshouldstillhittheirmarksafternormalpolicymd implementationlags. We now determinetheyear-to-yearstabilityof the DEA, SFA, and TFA efficiencyestimatesover time. Thethree DFA efficiencymeasures are excludedfrom this part of the analysisbecausetheymeasure only “core” efficiencythat persists overthe entiretimeperiod, and so we perfectlystable by cons~ction. We calculatedthe Spearman rank-order correlationsfor each of the six time-varyingefficiencymeasures between each pair of years. That is, we computedtherank-ordercorrelationbetweenDEA-S efficiencyin each year i, i=1977,...,1987, and DEA-S efficiency in each year j, j=1978,...,1988, with j>i to avoid redundancy,andthenrepeatedthisprocessforthefiveothertechniques. These396correlationswerepositive and statisticallysignificantin allcases. To summarizethis largeamountof informationinthe most useful way, Table 5 presentsthe averagecorrelationsby the numberofyears apart. Each figurein the One-Year- Apart first columnreports for a singleefficiencymethod,the averageof the correlationsof efficienciesin 1977with 1978, 1978with 1979,..., 1987with 1988,anaverageof 11correlationsinall. Eachfigureinthe nextcolumnreportstheaverageof 10two-year-apartcorrelations,1977with 1979,1978with 1980,..., 1986 with 1988. In general,the n-year-apartfiguresare averagesof the 12- n correlationsbetweenefficiencies that men years awayfromeachother. It istheseaveragesthatwouldseemto bemostusefulforregulatory analysisthatmustforecasttheeffectsoftheirpoliciesonfms inthefuture. Forexample,ifitisthoughtthat the policy and implementationlags are likely to take 3 years to work, then the three-year-apart average correlationsmaygivethebestindicatorastowhetherthepolicywillhittheintendedtargetbanks. The correlation coefficients decline over time, but remain surprisingly high ad statistically significantoveralltheavailablelagsfor allofthemethodsexamined. Afierthreeyems,thecorrelationsare between54.7Y0and75.97., suggestingthat allthemethodsarestable. Afterelevenyears,alltheefficiencies

25 still have statisticallysignificantcorrelationsbetween 16.2°Aand 31.5°4. This suggests thatmanyofthe “worstpractice” and “best practice” banks tend to remain inefficientoreficient, respectively,overtime.25 All oftheDEA, SFA,andTFAmethodsshownseemtoindicatethisstabili~. Thisalsolendssomesupport to thebasic assumptionof stabilitythat underliesthe DFA approach. Importantly,thereislittledifference inthestabilityofefficiencybetweentheparametricandnonpararnetricmethods. Theonlynotabledifference “amongthetechniquesisthattheDEAmethodsgenerallyshowslightlymorestabilitythantheSFAandTFA methods. . . co nsismcv Condition(v) -- Consistencv ofEfficiencieswith Market Com~etitiveCondltlons. As shown above inTable 2 and Figures 1 and 2, the parametricmethodsgenerallyyieldmeanefficiencies between about SOY.and 90Y0,with thevastmajorityof firmshavingrelativelyhigh efficiency,whereasthe nonparametricmethodsyieldmeanefficienciesbetweenabout20°Ato 400A,with thevastmajori~offms having relatively low efficiency. It seemsfairly clear that the parametric approaches are generallymore consistentwith what are generallybelievedtobe the competitiveconditionsin the banking industry. The relativelyhigh efficienciesfor the vastmajorityof banksseemsconsistentwiti a reasonablycompetitive industryinlocalmarketsthatallowedentrybybranchbanking(recallthatallthebanksinoursamplecome frombrmch-bankingstates).Moreover,allofthesefms survivedbranchingcompetitionoveratleasta 12year period of economicturbulencein the industry,whichwouldbe difficultto achievefor fms that consumedmanymoreinputsthanthebestpracticebanks. Incontrast,theDEAresultthatthevastmajorityoffms havemeasuredefficiencyoflessthan30Y0 doesnotseemtobeconsistentwithcompetitiveconditionsinthisindustry.Onepotentialexplanationofthis finding is that DEA doesnottakeaccountof randomerrorastheparametric approachesdo. As discussed above, the dispersionfromrandomerrorwotid likelyresultin loweraverageefficiency. If thereareafew firmswith very “lucky” outcomes,the fms that are comparedto themmay haveverylow measured efficiencybyDEA,andthismayhaveoccurredhere,sincetheDEAefficienciesaresomuchlowerthanthose 25ThestabilityshowninTable5ismuchlongerthanwasreportedbyEisenbeis,Ferrier,andKwan(1997) for asampleoflargemultibankholdingcompaniesovertheperiod 1986-91,wherestabilitywasstatistically significanftoraboutthreeandahalfyears. DeYoung(1997)foundeffectivelyan“optimal”stabilityofabout 6 yearsforuseinDFA malysis thatstruckabalancebetweenthebenefitsandcostsoftheextrainformation fromaddingamarginalyearofdata.

26 generatedby theparametricmodelsandsomuchlowerthan arelikelytobe allowedbymarketforces. Note that this problem likely is not as serious as a general concernas it appearshere, sincemost prior nonparametricstudiesofU.S.banksfmdmuchhigherefficiencies,onaverage12percentagepointslower thantheparametricstudies,asopposedtotheaveragedifferentialofabout53percentagepointshere.Inpart, the DEAefficiencyscoresheremaybe lowerthanmostof thosefoundinthebd efficiencyliterature, “-becauseourcost-basedDEAmethodsarebasedoneconomicefficiencyr,atherthantechnologicaelfficiency asmostoftie priorDEAstudiesare. Asnotedabove,technologicaelfficiencyscoreswilltendtobehigher thaneconomicefficiencyscoresbecauseeconomicefficiencysetsahigherstandardthatincludesallocative efficiency.In addition,ourDEAefficiencyscoresmaybe lowerthanthosein mostotherDEAfinancial institutionstudiesbecauseof the stepswetaketo reducethe self-identifierproblem(usinginputprices, speci~ingnoextraconstraints,andhavingalargenumberofobservationsrelativetoconstraints). . . . ConsistencvUltlon (vI).- Consi@cy with S@dard No-r PerformanceMeasurw. As indicated above, efficiency measures should be positively correlated with nofiontier measures of performancegenerallyused byre@ators, financialinstitutionmmagers,andindustryconsdtmts. Positive rank-ordercorrelationswiththesemeasureswouldgiveassurancethatthefi-ontiermeasuresarenotsimply artificialproductsof the assumptionsmaderegardingtheunderlyingoptimizationconcept(technological efficiencyversuseconomicefficiency)t,heshapeoftheefficientfrontier,theexistenceofrandomerror,and anydistributionalassumptionsimposedontheinefficienciesandrandomerror. Asalsoindicatedabove,the correlationsbetweenaccurateefficiencymeasuresandtheaccountingratiosofperformancearenotexpected be closeto 1.00,sincethe accountingratios embodynot onlythe efficiencies,but also the effectsof differencesininputpricesandotherexogenousvariablesovermanagershavenolittleornocontrol. Table6 showsthecorrelationsbetweentheefficienciesgeneratedbytheninetechniquesandfour notiontier measuresofperformance.Boththeefficienciesandthemorestandardratiosareaveragedover timetoreducetheeffectsofnoise. Thestandardperformancemeasuresarethereturnonassets(ROA),the negativeofthetotaloperatingandinterestcostperdollarofassets(-TC/TA),thenegativeoftotalcostper dollarofrevenue(-TC/TR),andthenegativeof laboremployedperbankingoffice(-Labor/Branch).The negativesignsareplacedonthelastthreeratiosto simplythediscussion-- allfourmeasuresarepositive indicators of performance which should be positively correlated with fi-ontierefficiency. Thesefour

27 performancreatiosandothersimilarmeasuresarewhatb~ managersandconstitantsusetogenerallyassess theirperformanceandrankthemselvesagainsttheirpeerswithinthe industry. Thefirstthreestandard performancemeasuresareindicatorsofeconomicoptimizationintermsofbankcostsandrevenues,whereas -Labor/Branchisameasureoftechnologicaolptimization.2G TheresultsinTable6suggestthattheparametric-basedef~cienciesaregenerallyconsistentwiththe ‘-standardperformancemeasures,buttheDEA-basedefficienciesaremuchlessso. LookingfirstattheDEA columns,onlyfour of the eightcorrelationsbetweenthe DEAmeasuresandthe standardmeasuresare positiveandstatisticallysignificantateitherthe5Y0or IYosignificancelevel,whiletwoothersarenegative andstatisticallysignificantatthe5°/0level.Thepositivecorrelationsaremostlyfatilylow,about10°/0orless, withonlyonecorrelationapproaching20°/0.Thetwonegative,statisticallysignificantcorrelationssuggest thatfms measuredasefficientbyDEAgenerallyusemorelaborperbranch,whichisgenerallyconsidered tobeasignalofpoorproductivityatthebranchlevelbybankersandconstants. Overall,thesimpleaverage of theseeightrank-ordercorrelationsof.053suggeststhatDEAefficiencyis atbestweaklyrelatedto our bting industryindicatorsoffirmperformance. Forthesevenparametrictechniques,all28correlationsarepositive(althoughsomearebtiely so), 21arestatisticallysignificantlydifferentfromzeroatthe IYolevelandanotherissignificantattie 5Y0level. The TFA frontierefficiencyestimatesarethemostconsistentwiththestandardperformanceratios,withall eightcorrelationsstatisticallysignificant,and averaging.218. Theresultsfor SFAare similar,withan averagerank-ordercorrelationwiththeperformancemeasuresof.205,butthecorrelationsfor-Labor/Branch ratiomeonlystatisticallysignificantatthe10°/0significancelevel(10°/0levelnotshownintable). Forthe threeDFAtechniques,theaveragecorrelationis similar,.199,butthereis lessconsistencythantheother parametricmeasures,with4ofthe12correlationsbelow10Yoandnotstatisticallysignificant. Overall,theparametricapproaches,SFA,TFA,andDFA,arefairlyconsistentwiththe standard 2bTheremaybe difficdties withusingthe quantityof laborinthislastratio. Ithasbeenshownthatthe ratiooflaborto costshaschangedconsiderablyovertime(BergerandHumphrey1992). Ithasalsobeen suggestedthatbankholdingcompanieshavemovedmanyoftheiroperationsintoaffiliatesoutsidetheb~ itself,so thatthelaborinputis consumed,butit is notcountedbecauseit is employedelsewherein the holdingcompany(Berger,Kashyap,andScalise1995). Thisislikelymoreofaproblemwithmeasuresof productivitygrowththanefficiency,becauseit is a changeovertime,butit doesintroducenoiseintoour measure.Weincludeittohaveatleastonemeasureoftechnologicaelfficiency.

28 nonfrontiermeasuresof performance and are statistically significantandpositivelycorrelatedwiththese measuresinthevastmajorityofcases. Thisgivesconfidencethatthegeneralmethodofteasingefficiencies fromrandomerrorintheresidualsfromcostfunctionsdoesnotdoexcessiveharmtothedata. Incontrast, there is a much weakerrelationshipbetweenthe DEA nonparametricmethodsand these same fm performancemeasures.Evenwhenthecorrelationsarestatisticallysignificantt,hemagnitudesaregenerally “muchsmaller.Thismightoccurinpti becausethesemethodsmayinadvertentlycountasubstantialamount oftherandomerrorasdifferencesinefficiency. VI. Conclusions This paper specifies a set of six consistency conditionsthat frontierefficiencyapproachesto measuringtheperformanceoffinancialinstitutionsshouldmeettobemostuseti forre@ato~ purposes. Thef~stthreeconditions--thattheefficienciesgeneratedbytheseapproachesbeconsistentwitheachother intermsoftheirefficiencylevels,rankings,andidentificationofbestandworstfms --helpdeterminethe degreetowhichthedifferentapproachesareconsistentwitheachother. Thelatterthreeconditions--that theefficienciesareconsistentovertime,consistentwithcompetitiveconditionsinthemarket,ad consistent withstandardnonfiontiermeasuresof performance-- helpdeterminethedegreeto whichtheefficiencies generatedbythedifferentapproachesareconsistentwithrealityandarebelievable,whichisnecessaryfor theefficiencyestimatesto beuseful. Theseconsistencyconditionswouldlikelybehelpti forevaluating efficiencyapproachesforotherpurposesandforfirmsoutsideofthefinancialinstitutionsindus@aswell, andweencourageotherstotryapplyingtheseconditionselsewhere. Weevaluatetheextenttowhichallfourofthemainapproachestoestimatingfi-ontierefficiencyor X-efficiencymeettheseconsistencyconditions.We employmultipletechniqueswithineachof the four approaches,usingsingle-periodandpanelmethods,foratotalofnineefficiencytechniquesevaluated.To be surethat the applicationsare comparable,all ninetechniquesuse the sameefficiencyconcept(cost efficiency)t,ie samesampleofbanks,sametimeinterval,samespecificationsofinputsandoutputs,and(for theparametricmethods)thesamefunctionalform. Ourdatasetconsistsofapanelof683largeU.S.banks (assetsover$100million)thatwereinoperationovertheentire12-yearperiodfrom1977-88,andoperated instatesthatallowedbranchbanking.Thiswasaperiodofmanyregulato~changesandmanychangesin marketconditions,makingit an ahnostidealperiodto determinehowthe differentfrontierapproaches

29 identi~ and measurebankefficiencyovera varietyof extremeconditions. It is also under such extreme conditions thatit is mostimportant for regulators to be able to determinethe effects of their actionson efficiency. Ourfindingsyieldsomemixedevidenceregardingtheconsistencyofthefourmainapproaches-nonparametridcataenvelopmenatnalysis(DEA),andtheparameticstochasticfrontierapproach(SFA),thick frontierapproach(TFA),anddistribution-freeapproach(DFA). Withregmdto thef~stthreeconsistency conditions,thesedatasuggestthattheparametricmethodsaregenerallyconsistentwithoneanotier,andthe nonparametricmethodsare generallyconsistentwithoneanother,but the parametricandnonpmametic methodsarenotgenerallymutuallyconsistent.TheSFA,TFA,andDFAparametricapproachestendtoyield aboutthesamedistributionsofefficiency(conditioni),rankbanksinroughlythesameorder(conditionii), andidentifimostlythesamebanksas“bestpractice”and“worstpractice”(conditioniii). Whilethereisalso consistencywithinthe nonparametricDEAmethods,the parametricandnonparametricmethodsare not consistentwitheachotherinthesedimensions.TheDEAmethodsyieldmuchloweraverageefficiencies, rankthebanksdifferently,andidenti~thebestandworstbanksdifferentlyfromparametricmethods.These restits suggestthat theremay be “fragility”in drawingregulato~ policyconclusionsthat may differ accordingtowhetherDEAversustheparametricapproachesarespecified. Possible“tie-breakers”-- or conditionswhichmayhelpchoosewhetherthenonparametricversus parametricmethodsmightbe“better”--arewhethertheefficienciesdrawnfromthedifferentapproachesare consistentwithrealityandarebelievable.Allofthemethodsarefoundtobeconsistentovertime(condition iv), buttheparametricmethodsappearto be moreconsistentwithwhataregenerallybelievedto be the competitiveconditionsinbankingmarkets(conditionv),andalsomoreconsistentwithnonfrontiermeasures of bankperformancesuch as returnon assetsor variouscostratiosthat are oftenusedby regulators, managers,andconsultants(conditionvi). SFA,TFA,andDFAyieldrelativelyhighefficienciesforthevast majorityoffirms,consistentwiththestateofcompetitioninbankingmarkets,whereasDEAyieldsrelatively lowefficienciesformostfins, perhapsreflectingtheconfoundingofrandomerrorandinefficiencyinthis approachthatusuallydoesnotaccountforrandomerror. Inaddition,theparametricmeasuresaregenerally highlypositivelycorrelatedwiththestandardnonfrontierperformancemeasures,whereasDEAmeasuresme muchlessstronglyrelatedtotheseotherindicatorsoffm performance.

30 The data also show a highdegree of consistencywithintheparametricmethodsmd withinthe nonparametricmethods.Thistendstosuggestthatre@atoV policyconclusionsmaynotbegreatlyaffected bythechoiceof SFAversusTFAversusDFA,orbythechoicebetweenpanelandsingle-yeartechniques (withthepossibleexceptionofsingle-yearTFA). Rather,theonlychoicethatappearstomattergreatlyfor regulato~policyconsiderationsis thechoicebetweentheparametricandnonparametricmethods,at least forthisdatasetandthesetechniques. Wehastento addthatthesearetheresdts of a singlestudy,andnopolicyconclusionsshodd be drawnfromasingleevaluationoftheseconsistencyconditions.Ourresdts aregenerallysupportedbypast research,butthereis stillrelativelylittleevidenceontheconsistencyconditions,md it is unknownhow robusttheseresultsare. Forexample,ourfindingsofverylowefficiencyfortheDEAmodelisnotvery typicalandmayreflectourinclusionof allocativeinefficiencyorsomethingelseaboutourspecificationor sample,somorerobustnesschecksareneededusingalternativespecificationsmddatasources. Asafinalpolicyconclusiono,urresdts suggestthatwhenperformingre~ato~ analysis--or really anyotheranalysisthatdependson frontierefficiencymeasurement-- theuseofmtitipletechniquesand specificationsislikelytobehelpti. Ifthesixconsistencyconditionsaremetfortwoormoreapproaches, thenonecanbemorecotildentintheconclusionsdrawn.

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i Figure1:AverageEfficiencyoverTimefortheNineTechniques 1.000 0.900 0.800 x “ m . . . “ . “ x 0.700 O.m ~ S - S 0.500 ~~FA-S *TFA-P -DEA-S --.- DEA-P 0.400 +DFA-P WITHIN — DFA-GPLS ‘DFP-P TRUNCATED o.3m 0.200 0.100 0.000 1976 1978 1980 1982 1984 1986 1988

Figure 2: T C Di t D G a D E S 1.0000 0.9000 0.8000 0.7000 0.6000 % o .g 0.5000 .- = 0.4000 0.3000 0.2000 0.1000 t , , , # 0.0000 m , , a I 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 ProportionofObservations

Table1: DescriptiveStatisticsoftheAggregateData(683bankso 12years:1977-88)1 _m Standard Maximum Mean Deviation Value Value TotalCost 2 9 8 2 OutDuQt uantities: .. Demand 6785700.65 3400066.25 1 1 Deposits RealEstate 323196.21 1215954.37 0 3 L Installment 214890.11 637394.20 0 1 Loans Commercial 755611.30 3780350.97 1 6 Loans bgl.ltOu-: LaboP 1 3 12.00 7 PhysicalCapital 3 1 495.32 6 TimeDeposits 6 1 605.65 3 PurchasedFunds 1 6 2727.43 1 buut prices: Labor 24.07 5.30 8 1 PhysicalCapital 76.71 9.96 4 1 TimeDeposits3 0.05 0.02 0 0 Purchased 0.08 0.04 0 1 Funds’ ‘A financialdataareannualrealvaluesin1000’sof1988dollars,urdessotherwiseindicated. zN fi eq e 3Prices Ii i a i r

Table2:DescriptiveStatisticsoftie EfficiencyS T D S S T T D D D W G T 0 0 0 0 0 0 0 0 I Median I 0.348 0,187 I 0.883 I 0.889 I 0.673 I 0,890 I 0.861 I 0.936 I 0.780 H 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I I Maximum I.000 0.888 ] 1.000 I 1.000 I 0.955 I 1,000 I 1.000 I 1.000 I 1 I I I I I I I 0 0 0 0 0 0 0 0 1 1 I 1 1 I 1 2 - - - - - - - 1 2 2 1 1 1 3 0 6 6 6 6 6 6 6 6 N T ef a ca u y d f 6 b ( o a t n t t a b t a ef f e b o t e s p D D En A @ u aS ( y d ar s D ——— D En A @ u t e 1 P ( d ar s S ——— St F A ( a t c f p v f e S ( y d — S St F A ( f t c t p ( t t i c o t 1 P ( d T —— T F A ( a t c f p v f e S ( y d T ——— T F A ( f t c f p ( t t i c o t 1 P d D ——— D A @ f t c f p is ( t t i c W o t 1 P ( d a e f d f i b ( m in D — D A @ f t c f p is ( t t i c I G o t 1 P d c f s c a e a r f i b ( m in D ——— D A ( a t c f pis v f e y d b f TR t in c f i b o t 1 P ( d e u t a re a t t e v a s l e v (

Table3:S Rank-OrderCorrelationsA t E S C V T D D S S T T D D D W G T D 1 0 0 O O 0 - - 0 D 1 0 0 0 0 0 0 0 S 1 0 0 0 ( 0 0 S 1 0 0 0 o 0 T 1 0 0 0 0 T 1 0 0 0 D 1 0 0 W D 1 0 G D 1 TR N r t e n t r co t averaveefficiencies t i b o t a re a t co u t e e f t s y a q s * Co st si d f z t 5°/0level,@o-sided. **Conflationisstatisticallysi@lcantlydtierentf z t 1°Alevel,two-sided.

Table4:Correspondence“ P a “ P B a T @ t s “ p a l t s “ p D DEA-P SFA-S S T T D D D W G T E 0.760** 0.357 0 0 0 0 0 0 D , , , 0 0.351 0 0 0 0 0 0.368 --- D 1 1 I I I 0 0.368 0 0 0 0 0 0 S 0 0.316 0.877** 0 0 0 0 0 0.374 0.725** 0 0 0 0 0 0.368 0.778** 0.790** 0.854** 0.591** 0.632** 0.813** T D 0 0.374 0.491** 0 0 0 0 0 W D 0 0.345 0.532** 0 0 0 0 0 G D 0 0.357 0.895** 0 0 0 0 0 I TR I N E n t u-. t a t .p- b t a i o t h e s t M ef 2 b t a a i t ~ e 2 t o t E n t l t t p b t a i o t h e s t ~ ef 2 b t a a id t ~ e 2 t o t * Co st si d f 0 t 5 l t Co st si d f 0 t 1 l t

Table5:T Pe Ef T Co n E O T T F F S S E N T E Y Y Y Y Y Y Y Y Y Y Y A A A A A A A A A A A 0 0 0 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 0 0 0 0 T 0 0 0 0 0 0 0 0 0 0 0 T N E e t t t a t co t n a e f as e t w o 1 t s f e t n r t a 1 nc F e t a 9d c f t 3 a co - w 1 1 w 1 . 1 w 1 A 3 t i c t u t av a st si d f z t I l t T t D ef m a e h t t b t m o “ e t p o t e t p a a pe pe co

Table6:Ef Co w S No P M D D S S T T D D D W G T O 0 0 0 0 0 0 0 0 R O 0 0 0 0 0 0 0 0 - - 0 1 0 0 0 0 0 0 0 0 - - - 0 0 0 0 0 0 1 0 - B N r t e n t r co t a—ve—ra~eefficiencieasnd~ standarndofiontierperformance me t in b o t a r a t c u t e e f t s y a q si * Co st si d f z t 5 l t Co st s d h z t 1 l t — ROA = r a -TCITA z n t co a - —— n t co r -L q n n e p b o