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A comparative study of alternative methods for efficiency measurement with applications to the transportation… Yu, Chunyan 1995

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A Comparative Study of Alternative Methods for Efficiency MeasurementWith Applications to the Transportation IndustrybyCHUNYAN YUB.Eng. Northern Jiaotong University, 1982M.Sc. The University of British Columbia, 1986A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREME1Sr FOR THE DEGREE OFDOCTOR OF PHILOSOPHYmTHE FACULTY OF GRADUATE STUDIESCOMMERCE AND BUSINESS ADMINISTRATIONWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJanuary, 1995© Chunyan Yu, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)__________________________Department of________________The University of British ColumbiaVancouver, CanadaDate /DE-6 (2188)AbstractThis thesis is concerned with measuring and explaining the productive efficiency offirms or organizations. In particular, the study compares three alternative methods ofmeasuring efficiency, namely, the deterministic frontier method, the stochastic frontiermethod, and the data envelopment analysis method (DEA).The dissertation consists of two parts. In Part I, the relative merits of the threemethods are discussed and evaluated through a Monte Carlo study under certain knownconditions. The study focuses on the effects of exogenous variables on efficiency estimates.The results show that the stochastic frontier method generally produces better efficiencyestimates than the other two methods. The DEA, however, has a slight advantage in caseswhere there are weak input substitution and large variations in input variables. In Part II,the three methods are examined empirically through their applications to a panel of 19railways in OECD countries and a panel of 36 international airlines. Comparison of thethree sets of efficiency estimates confirms that on average the stochastic frontier methodyields higher efficiency estimates than the other two methods, as indicated by the MonteCarlo results. The efficiency estimates by the two parametric methods are highly correlated,whereas there are considerable differences between the DEA estimates and those from theparametric methods. This is also consistent with the Monte Carlo results. By comparing thealternative efficiency estimates in the two applications, it is found that there is lessdiscrepancy among the three sets of efficiency estimates in the airline case than in therailway case. This can be partly attributed to the fact that there are fewer variations in the11operating environments in the airline case than in the railway case.The simulation results in Part I provide some general guidelines regarding the relativemerits of the three alternative methods under certain known conditions. The two applicationsof the three methods in Part II serve as examples of how these three methods can be appliedto practical problems where no a priori knowledge of either the production technology northe efficiency profile exists. They illustrate some of the problems that may be encounteredin empirical applications.111TABLE OF CONTENTSAbstractTable of ContentsList of TablesList of FiguresAcknowledgementOverview11ivvuxixli1Part I. A Monte Carlo Comparison of Alternative Methods for Efficiency MeasurementChapter 1 Introduction1.1 Background1.2 Research Issues .1.3 Objective of Part I1.4 Organization of Part I.• . 1010131718Chapter 2 Literature Review of Comparative Studies on Efficiency Measurement2.1 Studies based on Empirical Data19192.2 Studies based on Simulated DataChapter 3 Concept of Efficiency3.1 Farrell’s Definition of Efficiency3.2 Efficiency and Production FunctionAlternative Methods for Efficiency MeasurementParametric Approaches4.1.1. Deterministic Frontier Methods4.1.2. Stochastic Frontier Methods4.2 Nonparametric Approaches4.2.1 The CCR Ratio4.2.2 The BCC model4.3 The Effects of Exogenous Variables4.4 SummaryChapter 5 Design of the Monte Carlo Experiments5.1 Specification of the “true11 production technology24Chapter 44.12931364040434857606971757878iv5.2 Determination of Sample Size and Number of Replications 825.3 Generation of Inputs, Exogenous Variables, and Error Terms 835.4 Generation of the outputs 855.5 Conduct of the experiments 855.6 Evaluation Criteria 91Chapter 6 The Results of Monte Carlo Experiments 936.1 The Effects of Sample Size 946.2 The Effects of Input Range 1006.3 The Effects of Noise Level 1076.4 The Effects of Exogenous Variables 1146.5 The Effects of Outliers 1296.6 The Effects of Underlying Production Technology 133Chapter 7 Summary of Part I 145Part II. Applications to the Transportation IndustryChapter 8 The Efficiency of Passenger Railway Systems . . . . 1528.1 Introduction 1528.2 Model Specification and the Data 1558.3 The Results 1588.3.1 The DEA Results 1598.3.2 The Deterministic Frontier Results 1708.3.3 The Stochastic Frontier Results 1748.4 Comparison of Alternative Efficiency Estimates . . 1788.5 Summary and Concluding Remarks 181Chapter 9 The Efficiency of International Airline Industry . . . 1849.1 Introduction 1849.2 Model Specification and the Data 1899.3 The Results 1919.3.1 The DEA Results 1929.3.2 The Deterministic Frontier Results 2029.3.3 The Stochastic Frontier Results 2069.4 Comparison of Alternative Efficiency Estimates . . 2119.5 Summary and Concluding Remarks 213Chapter 10 Summary and Conclusions 216Bibliography 224VAppendix A Sample Computer Code 236Appendix B Data in Monte Carlo Experiments 241Appendix C Railway Data 250Appendix D Airline Data 260viList of TablesTable6.1.1 The Effects6.1.2 The Effects6.1.3 The Effects6.1.4 The Effects6.2.1 The Effects6.2.2 The Effects6.2.3 The Effects6.2.4 The Effects6.2.5 The Effects6.3.1 The Effects6.3.2 The Effects6.3.3 The Effects6.3.4 The Effects6.3.5 The Effects6.4.1 The Effects6.4.2 The Effects6.4.3 The Effects6.4.4 The Effects6.4.5 The Effectsof Exogenous Variables:of Exogenous Variables:of Exogenous Variables:of Exogenous Variables:95969899101102104105106107108110112113114115116119120123of Sample Size: Means of Efficiency Estimatesof Sample Size: The Mean Absolute Deviationsof Sample Size: Correlationof Sample Size: Rank Correlationof Input Range: Input Variable Statisticsof Input Range: Means of Efficiency Estimatesof Input Range: The Mean Absolute Deviationsof Input Range: Correlationof Input Range: Rank Correlationof Noise: Noise Term Statisticsof Noise: Means of Efficiency Estimatesof Noise: The Mean Absolute Deviationsof Noise: Correlationof Noise: Rank CorrelationExogenous Variable Statisticsof Exogenous Variables: Means of Efficiency EstimatesThe Mean Absolute DeviationsCorrelationRank Correlation6.4.6 The Effects of Exogenous Variables: Mean Efficiency Estimates with Twostep modelvii6.4.7 The Effects of Exogenous Variables:: The Mean Absolute Deviations byTwo-step Model 1246.4.8 The Effects of Exogenous Variables: Correlation by Two-Step Model . . . 1266.4.9 The Effects of Exogenous Variables: Rank Correlation by Two-Step Model 1276.5.1 The Effects of Outliers: Mean Efficiency Estimates 1296.5.2 The Effects of Outliers: The Mean Absolute Deviations 1306.5.3 The Effects of Outliers: Correlation 1326.5.4 The Effects of Outliers: Rank Correlation 1326.6.1 The Effects of Underlying Production Technology: Mean EfficiencyEstimates 1356.6.2 The Effects of Underlying Production Technology: Mean AbsoluteDeviations 1376.6.3 The Effects of Underlying Production Technology: Correlation 1406.6.4 The Effects of Underlying Production Technology: Rank Correlation 1428.1 Railways Efficiency: Definition of Policy and Uncontrollable Variables . . . 1578.2 Railways Efficiency: DEA Gross Efficiency Index 1608.3 Railways Efficiency: Tobit Regression Results 1648.4 Railways Efficiency: Residual Efficiency Index 1698.5 Railways Efficiency: Deterministic Frontier Production Function 1728.6 Railways Efficiency: Efficiency Estimates by Deterministic Frontier 1738.7 Railways Efficiency: Stochastic Frontier Production Function 1768.8 Railways Efficiency: Efficiency Estimates by Stochastic Frontier 1778.9 Railways Efficiency Comparison: Mean Efficiency Estimates 1788.10 Railways Efficiency Comparison: Correlation and Rank Correlation 179viii9.1 Airline Efficiency:9.2 Airline Efficiency:9.3 Airline Efficiency:9.4 Airline Efficiency:9.5 Airline Efficiency:9.6 Airline Efficiency:9.7 Airline Efficiency:9.8 Airline Efficiency:9.9 Airline Efficiency:Frontier Method9.10 Airline Efficiency9.11 Airline Efficiency:9.12 Airline Efficiency:Frontier Method9.13 Airline Efficiency:190193195196200202203205206207209210211212242246246247248249251List of Exogenous VariablesDEA Gross Efficiency IndexMean DEA Gross Efficiency by RegionTobit Regression ResultsThe Residual Efficiency IndexMean Residual Efficiency Estimates by RegionDeterministic Frontier Production FunctionEfficiency Estimates by Deterministic Frontier MethodRegional Mean Efficiency Estimates by DeterministicStochastic Frontier Production FunctionEfficiency Estimates by Stochastic Frontier Method . .Regional Mean Efficiency Estimates by StochasticAlternative Mean Efficiency Estimates9.14 Airline Efficiency: Correlation and Rank Correlation CoefficientsB. 1 Summary Statistics for Variables in Monte Carlo ExperimentsB.2 Determination of Number of Replications: Deterministic Frontier MethodresultsB.3 Hypothesis Test for Deterministic Frontier MethodB.4 Determination of Number of Replications: DEA resultsB.5 Hypothesis Test for DEAB.6 Parameter Values of the Underlying Production TechonologiesC.1 Lists of RailwaysixC.2 Railway Charateristics 253C.3 Railway Route Length and Traffic Volume 254C.4 Railway Cost Recovery Indicators 255D. 1 List of Airlines 261D.2 Airline Characteristics 262D.3 Airline Financial Outputs 264xList of FiguresFigure3.1 Production Function as Frontier 303.2 Farrell’s Definition of Efficiency 323.3 Production Frontier and Inefficiency 384.1 Diagrammatic Interpretation of DEA 66xiAcknowledgementsI am most grateful to my supervisor, Professor Tae Oum, for all the time, guidance, andhelp he has given me during all those years, and for his effort of pushing me to finish thisdissertation. I am also grateful to the members of my supervisory committee, ProfessorsTrevor Heaver, Maurice Queyranne, and Ken White, for their time, advice, and helpfulcomments, and to Professor Bill Waters for reading and providing comments on the draft ofthis dissertation and for his generous help throughout my study at UBC.I would like to thank my husband, Li, for his patience, support and encouragementthroughout the course of my study, and our children, Jimmy and Kathy, for all the joy theyhave brought into my life. I would like to thank my parents for their constant support andencouragement. This dissertation is dedicated to them.I gratefully acknowledge the financial support from the Tangkakji Fellowship and the UPSFoundation Fellowship.xii1OverviewThis thesis is concerned with measuring and explaining the productive efficiency offirms or organizations. In particular, the study compares three alternative methods tomeasure technical efficiency in situations where firms operate under different operatingconditions and are subject to the effects of factors which are beyond managerial control.The dissertation consists of two parts. In Part I, a Monte Carlo study is conductedto compare the relative merits of three alternative methods, i.e. the deterministic frontiermethod, the stochastic frontier method, and the data envelopment analysis (DEA) method,in measuring efficiency. The study focuses on the effects of exogenous variables and outlierson efficiency estimates. In Part II, the empirical performance of the three methods isexamined through their applications to two cases in the transportation industry. The firstapplication is to the passenger railways - a small sample case where the services are mostlyprovided by highly regulated, nationalized firms. The effects of exogenous variables areexpected to be significant, and outliers might be present in the sample. The secondapplication is to the international airlines - a medium sample case where the firms operatein a fairly competitive environment, and they have access to essentially the same technologyavailable even though there is a high degree of diversity in size.Since Farrell (1957)’s work, the development and refinements of various methods formeasuring efficiency have progressed rapidly. However, there are limited studies whichcompare the relative merits of the different methods. Most of these studies compare resultsfrom the application of different methods to the same empirical data (such as Banker, Conradand Strass, 1986, Bjurek, Hjalmarsson and Forsund, 1990). The main motivation for theseOverview 2comparative analyses is the desire for the alternative methods to tell consistent stories aboutthe efficiency performance of the firms or Decision Making Units (DMUs) under study. Inthis regard, these studies are applying the “methodology cross-checking” principle.However, further insight into the relative performance of the various methods can beobtained by applying different methods to a mixture of efficient and inefficient observationsgenerated using a “known” production technology model. Using this approach one canevaluate these different methods not only relative to each other, but also relative to the“known” underlying model.There have been a few studies which use simulated data to evaluate the relativeperformance of different methods with the knowledge of the “true” production andinefficiency structures, such as Banker, Charnes, Cooper and Maindiratta (1988), Gong(1987), Li (1991), Gong and Sickles (1992), and Banker, Gadh and Gorr (1993). However,none of these comparative studies have addressed the issue of outliers and data errorsexplicitly, although one of the main critiques of DEA is its sensitivity to outliers and dataerrors. Improved knowledge in this aspect could help provide guidelines for the selectionof methodology in practical applications. Another issue that has not been addressed in theseprevious studies is the effect of exogenous factors on efficiency measures. In most practicalapplications, one would be most likely to have a group of firms (or decision making units)which operate under different conditions. The observed productive performance of a firmis the result of both its productive efficiency and its environment. In making efficiencycomparisons, one must separate the effects of the production environment and the effects ofthe productive efficiency. The role of exogenous variables in measuring the efficiency ofOverview 3transportation firms is especially important because transportation production and cost dependheavily on network and market characteristics of the firm. In some previous empiricalstudies, researchers have attempted to take into account the effects of exogenous variables.In general, there are two basic procedures to account for the effects of exogenous variables:(1) an one-step procedure which includes the exogenous variables directly in estimating theparametric frontier production functions, and (2) a two-step procedure which first estimatesthe relative “gross” efficiencies using only inputs and outputs, then analyzes the effects ofthe exogenous variables on the “gross” efficiency. It is necessary to examine how well eachof these methods incorporates the exogenous variables in order to provide helpful guidelinesin selecting an appropriate method for a particular practical problem.In Part I of this thesis, Monte Carlo experiments are carried out to examine therelative performance of the three alternative methods with respects to the effects of thesample size, the variations in input values, the noise level, the exogenous variables, the dataoutliers, and the different (underlying) production structures. The results show that theperformance of all three methods (in terms of correlations) improves with the sample size,but deteriorates sharply with the noise level. The variations in input ranges do not appearto have much effects on the performance of the alternative methods in the cases where inputsubstitution is over one, however, the performance of the two parametric methods is foundto fall noticeably in the cases where weak input substitution exists. The results also showthat the magnitude of exogenous variables does not appear to have any significant effects onthe performance of the one-step deterministic frontier method and the one-step stochasticfrontier method as long as the exogenous variables are correctly identified and accounted for.Overview 4However, for the two-step procedure, especially for the DEA and the deterministic frontiermethod, the effects of the exogenous variables are significant. The stochastic frontiermethod, as expected, is quite robust with respects to outliers. By comparing theirperformance in terms of mean efficiency estimates, the DEA method appears to be morerobust to the outliers than the deterministic frontier method, although both methods havesimilar performance in terms of correlations. For the parameter range considered in thisstudy, the performance of the stochastic frontier method does not rely much on the structureof the underlying production technology. However, it is found that the performance of thedeterministic frontier method is affected by the presence of input complementarity but notby the returns to scale. The performance of the DEA method, on the other hand, is foundto deteriorate when the returns to scale increases and/or input complementarity exists. It isfound in general that the stochastic frontier method surpasses the other two methods in allaspects examined.The Monte Carlo study in Part I is an extension to the works by Gong (1987), Li(1991), and Banker, Gadh and Gorr (1993). It provides some useful information regardingthe relative merits of the alternative methods in the presence of exogenous factors andoutliers (data errors) as well as in situations where input complementarity and non-constantreturns to scale exist.The applications of the three alternative methods to the railways and the internationalairlines in Part II of this thesis provide empirical evidence on the relative performance ofthese methods. The studies of efficiency performance of the sample railways and airlinesalso provide useful information on the effects of some policy variables and some otherOverview 5factors on the efficiency performance of the industries.There are some previous studies examining the relative performance of the railwaysacross different countries, however, none of these studies has examined the effects of publicsubsidies on the efficiency performance of the railways. In addition, there has been nostudy comparing the results from applications of the DEA and the parametric methods to therailway industry. Chapter 8 measures the productive efficiency of the railway systems in 19OECD countries for the period of 1978-89, identifies the effects of government interventionand subsidization on the productive efficiency of those railways, and compares the alternativeefficiency estimates from the three methods.The empirical results from the railway study show that railway systems with highdependence on public subsidies are significantly less efficient than similar railways with lessdependence on subsidies, and railways with high degree of managerial autonomy fromregulatory authority tend to achieve higher efficiency. Therefore, the institutional andregulatory framework for railway industry must squarely address the question of railways’managerial freedom, and subsidy policy must encourage railways to use normal marketmechanisms for improving their cost recovery and to use the subsidies only for improvingtheir services. The empirical results also indicate that efficiency measures may not bemeaningfully compared across railways without controlling for the variations in railways’operating and market environments. Comparison of the efficiency estimates from the threealternative methods confirms that on average the stochastic frontier method yields higherefficiency estimates than the other two methods, as indicated by the simulation results in PartI. The efficiency estimates by the two parametric methods are highly correlated althoughOverview 6their mean values are different. There are substantial differences between the efficiencyestimates from the DEA-TOBIT and those from the parametric methods even though thepolicy implications from all three methods are consistent. The main reason for thedifferences is that the DEA-TOBIT is a two-step procedure whereas the paramethc methodsutilize an one-step procedure. As shown by the Monte Carlo simulation study in Part I, theone-step procedure would produce different estimates from those using the two-stepprocedure. Further, the DEA-TOBIT considers two outputs, i.e. passenger kilometers andfreight tonne-kilometers, while the one-step parametric production functions consider onlyone output, i.e. total train kilometers.The efficiency performance of 36 international airlines during the period of 19801992 is examined and analyzed in Chapter 9. Previous studies on the performance ofairlines are mostly based on late 1970s and early 1980s data. There has been no comparativestudy on alternative efficiency estimates for international airlines. This study includes moreup-to-date data to evaluate recent changes in efficiency performance of the airlines, andcompares the alternative efficiency estimates. Some other factors which have been ignoredin the previous studies are also included in this study, such as incidental services. Incidentalservices refer to catering services, ground handling services and maintenance servicesperformed for other airlines, etc. These services could account for up to 30 percent of totaloperating revenues for some airlines1. To properly reflect the “total” output of an airlines,1 Good, Nadiri, Roller, and Sickles (1993) does consider the incidental services as oneof airlines’ outputs. However, their studies are limited to the four largest European airlinesand their U.S. counterparts. They attempt to control for the effects of some operatingcharacteristics, but ignore the government ownership variable.Overview 7the incidental services are treated as one of the outputs in this study.The efficiency of the 36 sample airlines are measured and compared, and the effectson efficiency of government ownership and technical progress are identified. The empiricalresults show that technological progress has improved the overall performance of theinternational aviation industry, especially for some of the major Asian carriers. The airlineswith majority government ownership are shown to be less efficient than other airlines withsimilar operating characteristics. The results also indicate that the effects of network andmarket environments should be controlled for in order to measure productive efficiencymeaningfully comparable across airlines.Comparison of results from the alternative methods illustrates that although there arenoticeable differences in the levels of efficiency, the overall pattern of the efficiencyestimates from the two parametric methods are essentially the same, and the results by theDEA-TOBIT are broadly similar to those from the parametric methods. The policyimplications from all three methods are consistent.By comparing the relative performance of the alternative methods in the railway andthe airline cases, it is noted that there is much less discrepancy and hence higher correlationsamong the alternative efficiency estimates in the airline case than in the railway case. Thisis partly due to the better data quality for the airlines. The airlines’ operating environmentshave less variations than those for the railways. In addition, the airline data set has a largersample size and is collected in a more consistent manner than the railway data set. Thehigher correlations under these circumstances are consistent with the results from the MonteCarlo experiments in Part I which shows that when the magnitude of the exogenous factorsOverview 8and/or the random noise level are high the efficiency estimates by the three alternativemethods may deviate considerably from the “true” efficiency. Thus, larger discrepanciesamong the alternative efficiency estimates are expected for the railway case. One empiricallesson from this study is that extra care is warranted when examining the data and theestimation results in situations where large variations in the operating environments exist.Although the application of the stochastic frontier method is expected to yield better resultsin general, computational and data problems, such as multiple outputs, collinearity amongthe inputs, and the lack of necessary price data, may leave Data Envelopment Analysis asthe only choice for certain empirical situations. There is no definite answer to the questionof which method is the best method, the answer relies on the particular situation in question.The Monte Carlo results in Part I provide some general guidelines on the relative merits ofthe three alternative methods under certain known conditions. The applications in Part II, onthe other hand, demonstrate some of the problems that may be encountered and also serveas examples of how to deal with the practical problems where there is no prior knowledgeof either the production technology nor the efficiency profile.Part IA Monte Carlo Comparison of Alternative Methodsfor Efficiency Measurement910Chapter 1Introduction1.1 BackgroundSince the pioneering work of Farrell (1957), much has been accomplished in the fieldof efficiency measurement in terms of both theoretical studies and empirical applications.Various methods, such as Data Envelopment Analysis (DEA) methods, stochastic frontiermethods, etc., have been developed to measure productive efficiency. Different methodsoften yield different efficiency rankings among the firms being considered, and may lead todifferent policy implications on how to improve the efficiency of a particular firm and of theoverall industry. Each method has its strengths and weaknesses. Knowledge of thesestrengths and weaknesses will help researchers and policy analysts to choose the most“suitable” method for a particular situation, and thus to make more accurate measurementsof efficiency. These in turn will help policy makers to make appropriate policy decisions.Therefore, it is important to study the relative merits of different methods in terms of theirabilities to reveal the structure of production technology and the nature and extent ofinefficiency under different conditions.The idea of productive efficiency proceeds from the concept of the productionfunction. The production function specifies the maximum quantities of realizable output,given any level of inputs and, for any given level of output, the minimum quantities of inputsneeded for producing these outputs. That is, the production function describes a boundary,or a frontier which sets a bound on the range of possible observations. Any productionChapter 1 Introduction 11situation, given the definition of the frontier, is deemed inefficient if the outputs and inputsare represented by a point below the production frontier. On the other hand, a situation willbe deemed efficient if the point is on the production frontier itself’. The amount by whicha production situation lies below its production frontier provides a measure of efficiency.Thus, the production function acts as a criterion, or yardstick, serving as a base for assessingefficiency. Efficiency measurement, therefore, involves two aspects. The first is to definethe production frontier and the second is to measure the efficiency using the productionfrontier as a yardstick.In practice, the true production frontier is not likely to be known, thus one needs toconstruct an empirical production frontier using observations on inputs and outputs. Thesubsequent measurements of efficiency vary according to the chosen functional form anderror distribution of the frontier as well as the size and composition of the sample.Therefore, the construction of the production frontier is crucial in evaluating the relativeperformance of a group of firms in an industry. There are many ways of estimating aproduction frontier. The various estimation methods can be classified into two basicapproaches, parametric and nonparametric, depending on whether or not the frontier can bespecified as a function with constant parameters. The parametric approach estimates aparametric representation of the production frontier using econometric techniques2. Within‘Note that the idea of considering the production function as a frontier may be appliedto the cost function too. Efficiency can then be defined on the basis of the minimum costfunction.2 Some studies use linear programming techniques to estimate a parametric frontierfunction, such as Aigner and Chu (1968) and FOrsund and Hjalmarsson (1979).Chapter 1 Introduction 12the parametric approach, the estimation of production frontiers has proceeded along twogeneral paths: (1) deterministic frontiers, which force all observations to be on or below thefrontier so that any deviation from the frontier is attributed to inefficiency; and (2) stochasticfrontiers, where a deviation from the frontier is decomposed into a random componentreflecting measurement error and statistical noise, and a component reflecting inefficiency.The nonparametric approach, represented by the Data Envelopment Analysis (DEA)method (Charnes, Cooper and Rhodes, 1978), uses mathematical programming techniquesto envelop observed input-output vectors as tightly as possible without requiring a priorspecification of functional forms for the production frontiers. That is, it does not assumethat the underlying technology “belongs to a certain class of functions of a specific functionalform which depend on a finite number of parameters, such as the well-known Cobb-Douglasfunctional form” (Diewert and Parkan, 1983). It only requires the assumption of convexityof the production possibility set and disposability of inputs and outputs. It employs apostulated minimum extrapolation from observed input-output data in the sample. Theefficiency of a Decision Making Unit (DMU) is measured relative to all other DMUs in thesample.In the literature on measuring the performance of production units, the terms ofproductivity and efficiency are often used interchangeably. In this dissertation, productivityand efficiency are considered as separate but related concepts. Productivity refers to theratio of outputs to inputs. Efficiency involves the comparison between observed and optimalvalues of outputs and inputs of a production unit, and it is a component of productivity.With this distinction, this dissertation will not consider the methodologies for measuringChapter 1 Introduction 13productivity such as total factor productivity (TFP).1.2 Research IssuesThe development and refinements of the various methods for measuring efficiencyhave progressed rapidly during the last two decades. However, there are limited studieswhich compare the relative performance of the different methods in efficiency measurement.Most of these studies compare results from the application of different methods to the sameempirical data (such as Banker, Conrad and Strass 1986, Bjurek, Hjalmarsson and Försund1990). The main motivation for these comparative analyses is the desire for the alternativemethods to tell consistent stories about the efficiency performance of the firms or DMUsunder study. In this regard, these studies are applying the “methodology cross-checking”principle. However, further insight into the relative performance of the various methodscould be obtained by applying these different methods to a mixture of efficient and inefficientobservations generated using a “known” production technology model. In this way, it willbe possible to compare these different methods in terms of their accomplishments not onlyrelative to each other, but also relative to the “known” underlying model. There have beena few studies which use simulated data to evaluate the relative performance of differentmethods with the knowledge of the “true” production and inefficiency structures. Banker,Chames, Cooper and Maindiratta (1988) compares the performance of the DEA method anda deterministic translog frontier function under two “known underlying technologies” -piecewise loglinear and translog, first for a sample of 500 randomly generated observationsand then for the set consisting of the first 100 of the 500 observations. Banker, Gadh andChapter 1 Introduction 14Gorr (1993) extends the work of Banker, Charnes, Cooper and Maindiratta (1988) byintroducing measurement errors and different efficiency distributions. Li (1991) investigatesthe relative performance of the DEA and the stochastic frontier method in estimating firmspecific efficiency levels with regards to three aspects: (1) different inefficiency profiles, (2)different returns to scale, and (3) different noise levels. Gong and Sickles (1992) comparesthe performance of the stochastic frontier method (with panel data) and the DEA for thecases in which: (1) the complexity and structures of an underlying technology differ, (2) therelative size of technical inefficiency to statistical noise in the stochastic components differs,(3) the forms of the true structure of technical inefficiency vary, (4) input levels andtechnical inefficiency are allowed to have an arbitrary degree of correlation.One issue that is not considered in the previous comparative studies is the effect ofexogenous factors on the efficiency measures. In any realistic situation, the quantity ofoutput produced by a firm is determined by a large number of factors other than inputs.Usually only a few of these factors are controlled by the firm and can be varied at thediscretion of the decision makers. Other factors, not necessarily less important, areexogenously determined and serve as operating constraints within which production decisionsare to be made. For example, in the transportation industry, the output level, as measuredby revenue passenger kilometers and revenue tonne kilometres, is affected by shipment size,load factor, spatial pattern of networks, the general economic condition, the populationdensity, and the extent of development of other transport modes etc.. Some of these factors,Chapter 1 Introduction 15such as shipment size and load factor, are partially controllable by the carriers3. However,other factors, such as economic condition and population density, are beyond the carriers’control. Consider a firm whose output level is fixed4, the firm must minimize its inputconsumption in producing the given level of output in order to be productively efficient.When the actual level of input consumption exceeds the optimal level of input requirementas specified by the production function, inefficiency occurs. However, this observed“inefficiency” could be attributed to “bad” weather condition, which is out of the firm’scontrol, or to unnecessary waste of certain inputs, which could be avoided by efficientproduction. Thus, when evaluating public policies and management strategies designed toimprove performance, it is essential to separate the effects of the production environmentfrom the effects of productive efficiency. In making efficiency comparisons, one must takeinto account differences in the firms’ environments, otherwise the efficiency measures wouldreflect not merely the differences in efficiency but also the degree to which the environmentof a particular firm is favourable or unfavourable. The role of exogenous variables inmeasuring the efficiency of transportation firms is especially important because transportationproduction and cost depend heavily on network and market characteristics of the firm.The parametric methods can incorporate these exogenous variables directly inestimating the frontier function (Lee and Schmidt, 1993). This is referred to as the one-stepprocedure in this dissertation. However, the DEA method cannot account for the effects ofSome of these decisions are subject to governmental control.“In many cases output is essentially determined by the market condition, governmentalcontrol etc., therefore it can be considered as fixed.Chapter 1 Introduction 16exogenous variables when computing the efficiency indices5. It is necessary to use a secondstage, such as regression analysis with the DEA index as dependent variable, to account forthe variables beyond managerial control and to identify the effects of these exogenousvariables on the observed efficiency (Ray, 1991, and Oum and Yu, 1994). This two-stepprocedure can also be used with the parametric methods as well (Bruning, 1991, and Loeb,1994). In practice, one may face situations with different degrees of variations in the firms’operating environments. For example, one may have a sample of firms operating in verydifferent environments in the transportation industry. Under such conditions, the two-stepprocedure might be more useful to the decision makers since it relates the exogenousvariables directly to the firms’s performance. On the other hand, the differences in firms’operating environments in a manufacturing industry are likely to be to a much lesser degree.The production of a particular sets of products is more likely to follow a clear engineeringdesign where there is better information about the “true” production technology. In suchcases, the one-step procedure might work better. These arguments are mostly speculations,as there is not yet obvious methodological and empirical support. Knowledge about therelative merits of the alternative methods under such different situations would help theBanker and Morey (1986) takes into account non-discretionary inputs or outputs whichare not subject to managerial control. However, in applying their model to incorporate theeffects of the exogenous variables, one has to treat these exogenous variables either asnondiscretionary inputs or outputs, thus affecting the resulting efficiency scores. Anotherproblem with the Banker and Morey (1986) model is that it requires an assumption of freedisposability of the nondiscretionary inputs. That is, the decision-maker is allowed to underutilize any available nondiscretionary inputs. This is not necessarily a realistic assumption.For example, weather condition is a nondiscretionary input in transportation, since thecarriers have no control over the amount of “bad” weather. Thus, they can have neithermore nor less than the exogenously determined level of “bad” weather.Chapter 1 Introduction 17researcher to select the appropriate method for a particular problem. To my knowledge,there have not been any studies comparing the relative performance of the various methodsin measuring efficiency while taking into account the effects of exogenous factors. Chapter6 of this thesis will attempt to investigate this issue using Monte Carlo experiments wherethe underlying production technology is known.Another issue that is not considered in the previous comparative studies is the effectof outliers. One of the main criticisms of the DEA is its sensitivity to outliers and dataerrors (Sexton, Silkman, and Hogan, 1986, and Bauer, 1990) because it is an extremalmethod. However, there is no solid evidence on how sensitive the DEA is to outliers anddata errors, and how it compares to the parametric methods in this regard. Except forMensah and Li (1993), none of the previous comparative studies addresses the issue ofoutliers explicitly. Mensah and Li (1993) attempts to evaluate the effects of outliers onefficiency estimates by the DEA and the translog models. They first remove one efficientobservation to reestimate the efficiencies, and then compare the new estimates with theoriginal ones. Their study is based on observed empirical data. However, furtherinvestigation with the knowledge of the “true” production and efficiency structure willimprove our understanding of the effects of outliers and data errors, and this will bediscussed in Chapter 6 of this thesis.1.3 Objective of Part IThe objective of Part I is to evaluate the DEA and the parametric frontier methodsin measuring efficiency under certain less than ideal conditions, particularly, when there areChapter 1 Introduction 18considerable differences among the firms in question, and to provide some guidelines for theselection of methodology in practical applications. This study focuses on the effects ofexogenous variables and the effects of outliers (and data errors) on efficiency measurement.To accomplish this objective, the Monte Carlo technique is used. It allows for control of theunderlying production technology and the “operating environments”. In particular, weevaluate the comparative performance of the alternative efficiency measurement methods interms of the sample size, different functional forms for the underlying productiontechnology, and in the presence of exogenous variables and outliers. The comparisons aremade among three most popularly used models: the deterministic frontier method, thestochastic frontier methods, and the DEA method. This study serves as an extension to theworks by Li (1991), Gong and Sickles (1992), and Banker, Gadh and Gorr (1993).1.4 Organization of Part IThe previous comparative studies are reviewed in Chapter 2. The review is limitedto comparative studies because of the large volume of theoretical and empirical literature inthe field of efficiency measurement. Chapter 3 presents the basic defmitions of efficiencyadopted in this dissertation. Chapter 4 then describes the various efficiency measurementmethods and reviews the related theoretical literature. Chapter 5 outlines the general designof the Monte Carlo experiments. Chapter 6 discusses the experimental results. Chapter 7summarizes the general results and provides some guidelines for practical applications.19Chapter 2Literature Review of Comparative StudiesOn Efficiency MeasurementThe objective of this chapter is to give an overview of previous comparative studieson efficiency measurements. The studies are grouped into two categories: studies based onempirical data and studies based on simulated data. The related literatures on thedevelopment of various methods are reviewed in Chapter 4 where we discuss the alternativemethodologies.2.1 Studies based on empirical dataMost of the comparative studies on efficiency measurement are based on specificempirical data, and are ad hoc in nature, van den Broeck, Førsund, Hjalmarsson andMeeusen (1980) compares the deterministic and stochastic methods for estimation of afrontier production function, based on cross-section data for 28 Swedish dairy plants for theperiod of 1964-73. They estimate the stochastic frontier using a maximum likelihoodestimator, and estimate the deterministic frontier using a combined linear programming andmaximum likelihood approach. Their comparison focuses on characteristics of the estimatedfrontier functions, such as input elasticity and scale elasticity, as well as their relationshipsto the average production function. They did not compare the firm specific efficiencymeasures.Corbo and de Melo (1986) uses the 1967 Chilean manufacturing census data toChapter 2 Literature Review 20compare four different methods: (1) the modified linear programming approach proposed byAigner and Chu (1968)’; (2) the deterministic statistical frontier by corrected ordinary leastsquares (COLS) method (Richmond, 1974); (3) the deterministic statistical frontier bymaximum likelihood estimation method (MLE) (Greene, 1980); and (4) the stochasticfrontier approach using COLS. They focus their comparison on the sensitivity of the resultsto model selection, including the selection of functional form for the average productionfunction, and the selection of error structure and specific characteristics of the distributionof error terms. The study finds that within the deterministic frontier methods, the linearprogramming and the deterministic statistical model yield highly correlated measures oftechnical efficiency. It also finds that the estimates are sensitive to the selection betweendeterministic statistical and stochastic formulations, but the choice of error structures has avery small impact on the measurement of inefficiency. The general result of their study isthat the different methods to measuring technical efficiency yield broadly similar results.Although they carry out a rather comprehensive comparison of the frontier functionapproaches in measuring efficiency, the nonparametric methods are not included in theiranalysis.Banker, Conrad and Strauss (1986) applies a translog frontier cost function and theData Envelopment Analysis method to a sample of North Carolina hospitals to examineinferences about the hospital cost and production functions, and to compare the results fromthe two different methods. The translog results indicate the presence of constant returns to‘Schmidt (1976) shows that, in a logarithmic mode, the linear programming procedureis equivalent to the deterministic statistical model with an exponential error structure.Chapter 2 Literature Review 21scale, while the DEA estimates suggest both increasing and decreasing returns to scale indifferent segments of the production functions. The DEA efficiency estimates are found tobe closely related to the degree of capacity utilization, but no such relationship is found forthe translog estimates. The study is concerned only with the deterministic case, it does notinclude stochastic frontier models.Levitt and Joyce (1987) uses both conventional regression analysis and the DEA toexamine the relative performance of police authorities in Great Britain. They find that theDEA efficiency rankings and those from the regression analysis, “while not identical, havea good deal in common”. Their conclusions are derived from a direct comparison(observation by observation) of the rankings from the two methods.Bjurek, Hjamarsson and Forsund (1990) compares three different methods ofefficiency estimation on the basis of a sample of local social insurance offices of the Swedishsocial insurance system. The three methods considered are: (1) a Cobb-Douglasdeterministic frontier production function, (2) a Quadratic deterministic frontier productionfunction, and (3) the Data Envelopment analysis. They find that the differences in theefficiency estimates between the methods are very small. Stochastic frontier models are notincluded in their comparison.Ferrier and Lovell (1990) compares two methods for estimating production economiesand efficiencies. One is the stochastic frontier cost function method, and the second is avariant of the DEA method which constructs a production frontier through a series of linearprograms. They find that the two methods yield very similar results regarding returns toscale, but dissimilar results regarding estimated costs due to inefficiencies. Their comparisonChapter 2 Literature Review 22emphasizes the amount of increased costs due to inefficiency, rather than the firm specificefficiency measures.Land, Shutler and Kirthisingha (1990) conducts a comparison between theconventional regression method and the DEA in measuring the efficiency of the post officesin the United Kingdom. With the regression analysis, they measure the relative efficiencyusing the residuals from the regressions and the “unit residuals” which are the regressionresiduals divided by predicted values. Comparisons are made over two separate data sets,the 1987 data set and 1989 data set. Simple correlation, Spearman rank correlation andKendall rank correlation are used to compare the efficiency rankings from regression againstthe DEA efficiency rankings. The results do not show a clear pattern between the efficiencyestimates from the two methods. They believe that the two methods are complementary inmeasuring efficiency and assisting management to improve efficiency.Forsund (1992) estimates a non-parametric frontier and a deterministic parametricfrontier to seek out the most efficient firm among the observed Norwegian Ferries. Thestudy finds that the methods yield very similar results for individual efficiency scores. Thedifferences are mainly to be found at the extreme ends of the size distributions. Theestimated scale properties are different from the two methods. The study focuses onsimilarities or dissimilarities of the distributions of the efficiency scores, and scaleproperties. The study does not consider stochastic frontier models.Sickles and Streitierwer (1992) estimates the firm specific levels of efficiency of theU.S interstate natural gas transmission industry using a stochastic frontier model and theDEA. They find that the levels and relative rankings of efficiencies from the two methodsChapter 2 Literature Review 23are often not the same. However both methods indicate a downward trend in firm efficiencyover the sample period.Fecher, Kessler, Perelman and Pestieau (1993) examines the productive performanceof the French insurance industry using the DEA method and a stochastic frontier modelbased on a Cobb-Douglas production function. They find that there is a high correlationbetween estimated efficiency measures from the two methods, and the frequency distributionsof firm specific efficiency measures from the two methods also show similar pattern.In summary, all the comparative studies based on empirical data have one thing incommon, that is, they are mainly motivated by the desire for the alternative methods to tellconsistent stories about the efficiency performance of the firms or DMUs under study. Inthis regard, these studies are applying the “methodology cross-checking” principle since mostempirical work is hindered by the lack of knowledge of the true structure of production andefficiency. From the discussions above, it is noted that in most cases the alternative methodsyield similar estimates regarding the efficiency rankings among the firms or DMUs underconsideration, but the level of estimated efficiencies often depends on the choice of methodused. Further insight into the relative performance of the various methods could be obtainedby applying these different methods to a mixture of efficient and inefficient observationsgenerated using a “known” production technology model. This approach would make itpossible not only to compare these different methods in terms of their accomplishmentsrelative to each other, but also make it possible to compare what each accomplishes relativeto the “known” underlying model. There have been a few studies which use simulated datato compare the relative performance of different methods with the knowledge of the “true”Chapter 2 Literature Review 24production and inefficiency structures.2.2 Studies based on simulated dataA number of studies compare the performance of different methods based onsimulated data, thus they can compare the results from each of the alternative methods withthe “true” production technology. Bowlin, Charnes, Cooper and Sherman (1985) usesartificially generated data for a group of hypothetical hospitals to test the DEA relative tooutput-input ratio analysis (including index numbers) and regression approaches (linearregressions of cost against outputs in particular) in terms of locating sources and estimatingamounts of inefficiency in the observed inputs and outputs. Their study finds that the DEAmethod generally performs better than regression analysis and ratio analysis in identifyingsources and amounts of inefficiency in all except a very few cases. They carry out thecomparison under one specific “true” production situation, and consider only a deterministiccase.Using the same hypothetical hospitals, Thanassoulis (1993) extends Bowlin, Charnes,Cooper and Sherman (1985) by comparing DEA and regression analysis, on estimates notonly of relative efficiencies but also of marginal input or output values and target levels.The study concludes that DEA outperforms regression analysis on the accuracy of estimatesbut regression analysis offers greater stability of accuracy. Again the results are based ona “specific” sample, and are limited to a deterministic case.Banker, Charnes, Cooper and Maindiratta (1988) uses simulated data to compare theestimates of technical efficiency, returns to scale, and rates of substitution obtained by theChapter 2 Literature Review 25DEA method and a deterministic translog frontier function. Under two known underlyingtechnologies -piecewise loglinear and translog, the comparison is carried out first for asample of 500 randomly generated observations (from an uniform distribution) and then forthe set consisting of the first 100 of the 500 observations. The study concludes that DEAcould approximate the known underlying technology better than the translog frontier functionand the accuracy of DEA estimates improves for the larger sample size. However, it isfound that the DEA method tends to misclassify inefficient “corner” points2 as efficient. Thestudy does not consider the stochastic models.Banker, Gadh and Gorr (1993) extends the work of Banker, Charnes, Cooper andMaindiratta (1988) by introducing measurement errors in comparing the performance of thecorrected ordinary least squares method and the data envelopment analysis method inestimating efficiency. The study considers two piece-wise Cobb-Douglas technologies as theunderlying true production technology, four efficiency distributions (two half-normal and twoexponential), and four different sample sizes from 25 to 200. They use the mean absolutedeviation of true versus estimated efficiencies as the performance criterion of the twoalternative estimation methods. The study finds that COLS performs better with half-normalefficiency distribution when sample size is over 50, but DEA performs better in other cases.However, both methods perform poorly with high measurement errors. They also find thatCOLS fails to decompose deviations into efficiency and noise components in almost all cases.Because of the failure of the COLS procedure, the study is in fact limited to deterministic2 A “corner” point refers to an observation with a very small or very large quantity forat least one of the inputs or the outputs.Chapter 2 Literature Review 26methods.Gong and Sickles (1989) investigates the relative performance of three stochasticfrontier estimators in terms of their sensitivity to the complexity and structure of theunderlying technology. The three estimators considered are: maximum likelihood,generalized least squares and dummy variables (or the within estimator). They conclude thatthe ability of the stochastic frontier models to estimate firm-level efficiency is quite sensitiveto the complexity and structure of the underlying production technology. The study isconcerned only with the stochastic frontier models.Gong and Sickles (1992) conducts a rather comprehensive comparison of theperformance of the stochastic frontier method (with panel data) and the DEA in estimatingfirm specific technical inefficiency for cases in which: (1) the complexity and structure ofthe underlying technology differ, (2) the relative size of technical inefficiency to statisticalnoise in the stochastic components differs, (3) the form of the true structure of technicalinefficiency varies, and (4) input levels and technical inefficiency are allowed to have anarbitrary degree of correlation. Their results indicate that, in terms of correlation and rankcorrelation coefficients between the estimated efficiency level and the “true” efficiency level,DEA is dominated by the stochastic frontier models in most cases except for the case wherethere is high correlation between inputs and technical efficiencies. However, the studyconsiders only the case of constant returns to scale, and does not address the issues ofexogenous variables and outliers explicitly.Li (1991) examines the effectiveness of the DEA and the stochastic frontier modelin estimating firm specific efficiency levels with regards to three aspects: (1) different “true”Chapter 2 Literature Review 27inefficiency patterns; (2) different returns to scale specifications; and (3) various levels ofrandom noise. The study finds that the general performance difference between the twomethods is quite small, and both methods perform quite well. The study assumes a Cobb-Douglas function with one output and two inputs as the underlying production technology.The results are based on a simulated data set of 100 observation with 5 replications for eachexperiment. The artificially generated data set simulates a situation where firms or DMUsare fairly homogenous, since the allowed variations of the input variables and the randomnoise are rather small. Input variables are generated from an uniform distribution over theintervals of (10,20) and (20,30), respectively, and random noise is assumed to have a meanof zero, standard deviations ranging from 0.01 to 0.09, which may explain the smallvariations among the experimental results.In summary, there is a limited number of comparative studies based on simulateddata where the “underlying” production technology and efficiency structure are known.From the discussions above, it is clear that a researcher can learn more about theperformance of the alternative methods in measuring efficiency by applying the methods tohypothetical production situations. Since the characteristics of these hypothetical productiontechnology and structures of efficiency are artificially controlled, the researcher is able toachieve in depth understanding of the abilities of the alternative methods in a particularaspect, such as the effects of the sample size, the effects of random noise, etc. Thesecomparative studies have provided useful information about the relative merits of differentmethods in terms of their abilities to reveal the structure of production technology and thenature and extent of inefficiency under different conditions. However, none of the studiesChapter 2 Literature Review 28has looked at the effects of exogenous variables on efficiency estimates. In any realisticsituation firms operate under different conditions, and one should not neglect differences inthese firms’ environments in making efficiency measurement and comparison. Therefore,it is important to consider the effects of the exogenous factors on the observed productiveperformance of the firms. There is another point that should be mentioned here, that is, theproblem of outliers and data errors. Most of the studies in efficiency measurement haverecognized the problem of outliers and data errors, especially for the deterministic methods.However, there has been no explicit discussions about the magnitude of the potential effectsof outliers and data errors. It would be interesting to see some solid evidence on thesepotential effects.29Chapter 3The Concept of EfficiencyThe purpose of this chapter is to discuss the basic concept of efficiency. To measurethe relative efficiency performance of firms or organizations in an industry, we need first todefine what we mean by efficient production.The concept of productive efficiency proceeds from the concept of productionfunction. A production function specifies the maximal output obtainable from a given inputvector. Thus a production function can be interpreted as a frontier, delineating the limits ofwhat a firm or an organization can achieve. Figure 3.1 shows the concept of productionfrontier for a one-input one-output situation. The frontier production function is denoted byf(x) where x denotes the input level. All points on or below it, such as B, C, or D, aredeemed realizable, hence can be observed, whereas points beyond it, such as E, are neitherrealizable nor (in the absence of noise or measurement errors) can be observed. Thisfrontier interpretation of production function leads to the concept of productive efficiency.A productive situation is inefficient if its output-input point lies below the frontier, such asD and C in Figure 3.1, since it does not do as well as it could with the same inputs, asspecified by the production function. On the other hand, a situation is efficient if the output-input point is on the production frontier itself, such as B in Figure 3.1. The “distance” anobserved production point deviates from the frontier provides a measure of efficiency for thecorresponding firm or organization. Therefore, the frontier production function acts as acriterion, or norm, serving as a base for assessing efficiency.In his path-breaking paper Farrell (1957) gives an explicit definition of efficiency, andChapter 3 The Concept of Efficiency 30yf(x)xFigure 3.1 A Production Function as a FrontierChapter 3 The Concept of Efficiency 31provides a computational framework for subsequent studies on productive efficiency. In theremainder of this chapter, we first look at Farrell’s definition of efficiency, then relate theefficiency measurement to production function.3.1 Farrell’s Definition of EfficiencyConsider the simple case of a firm producing a single output, y, from two inputs, x1and x2. Suppose the firm’s production function (frontier) may be written as yf(x7,x2). Ifwe also assume that the firm produces under conditions of constant returns to scale, then theproduction function is homogeneous of degree 1 and the equation of the frontier can bewritten as 1 =f(x/y,x2/y). This means that we can represent the technical possibilities opento the firm in terms of a unit isoquant such as ACA “A” shown in Figure 3.2. Potential or“maximal” performance is defined along ACA ‘A11, the frontier. No firm could produce ata point below ACA “A” because this would not be technically feasible. On the other hand,a firm producing at any points above ACA ‘A” uses more of at least one input than that isneeded.Suppose now that the available budget is represented by the line PP’ which is tangentto ACA ‘A “, all points along PP’ have the same cost, that is, PP’ is the current isocost line.Its slope reflects the ratio of input prices. Therefore, the firm producing at a point on PP’is using the “optimal” input proportion.According to Farrell (1957), a firm is technically efficient if it chooses an input mixChapter 3 The Concept of Efficiency 32XjyA”Figure 3.2 Farrell’s Definition of EfficiencyChapter 3 The Concept of Efficiency 33on the unit isoquant, and a firm is allocatively efficient (price-wise efficient) if the marginalrate of substitution between the two inputs is equal to the input price ratio. In other words,technical inefficiency is due to excessive input usage (given the level of output); allocativeinefficiency results from employing inputs in the wrong proportions. Allocative efficiencyis significant in that it emphasises that the frontier production per se is not sufficient tominimize costs. Full efficiency requires simultaneous technical and allocative efficiencywhich is obtained at A”. Firm C is technically efficient, but is allocatively inefficient, whilefirm E is allocatively efficient but technically inefficient.Suppose a firm is observed using (x10, x) to producey°, let D in Figure 3.2 represent(x/y0, x20/y0), which cannot lie below ACA ‘A” by defmition. At point D the ratio ofx1 andx2 is identical to the ratio at all points along OD. On this OD line, C is the most technicallyefficient point. That is, if one could have chosen the best technology but kept the inputproportions of D, one would have chosen the technology represented by C. Comparing theamount of input requirements at C and D can obviously form a basis for efficiencycomparisons. Farrell (1957) defines the technical efficiency of D as:TE=2! (3.1)ODTE measures the proportion of (x, x2°) actually necessary to produce y°. It is easy to see thatFarrell’s defmition of technical efficiency requires all inputs of an inefficient firm to beChapter 3 The Concept of Efficiency 34reduced by the same proportion1. TE equals 1 if the firm is on the frontier ACA A”. Asthe observed performance of D worsens, the distance between C and D increases so that thetechnical efficiency ratio falls towards zero. Likewise, as performance improves, theefficiency ratio rises in value to unity. Thus in general:OTE1 (3.2)As mentioned above, firm C is technically efficient, but it costs too much because theunit of output could be produced at the cost of OB by substituting x1 for x2. Therefore, firmC has an allocative inefficiency (price inefficiency) caused by non-optimal input proportions.Farrell (1957) defines the allocative efficiency (price efficiency) of C as:PE = (3.3)OCAs C moves closer to A”, PE rises towards one. That is, PE also lies between zero andunity. Since D has the same input proportion as C, it has an allocative inefficiency of thesame amount PE=OB/OC. It is noted that measuring allocative efficiency requiresinformation on input prices which is not required for measuring technical efficiency.Combining the technical and allocative efficiency measures gives an overall measureof the efficiency of D. Following Farrell (1957), the productive efficiency (or economicefficiency) of D is defined as:1 This could cause mis-identifying an inefficient firm as efficient when the isoquant isnot everywhere downward sloping. An alternative efficiency measure, the “Russellmeasure”, was proposed to remedy this problem (see Fare, Grosskopf and Lovell, 1985 formore discussions). However this “Russell Measure” is not very easy to implement.Chapter 3 The Concept of Efficiency 35EE = TE*PE = (3.4)ODIt should be noted that the above analysis does not consider the optimality of the levelof production, since the optimal scale of production is indeterminate in the case of constantreturns to scale. However, if the technology has nonconstant returns to scale, then the scaleof production will be optimal if and only if at the chosen level of output, price is equal tomarginal cost. A firm is said to be scale efficient if it chooses a profit maximizing level ofproduction. Extensions to Farrell’s original approach to accommodate non-constant returnsto scale technologies appear to be cumbersome and without much success (Farrell andFieldhouse, 1962, and Seitz, 1971).In empirical studies, one often relies only on information on output and inputquantities, thus cannot measure allocative inefficiency. In those cases, both firms A” andC are considered as efficient, while firm E is considered as inefficient even though it isallocatively efficient and achieves the same degree of productive (overall) efficiency as firmC. This dissertation focuses on the technical efficiency as defined by Farrell (1957) althoughthe allocative efficiency is a very important aspect of efficiency analysis. The reason fordoing this is that (1) technical efficiency is always desired in any circumstances, andunderstanding of the structure of technical efficiency may serve as a starting point for furtherstudy of a production process including allocative efficiency2;and (2) in many sectors of theeconomy such as the public service sector, information on prices is either unavailable or2 Allocative efficiency has been investigated in numerous studies, including Schmidt andLovell (1979), Kopp and Diewert (1982), Kumbhakar (1987, 1989), Kalirajan (1990).Chapter 3 The Concept of Efficiency 36unreliable, thus studies of allocative efficiency are difficult if not impossible, whileenhancing technical efficiency appears to be the main avenue to improve productionperformance.3.2 Efficiency and Production FunctionThe technical efficiency (TE) can be defined directly in terms of the productionfunction3. When inefficiency is present, the production function may be written as aninequality:y f(X) (3.5)where y is observed output level of firm j, X is a vector of inputs of firm j, and f(.) is theproduction function and has the interpretation of a frontier, or At inefficientoperations, potential output (Ymax) will exceed observed performance y. Hence, technicalinefficiency implies e= y — y, a residual in the production function, is negative. Topreserve the frontier interpretation of f(.), the c are always non-positive and truncated atzero such that deviations are only possible below the production frontier. This ensures thatobserved output cannot exceed potential and that the distribution of the residuals is one-sided.Cost functions can also be used to estimate technical efficiency as well as allocativeefficiency. The estimation of cost functions requires data on input prices which may notalways available. For reasons mentioned in section 3.1, this dissertation is focused ontechnical efficiency estimated in relation to production frontier. However, the basic conceptis equally applicable to cost functions.Chapter 3 The Concept of Efficiency 37The addition of the efficiency residuals “balances” the production function4:y =fx3) — e,O for alif (3.6)This could be illustrated by Figure 3.3 where f(x) represents the production frontier.Firm j produces output y using input xj, denoted by point D, which is less than the frontieroutput ym for input x. The difference between actual and optimal output,€,is negative andhence production at firm j is relatively inefficient. The degree of technical efficiency at firmj may be measured by the ratio of observed to optimal output defined by the productionfrontier, and written as:TE. = (3.7)J Ax)Notice that efficient production implies that observed and frontier attainments coincide andthat the residual equals zero thus TE equals to unity. Equation (3.7) is the econometricversion of the Farrell’s measure of technical efficiency. The most notable differencebetween the Farrell efficient isoquant and the frontier functions is the assumption of aspecific functional form5.Assume that all the firms have access to the same technology, the deviations ofpoints, B, C, and D in Figure 3.3 from the efficient frontier could be interpreted in twoways. The first interpretation is that some firms are more successful in utilizing the availableFor now it is assumed that there are no statistical noises in the observed productionperformance.Kopp (1981) discusses how Farrell’s concept of efficiency measurement related to thefrontier functions.Chapter 3 The Concept of Efficiency 38yA— f(x)Yrnax-*cYj // B* Ixi xFigure 3.3 Production Frontier and InefficiencyChapter 3 The Concept of Efficiency 39technology than others, that is, all deviation from the frontier is attributed to inefficiency.Equation (3.7) is based on this interpretation, and it corresponds to the deterministic frontiermethod to be discussed in Chapter 4. The second interpretation is that all firms face thesame technology up to a random factor that takes into account the effects on production ofmeasurement errors in the output and input variables and other random shocks outside thefirm’s control. Thus, the resulting production frontier is stochastic and the departure fromthis frontier reflects technical inefficiency. This is the rationale for the stochastic frontiermethod described in the following Chapter.40Chapter 4Alternative Methods for Efficiency MeasurementAs stated in Chapter 1, there are two aspects in efficiency measurement: estimationof the production frontier and measurement of efficiency relative to the frontier. Indiscussing the basic concept of efficiency in Chapter 3, it is implicitly assumed that theproduction frontier is known. In practice, however, one has only data - a set of observationscorresponding to achieved output levels and input consumption. Thus, the initial problemis to construct an empirical production function or frontier based on the observed data.Then, the relative efficiency of the firms or organizations in question can be measured inrelation to this empirical frontier. The purpose of this chapter, therefore, is to describe anumber of alternative methods used for the estimation of frontiers and the subsequentmeasurement of efficiency.There are many ways of estimating a production frontier. The various methods canbe classified according to the assumptions made on the form of the production frontier. Adistinction is often made between the parametric and non-parametric methods, depending onwhether or not the frontier function can be specified by a particular functional form withconstant parameters, such as the Cobb-Douglas function, or the translog function and etc.4.1 Parametric ApproachesAigner and Chu (1968) made the first attempt to impose a parametric functional formon production frontier within the framework provided by Farrell. They specify aChapter 4 Alternative Methods 41homogeneous Cobb-Douglas production frontier. In estimating this frontier productionfunction, they force all observations to be on or beneath the frontier through the introductionof an one-sided error term. Aigner and Chu’s work introduces explicitly the concept of“frontier” production function into the production theory. Although the frontier productionfunction seems clearly to be in accord with the theoretical definition of a production function,standard statistical techniques had been used to estimate “average” production functions priorto Aigner and Chu’s work.To estimate the parametric production frontiers, Aigner and Chu (1968) suggestsminimizing the sum of absolute residuals from the logarithm of the production function (alinear programming problem), or alternatively minimizing the sum of squared residuals (aquadratic programming problem), while constraining all residuals to be non-negative. If wewrite the production function for the j-th firm as = f(x, ) - Cj, here f(x,f3) is themaximum output obtainable from xj, a vector of (non-stochastic) inputs, y is the observedoutput, c is the one-sided (non-negative) error term, and f is an unknown parameter vectorto be estimated. Then Aigner and Chu’s method can be expressed, in mathematical terms,as following:m Iy—f(x,P)I, s.t. y i=1,...,n (4.1.1)orChapter 4 Alternative Methods 42nun s.t. yf(x1,[3), i=1,...,n (4.1.2)If f(x) is linear in /, then equation (4.1.1) is a linear programming problem and equation(4.1.2) is a quadratic programming problem. This approach assumes all deviations from theproduction frontier to be the result of technical inefficiency and thereby constrains theresiduals to be of one sign. Their measure of technical efficiency can be specified asy1 I f(x3). With such mathematical programming techniques the choice of the functionalform for the production function has a strong influence on the conclusions reached about thedegree of technical inefficiency. This is clear from the consideration that the number ofobserved units that appear to be fully efficient technically is generally only as large as thenumber of parameters of the production function to be estimated. Another problem with thisapproach is that the estimated frontier is supported by a subset of the data and thus issensitive to outliers and data errors. This leads to the development of so-called“probabilistic” frontiers by Timmer (1971) which proposes a modified linear programmingproblem allowing a certain percentage (arbitrarily determined) of observations to lie abovethe frontier. The third problem with this mathematical programming approach is that theestimators lack identifiable statistical properties. This third problem could be solved bymaking further distributional assumptions about the one-sided errors and estimating theproduction frontier using statistical methods.Afriat (1972) was the first to impose a statistical assumption on the one-sided errors,Chapter 4 Alternative Methods 43and to introduce the concept of “distribution of technical inefficiency”, which makesstatistical estimation of frontier functions possible. Since Afriat’s work most of the studiesof the parametric frontiers have been focused on the statistical approach. In fact, Schmidt(1976) shows that if is exponential, then Aigner and Chu’s linear programming procedureis maximum likelihood, while their quadratic programming procedure is maximum likelihoodif c is half-normal. Therefore, this dissertation considers only the parametric frontier modelsusing econometric/statistical techniques. For more discussions on the various parametricfrontier models, readers are referred to the survey papers by Forsund, Lovell and Schmidt(1980), Lovell and Schmidt (1988), Bauer (1990), and Greene (1993).Within the parametric approach, a distinction is made between two different methods:the deterministic frontier method and the stochastic frontier method. The main differencebetween the two methods is that the deterministic method attributes all deviations from thefrontier to inefficiency while the stochastic method distinguishes the deviations into a randomcomponent capturing statistical noise and an inefficiency component. These two methodsare discussed in the following subsections.4.1.1 Deterministic Frontier MethodsThe deterministic frontier methods assume that the discrepancies between theestimated frontier function and the observed production situations exclusively capturetechnical inefficiencies, that is, the discrepancies are one-sided. In estimating thedeterministic frontiers, all observations are forced to be on or below the frontier so that alldeviation from the frontier is attributed to inefficiency. A basic model of the deterministicChapter 4 Alternative Methods 44frontier is specified as:y =f(x)e_1’ (4.1.3)where y is observed output, f(x) is the production frontier, and uO thus 0 e -u 1which captures the degree of inefficiency. This frontier function may be estimated by eitherthe maximum likelihood method (MLE) or the “corrected” ordinary least squares method(COLS).The maximum likelihood method was first suggested by Afriat (1972) and first usedby Greene (1980a) and Stevenson (1980). MLE requires the specification of a particulardistribution for the one-sided residual u, and it is then implemented by estimating all of theproduction parameters and the parameters of the distribution of u. The MLE frontierenvelops all observations, and the MLE residuals can then be used to provide a measure ofestimated efficiency:e. = —— = e (4.1.4)‘ fix)It should be noted that the choice of a distribution for u is important since the maximumlikelihood estimates depend on it in a fundamental way - different distributional assumptionslead to different estimates. There do not appear to be good a priori arguments for anyparticular distribution. Afriat (1972) assumes a two-parameter beta distribution for e” (aChapter 4 Alternative Methods 45gamma distribution for u). Greene (1980a) shows that a gamma distribution for u satisfiesthe conditions for the maximum likelihood estimators to be consistent and asymptoticallyefficient. Other distributions have also been assumed in the literature, such as the half-normal and exponential distributions (Aigner, Lovell, and Schmidt, 1977), truncated normaldistribution (Stevenson, 1980), and the two parameter gamma distribution by Greene (1990).Another way to estimate the parameters of the frontier function is by the correctedordinary least squares method (COLS). COLS was first proposed by Richmond (1974) andimproved by Greene (1980a,b). COLS first estimates the technology parameters of equation(4.1.3) by OLS, which gives unbiased and consistent estimates of the parameters except forthe constant term in log-linear form. The OLS intercept can be corrected by either of twoways. The first way1 to “correct” the OLS intercept requires the assumption of a specificfunctional form for the one-sided error term u. The OLS intercept is corrected by shiftingit up by adding the sample mean of u which can be estimated consistently from the secondor higher moments of the OLS residuals. The OLS residuals are consequently modified inthe opposite direction. There is no guarantee that this technique shifts the estimated interceptup far enough to cover all the observations, and if an observation has a sufficiently largepositive OLS residual, it is possible that e > 1.A second way to “correct” the OLS intercept is simply shifting it upward until allcorrected residuals are nonpositive and at least one is zero. With this method there is noneed to assume a specific distribution function for the one-sided error term, one needs only1 This technique is termed as “modified” ordinary least squares by Lovell (1993).Chapter 4 Alternative Methods 46assume that u is independently and identically distributed from an unspecified one-sideddistribution. This dissertation adopts this method in the estimation of the deterministicfrontier models because it is the easiest one to estimate and less restrictive.The basic procedure of the second variant of COLS is as follows: first, one estimatesan average production function through OLS estimation, then the constant term is “corrected”by shifting it up until no residual is positive and at least one is zero. This correctedproduction function is considered as the production frontier, and used as reference point inefficiency measurement. This method provides a consistent estimate of all the parametersof the frontier function. The efficiency is measured by the ratio of the actual output valueY divided by the “optimal” output value YJF obtained from the estimated production frontier,given a certain level of input. The firm (observation) which has the highest positive residualfrom the OLS regression is by definition 100% efficient.If we take the logarithm of both sides of equation (4.1.3), then an OLS estimationgives the following equation:lily. Infix)+ .(4.1.5)where cx is a vector of regression coefficients. Letting E = Max(s), one obtains thefollowing frontier function:hi F = fl) + E (4.1.6)Chapter 4 Alternative Methods 47which allows for measuring the efficiency of the firms (observations) as:F c.-E (4.1.7)e=yJy =e 1The deterministic frontier method has two major drawbacks. The first one is that thismethod is entirely deterministic with no allowance for noise, measurement error and the like,thus the estimated frontier depends on a small number of observations that might beinaccurately measured or otherwise abnormal. That is, it tends to be sensitive to outliers anddata errors. The second drawback is its inability to deal easily with multiple outputs unlessthe dual cost frontier can be estimated directly2.The deterministic frontier methods assume that all firms share a common productionfrontier, and all variations in firm performance are attributed to variations in firmsefficiencies relative to this common frontier. This assumption ignores the very realpossibility that a firm’s performance may be affected by factors entirely outside its control,such as bad weather, labour dispute, and so on, as well as by factors under its control(inefficiency). To distinguish the effects of “noise”, including exogenous random shocks,measurement errors, misspecification of production functions, and so on, from inefficiency,Aigner, Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977) introduce thestochastic frontier model which is discussed in the following section.2 All deviation from the cost frontier is attributed to cost inefficiency. Kopp and Diewert(1982) and Zieschang (1983) show how to decompose cost efficiency into its technical andallocative components.Chapter 4 Alternative Methods 484.1.2 Stochastic Frontier MethodsThe basic idea behind the stochastic frontier method is that the deviation from thefrontier is composed of two parts. A symmetric component permits random variation of thefrontier across firms and captures the effects of measurement error, statistical “noise”, andexogenous random shocks outside the firm’s control. An one-sided component captures theeffects of inefficiency. This method was first proposed by Aigner, Lovell and Schmidt(1977) and Meeusen and van den Broeck (1977), and has been extended by Jondrow, Lovell,Materov, and Schmidt (1982), and Battese and Coelli (1991) among others. Specificdistributional assumptions about the disturbance terms must be made in order to obtainestimates of individual firm efficiencies3. The statistical noise is generally assumed to beidentically independently distributed (iid) normal, while a number of distributions have beenassumed for the one-sided (inefficiency) term. Aigner, Lovell and Schmidt (1977) proposeshalf-normal and exponential distributions which have been widely used. Stevenson (1980)proposes a truncated normal distribution, while Greene (1990) proposes a two-parameterGamma distribution4. Further, other distributions could be constructed following Greene’s(1990) methodology.The basic stochastic frontier model is given by:When panel data are available, estimates of the inefficiency disturbances can beobtained without assuming a particular distribution for these terms (Schmidt and Sickles,1984). However, one must specify how efficiency changes over time instead. Most of thesestudies assume that inefficiency is time-invariant.‘ There are a number of survey papers which cover alternative frontier procedures, suchas Greene (1993), Bauer (1990), Lovell and Schmidt (1988), Schmidt (1986) and Forsund,Lovell and Schmidt (1980).Chapter 4 Alternative Methods 49y =f(x,)e1’e”, u 0 (4.1.8)where y represents output,f(x,j3) is the deterministic core of the frontier production function,f3 are the parameters to be estimated, v is a random variable that takes values over the range(-oo , + Go) and represents the effects of measurement errors, non-observable explanatoryvariables and random shocks, and u is a random variable that takes nonnegative values whichcaptures inefficiency. In other words, f(x,f3)e” is the stochastic frontier while e is theinefficiency. The condition u 0 ensures that all observations lie on or below thestochastic production frontier.The stochastic frontier method allows room for errors of observations, thus avoidsthe deterministic techniques’ high sensitivity to errors in the data. However, the problemremains of what distribution to assign to the residual components v and U5.Direct estimates of the stochastic frontier functions may be obtained by eithermaximum likelihood (MLE) or corrected least square (COLS). Olson, Schmidt andWaldman (1980) presents Monte Carlo evidence which indicates that COLS generallyperforms as well as MLE for the normal/half-normal case. Whether the model is estimatedby MLE or COLS, the distribution of u must be specified.The central limit theorem can be invoked to warrant assuming a normal distribution forv. However, no such consideration stands out to support any particular assumption aboutthe one-sided residuals u, and thus statistical convenience and general plausibility continueto rule the roost. Half normal and exponential distributions have often been used. Thereare limited evidence which suggests that these two assumptions give rise to quite similarparameters for the estimated production frontier (Aigner, Lovell, and Schmidt, 1977).Corbo and de Melo (1986) indicates that the estimated inefficiency levels based on the twoassumptions are highly correlated.Chapter 4 Alternative Methods 50The COLS is similar to the deterministic COLS as described earlier. We firstestimate the parameters, 13, in f(x,13) through OLS. We can then estimate the varianceand o.2 by:I 22_r I it, it (4.1.9)0uLI — )P3JN 2 it-42 ../ it—22 (4.1.10)itwhere and are the second and third moments of the OLS residuals. Next, we“correct” the intercept by adding to the OLS intercept the negative of the estimated biaswhich is the sample mean of e=v-u, I.L = Note that the COLS estimator of allelements of 13 except the intercept is the same as the OLS estimator. These estimates areconsistent and unbiased but not asymptotically efficient.The COLS is simpler in terms of computation. However, there is one problem withCOLS, that is, the estimates may not exist in some samples. If the third moment of the OLSresiduals turns out to be positive, then ô < 0, the estimation procedure breaks down. Theprobability of this occurrence depends on the value of the third moment of thedisturbance. When IL3 is near zero, the probability of being positive may be substantial.Chapter 4 Alternative Methods 51This is a problem mainly when ). = o, is small. According to Olson, Schmidt, andWaldman (1980), as X - 0 (cru2 — 0) the probability of being positive approaches(approximately) 1/2. There is another possibility when the estimation procedure may breakdown. This occurs when < ((it -2)/it)ô, which implies â < 0, This may occur withnon-negligible probability when X is large. Banker, Gadh, and Gorr (1993) finds that thetwo failure types are complementary, there is almost always one present. In their MonteCarlo experiments, only two out of 640 estimates have no failure. Because of theseproblems, COLS procedure is not used in this study.The maximum likelihood estimator is obtained by the (numerical) maximization ofthe log likelihood function of equation (4.1.8) with respect to the parameters in f(.) and thedistribution functions of v and u, after specific assumptions being made about thedistributions of the two error terms.Assume that the elements of v are identically independently distributed (lid) asN(0,o2) the elements of u are half-normal taking absolute values from variables which areiid as N(0,u2) and v and u are independent of each other and are also independent of x.Let o-2 = u + o2 X = u/u. The density function of u is given by:Au) = “ exp[———] U 0 (4.1.11)2aSuppose that after taking logarithms, equation (4.1.8) becomes linear in parameters and canChapter 4 Alternative Methods 52be specified as6:‘ =xp + (4.1.12)where c = v-u. Aigner, Lovell and Schmidt (1977) shows that the log likelihood functionof equation (4.1.12) is:N NL=!!In(2/it)—Nina +EIn[1 —F(e A a1)]_’EE (4.1.13)2 J=1 2cij=1where N is the total number of observations, j indexes the observations, and F is thecumulative distribution function of the standard normal distribution. Details may be foundin Aigner, Lovell, and Schmidt (1977). Note that the parameter X embodies the model ofinefficiency. The simple OLS regression model results from X = 0. The implication wouldbe that every firm operates on its frontier. The MLE estimates are obtained by themaximization of equation (4.1.13) with respect to the parameters (/3, X, o). The MLEestimates are consistent and asymptotically efficient.Several alternative distributional assumptions have been suggested in the literature.The log-likelihood function and associated results under some of these alternativedistributional assumptions, for example, may be found in Aigner, Lovell, and Schmidt(1977) for exponentially distributed inefficiency, Stevenson (1980) for truncated normaldistribution, and Greene (1980), Greene (1990) for gamma distributions among others.6 In nearly all applications, after transformation, the functional form of the model to beestimated is linear in the logs of output and a set of independent variables, so equation (4.12)is assumed as the general form of the production frontier.Chapter 4 Alternative Methods 53Since half-normal distribution is the most popular function among empirical works, this studyconsiders the normal/half normal situation.As mentioned at the beginning of this chapter, efficiency measurement involves twoaspects: estimation of the “empirical” production frontier and the measurement of efficiencyrelative to the estimated frontier. The COLS and MLE procedures described in the previousparagraphs accomplish the first aspect, that is, the estimation of the production frontier fromthe observed data. The next step is to measure the efficiency level of each observation inrelation to this estimated production frontier.Since the residuals from equation (4.1.12) or equation (4.1.8) estimate Ej = v -not u, the efficiency component of the composed error terms has to be estimated indirectly.Jondrow, Lovell, Materov and Schmidt (1982) suggests the use of either the mean or themode of the efficiency term, u, conditional on the estimate of E as a measure of observationspecific estimates of efficiency. They derive an explicit form for the half-normal model7as:ao fe.5/o) j (4.1.14)E(uJe.)= [ .‘ —s.—]‘ a2 1—F(Ja) awhere f(.) is the standard normal density function.In Chapter 3 the efficiency is defined as the ratio of observed output over frontieroutput for a given level of input consumption. Therefore, exp(-u) should be considered asthe efficiency measure instead of u. Again, in practice one needs to estimate the exp(-u.)Jondrow, Lovell, Materov and Schmidt (1982) also derives the expression for theexponential case. Greene (1993) presents the expressions for gamma models and truncatednormal model as well.Chapter 4 Alternative Methods 54given the residuals from equation (4.1.12). Battese and Coelli (1988) suggests the use ofconditional expectations of exp(-u), given c = as the firm specific efficiency estimate,where e is the residual from equation (4.1.12). With the half normal assumption, thisefficiency measure can be derived as follows.The density function of u is given by equation (4.1.11). As e = v - u, the jointdensity of u and c is given by:1 12122exp ——u ——(u ÷e +2ue) (4.1.15)2a 2aand the density function of c is given by:2 (1—F(e)./a)) exp [_!2] (4.1.16)2aTherefore, the conditional density of u given c is the ratio of equations (4.1.15) to (4.1.16),which can be written as:1 1 1 1L 1 .L 2 1 ‘ 2f(u/e) = exp —U ——ue——e uO (4.1.17)/°* 1-F(eA/a) 2a 202where Note that equation (4.1.17) is the density of a a) variabletruncated at zero, where = _GC/O2. Then the conditional expectation of exp (-u), given=e, is defined by:Chapter 4 Alternative Methods 55E(exp( —u)/e) =fexp( -u)f(u/e)du2 2(4.1.18)1—F(a—e/a) a,,,= exp I.L+—1-F(-qa) 2The operational estimator of equation (4.1.18) may be obtained by substituting therelevant parameters by their maximum likelihood estimators (or COLS estimators) as inequation (4.1.14).There are some problems with both COLS and MLE estimators as described above.The first problem is associated with the distributional assumptions of the error terms.Different distributional assumptions may lead to different efficiency measures. The secondproblem is that inefficiency and input levels are assumed to be independent which may notalways be the case. The third problem is that the estimator used to compute the firm specificinefficiency is not consistent although it is unbiased.To avoid assuming a particular distribution for the error terms, Schmidt and Sickles(1984) proposes a number of methods of estimating individual firm efficiency level given thatpanel data is available on sample firms. However, inefficiency is assumed to be timeinvariant in their panel data model while statistical noise is assumed to vary over firms andtime. In particular, they consider three estimators: (1) A “fixed effects” model, or the“within” model, which assumes that the inefficiency error term u is firm specific constant.The frontier function is estimated by using dummy variables in OLS. This model does notassume the independence between inefficiency and inputs, and it does not require theChapter 4 Alternative Methods 56normality assumption. Further, the estimated efficiency is consistent over the time period.(2) A generalized least squares (GLS) estimator, which makes no distributional assumptionon the inefficiency term, but assumes that the efficiency and inputs are independent. Thismodel allows for the inclusion of time-invariant firm specific attributes, such as the capitalstock of a firm which is not growing, location, or some other characteristics. (3) A MLEestimator, which makes both distributional and independent assumptions. Gong and Sickles(1989) compares the three estimators in a Monte Carlo study, and finds that efficiencyestimates from the three estimators are very similar.The time-invariant assumption may be plausible for short panels8. However, it wouldbe desirable to examine how the performance of a particular firm changes over time. Thusthe time-invariant assumption would appear to be restrictive. To allow efficiency to varyover time, Comwell, Schmidt and Sickles (1990) extends the models by Schmidt and Sickles(1984) to relax the time-invariant assumption by imposing certain structure on howinefficiency varies over time. Their model assumes that the one-sided firm effects are aquadratic function of time in which the coefficients vary over firms according to thespecification of a multivariate distribution. For large N, this model presents a fairlycumbersome problem of estimation since the number of additional variables to capture thetime-varying firm specific effects would be substantial.Battese and Coelli (1992) proposes a time-varying model for unbalanced panel datain which the efficiency is assumed to be distributed as truncated normal. This generalizes8 By short panel, we mean that production performance is observed over a short periodof time.Chapter 4 Alternative Methods 57a number of previous stochastic frontier models such as Aigner, Lovell and Schmidt (1977),Stevenson (1980), Pitt and Lee (1981), among others. This model assumes that theefficiency is an exponential function of time. This assumption appears to be restrictive.From the foregoing discussions, we can see that there is a trade-off between imposinga specific distributional assumption on the error terms and imposing an explicit functionalassumption on how the efficiency of a particular firm varies over time, while both bringingthe potential effects of misspecification. The choice of a specific model in any empiricalstudy will depend on the particular situation in question and the availability of necessarydata.For the Monte Carlo study in this dissertation, we consider the basic model proposedby Aigner, Schmidt and Lovell (1977) with the assumptions of normally distributed statisticalnoise and an efficiency term following a half-normal distribution. This model is chosen,because it is the most popular one used in empirical applications. The firm specific (orobservation specific) efficiency is measured by the conditional expectation of exp(u1), giventhe residuals from the estimated frontier function, as defined by equation (4.1.18).4.2 Nonparametric Approach - The Data Envelopment Analysis MethodThe nonparametric approach, represented by the Data Envelopment Analysis method,uses mathematical programming techniques to construct production frontiers. The DataEnvelopment Analysis (DEA) method, introduced in Charnes, Cooper and Rhodes (1978),involves an application of linear programming to observed data to locate frontiers which canthen be used to evaluate the efficiency of each of the firms or organizations responsible forChapter 4 Alternative Methods 58the observed output and input quantities. In DEA, the entities responsible for convertinginputs into outputs are referred to as decision making units (DMUs) which may represent anykind of firms and organizations or their subdivisions as long as they perform the same orsimilar tasks. DEA utilizes a sequence of linear programs, one for each DMU, to constructa piecewise linear production frontier, and to compute an efficiency index relative to thefrontier. Units that lie on the production frontier are deemed efficient. Units that do not lieon the production frontier are termed inefficient and the analysis provides a measure of theirrelative efficiency.DEA is non-parametric in the sense that it does not assume that the underlyingtechnology has a specific functional form with a finite number of parameters, such as Cobb-Douglas functional form. It only requires the convexity assumption about the productionfrontier. Note that DEA is also “non-statistical” because it makes no explicit assumption onthe probability distribution of the “errors” in the production function.DEA can accommodate multiple outputs and multiple inputs with each being statedin different units of measurement. The relative efficiency of a DMU is defined as the ratioof its total weighted output (virtual output) to its total weighted input (virtual input). Theweights (virtual multiplier) required to incorporate the multiple outputs and multiple inputsare determined by linear programming optimization. DEA assumes that each DMU willselect the weights that maximize its own efficiency score, that is, it will evaluate each inputand each output in such a way as to maximize the ratio of its own weighted output to its ownweighted input. Because different DMUs use different combinations of inputs to producedifferent combinations of outputs, they are expected to select sets of weights that reflect thisChapter 4 Alternative Methods 59variety. Generally, DMUs will place higher weights on the inputs that they use least andon the outputs they produce most. In this sense DEA shows each DMU in its best possiblelight. The efficiency ratio generated by DEA is consistent with a frontier interpretation ofperformance. A ratio of unity implies that observed and potential performance coincide inwhich case the corresponding DMU is said to be efficient or “best practice”. Whereobserved performance is lower than potential a DMU receives an efficiency ratio of less thanunity which implies that its performance is poorer than that of a combination of some of itspeer DMUs and so it is relatively inefficient.Extensions to the original Charnes, Cooper and Rhodes (CCR) model have resultedin a variety of alternative formulations all sharing the principle of envelopment. In this study,we consider two of these models9, the CCR ratio model (Charnes, Cooper and Rhodes,1978) and the BCC model (Banker, Charnes and Cooper, 1984). These two formulationshave been widely used in empirical applications. The CCR model yields an index of overallefficiency, it points out the observable differences in performance among DMUs’°. The BCCmodel takes into account the effect of “returns to scale” within the analyzed group of DMUs.It distinguishes between technical and scale inefficiencies by estimating pure technicalefficiency at a given scale of operation.For more discussions on the various DEA models, please refer to Charnes and Cooper(1985), Banker, Charnes, Cooper, Swarts and Thomas (1989), Seiford and Thrall (1990),and Ali and Seiford (1993).‘° Some of the subsequent models incorporate some of the explanations to the observedefficiency differences into models themselves.Chapter 4 Alternative Methods 604.2.1 The CCR ratioAssuming convexity of production possibility sets, Charnes, Cooper and Rhodes(1978) defines the DEA efficiency measure as the maximum of a ratio of weighted outputsto weighted inputs subject to the condition that similar ratios for every DMU be less thanor equal to unity. In mathematical form, the efficiency of the k-th DMU isUYh = Max r=1vxs (4.2.1)>UrYrjSt. r=1 1 j=1,2...nvixiUrVi 6 r=1,...,s; i=1,...,mwhere the Y, X are the known outputs and inputs of the j-th DMU, the u, v are theweights (virtual multipliers) to be determined by the solution of the problem, andcrepresents a small positive non-Archimedean quantity introduced to ensure that all of theobserved inputs and outputs will have “some” positive value assigned to them. This valueserves as a lower limit for the values that can be assigned to the weights u,. and v as shownby the final constraints in equation (4.2.1). In this model, h’ = 1 if and only if the k-thDMU is efficient relative to the other DMUs.Note that the above model is a fractional programming problem. This fractionalprogram is not used for actual computation of the efficiency scores because it has intractableChapter 4 Alternative Methods 61non-linear and non-convex properties (Charnes, Cooper and Rhodes, 1978). However, it canbe converted into a linear programming problem by using a simple transformation (Charnesand Cooper, 1962, 1973). Since > 0, and the problem is invariant under positivescalar multiplier, we may let m1=v,X1k 1, the problem can then be replaced by thefollowing linear programming problem:= MaxSt. - v1X, 0 j=1,...,n (4.2.2)= 1Ur v € r=1,...,s; i=1...,mand hence solutions can be obtained by repeated application of a linear programmingsoftware to each of the observations in the sample data.The dual for formulation (4.2.2) constructs a piecewise linear approximation to thetrue frontier by minimising the quantities of the m inputs required to meet stated levels ofthe r outputs. The dual program is formulated as:= Miii 0k - Or + s)s.t. Yrj’j-Gr = Yrk-+ S1 =0 (4.2.3)0, j-DMUs,j=1,...n0, r-outputs, r=1,...,s1 0, i—inputS, i=1,...,mNote that the choices of € > 0 are defined so that the optimal value of °k will not beChapter 4 Alternative Methods 62affected by any value that may be assigned to the slack variables associated with € in theobjective function of the dual. The dual problem has one variable X for each DMU in thesample. Variable 0k is unconstrained. It indicates the potential of a proportional reductionin all the inputs of the k-th DMU. Variables r,. are output slacks, and variables s, are inputslacks. If the k-th DMU is efficient, then the optimal dual solution will have all X, cry, ands equal to zero except for Xk, and 6 is equal to 1. When the k-th DMU is inefficient, thereis a hypothetical DMU on the empirical production frontier which serves as the referencepoint for the measurement of the inefficiency for the k-th DMU. The input and output levelsof this hypothetical DMU are linear combinations of the input and output levels of the DMUsin the efficient reference set of the k-th DMU. The optimal X from the dual problem areused as the coefficients for these linear combinations. To become efficient, all of the inputsof the inefficient DMUk must be reduced to where Ok* is less than 1. If, in addition,any input slack s, is not zero for DMUk then a further reduction by the amount of this slackmust also be made from the i-th input used by DMUk without altering any other inputs oroutputs.The DEA efficiency measures depend on the number of degrees of freedom that areavailable. There are m + s constraints to be satisfied in the dual formulation and nobservations, one for each of the j = 1,..., n DMUs, that form the possible combinationsfrom which efficiency estimation can be secured. From degrees of freedom considerations,the number of variables X used for the solutions in the problem on the left should be at leastas great as the number of constraints. Thus, the number of DMUs for which there areChapter 4 Alternative Methods 63observations should be greater than the number of constraints. According to Banker,Charnes, Cooper, Swarts and Thomas (1989), it is generally advisable to have n 3(m+s).The name Data Envelopment Analysis is obtained from the dual formulation in thefollowing manner. An optimal solution will envelop outputs of DMUk from above viaconstraints of the form Yrk Yrj”j with at least one of these r =1, ..., s constraintssatisfied as an equation. Thus, there will be at least one “touching” of an observed outputfor DMUk by the solution associated with an optimal choice of X* values. Similarly, theinputs of DMUk are enveloped from below via the constraints 6,x with at leastone of these input constraints satisfied as an equation. Thus, the term Data EnvelopmentAnalysis is used because the output and input data of DMUk are enveloped from above andbelow in the manner just described.An analogous linear programming formulation of equation (4.2.2) can be obtained bysetting the numerator of the objective function of fractional program equal to unity, andminimizing the weighted inputs for DMU k:Chapter 4 Alternative Methods 64MIN v1XSt.—UY,, 0 j=1,...,n (424)EUTY,.k = 1u, v € r=1,...,s; i=1,...,mThis formulation determines the output efficiency of a DMU for a given set of inputs.The corresponding dual ish = MAX + e(Ec, + Es1)S.t.425)j, CJ 0where Zk is unconstrained.In many circumstances outputs are partly determined by the market condition andgovernmental control. It may therefore not make much sense to suggest that output be raisedto increase efficiency. On the other hand, efficient use of inputs is desirable in any case.Therefore in this study we use the input minimization formulation given by equations (4.2.2)and (4.2.3). To have a better understanding of the basic idea underlying DEA, adiagrammatic interpretation of the dual for input minimization is provided.The estimated dual technology is not smooth but constructed out of a series ofintersecting linear facets. Each of these facets represents a constraint in the optimal solutionto the dual. Collectively they intersect to form a convex production set which is closed andChapter 4 Alternative Methods 65bounded from above. The frontier for efficiency comparisons is the lower convex hull ofthe possibility set which is shown in Figure 4.1. Figure 4.1 illustrates a hypothetical frontiertechnology based on 5 firms producing a single output, Y, from 2 inputs, X1 and X2. FirmsG, F and E, lying on the frontier, are the “best practice”. Thus, no other firms or linearcombination of firms in this sample can be identified which is producing the same level ofoutput for less of either or both inputs. These firms have unity efficiency scores and zeroslacks in the solution to the dual. For example, the solution of the dual for firm F:hF*= 1and the constraints are:input 1 X1PJZF -o- X1F).;input 2 X2FIZF* -o = X2)and on the output 1F + 0 = 17FThe left- hand side of the constraints defmes the “target”, which in this case is equalto actual performance on the right-hand side of the constraints because best-practice implies= 1. The peer group11 drops out of the RHS of the constraints and for an efficient firmis none other than that firm itself since ) = 1 and ç = 0, j * F.Firms B and D are inefficient relative to frontier performance. That is, for the same“ The peers are defined by those firms that have non-zero weights in the optimalsolution in the dual.Chapter 4 Alternative Methods 66GDX1Figure 4.1 Diagrammatic Interpretation of DEAChapter 4 Alternative Methods 67level of output it is possible to find a firm, or a linear combination of firms, which is usingless of at least one of the inputs. Consider firm B, for example, with an efficiency ratioOA/OB which is less than unity. This reflects the fact that a linear combination of firms Eand F is producing at least as much output as B with less of X1 and X2. That is, the existinginput consumption at firm B can be adjusted by the efficiency ratio to X’1 and X’2 in figure4.1 while maintaining its current level of output. Therefore, the peer group (or the referencefirms) for firm B are firms E and F. For firm B, the optimal solution in the dual is:h = OA/OB < 1and the constraints are:input 1 XlBhn -o = XlE)+ X.1•;input 2 Xh -o X2E) + X2F2and on output Y1B+O 17E”E +Target performance for B, X.B.h, i=1,2, is equal to a linear combination ofperformance at firms E and F where , ) > 0 and the weights on the other firms are allzero: = 0, j E, F.One can see that the solution for firm B has all input and output slacks equal to zero.However, firm D has a non-zero slack on input X1. The efficiency ratio for D is OC/ODwhich defines an initial radial contraction in both inputs. However at point C, firm E isChapter 4 Alternative Methods 68producing the same output for less of X1 and the same amount of X2. Hence D is not fullyefficient until it reduces its consumption of X1 by the horizontal distance C to E. Thisdistance is given by a non-zero slack s in the final solution of the dual for firm D:= OC/ODand the input constraints areinput 1 Xh s1’ = XlE)input 2 X2DJi - 0 =X2EAand on output 1D + 0 = 1’1EEThe target for D is radial contraction in both inputs given by h plus the additionalreduction in X1, given by s1. Its peer group is firm E alone since its target coincides exactlywith performance observed at this best-practice firm. Thus A = 1 and = 0 for j * E.From the above discussions, we can see that there are two aspects to the target in thedual. The input constraints define a radial (or equi-proportionate) reduction in inputs givenby the efficiency ratio, h, plus any further reductions in inputs suggested by non-zeroslacks. In addition, however, the presence of non-zero output slacks may requireadjustments to outputs.Chapter 4 Alternative Methods 694.2.2 The BCC modelThe basic CCR model assumes constant returns to scale. That is, proportionalchanges in all input levels result in changes of equal proportion in output level. In practice,one may find this assumption too restrictive. Banker, Charnes and Cooper (1984) extendsthe original CCR formulation to incorporate the effect of returns to scale on the efficiency.The BCC model adds an additional restriction to the envelopment requirements. It requiresthat the reference point on the production frontier for DMUk be a convex combination of theobserved efficient DMUs. It is formulated as:hk = Max uY -St. UrYr — — U0 0 j=1,...,n (4.2.6)= 1Ur v c r=1,...,s; i=1,...,mThe corresponding dual problem is formulated as:hk=Min Ok€(Gr+Si)S.t.-= Yrk-0kik + S —0i (4.2.7)2j =10, j-DMUs,j=1,...n=0, r-outputs, r=1,...,s0, i—inputs, i=1,...,mChapter 4 Alternative Methods 70Comparing formulation (4.2.3) with formulation (4.2.7), we can see that the only differencebetween the two formulations is the additional constraint in formulation (4.2.7). Thisadditional constraint ensures that a DMU is evaluated only by reference to original datapoints and their “convex combinations” on the efficiency frontier. The new variable, u0, inthe primal problem is unconstrained in sign, and is interpreted by BCC as an indicator ofreturns to scale. Banker, Charnes and Cooper (1984) shows that the returns to scale at thereferent efficient point are estimated by the sign of the variable u0: u0 < 0 indicatesincreasing returns to scale; u0 > 0 indicates decreasing returns to scale; u0 = 0 indicatesconstant returns to scale12.It should be noted that the returns to scale indicated by u0 are “local” in the sense thatthey are applicable only to the facet on the efficiency frontier where the reference point forthe efficiency evaluation is positioned.From the discussions above, a number of features of DEA become apparent. First,the DEA efficiency ratios are solely dependent on observed best practice in the sample. Oneof the consequences of measuring efficiency relative to observed best practice is that DEAis usually considered to be sensitive to extreme outliers and measurement errors. Secondly,since the weights for each DMU are chosen so as to give the most favourable efficiency ratiopossible subject to the specified constraints, DEA evaluates a DMU as efficient if it has thebest ratio of any one output to any one input. Therefore, the DEA efficiency ratios couldbe sensitive to the selection of inputs and outputs included in the analysis.12 Note that u0 is added to correct for the effect of non-constant returns to scale, but themagnitude of this parameter is not directly interpretable as a measure of returns to scale.Chapter 4 Alternative Methods 714.3 The Effects of Exogenous VariablesThere are mainly two motivations for studying the frontier techniques: (1) the desireto measure inefficiency; and (2) the desire to see how efficiency is related to observablecharacteristics of the firm and production environment. Both the parametric approach andthe non-parametric approach discussed in this chapter focus on how to construct a productionfrontier and how to measure efficiency relative to the estimated frontier. The resultingefficiency measures are based on observed units of outputs and inputs, and thus ignore theeffects of variations (except for the statistical noise with the stochastic frontier models) inthe market, operating, institutional and regulatory policy environments, and other specificfactors which may affect the observed production of the DMUs. Therefore, these efficiencyscores may represent factors other than efficiency. In order to make meaningful comparisonsabout the relative performance of the firms (DMUs), and to identify the sources of observedefficiency differentials, additional analysis would be necessary in order to purge theseefficiency scores of influences of these “exogenous variables”.As pointed out by Nerlove (1965), the observed differences in productionperformance among a group of firms may be attributed to three general sources: (1) abilityto maximize short-run profits, given a particular production function and in a givenenvironment; (2) the production function itself which summarizes the state of technicalknowledge and the possession of fixed factors; and (3) the environment. Within theframework of efficiency measurement discussed earlier we assume that the firms have thesame production technology, thus we can ignore the second source while the first source maybe broadly thought of as inefficiency. As for the environmental factors, some efforts mustChapter 4 Alternative Methods 72be made to standardize environment, ideally through the use of some meaningful quantitativevariables, in order to permit measurement of relative efficiency since firms cannot physicallybe transferred from one environment to another.The observed inefficiency may be related to various factors that could explain them,such as environmental conditions, administrative structures, social constraints, the quality ofproduction factors, etc. These factors are referred to as exogenous variables in thisdissertation since they are mostly beyond the control of the firms (DMUs) in question. Therelation between the observed efficiency level and the exogenous variables may be estimatedafter having assessed the production frontier and measured the observed efficiency. Thisapproach is often associated with the non-parametric methods such as in Ray (1991),McCarty and Yaisawarng (1993) and Oum and Yu (1994), but has also been used with theparametric methods as well such as in Bruning (1991) and Loeb (1994). This approach ofincorporating exogenous variables is referred to as the two-step procedure in thisdissertation. Alternatively, the relation may be estimated while assessing the productionfrontier in which case the frontier and the functional relationship explaining efficiencydifferences are simultaneously estimated. This approach is often associated with theparametric efficiency estimation methods for obvious reasons. An example of using thissecond approach, referred to as the one-step procedure, can be found in Lee and Schmidt(1993). Since with the one-step parametric methods, the exogenous variables can beincorporated directly in the estimation of the production function simply by introducingadditional variables, it is not necessary to make any methodological modifications withfrontier models described in section 4.1. Therefore, this section focuses on how to accountChapter 4 Alternative Methods 73for the effects of exogenous variables through the use of the two-step procedure, particularlywith the nonparametric methods. The two-step parametric methods follow the sameprocedure as the two-step DEA-regression procedure described below.Both CCR and BCC methods measure efficiency based on units of outputs and inputs,but ignore variations in the market, operating, institutional and regulatory policyenvironments, and other factors which may affect the observed performance of the DMUs.Therefore, the DEA efficiency indices from CCR and BCC models may represent factorsother than efficiency. To derive meaningful inferences about the relative performance of theDMUs, additional analysis is necessary to purge the DEA efficiency indices of influencesof these “exogenous variables”. A second stage regression analysis can be used to identifythe effects of these variables and to measure the “residual” efficiency’3which is a closerindicator of relative performance of the DMUs than the gross DEA efficiency index. Anumber of studies have used this DEA-regression procedure in empirical applicationsincluding Ray (1991), Fizel and Nunnikhoven (1992), and McCarty and Yaisawarng (1993).In the second stage, the DEA efficiency scores are used as dependent variable in aregression on the exogenous variables. There is a methodological problem with such aregression. As defined in sections 4.2.1 and 4.2.2, the DEA efficiency index falls betweenO and 1 (0 h*k 1), making it a limited dependent variable. Consequently, an OLSregression of h*k would produce biased and inconsistent parameter estimates. In order to‘ The term residual efficiency is used here instead of “true” efficiency, because inpractice one may not be able to identify all potential influential factors thus the efficiencymeasures from the regression analysis may still reflect the influences of some neglectedfactors.Chapter 4 Alternative Methods 74treat the limited dependent variable properly, the following form of the Tobit model(Tobin, 1958; Amemiya, 1985) is adopted 14:Lii = I ZJPJ + rj if Lii, < 0, fl,...,fl (4.3.1)i 1 0 otherwisewhere Lh is the logarithm of the DEA efficiency index for firm j, Z is a vector of thelogarithms of the variables potentially influencing the DEA efficiency scores, j3 is a vectorof coefficients to be estimated, and {} are assumed to be independently identicallydistributed error terms which can take on negative, zero, or positive values. The residualefficiency is defined as:Eff = EXP(-i.) (4.3.2)where E is the residual efficiency score for the j-th DMU, and max is the maximum of{m}• Note that the DMU with the largest positive will be considered as the most efficientDMU, and given a residual efficiency score of one. By definition, Efj also falls betweenzero and one.This two-step procedure accomplishes two tasks. First, it identifies the effects ofpotential influential factors on the DEA efficiency index. Second, it allows one to computethe “residual” efficiency index from the residuals of the Tobit regression.14 The Tobit Model allows one to incorporate only one bound on the dependent variablewhile the DEA efficiency index is constrained between zero and one. By taking thelogarithm of the DEA efficiency index, one can convert the dependent variable to have onlyone (upper) bound, zero.Chapter 4 Alternative Methods 754.4 SummaryThis section summarizes the comparative merits of the three alternative methodsbased on their theoretical and methodological differences.• The parametric methods use econometric techniques to estimate the frontier function,thus the estimators have identifiable statistical properties. However, they imposea particular functional form on the estimated frontier function and confounds theeffects of misspecification of functional form with inefficiency. On the other hand,the data envelopment analysis method does not impose an explicit functional form onthe frontier and is less prone to misspecification error in this regard. Therefore,DEA may have a comparative advantage over the parametric methods in situationswhere the underlying production technology does not meet the classical assumptions.The parametric methods are likely to have a comparative advantage when theclassical assumptions are met. Further, the DEA method assumes that the productionpossibility set is convex. This assumption is violated in the case of increasing returnsto scale. Therefore, the DEA method has an inherit disadvantage in situations withincreasing returns to scale.• The parametric methods are statistical methods, they normally require a relativelylarge sample size (depending on the particular functional form being estimated).DEA on the other hand is less restrictive in terms of the sample size since thefeasibility of the DEA analysis depends on only whether there are enoughobservations to span the convex cones in input and output space. However, all threemethods are expected to yield better estimates as the sample size increases.Chapter 4 Alternative Methods 76• The deterministic frontier method and the data envelopment analysis are bothdeterministic, and attribute all deviation from the frontier to inefficiency. Noallowance is made for noise, measurement error, and the like. Therefore, a singleerrant observation (efficient outlier) can have profound effects on the estimates.There is no a priori answer for which method is more sensitive to the effects ofoutliers and data errors. Any conclusion regarding the relative merits of these twomethods in this aspect would be drawn from empirical evidence. Unlike the twodeterministic methods, the stochastic frontier method attempts to distinguish theeffects of random noise from the effects of efficiency and is less sensitive to potentialoutliers or other extraordinary behaviour of observations. However, additionalstructure is imposed on the distribution of inefficiency. Thus, specification error ofthe efficiency term may affect the accuracy of the efficiency estimates. Nonetheless,the stochastic frontier method is expected to yield more robust efficiency estimatesthan the other two methods especially when there are large variations in the firms’operating environments.• The DEA method is designed to deal with production technologies with multipleinputs and multiple outputs. On the other hand, the parametric methods may havedifficulty estimating production frontier functions in situations involving multipleoutputs unless the dual cost frontier functions could be estimated.• The deterministic frontier method appears to combine the bad features of theeconometric and programming approaches to frontier construction: it is deterministicand parametric. It is still used in practical applications because it is simple and easyChapter 4 Alternative Methods 77to apply. The DEA method requires repeated solutions of linear programmingproblems, thus usually incurs more computational costs than the parametric methodsespecially for large samples. The stochastic frontier method involves complexestimation procedures and may not be easy to use. However, special computersoftware is available for estimating all three methods.• The one-step procedure for incorporating the effects of exogenous variables isexpected to produce efficiency estimates closer to the “true” values if the exogenousvariables can be correctly identified and accounted for, and specification errors(related to the exogenous variables) are not very serious. The two-step procedureis intuitively more appealing to decision makers since it relates firms’ performancedirectly to the potentially influential factors. However, cumulative specificationerrors from both steps are likely to have a negative effect on the accuracy ofefficiency estimates.The foregoing discussions indicate that one cannot always draw solid conclusionsregarding how well each of the methods performs in comparison with the other alternativemethods under different scenarios on the basis of their conceptual and methodologicaldifferences. Therefore, empirical evidence is necessary to help provide useful guidelinesfor selecting the appropriate methodology in practical applications. The Monte Carloexperiments described in the following chapter are intended to provide such information.78Chapter 5Design of the Monte Carlo ExperimentsThe Monte Carlo experiments are often used to “solve” those analytical problems thatare technically intractable. Monte Carlo experimentation solves these problems bysubstituting an equivalent stochastic problem and solving this simulated problem. Asdescribed in Chapter 1, Monte Carlo experiments are used herein to examine the relativeperformance of the three alternative efficiency measurement methods with respects to thesample size, the variations in input values, the noise level, the exogenous variables, the dataoutliers, and the different (underlying) production structures. This chapter outlines the basicsof the Monte Carlo experiments.For each Monte Carlo experiment, we need to specify: (1) the “true” underlyingproduction technology y = fix), (2) a sample size N, and (3) values for g2 and cr2, thevariances of the two error terms, and we also need to generate a N x j input data matrixin which I is the number of inputs specified in the “true” production technology.5.1 Specification of the “true’ underlying production technologyThe first step in conducting the Monte Carlo experiments is to specify the “true”underlying production technology. This “true” technology must satisfy the followingregularity conditions:(1) The production of an output always requires the use of at least one input.(2) Outputs are finite for all finite inputs.(3) An increase in inputs can not lead to a reduction in outputs.ChapterS Design of Experiments 79(4) A reduction in outputs remains producible with no change in inputs.(5) The production possibility set is closed.In practical situations, the choice of a particular functional form depends largely ona priori information about the underlying technology. Without such information, the choiceof functional form is usually based on its flexibility. In this study, we assume the CRESH(Constant Ratio of Elasticity of Substitution, Ilomothetic) production function as theunderlying production technology. The CRESH function was developed by Hanoch (1971).The well-known CES (Constant Elasticity of Substitution) function as well as its limitingforms (the Cobb-Douglas, Leontief, and linear functions) are special cases of CRESH. Thatis, CRESH function nests a number of commonly used functional forms. Moreover, theCRESH function allows different patterns of substitution or complementarity among threeor more inputs, and different returns to scales for changing input proportions. This makesit possible to examine the performance of the alternative methods under different degrees ofinput substitution and returns to scale. Because of these advantages, this functional form hasbeen used in a number of previous Monte Carlo studies as the underlying productiontechnology, including Guilkey and Lovell (1980), Guilkey, Lovell and Sickles (1983), Gongand Sickles (1989), and Gong and Sickles (1992). CRESH function is thus selected as theunderlying true production technology in this study. The CRESH production function isspecified as follows:ye°= ( )YIP (5.1)where U 0, ‘y > 0, > 0, for all i, and = 1. Function (5.1) has variable returnsChapter 5 Design of Experiments 80to scale r(x), and variable Allen-Uzawa partial elasticities of substitution o-(x), which aregiven by:n— Ptr(x) = ( fri (5.2)1+oyn-P1Ep81=1iD PkokX;Pka..(x) 1 k=1 ij (5.3)U (1+n)(1+n) n‘‘ ‘J l’k”k Pkk=1 1PkkUnder certain parametric restrictions function (5.1) becomes one of a number of wellknown production functions. For example, if 0 = 0 then equation (5.1) is an almosthomogeneous CRES (Constant Ratio of Elasticity of Substitution) function; if Pi = = p,it is a homothetic CES (Constant Elasticity of Substitution) function; if 0 = 0 and Pi=p,, it is a homogeneous CES function, and it has the homothetic Cobb-Douglas functionas a limiting form as (Pi = ... = p) — 0.The discussions above consider simply the relation between inputs and output. Inpractice, firms operate under different conditions. The observed productions are influencedby factors outside firms’ control. To simulate closer to a real situation, therefore, the effectsof operating environments should be incorporated into the “known” underlying productionChapterS Design of Experiments 81situation. Let us assume that all firms in an industry have the same underlying productionfunction, but operate in different environments with different levels of efficiency1. Thus theobserved production of a representative firm may be described by the following functions:Y = fix) exp() (5.4)= g(z) + V +p (5.5)where Y is output, x is a vector of inputs, f(x) is the industry’s underlying productiontechnology, represents the aggregate deviation from the deterministic core of theproduction frontier as specified by f(x), z is a vector of exogenous variables reflecting firmcharacteristics and operating environments, g(z) is the firm specific effects on the observedproduction2,v is a random disturbance which captures the statistical noises, and representsthe efficiency level of an individual firm.The industry underlying production technology f(x) is assumed to take the functionalform specified by function (5.1). An appropriate form for g(z) is not obvious. Herein weassume g(.) is a linear function of z. The disturbance v is typically assumed to be normally1 In this study, we focus only on technical efficiency thus assume that the firm isallocatively efficient. The definitions of allocative efficiency and technical efficiency aregiven in Chapter 3. See Farrell (1957) for more discussions on these concepts.2 Reifschneider and Stevenson (1989) consider g(z) as the systematic influences on thefirm’s inefficiency level. We assume here that the firm does not have direct control overthe variables z and we want to examine how the changes in z affect the estimated efficiencyscores while the “true” efficiency levels unchanged. Therefore, we choose to separate g(z)from the efficiency term u.Chapter 5 Design of Experiments 82distributed and represents the statistical noise which is not captured by g(z). The disturbanceis assumed to be a nonpositive error term reflecting technical inefficiency and follow thehalf normal distribution. The half-normal distribution is the most popular distributionalassumption for the efficiency component in the empirical literature on efficiencymeasurements (Lovell, 1993).5.2 Determination of Sample Size and Number of ReplicationsAs a general principle, the larger the sample size and the more the replications, thebetter the results. However, computation cost increases rapidly as the sample size andnumber of replications increase especially for the DEA models. The DEA method requiresthe solution of one linear programming problem for EACH observation point. For example,for a sample with 100 observations, DEA needs solving 100 different linear programmingproblems to produce a set of efficiency scores. In addition, each additional observationintroduces another variable into the dual formulation of the DEA model which furtherreduces the computation speed of the LP problem3. Due to these reasons, for most of theexperiments, the sample size is set at 250 observations which reflects the “real” sample sizeoften found in the empirical literature. However, a set of experiments are conducted toexamine the effects of sample size on the relative performance of the three alternativemethods. Each experiment is repeated 25 times. Appendix B tests the hypotheses whetherEach additional observation will add one more constraint in the primal formulation,which will have even greater effect on the computation speed of the LP program. SeeAppendix A for more discussions. Appendix A also lists a sample computer code forcomputing DEA efficiency index.ChapterS Design of Experiments 83or not there are any significant differences between the results with 25 replications and theresults with 50 and 100 replications. The test results indicate that there is no significantdifference. Therefore, 25 replications are adequate for this study.5.3 Generation of Inputs, Exogenous Variables and the Two-component ErrorsFor the sake of simplicity, we consider an one-output three-input productiontechnology as specified by equations (5.1), (5.4) and (5.5). The inputs are drawn randomlyand independently from a log-normal distribution, and fixed throughout an experiment. Thelog-normal assumption is considered as a reasonable approximation to reality when a randomvariable is regarded as representing the joint effects of a number of factors. It also has theadvantage that the possibility of generating negative values is avoided. It has been used forinput generation in previous simulation studies such as Wales (1977), Guilkey and Lovell(1980), and Gong and Sickles (1992). The procedure used to generate the log-normalrandom variables and the summary statistics of the inputs are given in Appendix B.One exogenous variable is assumed. The generality of the results will not be lost,since this variable may be considered as an aggregate indicator reflecting all identifiableexogenous factors. The exogenous variable is generated by z z where z is drawn fromN(0, 1), and the value for a varies according to the requirement of a particular experiment.Consequently, the effects of the exogenous variables on the observed production output, asreflected by exp(g(z)), follow a log-normal distribution. There have been very limitedsimulation studies which incorporate exogenous variables in the production or cost functions.Nerlove (1971) includes an uniformly distributed exogenous variable in his Monte CarloChapter 5 Design of Experiments 84study. Normal distribution is preferred to the uniform distribution here because of the samereasons as discussed for the input variables.The noise component of the error term is drawn randomly and independently froma normal distribution N(0,) over the range (-oo,+oo). The variance u2 varies acrossexperiments according to the requirement of a particular experiment. Normality is routinelyassumed for the statistical noise, and it is considered as a reasonable approximation to realitywhen the sample size is large enough to rely on the Central Limit Theorem. Theinefficiency component of the error term is drawn randomly and independently from a half-normal distribution which takes absolute values from N(0, 0.36). The half-normaldistribution is used here because it is the most popular distributional assumption for theinefficiency term in empirical studies. The values for this inefficiency term are fixedthroughout experiments. The choice of this particular value for cr2 would not likely to affectthe inferences from the experimental results in any significant way. This is examined byconducting additional experiments assuming u2= 1. The results show slight increases in thecorrelations between the “true” and the estimated efficiencies for all three methods (the meanefficiency estimates fall as expected). However, the relative performance of the alternativemethods does not appear to be affected by the changes in o2. This is also confirmed byGong and Sickles (1992). They show that the correlations between the true and estimatedefficiency rise as o increases (for given u2) for both the stochastic frontier method and theDEA method with the stochastic frontier method dominating DEA for all values ofexamined.Chapter 5 Design of &periments 855.4 Generation of the output observationsWe have fixed 0 = 0, -y =1,== 0.3, and ô3 = 0.4 in equation (5.1) throughout theexperiments. These values are assigned to the parameters mainly due to computationalconvenience. Other parameters are given different values for different experiments so thatthe underlying production technologies exhibit various degrees of factor substitution andreturns to scale. Therefore, it is adequate to fix 0, y, ô, at this particular set of values. Thisis done mainly to keep the amount of computation requirements within a manageable range.It should not have any dramatic effect on the generality of the experiment results.After the inputs, exogenous variable, and the two error terms are generated, thecorresponding values for the “observed” output can be generated by means of equations (5.4)and (5.5). This provides us with a sample data base with 250 observations on output, threeinputs and one exogenous variable.5.5 Conduct of the ExperimentsSix sets of experiments are conducted to examine the relative merits of the threealternative methods from different aspects. In the first five sets of experiments, we limit therange of technologies to those which exhibit constant returns to scale in order that thecomparison between the alternative methods is not distorted by the treatment of scaleeconomies. In particular, three production technologies are assumed with input substitutionat 3.03, 1.03 and 0.333 respectively. The BCC model is not included in these experimentssince this variant of the DEA methods is intended to deal with the problem of non-constantreturns to scale. In experiment six, non-constant returns to scale and input complementarityChapter 5 Design of Experiments 86are introduced into the underlying production technology to examine how each of the threealternative methods performs under complex technologies. In this last set of experiments,the BCC model is compared with the other three models. The experiments are described asfollows.Set 1: The Effects of Sample SizeThe first set of experiments examines whether sample size (N) has any effect on theperformance of the three alternative methods. In practice, people may encounter smallsamples with less than 100 observations, or large samples with over 500 observations. Ifthe relative performance of the alternative methods depends heavily on the sample size, thenpeople working with empirical data would need to take this into consideration in selectingthe appropriate method for their particular problems.In these experiments, the standard deviation for the noise term (cry ) is set at 0.15.To avoid the distortion caused by the effects of exogenous variables and to simplify thecomputation, the effects of exogenous variables are assumed to be negligible, that is, a isset to 0. Three sets of sample data are generated with sample size of 100, 250, and 500,respectively. The alternative methods are then applied to each of the three data sets.Set 2: The Effects of Input RangeThis set of experiments is intended to examine how sensitive the alternative methodsare to the homogeneity of the DMUs or observations in the sample. The less variations ininputs, the more homogenous the observations will be. In some data sets there are veryChapterS Design of Experiments 87small variations in the values of input variables among the sample firms, while in other datasets one may face observations with input levels varying over a wide range. If a method isvery sensitive to the homogeneity assumption, one may need to choose an alternative methodin situations where there are large variations among the DM1Js or observations. This set ofexperiments is also intended to verify whether the input specifications used in other sets ofexperiments are reasonable since the input variables are fixed throughout all the experimentsexcept for SET 2.Inputs (X) are assumed to follow a log-normal distribution. X are generated as X =e’, where w = 3 + rx0, and x0 is drawn from N(0, 1). The values of input variables for allother sets of experiments are generated assuming r equal to i. In this set of experiments,r is given the values of 0.1, 1, 1.5. Thus, three sets of sample data are generated. In allthree data sets, sample size (N) is fixed at 250, and the standard deviation for the noise term(u) is set at 0.15, and the exogenous variable is ignored by setting a equal to 0.Set 3: The Effects of Noise LevelSet 3 of the experiments examines the effects of noise level on the relativeperformance of the alternative methods. The more robust the method, the less the effectsof noise level on its performance. In situations where there are large variations in theoperating environments, one would probably need to use a method which is less sensitiveto the level of noise.‘ See Appendix B for more discussions and summary statistics for the input variables.ChapterS Design of Experiments 88Throughout set 3, the sample size (N) is fixed at 250, and as in SET 1, a is fixed at0 assuming the effect of quantifiable exogenous factors is negligible. By varying thestandard deviation of the noise term (u) at 0.03, 0.15, 0.25, 0.50, and 0.75, five sets ofsample data, with different levels of noise, are generated. The three alternative methodsare then applied to each of these data sets.Set 4: The Effects of Exogenous VariablesIn most practical situations, there is a large number of exogenous factors5 affectingthe productive performance of the firms or organizations. Some of these exogenous factorscan be quantitatively identified, some cannot. The statistical noise reflects the aggregateeffects of the exogenous factors which cannot be quantitatively identified. Experiment 4 isintended to examine how well each of the alternative methods deals with those exogenousfactors which can be quantitatively identified.In these experiments, sample size (N) is again fixed at 250, and the standard deviationfor noise (u) is set at 0.15. The extent of the effects of exogenous factors is reflected bythe values of a which is set at 0.05, 0.10, 0.25, 0.50, 0.75, 1.00, and 2.00.Set 5: The Effects of OutliersThis set of experiments examines how sensitive each of the three alternative methodsis to the “efficient” outliers or the measurement errors in the sample data. In measuring theFor the sake of computational convenience, it is assumed that productive performancehas no effects on those exogenous variables.Chapter 5 Design of Experiments 89relative efficiencies of the DMUs or organizations, the “best practice” DMUs are used asthe reference points for other DMUs. If one of the “best practice” DMUs happens to bean “efficient” outlier, then all the other DMUs whose performance is evaluated relative tothis “efficient” outlier would have a misleading performance rating. On the other hand, an“inefficient” outlier would only lead to an under-estimated performance rating for itself.An “efficient” outlier may be an indicator of using inappropriate input and outputvariables or model misspecification, or may be evidence of an extraordinary event orcondition, or may simply be due to data errors. There are two general strategies for dealingwith outliers. The first strategy requires that outliers be detected and their causesinvestigated. The offending observation can then often be corrected, deleted for goodreason, or otherwise given individual attention. The second strategy is to use robust methodsso that outliers have little influence over any inferences or conclusions. If the secondstrategy is chosen, it is necessary to know how robust the alternative methods are in orderto select the appropriate method. Both the DEA and the deterministic frontier method havebeen criticized as being sensitive to outliers. This set of experiments is intended to find outhow sensitive these methods actually are to the outliers.Again the sample size (N) is fixed at 250 in these experiments, and the standarddeviation of noise (o,) is set at 0.15. The exogenous variables are assumed to be negligible,thus a is set at 0. To minimize computational requirements, it is assumed that there is onlyone outlier in the data set. To create an “efficient” outlier, the output level of an efficientChapter 5 Design of Experiments 90observation is then increased by 5%, 25%, 50%, 75% and 100%6.Set 6: The Effects of the Underlying Production TechnologyThe first five sets of experiments consider only technologies of constant returns toscale, so that the relative performance of the alternative methods is not distorted by differenttreatments of returns to scale. This restriction is relaxed in this set of experiments. Theparameters of the underlying production technology are chosen such that the returns to scale,r(x), and the Allen-Uzawa partial elasticities of substitution, o(x), fall within the ranges of0.92 < ‘y < 1.57, and -0.27 < a;, < 0.54. These ranges for returns to scale and inputsubstitutions are determined from reviews of empirical studies on production and costcharacteristics in transportation industry. See, for example, Borger (1991) and McGeehan(1993) for railways, Gillen, Oum and Tretheway (1985) and Keeler and Formby (1994) forairlines. The transportation industry is considered here because in the second part of thisdissertation the alternative methods are applied to a railway data set and an airline data setto examine the efficiency performance of railways and international airlines.As in the other sets of experiments, the sample size N is fixed at 250, the standarddeviation of the noise term (o) is set at 0.15, a is set at 0 assuming the absence of theeffects of exogenous factors.6 Note that the observed “efficient” outlier may actually be inefficient.Chapter 5 Design of Experiments 915.6 Evaluation CriteriaThe performance of the three alternative methods is evaluated using the followingstatistics: the mean efficiency estimates, the mean absolute deviations (MAD) of theefficiency estimates from the true values, the correlation coefficient, and the rank correlationcoefficient between the estimated level of efficiency and the “true” level of efficiency.The mean efficiency estimates and the MAD examine the ability of the alternativemethods to approximate the actual level of efficiency. The MAD measures the degree ofdeviations, and the mean efficiency estimates provide an indicator as to which direction theestimates deviate from their true values, eg. overestimation vs underestimation.Pearson product-moment correlation coefficient is used here to measure the strengthof relatedness between the estimated and “true” efficiency levels. It is a number that variesfrom -l to + 1. A correlation of + 1 denotes a perfect positive relationship. A correlationof -1 denotes an inverse relationship. A zero value of correlation means that there is nolinear relationship. The Pearson correlation coefficient for the estimated efficiency ee andthe “true” efficiency EE is measured as:= Y(ee - )(EE - EE) (5.6)- &)2E(EE - EL)2where the a and EE are the average of ee and EE respectively.The rank correlation coefficient measures the correlation between the rankings of theestimated efficiency level and the “true’ efficiency level. In this study, the Spearman’s rankChapter 5 Design of &periments 92correlation coefficient is used, it is derived from the Pearson product-moment correlation byusing the ranks of the two variables considered instead of the raw data. The Spearman’srank correlation also falls within the interval of (-1, +1). A value of +1 indicates a strongpositive association between the rankings in which case the rankings of both groups aresimilar. A value of -1 indicates a strong negative association between the rankings wherethe rankings of the two groups would seem opposite. If the value is zero, one will observeno pattern between the rankings. The formula for calculating a Spearman’s rank correlationis as follows:Y =1—UL44 (5.7)n(n2-1)where n is the number of pairs being correlated, and d is the difference in the ranks of eachpair.93Chapter 6The Experiment ResultsThis chapter reports and discusses the results from the Monte Carlo experiments.The sample data for the experiments are generated by SHAZAM 6.2 (White, Wong,Whistler, and Haun, 1990) on a P.C. The deterministic frontier models are estimated usingSHAZAM on a UNIX computer. The DEA methods are also implemented on UNIX. Thestochastic frontier models are estimated on a P.C. using the program FRONTIER developedby Coelli (1991). Appendix A gives a more detailed discussion on the computer programfor the DEA and the stochastic frontier models. Appendix B gives a detailed description ofthe data generated.As mentioned in Chapter 5, the choice of a particular functional form mostly dependson its flexibility when there is no prior information on the true production technology1. Forthe deterministic frontier function and the deterministic core of the stochastic frontierfunction, therefore, we assume the following translog functional form:lily= pO+Eplnx.JEEP.lnxinx.+Eykhlzk (6.1)1=1 2=1j=1 ‘ k=1where y is the output, x, are the inputs, Zk are the exogenous variables, and i3, ‘y are thecoefficients to be estimated. The translog function has been shown to be quite flexible, andhas been widely used in empirical applications. In this study, we use a second order1 The simulated data sets in the Monte Carlo experiments are treated as “real world” datasets where information on the underlying production technology and error distributions isunknown.94approximation on inputs and a first order approximation on exogenous variables2.6.1 The Effects of Sample SizeThe first set of experiments is concerned with how well the three methods performwhen applied to samples of different sizes. Three sample sizes are examined: 100, 250, and500. For each of the three sample sizes, three sets of data are generated by assuming threedifferent underlying production technologies. The three technologies are specified by settingp in equation (5.1) equal to -0.67, -0.25, and +2.0, with the corresponding input substitutionat 3.03, 1.33, and 0.33, respectively. Thus, nine sets of sample data are generated, and thethree methods are applied to each of these nine data sets.The mean efficiency estimates are listed in Table 6.1.1 while the mean absolutedeviations (MAD) of the efficiency estimates are listed in Table 6.1.2. By comparing themean efficiency estimates by the stochastic frontier method and the true means in Table 6.1.1and looking at the corresponding MAD values in Table 6.1.2, we can see that the efficiencyestimates by the stochastic frontier method approximate the true efficiency levels very well,and remain quite stable over different sample sizes. By comparing the MAD values for theDEA and the deterministic frontier method (Table 6.1.2), it is found that the DEA efficiencyestimates are closer to the true efficiency values than those by the deterministic frontiermethod. It is also noted that the deterministic frontier method appears to underestimate theefficiency level for all three sample sizes, while the DEA method appears to overestimate2 Note that equation (6.1) specifies the frontier function to be estimated by the one-stepprocedure. As for the first stage of the two-step procedure, we simply ignore the exogenousvariables.Chapter 6 &periment Results 95Table 6.1.1The Effects of Sample SizeThe Means of Estimated EfficiencySample Size N=l00 N250 N=5001. o12=c113=o23=3.03. 0.6567 0.6561 0.6563True Mean (0.0186) (0.0157) (0.0092)Deterministic 0.5295 0.4983 0.4851(0.0469) (0.0281) (0.0281)Stochastic 0.6586 0.6566 0.6549(0.0330) (0.0218) (0.0129)DEA 0.6898 0.6257 0.59 13(0.0279) (0.0190) (0.0146)2. 1213r3—1.33 0.6567 0.6561 0.6563True Mean (0.0186) (0.0157) (0.0092)Deterministic 0.5293 0.5041 0.4861(0.0456) (0.0275) (0.0292)Stochastic 0.6639 0.6569 0.6556(0.0305) (0.0219) (0.0132)DEA 0.7141 0.6502 0.6138(0.0234) (0.0188) (0.0131)3.Or12=o3T,=O.33: 0.6567 0.6561 0.6563True Mean (0.0186) (0.0157) (0.0092)Deterministic 0.4899 0.4149 0.3853(0.0526) (0.07 15) (0.0562)Stochastic 0.6599 0.659 1 0.6574(0.0755) (0.0259) (0.0179)DEA 0.7219 0.6582 0.6197(0.0238) (0.0221) (0.0142)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 96Table 6.1.2The Effects of Sample SizeThe Mean Absolute DeviationsSample Size N=lO0 N250 N==5001. 02=dhi3—023=3.03.Deterministic 0.1435 0.1628 0.1737(0.0340) (0.0275) (0.0248)Stochastic 0.0836 0.0755 0.0695(0.0125) (0.0045) (0.0030)DEA 0.1113 0.1075 0.1097(0.0012) (0.0102) (0.0081)2.12=oi3r=1.33Deterministic 0.1431 0.1576 0.1726(0.0322) (0.0275) (0.0266)Stochastic 0.0808 0.0745 0.0685(0.0103) (0.0043) (0.0028)DEA 0.1164 0.1026 0.1004(0.0111) (0.0083) (0.0073)3.or12=u3_o20. 3:Deterministic 0.1807 0.2453 0.2727(0.0469) (0.0679) (0.0564)Stochastic 0.1047 0.0905 0.0833(0.0477) (0.0067) (0.0055)DEA 0.1143 0.0959 0.0941(0.0093) (0.0074) (0.0064)Note that standard deviations are in parenthesis.Chapter 6 &periment Results 97the efficiency level for the sample size of 100 but underestimate it for the sample size of500.As the sample size increases, the mean efficiency levels estimated by all threemethods tend to fall as shown in Table 6.1.1. The most noticeable drop is observed in theestimates by the DEA method. This is because that the percentage of efficient observationsin the sample would decrease as the sample size increases, and thus results in a decrease inthe mean efficiency level. In terms of the MAD values, both the DEA and the stochasticfrontier method show a slight improvement in their performance as the sample size increases.However, the opposite is true for the deterministic frontier method.By comparing the mean efficiency estimates and the MAD values under the threedifferent production technologies, it is noted that the performance of a particular methoddoes vary with the underlying production technology. The stochastic frontier method is theleast sensitive to the underlying production technology, it does not show any significantchanges when the underlying technology is changed. However, when the elasticity of inputsubstitution is less than unity the performance of the deterministic frontier methoddeteriorates substantially in terms of the MADs with the sample size. On the other hand,the performance of the DEA appears to improve slightly in terms of the MAD values as theelasticity of input substitution of the underlying production technology decreases. However,the relative performance of the three methods is not critically affected by assuming differentunderlying production technologies.The correlation coefficients between the estimated and the true efficiencies are listedin Table 6.1.3 while the rank correlations between the true and estimated efficiency levelsChapter 6 Experiment Results 98Table 6.1.3The Effects of Sample SizeCorrelation Between the True and Estimated EfficiencySample Size N=100 N=250 N=5001. 12—3-23M3Deterministic 0.8357 0.8609 0.8781(0.0271) (0.0195) (0.0119)Stochastic 0.8657 0.8900 0.9028(0.0353) (0.0128) (0.0087)DEA 0.7394 0.7818 0.8228(0.0619) (0.0451) (0.0234)2. (112 l3123—1.33Deterministic 0.8381 0.8644 0.8820(0.0253) (0.0193) (0.01 16)Stochastic 0.8718 0.8935 0.9066(0.0314) (0.0128) (0.0077)DEA 0.7464 0.7884 0.8281(0.0587) (0.0412) (0.0208)3. 12 13 230.33Deterministic 0.7845 0.7989 0.8201(0.0341) (0.0234) (0.0195)Stochastic 0.8304 0.8453 0.8610(0.0411) (0.0178) (0.0154)DEA 0.7659 0.8080 0.8407(0.0520) (0.0384) (0.0202)Note that standard deviations are in parenthesis.Chapter 6 &periment Results 99Table 6.1.4The Effects of Sample SizeRank Correlation Between the True and Estimated EfficiencySample Size N= 100 N=250 N=5001. I2 13 23—3.03Deterministic 0.8385 0.8714 0.8892(0.0365) (0.0223) (0.0111)Stochastic 0.8496 0.8764 0.8913(0.0372) (0.0168) (0.0095)DEA 0.7087 0.7889 0.8384(0.0765) (0.0427) (0.0212)2. 1213T23.Deterministic 0.8415 0.8742 0.8929(0.0347) (0.0220) (0.0104)Stochastic 0.8545 0.8804 0.8952(0.0348) (0.0171) (0.0083)DEA 0.7042 0.7855 0.8354(0.0743) (0.0395) (0.0195)3. uj2=o13o=O.3Deterministic 0.7972 0.8290 0.8471(0.043 1) (0.0266) (0.0165)Stochastic 0.8147 0.8453 0.8459(0.0461) (0.0178) (0.0161)DEA 0.7208 0.8015 0.8458(0.0689) (0.0385) (0.0193)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 100are listed in Table 6.1.4. Since we are more interested in the relative efficiency rankingsamong the DMUs, we rely more on the rank correlation coefficients in evaluating therelative performance of the three methods. The results in Table 6.1.3 and Table 6.1.4clearly show that in terms of correlation and rank correlations all three methods performbetter with a larger sample. The rank correlations for the stochastic frontier method and thedeterministic frontier method are very close. That is, the stochastic frontier method and thedeterministic frontier method would give very similar efficiency rankings of the DMUs. Therank correlations (and the correlation coefficients) for the DEA method are noticeably lowerthan those for the parametric methods, indicating that the two parametric frontier methodsperform better than the DEA method. However, the gaps in their performance becomesmaller as sample size increases. A closer look at the results in Table 6.1.3 and Table 6.1.4also shows that the performance of the two parametric methods deteriorates as the elasticityof input substitution of the underlying production technology departs from unity, but theperformance of the DEA method improves as the elasticity of input substitution decreases.6.2 The Effects of Input RangeIn this section, the relative performance of the three methods is examined in relationto the variations in input variables. The input variables, X, are generated as X= e”, wherew=3+rx0,and x0 is drawn from N(0,1). By varying the values of r, one can vary theranges over which input variables are taking values. In this set of experiments, r is giventhe values of 0.1, 1, and 1.5. The summary statistics for input variables are listed in Table6.2.1. All three input variables are assumed to vary over the same value ranges. Note thatChapter 6 Experiment Results 101the entries in Table 6.2.1 are the averages over 25 replications of the respective statistics.Table 6.2.1The Effects of Input RangeThe Statistics of Input Variablesr=0.1 r=1.0 Tl.SMean of X 20.17 33.51 60.82Standard Deviation of X 2.02 43.27 149.45Maximum of X 26.63 382.84 1674.0Minimum of X 15.08 1.28 0.40The mean efficiency estimates are listed in Table 6.2.2. For the deterministic frontiermethod, it can be seen that the mean efficiency level tends to decrease as r increases. Thisis because the efficiency estimates by the deterministic frontier method are measured againstthe largest OLS residual, and the OLS residuals are affected by the input variations. Thelarger the variations in the input variables, the larger the variations in the OLS residuals, andthus results in a lower average efficiency estimates by the deterministic frontier. On theother hand, the results in Table 6.2.2 shows that the mean efficiency estimates by the DEAmethod tend to rise with the increasing variations in input variables. This is because that inmaximizing the ratio of weighted output over weighted input for a particular observation, theDEA method might give higher efficiency ratings to those observations which have inputswith very small values even when there are excess consumptions of other inputs. Therefore,the average efficiency estimates by the DEA could be higher when there are large variationsin input variables. The mean efficiency estimates by the stochastic frontier method isChapter 6 Experiment Results 102Table 6.2.2The Effects of Input RangeThe Means of Estimated Efficiencyr0.1 r=1.0 r=1.51. 12—I3—23—3.03 0.6561 0.6561 0.6561True Mean (00157) (0.0157) (0.0157)Deterministic 0.5071 0.4983 0.4699(0.0364) (0.0281) (0.041 1)Stochastic 0.6528 0.6566 0.6597(0.0218) (0.0218) (0.0214)DEA 0.5962 0.6257 0.6407(0.0252) (0.0190) (0.0191)2. o2=o13=o23=1.33 0.6561 0.6561 0.6561True Mean (0.0157) (0.0157) (0.0157)Deterministic 0.5071 0.5041 0.499 1(0.0364) (0.0275) (0.0271)Stochastic 0.6528 0.6569 0.6614(0.0217) (0.0219) (0.0192)DEA 0.5968 0.6502 0.678 1(0.0251) (0.0188) (0.0185)3.o2=o13a0. 3 0.6561 0.6561 0.6561True Mean (0.0157) (0.0157) (0.0157)Deterministic 0.5071 0.4149 0.3 135(0.0363) (0.0715) (0.0721)Stochastic 0.6528 0.6591 0.6678(0.0218) (0.0259) (0.0789)DEA 0.5995 0.6582 0.6653(0.0249) (0.0221) (0.0199)Note that the standard deviations are in the parenthesis.Chapter 6 &periment Results 103relatively stable compared with those by the DEA and the deterministic frontier method.However, Table 6.2.2 indicates a slight increase in the mean efficiency estimates by thestochastic frontier method. One explanation for the slight increase in the mean efficiencyestimates by the stochastic frontier method is that when there are large variations in theinputs the production possibility set is scattered over a larger area (or space). As a result,the stochastic production frontier may have a wider “band”, and the resulting averageefficiency level would hence be higher.Table 6.2.3 presents the mean absolute deviations (MAD) of the efficiency estimatesfor different values of r. There are very small fluctuations in the MAD values for the DEAwhen T increases. Therefore, the DEA is considered as being rather robust to the variationsin input variables in terms of the MADs. Under the first two production technologies wherethe elasticity of input substitution is over one, the stochastic frontier method shows very littlechanges in its MAD values as the variations in input variables change, but the deterministicfrontier method sees slight rises in its MAD values as r increases. Under the thirdproduction technology where the elasticity of input substitution is less than one, however,the performance of the two parametric methods deteriorates noticeably when the variationsin input variables become larger.Examination of the correlation coefficients in Table 6.2.4 and the rank correlationsin Table 6.2.5 draws similar conclusions: (1) the DEA method does not appear to be verysensitive to variations in the input variables; (2) the two parametric methods yield relativelystable estimates when the elasticity of input substitution of the underlying production functionis over one; and (3) when the elasticity of input substitution is set at 0.33, the degree ofChapter 6 Experiment Results 104Table 6.2.3The Effects of Input RangeThe Mean Absolute Deviationsr0.1 r=1.01.Deterministic 0.1545 0.1628 0.1902(0.0287) (0.0275) (0.0366)Stochastic 0.0714 0.0755 0.0793(0.0049) (0.0045) (0.0054)DEA 0.1169 0.1075 0.1024(0.0149) (0.0102) (0.0097)2.u121323. 3Deterministic 0.1545 0.1576 0.1613(0.0287) (0.0276) (0.0222)Stochastic 0.0720 0.0745 0.0725(0.0055) (0.0043) (0.0051)DEA 0.1166 0.1026 0.1001(0.0148) (0.0083) (0.0095)3. 1213230.33Deterministic 0.1545 0.2453 0.3455(0.0286) (0.0679) (0.0703)Stochastic 0.0721 0.0905 0. 1214(0.0054) (0.0067) (0.0442)DEA 0.1150 0.0959 0.0987(0.0144) (0.0074) (0.0070)Note that the standard deviations are in the parenthesis.Chapter 6 Experiment Results 105Table 6.2.4The Effects of Input RangesCorrelation Between the True and Estimated EfficiencySample Size r = 0.1 r = 1.0 = 1.51. o12=r13=r23=3.03Deterministic 0.8777 0.8609 0.8434(0.0167) (0.0195) (0.0222)Stochastic 0.9005 0.8900 0.8754(0.0145) (0.0128) (0.0193)DEA 0.7865 0.7818 0.7909(0.0449) (0.0451) (0.0362)2. o12=o13=o23=1.33Deterministic 0.8777 0.8644 0.8701(0.0167) (0.0193) (0.0173)Stochastic 0.9005 0.8935 0.8984(0.0146) (0.0128) (0.0153)DEA 0.7869 0.7884 0.7992(0.0448) (0.0412) (0.0370)3. o12=o13=o23=0.33Deterministic 0.8777 0.7989 0.6914(0.0167) (0.0234) (0.0449)Stochastic 0.9005 0.8453 0.7443(0.0146) (0.0178) (0.0320)DEA 0.7886 0.8080 0.8022(0.0447) (0.0384) (0.0300)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 106Table 6.2.5The Effects of Input RangesRank Correlation Between the True and Estimated EfficiencySample Size r =0.1 r 1.0 T = 1.51. 12T3 33MDeterministic 0.8835 0.8714 0.8553(0.0201) (0.0223) (0.0250)Stochastic 0.8872 0.8764 0.8601(0.0180) (0.0168) (0.0238)DEA 0.8046 0.7889 0.7891(0.0386) (0.0427) (0.0432)2. 12 13 23—1.33Deterministic 0.8835 0.8742 0.8778(0.0201) (0.0220) (0.0218)Stochastic 0.8872 0.8804 0.8838(0.0181) (0.0171) (0.0213)DEA 0.8047 0.7855 0.7817(0.0387) (0.0395) (0.0466)3. 12=l3=0.33Deterministic 0.8835 0.8290 0.7544(0.0201) (0.0266) (0.0342)Stochastic 0.8873 0.8453 0.7444(0.0181) (0.0178) (0.0362)DEA 0.8051 0.8015 0.7905(0.0385) (0.0385) (0.033 1)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 107input variations has considerable negative effects on the performance of the two parametricmethods. In particular, at r = 1.5, the DEA method even outperforms the stochasticfrontier method under the third production technology whereas in all other cases the DEAis dominated by the stochastic frontier method.6.3 The Effects of NoiseThis section examines the effects of statistical noise on the relative performance ofthe three methods. As stated in Chapter 5, the noise term is generated from a normaldistribution with zero mean. By giving the noise term different standard deviations, we areable to create different noise levels. In particular, the standard deviation is given the valuesof 0.03, 0.15, 0.25, 0.50, and 0.75. The corresponding statistics for the noise term arelisted in Table 6.3.1. The entries in Table 6.3.1 are the averages over 25 replications of themean statistics from each replication. Note that the means of ev are sample means, thus theyare not necessarily equal to one.Table 6.3.1The Effects of NoiseThe Statistics of Noise Termr=O.O3 oO.15 Or=O.25 =O.5O o=O.75Meanofev 1.001 1.013 1.036 1.136 1.310St.Dev. of ev 0.030 0.154 0.267 0.594 1.080Maximum of ev 1.089 1.542 2.068 4.121 7.722Minimum of 0.915 0.670 0.512 0.249 0.126The mean efficiency estimates are shown in Table 6.3.2. As the noise levelChapter 6 Experiment Results 108Table 6.3.2The Effects of NoiseThe Means of Estimated Efficiencyy=0.03 o=0.15 O=0.25 o,,=0.50 o=0.75l.O213T—3.03 0.6561 0.6561 0.6561 0.6561 0.6561True Mean (0.0157) (0.0157) (0.0157) (0.0157) (0.0157)Deterministic 0.5818 0.4983 0.4005 0.2471 0.1627(0.0355) (0.0281) (0.0454) (0.0513) (0.0410)Stochastic 0.6674 0.6566 0.6720 0.6895 0.6804(0.0155) (0.0218) (0.0284) (0.1185) (0.1621)DEA 0.7051 0.6257 0.5403 0.3970 0.3032(0.0160) (0.0190) (0.03 14) (0.0362) (0.0377)2. l2I3= 1.33 0.656 1 0.656 1 0.656 1 0.6561 0.656 1True Mean (0.0157) (0.0157) (0.0157) (0.0157) (0.0157)Deterministic 0.5928 0.5041 0.4011 0.2497 0. 1624(0.0307) (0.0275) (0.0437) (0.0504) (0.0419)Stochastic 0.6715 0.6569 0.6717 0.6833 0.6829(0.0115) (0.0219) (0.0278) (0.1151) (0. 1668)DEA 0.7225 0.6502 0.5643 0.4153 0.3 162(0.0154) (0.0188) (0.0290) (0.0367) (0.0376)3.o12=3g20. 3 0.6561 0.6561 0.6561 0.6561 0.6561True Mean (0.0157) (0.0157) (0.0157) (0.0157) (0.0157)Deterministic 0.4458 0.4149 0.3652 0.2391 0. 1596(0.0722) (0.0715) (0.0566) (0.0568) (0.0405)Stochastic 0.6465 0.659 1 0.6780 0.7024 0.6824(0.0197) (0.0259) (0.0498) (0. 1345) (0. 1601)DEA 0.7273 0.6582 0.5724 0.4185 0.3154(0.0149) (0.0221) (0.0298) (0.0390) (0.0401)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 109increases, the mean efficiency estimates by the DEA and the deterministic frontier methodfall dramatically, but the mean efficiency estimates by the stochastic frontier method remainrelatively stable. This is expected since both the DEA and the deterministic frontier methodtreat all the deviation from the “best practice” frontier as inefficiency, thus the averageefficiency level will fall as the noise level rise. On the other hand, the stochastic frontiermethod allows random variation of the frontier across the observations, thus it is much lesssensitive to the noise level than the DEA and the deterministic frontier method. Anotherobservation from Table 6.3.2 is that the deterministic frontier method tends to underestimatethe “true” efficiency level in all cases. The DEA method appears to overestimate the “true”efficiency level when the noise level is low due to inner envelope property of the DEA’slinear frontier3,whereas it tends to underestimate the efficiency level when the noise levelis increased. The stochastic frontier method appears to overestimate the “true” efficiencylevel in almost all cases, however, the actual levels of mean efficiency estimates slightlyfluctuate as the noise level changes and do not follow a clear upward or downward pattern.As shown in Table 6.3.3, high noise level causes large MAD values for all threemethods, that is, the high noise level is shown to impose considerable negative effects on theaccuracy of the efficiency estimates, especially for the DEA and the deterministic frontiermethod. For all noise levels, the efficiency estimates by the stochastic frontier method havethe least absolute deviation from the “true” efficiency level, whereas those by thedeterministic frontier method have the largest absolute deviation.It is only relevant when the noise level is very low.Chapter 6 Experiment Results 110Table 6.3.3The Effects of NoiseThe Mean Absolute Deviationstr=0.03 o=0.15 =0.25 o=0.50 o=0.751 I2—0i3—023—3.03Deterministic 0.0807 0.1628 0.2600 0.4130 0.4973(0.0336) (0.0275) (0.0455) (0.0508) (0.0429)Stochastic 0.0319 0.0755 0.1055 0.1681 0.2042(0.0049) (0.0045) (0.0065) (0.0501) (0.0562)DEA 0.0534 0.1075 0.1806 0.3 178 0.4089(0.0088) (0.0102) (0.0251) (0.0297) (0.0339)2. 12 13 = 23 =1.33Deterministic 0.0695 0.1576 0.2592 0.4104 0.4976(0.0297) (0.0276) (0.0437) (0.0495) (0.0439)Stochastic 0.0282 0.0745 0.1049 0.1656 0.2068(0.0048) (0.0043) (0.0070) (0.0483) (0.0586)DEA 0.0682 0. 1026 0.1678 0.3042 0.3990(0.0089) (0.0083) (0.0221) (0.0283) (0.0333)3.I2=13=23=O.33Deterministic 0.2142 0.2453 0.2949 0.4210 0.5003(0.0681) (0.0679) (0.0565) (0.0561) (0.0418)Stochastic 0.0656 0.0905 0.1175 0.1778 0.2036(0.0056) (0.0067) (0.0187) (0.0589) (0.0547)DEA 0.0723 0.0959 0. 1594 0.2972 0.3965(0.0099) (0.0074) (0.0213) (0.0294) (0.0331)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 111The correlation coefficients under different noise levels are listed in Table 6.3.4 whilethe rank correlations are list in Table 6.3.5. In terms of the correlations and the rankcorrelations, the two parametric methods perform marginally better than the DEA method.The stochastic frontier method shows slight advantage over the deterministic frontier methodin terms of the correlation coefficients, however, their performances are very close in termsof the rank correlations. The DEA appears to be a rather competitive alternative to the twoparametric methods in terms of the rank correlations especially when the noise level is high.In general, the performances of all three methods deteriorate noticeably as the noiselevel increases. When the statistical noise level is relatively low ( u < 0.25), all threemethods perform reasonably well, and can be considered as reliable in estimating firm (orobservation) specific efficiencies. However, in the presence of high noise levels, the resultsfrom any one of the three methods would need careful examination to prevent misleadingresults.Chapter 6 Experiment Results 112Table 6.3.4The Effects of NoiseCorrelation Between The True and Estimated Efficiencycr=0.03 o=0.15 o=0.25 o=0.50 oc,,=0.751.12—Oj3—O23—3.03Deterministic 0.9576 0.8609 0.7352 0.4894 0.3 189(0.0156) (0.0195) (0.0299) (0.0558) (0.0514)Stochastic 0.9786 0.8900 0.7943 0.5724 0.3985(0.0051) (0.0128) (0.0278) (0.0562) (0.0599)DEA 0.9156 0.7818 0.6422 0.4018 0.2547(0.0355) (0.0451) (0.063 1) (0.0755) (0.0707)2.o12=r31.33Deterministic 0.9642 0.8644 0.7383 0.4917 0.3199(0.0157) (0.0193) (0.0288) (0.0552) (0.0516)Stochastic 0.9853 0.8935 0.7968 0.5737 0.3997(0.0045) (0.0128) (0.0267) (0.0545) (0.0594)DEA 0.9136 0.7884 0.6522 0.4133 0.2617(0.0321) (0.0412) (0.0614) (0.0735) (0.0704)3.or12=3i20. 3Deterministic 0.8769 0.7989 0.6977 0.4736 0.3 129(0.0235) (0.0234) (0.03 12) (0.0571) (0.0487)Stochastic 0.9088 0.8453 0.7638 0.5582 0.3926(0.0165) (0.0178) (0.0302) (0.0582) (0.0610)DEA 0.9210 0.8080 0.6685 0.4285 0.2710(0.0319) (0.0384) (0.0560) (0.0768) (0.0704)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 113Table 6.3.5The Effects of NoiseRank Correlation Between The True and Estimated Efficiencyi=0.O3 o=0.l5 u=0.25 o=0.50 o=0.751.j2—Oj3—O3—3.03Deterministic 0.9579 0.8714 0.7680 0.5460 0.3756(0.0171) (0.0223) (0.0367) (0.0660) (0.0587)Stochastic 0.9750 0.8764 0.7707 0.5471 0.3756(0.0058) (0.0168) (0.0362) (0.0656) (0.0582)DEA 0.9006 0.7889 0.6817 0.4748 0.3304(0.0391) (0.0427) (0.0575) (0.0735) (0.0636)2.12=o331.33Deterministic 0.9630 0.8742 0.7703 0.5472 0.3768(0.0167) (0.0220) (0.0357) (0.0650) (0.0585)Stochastic 0.9824 0.8804 0.7736 0.548 1 0.3767(0.0056) (0.0171) (0.0350) (0.0644) (0.0579)DEA 0.8954 0.7855 0.6798 0.4767 0.3324(0.0347) (0.0395) (0.0564) (0.0720) (0.0625)3.12=o13=o23=0.33Deterministic 0.899 1 0.8290 0.7372 0.5336 0.3700(0.0206) (0.0266) (0.0399) (0.0665) (0.0594)Stochastic 0.8995 0.8284 0.7383 0.5339 0.3701(0.0188) (0.0222) (0.0400) (0.0665) (0.0589)DEA 0.9038 0.8015 0.6918 0.4887 0.3411(0.0373) (0.0385) (0.0515) (0.0703) (0.0623)Note that standard deviations are in parenthesis.Chapter 6 Experiment Results 1146.4 The Effects of Exogenous VariablesIn the presence of exogenous variables, the observed productive performance reflectsthe combined outcome of both efficiency and environments. The usefulness of a particularefficiency measurement method depends on its ability to distinguish the effects ofenvironments from the effects of efficiency.This section examines the relative performance of the two-step DEA-TOBITprocedure and the parametric methods in measuring efficiency in the presence of exogenousvariables. The exogenous variable is generated as az where z is from N(0, 1). In this setof experiments, a is given the values of 0.05, 0.10, 0.25, 0.50, 0.75, 1.0, and 2.0. Someof these cases, such as when a =1, 2, are probably not realistic, they are included here justto show how dramatically the results might be affected. The statistics of the exogenousvariable are given in Table 6.4.1.Table 6.4.1The Effects of Exogenous VariablesThe Statistics of the Exogenous Variablesc=O.O5 y=O.1O a=O.25 a0.50 a0.75 a1.OO 2.OO1.003 1.005 1.035 1.238 1.320 1.615 7.6580.050 0.101 0.262 0.613 1.136 2.036 34.681.148 1.316 2.000 4.203 8.245 17.59 447.90.873 0.749 0.501 0.247 0.127 0.062 0.004First, we look at the one-step parametric methods in comparison with the DEA-Tobitprocedure. The mean efficiency estimates are presented in Table 6.4.2 and the meanChapter 6 Experiment Results 115Table 6.4.2The Effects of Exogenous VariablesThe Meinc of Efrwenev tmiteQft=O.OS ft=O.10 o=0.25 a=0.50 v0.75 a=11. j=3.03 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561True Mean (0.016) (0.016) (0.016) (0.016) (0.016) (0.016) (0.016)Determ 0.5006 0.4974 0.4946 0.4990 0.4959 0.4960 0.5004(0.03 1) (0.029) (0.026) (0.030) (0.028) (0.026) (0.027)Stoch 0.6563 0.6556 0.6546 0.6570 0.6551 0.6561 0.6575(0.025) (0.024) (0.023) (0.023) (0.024) (0.022) (0.022)DEAl 0.6194 0.6004 0.5317 0.3850 0.2864 0.2181 0.1026(0.020) (0.021) (0.029) (0.034) (0.036) (0.040) (0.026)DEA2 0.5751 0.5198 0.3715 0.1757 0.0889 0.0958 0.0488(0.040) (0.046) (0.050) (0.044) (0.033) (0.022) (0.094)2.o1.33 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561True Mean (0.016) (0.016) (0.016) (0.016) (0.016) (0.016) (0.016)Determ 0.5029 0.5040 0.5008 0.5046 0.5032 0.5030 0.5049(0.029) (0.028) (0.025) (0.029) (0.027) (0.025) (0.026)Stoch 0.6556 0.6552 0.6546 0.6572 0.6547 0.6565 0.6576(0.024) (0.023) (0.023) (0.022) (0.024) (0.022) (0.022)DEAl 0.6449 0.6268 0.5560 0.4028 0.2977 0.2243 0. 1033(0.020) (0.022) (0.029) (0.034) (0.034) (0.041) (0.027)DEA2 0.5971 0.5389 0.3882 0.1841 0.0926 0.0962 0.0495(0.040) (0.044) (0.054) (0.046) (0.035) (0.021) (0.010)3.orrO.33 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561True Mean (0.016) (0.016) (0.016) (0.016) (0.016) (0.016) (0.016)Determ 0.4145 0.4140 0.4138 0.4166 0.4132 0.4132 0.4169(0.071) (0.072) (0.072) (0.072) (0.070) (0.070) (0.071)Stoch 0.6603 0.6604 0.6589 0.6604 0.6596 0.6574 0.6583(0.031) (0.030) (0.028) (0.031) (0.030) (0.026) (0.026)DEAl 0.6538 0.6353 0.5667 0.4066 0.2989 0.2174 0.1009(0.023) (0.022) (0.035) (0.037) (0.031) (0.043) (0.027)DEA2 0.6058 0.5496 0.3965 0.1832 0.0918 0.0917 0.0473(0.040) (0.043) (0.055) (0.044) (0.034) (0.018) (0.010)Note that the standard deviations are in the parenthesis.DEAl-- Gross DEA efficiency measuresDEA2-- Residual DEA efficiency measures from DEA-TOBITChapter 6 Experiment Results 116Table 6.4.3The Effects of Exogenous VariablesThe Mein T)pviqtinnc— . £J3JILL —a=0.05 a=0.10 cz=0.25 a0.50 0.75 11. o,=3.03Determ 0.1612 0.1639 0.1662 0.1625 0.1650 0.1647 0.1613(0.029) (0.028) (0.027) (0.028) (0.026) (0.027) (0.027)Stoch 0.0760 0.0761 0.0757 0.0755 0.0760 0.0764 0.0757(0.005) (0.004) (0.004) (0.004) (0.005) (0.004) (0.005)DEAl 0.1126 0.1300 0.1952 0.3302 0.4245 0.4874 0.5940(0.011) (0.014) (0.023) (0.030) (0.033) (0.034) (0.026)DEA2 0.1293 0.1680 0.2974 0.4861 0.5733 0.6112 0.6479(0.022) (0.036) (0.052) (0.042) (0.040) (0.036 (0.020)2. = 1.33Determ 0.1588 0. 1578 0.1605 0. 1576 0. 1586 0. 1584 0. 1572(0.027) (0.028) (0.027) (0.028) (0.026) (0.025) (0.027)Stoch 0.075 1 0.0749 0.0749 0.0747 0.0750 0.0753 0.0746(0.005) (0.004) (0.004) (0.004) (0.005) (0.004) (0.005)DEAl 0.1068 0.1212 0.1823 0.3175 0.4154 0.4824 0.5930(0.009) (0.011) (0.020) (0.029) (0.031) (0.034) (0.027)DEA2 0.1182 0.1536 0.2813 0.4777 0.5697 0.6100 0.6502(0.018) (0.032) (0.056) (0.043) (0.041) (0.034) (0.014)3.01=0.33Determ 0.2458 0.2463 0.2466 0.2438 0.2471 0.2468 0.2435(0,067) (0.068) (0.067) (0.068) (0.065) (0.067) (0.068)Stoch 0.0911 0.0909 0.0901 0.0911 0,0907 0.0907 0.0907(0.007) (0.007) (0.006) (0.001) (0.007) (0.006) (0.007)DEAl 0.1000 0.1123 0.1746 0.3107 0.4107 0.4846 0.5935(0.008) (0.009) (0.020) (0.029) (0.030) (0.035) (0.028)DEA2 0.1088 0. 1418 0.2728 0.4786 0.5704 0.6126 0.6503(0.016) (0.030) (0.057) (0.042) (0.041) (0.030) (0.014)Note that the standard deviations are in the parenthesis.DEA 1 -- Gross DEA efficiency measuresDEA2 -- Residual DEA efficiency measures from DEA-TOBITChapter 6 Experiment Results 117absolute deviations are presented in Table 6.4.3. Recall from Chapter 4, the DEA-TOBITprocedure first estimates a gross efficiency index using the DEA method, and then uses theTOBIT regression to identify the effects of exogenous variables on this gross efficiency indexand to compute a residual efficiency index. The gross efficiency estimates and the MADsfrom the first stage are denoted by DEAl in Table 6.4.2 and Table 6.4.3, and the residualefficiency estimates and the MADs from the second stage are denoted by DEA2. Determ andStoch denote the one-step deterministic and stochastic frontier methods, respectively. Inparticular, the one-step procedure estimates the production frontier function specified byequation (6.1), and the influence of exogenous variables are controlled for while measuringefficiency.The results in these two tables draw essentially the same conclusion. They indicatethat the magnitude of exogenous variables does not have any significant effects on theefficiency estimates by the two parametric methods as long as the exogenous variables arecorrectly identified and are accounted for in the estimation. In such cases, there is nomisspecification related to those identifiable exogenous variables, hence, they do notcontribute to additional disturbance in measuring efficiency. On the other hand, the meangross efficiency estimates from the DEA-Tobit procedure (DEAl) fall dramatically as themagnitude of exogenous variable increases especially when cy > 0.25. The residualefficiency estimates from the DEA-TOBIT procedure (DEA2) is even more underestimatedthan the gross DEA efficiency measures (DEAl). This could partly be explained by the factthat in the second stage of the DEA-TOBIT procedure, only the observation with the largestpositive residual from the TOBIT regression is considered as efficient, and consequentlyChapter 6 Experiment Results 118average efficiency estimates would be lower. However, the DEA-Tobit procedure performsslightly better than the deterministic frontier when the magnitude of exogenous variables islow (a < 0.10).Table 6.4.4 lists the correlation coefficients and Table 6.4.5 lists the rank correlationsfor different magnitudes of exogenous variables. For the two parametric methods, there isessentially no change in the correlations and the rank correlations over the range of aexamined. That is, the exogenous variables have little effects on the efficiency estimates bythe two parametric methods. However, the performance of the DEA deteriorates rapidly interms of the correlations and the rank correlations as a increases, especially when a is largerthan 0.25. The second stage regression does not appear to be effective in improving theperformance of the DEA in terms of correlation coefficient when a is over 0.75, but it doesimprove the DEA’s performance considerably in terms of rank correlation. When themagnitude of exogenous variables is modest, the residual efficiency estimates from thesecond stage of the DEA-TOBIT procedure do approximate the “tru&’ efficiencies better thanthe gross DEA efficiency estimates. Overall, the DEA-Tobit procedure performs reasonablywell in terms of the rank correlations except for the cases where the magnitude of exogenousvariables is set unrealistically high (a =1, 2).In summary, if the exogenous variables can be correctly identified and incorporatedin estimating the production frontiers, the parametric methods have a natural advantage overthe DEA-TOBIT procedure in dealing with the effects of exogenous variables. The DEATOBIT procedure appears to be a reasonable competitor when the effects of the exogenousvariable are modest. When the magnitude of exogenous variables is high, the DEA-TOBITChapter 6 Experiment Results 119Table 6.4.4The Effects of Exogenous VariablesCorrelation Between The True and Estimated Efficiencyo=0.05 =0.10 a0.25 cr0.50 cv 0 .75 cv11.o =303Determ 0.8586 0.8582 0.8585 0.8587 0.8584 0.8589 0.8578(0.019) (0.020) (0.019) (0.020) (0.019) (0.019) (0.021)Stoch 0.8884 0.8877 0.8885 0.8888 0.8884 0.8880 0.8889(0.015) (0.015) (0.014) (0.013) (0.014) (0.013) (0.014)DEAl 0.7701 0.7348 0.5938 0.3705 0.2433 0.1812 0.0551(0.046) (0.053) (0.051) (0.073) (0.062) (0.064) (0.084)DEA2 0.7731 0.7527 0.6597 0.4051 0.2027 0.1677 0.0277(0.046) (0.055) (0.065) (0.108) (0.102) (0.087) (0.078)2.o,= 1.33Determ 0.8621 0.8617 0.8619 0.8622 0.8619 0.8623 0.8611(0.019) (0.020) (0.019) (0.019) (0.019) (0.019) (0.021)Stoch 0.8911 0.8907 0.8912 0.8908 0.8915 0.8917 0.8926(0.016) (0.015) (0.015) (0.013) (0.014) (0.013) (0.014)DEAl 0.7774 0.7432 0.6064 0.3816 0.2510 0.1864 0.0573(0.043) (0.049) (0.047) (0.069) (0.061) (0.063) (0.083)DEA2 0.7801 0.7610 0.6762 0.4239 0.2135 0. 1723 0.0298(0.043) (0.051) (0.058) (0.105) (0.105) (0.087) (0.076)3.uO.3Determ 0.7967 0.7959 0.7966 0.7969 0.7969 0.7975 0.7966(0.023) (0.023) (0.023) (0.024) (0.022) (0.023) (0.023)Stoch 0.8432 0.8436 0.8437 0.8438 0.8435 0.8439 0.8436(0.021) (0.021) (0.020) (0.020) (0.021) (0.018) (0.018)DEAl 0.7975 0.7655 0.6229 0.3999 0.2608 0.1899 0.0607(0.039) (0.045) (0.043) (0.061) (0.061) (0.064) (0.083)DEA2 0.8004 0.7837 0.6999 0.4470 0.2204 0. 1685 0.0282(0.039) (0.046) (0.056) (0.097) (0.112) (0.085) (0.075)Note that the standard deviations are in the parenthesis.DEAl -- Gross DEA efficiency measureDEA2-- Residual efficiency measure from DEA-TOBITChapter 6 Experiment Results 120Table 6.4.5The Effects of Exogenous VariablesRank Correlation Between The True and Estimated Efficiencya0.05 a=0.10 a0.25 a=O.50 a=0.75 a1 a=21 =3.03Detnn 0.8693 0.8688 0.8695 0.8693 0.8697 0.8694 0.8684(0.022) (0.022) (0.023) (0.022) (0.023) (0.022) (0.024)Stoch 0.8751 0.8742 0.8747 0.8753 0.8750 0.8745 0.8753(0.018) (0.019) (0.018) (0.017) (0.019) (0.017) (0018)DEAl 0.7808 0.7524 0.6320 0.4383 0.3271 0.2735 0. 1507(0.042) (0.047) (0.042) (0.065) (0.047) (0.064) (0.067)DEA2 0.7860 0.7775 0.7489 0.6653 0.6012 0.4478 0.2057(0.042) (0.046) (0.044) (0.073) (0.083) (0.085) (0.078)2. = 1.33Determ 0.8724 0.8717 0,8724 0.8723 0.8724 0.8720 0.8714(0.021) (0.022) (0.022) (0.022) (0.022) (0.022) (0.023)Stoch 0.8776 0.8773 0.8776 0.8772 0.8783 0.8784 0.8795(0.019) (0.019) (0.019) (0.017) (0.019) (0.017) (0.018)DEAl 0.7776 0.7495 0.6320 0.4420 0.3295 0.2757 0. 1515(0.041) (0.045) (0.039) (0.063) (0.047) (0.064) (0.066)DEA2 0.7821 0.7739 0.7477 0.6708 0.6111 0.4606 0.2057(0.041) (0.045) (0.040) (0.067) (0.082) (0.082) (0.075)3. u =0.33Determ 0.8270 0.8266 0.8269 0.8267 0.8271 0.8275 0.8260(0.027) (0.027) (0.027) (0.026) (0.027) (0.025) (0.027)Stoch 0.8266 0.8268 0.8267 0.8272 0.8271 0.8270 0.8268(0.024) (0.024) (0.023) (0.024) (0.025) (0.022) (0.023)DEAl 0.7932 0.7661 0.6418 0.4551 0.3367 0.2827 0. 1543(0.039) (0.043) (0.039) (0.059) (0.048) (0.064) (0.064)DEA2 0.7982 0.7923 0.7652 0.7099 0.6659 0.4949 0.2137(0.038) (0.042) (0.040) (0.063) (0.087) (0.071) (0.075)Note that the standard deviations are in the parenthesis.DEAl-- Gross DEA efficiency measureDEA2-- Residual efficiency measure from DEA-TOBITChapter 6 Experiment Results 121procedure performs poorly. One explanation for this is that when the effects of exogenousvariables are significant, the DEA gross efficiency scores are distorted so seriously’ that thesecond stage Tobit analysis could not adequately account for the effects of the exogenousvariables. Therefore, if it is necessary to use the DEA-TOBIT procedure in situationswhere there might be large variations in exogenous factors, one must be very cautious inmaking any inferences from the results.The one-step parametric methods clearly have a dominating advantage over the two -step DEA-TOBIT procedure for obvious reasons. The two-step procedure has also been usedwith the parametric methods. With the two-step parametric procedure, the variations inefficiency measures from the first stage are attributed to variations in the exogenous variablesin the second stage (see Kalirajan, 1990 for an example). This two-step parametric methodfirst estimates a frontier production function considering only output and input variables2,andthen uses the TOBIT regression to identify the effects of exogenous variables and to computethe residual efficiencies as in the second stage of the DEA-TOBIT procedure. This two-stepprocedure may be more appealing to policy and decision makers who are interested inimproving efficiency since it relates the exogenous factors (and other explanatory variables)directly to the efficiency performance. It would be interesting to see how the two-stepprocedure, when applied to the parametric methods, performs in comparison with the onestep parametric procedure discussed above.1 This can be confirmed by the extremely low average DEA scores from the DEAestimation.2 As specified by equation (6.1) after removing the exogenous variables.Chapter 6 Experiment Results 122For the sake of simplicity, the performance of the two-step parametric methods isexamined only under the second production function in previous tables where oij = 1.33(i j). This will not bias the comparative results since the results in Table 6.4.2 throughTable 6.4.5 indicate that the performance of all three methods in relation to the magnitudeof exogenous variables does not depend much on the underlying production structures, atleast not in the parameter range considered here.Table 6.4.6 lists the mean efficiency estimates and Table 6.4.8 lists the MAD valuesfrom the two-step methods. In both tables, DET1 denotes the first stage of the deterministicfrontier method, and DET2 refers to the second stage. Similarly, STOCH1 (Stoch2) denotesthe first stage (second stage) of the stochastic frontier method. DEAl and DEA2 are thesame as those in Table 6.4.2 and Table 6.4.3. Again, the stochastic frontier method is ableto filter out the effects of exogenous variables in the first stage fairly well and yieldefficiency estimates very close to their true values as indicated by its small MAD values.The residual efficiency estimates from the second stage of the stochastic frontier method tendto overestimate the mean efficiency level, especially when the effects of exogenous variablesare becoming larger. On the other hand, both the DEA and the deterministic frontier methodunderestimate the mean efficiency level in the first stage, and their abilities to reveal the realpicture of efficiencies in the first stage fall substantially as a increases. The incorporationof exogenous variables in the second stage helps improve the efficiency estimates by thedeterministic frontier method but not those by the DEA. In terms of the MADs, the DEAmethod performs better than the deterministic frontier method in the first stage, and also inthe second stage when a is small. However, when the magnitude of exogenous variablesChapter 6 Experiment Results 123Table 6.4.6The Effects of Exogenous VariablesMean Efficiency Estimates by Two Stage Procedure=0.05 a0.10 a0.25 0.50 y0.75 o12.u=1.33 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561 0.6561True Mean (0.016) (0.016) (0.016) (0.016) (0.016) (0.016) (0.016)DetI 0.4989 0.4776 0.3897 0.2432 0.1430 0.1018 0.0251(0.025) (0.030) (0.039) (0.041) (0.038) (0.037) (0.011)Det2 0.5030 0.5040 0.4923 0.4679 0.4304 0.3958 0.2719(0.028) (0.030) (0.037) (0.033) (0.068) (0.070) (0.094)Stochi 0.6553 0.6594 0.6674 0.7001 0.6785 0.6783 0.5758(0.024) (0.027) (0.050) (0.123) (0.144) (0.155) (0.227)Stoch2 0.6786 0.6804 0.7016 0.8005 0.8341 0.8640 0.8247(0.036) (0.045) (0.081) (0.113) (0.109) (0.114) (0.179)DEAl 0.6449 0.6268 0.5560 0.4028 0.2977 0.2243 0. 1033(0.020) (0.022) (0.029) (0.034) (0,034) (0.041) (0.027)DEA2 0.5971 0.5389 0.3882 0.1841 0.0926 0.0962 0.0495(0.040) (0.044) (0.054) (0.046) (0.035) (0.021) (0.010)Note that standard deviations are in the parenthesis.DET 1 -- efficiency estimates from first stage by deterministic frontier methodDET2 -- residual efficiency estimates from second stage by deterministic methodStoch 1 -- efficiency estimates from first stage by stochastic frontier methodStoch2 -- residual efficiency estimates from second stage by stochastic frontierDEAl -- gross DEA efficiency estimatesDEA2 -- residual efficiency estimates from DEA-TOBITChapter 6 Experiment Results 124Table 6.4.7The Effects of Exogenous VariablesThe Mean Absolute Deviations by Two Stage Procedurer=0.05 a0.lO crO.25 0.5O a=0.75 a12.u= 1.33Detl 0.1627 0.1842 0.2716 0.4175 0.5163 0.5579 0.6344(0.025) (0.025) (0.042) (0.041) (0.038) (0.038) (0.019)Det2 0.1587 0.1580 0.1694 0.1930 0.2302 0.2639 0.3875(0.026) (0.030) (0.037) (0.033) (0.068) (0.068) (0.096)Stochl 0.0765 0.0848 0.1166 0.1754 0.1922 0.2056 0.2584(0.043) (0.077) (0.021) (0.059) (0.050) (0.048) (0.058)Stoch2 0.0799 0.0843 0.1100 0.1831 0.2094 0.2387 0.2468(0.010) (0.017) (0.042) (0.087) (0.083) (0.084) (0.083)DEAl 0.1068 0.1212 0.1823 0.3175 0.4154 0.4824 0.5930(0.009) (0.011) (0.020) (0.029) (0.031) (0.034) (0.027)DEA2 0.1182 0.1536 0.2813 0.4777 0.5697 0.6100 0.6502(0.018) (0.032) (0.056) (0.043) (0.041) (0.034) (0.014)Note that standard deviations are in the parenthesis.DET 1 -- efficiency estimates from first stage by deterministic frontier methodDET2 -- residual efficiency estimates from second stage by deterministic methodStoch 1 -- efficiency estimates from first stage by stochastic frontier methodStoch2 -- residual efficiency estimates from second stage by stochastic frontierDEAl -- gross DEA efficiency estimatesDEA2 -- residual efficiency estimates from DEA-TOBITChapter 6 Experiment Results 125becomes large, the two-step deterministic frontier method appears to have a slight advantageover the DEA. Comparing the results in Table 6.4.2 and Table 6.4.3 with those in Table6.4.6 and Table 6.4.7, it is clear that technically it would be desirable to employ the one-step parametric methods since they produce more accurate efficiency estimates.The performance of the two-step models is also examined in terms of correlation andrank correlation (Table 6.4.8 and Table 6.4.9). The results confirm that the efficiencyestimates would not reflect the true picture of the relative efficiency performance of thesample firms if the effects of exogenous variables are ignored. This is indicated by theobservation that as a increases the correlations and the rank correlations from the first stagefall dramatically for all three methods. The residual efficiency estimates from the secondstage, on the other hand, approximate the true efficiency levels fairly well with the twoparametric methods as indicated by the fairly high correlation and rank correlationcoefficients. Furthermore, it is noted that although the stochastic frontier method performsbetter than the deterministic frontier method in the first stage, the opposite is true for theresidual efficiency estimates from the second stage when a is large. One explanation for thisis that the stochastic frontier method filters out some of the effects of exogenous variablesas noise in the first stage so that the first stage performs better. However, this leaves it withless explanatory power for the exogenous variables in the second stage regression, thusresults in the less satisfactory performance in the second stage.The DEA-TOBIT procedure does not do as well as the parametric methods over thevalues of a considered. However, it performs reasonably well in terms of rank correlationwhen the magnitude of exogenous variables is small, i.e. a 0.25.Chapter 6 Experiment Results 126Table 6.4.8The Effects of Exogenous VariablesCorrelation Coefficients by Two Stage Procedureo=0.05 a=0.10 a=0.25 0.50 a=0.75 1 a22.ql.33Deti 0.8556 0.8268 0.6867 0.4546 0.3026 0.2302 0.0560(0.019) (0.022) (0.038) (0.037) (0.045) (0.052) (0.073)Det2 0.8620 0.8616 0,8519 0.8202 0.7880 0.7537 0.5606(0.018) (0.019) (0.023) (0.034) (0.049) (0.068) (0.127)Stochi 0.8867 0.8620 0.7440 0.5418 0.4014 0.3325 0.1744(0.014) (0.020) (0.032) (0.046) (0.053) (0.062) (0.071)Stoch2 0.8909 0.8848 0.8599 0.8090 0.7656 0.7351 0.5524(0.014) (0.018) (0.016) (0.038) (0.042) (0.066) (0.119)DEAl 0.7774 0.7432 0.6064 0.3816 0.2510 0.1864 0.0573(0.043) (0.049) (0.047) (0.069) (0.061) (0.063) (0.083)DEA2 0.7801 0.7610 0.6762 0.4239 0.2135 0.1723 0.0298(0.043) (0.051) (0.058) (0.105) (0.105) (0.087) (0.076)Note that standard deviations are in the parenthesis.DET 1 -- efficiency estimates from first stage by deterministic frontier methodDET2 -- residual efficiency estimates from second stage by deterministic methodStoch 1 -- efficiency estimates from first stage by stochastic frontier methodStoch2 -- residual efficiency estimates from second stage by stochastic frontierDEAl-- gross DEA efficiency estimatesDEA2-- residual efficiency estimates from DEA-TOBITChapter 6 &periment Results 127Table 6.4.9The Effects of Exogenous VariablesRank Correlation by Two Stage Procedurecz=0.0S c0.10 cr=0.2S a0.50 0.75 1 c=22. = 1.33Deti 0.8669 0.8414 0.7175 0.5072 0.3735 0.3139 0.1728(0.022) (0.028) (0.037) (0.047) (0.050) (0.064) (0.063)Det2 0.8721 0.8724 0.8650 0.8362 0.8151 0.7851 0.6518(0.021) (0.022) (0.027) (0.034) (0.03 1) (0.062) (0.068)Stochi 0.8728 0.8450 0.7177 0.5076 0.3738 0.3138 0.1732(0.018) (0.024) (0.035) (0.047) (0.049) (0.063) (0.063)Stoch2 0.8774 0.8716 0.8507 0.8074 0.7690 0.7321 0.5644(0.017) (0.020) (0.020) (0.040) (0.034) (0.070) (0.105)DEAl 0.7776 0.7495 0.6320 0.4420 0.3295 0.2757 0. 1515(0.041) (0.045) (0.039) (0.063) (0.047) (0.064) (0.066)DEA2 0.7821 0.7739 0.7477 0.6708 0.6111 0.4606 0.2057(0.041) (0.045) (0.040) (0.067) (0.082) (0.082) (0.075)Note that standard deviations are in the parenthesis.DET 1 -- efficiency estimates from first stage by deterministic frontier methodDET2 -- residual efficiency estimates from second stage by deterministic methodStoch 1 -- efficiency estimates from first stage by stochastic frontier methodStoch2 -- residual efficiency estimates from second stage by stochastic frontierDEAl -- gross DEA efficiency estimatesDEA2 -- residual efficiency estimates from DEA-TOBITChapter 6 Experiment Results 128Comparing the residual efficiency estimates from the second stage with those fromthe one-step procedure reported in Table 6.4.2 to Table 6.4.5, it is clear that the two-stepprocedure is inferior to the one-step procedure except for the cases where a is small. Sincethe two-step procedure relates the exogenous variables directly to the observed efficiencyperformance of the sample firms, it sometimes might be more helpful to policy analyst anddecision makers in identifying the effects of exogenous variables on productive performance.If the effects of exogenous variables are not very strong, it may be desirable to employ thetwo-step procedure because of its intuitive appeal.In summary, the parametric methods have an obvious advantage over the DEAmethod in terms of correlations and rank correlations. However, the mean efficiencyestimates from the DEA are very close to the “true” efficiency level when the effects ofexogenous variables are not very strong. The abilities of the one-step parametric methodsdo not appear to be affected by the presence of exogenous variables as long as thesevariables can be correctly identified and accounted for. Omission and misspecification ofthe exogenous variables, however, would seriously affect the accuracy of the efficiencyestimates. The two-step procedure is able to yield reliable efficiency estimates when theeffects of exogenous variables are modest, but it is not reliable for cases where themagnitude of exogenous variables is high. However, they are logically and intuitively moreappealing for policy analysis and decision making. Before closing this section, one mustnote that some of the scenarios considered here are not realistic as mentioned in thebeginning of this section. Therefore, one should be careful in referring to the poorperformance of certain method when a is large which is intended to illustrate the significanceChapter 6 &periment Results 129of the effects of exogenous variables.6.5 The Effects of OutliersThis section examines the effects of outliers on the efficiency estimates. As statedin Chapter 5, only one outlier is considered to minimize computational requirements. The“efficient” outlier is generated by increasing the output level of an efficient observation3.The efficiency levels are then reestimated in order to see to what degree the outliers makethe efficiency estimates deviate from the true efficiency level. In particular, the output levelof an efficient observation is increased by 5%, 25%, 50%, 75%, and 100%.The results from earlier sets of experiments indicate that the relative performance ofthe three methods exhibits similar pattern with the three underlying production functionsTable 6.5.1The Effects of OutliersThe Mean Efficiency Estimates5% 25% 50% 75% 100%True Mean 0.6514 0.6514 0.6514 0.6514 0.6514(0.0157) (0.0157) (0.0157) (0.0157) (0.0157)Deterministic 0.4815 0.4086 0.3444 0.2981 0.2632(0.0259) (0.0215) (0.0188) (0.0176) (0.0171)Stochastic 0.6582 0.6686 0.6790 0.6881 0.6964(0.0237) (0.0225) (0.0230) (0.0242) (0.0256)DEA 0.6494 0.6441 0.6344 0.6225 0.6101(0.0187) (0.0184) (0.0220) (0.0294) (0.0375)Note that the standard deviations are in the parenthesis.The “efficient” outlier could in fact be inefficient. However, this should not affect theresults dramatically.Chapter 6 Experiment Results 130considered. Therefore, this section considers only the case where p = -0.25, that is, 2== = 1.33 (the second production function).Table 6.5.2The Effects of OutliersThe Mean Absolute Deviations5% 25% 50% 75% 100%Deterministic 0.1584 0.1628 0.1725 0.1886 0.2103(0.0288) (0.0362) (0.0470) (0.0608) (0.0749)Stochastic 0.0744 0.0753 0.0790 0.0840 0.0897(0.0048) (0.0069) (0.0093) (0.0115) (0.0137)DEA 0.1027 0.1046 0.1099 0.1179 0.1265(0.0083) (0.0093) (0.0141) (0.0214) (0.0291)Note that the standard deviations are in the parenthesis.The mean efficiency estimates are reported in Table 6.5.1 and the MAD values arereported in Table 6.5.2. The results show that the efficiency estimates by the DEA arerelatively stable, changing from 0.649 at 5% to 0.6 10 at 100%. However, the meanefficiency estimates by the deterministic frontier method fall sharply from 0.482 at 5% to0.263 at 100% as the scale of outliers rises. This seems to be contradictory to thepresumption that the DEA is more sensitive to outliers as stated by many authors inreviewing the various methodologies for efficiency measurements. However, these resultsare justifiable when we examine how each of the three methods compute efficiencymeasures. The deterministic frontier method evaluates the efficiency levels based on ONEsingle efficient” observation. If this observation is an (efficient) outlier or there is dataerror, e.g. inflated output level or deflated input level, associated with this observation, theChapter 6 Experiment Results 131efficiency estimates for all other observations would be reduced and the mean efficiencyestimates would be lower. The more inflated the output level, the lower the mean efficiencyestimates. On the other hand, the DEA method evaluates the efficiencies of the firms orobservations based on more than one “best practice” points. If one of the “best practice”points happens to be an outlier or has data error, only those observations which are evaluatedbased on this particular point will be affected. It would have little effects on the efficiencyestimates of other observations. Therefore, the DEA method is more flexible, in terms ofthe mean efficiency estimates, with regards to outliers and data errors than the deterministicfrontier method. This result confirms Mensah and Li (1993)’s conclusion that the DEA isrelatively more impervious to deletion of outliers than a deterministic frontier translogmodel. The efficiency estimates by stochastic frontier method are, as expected, rather stablebecause of its “built in” ability to buffer exogenous shocks.Both the DEA and the deterministic frontier method underestimate the efficiencylevels because the outliers inflate the output level of the reference point with which theproductive performance of other observations is compared. The stochastic frontier methodappears to overestimate the efficiencies, especially when the scale of outliers rises. This isbecause the estimated stochastic production frontier would have a wider “band” when thescale of outliers is higher, thus the average efficiency estimate would be higher.The correlation coefficients are presented in Table 6.5.3. The stochastic frontiermethod is again shown to be rather robust to the outliers. In terms of correlations, theoutliers have similar effects on the performances of the DEA and the deterministic frontiermethod, as indicated by the similar changing patterns of their correlation coefficients.Chapter 6 Experiment Results 132Table 6.5.3The Effects of OutliersCorrelation between True and Estimated Efficiencies5% 25% 50% 75% 100%Deterministic 0.8631 0.8559 0.8424 0.8249 0.8046(0.0194) (0.0201) (0.0212) (0.0227) (0.0245)Stochastic 0.8933 0.8943 0.8933 0.8913 0.8889(0.0145) (0.0155) (0.0160) (0.0163) (0.0165)DEA 0.7881 0.7843 0.7727 0.7568 0.7395(0.0412) (0.0425) (0.0505) (0.0626) (0.0755)Table 6.5.4The Effects of OutliersRank Correlation between True and Estimated Efficiencies5% 25% 50% 75% 100%Deterministic 0.8742 0.8745 0. 8745 0.8747 0.8748(0.0219) (0.0219) (0.0217) (0.0217) (0.0217)Stochastic 0.8795 0.8800 0.8799 0.8796 0.8793(0.0188) (0.0194) (0.0197) (0.0200) (0.0202)DEA 0.7856 0.7840 0.7763 0.7649 0.7514(0.0396) (0.0411) (0.0483) (0.0573) (0.0669)Note that the standard deviations are in the parenthesis.The rank correlations are presented in Table 6.5.4. In terms of the rank correlations,the two parametric methods appear to be rather flexible to the scale of outliers, while theperformance of the DEA tends to deteriorate as the scale of outliers increases. Theexplanation for the stochastic frontier method is obvious. However, some discussions on thedeterministic frontier method and DEA are warranted. With the deterministic frontierChapter 6 &periment Results 133method, the scale of the outliers would affect the actual levels of efficiency estimates of otherobservations (relative to the efficient outlier) but would have less effects on their relativerankings since these observations are evaluated on the basis of the same referenceobservation - the efficient outlier in this case. This partly explains the higher rankcorrelations enjoyed by the deterministic frontier method. On the other hand, the efficientoutlier would only affect the efficiency estimates of some of the observations with the DEAmethod, thus relative efficiency rankings of the observations would change. The larger thescale of outliers, the more significant the changes would be, thus the rank correlation forDEA fall noticeably as the scale of outliers rises.In summary, the stochastic frontier method is rather robust to the problem of outliersin terms of both the mean efficiency estimates (and the MADs) and the rank correlations(and the correlation coefficients). The presence of outliers does not have much effects onthe deterministic frontier method in terms of correlations and rank correlations, but doeshave significant effects in terms of the mean efficiency estimates. On the contrary, outliershave less effects on the DEA in terms of the mean efficiency estimates, but haveconsiderable effects in terms of the relative efficiency rankings.6.6 The Effects of the Underlying Production TechnologyIn the previous five sections, the constant returns to scale production technology isassumed in order not to distort the results of experiments by different treatment of returnsto scale. In this section, non-constant returns to scale and input complementarity are allowedinto the underlying production technology in order to examine how well the three alternativeChapter 6 Experiment Results 134methods deal with possible misspecification of functional forms of the production technology.Since the BCC model takes into account of non-constant returns to scale in the DEAcontext, it is decided to include the BCC model in the comparison together with the CCRmodel. Ten production functional forms with different degrees of returns to scale and inputcomplementarity are specified, consequently ten sample data sets are generated based onthese functions. The functions are labelled 4 to 13 in sequence to the three productionfunctions assumed earlier. The parameter values of these functions are listed in AppendixB. Table 6.6.1 presents the mean efficiency estimates and Table 6.6.2 presents the MADvalues from the four alternative methods. The stochastic frontier method is able to yieldstable and accurate efficiency estimates across all the functions considered. Therefore, thereturns to scale and input complementarity do not appear to have any significant effects onthe performance of the stochastic frontier method in terms of the mean efficiency estimates.The efficiency estimates by the deterministic frontier method underestimate the trueefficiency level in all the scenarios. The CCR model produces lower average efficiencyestimates than the BCC model. This is expected since the CCR ratios reflect the combinedeffects of technical efficiency and scale effects, and the BCC efficiency ratios are estimatedafter removing the effects of scale efficiency. Both models tend to underestimate the trueefficiency when input complementarity is present. When input complementarity is absent,the BCC model appears to overestimate the true efficiency level. The mean efficiencyestimates by both CCR ratio model and BCC model fall dramatically as the returns to scalerise. The results suggest that the BCC is not able to account adequately for the effects ofreturns to scale.Chapter 6 Experiment Results 135Table 6.6.1The Effects of Underlying Production TechnologyMean Efficiency EstimatesDET STOCH CCR BCCTnie 0.6561 0.6561 0.6561 0.6561(0.0157) (0.0157) (0.0157) (0.0157)4. y=O.92l 0.5048 0.6557 0.5894 0.6465o12—4L277 (0.0279) (0.0232) (0.0251) (0.0216)g130.540u12—0.5125. ‘y=O.927 0.5045 0.6556 0.5968 0.6515a12=0.399 (0.0280) (0.0229) (0.0223) (0.0190)U13 =0.454u12—-0. 1826.-y=O.929 0.4611 0.6558 0.6478 0.722112—0.356 (0.0437) (0.0286) (0.0235) (0.0190)a13=0.447u12—0.5087. ‘y=O.934 0.5046 0.6557 0.5892 0.646012-O.2O9 (0.0280) (0.0231) (0.0252) (0.0216)a13=0. 99g12—0.4548. yO.934 0.5047 0.6556 0.5886 0.6452a12—-0.091 (0.0279) (0.0231) (0.0252) (0.0216)U13 =0.399a12 = 0.453(a) and oij are the returns to scale and elasticities of substitution as defined by(5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f) The standard deviations are in the parenthesisChapter 6 Experiment Results 136Table 6.6.1(Cont.)The Effects of Underlying Production TechnologyMean Efficiency EstimatesDET STOCH CCR BCCTrue 0.6561 0.6561 0.6561 0.6561(0.0 157) (0.0157) (0.0157) (0.0 157)9. yl.2O9 0.5045 0.6556 0.4723 0.54520120,399 (0.0280) (0.0229) (0.0390) (0.0423)j13 = 0.454012 =-0. 18210. ‘yl.22’7 0.5047 0.6557 0.4379 0.5152012-0,209 (0.0280) (0.0231) (0.0519) (0.0509)013=0.399012 = 0.45411.-y=l.2139 0.4285 0.6562 0.6070 0.6924g=O.356 (0.0594) (0.0232) (0.0263) (0.0225)O3—0.447012—0.50812. y=l.286 0.5050 0.6557 0.3951 0.4736012—”’0.277 (0,0280) (0.0232) (0.0555) (0.0557)a13 =0. 5400120.5 1213. yl.573 0.5048 0.6556 0.2438 0.318803”0.091 (0.0277) (0.0230) (0.0549) (0.0587)03=0.399012 = 0.453(a) -y and oij are the returns to scale and elasticities of substitution as defined by(5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f) The standard deviations are in the parenthesisChapter 6 Experiment Results 137Table 6.6.2The Effects of Underlying Production TechnologyThe Mean Absolute DeviationsDET STOCH CCR BCC4. yO.927 0.1570 0.0744 0.1239 0.1278012—-0.277 (0.0277) (0.0044) (0.0153) (0.01 10)U3=0.540G12 —0.5125. 7=0.927 0. 1573 0.0743 0. 1222 0. 1276012—0.399 (0.0278) (0.0043) (0.0137) (0.01 10)a3=0.454012—-0. 1826. 7=0.929 0.1995 0.0826 0. 1083 0. 1227012—0.356 (0.0444) (0.0067) (0.0087) (0.0075)03 = 0.4470120,5087. 70.934 0.1572 0.0744 0.1237 0.1280012—-0.209 (0.0279) (0.0043) (0.0154) (0.0111)g13 0.399I2 =0.4548. 7=0.934 0.1571 0.0744 0. 1240 0. 12830120.091 (0.0276) (0.0044) (0.0156) (0.01 12)013 —0.3990120.453(a) and uij are the returns to scale and elasticities of substitution as defined by(5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f) The standard deviations are in the parenthesisChapter 6 Experiment Results 138Table 6.6.2(Cont.)The Effects of Underlying Production TechnologyThe Mean Absolute DeviationsDET STOCH CCR BCC9. yl.2O9 0.1573 0.0743 0.2239 0.2016u12—0.3 (0.0278) (0.0043) (0.0360) (0.0328)q13 =0.454u12 —-0. 18210. 7=1.227 0.1572 0.0744 0.2544 0.227812—-0.209 (0.0278) (0.0043) (0.0506) (0.0476)u13—O.399012—0.45411. 71.239 0.2315 0.0855 0.1304 0.1248012—0.356 (0.0565) (0.0057) (0.0142) (0.0092)03 =0.447012—0.50812. 7=1.286 0.1569 0.0744 0.2959 0.2666012—-0.277 (0.0275) (0.0044) (0.0545) (0.0526)u=0.540012—0.5 1213. 7=1.573 0.1570 0.0744 0.4443 0.4154012—-0.091 (0.0273) (0.0044) (0.0549) (0.0567)U3 =0.39912 = 0.453(a) -y and uij are the returns to scale and elasticities of substitution as defined by(5.2) and (5.3)(1,) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f) The standard deviations are in the parenthesisChapter 6 Experiment Results 139In summary, when inputs are all substitutes, and when there are decreasing returnsto scale, the deterministic frontier method produces the most conservative efficiencyestimates which underestimate the true efficiency level. However, when the underlyingproduction technology shows increasing returns to scale and there exists complementarityamong inputs, the DEA models would be more likely to produce the lowest efficiencyestimates. The poor performance of the DEA models in the cases where there are increasingreturns to scales is due to the fact that DEA assumes a convex production possibility set, andincreasing returns to scale technology is inconsistent with this convexity assumption(Petersen, 1990). Therefore, discretion is necessary in applying the DEA models wherethere is evidence suggesting increasing returns to scale.Regarding the correlations in Table 6.6.3 and the rank correlations in Table 6.6.4,we find that the stochastic frontier method dominates the competition among the four modelswith its high correlation and rank correlation coefficients. The performance of thedeterministic frontier method is very close to that of the stochastic frontier especially interms of rank correlations. The performance, in terms of correlations, of the two parametricmethods do not appear to be affected by the returns to scale. However, it is noticed that thecorrelation and rank correlation coefficients are slightly lower for cases where all inputs aresubstitutes (function 6 and 11). As for the two DEA models, the CCR model appears toperform better than the BCC model in all cases examined. The performance of the two DEAmodels deteriorates as the returns to scale increases (for the reasons given in the previousparagraph). Unlike the parametric methods, the two DEA models perform slightly betterwhen input complementarity is absent. In general, the parametric approach clearly has anChapter 6 Experiment Results 140Table 6.6.3The Effects of Underlying Production TechnologyCorrelation Between True and Estimated EfficienciesDET STOCH CCR BCC4. y=O.9Z1 0.8644 0.8927 0.7951 0.6785ai = -0.277 (0.0192) (0.0145) (0.0478) (0.0427)O3 =0.540012—M.5125. y=0.927 0.8644 0.8928 0.7621 0.6802oi=O.399 (0.0192) (0.0147) (0.0498) (0.0454)O3 =0.454012 —-0. 1826. y0.929 0.8327 0.8672 0.7734 0.7227012—0.356 (0.0175) (0.0177) (0.0399) (0.0359)j3 =0.447o12=0.5087. y=O.934 0.8643 0.8928 0.7662 0.678012 = -0.209 (0.0192) (0.0146) (0.0479) (0.0429)013—0.399o2=0.4548. y=O.934 0.8644 0.8927 0.7660 0.6774120.091 (0.0192) (0.0146) (0.0480) (0.0431)0130.399120.453(a) -y and are the returns to scale and elasticities of substitution as defined by (5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f The standard deviations are in the parenthesisChapter 6 &periment Results 141Table 6.6.3(Cont.)The Effects of Underlying Production TechnologyCorrelation between True and Estimated EfficienciesDET STOCH CCR BCC9. yl.209 0.8644 0.8928 0.6238 0.5551l2—0.399 (0.0192) (0.0147) (0.0647) (0.0576)13 =0.454z=°18210. y=1.227 0.8644 0.8928 0.5908 0.512012—0.209 (0.0192) (0.0146) (0.0790) (0.0801)=0.399012—0.45411. ‘y=l.239 0.8096 0.8520 0.7192 0.6940012—0.356 (0.0192) (0.0208) (0.0465) (0.0456)13 =0.44712—0.50812. yl.286 0.8643 0.8927 0.5258 0.4511I2”.0.277 (0.0192) (0.0146) (0.0832) (0.0850)0j3 =0.540I2—0.51213. y1.S73 0.8643 0.8927 0.2919 0.2336I2”0.091 (0.0192) (0.0147) (0.0837) (0.0838)Oj3 =0.399I20.453(a) -y and are the returns to scale and elasticities of substitution as defined by (5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH- Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f) The standard deviations are in the parenthesisChapter 6 Experiment Results 142Table 6.6.4The Effects of Underlying Production TechnologyRank Correlation Between True and Estimated EfficienciesDET STOCH CCR BCC4. ‘y0.92’l 0.8742 0.8790 0.7926 0.6837012 -0.277 (0.0219) (0.0188) (0.0420) (0.0447)c713—0.54012—0.5125. y=O.927 0.8743 0.8792 0.7837 0.6804i2—0.399 (0.0219) (0.0188) (0.0447) (0.0474)o3=0.454012— -0. 1826. y=O.929 0.8502 0.8537 0.7693 0.6832012=0.356 (0.0204) (0.0194) (0.0385) (0.0382)013 =0.447012—0.5087. y0.934 0.8742 0.8791 0.7939 0.6836012—0.209 (0.0218) (0.0187) (0.0421) (0.0448)013 =0.399012—0.4548. y=O.934 0.8743 0.8791 0.7939 0.6837012=0.091 (0.0219) (0.0188) (0.0422) (0.0447)0j3 =0.3992—0.453(a) ‘y and o are the returns to scale and elasticities of substitution as defined by (5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(f) The standard deviations are in the parenthesisChapter 6 Experiment Results 143Table 6.64(Cont.)The Effects of Underlying Production TechnologyRank Correlation between True and Estimated EfficienciesDET STOCH CCR BCC9. y=l.209 0.8742 0.8792 0.7141 0.6467012—0.399 (0.0218) (0.0187) (0.0535) (0.0540)13 =0.454120 18210. y1.22’7 0.8742 0.8791 0.7059 0.636712 = -0.209 (0.0218) (0.0187) (0.0502) (0.0535)13 =0.39912—0.45411. y 1.239 0.8339 0.8349 0.7284 0.6673012—0.356 (0.0204) (0.0236) (0.0414) (0.0416)013 =0.447012—0.50812. y=l.286 0.8743 0.8789 0.6637 0.6109012-”0.277 (0.0219) (0.0188) (0.0532) (0.0557)=0.540012—0.51213. y=l.S’73 0.8742 0.8790 0.4747 0.4650012—-0.091 (0.0219) (0.0188) (0.0597) (0.0606)013 =0.399012—0.453(a) y and are the returns to scale and elasticities of substitution as defined by (5.2) and (5.3)(b) DET - Deterministic Frontier Method(c) STOCH - Stochastic Frontier Method(d) BCC - the BCC Model(e) CCR - the CCR ratio(1 The standard deviations are in the parenthesisChapter 6 &periment Results 144advantage over the DEA method in distinguishing the effects of efficiency from the effectsof returns to scale. Both the parametric and the DEA methods are affected by the presenceof input complementarity, but the potential effects are in the opposite directions.145Chapter 7A Summary of Part IIn Part I of this dissertation, three alternative methods for efficiency measurements,namely, the deterministic frontier method, the stochastic frontier method, and the dataenvelopment analysis method, are first reviewed in relation to the basic definition ofproductive efficiency. A Monte Carlo study is then carried out to evaluate the relativemerits of the three methods in measuring efficiency under certain known conditions. Theresults are summarized as follows:• The performance of all three methods, in terms of correlations and rank correlationsbetween the true and the simulated efficiency estimates, improves as the sample sizeincreases. However, the mean efficiency estimates fall when the sample sizeincreases.• Non-homogeneity in inputs does not appear to have much effect on the performanceof the three methods in terms of correlations and rank correlations in cases where theelasticity of input substitution is greater than one. However, in the case of weakinput substitution the performance of the two parametric methods, in terms ofcorrelations and rank correlations, fall considerably as the degree of “nonhomogeneity” in inputs increases. The mean efficiency estimates by the DEA appearto increase with the variations in input values, but the opposite is true for thedeterministic frontier method. The mean efficiency estimates by the stochasticfrontier method meanwhile remain relatively stable regardless the variations in inputChapter 7 Summary 146variables.• Higher noise level reduces the efficiency estimates by the deterministic frontiermethod and the DEA, but tends to increase those by the stochastic frontier method.In terms of correlations and rank correlations, the performance of all threealternatives deteriorate sharply as the noise level rises.• The one-step parametric methods, both deterministic and stochastic, can adequatelyaccount for the effects of exogenous variables as long as the exogenous variables arecorrectly identified. On the other hand, the two-step procedure performs reasonablywell when the effects of exogenous variables are modest. However, when themagnitude of exogenous variables is large, the two-step procedure performs poorlywith all three methods.• The experiment results confirm that the stochastic frontier method is rather robustwith respects to outliers. In terms of mean efficiency estimates, the DEA appears tobe more flexible to outliers than the deterministic frontier method. However, theirperformances are quite similar in terms of correlations and rank correlations.Although both the DEA method and the deterministic frontier method are affected bythe presence of outliers, the extent of these effects is modest.• The performance of the stochastic frontier method does not depend much on thestructure of the underlying production technology over the parameter rangeconsidered. The deterministic frontier method appears to perform slightly better inthe presence of input complementarity, but its performance does not seem to beaffected by the returns to scale. The DEA method performs poorly when there areChapter 7 Summary 147increasing returns to scale, and when input complementarity is present.In general, the stochastic frontier method dominates the competition among the threealternative methods in all aspects examined. The deterministic frontier method performsbetter than the DEA method in most cases in terms of correlations and rank correlations,while the DEA method often outperforms the deterministic frontier method in terms of meanefficiency estimates and the MAD values. In drawing any conclusions from the Monte Carloresults, one must not ignore the fact that the production data are generated from a parametricproduction function which is expected to give the DEA method a comparative disadvantage.However, the DEA performs reasonably well in most situations where potential disturbanceis not very serious.It should be noted that the results summarized above, like results from otherexperimental studies, are based on a specific experimental design. Although it is believedthat the experimental design, as described in Chapter 5, is based on reasonable assumptionsabout possible empirical situations, at least for the transportation industry, there still existsthe possibility that the fmdings might be changed if the experimental design is altered.However, it is clear that the stochastic frontier method has an inherent advantage inmeasuring efficiency over the DEA and the deterministic frontier method in most cases.Therefore, it would probably be preferred to the other methods if data permit. Theestimation of stochastic frontier production functions might encounter difficulty in situationsinvolving multiple outputs, and the estimation of stochastic frontier cost functions wouldrequire multi-lateral price index for inputs which may be difficult, if not impossible, toobtain. In such cases, the DEA method would have a competitive advantage since it isChapter 7 Summary 148designed to deal with multiple inputs and outputs, and it does not require information onprices. In light of the Monte Carlo results, however, discretion would be necessary inapplying the DEA method in situations where there are large variations in the firms’operating environments and in firms’ production characteristics, and where there is evidencesuggesting possible increasing returns to scale. The following lists some specific guidelinesthat may be useful to empirical researchers in selecting an appropriate method for theparticular case on hand.• In situations where there is evidence indicating weak input substitution, and wherelarge variations in sample firms’ input variables are observed, the DEA method maybe a good choice. It is shown to perform well in terms of both rank correlations andthe actual level of efficiency estimates. Further, it requires less data than the twoparametric methods. However, if the sample size is very large, the computationalcosts of DEA might be higher compared to the other two methods.• When there are large differences in the operating environments of the sample firms,the stochastic frontier method is obviously the best choice. However, if the situationinvolves multiple outputs, and there is no consistent price data available, such as instudies involving cross-country comparisons, the application of the stochastic frontiermethod may not be practical.• The deterministic frontier method has the drawbacks of both the DEA method andthe stochastic frontier method: it is deterministic and parametric. However, it is theeasiest one to apply. If one is mainly interested in the relative efficiency rankingsrather than the actual efficiency levels, the deterministic frontier method is probablyChapter 7 Summary 149sufficient.• One should avoid the use of the DEA methods as much as possible in situationswhere there are increasing returns to scale in the production. Although the BCCmodel is developed to account for the effects of returns to scale, the Monte Carloresults show that the BCC model can not effectively deal with the increasing returnsto scale condition. Further, in situations where input complementarity is present,one should also try to use the parametric methods if data permit.• In dealing with the identifiable exogenous variables, technically it is desirable to usethe one-step stochastic frontier method. If the two-step procedure is preferred for itsintuitive appeal, the deterministic frontier method is a better choice if one is mainlyinterested in the relative efficiency rankings, while the stochastic frontier method ismore appropriate if the main concern is in the actual level of efficiency estimates.The DEA-TOBIT method is a reasonably good alternative in situations where theeffects of exogenous variables are not too strong.In the Monte Carlo experiments, the efficiency estimates are compared to the knownefficiency profile. In empirical situations, it is not possible to have an accurate picture aboutthe true efficiency performance of the sample firms. In Part II of this dissertation, the threealternative methods are applied to two real world data sets, namely, a railways data set, andan international airlines data set. These two data sets depict two fairly different situations.With the railways data, services are mostly provided by highly regulated, nationalized firmsthus the effects of exogenous variables are expected to be significant, and it is very likelyto have outliers in the sample. With the airline data set, firms operate in a fairly competitiveChapter 7 Summa7y 150environment, and they have access to essentially the same technology available even thoughthere is a high degree of diversity in size. The results from the empirical case studies areexamined to see if they are consistent with the observations from the Monte Carloexperiments. Moreover, the empirical results would also be useful in providing policyimplications for the industries examined.Part IIApplications in the Transportation Industry151152Chapter 8The Efficiency of Railway Systems in OECD CountriesIn this chapter, the three alternative methods are applied to a sample of 19 passengerrailways in OECD countries to examine the productive efficiency of the sample railways overthe period of 1978-1989. The first section provides an introduction to the problem and themotivation for the study. The second section describes the data and the model specification.The empirical results are reported and discussed in section 3. Section 4 compares therelative merits of the alternative methods in this application, and finally summary andconcluding remarks are given in Section 5.8.1 IntroductionIn recent years, the financial and economic performance of the passenger railwaysystem has attracted a great deal of attention due primarily to the mounting subsidies andalleged inefficiencies imbedded in the system. The economic efficiency of railways isbelieved to be influenced heavily by the degree of government intervention and subsidization(or taxation), and the institutional and regulatory setting within which the railways operate.The productive efficiency measured from observable data is also heavily influenced by themarket and operating environments to which the railways are subjected. These includefactors largely beyond managerial control such as topography and climate of the region, theextent of development of other transport modes, traffic density, average load, averagedistance of haul, the economic development stage of the nation, etc.The differences in policy adopted by different countries provide an excellentChapter 8 Efficiency of Railways 153opportunity to investigate the effects of policy choices on the economic efficiency of theindustry. Nash (1981) seeks to discover how much of the variation in the performance ofWestern European railways, measured by market share, traffic trends and support (subsidy)requirements, may be accounted for by government policies. He finds that, in the passengersector, the significant factors determining both market share and support requirements arethe prices charged and the mix of services offered. These decisions regarding prices andservices are almost entirely determined by government policy. Schwier, Jones and Pignal(1990) compares the performance and policy environment of regional rail passenger servicesoperated by VIA Rail Canada, British Railways (BR), French National Railways (SNCF) andAmtrak, with respect to the way they organize regional services, frequency and utilizationof services, and financial performance of the railways. Perelman and Pestieau (1988) andCompagnie, Gathon and Pestieau (1991) have analyzed the efficiency of railways, withrespect to their differences in the operating environments. Gathon and Perelman (1990) andCompagnie, Gathon and Pestieau (1991) introduce an index of regulatory and institutionalautonomy to correct for inefficiency caused by a lack of managerial autonomy.Different methodologies have been employed in studies on the performance ofrailways. Nash (1981) and Schwier, Jones and Pignal (1990) base their analysis mainly onsimple financial and performance indicators. Nash (1985), Jackson (1991), Thompson,Wood and Lures (1991), Jackson (1992), and Thompson and Fraser (1993) examine theperformance of railways through the use of some simple productivity measures, such aslabour, fuel, and rolling stock productivities. Caves, Christensen and Swanson (1980, 1981)and McGeehan (1993) measure railways’ productivities by estimating cost functions.Chapter 8 Efficiency of Railways 154Freeman, Oum, Tretheway and Waters (1987) uses the multilateral TFP index to examinethe performance of the two largest Canadian railways during the period of 1956-198 1.Perelman and Pestieau (1988), Deprins and Simar (1989), and Grabowski and Mehdian(1990) apply the deterministic frontier method to measure the efficiency of railways’.Gathon and Perelman (1990) and Compagnie, Gathon and Pestieau (1991) estimate stochasticfrontier functions to evaluate the productive performance of the European railways, and Jhaand Singh (1994) estimates zone specific technical efficiency in the Indian railways using astochastic frontier model. Bookbinder and Qu (1993) compares the performance of sevenClass I North-American railroads using the data envelopment analysis method.Some of these studies have considered the effects of exogenous variables, e.g.average trip length, average load, etc., on measuring the railways’ efficiency, such asPerelman and Pestieau (1988), Deprins and Simar (1989), and Gathon and Perelman (1990).However, none of the studies has examined the effects of public subsidies on the efficiencyperformance of the railways. There has been no study comparing the results fromapplications of the DEA and the parametric methods to the railway industry. Simar (1992)compares the deterministic frontier method, the stochastic frontier models with panel dataand a non-parametric method, the so-called Free Disposal Hull (FDH) method (Deprins,Simar, and Tulkens, 1984), in estimating railways’ efficiencies. However, he did notconsider the effects of the output attribute variables in applying the non-parametric approachwhich rated 50 percent of the observations as efficient.Grabowski and Mehdian (1990) is concerned with revenue efficiency where railroadrevenue is used as the output measure.Chapter 8 Efficiency of Railways 155This chapter attempts to measure productive efficiency of the railway systems ofnineteen OECD countries and analyze it in order to identify effects on efficiency of thepublic subsidy and degree of managerial autonomy. In the process, the study also attemptsto compare and reconcile the results of productive efficiency obtained by using the threealternative methods.8.2 Data and Model SpecificationIn this study, each observation in the data set, that is, combination of railway andyear, is regarded as an individual “decision-making unit” (DMTJ) in the terminology of thedata envelopment analysis. In total, there are 208 DMUs. The sample firms and theircharacteristics as well as the data sources are presented in Appendix C. More discussionscan be found in Oum and Yu (1991) and Oum and Yu (1994).For the DEA model, each DMU is assumed to produce two outputs: passengerservices and freight services. Two alternative sets of output measures are considered: (1)revenue output measures, as measured by passenger kilometres and freight tonne-kilometres,and (2) available output measures, as measured by passenger train kilometres and freighttrain kilometres. The available output measures indicate the level of capacity supplied, whilethe revenue output measures indicate the level of output consumed by users and the valuethey derive from them.For the parametric methods, there are problems in estimating multiple outputproduction functions due to the presence of more than one dependent variables (outputs).Estimation of the dual cost functions would require data on the railways’ input prices, forChapter 8 Efficiency ofRailways 156which there is no reliable data which is consistent across firms in different countries and overtime. Further, information on revenue shares is not available to aggregate passengerkilometres and freight kilometres, or to aggregate the train kilometres in a more sensibleway. Therefore, total train kilometres2is used as the output measure with the parametricmethods. This variable is certainly a crude measure of the production of a railway, but itoffers an aggregate measure of a railway’s activity. An attempt is made to account for thepotential effects of different service orientations by incorporating the percentage of passengertrain kilometres in total train kilometres, and to account for the degree of train utilization byincorporating the average load per train as additional explanatory variables.For both the DEA model and the parametric methods, seven inputs are considered:(1) labour, (2) energy consumption, (3) ways and structures, (4) materials, (5) the numberof passenger cars, (6) the number of freight wagons, and (7) the number of locomotives (seeAppendix C for more details).The observed productive performance of the railways are influenced by variations inthe market, operating, institutional and regulatory policy environment, on which railwayshave very limited control. The effects of these variables should be accounted for in makingefficiency comparison among the railways. Table 8.1 lists the definitions of such variablesexamined in this study.2 Deprins and Simar (1989) also uses total train kilometres as the output measure.Chapter 8 Efficiency ofRailways 157Table 8.1Definition of Policy and Uncontrollable VariablesVariables II DescriptionUncontrollable FactorsPDENSITY(KM) Passenger-km per route-kmFDENSITY(KM) Freight Tonne-km per route-kmPDENSITY(TR) Passenger Train Density: Passenger train-km per route-kmFDENSITY(TR) Freight Train Density: Freight train-km per route-km%PASSENGER Percentage of passenger train-km in total train-kmTRIP Average length of passenger trip (km)HAUL Average length of Haul of freight traffic (km)PLOAD Average passenger load per trainFLOAD Average freight load per trainTIME Time TrendControllable FactorsSUBSIDY/C Percentage of subsidy to total operating costs%ELECTRIC Percentage of Electrified Route MilesAUTONOMY Degree of managerial autonomy measured by Compagnie,Gathon and Pestieau (1991)** Since the information on the degree of management autonomy for theEuropean railways was available only at a single point in time (measured byCompagnie, Gathon and Pestieau, 1991), it was necessary to assume that therailways’ autonomy ratings remain unchanged during the study period. Inaddition, JNR is given the lowest autonomy rating.Chapter 8 Efficiency ofRailways 158The CCR model3 is applied to the data set, using the formulation (4.2.2), and thesecond stage regression analysis is accomplished using equations (4.3.1) and (4.3.2). Boththe deterministic frontier and the deterministic core of the stochastic frontier are specifiedas follows:lfly=+ E cL1flX1+E Pklnzkf (8.1)where y is the output in train kilometres for the j-th DMU, x, is the i-th input of DMU j,ZkJ are the variables listed in Table 8.1, and a, 13 are the coefficients to be estimated. Thedeterministic frontier function is estimated using the COLS, described by equations (4.1.5)to (4.1.7), and the stochastic frontier function is estimated using the Battese and Coelliprocedure as described in section 4.1.2 of Chapter 4.8.3 The ResultsThe study proceeds first with the DEA-TOBIT analysis. Then the deterministicfrontier method is used to measure the residual efficiencies of the railways, and to identifythe effects of the exogenous variables on the efficiency. Finally, the stochastic frontiermethod is applied to the sample data. The results from the deterministic frontier method andthe stochastic frontier method are briefly discussed in comparison with the DEA-TOBITresults.The BCC model is not used here, since the results in Chapter 6 indicate that it does notperform better than the CCR ratio even in cases of non-constant returns to scale.Chapter 8 Efficiency ofRailways 1598.3.1 The DEA ResultsWith the DEA-TOBIT procedure, we first estimate a gross efficiency index using theCCR ratio model, then use the TOBIT analysis to identify the effects of the exogenousvariables on the gross efficiency and to compute a residual efficiency index.8.3.1.1 The Computation and Comparison of Gross DEA Efficiency IndexThe DEA gross efficiency estimates are computed using a computer code slightlymodified from the one listed in Appendix A, and executed on a UNIX computer. Asmentioned in previous chapters, the LP problem in equation (4.2.2) has to be solved oncefor each DMU (observation) to compute the DEA gross efficiency ratings.Table 8.2 presents the gross efficiency estimates from the CCR model. These DEAindices reflect the combined outcome of true managerial and operational efficiency and theeffects of constraints imposed by the institutional, regulatory, market and operatingenvironment. Therefore, one cannot make inferences about true managerial and operationalefficiency from these indices without accounting for the variations caused by the variablesbeyond a firm’s control.There are some significant differences between the two sets of DEA gross efficiencyindices: the case of using revenue outputs versus the case of using available outputs. NSB(Norway) has one of the lowest DEA ratings when passenger-kilometres and tonnekilometres are used as outputs, while it has one of the highest DEA ratings when trainkilometres are used as outputs. This may be partiy explained by the fact that NSB’s(Norway) average loads per train are on the lower end for both passenger and freight trains.Chapter 8 Efficiency ofRailways 160Table 8.2DEA Gross Efficiency IndexOutput MeasureRailways Countries Revenue Outputs (I) Available Outputs (II)Passenger. -Km Passenger Train-Km && Tonne-Km Freight Train-Km1978 1989 1978 1989BR U.K. 0.89 1.00 1.00 1.00CFF-SBB Switzerland 0.69 0.73 0.94 0.89CFL Luxembourg 0.93 0.70 0.95 0.76CR Greece 0.52 0.50a 0.62 0.44aCIE Ireland - 1.00 - 1.00CP Portugal 0.89 1.00 0.72 0.85DB Germany 0.63 0.65 0.81 0.91DSB Denmark 0.54 0.75 0.66 0.82FS Italy 0.82 0.82 0.71 0.76JNR Japan 1.00 1 .OOb 0.96 1 .OObNS Netherlands 0.88 0.94 1.00 1.00NSB Norway 0.74 0.67 1.00 0.94OBB Austria 0.60 0.62 0.84 0.85RENFE Spain 0.76 0.77 0.97 1.00SJ Sweden 0.90 1.00 1.00 1.00SNCB Belgium 0.61 0.71 0.66 0.73SNCF France 0.77 0.84 0.95 0.99TCDD Turkey 0.88 0.94 0.91 0.64VR Finland 0.79 1.00 0.80 0.96a. for 1987;b. for 1986.Chapter 8 Efficiency ofRailways 161Similarly, TCDD (Turkey) has a quite high rating in terms of passenger-kilometres andtonne-kilometres, while it is rated low in terms of train-kilometres. This may be partlyexplained by the fact that TCDD (Turkey) has relatively higher average loads per train forboth freight and passenger services.Both sets of the DEA gross efficiency indices show that by 1989 BR (U.K.), SI(Sweden), JNR (Japan), CIE (Ireland), NS (Netherlands) and VR (Finland) have attained aposition close to the efficient production frontier. DSB (Denmark) and yR (Finland) haveachieved significant improvement during the period: 0.54 and 0.66 in 1978 to 0.75 and 0.82in 1989 for DSB (Denmark), and 0.79 and 0.80 in 1978 to 1.00 and 0.96 in 1989 for VR(Finland). CH (Greece) and SNCB (Belgium) are indicated as the least efficient railways byboth sets of the DEA gross efficiency indices. CFL (Luxembourg) and NSB (Norway) arethe two railways which exhibit a noticeable decline in their DEA gross efficiency indexduring the sample period.Overall, the two sets of the DEA gross efficiency indices are comparable. Thecorrelation coefficient between the two DEA gross efficiency indices is 0.624 while theirSpearman’s rank correlation coefficient is 0.6 15. These DEA gross efficiency results aregenerally consistent with the findings of Jackson’s surveys of European railway performance(Jackson, 1991 and 1992), the world bank surveys of railway performance (Thompson,Wood, and Lures, 1991, and Thompson and Fraser, 1993), and Compagnie, Gathon andPestieau, 1991.Chapter 8 Efficiency ofRailways 1628.3.1.2 The Effects of Exogenous VariablesAs mentioned earlier, the DEA gross efficiency indices are influenced by variationsin the market, operating, institutional and regulatory policy environment, variations overwhich railways have limited control. The effects of these variables are identified byanalyzing the DEA gross efficiency index. Subsequently, after controlling for the effects ofthese variables, the residual efficiency indices are computed and analyzed. This residualefficiency index is a closer indicator of managerial and technical efficiency than the DEAgross efficiency index.TOBIT regression as specified by equation (4.3.1) is used to identify the effects ofpolicy and other variables, and to measure the “residual” efficiency4of the railways. Thevariables listed in Table 8.1 are examined in the TOBIT analysis. Some of these variablesare considered controllable by government and/or regulatory agencies while others are not.Note that two policy variables are considered: SUBSIDY and AUTONOMY. SUBSIDY ismeasured by the ratio of subsidy to operating costs. Subsidy policy should ideally beexamined according to the types of subsidies and the ways in which they are provided (e.g.loss/balancing subsidy vs. a fixed sum subsidy, unconditional subsidy vs. paymentconditional on meeting a certain performance standards, etc.), which are likely to havesubstantial impacts on a firm’s efficiency. However, due to limited information, this study‘ We use the term “ residual efficiency” because there are some important factors leftout in our analysis, such as weather and climate, topography of the land, the generaleconomic condition of the country, the extent of development of other transport modes, andquality of services, etc. The presence of these factors (which are left out mainly due to thelack of data) makes it difficult to interpret the residual efficiency index as an indicator fortrue managerial and operational efficiency.Chapter 8 Efficiency of Railways 163examines the effects of subsidization with respect to the level of aggregate subsidy only. TheAUTONOMY variable is an index of regulatory and institutional environments, based on1988-1989 figures, which was constructed by Perelman and Gathon (1990) using theinformation collected through a survey of railways’ management. Its values range between40 and 100. The more autonomous management is, the higher the value of theAUTONOMY index. The degree of managerial autonomy is affected by a large number offactors including ownership form and managerial mandate. It is therefore very difficult toquantify managerial autonomy consistently across railways even with the best of efforts.Another problem with this variable is that this variable was based on only one year’sobservations. In using this variable it is therefore assumed that the institutional environmentof the railways had varied only minimally over our sample period, an assumption which isnot realistic for most railways. Although it is realized that the AUTONOMY variable is notideally defined, this variable is used in the analysis as it is the only information of its kindwhich had been collected systematically.Table 8.3 reports the two best log-linear TOBIT regressions: one with the passengerkilometres and freight tonne-kilometres as outputs (henceforth referred to as Model I), andthe other with the train-kilometres as outputs (henceforth referred to as Model II). Theresults can be interpreted as follows:(a) Effects of Uncontrollable Variables: Four of the variables used in the TOBITmodel (for both freight and passenger services) are largely beyond managerial orgovernmental control, i.e. traffic density (DENSITY: passenger and freight), averagedistance of the trip (TRIP or HAUL), load per train (LOAD: passenger and freight)Chapter 8 Efficiency ofRailways 164Dependent VariableTable 8.3Tobit Regression ResultsLog (DEA Efficiency Indices)Model (fl Model (IT)Passenger-Km & Passenger and FreightTonne-Km as Outputs Train-Kms as OutputsIndependent Variables Coef. (t-stat.) Coef. (t-stat.)DENSITY-Freight - - - -DENSITY-Pax - - - -HAUL - - - -TRIP - - - -%PASS - - - 0.567 (6.04)LOAD-Pax 0.26 (7.30) - 0.062 (1.96)LOAD-Freight - - - 0.418 (8.21)% ELECTRIC 0.018 (2.54) 0.043 (7.28)SUBSIDY -0.088 (6.27) -0.052 (5.20)AUTONOMY 0.452 (6.78) 0.163 (3.55)TIME 0.008 (2.04) 0.007 (2.71)CONSTANT -3.49 (9.81) 1.524 (4.82)No.ofObserv. 208 208LOG-Likelihood 27.8568 86.3713R2 0.4196 0.6067Note that all variables are in their logarithms, except for TIME. In this way,the coefficient for the variable lIME indicates the constant growth rate directly.Chapter 8 Efficiency of Railways 165and percentage of passenger train-kilometres in the total train-kilometres(%PASSENGER).The DENSITY and TRIP (HAUL) variables turn out to be statisticallyinsignificant in both models. The insignificance of the DENSITY variables may beexplained by the fact that: (i) railways with higher passenger traffic density do notnecessarily perform well in their freight services, and vice versa; and (ii) substantialcorrelations exist between traffic density and load per train (0.33 for passengerservices and 0.58 for freight services).Both LOAD variables, Passenger LOAD and Freight LOAD, are statisticallysignificant in the Model (II) while only Passenger LOAD is significant in Model (I).These results indicate that railways with higher average LOADs tend to attain ahigher DEA gross efficiency index when passenger-kilometres and freight tonnekilometres, rather than train-kilometres, are used as the output. These resultsconfirmed our hypotheses that: (i) the cost per train-kilometer increases as the loadper train increases; and (ii) the cost per passenger-kilometer (or tonne-kilometer)decreases as the load per train increases.The %PASSENGER variable has a significant negative coefficient in Model(II) while the same variable is statistically insignificant in Model (I). The negativecoefficient in Model (II) can be interpreted as meaning that railways with heavyconcentration in passenger services are at a disadvantage when their output ismeasured in terms of train-kilometres. Gathon and Perelman (1990) draws a similarconclusion.Chapter 8 Efficiency of Railways 166(b) Effects of Electrification: The %ELECTRIC variable has a statistically significant,positive coefficient in both Models (I) and (II). This indicates that electrificationimproves the performance of the railways. This may be due to the positive effectelectrification has on reducing energy consumption and the amount of labourrequired. The positive effect on labour productivity is also shown by Gathon andPerelman (1990).(c) Effects of Subsidy: The ratio of subsidy to the total operating expenses has astatistically significant negative coefficient in both models. This implies that heavilysubsidized railways tend to be less efficient than other railways5.The subsidization of a particular firm (or mode) puts its rivals in a weakercompetitive position, that is, the subsidized firm is protected from the pressure ofcompetition by the subsidy. As a result, these firms have little incentive to improvetheir productivities or minimize costs, thus becoming less innovative and efficient.The extent of those potential subsidy effects depends on how the subsidies areprovided. However, as stated earlier, our analysis does not distinguish amongst thetypes of subsidies and how they are provided due to the lack of information requiredAt first glance, the small coefficients for the SUBSIDY variable appear to indicate thatthe negative effect of subsidy on the efficiency is not that high. However, in terms of themagnitude of the effect on cost efficiency relative to the change in subsidy amount can berelatively large. For example, the Model (1) in Table 8.3 shows that doubling of the subsidywould reduce efficiency of the subsidized railway system (the freight and passenger servicestogether) by 8.8%. Suppose that the total size of a company of our interest is $2 billion andthe current subsidy level is $100 million. Then, doubling the subsidy to $200 million wouldcost the firm about $176 million (8.8% of $2 billion) due to the increased inefficiency.Chapter 8 Efficiency ofRailways 167for such an analysis6.(d) Effects of Managerial Autonomy: The managerial autonomy index constructed byGathon and Perelman (1990) has a statistically significant positive coefficient in bothmodels. This shows that efficiency is expected to improve if management is givenmore autonomy in making strategic and operational decisions. Higher degrees ofmanagerial autonomy enable management to response quickly to new opportunitiesand circumstances, thus keeping the firm competitive in the ever changing market.Managerial autonomy on the choice of the markets to serve (or abandon) andfrequency of services would heavily influence the productive efficiency of therailway. Moreover, higher degrees of managerial autonomy would requiremanagement to be more accountable for the firm’s performance, thus giving themgreater incentive to improve productive efficiency.(e) Effects of TIME: The TIME trend variable is intended to assess the overallimprovement of the industry’s technological progress and managerial efficiency (i.e.the efficiency improvement common across all of the railways in the sample). Thisvariable has a statistically significant positive coefficient, indicating that the industryexperienced technological progress at a rate of about 0.7% 0.8% each year duringthe study period from 1978 to 1989.Although there are some differences between the two TOBIT models, the overall6 It should be noted that subsidization and inefficiency are in fact inter-dependent.Subsidization causes inefficiency, and inefficiency is very likely to result in largerrequirements for subsidy. Since the main interest of this study is the comparativeperformance of the alternative methods, subsidy is assumed as an exogenous variable.Chapter 8 Efficiency of Railways 168results are consistent with our expectations. Certainly the differences in the signs ofregression coefficients are reconcilable. The regression results indicate that a railway’sefficiency performance may be significantly enhanced by an institutional and regulatoryframework which would provide railway management with greater freedom in decisionmaking. This is evident from the statistically significant positive coefficients for theAUTONOMY variable and the negative coefficients for SUBSIDY. These results are alsoconsistent with the general belief that a high degree of direct government intervention andhigh subsidy levels interfere with market mechanisms and encourage inefficiency to remainin railways’ operation.8.3.1.3 The Residual EfficienciesThe statistical significance of some of the variables beyond managerial control impliesthat these variables do influence the DEA gross efficiency index. This confirms our earlierstatement that the DEA gross efficiency index does not reflect true managerial andoperational efficiency. The results will be closer to the real level of efficiency only afterthe effects of these variables are removed from the DEA gross efficiency index.Residual efficiencies are computed from the residuals of the Tobit regressions.However, the reader is cautioned once again that these residual efficiency indicators may notreflect the true picture of the railways’ efficiency performance. The lack of adequate datamakes it impossible to control for such factors as quality of service, weather and climate,topography of the land, etc. However, the residual efficiency index is much closer to arailway’s true efficiency than the DEA gross efficiency index.Chapter 8 Efficiency of Railways 169Table 8.4Residual Efficiency IndexOutput MeasureRailways Countries Revenue Outputs (1) Available Outputs (II)Passenger-Km & Tonne- Passenger Train-Km &Km Freight Train-Km1978 1989 1978 1989BR U.K. 0.76 0.79 0.80 0.80CFF-SBB Switzerland 0.53 0.55 0.73 0.72CFL Luxembourg 1.00 0.72 1.00 0.71CH Greece 0.63 0.56a 0.82 0.61aCIE Ireland - 0.84 - 0.77CP Portugal 0.73 0.75 0.61 0.76DB Germany 0.59 0.55 0.69 0.76DSB Denmark 0.63 0.79 0.67 0.77FS Italy 0.65 0.64 0.66 0.67JNR Japan 0.74 0.71b 0.90 0.89bNS Netherlands 0.83 0.80 0.86 0.89NSB Norway 0.80 0.67 0.82 0.80OBB Austria 0.59 0.62 0.70 0.73RENFE Spain 0.69 0.69 0.83 0.85SJ Sweden 0.73 0.77 0.81 0.83SNCB Belgium 0.59 0.64 0.66 0.74SNCF France 0.60 0.61 0.80 0.82TCDD Turkey 0.68 0.68 0.84 0.64VR Finland 0.54 0.78 0.62 0.82a. for 1987;b. for 1986.Chapter 8 Efficiency ofRailways 170The residual efficiency indices are listed in Table 8.4. The variation of the residualefficiency index values are smaller than the DEA gross efficiency index values. This isbecause some of the variations in the DEA gross efficiency indices have been explained bythe variables included in the TOBIT regression model. The residual indices from Models(I) and (II) show that in 1989, BR (UK), NS (Netherlands), SJ (Sweden) and VR (Finland)were among the most efficient performers; CH (Greece) and OBB (Austria) were among theleast efficient performers.It is noteworthy to observe that the performance of CP (Portugal) and JNR (Japan),which are among the top performers in terms of the DEA gross efficiency index, are reducedto that of the “mid-range” performers when using the residual efficiency. On the other hand,the efficiency rating of DSB (Denmark), using the residual efficiency, is improved ascompared to its rating in terms of the DEA gross efficiency index. Furthermore, a greaternumber of railways join the ‘mid-range’ performers group when the residual efficiency indexis used. Finally, the performance of DSB (Denmark) and SNCB (Belgium) improves overthe years, in terms of residual efficiency, while the ratings of CFL (Luxembourg), CR(Greece), NSB (Norway), and TCDD (Turkey) noticeably deteriorate.8.3.2 The Deterministic Frontier Method ResultsIn this subsection, the results from the deterministic frontier method is examined.As stated in section 2 of this chapter, the total train-kilometer is used as the output measure.This is because it is difficult to deal with multiple outputs with the parametric productionfunction methods, and there is no consistent price data for estimating the alternative costChapter 8 Efficiency of Railways 171function. The input variables are the same as those used in the DEA analysis.The estimated “best practice” production function and resulting efficiency estimatesfrom the one-step deterministic frontier method are presented in Table 8.5 and Table 8.6,respectively. Recall from Chapter 6 that the one-step procedure includes the exogenousvariables directly in the estimation of the production frontier. Among the policy anduncontrollable variables, TRIP has a statistically significant positive coefficient, implying thatthe railways with longer average trip distance are expected to enjoy consistently higherefficiency ratings than other railways. As expected, %ELECTRIC has a statisticallysignificant positive coefficient reflecting its positive effects on labour and energyproductivities. The two LOAD variables have statistically significant negative coefficients.This is the case because the output is measured in train-kilometres, thus cost per train-kilometer increase as the average load per train goes up. %PASSENGER has a statisticallysignificant negative coefficient. This indicates that railways with larger portion of theirbusiness in passenger traffic are likely to have a lower efficiency rating when trainkilometers is used as the output measure. SUBSIDY has a significantly negative coefficientimplying that government financial support tends to decrease the railways’ efficiency. Thedegree of managerial freedom appears to have significant positive effects on the railways’performance as indicated by its statistically significant positive coefficient. As expected, theTIME trend variable is also statistically significant, indicating that technological progress hasimproved the overall efficiency performance of the industry. The results in Table 8.5essentially confirm those from the DEA-TOBIT analysis.Table 8.6 lists the efficiency estimates from the deterministic frontier method. ThereChapter 8 Efficiency of Railways 172Table 8.5Deterministic Production Frontier FunctionDependent VariableLog (Train-Kilometres)Independent Variables Coef. (t-stat.)LABOUR 0.062 (1.11)ENERGY 0.590 (17.63)WAYS AND STRUCTURAL 0.186 (12.91)FREIGHT CAR 0.131 (3.21)DENSITY-Freight - -DENSITY-Pax - -HAUL - -TRIP 0.258 (11.38)%PASS -0.791 (7.33)LOAD-Pax -0.232 (6.78)LOAD-Freight -0.603 (12.17)%ELECTRIC 0.116 (15.08)SUBSIDY -0.076 (8.55)AUTONOMY 0.299 (5.51)TIME 0.001 (3.45)CONSTANT 3.049 (7.86)No.of Observ. 208LOG-Likelihood 153.904R2 0.9919Note that all variables are in their logarithms, except for TIME.Chapter 8 Efficiency of Railways 173Table 8.6Efficiency Estimates by Deterministic FrontierRailways Countries Train-Kilometres as Output1978 1989BR U.K. 0.71 0.80CFF-SBB Switzerland 0.62 0.66CFL Luxembourg 0.89 0.64CH Greece 0.83 0.74aCIE Ireland - 0.76CP Portugal 0.65 0.83DB Germany 0.74 0.74DSB Denmark 0.73 0.73FS Italy 0.63 0.60JNR Japan 0.99 0.81bNS Netherlands 0.77 0.76NSB Norway 0.77 0.74OBB Austria 0.73 0.70RENFE Spain 0.74 0.70SI Sweden 0.86 0.90SNCB Belgium 0.74 0.72SNCF France 0.86 0.78TCDD Turkey 0.52 0.67VR Finland 0.54 0.82a. for 1987;b. for 1986.Chapter 8 Efficiency ofRailways 174are some noticeable differences between the results in Table 8.6 and those in Table 8.4 fromthe DEA-TOBIT analysis. For example, the results by deterministic frontier method indicatethat the performance of CP (Portugal) is among the top performers in 1989 while the DEATOBIT7results put it in the group of “mid-range” performers. In addition, the deterministicfrontier method also shows that there is significant improvement in BR’s performance in1989 from 1978, but the DEA-TOBIT results indicate that there is essentially no change inBR’s performance. Also, the deterministic frontier method indicates that JNR’s performancedeteriorates over time while the DEA-TOBIT results show that there is little change in JNR’sperformance between 1978 and 1986. One possible explanation for the difference is that thedeterministic frontier method follows an one-step procedure, while the DEA-TOBIT analysisis a two-step procedure, and the Monte Carlo results show that the two procedures yielddifferent efficiency estimates. Further, the DEA-TOBIT method considers two separateoutputs, but the deterministic frontier method consider only one aggregate output. However,both methods indicate that BR, JNR, SJ, and VR are among the most efficient performers,while CFL, FS and TCDD are among the least efficient performers in 1989.8.3.3 The Stochastic Frontier ResultsThe results from the application of the stochastic frontier method are reported andexamined in this section. As with the deterministic frontier method, total train-kilometer isIn comparing the results of the parametric methods and DEA, all references to theresidual efficiencies are directed to the second set of efficiency index in Table 8.4 using theavailable output measures so that they are consistent with the output measure used in theparametric methods.Chapter 8 Efficiency of Railways 175used as the output measure. Again, seven input variables are considered.The estimated deterministic core of the stochastic frontier production function, usingthe one-step procedure, is listed in Table 8.7. As one can see, the estimated frontierfunction is essentially the same as the one-step deterministic frontier production functionpresented in Table 8.5 except for the intercept term. This is not unexpected since theestimation procedure employed in the FRONTIER program starts with OLS estimates of thecoefficients which are all unbiased except for the intercept term (Coelli, 1991), the procedurethen proceeds with a two-phase grid search and an iterative process using a Quasi-Newtonmethod to obtain the final (approximate) maximum-likelihood estimates.The efficiency estimates by the one-step stochastic frontier method are presented inTable 8.8. The stochastic frontier efficiency estimates are considerably higher than both theresidual efficiencies from the DEA-TOBIT procedure (Table 8.4) and the deterministicfrontier efficiency estimates (Table 8.6). It is noteworthy to observe that some results inTable 8.8 are noticeably different from the earlier results: CH appears to be among the“upper-middle” group in terms of the stochastic frontier efficiency estimates, while it appearsto be among the “mid-range” group in terms of the deterministic frontier efficiencyestimates, and among the least efficient performers in terms of the DEA residual efficiencyestimates. JNR is among the efficient performers in terms of both the DEA residualefficiencies and the deterministic frontier efficiency estimates, however, it is shown to beamong the “mid-range” performer in terms of the stochastic frontier efficiency estimates.CP is ranked among the top performers by the two parametric methods but it is placedamong the mid-range group in terms of the DEA residual efficiency index in Table 8.4.Chapter 8 Efficiency ofRailways 176Table 8.7Stochastic Production Frontier FunctionDependent VariableLog (Train-Kilometres)Independent Variables Coef. (t-stat.)LABOUR 0.062 (1.27)ENERGY 0.602 (17.55)WAYS AND STRUCTURAL 0.183 (14.60)FREIGHT CAR 0.111 (3.60)DENSITY-Freight - -DENSITY-Pax - -HAUL - -TRIP 0.238 (11.82)%PASS -0.751 (7.93)LOAD-Pax -0.139 (4.63)LOAD-Freight -0.508 (11.43)%ELECTRIC 0.112 (17.33)SUBSIDY -0.094 (9.18)AUTONOMY 0.409 (9.22)TIME 0.003 (1.39)CONSTANT 1.911 (5.23)No.ofObserv. 208LOG-Likelihood 167.99Note that all variables are in their logarithms, except for TIME.Chapter 8 Efficiency of Railways 177Table 8.8Efficiency Estimates by Stochastic FrontierRailways Countries Train-Kilometers as Output1978 1989BR U.K. 0.84 0.97CFF-SBB Switzerland 0.71 0.79CFL Luxembourg 0.97 0.78CH Greece 0.96 0.91aCIE Ireland - 0.94CP Portugal 0.74 0.94DB Germany 0.90 0.94DSB Denmark 0.88 0.92FS Italy 0.73 0.75JNR Japan 0.99 0.88bNS Netherlands 0.91 0.92NSB Norway 0.95 0.95OBB Austria 0.88 0.92RENFE Spain 0.87 0.90SJ Sweden 0.97 0.99SNCB Belgium 0.87 0.89SNCF France 0.96 0.92TCDD Turkey 0.55 0.73VR Finland 0.56 0.96a. for 1987;b. for 1986.Chapter 8 Efficiency of Railways 178Similar to the results by the other methods, BR, SJ, and VR are again ranked among themost efficient railways in terms of the stochastic frontier efficiency estimates, while CFL,FS, and TCDD are among the least efficient performers in 1989.8.4 Comparison of the Alternative Efficiency EstimatesIn the previous section, comparison of the efficiency estimates from the three methodsis focused on the performance of some specific railways. In this section, however, the focusis on comparison of overall pattern of the alternative efficiency estimates.Table 8.9Means of the Railways’ Efficiency EstimatesDeterministic Stochastic DEA-TOBIT1Mean 0.7366 0.8729 0.7731Standard Deviation. 0.0835 0.0943 0.0759Minimum 0.5247 0.5529 0.5521Maximum 1.0000 0.9875 1.00001. With passenger and freight train kilometres as the output measures.Table 8.9 lists the mean efficiency estimates by the three alternative methods. Thedeterministic frontier method produces the most conservative average efficiency estimate,while the stochastic frontier method yields the highest average efficiency estimate. Thisresult is consistent with the results from the Monte Carlo experiments in Chapter 6.Furthermore, the difference between the stochastic frontier mean efficiency estimate and theChapter 8 Efficiency ofRailways 179mean of the DEA-TOBIT residual efficiencies is rather significant. The Monte Carlo resultsindicate that the gaps between the alternative efficiency estimates become larger when theeffects of the exogenous variables are significant. Therefore, the differences among the threesets of efficiency estimates confirm the initial hypothesis that exogenous factors havesubstantial effects on the observed productive performance of the railways. However, theranges of the efficiency estimates by the three methods are quite similar, with the railwaysTable 8.10Correlation and Rank Correlation CoefficientsBetween the Efficiency EstimatesCorrelation CoefficientDeterministic Stochastic DEA-TOBITDeterministic 1Stochastic 0.8859 1DEA-TOBIT 0.5204 0.4 150 1Rank Correlation CoefficientDeterministic 1Stochastic 0.8996 1DEA-TOBIT 0.5486 0.5152 1ranging from efficient performers to about 45 percent less efficient.Table 8.10 lists the correlation and rank correlation coefficients among the alternativeefficiency estimates. It is noted that the efficiency estimates from the two parametricmethods are highly correlated in terms of both the correlation coefficient and the rankChapter 8 Efficiency of Railways 180correlation coefficient. However, the residual efficiencies from the DEA-TOBIT are onlymarginally correlated with the residual efficiency estimates from the other two methods.This happens partly because the parametric methods employ an one-step procedure while theDEA-TOBIT is a two-step procedure, and the two procedures are shown by the Monte Carloexperiments to produce different efficiency estimates. In addition, the parametric methodsconsider only one output variable, while DEA considers two outputs.An examination of the efficiency estimates from the two-step parametric methods8shows that there are higher correlations between corresponding efficiency estimates: the rankcorrelation coefficient is 0.81 between the gross efficiency estimates by the stochastic frontiermethod and the gross DEA efficiency estimates, and 0.64 between the corresponding residualefficiency estimates; and is 0.83 between the gross efficiency estimates from thedeterministic frontier method and the gross DEA efficiency estimates, and 0.71 between thecorresponding residual efficiency estimates. This is also consistent with the simulationresults in Chapter 6 where the efficiency estimates from the two-step parametric methods arecloser to the DEA-TOBIT estimates than those using the one-step procedure.In general, the basic pattern of the efficiency estimates from the two parametricmethods are very similar, although there are noticeable differences in the actual levels ofestimated efficiency between the two sets of efficiency estimates. On the other hand, theefficiency estimates from the DEA-TOBIT analysis exhibit some significant differences fromthose obtained from the parametric methods. The main reason for this appears to be due toS The results from the two-step parametric methods are not reported in this chapter, theyare available from the author upon request.Chapter 8 Efficiency of Railways 181the uses of one-step vs two-step procedure, and the use of different output measures.However, the policy implications from all three methods are consistent.8.5 Summary and Concluding RemarksThis chapter attempts to identify the effects of government intervention andsubsidization on productive efficiency of those railways which derive a high proportion oftheir business from passenger services. In particular, the productive efficiency of the railwaysystems in 19 OECD countries are measured and analyzed in order to identify the effects ofboth public subsidies and degree of managerial autonomy on efficiency. A two-step DEATOBIT procedure is used along with two one-step parametric methods: the deterministicfrontier method and the stochastic frontier method. With the DEA-TOBIT procedure, theData Envelopment Analysis (DEA) method is first used to measure the gross efficiency indexfrom the panel data of the 19 railways over the 1978-89 period, and then the Tobitregression is used to identify the effects of the public subsidies and the extent of managerialautonomy after controlling for the effects of various operating characteristics and marketenvironments such as traffic density, average load per train, average distance hauled andpercentage of electrified route network, which are largely beyond managerial control. Withthe parametric methods, the policy and uncontrollable variables are incorporated directly inestimating the frontier production functions.The empirical results from all three methods show that railway systems with highdependence on public subsidies are significantly less efficient than similar railways with lessdependence on subsidies, and railways with high degree of managerial autonomy fromChapter 8 Efficiency of Railways 182regulatory authority tend to achieve higher efficiency. These two results together imply thatproductive efficiency of railway systems may be significantly enhanced by an institutionaland regulatory framework which provides a greater freedom for managerial decision making.Therefore, the institutional and regulatory framework for railway industry must squarelyaddress the question of railways’ managerial autonomy. Subsidy policies must encouragerailways to use normal market mechanisms to improve their cost recovery and to use thesubsidies only for improving services.In addition, the empirical results indicate that efficiency measures may not bemeaningfully compared across railways without controlling for the differences in operatingand market environments.Due to limited information and knowledge, this study examines the effects ofgovernment intervention and subsidization in a broad sense only. There are many alternativeand complementary means by which governments intervene in railway business. Theseinclude a partial or full ownership by a government, operation of railways by a governmentbranch, regulation of prices and service frequencies, and use of taxation and subsidy. Eachof these means of intervention has a distinct effect on the firm’s efficiency. Also, not onlythe amount of subsidy but also the method of subsidization affect management’s incentiveto improve efficiency. For example, a predetermined amount of subsidy is considered to bemore effective than subsidizing 100% of the financial loss. Therefore, an informed policymaking would require detailed studies for measuring the effects of various means ofgovernment intervention and alternative methods of subsidization on economic efficiency.In comparing the efficiency estimates from the three alternative methods, it is foundChapter 8 Efficiency ofRailways 183that the mean efficiency estimate by the deterministic frontier method is the lowest amongthe three sets of efficiency estimates while the stochastic frontier method produces thehighest average efficiency estimate. This is consistent with the Monte Carlo results inChapter 6. The study also shows that the efficiency estimates from the two parametricmethods exhibit essentially the same pattern. However, there exist noticeable differencesbetween the DEA-TOBIT efficiency estimates and those of the parametric methods althoughthe policy implications from all three methods are consistent. The main reason for thedifference between the DEA and the parametric methods is that the DEA-TOBIT is a two-step procedure while the parametric methods follow a one-step procedure which producesdifferent estimates as shown by the simulation results in Chapter 6. Further, the DEATOBIT considers two output variables whereas the parametric methods consider only oneaggregate output variable. The results from Monte Carlo experiments indicate that theefficiency estimates from the stochastic frontier method are closer to the “true” efficiencythan the DEA method if the inputs and exogenous variables are correctly identified andaccounted for. However, in this case study, we encounter the problem of collinearity amonginputs and exogenous variables in estimating the stochastic frontier production function (andthe deterministic frontier function). This is a rather common problem associated withestimating production functions. This may have caused misspecification of the “bestpractice” production function, consequently may have affected the accuracy of the efficiencyestimates. Therefore, the differences between the DEA-TOBIT estimates and the parametricestimates may be partly attributable to the potential mis-specification of the model as well.184Chapter 9The Efficiency of International AirlinesIn this chapter, the three alternative methods are applied to the international airlineindustry to examine the productive efficiency of 36 major international airlines over theperiod of 1980-1992. Section 1 gives a brief introduction and reviews selected literature onairline productivity and efficiency. Section 2 describes the variables and the modelspecifications. The efficiency estimates are reported and discussed in Section 3, and therelative merits of the three methods are compared in Section 4. Finally, a summary andsome concluding remarks are given in Section 5. Detailed descriptions of the sample airlinescan be found in Appendix D.9.1 IntroductionThe airline industry is characterized by failing unit costs and a growing demand,however, the profitability of the airline industry world wide has been marginal. Since the1978 deregulation of the U.S. domestic market, major institutional changes have beenintroduced in the world’s commercial air transport industry resulting in increased competitionin international aviation. As the market becomes more competitive, the ultimate ability ofa carrier to survive and prosper depends greatly on improvement in its efficiency andproductivity. The issue of efficiency and productivity will become increasingly importantfor airline industry since the differences in labour costs are likely to diminish over time asmore and more airlines practice global sourcing of their flight crews and maintenance work.Chapter 9 Airline Efficiency 185There have been numerous studies on airline productivity and efficiency. Caves,Christensen and Tretheway (1981) compares 11 U.S. trunk air carriers on the basis of levelsand rates of growth of outputs, inputs, and total factor productivity for the period of 1972-1977. They also examine the relationships between productivity and differences in outputs,average stage length, and load factor. Caves, Christensen, Tretheway and Windle (1987)compares the productive performance of a sample of U.S. and non-U.S. airlines over the1970-1983 period in terms of the growth rate of total factor productivity, focusing on theeffects of the U.S. deregulation. Gillen, Oum and Tretheway (1985, 1990) measure andcompare the productive performance of seven Canadian airlines in terms of total factorproductivity and unit cost. The study is conducted for the period of 1964 to 1981. Windle(1991) measures the productivity and unit costs of a set of U.S. and non-U.S. airlines andattempts to identify the factors which are most influential in explaining the observeddifferences in the airlines productivity. The study is based mainly on the 1983 data.Encaoua (1991) examines cost and productivity differences among European carriers, andfinds that the gap in productivity measures between the carriers are shrinking during the1981-1986 period.Barla and Perelman (1989) adopts the stochastic frontier method to measure theefficiency performance of 26 airlines from OECD countries during 1976 to 1986. Theyestimate the production frontier function using the one-step procedure, and use availabletonne-kilometre as output measure and consider two inputs, labour and capital, and fourexogenous variables. Their study compares the efficiency performance of American carriersoperating in deregulated markets with those of European and other airlines still subject toChapter 9 Airline Efficiency 186regulatory control, and finds that the deregulated airlines do not perform better thanregulated airlines. Bruning (1991) applies the two-step stochastic frontier (cost) functionmethod to a sample of 73 international airlines in 1987 to assess the relationship betweenoperating efficiency and measures of market competition. The study does not find anypositive effects of competition on airlines’ efficiency. Jha and Sahni (1992) applies the paneldata approach proposed by Cornwell, Schmidt and Sickles (1989) to the case of six Canadianairlines to measure these airlines’ technical inefficiency for the period of 1970 to 1986.They estimate two separate frontier production functions with available passenger kilometres(seat kilometres) and available tonne-kilometres as the output measures, respectively, usingthe same set of input variables in both estimations. Good, Nadiri, ROller, and Sickles (1993)compares the technical efficiency and productivity growth among the four largest Europeancarriers and eight U.S carriers using the stochastic frontier method. They estimate thepotential efficiency gains of the European aviation liberalization by comparing efficiencydifferences between the two carrier groups. The study is carried out for the period of 1976to 1986. Loeb, Bruning and Ru (1994) estimates a Cobb-Douglas total cost frontier functionto measure airlines’ relative inefficiency levels for the years 1982, 1985, 1988, and 1990.The inefficiency estimates are then compared across market groups and years using ANOVAanalysis. The study finds that the Pacific Rim carriers group is the least inefficient whilethe Northern European carriers are the most inefficient.Good and Rhodes (1991) uses the data envelopment analysis method to examine theproductive performance of 37 airlines from the Pacific region over the period of 1976 to1986, and finds that there is a strong correlation between the productive efficiency of aChapter 9 Airline Efficiency 187country’s national carrier and the restrictiveness of the country’s bilateral agreement with theU.S. Cooper and Gallegos (1991) employs a combination of the DEA method, theconventional regression analysis, and the frontier methods to examine the performance ofLatin American airlines, in particular, the effects of ownership and international competitionon airlines’ performance. Based on the same data set as in Cooper and Gallegos (1991),Ray and Ru (1993) examines the annual productivity growth rate for the airlines during 1981to 1988 using the data envelopment analysis method, and finds that the U.S. carriersexperience faster growth in productivity than the Latin American carriers mainly due totechnical progress. Distexhe and Perelman (1993) uses the data envelopment analysismethod to measure the technical efficiency and productivity gains for 33 airlines during1977-1988. They also find technological progress as a major source of productivity growth.Most of the previous studies are based on the data prior to mid-1980s. Significantchanges have occurred in the industry during the last few years such as consolidation of theairlines in the U.S. and Europe, regulatory liberalization of the industry in Europe and othercountries, continued liberalization of international air transportation, and privatization ofmany airlines. It would be interesting to examine if there are any significant changes in theairlines’ productive performance following the regulatory and institutional development inthe international aviation industry.In examining the efficiency performance of the airlines, most of the studies havetaken into consideration of the effects of some output attribute variables, mostly load factorand stage length. Although being considered as important in most studies the effects ofgovernment ownership are examined analytically only in Bruning (1991) and Loeb (1994).Chapter 9 Airline Efficiency 188Their results appear to indicate, in contrast to common arguments, possible positive effectsof government ownership on airlines’s efficiency. Further examination of this issue isnecessary.People usually distinguish airlines’ outputs as passenger, cargo and charter services.However, a closer examination of airlines’s revenue shares (see Table D.2 in Appendix D)shows that the so-called incidental services, including catering services, ground handlingservices and maintenance services performed for other airlines, account for up to 30 percentof total operating revenues for some airlines with an average of about 8 percent for thesample airlines in this study. It therefore is an important component of the airlines’business. However, this component of the airlines’ business has been largely ignored inprevious studies on airline performance. This study considers the incidental services as oneof the airlines’ output in order to properly reflect the total output of airlines.This chapter attempts to measure and compare the efficiency of 36 airlines over the1980-1992 period and analyze it in order to identify the effects on efficiency of governmentownership and technical progress. The study also attempts to control for the effects of otherfactors such as load factor, stage length, and output composition. The efficiencyperformance of the airlines is compared between airlines in different continents. In addition,the efficiency estimates from the three methods are compared.Chapter 9 Airline Efficiency 1899.2 The Data and Model SpecificationEach observation in the sample is treated as an individual “decision-making-unit”(DMU). There are 447 DMUs in total. The sample carriers and data sources are describedin Appendix D.Airlines produce several distinct outputs, including scheduled passenger, scheduledfreight, mail, non-scheduled, and incidental services. The DEA method is designed to dealwith multiple output problems. However, there are problems in estimating parametricfrontier production function’ with multiple outputs. In order to have consistent resultscomparable across the three alternative methods, the outputs are aggregated into output indexusing the translog multilateral index procedure proposed by Caves, Christensen and Diewert(1982). Revenue shares are used as the weights in computing the output index since thereis empirical evidence indicating the existence of constant returns to scale in the airlineindustry (Caves, Christensen, and Tretheway, 1984, Gillen, Oum and Tretheway, 1990,Cornwell, Schmidt, and Sickles, 1990).Five categories of inputs are considered: (1) labour, (2) fuel, (3) materials, (4) flightcapital, and (5) ground property and equipment. Appendix D provides details on definitionand construction of these input variables.Airlines operate over routes of varying size and length and in markets with differenttraffic densities. Therefore, a realistic characterization of the industry’s productiveThe Monte Carlo study in Part I considers only frontier production functions.Therefore, production frontier functions are estimated here instead of the frontier costfunctions in order to be consistent with Part I.Chapter 9 Airline Efficiency 190Table 9.1Definition of Exogenous VariablesVariables DescriptionLOAD Weight Load Factor for all trafficSTAGE Stage Length: average flight distance%PASS Percentage of Passenger RTK in total RTK%INTL Percentage of International RTK in total RTKMAJ A Dummy for majority government ownershipMIN A Dummy for minority government ownershipTIME Time trendperformance requires not only the incorporation of multiple outputs and multiple inputs, butalso the special characteristics of the route networks, output attributes, and technologicalchanges over time. Table 9.1 lists a number of variables which reflect some specialcharacteristics of airlines’ operation. Note that the variable MAJ is a dummy variable whichequals to 1 if government owns 50 percent or more of the airline. Similarly, the variableMIN is a dummy variable which equals to 1 if government’s ownership of the airline is lessthan 50 percent.The CCR ratio model of the DEA method, specified by equation 4.2.2, is used here.It is appropriate since earlier studies have found that there are roughly constant returns toscale for airlines over a rather broad ranges of network sizes. The CCR ratio modelestimates a gross efficiency index using the output index and the five input variables. TheTOBIT analysis is then used to examine the effects of the exogenous variables on the DEAChapter 9 Airline Efficiency 191gross efficiency index and compute a residual efficiency index using equations (4.3.1) and(4.3.2).The estimation of the parametric frontier functions assumes a Cobb-Douglasproduction technology with five inputs producing a single output2. The deterministic frontierproduction function is estimated by COLS (see Chapter 4 ), and the stochastic frontierproduction function is estimated using the Battese and Coelli (1988) procedure. Inparticular, the deterministic core of the production frontiers is specified as:lny =%+EcL.1nX+EPkInZ (9.1)where y is the output index for observation j, x1 are the input variables, Zkj are theexogenous variables listed in Table 9.1, and a, (3 are the coefficients to be estimated.9.3 The ResultsThis section reports and discusses the empirical results. The study proceeds first withthe DEA-TOBIT analysis. Then the one-step deterministic frontier method is used toexamine the airlines’ productive performance. Finally, the one-step stochastic frontiermethod is applied to the sample data.2 Madala (1979) noted that measurement of technological change and efficiency are quiteinsensitive to the choice of functional form of production since these measures are relatedto shifts of the isoquants rather than their shapes.Chapter 9 Airline Efficiency 1929.3.1 The DEA ResultsWe first discuss the DEA gross efficiency estimates, and then examine the resultsfrom the TOBIT regression. A residual efficiency index is computed after removing theeffects of the exogenous variables, and then compared with the DEA gross efficiency index.9.3.1.1 The Gross Efficiency EstimatesThe DEA gross efficiency estimates are computed using the CCR ratio model on aUNIX computer. The linear programming problem has to be solved once for each DMU(observation) in the sample, that is, a total of 447 linear programming solutions are requiredin order to compute the DEA gross efficiency index.The DEA efficiency estimates are given in Table 9.2g. As pointed out in the earlierchapters, these DEA indices measure the “observed” productive performance, thus theyreflect the combined effects of true managerial and operational efficiency and of theexogenous factors which are beyond the airlines’ managerial control. One can only makemeaningful inferences about the true efficiency performance of the airlines after controllingfor the effects of the potentially influential factors.In terms of the DEA gross efficiency index, Finnair, KLM, Singapore Airlines,Continental, TWA, JAL, and Cathay Pacific have achieved a position close to the productionfrontier in 1989-1991. Sabena, Finnair, Singapore Airlines, and Continental haveexperienced the most significant improvement in their performance during the sample period:The 1992 data are not very reliable, therefore, the discussions on the results arefocused on earlier years.Chapter 9 Airline Efficiency 193Table 9.2Gross DEA Efficiency Index51980-82 1983-85 1986-88 1989-91 1992Qantas 0.808 0.818 0.820 0.812 1.000AUA 0.364 0.385 0.419 0.456 0.463Sabena 0.776 0.882 0.806 0.927 0.913Air Canada 0.759 0.743 0.738 0.729 0.691Canadian 0.959 0.946 0.781 0.779 0.734Fiimair 0.734 0.741 0.831 0.944 1.000Air France 0.712 0.757 0.728 0.691 0.676UTA 0.932 0.984 0.990 0.964 -Lufthansa 0.730 0.773 0.804 0.772 0.815Air India 0.567 0.633 0.581 0.579 -Alitalia 0.640 0.776 0.732 0.759 0.832JAL 0.811 0.865 0.914 0.963 0.894MAS 0.634 0.618 0.619 0.465 -Mexicana 0.890 0.796 0.596 0.575 0.482KLM 0.916 0.886 0.907 0.985 0.994PTA 0.483 0.476 0.455 0.540 0.517PAL 0.490 0.754 0.688 0.639 0.562Tap Air 0.760 0.849 0.869 0.807 0.687KAL 0.767 0.744 0.722 0.733 0.726Saudia 0.768 0.896 0.838 0.827SAS 0.659 0.691 0.610 0.682 0.690SIA 0.783 0.856 0.947 0.986 1.000Iberia 0.584 0.669 0.774 0.744 0.630Swiss Air 0.811 0.818 0.874 0.872 1.000British Airways 0.636 0.602 0.561 0.635 0.728Cathay- .1.000z 0.944 1.000American 0.848 0.842 0.854 0.844 0.898US Air 0.573 0.606 0.656 0.633 0.683Continental 0.745 0.806 0.927 0.910 0.978Delta 0.776 0.744 0.801 0.807 0.794Northwest 0.955 0.992 0.812 0.867 0.914TWA 0.844 0.870 0.820 0.948 1.000United 0.967 0.912 0.889 0.873 0.893American W- 0.833 0.795 0.918 1.000Eastern 0.739 0.746 0.804 0.810 -Pan Am 0.824 0.894 0.852 0.967 -1. Table entries are all three-year average unless otherwise noted.2. for 1988. 3 for 1985.Chapter 9 Airline Efficiency 1940.776 in 1980-82 to 0.927 in 1989-91 for Sabena, 0.734 in 1980-82 to 0.944 in 1989-91 forFinnair, 0.783 in 1980-82 to 0.986 in 1989-91 for Singapore Airlines, and 0.745 in 1980-82to 0.9 10 in 1989-9 1 for Continental. The DEA gross efficiency index of British Airwaysimproved consistently after its privatization in December 1986. On the other hand,Mexicana has experienced noticeable decrease in its performance during the same period.The DEA gross efficiency index of Canadian Airlines International (formally CanadianPacific) has also declined considerably after 1986, the year in which PWA purchased CPAir and created CAT. In terms of these gross efficiency index, Austrian Airlines, Air India,and Pakistan Airlines appear to be among the least efficient carriers in this sample.Table 9.3 compares mean values of the DEA gross efficiency estimates by continentsof the airlines’ registry. Major Asian carriers, including JAL, KAL, Singapore Airlines, andCathay Pacific, are rated as the most efficient. Other Asian carriers, namely, Air India,Pakistan International Airlines, Philippines Airlines, and Malaysia Airlines, are rated as theleast efficient ones. The U.S. carriers are very close to the major Asian Carriers in termsof the DEA gross efficiency index while the Canadian carriers are rated slightly better thanthe European Carriers. There is apparently no significant difference between the majorEuropean carriers and other European carriers in the sample.Chapter 9 Airline Efficiency 195Table 9.3The Means of flEA Gross Efficiency IndexEurope Europe Asia Asia U.S. CanadaMajor other Major otherMean 0.756 0.752 0.860 0.581 0.834 0.797St. Dev 0.116 0.193 0.100 0.099 0.110 0.093Ob. No 91 77 44 50 121 269.3.1.2 The Effects of Exogenous VariablesThe DEA gross efficiency estimates are affected by the differences in the airlines’operating environments. A number of factors have been identified as potentially havinginfluences on the airlines’ observed performance. They are listed in Table 9.1 and arediscussed earlier. In the next step, we attempt to attribute the differences among the DEAgross efficiency estimates to the variations in these exogenous variables. A residualefficiency index is consequently computed after removing the effects of these variables.The Tobit analysis is used to examine the effects on the DEA gross efficiency ofthese exogenous variables. Table 9.4 reports the TOBIT regression results. The mainresults are discussed below:(a) The Effects of Output Characteristics: The weight load factor (LOAD) is a measureof output performance (or service quality). It has a marginally significant positivecoefficient, indicating that airlines with high load factor tend to achieve high DEAgross efficiency index. This occurs because when the load factor is high, moreoutputs are produced for a given level of input. That is, cost per unit output falls asChapter 9 Airline Efficiency 196the load factor increases.It should be noted that in the deregulated domestic markets, such as in the U.S.,Canada, Australia, New Zealand, etc., airlines can manage load factor by choosingflight frequency and aircraft size for specific routes. In these markets, therefore, theload factor is not an “exogenous variable”, but rather a policy variable. However,in regulated domestic markets, such as in Japan, China, India, etc, and many of theinternational markets governed by restrictive bilateral agreements between countries,airlines do not have direct control of the “load factor”. Since most of the airlines inTable 9.4Tobit Regression ResultsDependent Variable: LOG (DEA Index)Coefficient T-ValueLOAD 0.137 1.551STAGE 0.196 9.271%PASS -0.333 5.740%INTL -0.069 5.824MAJ -0.127 6.305T 0.003 1.323CONSTANT -2.370 6.147No. of Observations 447Log-Likelihood 94. 863R2 0.381Note that the first four variables are in logarithm form.Chapter 9 Airline Efficiency 197our sample operate in both markets, a part of the effect of load factor is attributableto good management while the rest of the effect are beyond managerial control.(b) The Effects of Network Characteristics: The stage length (STAGE) measures theaverage flight distance, and is an important dimension of an airline’s network.STAGE has a statistically significant positive coefficient, meaning that airlines flyinglonger routes are expected to achieve higher DEA gross efficiency index. This isexpected because unit cost per passenger kilometer or per revenue tonne-kilometerdecrease with average stage length. Since airport charges, station costs and otherground expenses do not vary with stage length, the costs spread over more passengerkilometers (or revenue tonne-kilometers) the longer the stage length is.(c) The Effects of Output Composition: Two variables describing the composition ofairline output, %PASS and %INTL, are included in the Tobit regression in order todetermine the direction and magnitude of their effects on the observed efficiencyperformance of the airlines. The variable %PASS is the percentage of passengerRTK in total RTK, and has a statistically significant negative coefficient. Thenegative coefficient indicates that, ceteris paribus, the airlines with larger share oftheir traffic in passenger services are likely to have a lower DEA efficiency rating.This is expected because passenger services are more input-intensive than freightservices. Therefore, an airline with smaller %PASS would be able to produce moreoutputs with a given level of inputs than an airline with higher %PASS. The variable%INTL is the share of international traffic in total traffic. It has a significantnegative coefficient indicating that an airline with a higher proportion of internationalChapter 9 Airline Efficiency 198services tends to have a lower DEA gross efficiency index. For a given load factorand stage length, international services may incur additional costs such as crewstopovers due to lower frequencies, costs associated with sales, ticketing and handlingof passengers, etc. Another reason for the negative coefficient of %INTL is thatrestrictions on airlines’ operations imposed by bilateral air services agreements causeinefficiency.(d) The Effects of Government Ownership: Two dummy variables are used to indicatethe extent of government ownership: MAJority and MINority. Since variableMINority is not statistically significant, it is not included in the final regression. Thisimplies that a minority government ownership would not have any significant positiveor negative effect on the productive performance of the airlines. On the other hand,variable MAJority has a statistically significant negative coefficient indicating airlineswith government majority ownership are likely to be less efficient. These airlinesoften have less managerial freedom for making strategic and operational changes inorder to improve efficiency. In addition, existence of subsidized services requiredby government makes airlines less efficient by reducing incentives to improveefficiency.(e) The Effects of Time: TIME variable is used to estimate the residual technical (ormanagerial) efficiency over time. TIME has a (marginally) statistically significantpositive coefficient, indicating that the airline industry experienced technologicalprogress at an annual rate of about 0.3% during the sample period, 1980-1992.Chapter 9 Airline Efficiency 1999.3.1.3 The Residual EfficiencyAs seen from the forgoing discussions, the exogenous factors do have a significanteffect on the DEA gross efficiency index. Therefore, it is necessary to purge the effects ofthese factors from the airlines’ observed efficiency levels in order to have a better pictureabout the “true” relative efficiency performance of the airlines.Table 9.5 presents the airlines’ residual efficiencies computed from the TOBITregression after removing the effects of the exogenous variables. It should be noted thatthese residual efficiencies may still be distorted by factors which are left out of theregression because of lack of information. However, it is overall a better indicator of theairlines’ productive performance than the DEA gross index.From Table 9.5, we can see that the differences in the residual efficiencies amongthe airlines are much smaller than the DEA gross efficiency index. Some of the variationsin the observed efficiency performance as measured by the DEA gross efficiency index arecaused by variations in the exogenous variables which are controlled for in the TOBITregression.Finnair, KLM, SIA, Continental, and TWA are still rated among the most efficientcarriers in 1989-91 in terms of the residual efficiency index, while the performance ofCathay Pacific and JAL have been reduced from the top performers in terms of the DEAgross efficiency index to that of slightly above average performers in terms of residualefficiency. The relative efficiency rankings of UTA, KAL, and American Airlines are alsoreduced considerably in terms of the residual efficiency index. On the other hand, MalaysiaAirlines, Saudia, SAS, and Iberia are given relatively higher ratings by the residualChapter 9 Airline Efficiency 200Table 9.5Residual Efficiency Index’1980-82 1983-85 1986-88 1989-91 1992Qantas 0.665 0.647 0.636 0.620 0.756AUA 0.432 0.452 0.488 0.501 0.493Sabena 0.611 0.674 0.613 0.718 0.760Air Canada 0.764 0.713 0.691 0.586 0.555Canadian 0.792 0.771 0.661 0.651 0.599Finnair 0.784 0.749 0.778 0.837 0.887Air France 0.644 0.648 0.607 0.572 0.556UTA 0.713 0.73 1 0.729 0.704 -Lufthansa 0.689 0.692 0.693 0.662 0.718Air India 0.471 0.529 0.470 0.456 -Alitalia 0.638 0.741 0.683 0.685 0.763JAL 0.603 0.625 0.648 0.684 0.644MAS 0.753 0.720 0.679 0.483 -Mexicana 0.854 0.779 0.564 0.531 0.448KLM 0.716 0.665 0.670 0.715 0.724PIA 0.490 0.479 0.454 0.533 0.513PAL 0.470 0.706 0.650 0.590 0.505Tap Air 0.742 0.803 0.825 0.765 0.656KAL 0.565 0.540 0.520 0.551 0.518Saudia 0.819 0.992 0.856 0.813 -SAS 0.712 0.742 0.656 0.727 0.738SL4 0.683 0.716 0.754 0.745 0.742Iberia 0.625 0.689 0.785 0.734 0.623Swiss Air 0.731 0.703 0.734 0.718 0.801British Airways 0.641 0.590 0.498 0.527 0.593Cathay - - 0.7272 0.679 0.711American 0.664 0.677 0.717 0.701 0.739US Air 0.521 0.522 0.532 0.536 0.596Continental 0.607 0.672 0.783 0.770 0.819Delta 0.683 0.646 0.688 0.688 0.683Northwest 0.814 0.806 0.680 0.714 0.745TWA 0.729 0.740 0.786 0.799 0.836United 0.688 0.693 0.721 0.716 0.732American W-O.699 0.658 0.718 0.774Eastern 0.635 0.639 0.684 0.699 -Pan Am 0.644 0.730 0.709 0.7891. Table entries are all three-year average unless otherwise noted.2. for 1988. 3. for 1985.Chapter 9 Airline Efficiency 201efficiency index. However, Air India, Pakistan International Airlines, and Austrian Airlinesare still rated as the least efficient carriers among the sample airlines.Continental and Sabena have had the most significant improvement in theirperformance over the 13-year time period, while the performance of Air Canada, CanadianAirlines International, Air France, Malaysia Airlines, and Mexicana have experiencednoticeable declines. Another observation from the results is related to British Airways: theefficiency ratings for British Airways declined consistently during the earlier 1980s, thenimproved consistently to the end of 1980s and earlier 1990s. This reflects the sluggishperformance of BA prior to 1986, and its ambitious restructuring process following itsprivatization in 1986.The mean DEA residual efficiency estimates by regions are shown in Table 9.6. Asone can see, there are very small differences in the mean residual efficiency estimatesbetween regions. Recall from Table 9.3 the regional mean DEA gross efficiency estimatesrange from 0.58 1 to 0.860 which is considerably reduced in terms of the DEA residualefficiency index to ranging from 0.565 to 0.7 19. The Major Asian airlines have the highestmean DEA gross efficiency rating, however, in terms of the DEA residual efficiency, theirperformance is reduced to that close to the lower end. The Other Asian carriers are still theleast efficient carrier group. It is also noted that the relative performance of EuropeanMajors and Other European Carriers, and the relative performance of Canadian carriers andU.S. carriers are reversed as compared to the DEA gross efficiency estimates.Chapter 9 Airline Efficiency 202Table 9.6Mean DEA Residual Efficiency Estimates by RegionEurope Europe Asia Asia U.S. CanadaMajor other major other[v1ean 0.669 0.688 0.643 0.565 0.692 0.694St. Dev. 0.075 0.122 0.081 0.110 0.083 0.079Obs No. 91 77 44 50 121 269.3.2 The Deterministic Frontier MethodThe deterministic frontier method is applied to the airline data set with the same setof output and input variables as used in the DEA-TOBIT analysis. The estimation of thefrontier production function follows the one-step procedure which includes the exogenousvariables directly in the estimation.The estimated “best practice” production function is reported in Table 9.7. The fourinput variables are all statistically significant and, as expected, have the positive coefficients.Except for dummy variable MIN, all the exogenous variables appear to be statisticallysignificant4. LOAD factor has a positive coefficient implying that airlines with high loadfactor achieve high efficiency ratings when their output is measured in terms of revenuegenerating output such as RPK, RTK, etc. STAGE length also has a significant positivecoefficient implying that airlines with longer average stage length are expected to achieve‘ Actually %INTL is not significant at the traditional 5%. However, it is included sinceit serves to confirm the results from the DEA-TOBIT procedure.Chapter 9 Airline Efficiency 203Table 9.7Deterministic Frontier Production FunctionDependent Variable: LOG (OUTPUT INDEX)Variables Coefficient T-ValueLABOUR 0.268 8.195FUEL 0.309 9.369MATERIALS 0.048 1.714FLIGHT EQUIPMENT 0.386 12.63LOAD 0.682 8.100STAGE LENGTH 0.218 11.18%PASS -0.239 4.834%INTL -0.014 1.292MAJORITY -0.118 6.475TIME 0.004 1.796CONSTANT -1.064 2.506R2 0.9719Log-likelihood Function 2 16.526No. of Observations 447Note that all variables are in logarithm form except for MAJ and TIME.Chapter 9 Airline Efficiency 204higher observed efficiency. On the other hand, a carrier with a large proportion ofpassenger traffic or a large proportion of international traffic is expected to achieve a lowerefficiency rating. The ownership variable MAJ has a negative coefficient of -0.12 which isvery similar to the DEA-TOBIT results. This indicates that airlines with governmentmajority ownership are about 11 to 12 % less efficient. The coefficient for the TIME isstatistically significant, and again similar to the DEA-TOBIT results, indicating that theannual technological progress in the international airline industry is about 0.3 to 0.4 %. Insummary, the results in Table 9.7 essentially confirm the DEA-TOBIT results.By following the procedure described in equations (4.1.5) to (4.1.7), the observationspecific efficiency estimates are estimated using the production function shown in Table 9.7.Table 9.8 lists the efficiency estimates from the deterministic frontier method. Since theexogenous variables are included in the estimation of the frontier production function, andtheir effects are removed in computing the efficiency estimates, these efficiency estimatesin Table 9.8 are closer to the DEA residual efficiency index than the DEA gross efficiencyindex. SIA, Continental, and TWA are still among the most efficient carriers in 1989-91,however, the performance of Finnair is reduced to that of mid-range. Air India, PakistanInternational Airlines, and Austrian Airlines are still among the least efficient carriers asindicated before by the DEA residual efficiency, but the performance of Philippines Airlinesis given a better rating. Sabena and Continental are shown to have experienced considerablyimprovement during the sample period, while the performance of Air Canada and CanadianAirlines International have declined during the same period.The mean efficiency estimates from the deterministic frontier method by regions areChapter 9 Airline Efficiency 205Table 9.8Efficiency Estimates by Deterministic Frontier Method11980-82 1983-85 1986-88 1989-91 1992Qantas 0.624 0.628 0.643 0.609 0.686AUA 0.568 0.518 0.528 0.525 0.579Sabena 0.682 0.689 0.670 0.830 0.853Air Canada 0.776 0.725 0.735 0.599 0.528Canadian 0.767 0.803 0.692 0.689 0.623Finnair 0.687 0.727 0.784 0.753 0.663Air France 0.621 0.596 0.589 0.556 0.717UTA 0.829 0.846 0.895 0.814 -Lufthansa 0.705 0.774 0.768 0.698 0.734Air India 0.551 0.532 0.489 0.475 -Alitalia 0.701 0.747 0.583 0.721 0.847JAL 0.590 0.586 0.552 0.585 0.569MAS 0.790 0.731 0.713 0.535 -Mexicana 0.773 0.842 0.651 0.647 0.654KLM 0.722 0.648 0.682 0.658 0.649PIA 0.527 0.495 0.518 0.582 0.565PAL 0.583 0.755 0.706 0.697 0.620Tap Air 0.803 0.773 0.873 0.837 0.730KAL 0.624 0.651 0.626 0.660 0.578Saudia 0.883 0.875 0.762 0.783 -SAS 0.616 0.787 0.722 0.724 0.737S1A 0.702 0.831 0.877 0.837 0.794Iberia 0.626 0.746 0.816 0.730 0.659Swiss Air 0.639 0.612 0.749 0.704 0.816British Airways 0.664 0.639 0.558 0.581 0.637Cathay - - 0.8132 0.706 0.691American 0.717 0.732 0.747 0.702 0.689US Air 0.586 0.629 0.600 0.570 0.612Continental 0.668 0.706 0.802 0.799 0.817Delta 0.698 0.751 0.727 0.691 0.692Northwest 0.692 0.741 0.651 0.718 0.726TWA 0.718 0.753 0.803 0.805 0.856United 0.704 0.706 0.738 0.708 0.700AmericanW - 0.685 0.617 0.689 0.682Eastern 0.644 0.655 0.717 0.592 -Pan Am 0.616 0.702 0.732 0.757 -1. Table entries are all three-year average unless otherwise noted. 2. for 1988. 3 for 1985.Chapter 9 Airline Efficiency 206given in Table 9.9. The average efficiency level is slightly higher than those of the DEAresidual efficiency index. Carriers in the “other Asia” group are again the least efficientcarriers. The average performance of major Asian carriers and major European carriers aresimilar at group level. Canadian carriers appear to perform slightly better than the UScarriers on average. The relative performance between the regions are similar to that interms of the DEA residual efficiency index. The efficiency gaps among the sample airlinesare fairly small as in the case of DEA residual efficiency estimates.Table 9.9Mean Efficiency Estimates by Deterministic FrontierEurope Europe Asia Asia US CanadaMajor other Major otherMean 0.677 0.727 0.680 0.612 0.701 0.712St. Dev 0.078 0.117 0.110 0.113 0.072 0.077Obs. 91 77 44 50 121 269.3.3 The Stochastic Frontier MethodThis section reports and discusses the results from the stochastic frontier method.Again the same set of input and output variables are considered, and the estimation of thefrontier production function follows the one step procedure which includes the exogenousvariables directly in the estimation.Table 9.10 presents the deterministic core of the stochastic frontier function. Thisfunction is very close to the deterministic frontier production function (Table 9.7) except thatChapter 9 Airline Efficiency 207Table 9.10Stochastic Frontier Production FunctionDependent Variable: LOG (OUTPUT INDEX)Variables Coefficient T-ValueLABOUR 0.263 9.043FUEL 0.253 10.79MATERIALS 0.113 4.056FLIGHT EQUIPMENT 0.342 13.35GROUND EQUIPMENT 0.019 1.647LOAD 0.574 8.539STAGE LENGTH 0.264 16.12%PASS -0.187 4.777%INTL -0.011 1.012MAJORITY -0.117 5.077MINORITY -0.042 1.511TIME 0.007 3.645CONSTANT -0.773 1.987Log-likelihood Function 239.966No. of Observations 447Note that all variables are in logarithm form except for MAJ and TIME.Chapter 9 Airline Efficiency 208the GPE and MINORITY variables here are statistically significant while they are not in thedeterministic model, and the variable %INTL is not statistically significant here while it ismarginally significant in the deterministic model.Similar to the results from previous production functions, the stochastic frontierresults indicate that airlines with higher load factor and longer stage length are expected tohave higher gross efficiency ratings, while airlines which have a larger percentage ofpassenger traffic and/or international traffic are expected to have lower gross efficiencyratings. Government ownership in airlines will have negative effects on the productiveefficiency of the airlines. Technological progress has improved the overall performance ofthe industry during the sample time period. This is particularly the case for some of themajor Asian carriers. They exhibit considerable improvement in terms of the grossefficiency measures5,but after removing the effects of the exogenous variables their “net”efficiency improvements over time appear to be reduced substantially. However, it is notedthat the annual rate of technological progress indicated by the stochastic frontier method ishigher than that from the deterministic frontier method or the DEA-TOBIT analysis. Thismay be partly due to the fact that the stochastic frontier method attempts to filter out thestatistical noise, thus attribute some of the “left over” positive variations in the efficiencyestimates to technological progress.The efficiency estimates from the stochastic frontier model are given in Table 9.11.Overall, these efficiency estimates are higher than those from the other two methods. ThisThe efficiency estimates from first stage of the two-step parametric (both deterministicand stochastic) procedure also show similar pattern.Chapter 9 Airline Efficiency 209Table 9.11Efficiency Estimates by Stochastic Frontier Method’1980-82 1983-85 1986-88 1989-91 1992Qantas 0.730 0.734 0.740 0.707 0.802AUA 0.626 0.584 0.588 0.593 0.651Sabena 0.851 0.854 0.820 0.954 0.967Air Canada 0.944 0.884 0.891 0.722 0.643Canadian 0.934 0.941 0.834 0.829 0.748Finnair 0.874 0.907 0.850 0.926 0.796Air France 0.785 0.762 0.750 0.707 0.846UTA 0.944 0.948 0.963 0.903 -Lufthansa 0.890 0.951 0.944 0.873 0.913Air India 0.572 0.595 0.549 0.524 -Alitália 0.848 0.910 0.939 0.876 0.970JAL 0.773 0.770 0.712 0.732 0.699MAS 0.936 0.875 0.856 0.678 -Mexicana 0.916 0.920 0.751 0.734 0.737KLM 0.922 0.834 0.856 0.833 0.826PIA 0.599 0.572 0.607 0.661 0.659PAL 0.715 0.914 0.837 0.836 0.734Tap Air 0.887 0.873 0.932 0.889 0.776KAL 0.761 0.784 0.750 0.797 0.711Saudia 0.965 0.955 0.895 0.910 -SAS 0.796 0.945 0.871 0.873 0.882SIA 0.845 0.953 0.964 0.937 0.909Iberia 0.772 0.906 0.955 0.863 0.770Swiss Air 0.830 0.796 0.913 0.885 0.966British Airways 0.820 0.788 0.685 0.706 0.768Cathay - - 0.9622 0.868 0.855American 0.891 0.910 0.931 0.875 0.860US Air 0.760 0.805 0.776 0.734 0.777Continental 0.836 0.872 0.964 0.959 0.966Delta 0.907 0.937 0.918 0.877 0.870Northwest 0.907 0.941 0.841 0.908 0.912TWA 0.896 0.921 0.963 0.961 0.977United 0.913 0.904 0.931 0.891 0.880American W - Ø•7553 0.786 0.867 0.853Eastern 0.841 0.846 0.903 0.774Pan Am 0.778 0.873 0.892 0.9221. Table entries are all three-year average unless otherwise noted2. for 1988. 3 for 1985.Chapter 9 Airline Efficiency 210observation is consistent with the results from Monte Carlo study in Chapter 6.Sabena, Finnair, SIA, Continental, and TWA are among the most efficient carriersin 1989-9 1 according to these efficiency estimates. Air India, Austrian Airlines and PakistanInternational Airlines are still among the least efficient carriers. Sabena and Continental areshown to have experienced substantial improvement in efficiency performance over the 13-year time period. Singapore Airlines, Iberia, and American Airlines had exhibitedconsiderable efficiency improvement during the first half of the 1980s, however, theirperformance show a downward trend since the mid-1980. The performance of Air Canada,Canadian Airlines International, and Malaysia Airlines appear to deteriorate throughout thesample period.Table 9.12Mean Efficiency Estimates Using Stochastic FrontierEurope Europe Asia Asia US CanadaMajor other major otherMean 0.876 0.846 0.820 0.718 0.877 0.859St. Dev 0.083 0.124 0.092 0.144 0.074 0.091Obs 91 77 44 50 121 26Table 9.12 lists the mean efficiency estimates by region. According to these resultsthe US carriers are the most efficient airlines on average followed by the Canadian carriersand the carriers in the “other European” group. The “other Asian” carriers are the leastefficient carriers.Chapter 9 Airline Efficiency 2119.4 Comparison of Efficiency Estimates by Alternative MethodsIn the previous section, the three alternative methods are applied to the same airlinedata set to examine the efficiency performance of the airlines. Earlier discussions about theresults are focused on the general policy implications and the performance of individualairlines. In this section, the three sets of efficiency estimates are compared in terms of themeans and correlations.Table 9.13Means of Airlines’ Efficiency EstimatesDeterministic Stochastic DEA-TOBITMean 0.690 0.834 0.669St. Dev 0.100 0.112 0.108Maximum 1.000 0.987 1.000Minimum 0.434 0.484 0.415Table 9.13 lists the means of the residual efficiency estimates by the three methods.The mean of the efficiency estimates by the stochastic frontier method is considerably higherthan those by the deterministic frontier method and the data envelopment analysis method.This observation is expected in view of the Monte Carlo results in Part I, which consistentlyshow higher mean efficiency estimates by the stochastic frontier method than the other twomethods. The lower means of the deterministic frontier method and the DEA are due to thefact that both methods attribute deviations from the frontier as inefficiency. Some of theinefficiencies indicated by these two methods are in fact due to the effects of someunidentifiable exogenous variables which are accounted for by the stochastic frontier method.Chapter 9 Airline Efficiency 212Table 9.14Correlation and Rank Correlation CoefficientsBetween the Alternative Efficiency Estimates of AirlinesCorrelation CoefficientsDeterministic I Stochastic DEA-TOBITDeterministic 1Stochastic 0.9262 1DEA-TOBIT 0.7945 0.8282 1Rank CorrelationsDeterministic 1Stochastic 0.9562 1DEA-TOBIT 0.7936 0.8277 1The correlations among the three sets of residual efficiency estimates are investigatedusing both Pearson correlation coefficients and Spearman’ rank correlation coefficients. Theresults aie reported in Table 9.14. Although there are some significant differences in termsof individual airlines’ efficiency estimates as mentioned in the previous section, there is avery high correlation between the residual efficiency estimates from the two parametricmethods, and the correlation between the DEA results and those from the two parametricmethods are also reasonably high. This indicates that for the present airline case the resultsare not very sensitive to the choice between the stochastic frontier method and thedeterministic frontier method, and the choice between the parametric methods and the DEAdoes not make any dramatic differences in the results either. The high rank correlationsindicate that the choice of different methods does not impinge much on the ranking of firmChapter 9 Airline Efficiency 213(observation) specific efficiency estimates among the airlines. Thus, the results in Table9.14 suggest that the three different methods yield broadly similar results in measuringairlines’ efficiency, at least for the functional form considered in this chapter. The MonteCarlo results in Part I indicate that when the variations in the firms’ environments are notvery large, the three alternative methods are expected to yield similar efficiency estimates.The high correlations and rank correlations in Table 9.14 are essentially consistent with theMonte Carlo results.9.5 Summary and Concluding RemarksThis chapter provides some empirical evidence on the comparative performance ofthe three alternative efficiency measurement methods, namely, the deterministic frontiermethod, the stochastic frontier method, and the data envelopment analysis method. Thethree methods are applied to a sample of 36 international airlines during the period of 1980to 1992 to measure the observation specific efficiency of the carriers and to identify theeffects on the efficiency performance of government ownership, operating characteristics,and technological progress.The two parametric frontier methods are estimated following an one-step procedurewhich incorporates the exogenous variables directly in the estimation of the frontierfunctions. The data envelopment analysis method utilizes a two-step procedure. At the firststage, the CCR ratio model is used to estimate a gross efficiency index of the airlines. Atthe second stage, the gross efficiency estimates are analyzed using the TOBIT regression toidentify the effects of exogenous variables and to compute a residual efficiency index. TheChapter 9 Airline Efficiency 214empirical results from all three models indicate that: (1) airlines with high load factor and/orlong stage length are expected to achieve higher gross efficiency ratings, (2) airlines with alarge proportion of passenger services and/or international traffic are likely to have lowerefficiency ratings, (3) airlines with majority government ownership are significantly lessefficient, implying that a greater degree of managerial freedom may enhance an airline’sproductive efficiency substantially, and (4) technological progress has significantly improvedthe overall performance of the industry during the sample period, especially for some of themajor Asian carriers. In addition, the discrepancy between the “gross” efficiency index andthe “residual” efficiency index indicates that the exogenous variables do have considerableeffects on the efficiency performance of the airlines.The comparison of the results from the three methods illustrates that although thereare noticeable differences in the actual levels of estimated efficiency, the overall pattern ofthe efficiency estimates from the two parametric methods are essentially the same, and theresults by the DEA-TOBIT are broadly similar to those from the parametric methods. Thepolicy implications from all three methods are consistent.By comparing the results from Chapter 8 and Chapter 9, it is found that the threealternative methods give more consistent results in the airline case than in the railway case.This is because that the airline data set is considered as a ‘better” data set for application ofthe three methods for the following reasons: (1) the data are fairly consistent across theairlines, (3) there are more observations, and (3) there is less degree of variations in thecarriers’ operating environments. This is consistent with the Monte Carlo results in Part Iwhich indicate that the three alternative methods are expected to yield similar efficiencyChapter 9 Airline Efficiency 215estimates when the variations in firms’ environments are not very large. The empiricallesson from this is that in situations where there are large variations in the DMUs’ operatingenvironments and production characteristics the results would be sensitive to the choice ofmethod, therefore, careful examination of the situation and the estimation results isnecessary.216Chapter 10Summary and ConclusionsProductive efficiency is a performance measure to evaluate production units, and isall indicator of success. Measuring efficiency and identifying the sources of efficiencydifferentials are essential for designing public and corporate policies to improve performance.During the last three decades, various methods have been developed to measure productiveefficiency. Different methods often yield different efficiency rankings among the firms beingconsidered, and may lead to different policy implications on how to improve the efficiencyof a particular firm and of the overall industry. Each method has its strengths andweaknesses. Knowledge of these strengths and weaknesses will help researchers and policyanalysts to choose the most “suitable” method for a particular situation, and thus to makeaccurate measurements of efficiency. These in turn will help policy makers to makeappropriate policy decisions. Therefore, it is important to study the relative merits ofdifferent methods in terms of their abilities to reveal the structure of production technologyand the nature and extent of inefficiency under different conditions. This study comparesthree alternative methods in measuring firm specific efficiency, namely the deterministicfrontier method, the stochastic frontier method, and the data envelopment analysis method.In Part I, Monte Carlo experiments are carried out to examine the relative merits ofthe three methods where the underlying production technology and efficiency profile areknown. The Monte Carlo results and their methodological implications are summarized asfollows:Summary and Conclusions 217• The performance of all three methods improve as the sample size increases.Therefore, it is desirable to have a large sample whenever possible. However, itshould be noted that when the sample size is too large computational cost of the DEAmethod becomes high.• The variations in inputs do not appear to have much effect on the performance of thethree methods in cases where the elasticity of input substitution is greater than one.However, in the case of weak input substitution, the performance of the deterministicfrontier method and the stochastic frontier method fall noticeably with the variationsin input variables. Therefore, the DEA method would be a good choice in situationswhere there are evidences indicating weak input substitution, and where largevariations in sample firms’ input variables are observed.• The performance of all three methods deteriorates sharply as the level of noise rises,especially the DEA and the deterministic frontier method. The stochastic frontiermethod is expected to produce better estimates than the other two methods when thenoise level is high.• The magnitude of exogenous variables does not appear to have any significant effectson the performance of the one-step parametric methods as long as the exogenousvariables can be correctly identified and accounted for. On the other hand, theperformance of the two-step procedure, especially for the DEA and the deterministicfrontier method, is very sensitive to the magnitude of the exogenous variables.Technically, it is desirable to use the one-step stochastic frontier method. However,the two-step procedure relates the exogenous variables directly to efficiencySummaiy and Conclusions 218performance, and thus may be appealing to the policy and decision makers. TheDEA-Tobit method is a reasonably good alternative in situations where variations inthe firms’ operating environments are not very large.• The stochastic frontier method is rather robust with respects to outliers. Both theDEA method and the deterministic frontier method are affected by the presence ofoutliers, but the extent of the effects are not as significant as expected. If it ispossible, one should try to identify and investigate the causes of any potentiallyinfluential outliers before applying the DEA method and the deterministic frontiermethod to a particular data set.• The performance of the stochastic frontier method is not heavily influenced by thestructure of the underlying production technology, while that of the deterministicfrontier method is affected by the presence of input complementarity, but not by thereturns to scale. The performance of the DEA method deteriorates as the returns toscale increases, and in the presence of input complementarity. The use of DEAmethods should be avoided as much as possible in situations where there areincreasing returns to scale, and/or high complementarity among inputs. Although theBCC model is developed to account for the effects of returns to scale, our MonteCarlo results show that it can not effectively deal with the increasing returns to scalecondition.The simulation results in Part I provide some general guidelines regarding theperformance of the three alternative methods under certain known conditions. Since inpractice the “true” underlying production technology is not likely to be known, the choiceSummary and Conclusions 219of methods would depend on careful examination of available information and a goodunderstanding of the production situation. The two case studies in Part II serve as examplesof how the three methods can be applied to a real world problem. Further, researchers mayencounter problems which are not considered in simulation studies. The two case studiesalso illustrate some of the problems one may encounter in empirical situations where thereis no prior knowledge of either the production technology nor the efficiency profile.Chapter 8 reports on a case study of efficiency performance of railways from 19OECD countries during the period of 1978-89. The three alternative methods are appliedto the sample data to measure the productive efficiency of the railways and to identify theeffects of government intervention and subsidization on the productive efficiency. Theresults show that: (1) railway systems with high dependence on public subsidies aresignificantly less efficient than similar railways with less dependence on subsidies, and (2)railways with high degree of managerial autonomy from regulatory authority tend to achievehigher efficiency. The empirical results also confirm that efficiency measures may not bemeaningfully compared across railways without controlling for the variations in railways’operating and market environments. In addition, comparison of the efficiency estimates fromthe three alternative methods confirms the Monte Carlo result in Part I that the stochasticfrontier method yields higher efficiency estimates, on average, than the other two methods.The efficiency estimates by the two parametric methods are highly correlated although theirmean values are different. There are substantial differences between the efficiency estimatesobtained from the DEA-TOBIT analysis and those obtained by the parametric methods.However, the policy implications from all three methods are consistent. The main reasonSummary and Conclusions 220for the differences is that the DEA-TOBIT analysis is a two-step procedure while theparametric frontier production functions are estimated using an one-step procedure. TheMonte Carlo study in Part I has shown that these two procedures often produce differentefficiency estimates. Further, the DEA-TOBIT analysis considers two output variables,while the parametric methods consider only one output variable.The second case study is reported in Chapter 9. It measures and compares theefficiency of 36 international airlines during the period of 1980-1992, and identifies theeffects on efficiency of government ownership and technical progress. The empirical resultsshow that technological progress has improved the productive efficiency of the airlineindustry over time, especially for some of the major Asian carriers. The airlines withmajority government ownership are shown to be less efficient than other airlines with similaroperating characteristics. The results also indicate that the effects of network and marketenvironments should be controlled for in order to measure productive efficiency meaningfullycomparable across airlines.These two cases represent two rather different situations. In the railway case, theservices are mostly provided by highly regulated, nationalized firms. The firms operate invery different environments. The noise level and the effects of exogenous variables areexpected to be high, and it is very likely to have outliers in the sample. In the case ofinternational airlines, firms operate in a fairly competitive environment. Although there isa high degree of diversity in size, the firms have access to essentially the same technologies.They acquire basic capital inputs (aircrafts), fuel and other inputs from the internationalmarket. The noise level and the magnitude of the effects of exogenous variables are likelySummary and Conclusions 221to be lower than those in the railway case. Also, the sample size in the airline case is largerthan that in the railway case. Therefore, in view of the Monte Carlo results in Part I weexpect more consistent results from the three alternative methods in the airline case. Thereis much less discrepancy and higher correlations among the alternative efficiency estimatesin the airline case than in the railway case. This is consistent with the results of the MonteCarlo study in Part I.Both cases are affected by statistical noise and identifiable exogenous variables.According to the Monte Carlo results, the one-step stochastic frontier method would producebest estimates if one has a clear picture of the production situation and the environment suchthat the model can be correctly specified. However, there are some practical limitations tothe parametric methods. First, data on input and output prices are often required to applythe parametric methods, especially for the cases involving multiple outputs. However,consistent price data across sample firms may not be available. Further, the intrinsiccollinearities among the explanatory variables pose another problem for estimation of theparametric frontier functions. When these problems cannot be overcome, the DataEnvelopment Analysis method may be the only choice available.It is worth noting some of the limitations of this study. (1) The results from theMonte Carlo experiments in Part I are based on a specific experimental design. Althoughit is believed that the experimental design as described in Chapter 5 is based on reasonableassumptions about possible empirical situations, at least for the transportation industry, therestill exists a possibility that the findings may change if the experimental design, such as theranges of exogenous variables and outliers, is altered. (2) Total train kilometres is used asSummary and Conclusions 222the output measure in estimating the parametric frontier productions in the railway casebecause data required to aggregate the multiple outputs into a more sensible single outputmeasure are not available. (3) The number of employees is used as the labour input in bothapplications, because of the lack of better data such as number of labour hours by jobcategories. (4) For convenience, all exogenous variables are assumed to affect only theobserved production output level, thus are treated in the same manner. Further study canbe conducted to address this issue under the assumption that some of these variables wouldaffect only the observed production level (input consumption and output level), while otherswould affect the “true” efficiency level and consequently affect the observed productionlevel. Furthermore, exogenous variables are assumed to be independent of the efficiencyperformance. In practice, however, there may exist certain causal relationship between some“exogenous variables” and efficiency, such as efficiency and subsidization. How to treatsuch causal relationship properly in making efficiency measurement requires further research.This thesis is concerned only with technical efficiency. Allocative efficiency andprofitability are not considered in the thesis. Technical efficiency measures only the relationbetween output and input quantities, it does not consider the costs of the inputs nor the(monetary) values of the outputs. That is, there is no obvious relationship between efficiencyand profitability. This is confirmed by examining the efficiency performance of the railwaysand airlines and their financial performance in terms of operating ratios. For example,British Airways is considered as a successful example by the international aviation industry,it has made profits when most airlines have been losing money. 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(1983), “A Note on the Decomposition of Cost Efficiency into Technicaland Allocative Components”, Journal of Econometrics, 23, pp.401-5236Appendix ADescription of Computer ProgramsA.1 Computer Program for DEAIn actual computation, the dual formulation of the DEA method is more tractable thanthe primal. In the primal formulation, the constraints are indexed on all DMUs, while inthe dual formulation the constraints are indexed on inputs and outputs and sum over DMUs.For the problems in this thesis, the number of inputs and outputs is far less than the numberof DMUs. Phillips, Ravindran and Solberg (1987) have shown that the computationalefficiency of the simplex method falls with increases in the size of the constraint set. Hencethe dual program with only (m + s) constraints on inputs and outputs is computed inpreference to its primal with n+ 1 constraints.Implementation of the DEA method in the Monte Carlo experiments is illustrated bythe following sample FORTRAN program1 which follows the one step non-Archimedeanmodel procedure where E, the non-Archimedean scalar, is given the value of 106. TheDEA method requires the solution of linear programming problems. The user-callablesubroutines from the XMP Mathematical Programming Library (Marsten, 1981) are used.Note that the linear programming problem is solved for each of the n DMUs.The computer codes for BCC model and for railway application requires only minorchanges, so they are not listed here.Appendix A Computer Programs 237CC THIS IS THE SAMPLE FORTRAN CODE FOR THE DEA METHODC USING THE DUAL FORMULATION OF THE CCR MODEL(1978)C For use on a P.C.CIMPLICIT REAL*8 (A—H,O-Z)CHARACTER*24 FIN, FOUT, INXWRITE(*,999)READ(*,998) FINWRITE(*, 997)READ(*,998) FOUTWRITE (*, 996)READ(*,998) INX999 FORMAT(1OX,’ INPUT DATA FILE NAME=’)997 FORMAT (lOX,’ OUTPUT FILE NANE=’)996 FORMAT(lOX,’ INDEX FILE NANE=’)998 FORMAT(A24)OPEN(5, FILE=FIN, STATUS=’OLD’)OPEN(6, FILE=FOUT)OPEN(7, FILE=INX)CCALL MATRXSTOPENDCCSUBROUTINE MATRXCCC THE PURPOSE OF THIS SUBROUTINE IS TO GENERATE XMP MATRIXC FROM THE ORIGINAL DATA FILE WHICH IS ORGANIZED VARIABLE BYC VARIABLE,CC DEFINITION OF VARIABLESC THE OBSERVED OUTPUT LEVELC Xl: INPUT 1C X2: INPUT 2C X3: INPUT 3CCC DECLARATIONS FOR THE PROBLEM DATA.CREAL DATA(4,250)REAL RATIO(250)Appendix A Computer Programs 238REAL WEIGHT(4,250)CCC DECLARATIONS FOR THE XMP VARIABLES AND ARRAYS.CDOUBLE PRECISION B(4) ,BASCB(4) ,BASLB(4) ,BASUB(4),X BOUND,CANDA(4,6),CANDCJ(6),CJ,COLA(4),X LJ,MEMORY(20000),UJ,UZERO(4),XBZERO(4),YQ(4),X Z,ZTEMP,UTEMPDOUBLE PRECISION BLOW(4)INTEGER BNDTYP, COLLEN, COLMAX,X ERROR, FACTOR, IOERR, IOLOG, ITER,X ITER1,ITER2,LENMA,LENMI,LENMY,LOOK,M,MAPA(20),X MAPI(20) ,MAXA,MAXN,MAXN,N,NTYPE2,PICK,PRINT,TERMIN,X UNBDDQC THE NEXT STATEMENT SHOULD SPECIFY HALF-WORDS IF POSSIBLE.INTEGER*2 CAND(6) ,CANDI(4,6) ,CANDL(4) ,BS(4),X COLI(4),ROWTYP(4),STATUS(259),MINMAX,BASIS(4)CC MINMAX = -1 MEANS MINIMIZATIONC +1 MEANS MAXIMIZATIONCC Note that the convention in XMP is maximizationCCINTEGER IDNO,K,I,JCC SET VALUES FOR I/O UNITSCIOIN=5IOLOG=6IOERR=6CCC INITIALIZE XMP PARAMETERS.CMINMAX= 1MAXA=1O11MAXM=4MAXN=2 59COLMAX=4PICK=6LOOK=5 0FACTOR=5 0LENNY=2 0000CDO 310 I=1,N310 STATUS(I)=0CAppendix A Computer Programs 239C UP TO HERE WE FINISH DEFINING THE DATA STRUCTURECCC THE FOLLOWING SUBROUTINE IS TO SET UP A STARTING BASISCCALL XSLACK (B, BASCB,, BASLB, BASUB, BLOW,X BNDTYP, BOUND, COLA, COLI, COLMAX, IOERR,X LENNA,LENMI,LENNY,M,MAPA,MAPI,MAXM,MAXN,MEMORY,X N,ROWTYP, STATUS,UZERO,XBZERO, Z)CC THE FOLLOWING SUBROUTINE IS TO SOLVE THE LP BY THEC PRIMAL SIMPLEX METHODCCALL XPRIML (B, BASCB, , BASLB, BASUB, BNDTYP,BOUND,X CAND, CANDA, CANDCJ, CANDI, CANDL, COLA, COLI, COLMAX,X FACTOR, IOERR, IOLOG, ITER1, ITER2 , LENMA,LENMI ,LENNY,X LOOK,M,MAPA,MAPI,MAXM,MAXN,MEMORY,N,NTYPE2,PICK,X PRINT,STATUS,TERMIN,UNBDDQ,UZERO,XBZERO,YQ, Z)WRITE (6 , 650)650 FOPMAT(5X, ‘XPRIMAL’)CCC “Z” IS THE VALUE OF THE OBJECTIVE FUNCTION.CZTEMP=ZIF (MINMAX .EQ. -1) ZTEMP=-ZTEMPRATIO (K) =1/ZTEMPCC PRINT OUT SOLUTIONS: IT INCLUDES THE CURRENT BASICC SOLUTION AND THE OBJECTIVE FUNCTION VALUECCALL XPRINT(BASIS,BNDTYP, BOUND, IOERR, IOLOG,X LENMA,LENNY,M,MAPA,MAXN,MAXN,MEMORY,N,NTYPE2,X STATUS,XBZERO, ZTEMP)CC PRINT OUT THE VALUES OF THE DUAL VARIABLESCWRITE(6 ,910)910 FORMAT(1HO///5X,28HVALUES OF THE DUAL VARIABLES)WRITE(6,911)911 FORMAT(1H0, 5X, 1OHCONSTRAINT, 12X, 14H DUAL VARIABLE)C FLIP THE SIGN IF WE ARE MINIMIZINGWEIGHT (1, K) =-UZERO ( 1)DO 300 I=2,MUTENP=UZERO (I)WEIGHT(I,K)=UTEMPIF(MINMAX.EQ.-1) WEIGHT(I,K)=-UTENPAppendix A Computer Programs 240300 CONTINUEWRITE(7, 799) K, RATIO(K) ,WEIGHT(1,K), WEIGHT(2 ,K),X WEIGHT(3,K), WEIGHT(4,K)799 FORMAT(2X,18,5F12.6)CCC END OF ONE ROUNDC800 CONTINUECWRITE(*,803) KN803 FORMAT(2X,” NUMBER OF SAMPLE”, F8.0)999 CONTINUECC END OF THE SUBROUTINECRETURNENDA.2 The Program for Stochastic Frontier ModelThe stochastic frontier models are estimated using the program FRONTIER version2.0 developed by Coelli (1991). The program provides maximum likelihood estimation ofa wide variety of stochastic frontier production function model formulations. The programfollows a three-step procedure in estimating the maximum likelihood estimates of theparameters of a stochastic frontier production function. The first step obtains the OLSestimates of the production function; then the second step conducts a two-phase grid searchof the parameters in the log likelihood function with the OLS estimators excepting theintercept; finally the values selected in the grid search are used as starting values in aniterative procedure using Davidon-Fletcher-Powell Quasi-Newton method to obtain the final(approximate) maximum-likelihood estimates.241Appendix BData in Monte Carlo ExperimentsB.1 Data GenerationThe inputs, the exogenous variable and the error terms are generated using therandom number generators in SHAZAM version 6.2.The inputs X are drawn from lognormal distributions with the following p.d.f.:1exp[-’], Oxoof(x)=2t B.1otherwise=3, t=1Since SHAZAM version 6.2 has only uniform random number generator and normal randomnumber generator, the lognormal random numbers are generated using the followingalgorithm2:1. Generate x0 from N(O,1)2. W - j + T X03. X E— e’’4. Deliver X.The error term for inefficiency u is from a half-normal distribution which takesabsolute values from N(O, 0.36).2 Rubinstein (1981) gives a very good description about the methods for generatingrandom variables and random vectors from different probability distributions.Appendix B Monte Carlo Experiments 242The values for inputs and inefficiency term are fixed throughout the experimentation.Table B. 1 lists the summary statistics for X and e”. Note that the values listed here are theaverages from 25 replications.Table B.1Summary Statistics for Variables in Monte Carlo ExperimentsMeans St.dev. Minimum MaximumXl 34.04 42.675 1.2851 358.48X2 33.45 44.948 1.1938 410.64X3 33.02 42.172 1.3541 379.39e 0.6561 0.2034 0.1641 0.9979The values for exogenous variables and the noise term vary from experiment toexperiment, and given in Chapter 6.B. 2 Determination of the Number of ReplicationsAs mentioned in Chapter 5, the more replications, the better the results. However,computation costs increase significantly with the number of replications. Therefore, it isnecessary to examine whether or not it makes any significant differences in the simulationresults to use different number of replications. In particular, we test the hypothesis thatAppendix B Monte Carlo Experiments243there are no significant differences among the results with 25, 50, and 100replications, foreach of the three alternative methods, in terms of mean efficiency levels and correlationcoefficients. This is accomplished by doing three sets of experiments, with 25, 50 and 1(X)replications respectively, and comparing the results between the different sets of experiments.Here the underlying production technology is specified by functions (5.1), (5.4) and (5.5)with 0=0, ‘y=l,ô1=ô20.3, 3—0.4, cx=O, and u = 0.15. With each experiment, thesample size is set at 100, and p is setat -0.25.In comparing the results from different sets of the experiments, we consider each setof experiments as simulating a sample representing a particular population. Consequently,hypotheses tests are conducted aboutthe differences among the means ofthree populations.Each replication is considered as a sample observation from that particular population, thuswe have three separate samples with 25, 50 and 100 observations respectively, represeiltingthree different populations.B.2. 1 Stochastic Frontier MethodFirst we look at the average efficiency estimates. The following lists the resultsfrom three sets of experiments with25, 50 and 100 replications, respectively.I25 = 0.53614= 0.020749I5O = 0.54150= 0.025803B.2= 0.5423 1 s1 = 0.034441where is the mean of averageefficiency estimates with n replications, s, is thecorresponding standard deviations. To determine whether the number ofreplications makesAppendix B Monte Carlo Experiments 244any significant differences in the simulation results, we test the following hypothesis:Hi0:= P’50 B.3H20: = I.L100using the test statistic z:(--(-)B412ii’ /n1+s In2- P25 P’s0- 0.53614-0.54150 - -zstol-_______-______________-1/s/25+s2°/50 1/0.0207492/25+0.0258032/50- P2sP’ioo - 0.53614-0.54231 - -zst02-_-________-1/s/25 +s 21/i00 1/0.0207492/25+0.0344412/100since -1.96 < z01 < 1.96, and -1.96 < z < 1.96, we can not reject either Hi0 orH20. That is, there is no significant differences in mean efficiency estimates between 25replications and 50 replications, and between 25 replications and 100 replications.Next we look at correlation coefficients and rank correlation coefficients. The meansof average correlation coefficients and rank correlation coefficients (between the estimatedefficiency levels and the true efficiency levels) and the corresponding standard deviations areas follows:Appendix B Monte Carlo Experiments 245= 0.76653 s = 0.34063C50 = 0.77842 = 0.31610ioo = 0.79202 c1OO = 0.30153= 0.76761 s = 0.33988= 0.77763 = 0.3 1725= 0.79273 SRl = 0.30202Following the same procedure used for the mean efficiency estimates, the followinghypotheses are tested:Hi0: C = C50 = R50 B.5H20: C = C1 =Using B.4 , the test statistics are computed as:z1 = -0.1459, z2 = -0.342zRl = -0.123, z = -0.3377All of the four test statistics fall within the interval of (-1.96, + 1.96), thus the hypothesesstated in B.5 can not be rejected. There is no significant differences, in terms of averagecorrelation coefficients and rank coefficients, between 25 replications and 50 replications,and between 25 replications and 100 replications.From the above test results, we can conclude that it is sufficient to use 25 replicationsfor the stochastic frontier models.Similar hypothesis tests are conducted for the deterministic frontier method and theDEA. These tests are summarized in the following tables.Appendix B Monte Carlo Experiments 246B.2.1 Deterministic Frontier MethodTable B.2Determination of Number of ReplicationExperiment Results by Deterministic Frontier MethodNo. of Replications Sample Mean Standard DeviationAverage Efficiency 25 0.4203 1 0.04160650 0.41362 0.063023100 0.40822 0.058574Correlation 25 0.87340 0.03217950 0.87312 0.034146100 0.87182 0.039585Rank Correlation 25 0.89287 0.03354850 0.89845 0.028357100 0.89767 0.032981Table B.3Hypothesis Test for Deterministic Frontier MethodNull Hypothesis Test StatisticAverage Efficiency t25=z1 = 0.5487P25 = z2=1.1881Correlation C25 = C50 z1 =0.0348C25 = C1 z_=0.2091Rank Correlation R25 = R50 ZR1 = -0.7139R25 = Rico z=-0.6420Appendix B Monte Carlo Experiments 247As one can see from Table B.3, all test statistics are within the interval of (-1.96,+ 1.96), that is the null hypotheses can not be rejected. Therefore, the use of 25 replicationsfor the deterministic frontier methods will not give different results from using 50replications, or 100 replications.B.2.1 Data Envelopment Analysis MethodTable B.4Determination of Number of ReplicationsExperiment Results by Data Envelopment AnalysisNo. of Replications Sample Mean Standard DeviationAverage Efficiency 25 0.61645 0.02864950 0.61954 0.032030100 0.62000 0.030646Correlation 25 0.79608 0.05 196150 0.79139 0.067507100 0.79254 0.060102Rank Correlation 25 0.77979 0.05923 150 0.77669 0.073567100 0.77907 0.065774Appendix B Monte Carlo Experiments 248Table B.5Hypothesis Test for Data Envelopment AnalysisNull Hypothesis Test StatisticAverage Efficiency IL25 = A5 z1 = -0.42311L25 = ILioo z2=-O.5463Correlation C25 = C50 z1 =0.3323C25 = C1 z =0.2949Rank Correlation R25 = R50 ZR1 = -0.1966R25 = R10 z=-0.0531Again all test statistics fall within the interval of (-1.96, + 1.96). Therefore, it issufficient to use 25 replications for the DEA method.In summary, the sample results indicate that there is no significant difference betweenthe results using 25 replications and 50 or 100 replications. Therefore, it is sufficient toconduct 25 replications for all three alternative methods.B. 3 Specification of Functional Forms of Underlying Production TechnologyIn the first five sets of Monte Carlo experiments, the underlying productiontechnology is assumed to be constant returns to scale. In the last set of Monte Carloexperiments, non-constant returns to scale and input complementarity are allowed for theunderlying production function. Table B.7 lists the related parameter values of theproduction function. The basic functional form is the CRESH function as specified byequation (5.1). The first three functions are the constant returns to scale technologies usedAppendix B Monte Carlo Experiments 249in the first five sets of experiments. Function 4 to 13 are the functions considered in the lastset of experiments.Table B.6Parameter Values for Underlying Production Function‘Y 12 013 023 P Pi P2 P31 1 3.03 3.03 3.03 -0.67 -0.67 -0.67 -0.672 1 1.33 1.33 1.33 -0.25 -0.25 -0.25 -0.253 1 0.33 0.33 0.33 +2.00 +2.00 +2.00 +2.004 0.927 -0.277 0.540 0.512 -2.15 +0.85 +0.95 -2.005 0.927 0.399 0.454 -0.182 -2.15 -2.00 +1.50 1.206 0.929 0.356 0.447 0.508 +1.00 +1.50 +1.20 +0.757 0.934 -0.209 0.400 0.454 -2.30 +1.50 +1.20 -2.158 0.934 -0.091 0.399 0.453 -1.60 +1.50 +1.20 -1.509 1.209 0.399 0.454 -0.182 -1.65 -2.00 +1.50 +1.2010 1.227 -0.209 0.400 0.454 -1.75 +1.50 +1.20 -2.1511 1.239 0.356 0.447 0.508 -0.75 +1.50 +1.20 +0.7512 1.286 -0.277 0.540 0.512 -1.55 +0.85 +0.95 -2.0013 1.573 -0.091 0.399 0.453 -0.95 +1.50 +1.20 -1.50-y , and o are the returns to scale and Allen-Uzawa partial elasticities of substitutionas specified by equation (5.2) and equation (5.3) in Chapter 5.250Appendix CRailway Sample DataC. 1 Sample Firms and Their CharacteristicsAn annual panel of 19 selected railways for the period of 1978-1989 forms theprimary data base for the case study of railways. The sample railways are selected becausetheir data are available in relatively consistent form over the sample period. The samplefirms represent different institutional settings and operate in a variety of environments. Thenames of these railways and the years for which data are collected for this study are listedin Table C.13.The management of some of the railways such as British Rail (BR), and NetherlandsRailways (NS) enjoy substantial freedom for making strategic and operational decisionswithout government intervention, provided that the predetermined minimum performancecriteria are met. On the other hand, some of the railways such as the Finnish State Railways(VR), Austrian Federal Railways (OBB), and Norwegian State Railways (NSB) are subjectto strict governmental control. Of the sample railways, only German Federal Railways(DB), Italian State Railways (FS prior to 1985 only) and Danish State Railways (DSB priorto 1986) are run by governmental agencies while French National Railways (SNCF prior to1982 only) is organized as quasi-public firms4. The rest of the railways are organized aspublic firms (crown corporations).More discussions for some of the railways can be found in Oum and Yu (1991).‘ The term uquasipublic firm11 is used to indicate mixed ownership by public andprivate interests.Appendix C Railway Data 251There are large variations in operating environments across the selected railways.Japanese National Railways (JNR, however since 1987 it is known as IR) enjoys very highTable C.1List of Sample Railways, Abbreviations, & Years UsedBR : British Railways (UK) 1978-1989CFF: Swiss Federal Railways (Switzerland) 1978-1989CFL: Luxembourg National Railway Company 1978-1989CH: Hellenic Railways Organization (Greece) 1978-1987CIE: Irish Transport Company 1988-1989CP: Portuguese Railways 1978-1989DB: Deutsche Bundesbahn-German Federal Railways 1978-1989DSB: Danish State Railways (Denmark) 1978-1989FS: Italian State Railways 1978-1989JNR: Japanese National Railways 1978-1986NS: Netherlands Railways 1978-1989NSB: Norwegian State Railways 1978-1989OBB: Austrian Federal Railways 1978-1989RENFE: National System of Spanish Railways 1978-1989SJ: Swedish State Railways 1978-1989SNCB: Belgium National Railways 1978-1989SNCF: French National Railways 1978-1989TCDD: Turkish Republic State Railways 1978-1989VR: Finnish State Railways 1978-1989passenger traffic density5 (9.6 million passengers as an average density in 1987) while NSBhas much lower passenger traffic density (528,000 in 1989). Hellenic Railways (CH) ofGreece serves relatively long distance passenger traffic, while most other railways have highTraffic density is measured in passenger-kilometers per route-kilometers.Appendix C Railway Data 252percentage of their passenger services in short suburban and commuter traffic (see theaverage passenger trip length in Table C.2). JNR (Japan) and TCDD (Turkey) enjoy muchhigher average passenger load per train than the other railways. SNCF (France), VR(Finland) and TCDD (Turkey) exhibit a fair balance between passenger and freight services,while NS (Netherlands) and DSB (Danish State Railways) provide primarily passengerservices. CFF-SBB (Swiss) is fully electrified while CH (Greece), CIE (Ireland), DSB(Denmark), and TCDD (Turkey) still operate fuel traction over nearly their entire network.The sizes of the railways as indicated by length of route range from a 270 kilometer(Table C.3) network of the Luxembourg Railway (CFL) to larger firms such as the FrenchNational Railway (SNCF) that serves rail lines of over 30,000 kilometers. Traffic volumesas measured by passenger kilometers, freight tonne kilometers and train kilometers also varyover a wide range. JNR produces over 200 billion passenger kilometers each year whileCFL produces only a little over 200 million passenger-kilometers (Table C.3).Cost recovery conditions vary greatly among the sample railways. Table C.4presents the overall operating cost recovery ratios and the ratio of direct subsidy to totaloperating cost. It is noted that the subsidy ratio is not necessarily equal to:(1 - operating cost recovery ratio)This is because most countries do not provide balancing subsidies to their railways6. Otherrailways, such as JNR (Japan), before its reorganization, and DB (Germany), finance partof their deficits by raising debt. Among the 19 OECD railways, the Italian State Railways6 Most of the railways receive payments from governments for specific services. Somerailways, such as CH, DSB and NSB, get their operating losses subsidized from the state.aAveragepassengerload:numberofpassengerspertrainbPassengertrafficdensity:passenger-kilometersperroute-kilometercAveragetriplengthofrailpassengersdPercentageofpassengertram-kilometersintotaltrain-kilometersePercentageofelectrifiedroutemileageintotalroutemileagefAveragefreightload:numberoftonnespertraingFreight trafficdensity:freighttonne-kilometersperroute-kilometerhAveragelengthofhauloffreighttrafficiPassengertraindensity:passengertrain-kilometersperroute-kilometerjFreighttraindensity:freighttrain-kilometersperroute-kilometerTableC.2RailwayCharacteristics(MeanValueofthevariablesduringthesampleperiod)LoadaDensity”Trip0%PASSd%ELEC°LoadDensityHaul”Pden’Fden’RailwaysCountries(Pax)(Pax)(Kin)(%)(%)(FRE)(FRE)(Km)(000)(000)(000)(000)BRUK9318624478232321019119204CFFSwitzerland12331664173992572358157269CFLLuxembourg79868216559424225338115CHGreece12567015281025429520151CIEIreland1185285765113730917842CPPortugal194160826791317435225482DBGermany10314413866403032142207147DSBDenmark99178134837201742239184FSItaly173246710277553041079336144JNRJapan37289502881402411391318246NSNetherlands9132014489642431080159355NSBNorway9852762675826265111252OBBAustria11612724562532981880205116RENFESpain151117181684626491835083SJSweden102552846062410144332854SNCBBelgium9316784376473752070118185SNCFFrance1951686796132312170933895TCDDTurkey2637574857338081249632VRFinland129527785721423129026643I cMAppendix C Railway Data 254Table C. 3Railways Route Length and Traffic Volume*Railway Country Route Train-Km Train-Km Pax-Km Ton-Km(KM) (Pax) (Freight) (000000) (000000)(000) (000)BR UK 16588 362430 77232 33323 16742CFF Switzerland 2994 91063 27366 11021 8161CFL Luxembourg 272 3083 1402 224 669CH Greece 2479 13711 2313 2011 657CIE Ireland 1944 9541 4136 1220 560CP Portugal 3064 29559 6939 5908 1719DB Germany 27045 397437 196029 41144 61109DSB Denmark 2344 42410 7350 4649 1677FS Italy 16030 236451 65660 44443 18650JNR Japan 20341 655095 91164 222670 24752NS Netherlands 2828 106664 11479 10164 3108NSB Norway 4044 21461 9561 2136 2749OBB Austria 5641 71297 37562 8445 11849RENFE Spain 12565 109853 48214 14715 14048SJ Sweden 11022 59171 40620 6060 18532SNCB Belgium 3513 71285 21041 6400 9275SNCF France 34322 312348 167964 64256 52449TCDD Turkey 8430 26959 16690 6844 7564VR Finland 5884 22027 17239 3208 7958* 1989 dataSource: International Railway StatisticsAppendix C Railway Data 255(FS) has the lowest revenue-cost ratio (24% in 1989) while the operating revenue of the JRsystem (Japan) exceeded costs in 1988. It is worth noting that the most heavily subsidizedrailways including FS (Italy), SNCB (Belgium), and CR (Greece) provide primarilypassenger services with passenger train-kilometers accounting for over 75% of their totaltrain-kilometers.Table C.4Cost Recovery IndicatorsRailways Countries Revenue/Costs Subsidy/Costs1980 1989 1980 1989BR U.K. 0.74 0.90 0.24 0.18CFF Switzerland 0.74 0.73a 0.08 0. 12aCFL Luxembourg 0.31 0.23 0.66 0.75CH Greece 0.54 0.28b 0.47 0.72bCIE Ireland 0.68 0.76 0.31 0.25CP Portugal 0.46 0.46 0.45 0.29DB Germany 0.58 0.60 0.30 0.28DSB Denmark 0.68 0.62 0.32 0.38FS Italy 0.29 0.24 0.45 0.71JNR Japan 0.70 1.10 0.07 0.01NS Netherlands 0.56 0.57 0.43 0.45NSB Norway 0.72 0.60 0.28 0.40OBB Austria 0.67 0.64 0.14 0.40RENFE Spain 0.55 0.47 0.45 0.48SJ Sweden 0.83 0.81c 0.13 0.15cSNCB Belgium 0.44 0.42 0.55 0.58SNCF France 0.66 0.68 0.30 0.29TCDD Turkey 0.45 0.73 0.41 0.13VR Finland 0.77 0.72 0.02 0.02a. The Government has been paying the infrastructure costs of CFF since 1987, on the other hand, CFFpay the government a contribution towards the infrastructure expenses. To be consistent with the earlierdata, government payment for infrastructure and CFF’s contribution towards infrastructure expenditureare excluded in calculating the financial ratios.b. for 1988.c. for 1988. SJ was separated into two organizations SJ and BV in 1989.Sources: UIC: International Railway Statistics; EC: COM(88) 12, Com(89) 364, 564Appendix C Railway Data 256C. 2 Source of Data and Description of VariablesThe annual data are compiled from the International Railway Statistics, railways’annual reports, and the (statistical) Yearbooks of the respective countries as well aspublications by the Commissions for the European Community.There are two sets of variables collected for the railways in order to measure theproductive performance of the railways: (1) outputs and inputs, and (2) the variablesdescribing the operating environment and production characteristics of the railways.Two alternate sets of output measures are considered: (i) revenue output measures(passenger-kilometers and freight tonne-kilometers); and (ii) available output measures(passenger train kilometres and freight train-kilometres). The available output measuresindicate essentially the level of capacity supplied while the revenue output measures indicatethe level of output consumed by users, and the value they derive from them. These outputdata are collected from International Railways Statistics.Seven input measures are used: (i) labour; (ii) energy consumption; (iii) ways andstructures; (iv) materials; (v) the number of passenger cars; (vi) the number of freightwagons; and (vii) the number of locomotives.The number of employees from International Railways Statistics is used as themeasure of labour input as uniform statistics on labour hours are not available for all firmsin the sample. Energy input is measured by the total BTU consumed, using the followingconversion factors:Diesel oil : 0.1657 million Btu I imperial gallonsAppendix C Railway Data 257Fuel oil: 0.1801 million Btu/ imperial gallonsElectricity: 3412 BtuIKWHCoal: 26.2 million Btu/ton1 ton of diesel oil = 1.571 tons of coalRailways’ energy consumption for 1978-1984 are obtained from InternationalRailways Statistics. Energy data of the European railways for 1985-1987 are obtained fromStatistical Trends in Transport 1965-1989. JNR’s energy consumption for 1985-1986 areobtained from Japan Statistical Yearbook.The ways and structures input is determined, using the perpetual inventory method(Christensen and Jorgenson, 1969), by the amount of land and infrastructure capital stockreported by International Railways Statistics. The land and infrastructure assets are firstconverted into U.S. currency using the Purchasing Power Parity (PPP) index for GDP(OECD, 1992), and then the following perpetual inventory method is applied to constructthe ways and structures capital stock:K= I + (1—8)K_ C-iwhere K is the capital stock at year t, 1 is the real value of the net investment in year t.3 is the depreciation rate. The U.S. GDP deflator and 3% annual depreciation rate are usedto create the real capital stock series for each railway.Finally, the railways’ expenditures on services provided by third parties andOECD publication prepared by the European Conference of Ministers of Transport(ECMT).Appendix C Railway Data 258purchased materials are used as a measure of materials input. These expenditures are firstconverted into U.S. currency using the Purchasing Power Parity (PPP) index for GDP(OECD, 1992) and are then deflated to constant 1985 value using the U.S. GDP deflator.The environmental and production characteristics variables include: (1) passenger andfreight traffic densities, measured by passenger kilometers (freight tonne-kilometers) perroute kilometer and train-kilometers per route kilometers (Table C.2), these variables reflectsin part the market demand condition of the railways; (2) average load per train, it indicatesthe level of vehicle utilization; (3) average length of trip and average length of haul, theyindicate the type of traffic, e.g. long distance vs short distance commuter traffic; (4)percentage of passenger train kilometers in total train kilometers, this variable is consideredas an indicator of the importance of passenger services in the railways’ overall operation;(5) electrification rate, as measured by the percentage of electrified lines in total rail linesoperated, is intended to reflect the state of technology being employed and the extent ofinfrastructure investment.In addition, two policy variables are considered: SUBSIDY and AUTONOMY.SUBSIDY is measured by the ratio of subsidy to operating costs. Subsidy policy shouldideally be examined according to the types of subsidies and the ways in which they areprovided (e.g. loss/balancing subsidy vs. a fixed sum subsidy, unconditional subsidy vs.payment conditional on meeting a certain performance standards, etc.), which are likely tohave substantial impacts on a firm’s efficiency. However, due to limited information, thisstudy considers only the aggregate subsidy. The AUTONOMY variable is an index ofregulatory and institutional environments, based on 1988-1989 figures, which wasAppendix C Railway Data 259constructed by Perelman and Gathon (1990) using the information collected through a surveyof railways’ management. Its values range between 40 and 100. The more autonomousmanagement was, the higher the value of the AUTONOMY index. The degree ofmanagerial autonomy is affected by a large number of factors including ownership form andmanagerial mandate. It is therefore very difficult to quantify managerial autonomyconsistently across railways even with the best of efforts. Another problem with thisvariable is that it was based on only one year’s worth of observations. In using this variable,it is assumed that the institutional environment of the railways had varied only minimallyover the sample period, an assumption which is not realistic for most railways. Moreover,there is no index available for JNR, it is given the lowest autonomy rating in light of JNRwas under strict governmental control for the sample period. Although it is realized that theAUTONOMY variable is not ideally defined, it is used here as it was the only informationof its kind which had been collected systematically.260Appendix DAirline Sample DataD.1 The Sample Airlines and Data SourcesThe airline data base contains 36 airlines observed over the period 1980-1992. Theairlines are selected mainly due to data availability, however, they must be international aircarriers, and have a significant involvement in scheduled passenger services. That is, carriersthat are mainly engaged in cargo or charter services are excluded. The data are mainlycollected from Digest of Statistics published by the International Civil Aviation Organization(ICAO), in particular, the annual series on Traffic, Fleet-Personnel and FinancialStatistics. A primary advantage of using the ICAO statistics lies in the fact that, for themost part, they are complied in a consistent manner across airlines. Additional data areobtained from airlines’ annual reports, AVMARK and directly from airlines. The names ofthe airlines, their countries of origin, and the years for which the data are collected are listedin Table D.1.The sample includes airlines from Europe, North America and Asia-Pacific. Someof the airlines are 100 percent stated owned, some are private companies, while others havemixed ownership. For example, Air France, Air India, Pakistan International Airlines are100 percent government owned, while the US carriers are all private companies. There arelarge variations in size, traffic mix, and other operating characteristics among the sampleairlines (Table D.2). The size of the airlines as measured by the length of the route networkrange from 92 thousand kilometers for Finnair to over 940 thousand kilometers for LufthansaAppendix D Airline Data 261Table D.1List of Airlines & Years UsedQantas (Australia) 1980-1992AUA (Austrian Airlines) 1980-1992SABENA (Belgium) 1980-1992Air Canada (Canada) 1980- 1992CP (Canadian Airlines International) 1980- 1992Finnair (Finland) 1980- 1992Air France (France) 1980-199 1UTA (France) 1980- 1991Lufthansa (Germany) 1980- 1992Air India (India) 1980-199 1Alitalia (Italy) 1980-1992JAL (Japan Airlines) 1980- 1992MAS (Malaysia Airlines) 1980-199 1Mexicana (Mexico) 1980-1992KLM (Netherlands) 1980-1992PTA (Pakistan International Airlines) 1980-1992PAL (Philippines Airlines) 1980-1992Tap Air (Tap Air Portugal) 1980-1992KAL (Korean Air) 1980-1992SAS (Scandinavian Airlines Systems) 1980-1992SAUDIA (Saudi Arabian Airline) 1980-199 1SIA (Singapore Airlines) 1980-1992Iberia (Spain) 1980-1992Swiss Air (Switzerland) 1980-1992British Airways (U.K.) 1980-1992Cathay Pacific (Hong Kong) 1988-1992American Airlines (U.S.) 1980-1992US Air (U.S.) 1980-1992American West (U.S.) 1985-1992Continental (U.S.) 1980-1992Delta (U.S.) 1980-1992Eastern (U.S.) 1980-1990Northwest (U.S.) 1980-1992Pan American (U.S.) 1980-1990TWA (Trans-World Airlines, U.S.) 1980-1992United (U.S.) 1980-1992Appendix D Airline Data 262Table D2Airline Characteristics’Route Stage Length Load %PASS2 %INT’L3(000’km) (km) (%) (%) (%)Qantas 452 4368 65 67 99.95AUA 95 1051 50 78 100Sabena 190 988 62 58 100Air Canada 329 1485 54 73 60Canadian 268 1514 54 77 68Finnair 92 1072 57 85 93Air France 858 1585 61 49 89UTA 224 3841 68 48 100Lufthansa 942 1063 65 48 95Air India 292 2712 58 69 96Alitalia 486 1173 66 63 94JAL 492 2476 64 60 85MAS- 672 64 67 87Mexicana- 1131 48 96 61KLM 492 1919 70 53 100PIA 263 978 57 68 80PAL 117 1045 63 73 88Tap Air 136 1504 59 80 87KAL- 1292 75 41 94Saudia 310 1103 49 72 75SAS 212 786 59 77 85SIA 719 4039 67 58 100Iberia 326 1191 55 78 74Swiss Air 309 1243 61 55 99British Airways 581 1564 66 70 98Cathay 243 2901 67 58 100American 798 1493 54 86 30USAir- 832 49 93 9Continental 484 1370 51 86 30Delta 849 1184 52 87 27Northwest 735 1397 55 74 49TWA 304 1338 57 85 35United 778 1551 57 84 39American W 140 1015 53 92 2Eastern- 1009 51 92 9Pan Am 289 1735 63 78 771. The most recent year’s (in the sample) data; 2. Percentage of passenger RTK including non-scheduled service.3. Percentage of international traffic in terms of RTK (including non-scheduled services)Appendix D Airline Data 263of Germany8. Aside from the US carriers, most of the airlines in the sample provide mainlyinternational services, some do not provide domestic services at all. Qantas, SingaporeAirlines, Cathay Pacific, and JAL serve mostly inter-continental traffic as indicated by thelong stage length, while other airlines have a large proportion of their business in intracontinental traffic. KLM achieved the highest average (weight) load factor at 70 % , on theother hand, the average loads for Mexicana, Saudia and US Air are below 50%. Air Franceand Lufthansa have a rather significant proportion of their business in cargo services, whilethe U.S. carriers, Finnair, and Mexicana provide primarily passenger services.The size of the airlines in terms of annual total operating revenue ranges from 780million US dollars for Austrian Airlines to over 13,000 million US dollars for AmericanAirlines (Table D.3). There are also large variations in terms of revenue shares. Passengerrevenue accounts for over 90 percent of operating revenues for Austrian Airlines, US Air,and Delta, while accounts for less than 60 percent for Sabena and Korean Air. Air France,Korean Air, and Singapore Airlines’ cargo services contribute about 18 percent of theiroperating revenues. The revenue shares of incidental services range from 0.3 percent forBritish Airways to 41 percent for UTA and 30 percent for Sabena.Cost recovery conditions vary greatly among airlines (Table D.3). In 1992, aboutone third of the airlines were able to recover their operating expense. On the other hand,none of the North American carriers had a revenue cost ratio over 1.00. Iberia incurred thelowest revenue cost ratio of 79 percent compared to 119 percent achieved by Sabena,S Airlines’ route kilometers are obtained from World Air Transport Statistics publishedby International Air Transport Association (IATA).Appendix D Airline Data 264Table D.3Airline Financial Performance1Revenue %Pax Rev2 %Fre Rev2 %Inc Rev2 Rev/ Exp3__________________(Mill.US$) (%) (%) (%)Qantas 2614 77 10 6 1.08AUA 787 90 6 3 0.94Sabena 1424 58 11 30 1.19Air Canada 2226 83 9 3 0.92Canadian 2011 84 8 3 0.89Fmnair 982 70 5 9 1.19Air France4 5284 77 18 3 0.98UTA4 1399 42 16 41 0.99Lufthansa 9433 67 14 17 0.99Air India 799 76 14 7 1.09Alitalia 4105 72 8 19 1.02JAL 7858 78 14 5 0.95MAS 1390 76 13 7 0.98Mexicana 1044 86 6 6 0.87KLM 3402 67 16 16 0.84PIA 824 85 11 3 1.08PAL 1204 84 11 3 1.08Tap Air 1123 74 7 18 0.83KAL 2966 58 23 15 0.99Saudia4 2093 68 7 13 0.98SAS 3756 74 5 20 1.05SIA 3153 78 18 3 1.12Iberia 3298 86 7 6 0.79Swiss Air 3392 66 10 23 0.97British Airways 8444 89 7 0.3 1.07Cathay 2902 77 16 5 1.19American 13581 88 3 8 0.99US Air 6236 93 1 4 0,94Continental 5210 89 3 5 0.96Delta 11639 92 4 2 0.93Northwest 7964 87 7 3 0.96TWA 3570 83 3 12 0.91United 12725 88 5 5 0.96American W 1303 92 2 3 0.95Eastern5 2182 89 1 7 0.80Pan Am5 3931 86 5 6 0.891. all entries are 1992 data except for noted otherwise; 2. revenue shares for passenger, freight and incidental;5.3. ratio of operating revenue over operating expenses; 4. 1991 data 1990 dataAppendix D Airline Data 265Finnair, and Cathay Pacific.D.2 Description of VariablesThere are two sets of variables collected for the airlines: (1) inputs and outputs; and(2) the variables describing the operating environments and production characteristics of theairlines.The outputs are distinguished by the following five categories: scheduled passengerservice, scheduled freight service, mail service, non-scheduled (charter) services, andincidental services. Incidental services include, among other things, equipment leasing,maintenance provided to other carriers’ equipment, and catering. They have been ignoredin most of previous studies9,even though they account for upto 30 to 40 percent of revenuesfor some airlines such as UTA and Sabena (see Table D.3). The first four outputs aremeasured by revenue tonne-kilometers (RTK), while incidental output is measured by aquantity index which is constructed by deflating the incidental revenue using the PurchasingPower Parity index for consumption obtained from Summers and Heston (1991) and U.S.consumer price index.Five categories of inputs are used: labour, fuel, material, flight equipment, andground property and equipment (GPE). Labour input is measured by total number ofemployees, since labour compensation data required for constructing multilateral index is notavailable for some airlines in the sample. Fuel input is measured by number of gallons (USGood, Nadiri, Roller, and Sickles (1993) includes incidental service output.Appendix D Airline Data 266gallons) of fuel consumed. ICAO reports fuel expense data, but does not report fuel priceor quantity. The data on fuel quantity for some airlines are provided by the airlines directly,those of other airlines are estimated on the basis of fuel expense data and estimated fuelprices.Flight equipment is a multilateral index of 14 types of aircraft constructed using thefollowing translog multilateral index procedure proposed by Caves, Christensen, and Diewert(1982):LnQPLJh = &(W+Wk)(LnQ-LnQk) D-lwhere Q, is number of type k aircraft, W1’ is the weight for aircraft of type k for j-theobservation, the bar indicates the arithmetic mean over all the observations, and QPLJk is themultilateral index in comparison with a hypothetical representative observation (h) withaircraft number vector LnQk, and weights Wk. Aircrafts’ leasing values are used as theweights. The data on leasing prices for the period of 1986-1992 is provided by AVMARK,the leasing prices for earlier years are obtained from Professor Tretheway for some of theaircraft types and rest are estimated based on similar aircrafts.GPE input is estimated following the Christensen and Jorgenson (1969) procedure.The method assumes that the flow of capital services is proportional to the stock. The netGPE investment series for the airlines are collected from ICAO account data. The 1980’sGPE stock is considered as the initial GPE stock. Both the initial GPE stock and netinvestment series are delated by the PPP investment price index (Summers and Heston, 1991)Appendix D Airline Data 267in order to convert them into real quantity series comparable across countries and over time.The following perpetual inventory method is used to create the real GPE capital stock seriesfor each firm:K = + (1—8)K (D-2)where K1 is the GPE capital stock at year t, I, is the real value of the net investment in yeart, and (5is the depreciation which is assumed to be 7%.The last category of input is materials. The materials input is an catch-all-expensecategory, it includes all other inputs or material costs which are not included in the otherinput categories. Material cost is defined as the part of total operating cost which is notattributable to labour, fuel or capital, and is computed as follows: materials cost = carrier’soperating cost (reported in ICAO pub.) - labour cost - fuel cost - reported rental anddepreciation of flight equipment - depreciation of GPE. The materials input is a quantityindex constructed by deflating the materials cost with PPP Consumption Price Index(Summers and Heston, 1991).The second sets of variables describe the operating environment and productioncharacteristics. They include that: (1) average stage length which is the average distancebetween takeoffs and landings, and serves as an indicator how wide an airline’s network is;(2) average load factor which indicates the level of aircraft utilization; (3) revenue shares offreight, mail, non-scheduled services, and incidental services which are to reflect service mixof a particular airline; (4) government ownership dummy variables, one for majoritygovernment ownership where government ownership is over 50 percent, one for minorityAppendix D Airline Data 268government ownership where government ownership is between 20 and 50 percent, anddefault is private ownership; (5) geographic dummy variables, distinguishing the airlinesaccording to their home counthes geographic locations, and their importance in that region,in particular, seven groups are classified: Europe, Europe Major, Asia, Asia Major, Canada,other, and the default is U.S. carriers; (6) technical changes over time: yearly dummyvariables are used to reflect any potential technical changes occurring over time.

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