Fficiency of Public Spending in Developing Countries

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Efficiency of Public Spending in Developing Countries:

A Stochastic Frontier Approach

William Greene

Stern School of Business, New York University

May, 2005

Since public spending in developing countries amounts to significant shares of GDP, there is acompelling interest in measuring and understanding (in)efficiency in the pursuit of the government sobjectives Recent work done in the World Bank analyzes the efficiency of attainment of these goals with awidely used body of programming techniques generally known as Data Envelopment Analysis (DEA). Elsewhere [Greene (1993, 2004a,b, 2005a,b)], I have argued that the stochastic frontier methodology is an

attractive alternative to DEA for various reasons, specifically: (a) the stochastic aspect of the modelallows it to handle appropriately measurement problems and other stochastic influences that wouldotherwise show up as causes of inefficiency, (b) particularly with respect to cross country data such asthose we will analyze here, the stochastic frontier framework provides a means of accommodatingunmeasured but surely substantial cross country heterogeneity, (c) the frontier model provides a means toemploy information on measured heterogeneity, such as per capita gdp, in the model.

This study will present the stochastic frontier methodology and revisit the World Bank data whileemploying this alternative set of methods. The first chapter of the presentation describes the stochasticfrontier methodology. This is a body of techniques that has evolved continuously since the pioneeringpaper of Aigner et al. (1977). Then , the methods will be illustrated in two applications, Christensen andGreene s (1976) data on electricity production, which is a standard platform for studying the stochasticfrontier model and second, to a previously examined data set from the World Health Organization on healthcare and education attainment, allowing for both measured and unmeasured heterogeneity (cross countrydifferences). In our second chapter, we will apply the techniques developed to the World Bank data onhealth care and education. This section will illustrate a variety of computations based on the stochasticfrontier mothodology. The stochastic frontier production model has provided the standard platform forsingle output analysis of technical inefficiency in most studies. Where there are multiple outputs, cost dataare typically required. Recent studies have established the usefulness of the distance function approach foranalyzing multiple output - multiple input data such as we have here. As part of my analysis, I will extendthe production function methods to a multiple output distance function.


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Chapter 1. The Stochastic Frontier Model

1 Introduction

1.1 Modeling Production1.2 History of Thought1.3 Empirical Antecedents1.4 Organization of the Survey

2 Production and Production Functions

2.1 Production2.2 Modeling Production2.3 Defining Efficiency

3 Frontier Production and Cost Functions

3.1 Least Squares Regression Based Estimationof Frontier Functions

3.2 Deterministic Frontier Models3.3 Data Envelopment Analysis (DEA)

4 The Stochastic Frontier Model

4.1 Implications of Least Squares4.2 Forming the Likelihood Function4.3 The Normal Half Normal Model4.4 The Truncated Normal Model4.5. Estimation by Corrected Ordinary Least Squares

Method of Moments Estimators5 Stochastic Frontier Cost Functions, Multiple

Outputs, and Distance and Profit Functions: Alternatives

to the Production Frontier5.1 Multiple Output Production Functions5.2 Stochastic Frontier Cost Functions5.3 Multiple Output Cost Functions5.4 Distance Functions

6 Heterogeneity in Stochastic Frontier Function Models6.1. Heterogeneity6.2 One Step and Two Step Models6.3 Shifting the Production and Cost Function6.4 Parameter Variation and Heterogeneous Technologies6.5 Location Effects on the Inefficiency Model

6.6 Shifting the Underlying Mean of ui6.7. Heteroscedasticity

7. Panel Data Models

7.1 Time Variation in Inefficiency7.2 Distributional Assumptions


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7.3 Fixed and Random Effects and Bayesian Approaches8 Estimation of Technical Inefficiency

8.1 Estimators of Technical Inefficiency in the StochasticFrontier Model

8.2. Characteristics of the Estimator

8.3 Does the Distribution Matter?8.4 Confidence Intervals for Inefficiency8.5 Fixed Effects Estimators

9 Applications

9.1 Computer Software9.2 The Stochastic Frontier Model: Electricity Generation9.3 Heterogeneity in Production: World Health Organization Data

Chapter 2. Analysis of Efficiency in DevelopingCountries

1 World Bank Data2 Stochastic Frontier Models for Health and Education Outcomes

2 .1 Stochastic Frontier Models for Health Outcomes2..2 Educational Outcomes


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List of Tables

Table 1 Descriptive Statistics for Christensen and GreeneElectricity Data. 123 Observations

Table 2 Estimated Stochastic Frontier Production Functions.

Table 3 Descriptive Statistics for Estimates of E[ui|ei]. 123 ObservationsTable 4 Pearson and Spearman Rank Correlations for Estimatesof E[ui|ei]

Table 5 Estimated Stochastic Cost Frontiers.Table 6 Method of Moments Estimators for Efficiency Distribution

for Deterministic Frontier Model Based on Ordinary LeastSquares Residuals

Table 7 Method of Moments Estimates for Stochastic Frontier ModelsBased on Ordinary Least Squares Residuals

Table 8 Descriptive Statistics for J LMS Estimates of E[ui|ei] Based onMaximum Likelihood Estimates of Stochastic Frontier Models

Table 9 World Health Organization Data on Health Care Attainment.Table 10 Estimated Heterogeneous Stochastic Frontier Models for lnDALETable 11 Estimated Heteroscedastic Stochastic Frontier ModelsTable 12 Estimated Latent Class Stochastic Frontier ModelTable 13 Second Step Regression of Estimates of E[u|e] on CovariatesTable 14 Descriptive statistics for Estimates of E[u|e]Table 15 Variables Used in Stochastic Frontier AnalysisTable 16 Country Codes, Countries and Country Selection by Herrera

and PangTable 17 Estimated Stochastic Frontier Models for Health Related OutcomesTable 18 Rank Correlations for Efficiency Ranks

Table 19 Estimated Efficiency: Year 2000 Values, Four Health OutcomesTable 20 Estimated Stochastic Frontier Models for Health Related OutcomesTable 21 Estimated Inefficiencies from Random Effects ModelTable 22 Estimated Output Distance Function for Health OutcomesTable 23 Estimates of Technical Efficiency from Stochastic Frontier

Distance Function for Health OutcomesTable 24 Linear Regression of Distance (Multiple Output) Efficiency

on CovariatesTable 25 Estimated Stochastic Frontier Models for Educational Attainment

OutcomesTable 26 Estimated Inefficiency for Educational Attainment


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List of Figures

Figure 1 Input RequirementsFigure 2 Technical and Allocative InefficiencyFigure 3 OLS Production Frontier Estimators

Figure 4. Density of a Normal Minus a Half NormalFigure 5 Truncated Normal DistributionsFigure 6 Scatter Plots of Inefficiencies From Various SpecificationsFigure 7 Kernel Density Estimator for Mean Inefficiency

Based on Normal-Gamma Stochastic Frontier ModelFigure 9 Estimates of E[ui|ei]Figure 10 Confidence Limits for E[ui|ei]Figure 11 Cost and Production InefficienciesFigure 12 Kernel Density Estimates for Inefficiency DistributionsFigure 13 Kernel Density for Inefficiencies based on Doubly Heteroscedastic

Model

Figure 14 Scatter plot of Estimated InefficienciesFigure 15 Plot of Ranks of Group Means of InefficienciesFigure 16 Kernel Density Estimators for Estimates of E[u|e]Figure 17 Scatter plot of Efficiencies: DALE vs. Life Expectancy at BirthFigure 18 Scatter plot of Efficiencies: DPT Immunization vs. Measles

ImmunizationFigure 19 Efficiency Measures from Normal Half Normal Distance FunctionFigure 20 Inefficiency Measures from Normal Truncated Normal Distance

FunctionFigure 21 Estimated Distribution of E[u|e] for Educational Attainment


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Chapter 1. The Stochastic Frontier Model

1 Introduction

This chapter presents an overview of techniques for econometric analysis oftechnical (production) and economic (cost) efficiency. The stochastic frontier model ofAigner, Lovell and Schmidt (1977) is now the standard econometric platform for this type ofanalysis. We will survey the underlying models and econometric techniques that havebeen used in studying technical inefficiency in the stochastic frontier framework, andpresent some of the recent developments in econometric methodology. Applications thatillustrate some of the computations are presented at the end of the chapter.

1.1 Modeling Production

The empirical estimation of production and cost functions is a standard exercise in

econometrics. Thefrontier production function

orproduction frontier

is an extension ofthe familiar regression model based on the theoretical premise that a productionfunction, or its dual, the cost function, or the convex conjugate of the two, the profitfunction, represents an ideal, the maximum output attainable given a set of inputs, theminimum cost of producing that output given the prices of the inputs or the maximumprofit attainable given the inputs, outputs, and prices of the inputs. The estimation offrontier functions is the econometric exercise of making the empirical implementationconsistent with the underlying theoretical proposition that no observed agent canexceed the ideal. In practice, the frontier function model is (essentially) a regressionmodel that is fit with the recognition of the theoretical constraint that all observations liewithin the theoretical extreme. Measurement of (in)efficiency is, then, the empirical

estimation of the extent to which observed agents (fail to) achieve the theoretical ideal.Our interest in this study is in this latter function. The estimated model of production,cost or profit is the means to the objective of measuring inefficiency. As intuition mightsuggest at this point, the exercise here is a formal analysis of the residuals from theproduction or cost model. The theory of optimization, and production and/or costprovides a description of the ultimate source of deviations from this theoretical ideal.

1.2 History of Thought

The literature on frontier production and cost functions and the calculation of efficiencymeasures begins with Debreu (1951) and Farrell (1957) [though there are intellectual

antecedents, such as Hicks s (1935) suggestion that a monopolist would enjoy theirposition through the attainment of a quiet life rather than through the pursuit ofeconomic profits, a conjecture formalized somewhat more by Leibenstein (1966, 1975)].Farrell suggested that one could usefully analyze technical efficiency in terms ofrealized deviations from an idealized frontier isoquant. This approach falls naturally


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into an econometric approach in which the inefficiency is identified with disturbances ina regression model.

The empirical estimation of production functions had begun long before Farrell swork, arguably with the papers of Cobb and Douglas (1928). However, until the 1950s,

production functions were largely used as devices for studying the functionaldistribution of income between capital and labor at the macroeconomic level. Thecelebrated contribution of Arrow et al. (1961) marks a milestone in this literature. Theorigins of empirical analysis of microeconomic production structures can be morereasonably identified with the work of Dean (1951, a leather belt shop), J ohnston(1959, electricity generation) and, in his seminal work on electric power generation,Nerlove (1963). It is noteworthy that all three of these focus on costs rather thanproduction, though Nerlove, following Samuelson (1938) and Shephard (1953),highlighted the dual relationship between cost and production.1 Empirical attention toproduction functions at a disaggregated level is a literature that began to emerge inearnest in the 1960s (see, e.g., Hildebrand and Liu (1965) and Zellner and Revankar

(1969).

1.3 Empirical Antecedents

The empirical literature on production and cost developed largely independently of thediscourse on frontier modeling. Least squares or some variant was generally used topass a function through the middle of a cloud of points, and residuals of both signswere, as in other areas of study, not singled out for special treatment. The focal pointsof the studies in this literature were the estimated parameters of the productionstructure, not the individual deviations from the estimated function. An argument was

made that theseaveraging estimators were estimating the average, rather than the best practice technology. Farrell s arguments provided an intellectual basis for

redirecting attention from the production function specifically to the deviations from thatfunction, and respecifying the model and the techniques accordingly. A series ofpapers including Aigner and Chu (1968) and Timmer (1971) proposed specificeconometric models that were consistent with the frontier notions of Debreu (1951) andFarrell (1957). The contemporary line of research on econometric models begins withthe nearly simultaneous appearance of the canonical papers of Aigner, Lovell andSchmidt (1977) and Meeusen and van den Broeck (1977), who proposed the stochasticfrontier models that applied researchers now use to combine the underlying theoreticalpropositions with a practical econometric framework. The current literature on

production frontiers and efficiency estimation combines these two lines of research.

1 Some econometric issues that arise in the analysis of primal productions and dual cost functions arediscussed in Paris and Caputo (2004).


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1.4 Organization of the Survey

This survey will present an overview of this literature. We proceed as follows:

Section 2 will present the microeconomic theoretical underpinnings of theempirical models. As in the other parts of our presentation, we are only able to give acursory survey here because of the very large literature on which it is based. Theinterested reader can find considerable additional detail in the first chapter of this bookand in a gateway to the larger literature, Chapter 2 of Kumbhakar and Lovell (2000).

Section 3 will construct the basic econometric framework for the econometricanalysis of efficiency. This section will also present some intermediate results on deterministic (orthodox) frontier models that adhere strictly to the microeconomictheory. This part will be brief. It is of some historical interest, and contains some

useful perspective for the more recent work. However, with little exception, currentresearch on the deterministic approach to efficiency analysis takes place in theenvironment of Data Envelopment Analysis (DEA), which is the subject of the nextchapter of this book.

2This section provides a bridge between the formulation of

orthodox frontier models and the modern stochastic frontier models.

Section 4 will introduce the stochastic production frontier model and presentresults on formulation and estimation of this model. Section 5 will extend the stochasticfrontier model to the analysis of cost and profits, and will describe the importantextension of the frontier concept to multiple output technologies.

Section 6 will turn to a major econometric issue, that of accommodatingheterogeneity in the production model. The assumptions made in Sections 4 and 5regarding the stochastic nature of technical inefficiency are narrow and arguablyunrealistic. Inefficiency is viewed as simply a random shock distributedhomogeneously across firms. These assumptions are relaxed at the end of Section 5and in Section 6. Here, we will examine models that have been proposed that allow themean and variance of inefficiency to vary across firms, thus producing a richer, albeitconsiderably more complex formulation. This part of the econometric model extendsthe theory to the practical consideration of observed and unobserved influences thatare absent from the pure theory but are a crucial aspect of the real world application.

The econometric analysis continues in Section 7 with the development of modelsfor panel data. Once again, this is a modeling issue that provides a means to stretchthe theory to producer behavior as it evolves through time. The analysis pursued here

2 Some current research is directed at blurring this distinction by suggesting a statistical underpinning forDEA. As DEA is the subject of subsequent chapters in this book, we will not visit the topic here.


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goes beyond the econometric issue of how to exploit the useful features of longitudinaldata. The literature on panel data estimation of frontier models also addresses thefundamental question of how and whether inefficiency varies over time, and howeconometric models can be made to accommodate the theoretical propositions.

The formal measurement of inefficiency is considered in Sections 8. The use ofthe frontier function model for estimating firm level inefficiency which was suggested inSections 3 and 4 will be formalized in the stochastic frontier model in Section 8.

Section 9 will describe contemporary software for frontier estimation and willillustrate some of the computations with live data sets.

2 Production and Production Functions

We begin by defining a producer as an economic agent that takes a set ofinputs andtransforms them either in form or in location into a set of outputs. We keep the

definition nonspecific because we desire to encompass service organizations such astravel agents or law or medical offices. Service businesses often rearrange orredistribute information or claims to resources, which is to say move resources ratherthan transform them. The production ofpublic services provides one of the moreinteresting and important applications of the techniques discussed in this study. (See,e.g., Pestieau and Tulkens (1993).)

2.1 Production

It is useful to think in terms of a producer as a simple machine. An electric motorprovides a good example. The inputs are easily definable, consisting of a lump of

capital, the motor, itself, and electricity that flows into the motor as a precisely definedand measurable quantity of the energy input. The motor produces two likewiseprecisely measurable (at least in principle) outputs, work, consisting of the rotation ofa shaft, and heat due to friction, which might be viewed in economic terms as waste, ora negative or undesirable output. (See, e.g., Atkinson and Dorfman (2005).) Thus, inthis setting, we consider production to be the process of transforming the two inputsinto the economically useful output, work. The question of usefulness is crucial to theanalysis. Whether the byproducts of production are useless is less than obvious.Consider the production of electricity by steam generation. The excess steam from theprocess might or might not be economically useful (it is in some cities such as NewYork and Moscow), depending, in the final analysis on relative prices. Conceding the

importance of the distinction, we will depart at this point from the issue, and focus ourattention on the production of economic goods which have been agreed upon a priorito be useful in some sense.

The economic concept of production generalizes from a simple, well defined


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engineering relationship to higher levels of aggregation such as farms, plants, firms,industries, or, for some purposes, whole economies that engage in the process oftransforming labor and capital into GDP by some ill-defined production process.Although this interpretation stretches the concept perhaps to its logical limit, it is worthnoting that the first empirical analyses of production functions, by Cobb and Douglas

(1928), were precisely studies of the functional distribution of income between capitaland labor in the context of an aggregate (macroeconomic) production function.

2.2 Modeling Production

The production function aspect of this area of study is a well-documented part of themodel. The function, itself, is, as of the time of the observation, a relationshipbetween inputs and outputs. It is most useful to think of it simply as a body ofknowledge. The various technical aspects of production, such as factor substitution,economies of scale, or input demand elasticities, while interesting in their own right, are

only of tangential interest in the current context. To the extent that a particularspecification, Cobb-Douglas vs. translog, for example, imposes restrictions on thesefeatures which then distort our efficiency measures, we shall be interested in functionalform. But, this will not be the primary focus.

The Cobb-Douglas and translog models overwhelmingly dominate theapplications literature in stochastic frontier and econometric inefficiency estimation. (Incontrast, we note, the received literature in DEA by construction is dominated bylinear specifications.) The issue of functional form for the production or cost function(or distance, profit, etc.) is generally tangential to the analysis, and not given muchattention. There have been a number of studies specifically focused on the functional

form of the model. In an early entry to this literature, Caves, Christensen (one of thecreators of the translog model) and Trethaway (1980) employed a Box-Cox functionalform in the translog model to accommodate zero values for some of the outputs.3 Thesame consideration motivated Martinez-Budria, J ara-Diaz and Ramos-Real (2003) intheir choice of a quadratic cost function to study the Spanish electricity industry.Another proposal to generalize the functional form of the frontier model is the Fourierflexible function used by Huang and Wang (2004) and Tsionas (2004).

In a production (or cost) model, the choice of functional form brings a series ofimplications with respect to the shape of the implied isoquants and the values ofelasticities of factor demand and factor substitution. In particular, the Cobb-Douglas

production function has universally smooth and convex isoquants. The implied costfunction is likewise well behaved. The price to be paid for this good behavior is thestrong assumption that demand elasticities and factor shares are constant for given

3 This does not fully solve the problem of zero values in the data, as the appropriate standard errors forthe Box-Cox model still require the logarithms of the variables. See Greene (2003a, p. 174.)


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input prices (for all outputs), and Allen elasticities of factor substitution are all -1. Costfunctions are often used in efficiency analysis because they allow the analyst to specifya model with multiple inputs. This is not straightforward on the production side, thoughdistance functions (see below, Section 2.5.4) also provide an avenue. The Cobb-Douglas multiple output cost function has the unfortunate implication that in output

space, the output possibility frontiers are all convex instead of concave thus implyingoutput specialization. These considerations have generally motivated the choice offlexible (second order) functional forms, and in this setting, the translog productionmodel for one output and Kinputs,

= = == a + b + g 121 1 1ln ln ln ln

K K K

k k km k mk k my x x x ,

or the translog multiple output cost function for Kinputs and L outputs,

= = =

= = =

= =

= a + b + g

+ d + f

+ q

121 1 1

L L1st21 s 1 t 1

1 1

ln ln ln ln

ln ln ln

ln ln ,

K K K

k k km k mk k m

Ls s s ts

K L

ks k sk s

C w w w

y y y

w y

are most commonly used. (See, e.g., Kumbhakar (1989).) These models do relax therestrictions on demand elasticities and elasticities of substitution. However, thegenerality of the functional form produces a side effect; they are not monotonic orglobally convex, as is the Cobb-Douglas model. Imposing the appropriate curvature ona translog model is a generally challenging problem. (See Salvanes and Tjotta (1998)for methodological commentary.) Researchers typically (hopefully) check the

regularity conditions after estimation. Kleit and Terrell (2001) in an analysis of the U.S.electricity industry, have used a Bayesian estimator which directly imposes thenecessary curvature requirements on a two output translog cost function. Thenecessary conditions, which are data dependent they will be a function of the specificobservations are: (1) Monotonicity; sk = lnC/lnwk = bk+Smgkmlnwm > 0, k=1,...,K(nonnegative factor shares) and (2) Concavity; G S +ssT negative semidefinite whereG = [gkm], S = diag[sk] and s = [s1,s2,...,sk]

T. Monotonicity in the outputs requires

lnC/lnys = ds + Srfsrlnyr> 0. As one might imagine, imposing data and parameterdependent constraints such as these during estimation poses a considerablechallenge. In this study, the authors selectively cull the observations during estimation,retaining those which satisfy the restrictions. Another recent study, O Donnell and

Coelli (2005) also suggest a Bayesian estimator, theirs for a translog distance functionin which they impose the necessary curvature restrictions a priori, parametrically.Finally, Griffiths, O'Donnell, and Cruz (2000) impose the regularity conditions on asystem of cost and cost -share equations.


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The preceding is an issue that receives relatively little attention in the stochasticfrontier applications, though it is somewhat more frequently examined in the moreconventional applications in production and cost modeling (e.g., Christensen andGreene (1976)).4 We acknowledge this aspect of modeling production and cost at thispoint, but consistent with others, and in the interest of brevity, we will not return to it in

what follows.

2.3 Defining Efficiency

The analysis of economic inefficiency stands at the heart of this entire exercise. If onetakes classical microeconomics at face value, this is a fruitless exercise, at least asregards competitive markets. Functioning markets and the survivor principle simply donot tolerate inefficiency. But, this clearly conflicts with even the most casualempiricism. We also note that analysis of regulated industries and governmententerprises (including buses, trains, railroads, nursing homes, waste hauling services,sewage carriage, and so on) have been among the most frequent recent applications

of frontier modeling. Since the orthodoxy of classical microeconomics plays a muchlesser role here, the conflict between theory and practice is less compelling. We willeschew a philosophical discussion of the concept of inefficiency, technical, allocative,or otherwise. [For a very readable, if somewhat glib discussion, the reader may begintheir survey with Frsund, Lovell, and Schmidt (1980, pp. 21-23).5] Alvarez Arias andGreene (2005) also pursue this issue from an econometric perspective. In whatfollows, producers will be characterized as efficient if they have produced as much aspossible with the inputs they have actually employed or whether they have producedthat output at minimum cost. We will formalize the notions as we proceed.

By technical efficiency, we shall mean to characterize the relationship between

observed production and some ideal, or potential production. In the case of a singleoutput, we can think in terms oftotal factor productivity, the ratio of actual output to theoptimal value as specified by a production function. Two crucial issues, which willreceive only occasional mention in this essay are the functional form of the productionfunction and the appropriate list of inputs. In both cases, specification errors can bringsystematic errors in the measurementof efficiency.

4 A few other applications do note the idea, including Koop, Osiewalski and Steel (1994, 1997), Tsionas(2002) and Kumbhakar and Tsionas (2005). Mention of the regularity conditions (to be kept distinctfrom the regularity conditions for maximum likelihood estimators) is common in the frontier applications,though relatively few actually impose them. It is more common to check the conditions after estimation.For example, Farsi and Filippini (2003) estimated a translog cost frontier for Swiss nursing homes, andobserved ex post that the estimated parameters did not satisfy the concavity conditions in the inputprices. This result was attributed to the price setting mechanism in this market.5 The authors argue that economic dogma has essentially painted its proponents into a corner. Everyaction taken by an economic agent must be efficient; else it would not have been taken. This takes a bitof artful footwork in some cases.


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We will define production as a process of transformation of a set of inputs,

denoted x K+

into a set of outputs, y M+

. Henceforth, the notation, z, in boldface,

is intended to denote a column vector of the variables in question, whereas the samesymbol, z, in italics and not in boldface, will denote a scalar, or a single input or output.The process of transformation (rearrangement, etc.) takes place in the context of a

body of knowledge called the production function. We will denote this process oftransformation by the equation T(y,x) =0. (The use of 0 as the normalization seemsnatural if we take a broad view of production against a backdrop of the laws ofconservation ify is defined broadly enough neither energy nor matter can becreated nor destroyed by the transformation.)

We should emphasize that the production function is not static; technical changeis ongoing. It is interesting to reflect that given the broadest definition, the force oftechnical change would only be to change the mix of outputs obtained from a given setof inputs, not the quantities in total. The electric motor provides a good example. A more efficient motor produces more work and less heat (at least by the yardstick that

most people would use), but, in fact, the total amount of energy that flows from it will bethe same before and after our hypothetical technical advance. The notion of greaterefficiency in this setting is couched not in terms of total output which must be constant,but in terms of a subset of the outputs which are judged as useful against the remainingoutputs which arguably are less so.

The state of knowledge of a production process is characterized by an inputrequirements set

( ) = { : ( , ) is producible}.L y x y x

That is to say, for the vector of outputs y, any member of the input requirements set issufficientto produce the output vector. Note that this makes no mention of efficiency,nor does it define the production function per se, except indirectly insofar as it also

defines the set of inputs that is insufficient to produce y (the complement ofL(y) in K+

)

and, therefore, defines the limits of the producer s abilities. The production function isdefined by the isoquant

( ) = { : ( ) and ( ) if 0


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The distinction between these two similar definitions can be seen in Figure 1. Note that

xA

=( 1 2,A Ax x ) is on the isoquant, but is not in the efficient subset, since there is slack in

2 .Ax But, x

B is in both I(y) and ES(y). When the input requirements set is strictly convex,

as is commonly assumed in econometric applications, the distinction disappears. But,the distinction between these two sets is crucially important in data envelopmentanalysis.

Shephard's (1953) input distance function is

1( , ) = Max : ( ) .

l l

ID Ly x x y

It is clear that DI(y,x) 1 and that the isoquant is the set ofxs for which DI(y,x) =1.The Debreu (1951) - Farrell (1957) input based measure of technical efficiency is

( , ) = Min { : L( )}.TE q q y x x y

From the definitions, it follows that TE (y,x) 1 and that TE(y,x) = 1/DI(y,x). TheDebreu-Farrell measure provides a natural starting point for the analysis of efficiency.

The Debreu-Farrell measure is strictly defined in terms of production, and is ameasure of technical efficiency. It does have a significant flaw in that it is wedded toradial contraction or expansion of the input vector. Consider in Figure 2, the impliedinefficiency of input vector XA. Figure 2 is a conventional isoquant/isocost graph for asingle output being produced with two inputs, with price ratio represented by the slopeof the isocost line, ww. With the input vector XA normalized to length one, the Debreu-

Figure 1 Input Requirements


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Farrell measure of technical efficiency would be . But, in economic terms, thismeasure clearly understates the degree of inefficiency. By scaling back both inputs bythe proportion , the producer could reach the isoquant and, thus, achieve technicalefficiency. But, by reallocating production in favor of inputx1 and away fromx2, thesame output could be produced at even lower cost. Thus, producer A is both

technically inefficient and allocatively inefficient. The overall efficiency or economicefficiency of producer A is only a. Empirically decomposing (observable) overallinefficiency, 1-a into its (theoretical, latent) components, technical inefficiency, (1-q)and allocative inefficiency, (q - a), is an ongoing and complex effort in the empiricalliterature on efficiency estimation.

3 Frontier Production and Cost Functions

The theoretical underpinnings of a model of production are suggested above. We willtake as given the existence of a well defined production structure characterized bysmooth, continuous, continuously differentiable, quasi-concave production or

transformation function. Producers are assumed to be price takers in their inputmarkets, so input prices are treated as exogenous. The empirical measurement ofTE(y,x) requires definition of a transformation function. For most of this analysis, wewill be concerned with a single output production frontier. Let

( )y f x

denote the production function for the single output, y, using input vector x. Then, an

output based Debreu-Farrell style measure of technical efficiency is

( ) = 1.( ) y

TE y, fx x

Note that the measure is the conventional measure of total factor productivity, TFP, andthat it need not equal the input based measure defined earlier.

Our econometric framework embodies the Debreu-Farrell interpretation as wellas the textbook definition of a production function. Thus, we begin with a model such as

( , ) ,i ii = f TE y x b


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Figure 2 Technical and Allocative Inefficiency

where 0 0 is a measure oftechnical inefficiencysince lni i iu - TE 1 TE .= - Note that

TEi=exp(-ui).

(See J ondrow et al. (1982) and Battese and Coelli (1992) for discussion and analysis ofthe distinction between these two measures.) The preceding provides the central pillarof the econometric models of production to be described below.

Formal econometric analysis of models of this sort as frontier productionfunctions begins with Aigner and Chu's (1968) reformulation of a Cobb-Douglas model.A parallel literature is devoted to the subject of data envelopment analysis, or DEA.

The centerpiece ofDEA is the use of linear programming to wrap a quasi-convex hullaround the data in essentially the fashion of Farrell's efficient unit isoquant. The hulldelineates the efficient subset defined earlier, so, by implication, points observed insidethe hull are deemed observations on inefficient producers. DEA differs fundamentallyfrom the econometric approach in its interpretation of the data generating mechanism,


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but is close enough in its philosophical underpinnings to merit at least someconsideration here. We will turn to the technique of DEA in the discussion ofdeterministic frontiers below.

6

3.1 Least Squares Regression Based Estimation of Frontier Functions

In most applications, the production model, f (xi,) is linear in the logs of theinputs or functions of them, and the log of the output variable appears on the left handside of the estimating equation. It is convenient to maintain that formulation and write

ln +a ei ii = +y xTb

where i = -ui and xi is the set of whatever functions of the inputs enter the empiricalmodel. We assume that i is randomly distributed across firms. An importantassumption, to be dropped later, is that the distribution of i is independent of allvariables in the model. For current purposes, we assume that i is a nonzero (negative)

mean, constant variance, and otherwise ordinary regression disturbance. Theassumptions thus far include E[i|xi] < 0, but, absent any other special considerations,this is a classical linear regression model.

7The model can thus be written

ln ( [ ]) ( [ ]) * *.= a + e + + e - e = a + + ei i i i i i iy E Ex xT Tb b

This defines a classical linear regression model. Normality of the disturbance isprecluded, since * is the difference between a random variable which is alwaysnegative and its mean. Nonetheless, the model's parameters can be consistentlyestimated by ordinary least squares (OLS) since OLS is robust to nonnormality. Thus,

the technical parameters of the production function, with the exception of the constantterm, can be estimated consistently, if not efficiently by OLS. If the distribution ofwere known, the parameters could be estimated more efficiently by maximum likelihood(ML). Since the constant term usually reveals nothing more than the units ofmeasurement of the left hand side variable in this model, one might wonder whether allof this is much ado about nothing, or at least very little. But, one might argue that in thepresent setting, the constant is the only parameter of interest. Remember, it is theresiduals and, by construction, now E[ui|xi] that are the objects of estimation. Threeapproaches may be taken to examine these components.

6 There is a tendency on the part of many authors in economics to equate an estimation technique with amodel. In few places is this more evident than in the literature on DEA.7A crucial assumption which was discussed early in this literature, but is now implicit, is that there is nocorrelation betweenxi and i in the model. We refer to Zellner, Kmenta, and Dreze (1966) for discussionof the proposition that deviations of the observed factor demands xi from the cost-minimizing or profit-maximizing values could be uncorrelated with the deviation ofyi from its ideal counterpart as specifiedby the production function. Further discussion and a model specification appear in Sickles andStreitwieser (1992).


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Since only the constant term in the model is inconsistent, any information usefulfor comparing firms to each other that would be conveyed by estimation ofui from theresiduals can be obtained directly from the ordinary least squares residuals,

ln * [ ]= - - = - +i i i i ie y a u E ub x

T

where b is the least squares coefficient vector in the regression of lnyi on a constantand xi. Thus, for example, ei em is an unbiased and pointwise consistent estimator ofuj um. Likewise, the ratio estimator exp(ei)/exp(em) estimates

exp( [ ])

exp( [ ])i ii

m mm

EuTE TE =

EuTE TE

consistently (albeit with a finite sample bias because of the nonlinearity of the function).

For purposes only of comparison of firms, one could simply ignore the frontier aspectof the model in estimation and proceed with the results ofOLS. This does precludeany sort of estimator ofTEior ofE[ui], but for now, that is not consequential.

Since the only deficiency in the OLS estimates is a displacement of the constantterm, one might proceed simply by fixing the regression model. Two approaches havebeen suggested. Both are based on the result that the OLS estimator of the slopeparameters is consistent and unbiased, so the OLS residuals are pointwise consistentestimators of linear translations of the original uis. One simple remedy is to shift theestimated production function upward until all residuals except one, on which we hangthe function, are negative. The intercept is shifted to obtain the corrected OLS (COLS)

constant,

aCOLS = a* +maxiei.

All of the COLS residuals,

ei,COLS = ei maxiei,

satisfy the theoretical restriction. Proofs of the consistency of this COLS estimator,which require only that in a random sample drawn from the populationui, 0i iplim min u = ,appear in Gabrielsen (1975) and Greene (1980a). The logic of the

estimator was first suggested much earlier by Winsten (1957). A lengthy applicationwith an extension to panel data appears in Simar (1992). In spite of the methodologicalproblems to be noted below, this has been a popular approach in the analysis of paneldata. (See, e.g., Cornwell, Schmidt, and Sickles (1990) and Evans et al. (2000a,b).)


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The received literature contains discussion of the notion of an average, frontier(an oxymoron, perhaps) as opposed to the best practice frontier, based on thedistinction between OLS and some technique, such as maximum likelihood, whichtakes account of the frontier nature of the underlying model. One could argue that theformer is being defined with respect to an estimator, OLS, rather than with respect to a

definable, theoretically specified model. Whether the distinction is meaningful in aneconomic sense is questionable. There is some precedent for raising the question ofwhether the technology in use at the frontier differs from that in the middle of the pack,so to speak (see Klotz, Madoo, and Hansen (1980)), but the simple scaling of aloglinear production function is unlikely to reveal much about this possibility.Indeed, the implied radial expansion of the production function thus formulated mightreveal nothing more than different rates of adoption of Hicks neutral technicalinnovations. But, Frsund and J ansen (1977) argue that this difference, or moregenerally, differential rates of adjustment of capital stocks across firms in an industry docreate a meaningful distinction between average and best practice production frontiers.Some firms in an industry might be achieving the maximum output attainable, i.e., be

locating themselves on the frontier that applies to them, but might not have completelyadjusted their capital stock to the most up to date, technically advanced available.Thus, the best practice frontier for an industry which reflects this full adjustment wouldlie outside the frontiers applicable to some of the constituent firms. (See Frsund andHjalmarsson (1974) for additional discussion.) The description, as we shall see later, isakin to the motivation for the stochastic frontier model. However, the positeddifferences between firms are more systematic in this description.

3.2 Deterministic Frontier Models

Frontier functions as specified above, in which the deviation of an observation from the

theoretical maximum is attributed solely to the inefficiency of the firm, are labeleddeterministic frontier functions. This is in contrast to the specification of the frontier inwhich the maximum output that a producer can obtain is assumed to be determinedboth by the production function and by random external factors such as luck orunexpected disturbances in a related market. Under this second interpretation, themodel is recast as a stochastic frontier production function.

Aigner and Chu (1968) suggested a log-linear (Cobb-Douglas) productionfunction,

1 2

1 2i i i i= AX X U Yb b

in which Ui (which corresponds to TEi) is a random disturbance between 0 and 1.Taking logs produces


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1

1

ln =

a + +b e

a b

K

i ki ik

k

K

ki ik

k=

= xY

= + - ,x u

where =lnA,xki =lnXki, and i =lnUi. The nonstochastic part of the right hand side isviewed as the frontier. It is labeled deterministic because the stochastic component ofthe model is entirely contained in the (in)efficiency term, -ui. Aigner and Chu (1968)suggested two methods of computing the parameters that would constrain the residualsuito be nonnegative, linear programming;

, 1ln 0a = e - a - "

N

i i iiMin subject to y ix

Tb b

and quadratic programming;

2

, 1 ln 0a = e - a - "N

i i iiMin subject to y ix

T

b b .

In both applications, the slack variables associated with the constraints produce theestimates of -ui. A number of early applications, such as Forsund and J ansen (1977),built upon this approach both to study the technical aspects of production and toanalyze technical efficiency.

The Aigner-Chu (1968) approach satisfies the original objective. One cancompare the individual residuals based on the programming results,

- ln ,ai i iu = + yxTb to each other or to an absolute standard to assess the degree of technical (in)efficiencyrepresented in the sample. A summary measure which could characterize the entiresample would be the

N ii=1

1= u

N.average technical inefficiency

Another useful statistic would be the

iN

-u

i=1

1= eN

average technical inefficiency = [ ]iE TE This semiparametric approach was applied in a series of studies including

Frsund and Hjalmarsson (1979), Albriktsen and Frsund (1990) and Frsund (1992).


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In these studies, the authors specified the generalized production function proposed byZellner and Revankar (1969),

0 1ln ag g bK

kiki i

k=1

+ = +y y x

and minimized the sum of the residuals subject to the additional constraints Skbk =1and (0,1,k,k=1,...,K) >0. The foci of the applications are economies of scale andtechnical efficiency.

3.2.1 Deterministic Cost Frontiers

Frsund and J ansen (1977) formulated a hybrid of the linear programming approachesand the parametric model above to extend the analysis to costs of production. TheFrsund and Jansen specification departs from a homothetic production function,

9

[ ( )], [ ( )] > 0, ( ) = ( ) .i i i i ii = F f F f f t tf y "x x x x x

The empirical model is obtained by specifying

[ ( ) ]iii = F fy vx

where vihas a beta density (with parameters q+1 and 1)

( ) (1 ) , 0 1, 0i i ih .v v vq= + q < < q >

The cost function that corresponds to this production function is

lnCi = lnF-1(yi) + lnc(wi) - lnvi

where wi is the vector of input prices and c(wi) is the unit cost function. The authorsthen derive the corresponding log-likelihood function. The parameters of the

production function are obtained by using linear programming to minimize S1

N

i =lnCi

subject to the constraints that observed costs lie on orabove the cost frontier.10

Thereare three aspects of this approach that are of interest here. First, this construction isan alternative attempt to derive the linear programming criterion as the solution to a

maximum likelihood problem.11

Second, this is one of the first applications to estimate

9See Christensen and Greene (1976) for discussion. The outer transformation is strictly monotonic andthe inner function is linearly homogeneous.10 The constrained linear programming solution is not the maximizer of the log likelihood function.11 This is an antecedent to the recent DEA literature, e.g., Bankar (1993, 1997) which has attempted to


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a cost frontier instead of a production frontier. There is some controversy to thisexercise owing to the possible contribution of allocative inefficiency to the observedestimates of firm inefficiency. Lastly, there is a subtle sleight of hand used informulating the cost function. If the technical inefficiency component, vi were to enterthe production function more naturally, outside the transformation of the core function,

the form in which it entered the cost frontier would be far more complicated. On theother hand, if the inefficiency entered the production function in the place ofvixi, insidethe homogeneous kernel function (in the form of input oriented inefficiency), then itsappearance in the cost function would be yet more complicated. [See, for example,Kumbhakar and Tsionas (2005a) and Kurkalova and Carriquiry (2003).]

3.2.2 Corrected and Modified Ordinary Least Squares Estimators

The slope parameters in the deterministic frontier models can be estimatedconsistently by OLS. The constant term can be consistently estimated simply by

shifting the least squares line upward sufficiently so that the largest residual is zero.The resulting efficiency measures are i i i i-u = e - max e . Thus, absolute estimators of theefficiency measures in this model are directly computable using nothing more elaboratethan OLS. In the gamma frontier model, a, the OLS estimate of convergesto [ ] ( / )a a qiplim a = - E u = - P . So, another approach would be to correct the

constant term using estimates ofPand . The gamma model also produces individualestimates of technical efficiency. A summary statistic which might also prove useful is

[ ] / miE u = P = ,q which can be estimated with the corrected residuals. Likewise, an

estimate of 2 2[ ] /i uVar u = P =q s will be produced by the least squares residual

variance. Combining the two produces a standardized mean / u Pm s = . Here, aselsewhere, functions of the OLS parameter estimates and residuals can be used toobtain estimates of the underlying structural parameters. Consistent estimators ofq =P/m and P = qm are easily computed. Using this correction to the least squaresconstant term produces the MOLS estimator. Another useful parameter to estimate isE[exp(-ui)] =[q/(1+q)]

P. A similar approach is taken by Afriat (1972), who suggests that

ui be assumed to have a one parameter gamma distribution, with = 1 in the

preceding. Richmond (1974) builds on Afriat's model to obtain the distribution of e-ui,then derives E[exp(-ui)] and other population moments.

12 Both authors suggest that theOLS residuals be used to estimate these parameters. As Richmond demonstrates, Pcan be consistently estimated simply by using the standard deviation of the OLS

cast the linear programming approach as the maximizer of a log likelihood function. An application,among many, which compares econometric approaches to this linear programming methodology isFerrier and Lovell (1990).12As we can see from (3.39), E[e-ui] is 2-P, which is Richmond's result.


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residuals.

3.2.3 Methodological Questions

A fundamental practical problem with the gamma and all other deterministic frontiers is

that any measurement error and any other outcome of stochastic variation in thedependent variable must be embedded in the one sided disturbance. In any sample, asingle errant observation can have profound effects on the estimates. Unlikemeasurement error in yi, this outlier problem is not alleviated by resorting to largesample results.

There have been a number of other contributions to the econometrics literatureon specification and estimation of deterministic frontier functions. Two important paperswhich anticipated the stochastic frontier model discussed in the next section areTimmer (1971), which proposed a probabilistic approach to frontier modeling thatallowed some residuals to be positive, and Aigner, Amemiya, and Poirier (1976), who,

in a precursor to Aigner, Lovell, and Schmidt (1977), focused on asymmetry in thedistribution of the disturbance as a reflection of technical inefficiency. Applications ofthe parametric form of the deterministic frontier model are relatively uncommon. Thetechnical problems are quite surmountable, but the inherent problem with the stochasticspecification and the implications of measurement error render it problematic. Thenonparametric approach based on linear programming has an intuitive appeal and nowdominates this part of the literature on frontier estimation.

3.3 Data Envelopment Analysis (DEA)

DEA is a body of techniques for analyzing production, cost, revenue, and profit data,essentially, without parameterizing the technology. This constitutes a growth industryin the management science literature and appears with some frequency in economicsas well.

13We begin from the premise that there exists a production frontierwhich acts

to constrain the producers in an industry. With heterogeneity across producers, theywill be observed to array themselves at varying distances from the efficient frontier. Bywrapping a hull around the observed data, we can reveal which among the set ofobserved producers are closest to that frontier (or farthest from it). Presumably, thelarger is the sample, the more precisely will this information be revealed. In principle,the DEA procedure constructs a piecewise linear, quasi-convex hull around the datapoints in the input space. As in our earlier discussions, technical efficiency requiresproduction on the frontier, which in this case is the observed best practice. Thus, DEAis based fundamentally on a comparison of observed producers to each other. Onceagain, to argue that this defines or estimates an ideal in any sense requires the analystto assume, first, that there exists an ideal production point and, second, that producers

13See Lovell (1993), Ali and Seiford (1993) and Chapter 3 of this volume for discussion.


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strive to achieve that goal. Without belaboring the obvious, it is not difficult to constructsituations in which the second of these would be difficult to maintain. The servicesectors of the recently dismantled centrally planned economies of eastern Europecome to mind as cases in point.

There are many survey style treatments of DEA, including Chapter 3 of thisbook. As this survey is devoted to econometric approaches to efficiency analysis, wewill eschew presentation of any of the mathematical details. A brief (tight) and veryreadable sketch of the body of techniques is given in Murillo-Zamorano (2004, pp. 37-46).

The DEA method of modeling technical and allocative efficiency is largelyatheoretical. Its main strength may be its lack of parameterization; it requires noassumptions about the form of the technology. The piecewise linearity of the efficientisoquant might be problematic from a theoretical viewpoint, but that is the price for thelack of parameterization. The main drawback is that shared with the other deterministic

frontier estimators. Any deviation of an observation from the frontier must be attributedto inefficiency.14

There is no provision for statistical noise or measurement error in themodel. The problem is compounded in this setting by the absence of a definable set ofstatistical properties. Recent explorations in the use of bootstrapping methods hasbegun to suggest solutions to this particular shortcoming. [See, for example, Xue andHarker (1999), Simar and Wilson (1998, 1999) and Tsionas (2001b). The last of theseused efficiency measures produced by a DEA as priors for inefficiency in a hierarchicalBayes estimation of a stochastic frontier.]

As we will not return to the subject of data envelopment analysis in this study, wetake the opportunity at this point to note a few of the numerous comparisons that have

been made between (nonparametric) DEA and statistical based frontier methods, bothdeterministic and stochastic. There have been several studies that have analyzed datawith both DEA and parametric, deterministic frontier estimators. For example, Bjurek,Hjalmarsson and Forsund (1990) used the techniques described above to study theSwedish social insurance system. Forsund (1992) did a similar analysis of Swedishferries. In both studies, the authors do not observe radical differences in the resultswith the various procedures. That is perhaps not surprising since the main differencesin their specifications concerned functional form Cobb-Douglas for the parametricmodels, piecewise linear for the nonparametric ones. The differences in the inferencesone draws often differ more sharply when the statistical underpinnings are made moredetailed in the stochastic frontier model. But, even here, the evidence is mixed. Ray

and Mukherjee (1995) using the Christensen and Greene (1976) data on U.S.electricity generation find a good agreement between DEA and stochastic frontier

14For extensive commentary on this issue, see Schmidt (1985). Banker and Maindiratta (1988) showhow DEA gives an upper bound for efficiency. With input price data, one can also use the technique tocompute a lower bound.


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based estimates. Murillo-Zamorano and Vega-Cervera (2001) find similar results for alater (1990) sample of U.S. electricity generators. Cummins and Zi (1998) also foundconcordance in their analysis of the U.S. insurance industry. Finally, Chakraborty,Biswas and Lewis (2001) find in analyzing public education in Utah that the empiricalresults using the various techniques are largely similar. These studies do stand in

contrast to Ferrier and Lovell (1990) who found major differences between DEA andstochastic frontier based inefficiency estimates in a multiple out distance function fit ina large sample of American banks. Bauer et al. (1998) likewise found substantialdifferences between parametric and nonparametric efficiency estimates for a sample ofU.S. banks. In sum, the evidence is mixed, but it does appear that quite frequently, theoverall picture drawn by DEA and statistical frontier based techniques are similar. Thatthe two broad classes of techniques fail to produce the same pictures of inefficienciesposes a dilemma for regulators hoping to use the methods to evaluate theirconstituents (and, since they have the same theoretical underpinning, casts suspicionon both methods). As noted above, this has arisen in more than one study of thebanking industry. Bauer, et al. (1998) discuss specific conditions that should appear in

efficiency methods to be used for evaluating financial institutions, with exactly thisconsideration in mind.

4 The Stochastic Frontier Model

The stochastic production frontier proposed by Aigner, Lovell, Schmidt (1977) andMeeusen and van den Broeck (1977)

15is motivated by the idea that deviations from the

production frontier might not be entirely under the control of the firm being studied.Under the interpretation of the deterministic frontier of the preceding section, forexample, an unusually high number of random equipment failures, or even badweather, might ultimately appear to the analyst as inefficiency. Worse yet, any error or

imperfection in the specification of the model or measurement of its componentvariables, including the (log) output, could likewise translate into increased inefficiencymeasures. This is an unattractive feature of any deterministic frontier specification. Amore appealing formulation holds that any particular firm faces its own productionfrontier, and that frontier is randomly placed by the whole collection of stochasticelements which might enter the model outside the control of the firm. (This is a similarargument to Frsund and Jansen's (1977) rationale for an average vs. best practicefrontier function.) An appropriate formulation is

( ) iviii = fy eTEx

where all terms are as defined earlier and vi is unrestricted. The latter term embodiesmeasurement errors, any other statistical noise, and random variation of the frontier

15An application which appeared concurrently is Battese and Corra (1977).


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across firms. The reformulated model is

ln + = + .a - a ei i i i ii = + v u +y x xT Tb b

(The issue of functional form was considered earlier. We will use the linear

specification above generically.) As before, ui > 0, but vi may take any value. Asymmetric distribution, such as the normal distribution, is usually assumed for vi. Thus,the stochastic frontieris a+bTxi+viand, as before, uirepresents the inefficiency.

We note before beginning this lengthy section, our ultimate objective in theeconometric estimation of frontier models is to construct an estimate of ui or at least

ui - miniui. The first step, of course, is to compute the technology parameters, a, b, suand sv (and any other parameters). It does follow that if the frontier model estimatesare inappropriate or inconsistent, then estimation of the inefficiency component ofei,that is, ui, is likely to be problematic as well. So, we first consider estimation of thetechnology parameters. Estimation ofui is considered in detail in Section 8.

4.1 Implications of Least Squares

In the basic specification above, both components of the compound disturbanceare generally assumed to be independent and identically distributed (iid) acrossobservations.16 So long as E[vi - ui] is constant, the OLS estimates of the slopeparameters of the frontier function are unbiased and consistent. The averageinefficiency present in the distribution is reflected in the asymmetry of the distribution, aquantity that is easily estimable, even with the results ofOLS, with the third moment ofthe residuals,

3

1

1 ( - [ ])=

e eN

3 i i

i

= EmN

however estimated, so long as the slope estimators are consistent. By expanding

[ ]3

( ) m i i i3 = E - - E ,v u u

we see that, in fact, the skewness of the distribution of the estimable disturbance, i, issimply the negative of that of the latent inefficiency component, ui. So, for example,

16Recent research has begun to investigate the possibility of correlation across the two components ofthe composed disturbance. The econometric issues are considerable for example, identification is aproblem. The underlying economics are equally problematic. As of this writing (mid 2005), the returnson this model extension are far from complete, so we will eschew further consideration of the possibility.


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regardless of the assumed underlying distribution, the negative of the third moment ofthe OLS residuals provides a consistent estimator of the skewness of the distribution ofui. Since this statistic has units of measurement equal to the cube of those of the log ofoutput, one might, as a useful first step in any analysis, examine the conventional

normalized measure, 33 /3 = - m sb where s is the sample standard deviation of the

residuals. Values between 0 and 4 are typical. A Wald test of the hypothesis of nosystematic inefficiency in the distribution could be based on the familiar chi-squaredtest,

17

1

6

c

2

2 3

1 3

-m= .

s

The skewness coefficient of the least squares residuals in any finite sample could havethe wrong sign (positive in this case). This might cast doubt on the specification of thestochastic frontier model and suggest that the Wald test is meaningless.18 Other tests

of the stochastic frontier specification are presented in Schmidt and Lin (1984). Theskewness of the residuals turns out to be an important indicator of the specification ofthe stochastic frontier model. We emphasize, however, that this is merely a samplestatistic subject to sampling variability. The skewness is only suggestive m3 could bepositive even if the stochastic frontier model is correct. Indeed, for a non-normalspecification of the random components, m3 could be positive in the population.

4.2 Forming the Likelihood Function

We begin with a general formulation of the model, then narrow the specification to theparticular models that have been proposed in the contemporary literature. The generic

form of the stochastic frontier is

ln +

= + .

a -

a ei i ii

i i

= + v uy

+

x

x

T

T

b

b

It is convenient to start with the simplest assumptions, that

(a) fv(vi) is a symmetric distribution;(b) viand uiare statistically independent of each other;

17See Schmidt and Lin (1984) for this test and Coelli (1995) for a slightly different form of the test. Thestatistic appears in a slightly different form in Pagan and Hall (1983).18 This need not be the case. The skewness ofi is entirely due to uiand, so long as ui is positive, in fact,the skewness could go in either direction. Nonetheless, in the most common formulations of thestochastic frontier model, involving the normal distribution, the skewness provides an importantdiagnostic check on the model specification.


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(c) viand uiare independent and identically distributed across observations.

Thus, our starting point has both error components with constant means 0 and m andvariances sv

2 and su2, respectively, over all observations. To form the density of lnyi that

underlies the likelihood function, we use these assumptions to obtain the joint density of the

components,

fv,u(vi,ui) = fv(vi)fu(ui).

Then, ei=vi ui, so

fe,u(ei,ui) =fu(ui)fv(ei+ui)

(The J acobian of the transformation from (v,u) to (e,u) is 1 10 1

1

det-

=1.) Finally, to obtain

the marginal density ofei, we integrate uiout of the joint density;

fe(ei) =0

( ) ( )u i v i i if u f u du

e + .

The final step gives the contribution of observation ito the log likelihood

lnLi(a,b,su2,sv

2 | lnyi,xi) = ln fe(yi- a - bTxi|su

2,sv2).

In several important cases that we will examine below, the integral has a convenientclosed form, so that estimation of the model by maximum likelihood or throughBayesian methods based on the likelihood is straightforward. We do note, however,

that with current techniques of simulation based estimation, closed forms for integralssuch as this are not always necessary for estimation. 19

The derivation above requires a trivial modification for a cost frontier. In thiscase,

lnCi = a + bTxi + vi+ui.

(For convenience here, we retain the symbol x for the variables in the frontier function,though in a cost function, they would be output and the input prices, not the inputs.)Then, ei =vi +ui and fe,u(ei,ui) =fu(ui)fv(ei- ui). Since vi is assumed to have a symmetric

distribution, the second term may be written fv(ei - ui) = fv(ui - ei). Making this simplechange, we see that in order to form the density for log cost for a particular model inwhich observations lie above the frontier, it is necessary only to reverse the sign ofei

19 See Train (2003), Greene (2003a, Section 17.8 and 2005) and Greene and Misra (2003).


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where it appears in the functional form. An example below will illustrate.

4.3 The Normal - Half Normal Model

The compound disturbance in the stochastic frontier model, while asymmetrically

distributed, is, for most choices of the disturbance distributions, otherwise well-behaved. Maximum likelihood estimation is generally straightforward. The literature onstochastic frontier models begins with Aigner, Lovell and Schmidts (1977) normal-halfnormal model in which

fv(vi) = N[0,sv2] = (1/sv)f(vi/sv), -


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parameter l =su/sv characterizes the distribution. Ifl +, the deterministic frontierresults. Ifl 0, the implication is that there is no inefficiency in the disturbance, andthe model can be efficiently estimated by ordinary least squares.

Figure 4. Density of a Normal Minus a Half Normal

With the assumption of a half-normal distribution, we obtain [ ] 2 /uE u = s p and

[ ] [( 2) / ]2uiVar u = -p ps . A common slip in applications is to treat u2 as the variance of

ui. In fact, this overstates the variance by a factor of nearly 3! Since u is not thestandard deviation of ui, it gives a somewhat misleading picture of the amount ofinefficiency that the estimates suggest is present in the data. Likewise, although is

indicative of this aspect of the model, it is primarily a convenient normalization, notnecessarily a directly interpretable parameter of the distribution. It might seem that thevariance ratio u

2/2 would be a useful indicator of the influence of the inefficiency

component in the overall variance. But again, the variance of the truncated normalrandom variable ui is Var[Ui|Ui>0] =[(-2)/]u

2, not u

2. In the decomposition of the

total variance into two components, the contribution ofui is

2

2 2

[( 2) / ][ ]

[ ] [( 2) / ]u

u v

Var u

Var

p - p s=

e p - p s + s.

Further details on estimation of the half-normal model may be found in Aigner,Lovell, and Schmidt (1977) and in Greene (2003a). The parameter is the inefficiencycomponent of the model. The simple regression model results if equals zero. Theimplication would be that every firm operates on its frontier. This does not imply,however, that one can test for inefficiency by the usual means, because the polarvalue, =0, is on the boundary of the parameter space, not in its interior. Standard


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tests, such as the LM test are likely to be problematic.22

The log-likelihood function for the normal - half normal stochastic frontier modelis

2N

i=1

- 1Ln , , ) = - ln - constant + ln -2

i iL( , N e l e a s l s F s s b

where2 2 2ln , / , ,i i u v u vi= =ye - a - l s s = s + sx s

Tb

Fand = the standard normal CDF The log likelihood function is quite straightforwardto maximize, and has been integrated into several contemporary commercial computerpackages, including Frontier4 (Coelli (1996)), LIMDEP (Greene (2000)), Stata (Stata(2005)) and TSP(TSP International (2005)). (See, also, Greene (2003) for discussionof maximizing this log likelihood.) The normal-half normal model has an intriguing and

useful feature. Regarding an earlier point about the incorrect skewness of the leastsquares, Waldman (1982) has shown that in estimation of a stochastic production(cost) frontier with the normal-half normal model, if the OLS residuals, ignoring thefrontier function altogether, are positively (negatively) skewed (that is, in the wrongdirection), then the maximizers of the log likelihood are OLS for (a,b,s2) and zero forl.23 This is a very useful self diagnostic on specification and estimation of this frontiermodel.

24

4.4 The Truncated Normal Model

22The standard statistics, LM, Wald and LR. are quite well defined, even at = 0, which presents

something of a conundrum in this model. There is, in fact, no problem computing a test statistic, butproblems of interpretation arise. For related commentary, see Breusch and Pagan (1980). Thecorresponding argument as regards testing for a one-sided error term would be the same. In this case,the parametric restriction would be + or (1/l) 0, which would be difficult to test formally. Moreencouraging and useful results are given in Coelli (1995), who shows that the likelihood ratio statistic hasa limiting distribution that is a tractable mix of chi-squared variates.23 The log likelihood for the normal-half normal model has two roots, one at OLS with l = 0, and one atthe MLE with positive l. In the event noted, the first solution is superior to the second.24 It does create a bit of a dilemma for the practitioner. In itself, the result is an important diagnostic forthe model specification. However, it does not carry over to other model formulations, and moreelaborate extensions. As such, one might choose to proceed in spite of the warning. Then again, someof the estimators of these elaborate models use the plain vanilla ALS frontier estimates as startingvalues for the iterations. In this case, at least the warning will be heard. We note for the benefit of thepractitioner, the occurrence of this result is not indicative of a problem with the data or their software itsignals a mismatch between the model and the data. The appropriate conclusion to draw is that the datado not contain evidence of inefficiency. A more encouraging result, however, is that this result is specificto the half normal model above. Other formulations of the model, or more highly developedspecifications might well reveal the presence of inefficiency. That is, this finding can emerge fromseveral sources.


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Stevenson (1980) argued that the zero mean assumed in the Aigner, et al. (1977)model was an unnecessary restriction. He produced results for a truncatedas opposedto half-normal distribution. That is, the one-sided error term, ui is obtained bytruncating at zero the distribution of a variable with possibly nonzero mean. The

complete parameterization is[ ]

[ ] | |

2i v

2i u i i

~ N 0, ,v

~ N , , = ,U u U

s

m s

For convenience, we will use the parameterizations given earlier for and . Then,the log-likelihood is

2

1

1Ln , , , ) = - ln + ln 2 + ln ( / ) +

2

1

ln2

u

N i i

i

L( , N

=

F

+

- + F -

a b s l s p m sm

e m e lm

s sl s

where 2/ 1us = ls + l . [A derivation appears in Kumbhakar and Lovell (2000).]

Starting values for the iterations in the stochastic frontier models are typically obtainedby manipulating the results of OLS to obtain method of moments estimators for theparameters of the underlying distribution. There does not appear to be a convenientmethod of moments estimator for the mean of the truncated normal distribution. Butmaximum likelihood estimation presents no unusual difficulty. The obvious startingvalue for the iterations would be the estimates for a half-normal model and zero for .The benefit of this additional level of generality is the relaxation of a possibly erroneousrestriction. A cost appears to be that the log-likelihood is sometimes ill-behaved whenis unrestricted. As such, estimation of a nonzero often inflates the standard errorsof the other parameter estimators considerably, sometimes attends extreme values ofthe other parameters, and quite frequently impedes or prevents convergence of theiterations. It is also unclear how the restriction ofto zero, as is usually done, wouldaffect efficiency estimates. The Bayesian applications of this model, e.g., Tsionas(2001a) and Holloway et al. (2005) have apparently encountered less difficulty inestimation of this model.

The parameters of the underlying distribution ofui provide a mechanism forintroducing heterogeneity into the distribution of inefficiency. The mean of thedistribution (or the variance or both) could depend on factors such as industry, location,capital vintage, and so on. One way such factors might be introduced into the modelcould be to use

mi = m0 +m1Tzi,


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where zi is any variables that should appear in this part of the model. As noted, we willrevisit this possibility later.

4.5 Estimation by Corrected Ordinary Least Squares

Method of Moments Estimators

The parameters of the stochastic frontier model can be estimated using the second andthird central moments of the OLS residuals, m2 and m3. For the half-normal model, themoment equations are

2 22 u v

33 u

- 2= +m

2 4= 1 - .m

ps s

p

sp p

(Note that m3 is negative, since the offset in i by ui is negative.) Thus, u and v areeasily estimable. Since E[ui] = (2/)

u, the adjustment of the OLS constant term is /u= a + 2 .a ps These MOLS estimators are consistent, but inefficient in comparison

to the MLEs. The degree to which they are inefficient remains to be determined, but itis a moot point, since with current software, full maximum likelihood estimation is nomore difficult than least squares.

Waldman (1982) has pointed out an intriguing quirk in the half-normal model.

Normally, there are two roots of the log-likelihood function for the stochastic frontiermodel, one at the OLS estimates, and another at the MLE. In theory, the distribution ofthe compound disturbance is skewed to the left. But, if the model is badly specified, itcan occur that the OLS residuals are skewed in the opposite direction. In this instance,the OLS results are the MLEs, and consequently, one must estimate the one-sidedterms as 0.0.25 (Note that if this occurs, the MOLS estimate of is undefined.) Onemight view this as a built-in diagnostic, since the phenomenon is likely to arise in abadly specified model or in an inappropriate application. This failure we use theterm advisedly here, since analysts might differ on whether the estimation tools or theanalyst has failed occurs relatively frequently. Coelli's (1995) formulation may bemore convenient in this regard. (See footnote 26 above.) He suggests the moment

estimators

25The results are extended in Lee (1993). Lee addresses the issue of inference in the stochastic frontiermodel at the boundary of the parameter space, = 0.


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p p s = + p p

2

32

3

2

2 - 42 2m m

p p g = ps

2

3

321

2 - 4m

gsa =22+

2a

As before, the wrong sign on m3 can derail estimation of , but in this instance, aconvenient place to begin is with some small value; Coelli suggests 0.05. As notedearlier, there is no obvious method of moments estimator for in Stevenson's truncatednormal model.

The MOLS estimators for the exponential model are based on the moment

equations m2 =2 3

3and 22v u u+ m = - .s ss Thus,

[ ] 2 , ,/ 1 23 v 23u u u2= = - = a +- mms s a ss .For the gamma model, the MOLS estimators are

3 2 ( ) /( ), /( ), ,2 24 2 3 3 v 2u u u u= - 3 3 P = - 2 = - P = a + P .m m m m m-s s s a ss Any of these can be used to obtain a full set of estimates for the stochastic frontiermodel parameters. They are all consistent. Thereafter, estimates of the efficiencydistributions or of the individual coefficients, -ui or TEi, can be computed just byadjusting the OLS residuals. There is a question of the statisticalefficiency of theseestimators. One specific result is given in Greene (1980a) for the gamma distributed,deterministic frontier model, namely that the ratio of the true variance of the MLEof anyof the slope coefficients in the model to its OLS counterpart is (P-2)/P. Thus, thegreater is the asymmetry of the distribution the gamma density tends to symmetry asPincreases the greater is the efficiency gain to using MLE. (See Deprins and Simar(1985) for further results.) Of course, efficient estimation of the technical parameters isnot necessarily the point of this exercise. Indeed, for many purposes, consistency is all

that is desired. As noted, estimation of all of these models is fairly routine withcontemporary software. The preceding are likely to be more useful for obtaining startingvalues for the iterations than as estimators in their own right.


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5 Stochastic Frontier Cost Functions, Multiple Outputs, and Distance and ProfitFunctions: Alternatives to the Production Frontier

Here, we consider a variety of specifications that model production and (in)efficiency infunctional forms that differ from the single output production function examined up to

this point.

5.1 Multiple Output Production Functions

The formal theory of production departs from the transformation function that links thevector of outputs, y to the vector of inputs, x;

T(y,x) =0.

As it stands, some further assumptions are obviously needed to produce the framework

for an empirical model. By assuming homothetic separability, the function may bewritten in the form

A(y) =f(x).

(See Fernandez, Koop and Steel (2000) for discussion of this assumption.) Thefunction A(y) is an output aggregator that links the aggregate output to a familiarproduction function. The assumption is a fairly strong one, but with it in place, we havethe platform for an analysis of (in)efficiency along the lines already considered.Fernandez, Koop and Steel (2000) proposed the multiple output production model,

( )= a = + -1 / q

T

1xM q qm i,t,m it it itm

y v ub

Inefficiency in this setting reflects the failure of the firm to achieve the maximumaggregate output attainable. Note that the model does not address the economicquestion of whether the chosen output mix is optimal with respect to the output pricesand input costs. That would require a profit function approach. The authors apply themethod to a panel of U.S. banks the 798 bank, ten year panel analyzed by Berger(1993) and Adams et al. (1999).

26Fernandez et al. (1999, 2000, 2002a,b) have

extended this model to allow for bads, that is undesirable inputs. Their model consistsof parallel equations for the goods (dairy output of milk and other goods in Dutch dairy

farms) and bads (nitrogen discharge). The two equations are treated as a SURsystem, and are fit (as is the banking model) using Bayesian MCMC methods. The

26Koop (2001) also applied this approach to the output of major league baseball players were the fouroutputs are singles, doubles and triples, home runs and walks, and the inputs are time, team and leaguedummy variables. Illustrative of the technique, but perhaps of limited policy relevance.


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study of the electric power industry by Atkinson and Dorfman (2005) takes a similarapproach, but fits more naturally in Section 5.4, where we examine it in a bit moredetail.

5.2 Stochastic Frontier Cost Functions

Under a set of regularity conditions (see Shephard (1953) or Nerlove (1963)), analternative representation of the production technology is the cost function,

T( ) = min{ : ( ) }C y, f yw w x x

where w is the vector of exogenously determined input prices. The cost function givesthe minimum expenditure needed to produce a given output, y. If a producer istechnically inefficient, then their costs of production must exceed the theoretical

minimum. It seems natural, then, to consider a frontier cost function as an alternative tothe frontier production function model. The interpretation of the inefficiency terms in anempirical model is complicated a bit by the dual approach to estimation, however.Suppose that on the production side of the model, the representation of a one-sidederror term as reflective purely of technical inefficiency is appropriate. The computationis conditional on the inputs chosen, so whether the choice of inputs is itself allocativelyefficient is a side issue. On the cost side, however, anyerrors in optimization, technicalorallocative, must show up as higher costs. As such, a producer that we might assessas operating technically efficiently by a production function measure might still appearinefficient viz-a-viz a cost function.

Similar arguments would apply to a profit function. This does not preclude eitherformulation, but one should bear in mind the possible ambiguities in interpretation inthese alternative models. It might make more sense, then, to relabel the result on thecost side as cost inefficiency. The strict interpretation of technical inefficiency in thesense of Farrell may be problematic, but it seems counterproductive to let this be astraightjacket. The argument that there is a cost frontier that would apply to any givenproducer would have no less validity. Deviations from the cost frontier could then beinterpreted as the reflection of both technical and allocative inefficiency. At the sametime, both inefficiencies have a behavioral interpretation, and whatever effect is carriedover to the production side is induced, instead. The same logic would carry over to aprofit function. The upshot of this argument is that estimation techniques which seek to

decompose cost inefficiency into an allocative and a true Farrell measure of technicalinefficiency may neglect to account for the direct influence of output, itself, on theresidual inefficiency once allocative inefficiency is accounted for.

We will begin by examining the costs of production of a single output


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conditioned on the actual input choices. That is, neglecting the first-order conditionsfor optimality of the input choices, we consider the implications for the costs ofproduction of technical inefficiency. For simplicity, we assume constant returns toscale. The production function, f(x) is linearly homogeneous and therefore homothetic.For homothetic production functions,

27

[ )]y = F f(x

where F(t) is a continuous and monotonically increasing function when t is positive.We have the fundamental result (from Shephard (1953)) that the corresponding costfunction is

-1( ) = ( ) ( )C y, y cFw w

where c(w) is the unit cost function. For the stochastic frontier production function,then

) e iviii = f(y TEx ,

so that the cost function is

-1( ) ( )

i-vi i

i

1= y cC eF

TEw

This corresponds to Farrell's (1957) original efficiency measure, i.e., the cost savingsthat would be realized if output were produced efficiently. The theoretical counterpart

would be the input-based measure above. In logs, then,

-1ln ( ) + ln ( ) - ln - .lni i i iC = y c TE vF w

In terms of our original model, then, the stochastic cost f

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