Recent developments in the econometric estimation of frontiers

Journal

of Econometrics

46 (1990) 39-56.

North-Holland

RECENT DE~LOPMENTS IN THE ECONOMETRIC ESTIMATION OF FRONTIERS Paul W. BAUER* Federal Reser1.e Bank, Cleveland, OH 44114, USA A number of techniques have been developed that expand the range of options available to researchers for estimating frontiers. This paper discusses recent developments in the econometric approach to the estimation of stochastic frontiers such as production, costs, and profit functions. Areas requiring further work are also noted.

1. Introduction

The use of frontier models is becoming increasingly widespread for a variety of reasons. First, the notion of a frontier is consistent with the underlying economic theory of optimizing behavior. Second, deviations from a frontier have a natural interpretation as a measure of the efficiency with which economic units pursue their technical or behavioral objectives. Finally, information about the structure of the frontier and about the relative efficiency of economic units has many policy appIications. There are two competing paradigms on how to construct frontiers. One uses mathematical programming techniques, the other employs econometric techniques.’ The chief advantage of the mathematical pro~amming or ‘Data Envelopment Analysis’ (DEA) approach is that no explicit functional form need be imposed on the data. However, the calculated frontier may be warped if the data are contaminated by statistical noise. The econometric approach can handle statistical noise, but it imposes an explicit, and possibly overly restrictive, functional form for technology. Unless panel data are available, an explicit distribution for the inefficiency term must be imposed as well. Recent developments in the mathematical programming approach are discussed in Seiford and Thrall (1990). This paper samples recent develop*The author Peter Schmidt author.

would like to thank Gary Ferrier, William H. Greene, for helpful comments. All errors and omissions remain

CA. Knox Love& and the responsibility of the

‘A complementary literature that allows for only allocative inefficiency is not examined. In this approach; allocative ine~c~ency is modeled in a parametric manner: For a sampling of this literature see Lau and Yotopoulos (1971), Toda (1976), Atkinson and Halvorsen (1980, 1984). Love11 and Sickles (1983), and Sickles, Good, and Johnson (1986).

0304-4076/90/$3.50

0 1990, Elsevier

Science

Publishers

B.V. (North-Holland)

40

P. W. Bauer, Econometric estimation offiontiers

ments in the econometric approach. 2 My aim is not to be exhaustive, but to present a selective evaluation of some of the more significant advances in the field. The trade-offs between imposing more or less structure are discussed in section 2. The techniques common to most of the single-equation econometric models are presented in section 3. Various attempts to estimate stochastic frontier cost systems are analyzed in section 4, while the following section looks at estimation techniques for other stochastic frontier systems, such as systems based on profit or distance functions. Estimation techniques which strive to minimize the assumptions that have to be made concerning the disturbance terms and tests of those assumptions are examined in section 6. The final section summarizes the recent advances in stochastic frontier modeling and makes some recommendations for future research. 2. Structure

versus flexibility

A recurring theme in econometrics is the conflict between structure and flexibility. The more structure we impose on a model the better our estimates - provided the structure we impose is correct. When estimating frontiers, we face the usual problems of selecting a functional form and deciding whether to estimate a single equation or a system of equations (when applicable), but we also face a myriad of choices about how to model inefficiency, particularly when panel data are employed. Ideally, either we know the correct structure to impose a priori or we estimate a sufficiently flexible model so that possible restrictions can be tested. Consider the choice of whether to estimate a deterministic or stochastic frontier. Usually a stochastic frontier is selected because it allows for statistical noise resulting from events outside the firm’s control, such as luck and weather. Employing a stochastic frontier can also be seen as allowing for some types of specification error and for omitted variables uncorrelated with the included regressors. However, obtaining individual firm estimates of efficiency is more invofved with a stochastic frontier than with a deterministic (or full) frontier, which directly yields estimates of individual firm efficiency as the residuals from estimation. Now consider the problem of choosing a functional form and deciding whether to estimate a single equation or a system of equations (when applicable). Flexible functional forms allow for a more sophisticated technology, and estimation of systems of equations hold the promise of more asymptotically efficient estimates of technology and efficiency. However, as we will see in section 4, once one moves very much beyond Cobb-Douglas ‘Forsund, Lovell, and Schmidt (19&O), Schmidt (19851, and Love11 and Schmidt valuable discussions of earlier work on the estimation of frontiers.

(1988)

provide

P. W. Bauer, Econometric estimation of frontiers

41

functional forms or perhaps other self-dual forms, the estimation of a system of equations becomes much more difficult. In addition, statistical efficiency is lost by estimating an overly flexible functional form. A final set of choices involves the modeling of inefficiency, particularly when panel data are employed. In stochastic frontier models, inefficiency is modeled by a ‘composed’ error term, composed of statistical noise and a one-sided disturbance to allow for ineficiency. Unless one has panel data, specific distributional assumptions about the one-sided component of the disturbance term must be made in order to obtain estimates of individual firm efficiencies. When panel data are available, specific distributional assumptions may be avoided, although then a model of how efficiency varies over time must be imposed. If efficiency does not vary over time, the researcher has a strong lever for prying out estimates of firm-specific measures of efficiency, but this is a very strong assumption, particularly as the number of time-series observations increases. In short: ‘Stronger assumptions generate stronger results, but they strain one’s conscience more.’ 3 In general, the choice of maintained assumptions (ones required to obtain estimates of the frontier and firm-specific measures of efficiency) are broadened by the techniques discussed in the following sections. The appropriate structure to impose can only be determined by a careful consideration of the data and the characteristics of the industry under study. Unfortunately, there are not always statistical tests to guide the way.

3. Building

blocks

The econometric approach to estimating frontiers uses a parametric representation of technology along with a two-part composed error term. This approach was first proposed by Aigner, Love& and Schmidt (1977), Meeusen and van den Broeck (19771, and Battese and Corra (1977). One part of the composed error term represents statistical noise and is generally assumed to follow a normal distribution4 The other part represents inefficiency and is assumed to follow a particular one-sided distribution. Most of the models discussed here employ measures of efficiency derived from Farrell (1957). While the basic set of. econometric estimation techniques has changed relatively little in recent years, there have been some useful combinations and extensions of these basic techniques. ‘Anonymous referee. 4Kopp and Mullahy (1990) use a generalized method of moments estimation strategy to relax the normality assumption for statistical noise (although specific distributional assumptions still must be maintained for the inefficiency term). Only mild restrictions are placed on the statistical noise component: it must be independent of the inefficiency component; it must be symmetric about zero; and it must have finite higher moments (at least up to a predetermined order).

42

P. u! Bauer, Econometric estimation offrontiers

The following one-sided distributions have been employed: the half-normal and exponential distributions proposed by Aigner, Lovell, and Schmidt (1977) (among others), the truncated normal proposed by Stevenson (1980), and the two-parameter Gamma distribution proposed by Greene (1990). Other composed-error distributions could be constructed following Greene’s (1990) methodolo~. Tests of the appropriateness of these various distributions can be made using Lagrange multiplier techniques proposed by Lee (1983) and Schmidt and Lin (1984). To illustrate the basic econometric approach, consider the single-equation stochastic cost function model InCi=InC(y,,wi)

+ui+ui,

(3.1)

where Ci is observed cost, yi is a vector of outputs, wi is an input price vector, ui is a one-sided disturbance (positive for cost frontiers) capturing the effects of inefficiency, ui is a two-sided disturbance capturing the effects of noise. The deterministic kernel of the cost frontier is C(y,, wi), and the stochastic cost frontier is C(yj, w,)exp(u,). Following Aigner, LoveIl, and Schmidt (1977), let ui ++N(0, a:) and ui IN(0, gU2>I.The likelihood function for this system can be written as

lnJY=~ln(2/7r-Nlnrr+

cln i=,

[

1-Q

(

-Eih a j] -$!,“i, (3.2)

where N is the number of observations, &i= ui + c’,, u2 = flu2+ Ef,2, A = a,,/~,., and @(.> is the standard normal distribution function. Estimates of this model can be obtained using corrected ordinary least squares (COLS) or by maximizing the likelihood function directly. Olson, Schmidt, and Waldman (1980) used a Monte Carlo approach to examine the relative advantages of these two estimation techniques. MLE tended to outperform COLS in sample sizes larger than 400, whereas COLS tended to outperform MLE in sample sizes of less than 400. Observation-specific estimates of inefficiency, u (subscripts can safely be omitted here), can be obtained by using the distribution of the inefficiency term conditional on the estimate of the entire composed error term, as suggested by Jondrow, Lovell, Materov, and Schmidt (1982) and Kalirajan and Flinn (1983).’ One can use either the expected value or the mode of this ‘Battese developed

and Coelli (1988), Kumbhakar (1988b), and Bat&e, extensions of this technique for use with panel data.

Coelli,

and Colby (1989) have

P. W. Eauer, Econometric estimation

offro&em

43

conditional distribution as an estimate of u:

(3.3) M( u I&) = +u’,2)

if

E 2

if

E
0,

(3.4) = 0

where +(.I is the standard normal density function. Unfortunately, these estimates cannot be shown to be consistent estimates of u, since the variability of the conditionai distribution u given E is independent of sample size (i.e., E contains only imperfect information about u). The models discussed below can be seen as enhancements or modifications, in one way or another, of the framework outlined above. 4. Cost system approaches Estimating cost systems that impose as few assumptions as possible (particularly ones that minimize arbitrary assumptions about functional forms, the distributions of disturbances, and independence of the level of inefficiency with the regressors) has proven to be a di~cult task. Using FarreII (1957) definitions for technical and allocative inefficiency, cost systems that allow for cost inefficiency can be written as InCi=InC(yi,wi) sij=s,(yi,wi)

+ln7;.+lnAi+ui, +eij

for

j=l,...,M-

(4.1) 1,

(4.2)

where Ci is the observed cost, C(J is the deterministic minimum cost frontier, yi is a vector of outputs, wi is a vector of input prices, In 1; is a nonnegative term reflecting the increase in cost due to technical inefficiency, In Ai is a nonnegative term reflecting the increase in cost due to allocative inefficiency, ui represents statistical noise, sjj is the observed share of the jth input, s,(.) is the efficient share of the jth input, eij is the disturbance on the jth input share equation (a mixture of allocative inefficiency and noise), M is the number of inputs, and N is the number of observations. This system has the following characteristics. First, in the cost equation the disturbances representing technical and allocative inefficiency increase observed cost, whereas statistical noise can either increase or decrease observed cost. Second, in the input share equations, allocative inefficiency and noise may increase or decrease a given input’s cost share. Technical inefficiency does not appear in the input share equations when considered from a cost

44

P. K ISaw& Eco~omet~~ estimation of frontiers

perspective, since output is exogenously determined in this ~amework.6 Last, the allocative inefficiency disturbance in the cost equation is related to the allocative inefficiency disturbances in the input share equations. Many researchers have employed such a cost system, usually employing a translog cost function. Within such systems, a key problem is how to model the relationship between the two-sided disturbances on the input share equations (which are composed in part of allocative inefficiency, i.e., over- or underemployment of a given input) with the nonnegative allocative inefficiency disturbance in the cost equation. This problem (sometimes referred to as ‘the Greene problem’) was first noted by Greene (1980, pp. 104-1051, and was also discussed in Nadiri and Schankerman (1981, fn. 3, pp. 226-227). There are three possible routes to take in modeling the Greene problem: 1) find the analytic relationship among the allocative inefficiency disturbances, eij and In Ai [e.g., Schmidt and Love11 (1979) or Kumbhakar (198911; 2) model the relationship using an approximating function, imposing ail the structure one knows a priori [e.g., Schmidt (198411; 3) ignore the relationship among the disturbances in the cost and input share equations [e.g., Greene (1980) treats these disturbances as independent]. The first approach is generally to be preferred, since one derives the exact analytic representation of the relationship. However, an analytic relationship can be found only when fairly restrictive functional forms are imposed, such as the Cobb-Douglas or perhaps other self-dual forms. The second approach enables one to use more flexible functional forms, and it employs information about how allocative inefficiency links the disturbances. Unfortunately, this approach is limited by how well the approximation models the true relationship between the allocative inefficiency term in the cost equation and the disturbances on the share equations. The last approach foregoes trying to model the relationship, even imperfectly, but by ignoring an important link among the disturbances, the estimation technique fails to employ all available information. Each of these approaches is discussed in more detail below. 4.1. Analytic solutions When there exists a closed-form representation of both the cost and production functions, as with the Cobb-Douglas functional form, an analytic %ome authors, e.g., Schmidt (1984, 1985-86), have argued that in cost systems, the input share equations will involve technical inefficiency if the cost function is not homothetic, since the cost-efficient input shares will vary with output. This is certainly true if output based measures of technical and allocative inefficiency are employed. However, if input-based measures of technical and allocative inefficiency are employed - which is a more natural basis to estimate cost systems, since output is assumed to be exogenous - then the issue does not arise. The decomposition of overall cost inefficiency into technical and allocative inefficiency is essentially arbitrary to begin with and may depend on whether radial [Farrell (195711or nonradial measures [for example, the Russell measure introduced by Fare and Love11(1978)I are employed.

45

P. W. Bauer, Econometric estimation offrontiers

representation of the relationship among the allocative inefficiency disturbances in (4.1)-(4.2) is possible. Schmidt and Love11 (1979, 1980) were the first to develop this- systems approach. For a Cobb-Douglas production function of the form M

y=anxpe’,

(4.3)

j=l

the system of cost and factor demand equations can be written as InC=K+lny/r+

(4.4)

~aj/rInw,-(~+u)/r+(E-Inr),

j=l

1

M

In x1 = In k, + l/r

In y + In n

pzm/‘/pl

[ m=l + l/r(

U - u) + E

a,/rE,,

(4.5)

j=2

ln xi = In kj

+ l/r

In y + In

- l/r(u+u)-ej+

E ff,/rE,

m=2

t

j=2

>..-, M,

(4.6)

where y is a scalar measure of output, x is a vector of n inputs, r = cz= 1cy, (the returns to scale), A4 is the number of inputs, k. = cxj[an~=lcx$-l’r, K= ln[C~_,k,], and E = czJcx,/r)E, + ln[a, + ~~=2~,lePBml. The errors, both noise (u) and inefficiency (u, nonpositive in the case of a production frontier), from each equation find their way into every equation in the system in a rather involved way. The Greene problem is ‘solved’ in that the disturbances in the factor demand equations are functionally mapped into the allocative inefficiency term in the cost equation (In A = E - In r). Note that E is a rather involved function of the Cobb-Douglas production function coefficients and the disturbances on the factor demand equations. Technical inefficiency is simply a function of the returns to scale and the one-sided disturbance in the production frontier (In T = -u/r). Kumbhakar (1988a) generalizes this approach to allow for multiple outputs and fixed inputs and for the identification of what he defines to be ‘inputspecific technical inefficiency’, a nonradial measure of efficiency. This ap-

46

P. W. Bauer, Economettic estimation of frontiers

preach deviates from Farrell’s (1957) radial definitions of technical efficiency by explicitly allowing some inputs to be used more efficiently than others. Kumbhakar (1988b) uses the generalized production function (GPF) developed by Zellner and Revankar (1969), which generalizes the Cobb-Douglas functional form by allowing the scale elasticity to vary across firms. Another analytic approach to dealing with the Greene problem was developed by Kumbhakar (1989). Kumbhakar finesses the Greene problem by estimating the cost-minimizing input demand equations from a symmetricgeneralized-McFadden (SGM) cost function [see Diewert and Wales (198711 and by again using his input-specific technical inefficiency measures. Since the cost equation is not needed to identify all the parameters for this functional form and one equation must be dropped from the system in any case, the cost equation is the obvious choice. The cost-minimizing input demand equations can be written as

+

bii + hi/Y + zkdikqk/Y + skaikqk/Y

+ PiY (4.7)

j,r=1,2

,...,

n,

k,1=1,2

,..., m,

where y is a scalar measure of output, x is a vector of II variable inputs with corresponding input price vector w, q is vector of m fixed inputs, ui represents noise that is allowed to follow a first-order autoregressive process, and ri is the one-sided error (nonpositive here). With only the input demand equations being estimated, there is no problem in relating inefficiency in the input share equations to the cost equation. The efficiency or ‘effectiveness’ of the ith input can be defined as TE, =x:/xi

= 1 + ri/( yx,),

(4.8)

where xi* =xi + 7i + ui is the minimum quantity of input i required to produce a given level of output keeping all other inputs unchanged. Thus a measure of overall efficiency is (4.9)

Allocative inefficiency is not explicitly ‘apparent allocative inefficiency’ in the the cost-efficient input demands, those Shepherd’s lemma, are not necessarily usage.

included in this model, but there is model in the sense that the ratio of derived from the cost function using equal to the ratio of observed input

P. W. Bauer, Ecorwmettic estimation of frontiers

47

4.2. Approximate solutions

For systems such as (4.1)-(4.2), Schmidt -(1984) proposes modeling the relationship between allocative inefficiency in the cost and input share equations as In Ai = e;Fe,,

(4.10)

where ej = (e,r, ei2,. . . , eiM)I and F is an M x M positive semi-definite matrix. This specification ensures that In A, = 0 when ei = 0 and that In Ai and (eijl are positively correlated for all j. The problem is how to choose F. Schmidt (1984) suggests F=@/‘M-1)x+,

(4.11)

where e, N N(0, Z), D is the product of the positive eigenvalues of Z, and 1?+ is the generalized inverse of _I$(the covariance of the input share equation disturbances). Given these assumptions, In Aj is distributed as a chi-squared random variable and is positively correlated with the variances of the disturbances on the input share equations. The likelihood function for this system can be derived after positing an appropriate one-sided distribution for In Ti, such as the half-normal, and letting ui - N(0, c$>. To date, no one has used this model to obtain empirical estimates, since the resulting likelihood function would be fairly formidable to maximize. The Kopp and Mullahy (1990) generalized method of moments (GMM) approach (which is less demanding computationally than MLE) may be of use here. Melfi (1984) simplifies Schmidt’s specification to obtain a more tractable maximum-likelihood procedure. Most of the complexity of the above model comes from the assumptions required to ensure that In Ai follows a known distribution. Melfi (1984) demonstrates how the likelihood function for the system can be derived given the relation of the disturbances in the input share equations to the allocative inefficiency term in the cost equation by noting f(ln & + In Ai + s) = g(ln K + C:l/ei)h( ei),

(4.12)

where f, g, and h are the density functions for (In 7; + In Ai + ui), (In 7;.+ ui), and ei. Since in this model the aliocative inefficiency term, In Ai, is a function of ei, then In Ai is fixed given ei. Thus, distributional assumptions need only be made for In T,, ui, and ej, which are assumed to follow half-normal, normal, and multivariate-normal distributions. For tractability, Melfi (1984) modeled F = IM and assumed no cross-equation correlation among the input share equations, so that In A, is the sum of the squared errors on ail the share equations. A drawback of this specification is that the estimates of allocative inefficiency are forced towards zero. The input share residuals are less than one in

48

I? W. Bauer, Econometric

estimation offrontiers

absolute value and so the sum of the squares will almost necessarily be small. For example, if there are two inputs with input share disturbances of 0.1 and -0.1, then the allocative inefficiency implied by (4.10) with F = I is 0.02, which may or may not be plausible. One way of overcoming this problem is to set F = cZ, where c is a scalar to be estimated. The sum of squared errors from the input share equations can be scaled up (or down) by c in order to more flexibly model the effects of allocative inefficiency. Bauer (1985) makes several extensions to Melfi’s approach in order to develop a more flexible estimation technique. First, F is modeled as a positive semidefinite diagonal matrix whose elements are treated as parameters to be estimated. Strictly speaking, F would only have to be required to be positive semidefinite. Second, Bauer modeled ei N N(cr, _I$),allowing ei to have a nonzero mean.’ This enables a firm to persistently (and still transiently) over- or underemploys a given input relative to its cost-minimizing employment. The likelihood function for this model can be written as TNM

in_..&=Nln2--

2

-

ln(ai? + TV:) - y

- :

_ +i t=1

+-$

N(T-

ln(2r)

$ ( n=l

“2’

CSint -

1)

2

In a,2

~~‘ln ai 1=l si( Y;;wnt)

-

ffi>z

1

i=l

5 (~nC,t-lnC(y,t,ytt~ L’ [ t=1

--Ant)”

Ii x

i t=1

(InC,t-~nCty,t,~,t)

‘Schmidt and Love11 (1979) in section in this way.

4 of their paper

--At)

modeled

(4.13)

f

the factor

demand

equations

P. W. Bauer, Economettic

49

estimation of frontiers

where T is the number of time-series observations, N is the number of firms, In A, defined as in (4.101, where the diagonal elements of F are parameters to be estimated and the off-diagonal elements are set to zer-0.s Note that ui is assumed to be constant over time. The likelihood function for Melfi’s model can be obtained by restricting the cy,‘s to be equal to zero and forcing FiI equal to a common value. Despite these modifications, some problems remain. First, even for small number of outputs and inputs, there is a large number of parameters to estimate. Some of these parameters, such as the off-diagonal elements of F and 2, would be very difficult to estimate in practice without imposing additional structure. For example, Bauer (1985) restricted both F and 2 to be diagonal matrices. Second, solving the Greene problem by flexibly modeling the relationship among the allocative inefficiency disturbances does not necessarily lead to better estimates of the cost frontier. Ignoring these relationships (as discussed in the following section) may yield better estimates than imperfectly modeling them. 4.3. Qualitative solutions Estimation techniques can be developed that ignore the link among the allocative inefficiency disturbances across the equations in the system. Greene (1980) first proposed this approach in a full frontier framework where he constructed a translog cost system using a Gamma distribution for the cost inefficiency disturbance. The disturbances on the input share equations were assumed to follow a multivariate normal distribution with mean zero. He recognized the relationships among these disturbances, but treated them as statistically independent of the inefficiency term in the cost equation in deriving the likelihood function. Another illustration of this approach would be to take the cost system developed in Bauer (1985) and sever the link among the allocative inefficiency disturbances by restructuring the disturbances in the cost equation so that (In T, + In A ;) instead of just In T, is modeled as a half-normal random variable. The disturbances in the input share equations remain unchanged, still allowing for persistent and transient allocative inefficiency. There is no longer any linkage, such as (4.101, because the link among the allocative disturbances is being ignored. Such a system could be written as lnCj=lnC(y,,wi)

+ui+cl,

sij=sj(yi,wi) +eij,

‘Note

that u, is assumed

j=l

to be constant

(4.14) ,...,

M-l,

i=l,...,

over time for each firm

N,

50

P. W Bauer, Econometric estimation of frontiers

where the one-sided disturbance is ui = In Ti + In Ai. Using Kopp and Diewert (1982) and Zieschang (1983), the estimate of ui [which would come from an approach like Jondrow, Lovell, Materov, and Schmidt (1982)l can be decomposed into estimates of In q and In Ai. The resulting likelihood function is similar to (4.13) above. This approach is not fully efficient statistically in that information about the relationship among the allocative inefficiency disturbances is being ignored, but it does not necessarily yield worse results than an approach which models the relationship incorrectly. 5. Other system approaches

(profit and distance

functions)

Frontier estimation techniques have also been developed for relationships other than cost and production functions. Applying production or cost frontier techniques is straightforward only when a single equation is considered. For example, Ali and Flinn (1983) estimate profit efficiency using a single-equation approach with a composed error term (here the inefficiency term is nonpositive as with a production frontier). When estimating other systems (such as a profit function, output supply equations, and input demand equations), problems similar to those faced in developing cost system estimation techniques in integrating the error structures are found as well. Paralleling the Schmidt-Love11 approach for cost systems, Kumbhakar (1987b) extended the use of frontier production models to firms under the behavioral goal of profit maximization for the single-product firm. Using a Cobb-Douglas production function and the first-order conditions for profit maximization, a system with composed error terms is constructed and estimated using maximum-likelihood estimation. Kumbhakar (1987~) extends this framework to multi-product firms facing constant elasticity of transformation (CET) output [introduced by Powell and Gruen (1968)] and Cobb-Douglas input functions and constructs the likelihood function for the system composed of the production function (with a composed error term made up of technical inefficiency and noise) and the first-order conditions for profit maximization (which contain allocative inefficiency). As in the cost function approach, a drawback to the analytic approach in the stochastic-frontier profit system is the relatively inflexible functional forms one is forced to work with in order to obtain closed-form solutions. It is possible to develop stochastic-frontier profit systems using flexible functional forms in a manner similar to that discussed in section 4.3 above. Similar techniques can also be developed to extend the stochastic frontier literature to the estimation of distance functions, introduced by Shephard (1953). Unlike production functions, but like profit and cost functions, the distance function can easily handle multiple outputs. Also, the distance

51

P. W. Bauer, Economettic estimation of frontiers

function contains the same information about technology as does the cost function. Moreover, the distance function is the reciprocal of Farrell’s radial measure of technical efficiency. The distance function may have some advantages econometrically over the cost function if, for example, input prices are the same for firms, but input quantities vary across firms. In situations where it is the inputs, not the input prices that are fixed, the distance function will be preferred to the cost function. A system of the distance function and the cost-minimizing input share equations can be estimated using the Ferrier and Love11 (1987) full frontier estimation approach. 6. Avoiding and testing disturbance

term assumptions

Pitt and Lee (1981) first developed techniques to use panel data to estimate frontier functions, but their maximum-likelihood approach imposes a great deal of structure. Schmidt and Sickles (1984) show that when panel data are available a number of restrictive assumptions can be relaxed. Their production function model can be written as yi, = ff +

xup + ui

+

Vi,)

i=l

,..., N,

t=l,..,,

T.

(6.1)

Here i indexes firms and t indexes time periods. Since inefficiency is assumed to be time-invariant, the model can be transformed as follows to give (6.1) a disturbance that is i.i.d. with mean zero: Yir =

ai + x,tP + uit where

(Y,= LY+ ui.

(6.2)

If there are no time-invariant regressors, the within estimator enables researchers to relax the assumption that the inefficiency disturbances are independent of the regressors, as is required for MLE. If only a subset of the regressors are correlated with the inefficiency disturbances, then a Hausman-Taylor (1981) estimator can be employed. Estimates of the inefficiency disturbances can be obtained without assuming a particular distribution for these terms. Schmidt and Sickles (1984) treat the most efficient firm in the sample as 100 percent efficient (ui = 0 for that firm), so that the estimate of (Yis just the largest of the estimated (Y~.All that is required for these estimates to be consistent as the number of firms (N) approaches infinity is that the density of inefficiency disturbances are nonzero in some neighborhood (0, 5) for some 5 > 0. If one finds the assumption that inefficiency is time-invariant untenable (and it becomes increasingly so as the number of time-series observations becomes larger), inefficiency could be modeled as being statistically independent over time. However, then panel data cease to have any qualitative advantage over time-series or cross-sectional data.

P. W: Bauer, Econometric estimation of frontiers

52

Cornwell, Schmidt, and Sickles (1990) develop an approach that strives to bridge these two extremes by imposing some structure on how inefficiency varies over time. In their model, the intercept as well as the slope coefficients are allowed to vary over firms and time, allowing the levels of efficiency to vary over time by firm. They generalize Schmidt and Sickles (1984) by replacing the firm effects, ui, by Uir = 8j,

+

l&t +

oi3t2.

(6.3)

This allows the efficiency levels to vary over firms and time. This approach can also be seen as an extension of the Hausman-Taylor estimator in that it allows for cross-sectional variation in the slopes as well as the intercepts in a panel data model. The model can be estimated by within, GLS, Hausman-Taylor, or MLE, depending on the number of assumptions the researcher is wilhng to make about the independence and the distribution of the firm effects. Kumbhakar (1990) starts with an equation similar to (6.11, but proposes the following formulation for air: Uit =

7(t)z$,

t= I,2 ,..., T,

(6.4)

where (6.5) where b and c are coefficients to be estimated. The resulting system must be estimated using MLE. 7. Summary and suggestions

for future research

The estimation of any type of a frontier, and hence efficiency relative to that frontier, is a challenging endeavor. The efficiency of a firm is inherently a residual concept: ‘ . , . when we remove the effects of differences in certain measured inputs, some firms still produce more or less than others; and this we call inefficiency.’ 9 Over the last few years, researchers have developed a myriad of techniques expanding the range of options available for estimating models that allow for inefficiency. First, there has been great progress towards using more flexible functional forms. Also, systems of equations based on cost, profit, and even distance functions have been developed in order to incorporate all available information into the estimates of technology and efficiency. Cost, revenue, ‘Schmidt

(1985-86).

P. W. Bauer, Economettic estimation offiontiers

53

profit, and distance functions also enable us to deal easily with multiple outputs, thereby deflecting a common criticism of the econometric approach to frontier estimation by DEA proponents. Since Christensen and Greene (19761, nonfrontier estimation of cost systems have almost all used techniques that employed flexible functional forms and systems of equations. It has taken a while for the econometric frontier literature to catch up. Next, using ideas developed by Jondrow, Lovell, Materov, and Schmidt (1982) and Kalirajan and Flinn (19831, obse~ation-specific estimates of inefficiency can be obtained. The price of their advances is that one must impose specific distributional assumptions on both the noise and inefficiency terms. Last, estimation techniques specifically tailored to the unique characteristics of panel data are being developed that enable the researcher to relax many of the more restrictive assumptions about the inefficiency disturbances. In particular, with panel data, researchers no longer have to assume that the level of inefficiency is independent of the regressors and no longer have to impose a particular distribution for the inefficiency terms, making these restrictions testable propositions. A number of problems remain. First, while the nonindependence of inefficiency and regressors can be handled when the researcher is fortunate enough to have panel data, it would be desirable to develop techniques for handling this problem with cross-sectional or time-series data. Second, very little work has been done on the sensitivity of the results to the stochastic assumptions. The half-normal distribution has become the standard choice for the one-sided distribution, even though more flexible distributions are available, with very little attempt at either empirical or theoretical justification. In addition to Lee’s (1983) approach to testing distributional assumptions, the Kappa criterion, used by Molina and Slottje (1987) to test distributional assumptions about income distributions, could be adapted to test for the appropriateness of frontier disturbances. Next, few researchers have focused much attention on the relationship between technical and allocative efficiencies. An exception is Schmidt and Love11 (1980) who allow technical and allocative inefficiency to be correlated. Another exception is Kalirajan and Shand (1988). They test for the direction of causality between technical and allocative efficiency using a test developed by Sims (1972). While noting that the econometric test used is ‘not without criticism’, they find that there is no bidirectional causality between technical efficiency and allocative efficiency, but that technical efficiency does influence allocative efliciency. More work in this area is required to determine whether this is a general result or merely specific to the data they employed. More work also needs to be done in the area of GMM [pioneered in the frontier field by Kopp and Mullahy (199011, particularly towards weakening the assumptions that must be made concerning the inefficiency disturbance.

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The GMM approach is less computationally intensive than the MLE techniques that have been one of the mainstays of the frontier field. Last, by construction, stochastic frontier models with composed errors terms are constrained in how far they may deviate from ordinary least-squares results (and estimated technology) without a Hausman-type test rejecting the assumptions of the model. In other words, there is very little difference in what many authors describe as best- vs. average-practice technology in these types of models (except for shifts in the intercept term). ~though this is an issue that Schmidt (1985-86) raised in his last survey of the literature, no progress has since been made in this area.

References Aigner, Dennis, C.A. Knox Love& and Peter Schmidt, 1977, Formulation and estimation of stochastic frontier production function models, Journal of Econometrics 6, 21-37. Ah, Mubarik and John C. Flinn, 1987, Profit efficiency among Basmati rice producers in Pakistan Punjab, American Journal of Agricultural Economics 71, 303-310. Atkinson, S. and R. Halvorsen, 1980, A test of relative and absolute price efficiency in regulated utilities, Review of Economics and Statistics 62, 81-88. Atkinson, S. and R. Halvorsen, 1984, Parametric efficiency tests, economies of scale, and input demand in U.S. electric power generation, International Economic Review 25, 647-661. Battese, George E. and Tim J. Coelli, 1988, Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data, Journal of Econometrics 38, 387-399. Battese, George E., Tim J. Coelli, and T.C. Colby, 1989, Estimation of frontier production functions and the efficiencies of Indian farms using panel data from ICRISAT’s village level studies, Journal of Quantitative Economics 5, 327-348. Battese, George E. and Greg S. Corra, 1977, Estimation of a production frontier model with application to the pastoral zone of eastern Austraha, Australian Journal of Agricultural Economics 21, 169- 179. Bauer, Paul W., 1985, An analysis of multiproduct technology and efficiency using the joint cost function and panel data: An application to the U.S. airline industry, Unpublished doctoral dissertation (University of North Carolina, Chapel Hill, NC). Christensen, L.R. and William H. Greene, 1976, Economies of scale in U.S. electric power generation, Journal of Political Economy 84, 655-676. Cornwell, Christopher, Peter Schmidt, and Robin C. Sickles, 1990, Production frontiers with cross-sectional and time-series variation in efficiency levels, Journal of Econometrics, this issue. Diewert, W.E. and T.J. Wales, 1987, Flexible functional forms and global curvature conditions, ~conometrica 25, 43-68. Fare, R. and C.A. Knox Lovell, 1978, Measuring the technical efficiency of production, Journal of Economic Theory 19,150-162. Farrell, M.J., 1957, The measurement of productive efficiency, Journal of Royal Statistical Society A 120, 253-281. Ferrier, Gary and C.A. Knox Love& 1987, Estimating economies and efficiencies in production: An evaluation of alternative approaches, in: Proceedings of the Business and Economic Statistics Section (American Statistical Association, Alexandria, VA). F$rsund, Finn, CA. Knox Lovell, and Peter Schmidt, 1980, A survey of frontier production functions and of their relationship to efficiency measurement, Journal of Econometrics 13, 5-25. Greene, William H., 1980, On the estimation of a flexible frontier production model, Journal of Econometrics 13, 101-115.

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William H., 1990, A gamma distributed stochastic frontier model, Journal of Econometrics, this issue. Hausman, J.A. and W.E. Taylor, 1981, Panel data and unobservable individual effects, Econometrica 49, 1377-1398. Jondrow, James, C.A. Knox Lovell, Ivan S. Materov, and Peter Schmidt, 1982, On the estimation of technical inefficiency in the stochastic frontier production function model, Journal of Econometrics 19, 233-238. Kalirajan, K.P. and J.C. Flinn, 1983, The measurement of farm-specific technical efficiency, Pakistan Journal of Applied Economics 2, 167-180. Kalirajan, K.P. and R.T. Shand, 1988, Testing causality between technical and allocative efficiencies, Oxford Economic Papers, forthcoming. Kopp, Raymond J. and W. Erwin Diewert, 1982, The decomposition of frontier cost function deviations into measures of technical and allocative efficiency, Journal of Econometrics 19, 319-331. Kopp, Raymond J. and John Mullahy, 1990, Moment-based estimation and testing of stochastic frontier models, Journal of Econometrics, this issue. Kumbhakar, Subal C., 1987a, Production frontiers and panel data: An application to U.S. class 1 railroads, Journal of Business and Economic Statistics 5, 249-255. Kumbhakar, Subal C., 1987b, The specification of technical and allocative inefficiency in stochastic production and profit frontiers, Journal of Econometrics 34, 335-348. Kumbhakar, Subal C., 1987c, The specification of technical and allocative inefficiency of multiproduct firms in stochastic production and profit frontiers, Journal of Quantitative Economics 3, 213-223. Kumbhakar, Subal C., 1988a, Estimation of input-specific technical and allocative inefficiency in stochastic frontier models, Oxford Economic Papers 40, 535-549. Kumbhakar, Subal C., 1988b, On the estimation of technical and allocative inefficiency using stochastic frontier functions: The case of U.S. class 1 railroads, International Economic Review 29, 727-744. Kumbhakar, Subal C., 1989, Estimation of technical efficiency using flexible functional form and panel data, Journal of Business and Economic Statistics 7, 253-358. Kumbhakar, Subal C., 1990, Production frontiers, panel data, and time-varying technical inefficiency, Journal of Econometrics, this issue. Lau, L. and P. Yotopoulos, 1971, A test for relative efficiency and an application to Indian agriculture, American Economic Review 61, 94-109. Lee, Lung-Fei, 1983, A test for distributional assumptions for the stochastic frontier functions, Journal of Econometrics 22, 245-267. Lovell, CA. Knox and Peter Schmidt, 1988, A comparison of alternative approaches to the measurement of productive efficiency, in: Ali Dogramaci and Rolf Fare, eds., Applications of modern production theory: Efficiency and production (Kluwer Academic Publishers, Boston, MA). Lovell, C.A. Knox and Robin C. Sickles, 1983, Testing efficiency hypotheses in joint production: A parametric approach, Review of Economics and Statistics 65, 51-58. Meeusen, Wim and Julien van den Broeck, 1977, Efficiency estimation from Cobb-Douglas production functions with composed error, International Economic Review 8, 435-444. Melfi, C.A., 1984, Estimation and decomposition of productive efficiency in a panel data model: An application to electric utilities, Unpublished doctoral dissertation (University of North Carolina, Chapel Hill, NC). Molina, David and Daniel J. Slottje, 1987, The gamma distribution and the size distribution of income reconsidered, Atlantic Economic Journal 15, 86. Nadiri, M. Ishaq and Mark A. Schankerman, 1981, The structure of production, technological change, and the rate of growth of total factor productivity in the U.S. Bell System, in: Thomas G. Cowing and Rodney E. Stevenson, eds., Productivity measurement in regulated industries (Academic Press, New York, NY). Olson, J.A., Peter Schmidt, and Donald M. Waldman, 1980, A Monte Carlo study of estimators of the stochastic frontier production function, Journal of Econometrics 13, 67-82. Pitt, Mark M. and Lung-Fei Lee, 1981, The measurement and sources of technical inefficiency in the Indonesian weaving industry, Journal of Development Economics 9, 43-64. Greene,

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Powell, AA. and F.H.G. Gruen, 1968, The constant elasticity of substitution production frontier and linear supply system, International Economic Review 9, 315-328. Schmidt, Peter, 1984, An error structure for systems of translog cost and share equations, Econometrics workshop paper 8309 (Department of Economics, Michigan State University, MI). Schmidt, Peter, 1985-86, Frontier production functions, Econometric Reviews 4, 289-328. Schmidt, Peter and Tsai-Fen Lin, 1984, Simple tests of alternative specifications in stochastic frontier models, Journal of Econometrics 24, 349-361. Schmidt, Peter and CA. Knox Love& 1979, Estimating technical and allocative inefficiency relative to stochastic production and cost frontiers, Journal of Econometrics 9, 343-366. Schmidt, Peter and CA. Knox Lovell, 1980, Estimating stochastic production and cost frontiers when technical and allocative inefficiency are correlated, Journal of Econometrics 13, 83-100. Schmidt, Peter and Robin Sickles, 1984, Production frontiers and panel data, Journal of Business and Economic Statistics 2, 367-374. Seiford, Lawrence M. and Robert Thrall, 1990, Recent developments in DEA: The mathematical programming approach to frontier analysis, Journal of Econometrics, this issue. Shephard, Ronald W., 1953, Cost and production functions (Princeton University Press, Princeton, NJ). Sickles, Robin C., David Good, and Richard L. Johnson, 1986, Allocative distortions and regulatory transition of the U.S. airline industry, Journal of Econometrics 33, 143-163. Sims, C.A., 1972, Money, income, and causality, American Economic Review 62, 540-552. Stevenson, Rodney E., 1980, Likelihood functions for generalized stochastic frontier estimation, Journal of Econometrics 13, 57-66. Toda, Y., 1976, Estimation of a cost function when cost is not minimum: The case of Soviet manufacturing industries, 1958-1971, Review of Economics and Statistics 58, 259-268. Zellner, A. and N.S. Revankar, 1969, Generalized production functions, Review of Economic Studies 36, 241-250. Zieschang, Kimberly D., 1983, A note on the decomposition of cost efficiency into technical and allocativc components, Journal of econometrics 23, 401-405.