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Predicting technical effciency in stochastic productionfrontier models in the presence of misspecification: aMonte-Carlo analysisKonstantinos Giannakas a , Kien C. Tran b & Vangelis Tzouvelekas ca Department of Agricultural Economics, University of Nebraska-Lincoln, 216 H.C. FilleyHall, Lincoln, NE 68583-0922, USAb Department of Economics, University of Saskatchewan, 9 Campus Drive, Saskatoon,Saskatchewan, S7N 5A5 Canadac Department of Economics, University of Crete, Crete, GreecePublished online: 05 Oct 2010.

To cite this article: Konstantinos Giannakas , Kien C. Tran & Vangelis Tzouvelekas (2003) Predicting technical effciency instochastic production frontier models in the presence of misspecification: a Monte-Carlo analysis, Applied Economics, 35:2,153-161, DOI: 10.1080/0003684022000015964

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Predicting technical efficiency in stochastic

production frontier models in the presence

of misspecification: a Monte-Carlo analysis

KONSTANTINOS GIANNAKAS*, KIEN C. TRAN{ andVANGELIS TZOUVELEKAS}

Department of Agricultural Economics, University of Nebraska-Lincoln, 216 H.C.Filley Hall, Lincoln, NE 68583-0922, USA, {Department of Economics, Universityof Saskatchewan, 9 Campus Drive, Saskatoon, Saskatchewan, S7N 5A5 Canada and}Department of Economics, University of Crete, Crete, Greece

This paper provides a theoretical explanation for the sensitivity of technical effi-ciency measures to the choice of functional specification in stochastic productionfrontier models. It is shown that inappropriate functional specifications translateinto a misspecification in the conditional mean of the stochastic frontier regressionmodel. This misspecification, in turn, results in estimates of technical efficiency,confidence intervals and production elasticities being biased, even asymptotically.Monte-Carlo simulations reveal that the severity of the bias depends on the func-tional specification and the percentage contribution of the variance of technicalinefficiency to the total variance of the composed errors.

I . INTRODUCTION

The stochastic production frontier model introduced inde-pendently by Aigner et al. (1997) and Meeusen and Vanden Broeck (1977) has dominated the empirical literatureon technical efficiency measurement in both developed anddeveloping countries. Within this framework, severalmodels for estimating technical efficiency have been pro-gressively developed, extending the stochastic frontiermethodology to account for different theoretical andempirical issues (for detailed reviews of the research workin this area see Coelli et al., 1998; Greene, 1999; andKumbhakar and Lovell, 2000).

An attractive feature of the stochastic frontier model isthe separation of the impact of exogenous shocks on out-put from the contribution of variation in technical effi-ciency. Specifically, the disturbance term in the model, ",consists of two independent and identically distributed

across observations elements, v and u, so that " � v� u.The component v is a symmetric normally distributederror term that represents factors that cannot be controlledby production units (e.g. weather), measurement errors,and left-out explanatory variables. On the other hand,the component u is a one-sided non-negative errorterm representing the stochastic shortfall of producer i’soutput from his/her production frontier due to technicalinefficiency. In this context, technical efficiency is definedin an output-expanding manner (Debreu-type) and revealsthe maximum amount by which output can be increasedusing the same level of inputs and technological con-ditions.1

After the imposition of appropriate distributionalassumptions concerning the compound disturbance term,the model can be estimated using standard maximum like-lihood (ML) techniques. The information on the technical(in)efficiency of production units (contained in the com-

Applied Economics ISSN 0003–6846 print/ISSN 1466–4283 online # 2003 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/0003684022000015964

Applied Economics, 2003, 35, 153–161

153

* Corresponding author.1 An input-based measure of technical inefficiency (Shephard-type) is defined as the ratio of best practice input usage to actual usage, withoutput held constant. Fare and Lovell (1978) have shown that these two measures of technical inefficiency are equal under constantreturns to scale. Under decreasing (increasing) returns to scale the output-based measure is greater (less) than the input-based measure oftechnical inefficiency.

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pound disturbance term) can then be obtained using the

predictor developed by Jondrow et al. (1982), which is

based on the conditional distribution of u given ".2 The

individual technical efficiency levels using this predictor

are derived under the assumption that the model is cor-

rectly specified, however. This includes the correct specifi-

cation of the distribution for the technical inefficiency term

(u), the (conditional and unconditional) variance functions,

and the mean function.

Concerning the distribution of u, the assumption of a

half-normal distribution has dominated the empirical lit-

erature on stochastic frontier modelling.3 On the other

hand, the symmetric component v is customarily assumed

to be normally distributed. While normality is a rather

natural assumption for v (based on central limit argu-

ments), particular distributions for u are less easily moti-

vated (for a thorough discussion on this issue see Greene,

1990, 1999). In fact, there is plenty of empirical evidence

suggesting that sample mean technical efficiencies are sen-

sitive to the statistical distribution assigned to the one-

sided error component.

Regarding the specification of the variance functions,

Caudill et al. (1995), Hadri (1999) and Lothgren (1999)

examine stochastic frontier models that allow for the pres-

ence of heteroscedasticity in the compound disturbance

term. The first two papers examine stochastic cost frontier

models with unconditional heteroscedasticity, while the

later considers a stochastic production frontier with

dynamic conditional heteroscedasticity. All three papers

show that a misspecification of the variance functions can

lead to incorrect measures of inefficiency and invalid infer-

ences concerning the estimated parameters.

Other than the distribution of the one-sided error com-

ponent and the specification of the variance functions,

recent work by Gong and Sickles (1992), Zhu et al.

(1995), Battese and Broca (1997), and Giannakas et al.

(2002) shows that technical efficiency measures are also

highly sensitive to the choice of functional specification

for the mean function. In general, the use of inappropriate

functional specifications can be thought of as a misspecifi-

cation in the conditional mean of the stochastic frontier

regression. Since the measurement of technical inefficiency

within the stochastic frontier model is based on the resi-

duals obtained from the econometric estimation of the

model, and since these residuals are sensitive to the func-

tional specification of the frontier, it is likely that this sen-sitivity will pass on to the individual efficiency estimates.

The purpose of this paper is to provide a theoreticalexplanation and some Monte-Carlo results for the sensitiv-ity of the estimates of technical efficiency predictor, con-fidence interval, and production elasticities to the choice offunctional form in stochastic production frontier models.The rest of the paper is organized as follows. Section IIformalizes the problem of estimating technical efficiencywhen the conditional mean is misspecified. Section III pre-sents the Monte-Carlo simulation design. The simulationresults are discussed in Section IV, and Section V concludesthe paper.

II . PREDICTING TECHNICAL EFFICIENCYUNDER MISSPECIFICATIONS

Suppose we have observed panel data (repeated observa-tions on each production unit) on (log) output, yit, and avector of inputs, xit, ðyit 2 <; xit 2 <kþ : i ¼ 1; 2; . . . ;N;t ¼ 1; 2 . . .TÞ, with the stochastic production frontierbeing specified as:

yit ¼ f ðxit;�Þ þ vit � uit ¼ mit þ "it ð1Þ

where mit denotes the unknown production frontier,f ðxit;�Þ, � is a vector of unknown coefficients, and "it isthe stochastic compound disturbance term. Note that, inthis setting, the technical inefficiency term is allowed to be atime-dependent random variable.4

Assuming that uitiid jNð0; �2

uÞj and vitiid Nð0; �2

vÞ,we can then predict producer- and time-specific technicalinefficiency, expð�uitj"itÞ, as (Battese and Coelli, 1988):

E½expð�uitj"itÞ� ¼ exp½ð��it þ 0:5�2 Þ��½�it=� Þ � � �

Fð�it=� Þð2Þ

where �it ¼ �"it, ¼ �2u=ð�2

u þ �2vÞ, �2

¼ ð1 � Þ�2,�2 ¼ �2

u þ �2v , and �ð�Þ and Fð�Þ are the probability density

and distribution functions of a standard normal randomvariable, respectively.

Horrace and Schmidt (1996) derive the expressions forð1 � Þ100% confidence intervals ðLBit;UBitÞ forexpð�uitj"itÞ based on monotonic transformations of the=2 and ð1 � =2Þ quantiles of the distribution of uit con-ditional on "it as:5

154 K. Giannakas et al.

2 The predictor of Jondrow et al. (1982) is utilized when cross-section data is available. The predictor of producer-specific technicalefficiency in a panel data setting is derived by Kumbhakar (1987) and Battese and Coelli (1988).3 Alternative statistical distributions utilized include the truncated normal, exponential and gamma (see Kumbhakar and Lovell, 2000,pp. 74–90).4 Several approaches have been developed to account for time-varying technical inefficiency. For a review of alternative model specifica-tions of time-varying technical efficiency in the context of panel data see Kumbhakar et al. (1997) and Kumbhakar and Lovell (2000, pp.108–15).5 As noted by Greene (1999, p. 108), since these confidence intervals are conditioned on known values of the parameters ignoring anyvariation in the parameter estimates used to construct them, they should be regarded as minimal width intervals.

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LBit ¼ expð��it � ZU� Þ and UBit ¼ expð��it � ZL� Þð3Þ

where

ZL ¼ F�1½1 � ð1 � =2Þf1 � Fð��it=� Þg� ð3aÞand

ZU ¼ F�1½1 � ð=2Þf1 � Fð��it=� Þg� ð3bÞIt is important to recognize that both the technical ineffi-ciency predictor in Equation 2 and the confidence intervalestimator in Equation 3 are derived under the assumptionthat the model is correctly specified. With respect to themean function, the underlying assumption is that the pro-duction frontier f ð�Þ is known, which implies that the com-posed errors are also known.

However, the underlying production technology is rarelyknown a priori. In practice, we empirically estimate thefollowing model:

yit ¼ gðxit; �Þ þ !it � �it ¼ it þ eit ð4Þwhere it is a presumed specification of the productionfrontier. In most empirical applications, the mean function it is chosen to be either Cobb–Douglas or Translog.

Let us denote the estimated residuals by

eeit ¼ ðmit � itÞ þ "it ¼ it þ "it ð5Þwhere it is the ML estimate of it. If the production fron-tier function is correctly specified (i.e. it � mit), then it ! mit a.s. under mild regularity conditions. It followsthat eit � "it and eeit ¼ "it þ opð1Þ. In such a case, any pre-diction of uit based on the estimated residual eeit is asymp-totically equivalent to the prediction of uit based on "itimplying that the estimated residuals are appropriate forpredicting the inefficiency parameter uit. Recent simulationwork by Kumbhakar and Lothgren (1998) for cross-sectiondata, and Giannakas et al. (2000a) for panel data verifiesthis contention.

Consider now the case where the model is misspecified inthe sense that it != pmit with eeit ¼ it þ "it. The model mis-specification may arise due to the imposition of an inap-propriate functional form for the production frontier,violation of the assumption of independence between thevector of inputs and technical inefficiency, and measure-ment errors on the factors of production. To see how themisspecification in the production frontier function affectsthe point prediction and/or the interval estimation of tech-nical efficiency, note that, because "it is not observable, weinadvertently use eeit to compute the point prediction inEquation 2. It is then clear from Equation 5 that thepoint predictor of technical efficiency will be biased, evenasymptotically, due to incorrect information contained in

eeit. The error in measuring technical (in)efficiency, regard-less of the type of misspecification, can be easily verifiedusing Equation 2.

III . MONTE-CARLO DESIGN

To examine the severity of bias of the point predictor andthe confidence interval estimator of technical efficiencywhen the production frontier is misspecified, Monte-Carlo experiments are conducted. The data are generatedaccording to the Generalized Quadratic Box–Cox(GQBC)6 frontier of the form:

yð�Þit ¼ �0 þ

XJ

j¼1

�jxð�Þjit þ 1

2

XJ

j¼1

XJ

k¼1

�jkxð�Þjit x

ð�Þkit þ "it

"it ¼ vit � uit ð6Þ

where yð�Þit ¼ ðy2�

it � 1Þ=2� and xð�Þit ¼ ðx�it � 1Þ=�. The main

reason for the choice of GQBC as the true productiontechnology is the fact that it nests six other functionalforms, including the familiar and commonly used Cobb–Douglas and Translog specifications. Table 1 summarizesthe parametric restrictions on GQBC that lead to the vari-ous nested functional specifications considered in thispaper. Obviously, the misspecification in the productionfrontier is equivalent to imposing the incorrect restrictionson the parameters of the GQBC frontier.

To limit the simulation study to a manageable scale,we fix the parameter � ¼ 0 and consider three differentvalues for � ¼ f0:25; 0:5; 2g. In addition, the case is con-sidered where there are only two inputs and a set ofvalues for the parameter vector f�0; �1; �2; �11; �22; �12g ¼f1:0; 0:5; 0:5; 0:25; 0:25; 0:5g. The two inputs, labour andfertilizer, are taken from the balanced panel data set usedby Giannakas et al. (2000b) in examining the efficiency of125 Greek olive-growing farms during the period 1987–1993.

Predicting technical efficiency in stochastic production frontier models 155

6 The generalization of the Box–Cox transformation function to allow for a variety of functional forms to be nested within this functionis due to Appelbaum (1979) and Berndt and Khaled (1979).

Table 1. Functional specifications of the Production FrontierModel

Frontier specifications Parameter restrictions

GQBC NoneTranslog (TL) � ¼ 0; � ¼ 0Generalized Leontief (GL) � ¼ � ¼ 0:5Normalized quadratic (NQ) � ¼ 0:5; � ¼ 1Square-rooted quadratic (SRQ) � ¼ � ¼ 1Non-homothetic CES jk ¼ 0; 8j; kCobb–Douglas (C–D) � ¼ 0, � ¼ 0 and jk ¼ 0, 8j; k

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Due to the invariance results noted by Olson et al. (1980)only one value of �2 is considered, namely,�2 ¼ �2

u þ �2v ¼ 1. The variance ratio is controlled for

that reflects the percentage contribution of the varianceof uit to the total variance of the error term "itð� vit � uitÞin the data generating process. This variance ratio isdefined as ¼ ð1 � 2=�Þ�2

u=bð1 � 2=�Þ�2u þ �2

vc ¼=f þ ð1 � Þ½�=ð�� 2Þ�g where ¼ �2

u=ð�2u þ �2

vÞ (seeCoelli (1995)). Eleven values of the variance ratio ,which lies between zero and one, inclusive, in incrementof 0.1 are considered.

The random error terms vit are generated from i.i.d.Nf0; ð1 � Þg and the technical inefficiency terms uit aregenerated from i.i.d. Nð0; Þ truncated at zero from below,using rndnðÞ function in Gauss Version 3.27 with seed equalto 3157. The simulations are based on 1000 replications.The numerical maximizations are implemented usingMAXLIK library Version 4.034/5 with BFGS-algorithm.The expressions for the likelihood function along withthe gradients can be found in Battese and Coelli (1993).

For each Monte-Carlo experiment, the bias and meansquared errors (MSE) of the point predictor of technicalefficiency in Equation 2 are computed, and as a secondaryinterest, the production elasticities. Furthermore, in eachreplication the firm-specific confidence interval for techni-cal efficiency is computed using Equation 3, and then thenumber of times in 1000 replications that the confidenceinterval includes the true efficiency is calculated.

IV. SIMULATION RESULTS

Before discussing simulation results, note that the finitesample properties of the ML estimator, confidence inter-vals and point predictors of technical inefficiency have beenpreviously examined by Coelli (1995) and Kumbhakar andLothgren (1998). The reader is referred to these studies fora detailed discussion on these issues. To further conserve

space, the discussion of the Monte-Carlo experiment

results will focus on the case where � ¼ 0:25.7 The full

set of results is available from the authors upon request.

Table 1(a) and Fig. 1(a) display the bias in ML estimator

of for the different specifications of the production fron-

tier. Examining Table 1(a) and Fig. 1(a) closely, it is seen

that several patterns emerge. First, for all functional speci-

fications, there is a significant positive bias when ¼ 0.

Second, when the production frontier is correctly specified

(i.e. GQBC), the bias is small and negative for 5 0:1.

This negative bias increases to a maximum at ¼ 0:2 and

then diminishes (to almost zero) as approaches one.

Third, when the production frontier is misspecified, the

bias becomes significant. For the TL, the bias pattern is

very similar to that of the GQBC except that the bias is

much larger (approximately 10 times larger). For GL and

NQ the bias is significantly positive. This positive bias

increases to a maximum at ¼ 0:2 and then decreases

slightly to a local minimum at ¼ 0:4 before increasing

monotonically as approaches one. A similar pattern is

observed also for SRQ except that the bias is significantly

negative. For CES and C–D, the bias is significantly nega-

156 K. Giannakas et al.

7 The results for values of � of 0.5 and 2 are found to be similar except that the bias for the TL specification increases with an increase in �.

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

Bia

s

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 1 (a). Bias in the ML estimator of (N ¼ 125, T ¼ 7 and� ¼ 0:25)

Table 1(a). Bias in the ML estimator of (N ¼ 125, T ¼ 7 and � ¼ 0:25)

GQBC TL GL NQ SRQ CES C-D

0.0 0.1299 0.0918 0.1512 0.1525 0.1305 0.1628 0.16340.1 70.0219 70.0799 0.0113 0.0119 70.0168 70.1238 70.12710.2 70.0322 70.1239 0.0234 0.0276 70.1583 70.1578 70.15990.3 70.0156 70.1153 0.0123 0.0163 70.1202 70.1318 70.13460.4 70.0072 70.0993 0.0098 0.0107 70.1048 70.1287 70.12990.5 70.0039 70.0865 0.0101 0.0134 70.1132 70.1278 70.12840.6 70.0020 70.0813 0.0107 0.0141 70.1161 70.1218 70.12360.7 70.0008 70.0753 0.0119 0.0149 70.1182 70.1135 70.11830.8 70.0003 70.0710 0.0130 0.0157 70.1191 70.1102 70.11390.9 70.0001 70.0686 0.0137 0.0162 70.2001 70.1084 70.11011.0 70.0000 70.0636 0.0139 0.0164 70.2008 70.1037 70.1073

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tive and increasing to a maximum at ¼ 0:2 and thenmonotonically decreasing as approaches one.

Interesting patterns also emerge in the MSE of the MLestimator of , which are presented in Table 1(b) and Fig.1(b). For GQBC, TL, GL and NQ specifications, the MSEis largest when ¼ 0 and it declines monotonically as

increases. For the CES and C–D, the MSE declines to aminimum at ¼ 0:1 and then increases to a local maxi-mum at ¼ 0:2 before decreasing monotonically as

increases.

Table 2(a) and Fig. 2(a) show the bias in the estimateof the mean of technical efficiency predictor (TE). Theestimates of TE were calculated using Equation 2 withthe unknown parameters replaced by their ML estimates.Note that when ¼ 0 all specifications are fully efficientand the bias is naturally negative. The magnitude ofthe bias (in absolute terms) ranges from 13% to 30%depending on the specification. This finding suggeststhat if a large number of fully efficient firms are examinedusing the stochastic frontier approach, the average level

Predicting technical efficiency in stochastic production frontier models 157

Table 1(b). MSE of the ML estimator of (N ¼ 125, T ¼ 7 and � ¼ 0:25)

GQBC TL GL NQ SRQ CES C-D

0.0 0.0460 0.0112 0.0763 0.0786 0.0679 0.0819 0.08720.1 0.0279 0.0556 0.0552 0.0559 0.0569 0.0635 0.06750.2 0.0198 0.0526 0.0331 0.0354 0.1247 0.1241 0.11810.3 0.0117 0.0354 0.0156 0.0167 0.1107 0.1137 0.11630.4 0.0065 0.0264 0.0083 0.0101 0.0994 0.1075 0.10990.5 0.0041 0.0212 0.0056 0.0083 0.0964 0.1046 0.10620.6 0.0028 0.0175 0.0042 0.0063 0.0912 0.1007 0.10140.7 0.0018 0.0139 0.0031 0.0049 0.0867 0.0988 0.09930.8 0.0010 0.0109 0.0021 0.0035 0.0831 0.0821 0.08290.9 0.0003 0.0078 0.0012 0.0029 0.0799 0.0790 0.07981.0 0.0000 0.0020 0.0010 0.0015 0.0772 0.0763 0.0768

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

MSE

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 1 (b). MSE of the ML estimator of (N ¼ 125, T ¼ 7 and� ¼ 0:25)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

Bia

s

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 2 (a). Bias in the estimate of the mean of TE predictor(N ¼ 125, T ¼ 7 and � ¼ 0:25)

Table 2 (a). Bias in the estimate of the mean of TE predictor (N ¼ 125, T ¼ 7 and � ¼ 0:25)

GQBC TL GL NQ SRQ CES C-D

0.0 70.1728 70.1301 70.1967 70.2007 70.3010 70.2121 70.23150.1 0.0595 0.1142 70.0217 70.0819 0.1415 0.1856 0.18790.2 0.0271 0.0923 70.1409 70.1517 0.1672 0.2493 0.25150.3 0.0081 0.0572 70.2052 70.1918 0.2013 0.2852 0.28710.4 0.0034 0.0363 70.2410 70.2216 0.2261 0.2101 0.21170.5 0.0021 0.0250 70.3016 70.2872 0.2304 0.3255 0.32740.6 0.0014 0.0213 70.2877 70.2921 0.2393 0.3387 0.33990.7 0.0010 0.0172 70.2931 70.3107 0.2465 0.3521 0.35180.8 0.0008 0.0151 70.3172 70.3182 0.2512 0.3592 0.35940.9 0.0005 0.0147 70.3355 70.3276 0.2527 0.3642 0.36541.0 0.0015 0.0135 70.3436 70.3363 0.2568 0.3718 0.3728

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of TE estimates will range from 70% to 87%, depending on

the specification, when in fact it is close to 100%. For the

range of values of 5 0:1, the pattern of bias for all

specifications is found to be similar to that observed for

, except that the bias in the estimate of TE predictor is of

the opposite sign to the bias in . In addition, it is observed

that the magnitude of the bias is very small when the

frontier is correctly specified. However, when the model

is misspecified, the bias can be fairly large. The largest

bias is found in the C–D specification, followed by the

CES, NQ, GL, SRQ and TL. For example, when

takes values between 0.5 and 0.9, the TE will be overesti-

mated by about 33% and 37% for the C–D and CES,

respectively, and will be underestimated by about 28%

and 34% for the GL and NQ, respectively. These findings

are consistent with the theoretical explanation provided in

Section II.

Table 2(b) and Fig. 2(b) show that, for all functional

specifications, the MSE for the technical efficiency predic-

tor traces the bias pattern of the estimated technical effi-

ciency predictor quite well, that is, the MSE rises (falls)

whenever the bias rises (falls). Interestingly, the variance

appears to be the main contributor to the MSE.

Next the results associated with the confidence intervals

of the technical efficiency predictor are reported. The

empirical coverage is computed based on whether the esti-

mated interval includes the true value of technical effi-

ciency, e�u, for confidence levels of 0.90, 0.95 and 0.99.

To preserve space, only the results for the confidence

level of 0.95 are reported (see Table 3 and Fig. 3). The

results indicate that when the production frontier is

misspecified, the empirical coverage accuracy of the

confidence intervals is, on average, significantly lower

than the theoretical confidence level. For GL and NQ,

the maximum coverage is approximately 30% at ¼ 0:4while for SRQ, CES and C–D, the relevant maximum

coverage is approximately 14%. When the production

frontier is correctly specified, the mean coverage accuracy

is very close to the theoretical confidence levels for values

of between 0.4 and 0.9. These results suggest that

there is a greater risk in drawing inference on technical

efficiency using confidence interval procedure than using

158 K. Giannakas et al.

Table 2 (b). MSE of the estimate of the mean of TE predictor (N ¼ 125, T ¼ 7 and� ¼ 0:25)

GQBC TL GL NQ SRQ CES C-D

0.0 0.0340 0.0179 0.0751 0.0753 0.0994 0.1214 0.09890.1 0.0440 0.0909 0.0772 0.0762 0.1171 0.1099 0.11060.2 0.0398 0.0484 0.1162 0.0413 0.1672 0.1519 0.16320.3 0.0383 0.0424 0.1678 0.1321 0.1997 0.1939 0.19020.4 0.0355 0.0389 0.2181 0.1963 0.2301 0.2128 0.21810.5 0.0321 0.0354 0.2588 0.2570 0.2417 0.2323 0.23250.6 0.0278 0.0314 0.2991 0.2973 0.2542 0.2422 0.24230.7 0.0217 0.0260 0.3459 0.3407 0.2635 0.2561 0.25560.8 0.0156 0.0209 0.3846 0.3779 0.2664 0.2676 0.26540.9 0.0090 0.0158 0.4190 0.4019 0.2692 0.2692 0.26781.0 0.0001 0.0098 0.4981 0.4513 0.2741 0.2780 0.2770

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

MSE

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 2 (b). MSE of the estimate of the mean of TE predictor(N ¼ 125, T ¼ 7 and � ¼ 0:25)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

95%

Cov

erag

e

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 3. 95% empirical coverage for the mean of TE predictor(N ¼ 125, T ¼ 7 and � ¼ 0:25)

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point predictors. This is especially true when is less than

0.4.

In addition to the bias and MSE of the estimates of

technical efficiency predictor and confidence interval, the

bias and MSE of the (labour) production elasticity are

also reported, as a secondary interest.8 These results are

displayed in Table 4(a), Fig. 4(a), Table 4(b) and Fig.

4(b). The bias in the (labour) production elasticity is

found to be significantly positive and constant for all values

of for the TL, SRQ, CES and C–D. For the GL and NQ

Predicting technical efficiency in stochastic production frontier models 159

8 To conserve space, the results for the other input elasticity are not reported. Similar patterns are found and the results are availableupon request.

Table 3. 95% empirical coverage for the mean of TE predictor ðN ¼ 125, T ¼ 7and � ¼ 0:25Þ

GQBC TL GL NQ SRQ CES C-D

0.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.1 0.6970 0.5376 0.0903 0.1013 0.0569 0.0490 0.3830.2 0.8760 0.7118 0.1856 0.1907 0.0847 0.0871 0.08540.3 0.9345 0.8353 0.2735 0.2842 0.1124 0.1276 0.12190.4 0.9442 0.8938 0.2968 0.2997 0.1317 0.1437 0.13970.5 0.9459 0.9217 0.2711 0.2811 0.1246 0.1375 0.13560.6 0.9464 0.9299 0.2208 0.2364 0.1071 0.1194 0.11610.7 0.9460 0.9410 0.1387 0.1542 0.0964 0.1073 0.10440.8 0.9450 0.9487 0.0540 0.0993 0.0921 0.1067 0.10420.9 0.9426 0.9538 0.0008 0.0821 0.0603 0.0954 0.09361.0 0.8962 0.9614 0.0001 0.0607 0.0597 0.0923 0.0906

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

Bia

s

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 4 (a). Bias in the estimate of labour production elasticityðN ¼ 125, T ¼ 7 and � ¼ 0:25Þ

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

*

MSE

GQBC TL GL NQ

SRQ CES C-D

g

Fig. 4 (b). MSE of the estimate of labour production elasticityðN ¼ 125, T ¼ 7 and � ¼ 0:25Þ

Table 4 (a). Bias in the estimate of labour production elasticity ðN ¼ 125, T ¼ 7 and� ¼ 0:25Þ.

GQBC TL GL NQ SRQ CES C-D

0.0 0.0006 0.1681 70.2271 70.2631 0.1817 0.1127 0.17860.1 0.0002 0.1677 70.2269 70.2602 0.1801 0.1096 0.17140.2 0.0000 0.1677 70.2269 70.2624 0.1797 0.1094 0.17120.3 0.0000 0.1678 70.2270 70.2681 0.1792 0.1093 0.17100.4 0.0000 0.1679 70.2271 70.2701 0.1789 0.1092 0.17080.5 0.0001 0.1681 70.2271 70.2714 0.1782 0.1091 0.17090.6 0.0001 0.1682 70.2272 70.2719 0.1779 0.1090 0.17080.7 0.0000 0.1683 70.2272 70.2729 0.1777 0.1089 0.17070.8 0.0000 0.1681 70.2773 70.2731 0.1775 0.1086 0.17060.9 0.0002 0.1675 70.2273 70.2734 0.1773 0.1085 0.17051.0 0.0004 0.1651 70.2275 70.2751 0.1770 0.1084 0.1703

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the pattern of bias is similar except in the opposite direc-tion. For the GQBC, however, there is virtually no bias forall values of .

The evidence of constant (absolute) bias might be due tothe inconsistency of the ML estimator. The largest bias (inabsolute values) is found in NQ, followed by GL, SRQ,C–D, TL and CES functional specification. The constantbias observed is also reflected in the MSE of the productionelasticity.

In summary, these results reveal that the imposition ofan inappropriate functional specification can cause severebias in the estimates of technical efficiency, their confidenceinterval and the production elasticities. Consequently, amisspecification of the production frontier can result inseriously misleading policy recommendations regardingefficiency improvements.

V. CONCLUDING REMARKS

This paper provides a theoretical explanation for the sen-sitivity of technical efficiency measures to the choice offunctional specification in stochastic production frontiermodels. An inappropriate specification of the productionfrontier can be thought of as a misspecification in the con-ditional mean of the stochastic frontier regression. It isshown that if the conditional mean is misspecified, the esti-mates of technical efficiency predictor and confidence inter-val will be biased, jeopardizing the validity of the resultsand, thus, the relevance of model predictions.

The extent of this bias is examined using Monte-Carloexperiments. Simulation results indicate that a misspecifi-cation in the conditional mean can cause severe bias in theestimates of technical efficiency, confidence interval andproduction elasticities. The magnitude of this bias is speci-fic to the functional form used to approximate the trueproduction technology. Since functional forms are bothdata and model specific, and since policy recommendationscan be drawn from studies of productive efficiency, diag-

nostic tests on an array of alternative functional specifica-tions become essential to identify the proper productiontechnology for the case under study.

At this point, it is important to recognize that all speci-fications considered in this paper are nested within theGeneralized Quadratic Box–Cox transformation function.In such a case, it is not actually a problem that differentfunctional forms give different efficiency measures since the(relative) statistical fitness of the alternative functional spe-cifications can be determined relatively easily by means ofstandard hypothesis testing.

A problem might arise however, when different func-tional specifications that result in different efficiency pre-dictions are statistically indistinguishable, i.e. the usualtesting or model selection procedures fail to favour onemodel over the other. Such a situation might arise due todifferences in the distributional assumptions about thecomposed error terms, for instance. In such a case, testssuch as Lee (1983) can be used to determine the validity ofthe half-normal (or truncated-normal) assumption on thetechnical inefficiency component and identify the appropri-ate formulation of the problem under study.

ACKNOWLEDGEMENTS

An earlier version of this article was presented at the CEA/CAES joint meeting in Vancouver, Canada, 1–4 June 2000,and at the NAPW, Union College, Schenectady, NY, 14–18 June 2000.

The authors wish to thank Robert Romain, WilliamGreene and the participants at these conferences for usefulcomments and suggestions. The usual caveats with respectto opinions expressed in the paper apply.

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Table 4 (b). MSE of the estimate of labour production elasticity ðN ¼ 125, T ¼ 7and � ¼ 0:25Þ

GQBC TL GL NQ SRQ CES C-D

0.0 0.0008 0.0290 0.0515 0.0552 0.0423 0.0136 0.03110.1 0.0007 0.0288 0.0514 0.0551 0.0416 0.0134 0.03080.2 0.0006 0.0287 0.0515 0.0553 0.0413 0.0133 0.03060.3 0.0005 0.0287 0.0515 0.0554 0.0410 0.0132 0.03050.4 0.0005 0.0287 0.0516 0.0558 0.0407 0.0132 0.03040.5 0.0004 0.0287 0.0516 0.0559 0.0403 0.0131 0.03040.6 0.0004 0.0287 0.0516 0.0561 0.0401 0.0131 0.03030.7 0.0003 0.0287 0.0516 0.0562 0.0401 0.0130 0.03030.8 0.0003 0.0286 0.0517 0.0562 0.0400 0.0130 0.03020.9 0.0002 0.0283 0.0517 0.0562 0.0399 0.0129 0.03021.0 0.0000 0.0275 0.0518 0.0565 0.0401 0.0127 0.0301

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