Empirical Economics (2007) 33:531–540DOI 10.1007/s00181-006-0116-z

O R I G I NA L PA P E R

A flexible time-varying specification of the technicalinefficiency effects model

Giannis Karagiannis · Vangelis Tzouvelekas

Received: 15 October 2005 / Accepted: 15 August 2006 / Published online: 16 January 2007© Springer-Verlag 2007

Abstract The temporal pattern of technical efficiency in the technical ineffi-ciency effects model, as modeled by Battese and Coelli (Empir Econ 20:325–332,1995), is rather restrictive. Specifically, it a priori imposes a common patternupon all firms in the sample, which in addition is monotonic over time. Obvi-ously this is an undesirable implication of the model especially when there isevidence of strong firm heterogeneity and/or a long time span. To overcomethis shortcoming, the present paper incorporates the Cornwell et al. (J Econom46:185–200, 1990) flexible specification of the temporal pattern of technical effi-ciency into technical inefficiency effects model. The proposed formulation isthen applied to the agricultural sector of the EU and US, during the period1973–1993. The empirical result support the proposed formulation as quitedifferent temporal patterns of technical efficiency have been found for the tencountries included in the analysis.

Keywords Technical inefficiency effects · Time-varying technical efficiency ·Agriculture · Panel data

We would like to thank an anonymous referee and an associate editor for valuable suggestions inan earlier version of the paper.

G. KaragiannisDepartment of Economics, University of Macedonia, 156 Str,Thessaloniki 540 06, Greecee-mail: karagian@uom.gr

V. Tzouvelekas (B)Department of Economics, Faculty of Social Sciences,University of Crete, University Campus, 74100 Crete, Greecee-mail: vangelis@econ.soc.uoc.gr

532 G. Karagiannis, V. Tzouvelekas

1 Introduction

The technical inefficiency effects model is perhaps the most widely used modelin stochastic frontier analysis.1 Its main advantage is that it can simultaneouslyprovide firm-specific estimates of technical efficiency scores and associate varia-tion in firm performance with variation in exogenous or conditioning variables(e.g., managerial ability, socioeconomic characteristics, ownership form, etc.)characterizing the environment in which production occurs. Another usefulaspect of the technical inefficiency effects model, apparent though only withpanel data, is that it permits the identification of the effects of technical changeand of time-varying technical efficiency, even if both are modeled via a simpletime trend (Battese and Coelli 1995).2 This is true as long as the inefficiencyeffects are stochastic and follow a truncated distribution. The distributionalinformation in the technical inefficiency effects model identifies the two effectsvia a non-linear specification of the mean of technical efficiency and a linearspecification of the mean of technical change.3

On the other hand, a shortcoming of the technical inefficiency effects modelseems to be the rather restrictive specification of the temporal pattern of techni-cal inefficiency, at least as modeled by Battese and Coelli (1995). In their speci-fication, the effect of the passage of time on technical inefficiency is necessarilymonotonic and whenever is time-varying, it may be either efficiency-enhancingor efficiency-impending, but not both (Wang 2002). This monotonicity assump-tion implies further that it would be the same for all firms in the sample.4

While the assumption that the temporal pattern of technical inefficiency is thesame for all firms is restrictive, it is not unreasonable for putty-clay industries(Kumbhakar et al. 1997). In contrast, in samples with strong firm heterogene-ity, it is likely that some firms will tend to improve their technical efficiency

1 Deprins and Simar (1989) first presented it in a deterministic setting, Reifschneider and Stevenson(1991) and Kumbhakar et al. (1991) extended it into a stochastic frontier framework for cross sectiondata, and Battese and Coelli (1995) developed the panel data version of the model.2 Since the pioneering work of Nishimizu and Page (1982), a great number of studies have consid-ered the effect of technical efficiency changes into productivity growth using either the Tornqvist orthe Malmquist index (see Kumbhakar and Lovell 2000, pp. 279–309; Färe et al. 1998 for extensivereviews). In that respect, what really matters is not the degree of technical efficiency per se but itschanges over time. Specifically, if technical efficiency is time invariant it makes no contribution toproductivity growth, whereas if is time-varying it affects productivity growth positively (negatively)when is associated with movements towards (away from) the production frontier. Cornwell et al.(1990) and Kumbhakar (1990) were the first to model time-varying technical efficiency. For a reviewof time-varying efficiency models see Kumbhakar and Lovell (2000, pp. 108–115).3 Without making distributional assumptions, there are two approaches to separate the effect oftechnical change from that of time-varying technical inefficiency. First, the Cornwell et al. (1990)efficient IV estimator may be used, which under selected orthogonality conditions can accomplishthis goal when the time trend represents (imbedded) technical change shared by all firms in theindustry. Second, the two effects may be modeled via different variables, as in Karagiannis et al.(2002), where the general index of Baltagi and Griffin (1988) is used to model technical change anda time trend is used to capture time-varying technical inefficiency.4 Even though all firms in the sample share the same temporal change in technical inefficiency, theinitial efficiency levels differ across firms.

Time-varying technical inefficiency and panel data 533

scores over time, others will tend to deteriorate them, and some will leave themunaffected. Even though all these outcomes are equally possible at the outset,it is impossible to take them into account appropriately with the specification ofthe temporal pattern of technical inefficiency used by Battese and Coelli (1995).

Nevertheless, the relative contribution of technical efficiency changes intoproductivity growth is non-monotonic because of its dependency on an adjust-ment function (defined as the ratio of the conditional to unconditional varianceof the one-sided error term), which differs across observations. That is, the rel-ative importance of technical efficiency changes as a source of growth differsacross firms. But since the adjustment function is always positive for the tech-nical inefficiency effects model (Wang 2002), the effect of technical efficiencychanges would be positive or negative according to the sign of the (estimated)time coefficient in the technical inefficiency effect function. And this sign isthe same for all observation in the sample. Thus, with the Battese and Coelli(1995) specification of the temporal pattern of technical inefficiency, the effectof technical efficiency changes into productivity growth is qualitatively similarfor all firms in the sample but it is quantitatively different.

The objective of this paper is to incorporate a flexible specification of time-varying technical efficiency into the technical inefficiency effect model. For thispurpose, the Cornwell et al. (1990) specification is used. Its main advantagesare that allow for firm-specific patterns of temporal variation in technical effi-ciency and more importantly, for testing for the existence of a common temporalpattern across firms. Consequently, the temporal specification of technical effi-ciency used by Battese and Coelli (1995) can be obtained as a special case.Moreover, it allows technical efficiency to vary through time employing a qua-dratic specification.5 Thus the proposed formulation attempts to combine theadvantages of the Cornwell et al. (1990) specification (i.e., analyzing flexibletemporal patterns of technical efficiency changes) with those of the technicalinefficiency effects model (i.e., explaining efficiency differentials).

The proposed formulation is used to analyze the temporal pattern of tech-nical efficiency for USA and nine European countries (i.e., Germany, France,Italy, Belgium, Netherlands, UK, Ireland, Denmark and Greece). The empiricalresults indicate that the evolution of technical efficiency in these countries hasbeen different during the period 1973–1993, and for only two countries (i.e.,Denmark and Greece) technical efficiency had been time invariant. This in turnmeans that technical efficiency cannot be considered as a source of growth forthese two countries during the period under consideration. On the other hand,it is found that technical efficiency changes have contributed positively to pro-ductivity growth in France, Italy, Ireland and USA and negatively in Germany,Netherlands, Belgium and UK. These quite different temporal patterns of tech-nical efficiency changes could not be captured by the Battese and Coelli (1995)specification.

5 Other specifications regarded as flexible (i.e., Lee and Schmidt 1993; Cuesta 2000) do not allowfor variation of (firm-specific) technical efficiency changes over time.

534 G. Karagiannis, V. Tzouvelekas

2 Empirical model

Consider the following translog production frontier:

yit = β0 + βTt + 12βTTt2 +

n∑

j=1

βjxjit + 12

n∑

j=1

l∑

k=1

βjkxjitxkit +n∑

j=1

βjtxjitt + eit,

(1)

where yit is the logarithm of the observed output produced by the ith firm atyear t, xjit is the logarithm of the quantity of the jth input used by the ith firmat year t, t is a time index that serves as a proxy for technical change, β is avector of parameters to be estimated after imposing symmetry (i.e., βjk = βkj),and eit = vit − uit is a stochastic composite error term. The vit term corre-sponds to statistical noise that is assumed to be independently and identicallydistributed, and the uit term is a non-negative random variable associated withtechnical inefficiency. It is further assumed that vit and uit are independently dis-tributed from each other. In the technical inefficiency effects model, uit, couldbe replaced by a linear function of explanatory variables reflecting firm-andtime-specific characteristics. Specifically,

uit = δ0 +M∑

m=1

δmzmi + ωit, (2)

where zml are farm- and time-specific explanatory variables associated withtechnical inefficiency; δ0 and δm (m = 1, . . . , M) are parameters to be esti-mated; and ωit is an independently and identically distributed with N(0, σ 2

u )

random variable truncated at −(δ0 + ∑

δmzmi)

from below. The latter impliesthat uit ∼ N

(δ0 + ∑

δmzmi, σ 2u)

truncated at zero from below.For the purposes of this paper, the inefficiency effect model is specified only

in terms of a simple time trend, although several demographic, socioeconomicetc. variables could have also been easily included. In particular, followingCornwell et al. (1990), (2) is specified as

uit = δi0 + δi1t + δi2t2 + ωit (3)

where δi0, δi1, and δi2 (i = 1, . . . , n) are firm-specific parameters to be estimated.Thus this specification allows for firm-specific patterns of temporal variation oftechnical efficiency and captures effects not visible in those models that assumea common pattern of technical efficiency. In addition, we can test (i) for theexistence of a common temporal pattern for all firms in the sample (i.e., δi1 = δ1and δi2 = δ2 for all i = 1, . . . , n), and (ii) the hypothesis of time-varying tech-nical efficiency for all or some of the firms in the sample (i.e., δi1 = δi2 = 0 forall i = 1, . . . , n). That is, it is possible to test the hypothesis of time invarianttechnical efficiency for each firm by performing tests over each of the δi1, and

Time-varying technical inefficiency and panel data 535

δi2. Moreover, the Battese and Coelli (1995) specification is viewed as a specialcase in the proposed formulation, which results as a nested model under thehypothesis that δi0 = δ0, δi1 = δ1 and δi2 = δ2 for all i = 1, . . . , n.

After substituting (3) into (1) the resulting model is estimated by a single-equation estimation procedure using the maximum likelihood method. Thevariance parameters of the likelihood function are estimated in terms of σ 2

s =σ 2

v + σ 2 and γ = σ 2/σ 2s , where σ 2 is the variance of the normal distribution

that is truncated at zero to obtain the distribution of uit and the γ -parameterhas a value between zero and one. Then, farm-specific estimates of the output-oriented measure of technical efficiency can be obtained from the conditionalexpectation of exp(−uit) given eit (Battese and Coelli 1988).

3 Empirical results

The empirical results are based on a data set developed recently by Ball etal. (2001). This data set contains multilateral data on agricultural output, land,labor, capital, and intermediate inputs for ten countries (Germany, France, Italy,Belgium, Netherlands, UK, Ireland, Denmark, Greece and USA) during theperiod 1973–1993. Details on data sources, definition and construction of allrelevant variables, as well as descriptive statistics are given in Ball et al. (2001).

The estimated parameters of the translog production frontier are presentedin Table 1. The first-order parameters (βj) have the anticipated (positive) signand magnitude (being between zero and one), and the bordered Hessian matrixof the first- and second-order partial derivatives is negative semi-definite indi-cating that all regularity conditions (namely, positive and diminishing marginalproducts) are valid at the point of approximation (i.e., the sample mean). Onthe other hand, the ratio-parameter, γ , is positive and statistically significant atthe 5% level of significance, indicating that the technical inefficiency is likelyto have an important effect in explaining output variability among farms in thesample. According to the estimated variances, output variability is mainly dueto technical inefficiency rather than to statistical noise.

Hypotheses testing regarding model specification are reported on Table 2.6

The null hypotheses that γ = δi0 = δi1 = δi2 = 0 and γ = 0 for all i are bothrejected at 5% level of significance indicating, respectively, that the technicalinefficiency effects are in fact present and stochastic in nature.7 Consequently,

6 The generalized likelihood-ratio test statistic, λ = −2{ln L(H0) − ln L

(H1

)}is used for these

purposes, where L(H0) and L(H1) denote the values of the likelihood function under the null (H0)

and the alternative (H1) hypothesis, respectively. If the given null hypothesis is true, λ has approxi-mately a chi-square distribution, except cases where the null hypothesis involves also γ = 0. In thiscase, the asymptotic distribution of λ is a mixed chi-square and the appropriate critical values areobtained from Kodde and Palm (1986).7 In the latter case, the variance of the inefficiency effects is zero and the model reduces to a tradi-tional response function, in which country-specific intercept terms and time variables are includedin the production function. Then, the parameters γ and δi0, δi1 and δi2 for one i cannot be identified.In our case, the critical value to test the null hypothesis is obtained from the χ2

(4)-distribution.

536 G. Karagiannis, V. Tzouvelekas

Table 1 Parameter estimates of the translog production frontier

Parameter Estimate SE Parameter Estimate SE

β0 0.0259 0.0107∗∗ ρG0 −0.3287 0.1090∗βC 0.0342 0.0323 ρF0 −1.6033 0.1326∗βA 0.4337 0.0777∗ ρI0 −1.6410 0.3435∗βL 0.1846 0.0474∗ ρN0 −1.1570 0.2743∗βE 0.3659 0.0910∗ ρB0 −0.2462 0.0955∗βT 0.4218 0.0183∗ ρK0 −0.7355 0.1976∗βCA 0.2219 0.0460∗ ρR0 −0.5868 0.1600∗βCL −0.0999 0.0470∗∗ ρD0 −0.9307 0.1657∗βCE −0.2179 0.1039∗∗ ρH0 −0.1005 0.0868βCT −0.0136 0.0192 ρU0 −0.9290 0.1922∗βCC −0.0219 0.0994 ρG1 1.1558 0.1100∗βAL −0.0376 0.0723 ρF1 −0.7375 0.1752∗βAE −0.0028 0.0489 ρI1 −0.7260 0.3417∗∗βAT −0.0350 0.0148∗∗ ρN1 0.5075 0.2129∗∗βAA −0.0018 0.0426 ρB1 2.1323 0.1484∗βLE −0.0300 0.0976 ρK1 −0.1724 0.3652βLT 0.0282 0.0137∗∗ ρR1 −0.9864 0.2353∗βLL 0.1109 0.0240∗ ρD1 0.2281 0.2519βET 0.0969 0.0211∗ ρH1 −0.0284 0.1652βEE 0.0537 0.0230∗∗ ρU1 −0.0633 0.2558βTT 0.0837 0.0050∗ ρG2 0.2280 0.0315∗

ρF2 −0.2685 0.0625∗ρI2 −0.0362 0.1519ρN2 0.2903 0.1432∗∗ρB2 0.7403 0.0619∗ρK2 −0.2131 0.0749∗ρR2 0.1901 0.0863∗∗

σ 2 0.0456 0.0053∗ ρD2 0.0049 0.1283γ 0.9980 0.0011∗ ρH2 −0.0213 0.0779Ln(θ) −357.967 ρU2 −0.0977 0.0358∗∗

Subscripts: A land, L labor, C capital, E intermediate inputs, T time trend G Germany, F France, IItaly, N Netherlands, B Belgium, K UK, R Ireland, D Denmark, H Greece, U USA*(**) indicates statistical significance at the 1 (5) % level

most of the countries in the sample operates below the production frontierand thus, a significant part of output variability among them is explained bythe existing differences in the degree of technical efficiency. As a result, thetraditional average production does not seem to be an adequate representationthe production technology. More importantly, rejection of the above hypothe-ses implies that technical change can be separated from time-varying technicalinefficiency even though both are modeled via a simple time trend.

Concerning now the temporal variation of technical efficiency, the resultsin Table 2 indicate that the common pattern specification (i.e., δi1 = δ1 andδi2 = δ2 for all i) used in Battese and Coelli (1995) is rejected at 5% levelof significance.8 This implies that the countries considered in the analysis

8 In addition, the hypothesis that the sample countries share a common temporal pattern of tech-nical efficiency, along with a common intercept in the inefficient effect model, is rejected at 5%level of significance (see Table 2).

Time-varying technical inefficiency and panel data 537

Table 2 Model specification test

Hypothesis LR-test Critical value (α=0.05)

γ = δi0 = δi1 = δi2 = 0 ∀ i 369.8 χ2(31)

= 43.19a

γ = 0 25.9 χ2(4)

= 8.76a

δi0 = δ0 ∧ δi1 = δ1 ∧ δi2 = δ2 ∀ i 77.6 χ2(30)

= 43.77

δi1 = δ1 ∧ δi2 = δ2 ∀ i 52.7 χ2(20)

= 31.41

δi1 = δi2 = 0 ∀ i 42.7 χ2(20)

= 31.41

δG1 = δG2 = 0 24.6 χ2(2)

= 5.99

δF1 = δF2 = 0 34.3 χ2(2)

= 5.99

δI1 = δI2 = 0 26.5 χ2(2)

= 5.99

δN1 = δN2 = 0 43.5 χ2(2)

= 5.99

δB1 = δB2 = 0 21.8 χ2(2)

= 5.99

δK1 = δK2 = 0 18.1 χ2(2)

= 5.99

δR1 = δR2 = 0 33.1 χ2(2)

= 5.99

δD1 = δD2 = 0 2.9 χ2(2)

= 5.99

δH1 = δH2 = 0 0.7 χ2(2)

= 5.99

δU1 = δU2 = 0 26.6 χ2(2)

= 5.99

δi2 = 0 ∀ i 30.7 χ2(10)

= 18.31

Subscripts: G Germany, F France, I Italy, N Netherlands, B Belgium, K UK, R Ireland, D Denmark,H Greece, U USAa Critical values are taken from Kodde and Palm (1986, Table 1)

have followed different patterns of temporal variation in technical efficiency.Consequently, the contribution of technical efficiency changes into productivitychanges is expected to vary across countries, both in terms of its direction andmagnitude. This is an expected result since there is evidence of strong heteroge-neity in the sample. Ball et al. (2001) have documented substantial differencesamong the sample countries in output produced, capital-labor and land-laborratios, as well as changes of input quantities over time. The most remarkabledifferences are reported for land and the less significant for labor. Furthermore,the patterns of change for labor input bear little resemblance to those of land,capital and intermediate inputs, which increased in both absolute and relative(to US) terms.

The hypothesis of time invariant technical efficiency (i.e., δi1 = δi2 = 0 forall i) is also rejected at 5% level of significance, when all countries are consid-ered as a whole (see Table 2). However, the picture changes significantly whenthe relevant test is conducted on a country basis. The results reported in Table 2indicate that technical efficiency is found to be time invariant for Denmark andGreece, while it is time varying for the rest of the countries in the sample. Thus,for Denmark and Greece, technical efficiency changes cannot be considered asa source of productivity changes. The estimated values of the δi1 and δi2 parame-ters (see Table 1) imply that technical efficiency changes contributed positivelyto productivity growth in France, Italy, Ireland and US, whereas they negativelyaffected productivity in Germany, Netherlands, Belgium and UK. On the other

538 G. Karagiannis, V. Tzouvelekas

Table 3 Technical efficiency scores and technical efficiency change estimates, 1973–1993 (averagevalues)

Country Technical efficiency Technical efficiency change(average value) (average annual growth rate)

Germany 94.5 −1.06France 97.1 1.65Italy 95.6 1.28Netherlands 96.7 −0.13Belgium 93.8 −1.72UK 95.0 −0.24Ireland 92.9 0.88Denmark 90.8 0.00Greece 93.3 0.00USA 95.9 0.18

hand, the hypothesis that technical efficiency varies through time (i.e., δi2 = 0for all i) cannot be rejected at 5% level of significance (see Table 2). From thestatistical significance of the estimated δi2 parameters reported in Table 1, itcould be seen that, with the exception of Italy, this is true for all countries thatexhibited time-varying technical efficiency.

By taking only statistically significant parameters into account and followingBattese and Broca (1997), the annual rate of change in technical efficiency maybe calculated as

∂uit

∂t= −[δi1 + 2δi2t]

⎡

⎣1 − 1σu

⎧⎨

⎩φ(

µitσu

− σu

)

(

µitσu

− σu

) −φ(

µitσu

)

(

µitσu

)

⎫⎬

⎭

⎤

⎦, (4)

where σ 2u =µ2

it[ (−ρ)]−1{2 − [ (−ρ)]−1}/4+σ 2[ (−ρ)]−1{π−[ (−ρ)]−1}/2π

(Kumbhakar and Lovell 2000, p. 84) µit = (δ0 + ∑δmzmi), ρ = µit/σ , π ≈ 3.14,

and φ(·) and Φ(·) represent, respectively, the density and the cumulative den-sity function of the standard normal random variable. The results are pre-sented in Table 3 along with averages of technical efficiency over the 1973–1993period. From the countries that exhibited time varying technical efficiency,France achieved the faster improvement in efficiency and Belgium the fasterdeterioration. The corresponding slower changes in technical efficiency haveoccurred in the US and the Netherlands. It can also be seen from the resultsin Table 3 that, with the exception of Ireland, the countries, which on averageachieved relatively lower efficiency scores, exhibited either efficiency deterio-ration or time invariant efficiency. In contrast, with the exception of the Nether-lands, the countries that on average achieved relatively higher efficiency scores,exhibited efficiency improvements.

Time-varying technical inefficiency and panel data 539

4 Concluding remarks

During the last 15 years or so, an increasing number of empirical studies haveconsidered the effect of technical efficiency changes into productivity growthusing either parametric or non-parametric methods. The apparent advantage ofemploying the parametric approach in such studies is the capability of testingseveral statistical hypotheses concerning the existence and the magnitude of thevarious sources of productivity changes. Among other things, there is generalcohesion that in samples with strong firm heterogeneity and long time span itis undesirable to model the contribution of technical efficiency changes intoproductivity changes as being the same across firms and/or invariant over time.

This provided the motivation for incorporating the Cornwell et al. (1990)flexible time-varying specification of technical efficiency into the widely usedtechnical inefficiency effects model. In the form used by Battese and Coelli(1995), the technical inefficiency effects models is perhaps the best alterativeavailable for simultaneously explaining efficiency differentials and separatingtechnical change from technical efficiency changes, but it has the disadvantageof imposing the same temporal pattern of technical efficiency for all units inthe sample. In contrast, the proposed formulation allows for firm-specific pat-terns of temporal variation in technical efficiency and more importantly, fortesting for the existence of a common temporal pattern across firms and of timeinvariant technical efficiency. The empirical result presented above supportthe proposed formulation as quite different temporal patterns of technical effi-ciency have been found in the agricultural sector of the ten countries included inthe analysis. Two of them (i.e., Denmark and Greece) exhibited time invarianttechnical efficiency; four (i.e., France, Italy, Ireland and USA) improved theirefficiency over the period 1973–1993, while four others (i.e., Germany, Nether-lands, Belgium and UK) deteriorated their performance in terms of technicalefficiency.

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