Group-specific stochastic production frontier models with parametric specifications

European Journal of Operational Research 200 (2010) 508–517

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

Group-specific stochastic production frontier models with parametric specifications Young Hoon Lee * Sogang University, Sinsoo-dong 1, Mapo-gu, Seoul 121-742, South Korea

a r t i c l e

i n f o

Article history: Received 15 November 2007 Accepted 13 January 2009 Available online 30 January 2009 Keywords: Group-specific temporal pattern Panel data Stochastic frontiers Time-varying technical efficiency

a b s t r a c t This paper develops a stochastic frontier model that not only focuses more on group-specific temporal variations in technical efficiency rather than individual temporal variations, but also allows for a parametric function of the time-varying coefficient of the efficiency factor. We derived the concentrated least squares estimator and its asymptotic properties. When applied to the Penn World data set, the groupspecific models yield much more variation in the temporal patterns of efficiency across countries. This application demonstrates the feasibility of applying a group-specific stochastic frontier model with a parametric function of temporal pattern to a real empirical analysis.  2009 Elsevier B.V. All rights reserved.

1. Introduction Stochastic frontier models are popular econometric tools for estimating the technical efficiency of individual firms. Time-varying models allow us to estimate not only differences in technical efficiency between firms, but also the temporal variations in efficiency of each firm. Schmidt (1986), Cornwell and Schmidt (1996), Greene (1997), and Kumbhakar and Lovell (2000) provided reviews of the literature. However, in many applications, the main purpose of analysis is to compare temporal variations in efficiency between groups, but not between individual firms. For example, Good et al. (1995) examined the performance of European and American airlines to analyze the effects of deregulated market structures on individual carriers. Kim and Lee (2006) compared temporal changes in efficiency across several regions. Their study examined how the productivities of East Asian countries such as Japan, Korea, and Taiwan have changed over a long period. They conjectured that the efficiency levels of East Asian countries may have changed in a different manner from those of countries in other regions. Recently, Lee (2006) developed a stochastic frontier model with group-specific temporal variations in technical efficiency (henceforth, ‘group-specific model’) by extending findings Lee and Schmidt (1993, LS). This model allows group-specific patterns of temporal change in technical inefficiency to be analyzed without imposing any specific form of temporal pattern. The group-specific pattern means that firms from different groups are allowed to have different temporal patterns, but that firms from the same group are restricted to having the same pattern. Therefore, this model is suitable for the aforementioned applications. Moreover, it is reasonable to conjecture that firms in the same industry share similar temporal patterns of efficiency, but firms from different industries have different temporal patterns. When applying a panel dataset including firms from various industries, this group-specific model would be more suitable than time-varying models with an assumption of identical temporal variations in efficiency, such as LS. Stochastic frontier models have been applied extensively to agricultural industries. Farm-level efficiency is influenced by geographical characteristics. Therefore, farms in the same region, again, could have similar temporal patterns of efficiency, but farms from different regions could have different temporal patterns. The group-specific model allows each region’s temporal patterns of efficiency to be estimated separately, as well as each farm’s efficiency level. However, the asymptotic property of the group-specific model developed by Lee (2006) applies only when N ? 1 with a fixed T. Thus, the panel data in which the number of firms is small and the number of time-series observations per firm is relatively large cannot be applied to this group-specific model. Like LS, this model does not impose any specific form of temporal pattern. This is an advantage theoretically because it nests the specifications of the temporal patterns proposed by Kumbhakar (1990) and Battese and Coelli (1992, BC).1 * Tel.: +82 2 705 8772; fax: +82 2 704 8599. E-mail address: [email protected] 1 The specification of temporal pattern of technical inefficiency in LS nests Kumbhakar (1990) and BC if we consider them in the fixed effects approach. LS developed a minimum chi-square statistic to test both the Kumbhakar and the BC specifications. Since the group-specific model is a general version of the LS model, its specification also nests the specifications of Kumbhakar (1990) and BC. 0377-2217/$ - see front matter  2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.01.030

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

509

However, applications using unrestricted temporal patterns have occasionally yielded temporal patterns of efficiency that seem unreasonably variable. In this paper, we present a group-specific model with a time-varying parametric function of the temporal pattern of efficiency. The assumption of parametric function reduces the number of parameters relevant to temporal pattern of efficiency to a certain fixed number, and then the panel dataset with large time-series observations can also be applied to this model. Moreover, the assumption reduces the possibility that temporal variations in efficiency are estimated to be unreasonably variables due to the effect of smoothing. BC and Kumbhakar (1990) proposed specific parametric functions of temporal variations in efficiency. When the exponential functional form of BC is assumed, this group-specific specification is a generalization of BC as well as a restriction of Cuesta (2000) which allows for individual-specific temporal variations in technical efficiency. Han et al. (2005) extended Ahn et al. (2001) to allow a parametric function for time-varying coefficients of the individual effects. This provides a fixed-effects treatment of models proposed by Kumbhakar (1990) and BC. They present several of the generalized methods of moment estimators and compare their performance in the empirical application of data from Spanish savings banks. Like Han et al. (2005), this paper also considers a stochastic frontier model with a parametric function of temporal variations in efficiency and a fixed-effects treatment. We begin the presentation of our research methodology with a discussion of our proposed model, along with previous time-varying models. An empirical analysis and the econometric results follow. The conclusions that we derived from the results of our findings are also included. 2. Model The time-varying stochastic frontier production model is defined by

yit ¼ at þ xit b þ v it  uit ¼ xit b þ ait þ v it ;

ð1Þ

where i indexes firms or production units and t indexes time periods. yit is the dependent variable that represents the logarithm of output for firm i (i = 1, . . . , N) in the period t (t = 1, . . . , T), xit is 1  k vector of inputs, b is a k  1 vector of coefficients, and vit is an i.i.d. N(0, r2). The variable uit is the nonnegative technical inefficiency term for firm i in period t. The time-varying parameter at is the frontier intercept term at time t (no overall intercept is included in b). Accordingly, ait = at  uit represents firm i’s efficiency level at time t. Note that uit P 0, so ait 6 at. This is a standard setup. In the case that the ait (or equivalently, uit) are ‘‘fixed effects”, the number of parameters (NT + K) exceeds the number of observations. Therefore, different time-varying models have emerged as different choices for the form of ait (or equivalently, uit) for the same purpose of reducing a number of parameters. LS proposed a flexible alternative but with the assumption of a common temporal pattern in technical inefficiency across different firms as follows:

yit ¼ xit b þ ht ai þ v it ;

i ¼ 1; . . . ; N; t ¼ 1; . . . ; T:

ð2Þ

The ai and ht represent an individual firm’s efficiency and temporal variations in efficiency, respectively. Since the ht are treated as parameters, the number of the relevant parameters is (T  1) (note the normalization of h1 = 1). This model also assumes an identical temporal pattern of efficiency across different firms since h has only t but not i as a subscript. On the other hand, BC and Kumbhakar (1990) considered the models with specific parametric functions:

yit ¼ xit b þ ht ðgÞai þ v it ;

i ¼ 1; . . . ; N; t ¼ 1; . . . ; T:

ð3Þ

The distinctive feature of the regression Eq. (3) is the interaction between the time-varying parametric function ht(g) and the individual effect ai. Specifically, Kumbhakar (1990) considered the case of ht(g) = [1 + exp(g1t + g2 t2)]1 and BC considered the case of ht(g) = exp (g(t  T)). Both of these models assume not only random effects, but also specific one-sided distributions for the ai and then provide likelihood functions for estimation. The identical temporal assumption imposed by LS, BC, and Kumbhakar (1990) may be unrealistic, especially when a panel dataset includes relatively long time-series, since it implies that ranking of every firm’s technical efficiency is fixed over long sample periods. Therefore, Cuesta (2000) extended BC in the way to relax the identical temporal assumption: ht (gi) = exp(gi(t  T)). Since each firm has a different parameter representing temporal pattern of efficiency, gi, ranking of an individual firm’s efficiency may fluctuate over time. This model is particularly useful when it is applied to a panel dataset with a small N and a large T since the consistency of gi estimator requires the condition of T ? 1 with a fixed N. Recently, Lee (2006) proposed a group-specific model by relaxing the identical temporal pattern assumption imposed on the model of (2). This model allows for group-specific temporal patterns of efficiency as follows:

yit ¼ xit b þ hgt ai þ v it

g ¼ 1; 2; . . . ; G;

and i 2 Group g:

ð4Þ

PG

Here, Group g has Gg number of firms so that N ¼ g¼1 Gg . Therefore, the number of parameters relevant to the temporal pattern of efficiency is G(T  1) (note again the normalization of hg1 = 1,g = 1, . . . , G). The purpose of this paper is to apply the BC model (and likewise the Kumbhakar (1990) model) to allow for group-specific temporal variations in efficiency and to provide a fixed effects treatment of the model. Then the model of (3) becomes

yit ¼ xit b þ ht ðgg Þai þ v it ;

i ¼ 1; . . . ; N; t ¼ 1; . . . ; T;

ð5Þ

where ht(gg) = exp(gg(t  T)) from BC and ht (gg) = [1 + exp(g1gt + g2gt2)]1 from Kumbhakar (1990). Therefore, the group-specific model with the BC specification proposed in this paper is a restricted version of the Cuesta (2000) specification in the fixed effect treatment (gi = gj = gg, "i,j 2 group g). This restriction reduces the number of parameters relevant to the temporal pattern of efficiency. The parametric function in the group-specific model of (5) includes only G parameters in the extension of BC or in the restriction of Cuesta (2000). Since G is fixed, the consistency of gg estimator is not hurted as N ? 1.

510

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

This model requires a priori information in order to compose firms into groups. However, we can test the preassigned grouping. The hypothesis of gg = gh for g – h can be tested to see whether Groups g and h have an identical temporal pattern of technical inefficiency. We can test not only the identical temporal pattern assumption of BC by testing the hypothesis of g1 = g2 =    = gG, but also time-invariance hypothesis of any specific group, gg = 0. 3. Estimation The direct generalization of the BC model is the model of (5) with ht(gg) = exp(gg(t  T)). We write the T observations for firm i as

yi ¼ X i b þ hðgg Þai þ v i ;

ð6Þ 0

where yi = (yi1, . . . , yiT)0 , Xi = (Xi1, . . . , XiT)0 , vi = (vi1, . . . , viT)0 , and h(gg) is a vector of [h1(gg),h2(gg), . . . , hT (gg)] . In this paper, we modify the parametric function to ht(gg) = exp(gg(t  1)), following Han et al. (2005). This parametric function is virtually the same as the specification from BC, but allows h1(gg) = 1, which is comparable to the normalization of Lee (2006), hg1 = 1, g = 1, . . . , G. We will now consider the fixed-effects case, where ai is treated as a parameter. The advantage to considering fixed effects is that consistency does not hinge on the assumption that the inputs are uncorrelated to efficiency. It also does not depend on the distribution of the technical inefficiency as in considering it fixed, but simply proceeds conditionally from whatever the realizations may be. Estimation of this model is nontrivial. As Lee (2006) derived the concentrated least squares (CLS) estimator as the solution to an eigenvalue problem, we provide the first-order conditions of CLS and its asymptotic properties. Unlike Lee (2006), we do not necessarily need to treat T as fixed, since the number of parameters relevant to ht (gg) is fixed and independent of T. Here, ‘‘asymptotic” means as N ? 1. Ahn et al. (2001, ALS), considered GMM to obtain a more efficient estimator for the LS model and Han et al. (2005) extended ALS to allow a parametric function for time-varying coefficients of technical inefficiency from the BC specification. They compared several GMM estimators, along with the CLS estimator and the ML estimator of the BC model by using data from Spanish saving banks. The comparison results, along with the simulation results of ALS, in small samples, suggest CLS performs almost as well as the more efficient GMM. Even though, we derive only the CLS estimator in this paper, the GMM approach adopted by ALS and Han et al. (2005) can also be applied to this groupspecific model. Since we cannot assert the standard asymptotic results (consistency and asymptotic normality) for least squares or MLE because of the ‘‘incidental parameters problem,” (i.e., the number of a’s grows with sample size; Neyman and Scott, 1948; Chamberlain, 1984), we have to establish them directly. However, they comprise a straightforward extension to LS and Lee (2006). We transform Eq. (6) by multiplying the idempotent matrix Mg = IT  Pg, where Pg = h(gg)[h (gg)0 h(gg)]1h(gg)0

Mg yi ¼ M g X i b þ M g v i :

ð7Þ

By this transformation, the effects (ai) have been removed since Mgh(gg) = 0. We also define the sum of squared errors (SSE) associated with Eq. (7):

SSE ¼

X

ðyi  X i bÞ0 Mg ðyi  X i bÞ:

ð8Þ

i

Like Lee (2006), we define our estimates of b and gg as the values that minimize SSE as given in Eq. (8). By taking derivatives of Eq. (8) with respect to b and (g1, . . . , gG), the first order conditions are obtained as N X @SSE X 0i M g ðyi  X i bÞ ¼ 0; i 2 Group g and g ¼ 1; 2; . . . ; G; ¼ 2 @b i¼1 2 3 X X dhðgg Þ @SSE 2 0 0 4 ¼ e hðgg Þei  ei Pg ei hðgg Þ5 ¼ 0; dgg @ gg hðgg Þ0 hðgg Þ i2Gg i i2Gg

ð9Þ

ð10Þ

^ which is a function of g ^ g as where ei = yi  Xib. From Eq. (9), we derive the solution b,

^¼ b

X i

!1 b g Xi X 0i M

X

! b gy : X 0i M i

ð11Þ

i

^ and g ^ g by using Eqs. (11) and (8). Instead of minimizing the objective function (8) with We cannot simplify Eq. (10), but we can calculate b ^ minimize the ^ and g ^ g , the numerical minimization basically follows these two steps: (i) with any consistent initial value of b, respect to b ^ g from (i) to the solution of Eq. (11). Then, the two steps should be ^ g ; (ii) substitute value of g objective function (8) with respect to only g iterated until convergence is achieved. The GAUSS code of this numerical iteration is available upon request to the author. By using results of optimization estimators from Amemiya (1985), the estimates are consistent and asymptotically normal under the ^ and regularity conditions, as listed in Lee (1991, pp. 14–17). These asymptotic properties are applied when N ? 1 for b g^ g ð8g ¼ 1; . . . ; GÞ. Note that Gg ? 1 because of a fixed number of groups. The covariance matrix of the estimates is not the form of standard expression as

pffiffiffiffi ^0 ; g ^1 ; . . . ; g b G Þ0  ðb0 ; g1 ; . . . ; gG Þ0 ! Nð0; A1 BA1 Þ; N½ðb

ð12Þ

^ and g ^ g ð8g ¼ 1; . . . ; GÞ, and B is the probwhere A is the probability limit of the matrix of second-order derivatives of SSE with respect to b ability limit of the matrix of the cross-product of first derivatives of SSE (see Appendix A for the specific forms of A and B). Here, A is not equal to B. Certain hypotheses about hg are of interest. The most obvious is the hypothesis that all gg are equal, in which case this model is reduced to the BC model. Following LS and Lee (2006), who developed test statistics similar to Gallant (1985), the generalized likelihood ratio (LR) test is derived as

511

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

LR ¼

SSER  SSEU ; r^ 2

ð13Þ

where SSEU can be the SSE of Eq. (8), and where SSER is the same as SSE, but estimated, and calculated under the restriction that g1 = g2 =    = gG (i.e., from the BC model). This statistic is asymptotically distributed as chi-square with (G  1) degrees of freedom. We can also test the hypotheses that parts of gg ("g = 1, . . . , G) are equal to each other, for example, gg = gh or gg = gh = gf. In the case of gg = gh, SSER is SSE of the regression model in which Groups g and h are combined into one group. Another example of a hypothesis that may be of interest is offered by the following: gg = 0, which is the time-invariance hypothesis of the first group. Suppose G = 2, so that there are only two groups and we test g1 = 0. Therefore, h(g1 = 0) = eT, where eT is a vector of ones. In this restricted model, P 1 ¼ eT e0T =T and P2 = h(g2)[h (g2)0 h(g2)]1h(g2)0 . By substituting these idempotent matrices into Eq. (7), we can obtain the estimates of b and g2 through the aforementioned iteration method, and then calculate SSER. 4. Empirical results The dataset used to compare growth and efficiency among the countries was derived from the Penn World Tables of Summers and Heston (1991) over the period 1965–1990. The descriptive statistics are presented in Table 1. Forty-nine countries are included in the statistics: four from Africa, eight from North and Central America, six from South America, 11 from Asia, 18 from Europe, and two from Oceania. This dataset was previously analyzed by Kim and Lee (2006), who applied the group-specific stochastic frontier model of Lee (2006). Their main focus was on the temporal variations in total factor productivity in East Asian countries. However, our motivation is not to undertake a serious analysis of economic growth and productivity changes, but rather to examine how our group-specific model performs with a real dataset. Therefore, we are more interested in analyzing how the various stochastic frontier models compare. For empirical analysis, a translog stochastic frontier production function is assumed to specify the technology in countries, as follows:

ln yit ¼

X j

aj ln xjit þ

XX j

bjl ln xlit ln xjit þ v it þ ht ðgg Þ ai j; l ¼ L; K; trend;

ð14Þ

l

where y is the real GDP and (L, K, trend) = (labor, capital stock, time trend). To categorize the 49 countries into a number of groups, we followed the method of Kim and Lee (2006) and divided them into four groups: G6 countries (Japan excluded); Western countries; East Asian countries; and the remaining countries, including Middle Eastern, South American, and African countries. The data from the four groups correspond to 6, 16, 6, and 21 countries, respectively. However, we will check the plausibility of this grouping through several hypothesis tests. First, we adopted the minimum chi-square (MCS) statistic to test the parametric function proposed by BC via use of the Lee (2006) model. The top rows of Table 1 include the test results. The MCS statistic is 74.44 of which the p-value is 0.399. Thus, the exponential form of restriction implied by BC cannot be rejected at a 10% level when each of the four groups is allowed to have different parameters in the exponential function. Therefore, the dataset supports the BC specification in the group-specific stochastic frontier model. Next, we tested several hypotheses in order to finalize our grouping. Table 2 also includes various test results from using both Lee (2006) model and this group-specific model with the BC specification. When we apply Lee (2006), the test results suggest three groups: Group 1 (G6 + Western countries), Group 2 (East Asian countries), and Group 3 (the other countries). The first hypothesis of the LS model in which all countries in the four groups have identical temporal variations in technical efficiency is rejected at a 1% level. The second hypothesis is that countries except the East Asian countries have identical temporal variations in efficiency. The fourth and fifth hypotheses are that the third group and the fourth group, and the second group and the fourth group, have identical temporal patterns. All of these hypotheses are strongly rejected at the 1% level. On the other hand, the fourth hypothesis that G6 and Western countries share the same temporal pattern of efficiency cannot be rejected. Its LR statistic is only 16.23 and the p-value is 0.908. However, our group-specific model with the BC specification suggests that all four groups have different temporal patterns since all five hypotheses are rejected at the 1% level. Since the BC parametric function is not rejected at the earlier test, we keep in mind that our final choice of regression is the result from this group-specific model with four groups. The parameter estimates for the production frontiers are presented in Table 3. The eight different models are used for comparison. SS is the estimator of the time-invariant model of Schmidt and Sickles (1984). LS and BC are already denoted above. G-LS1 shows the estimators of the group-specific Lee (2006) model with the four groups while G-LS2 shows the estimators of the same model with the three groups. G-BC1 and G-BC2 are the estimators of the model this paper proposes with the four groups and the three groups, respectively. Cuesta is the estimator of the firm-specific Cuesta (2000) model which is identical to G-BC with 49 groups.

Table 1 Descriptive statistics of the output and inputs. G6

Western

East Asia

Other

GDP

Mean SD Maximum Minimum

994.28 1090.73 4520.22 170.49

90.28 64.57 373.34 11.50

245.51 403.90 1770.41 12.51

79.94 149.49 1073.79 2.57

Labor

Mean SD Maximum Minimum

35.45 31.27 122.93 7.66

5.28 4.69 24.31 1.01

22.22 22.85 78.25 1.92

16.88 54.91 331.93 0.44

Capital

Mean SD Maximum Minimum

937.56 938.26 4266.25 141.23

95.39 67.59 386.60 7.72

297.82 589.09 2854.69 12.95

57.60 98.32 645.93 1.75

512

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

Table 2 Hypothesis test results. Hypothesis

MCS statistic

BC specification

1. h1 = h2 = h3 = h4 2. h1 = h2 = h4 3. h1 = h2 4. h3 = h4 5. h2 = h4 6. g1 = g2 = g3 = g4 7. g1 = g2 = g4 8. g1 = g2 9. g3 = g4 10. g2 = g4

Degrees of freedom

p-Value 0.399

74.44

72

LR statistic

Degrees of freedom

p-Value

292.92 117.92 16.23 269.82 75.61 270.55 87.66 14.23 264.00 54.83

75 50 25 25 25 3 2 1 1 1

0.000 0.000 0.908 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 3 Production function estimation results. LSa

SS

ln L ln K Trend (ln L)b (ln K)b (Trend)b ln L ln K ln L trend ln K trend

G-LS1a

G-LS2a

BC

G-BC1

G-BC2

t-Stat.

Coef.

t-Stat.

Coef.

t-Stat.

Coef.

t-Stat.

Coef.

t-Stat.

Coef.

t-Stat.

Coef.

t-Stat.

Coef.

t-Stat.

0.167 1.162 0.131 0.046 0.034 0.020 0.058 0.001 0.014

0.42 5.39 0.96 1.67 3.16 2.58 2.14 0.10 1.83

0.044 1.222 0.714 0.009 0.033 0.002 0.037 0.047 0.074

0.19 9.40 6.03 0.55 3.75 0.06 1.69 3.55 7.77

0.804 1.623 0.188 0.036 0.049 0.002 0.052 0.065 0.047

2.95 10.70 2.54 1.82 5.66 0.17 2.32 7.15 6.02

0.640 1.546 0.259 0.027 0.047 0.006 0.052 0.058 0.048

2.41 10.45 3.60 1.36 5.49 0.48 2.32 6.41 6.07

0.439 1.463 1.346 0.026 0.037 0.017 0.027 0.014 0.069 0.017

1.67 8.71 4.08 1.49 4.04 0.92 1.19 1.03 7.73 10.80

0.938 1.690 0.048 0.064 0.046 0.039 0.031 0.060 0.040 0.002 0.003 0.038 0.012

0.722 1.586 0.124 0.053 0.042 0.041 0.029 0.049 0.041 0.002

2.66 10.57 1.95 2.70 4.89 5.97 1.30 5.89 5.50 1.18

0.896 0.576 0.326 0.033 0.014 0.0002 0.079 0.019 0.009

2.27 2.04 8.68 0.84 1.17 1.72 2.39 5.81 5.35

0.036 0.012

7.13 8.10 11.608

4.190

0.650 0.377 1.027

0.195 0.288 0.093

Obj. function

16.070

14.059

10.369

10.522

14.769

3.40 11.10 0.73 3.24 5.24 5.65 1.39 6.83 5.55 0.97 1.99 7.18 8.51 11.472

eL eK

0.399 0.463 0.862

0.454 0.464 0.918

0.651 0.381 1.032

0.668 0.377 1.044

0.455 0.464 0.918

0.634 0.379 1.013

g1 g2 g3 g4

RTS a b

Cuesta

Coef.

The estimates of h or hg are not reported, but they are available upon request. The 49 estimates of gi are not reported, but they are available upon request.

The results of SS, LS, and BC are not substantially different other than the t-values of some coefficients. The trend coefficient of SS is significant while the trend-squared is not, but the opposite results occur in LS and BC. The coefficient of (lnLlnK) is significant in SS but not in LS and BC. This implies that the time-varying models with the identical temporal pattern assumption do not lead to substantially different estimates while leading to different t-values of several coefficients, as compared to the time-invariant model. The coefficient estimates from the four group-specific models are similar to each other. Minor differences are in the t-values. For example, the trend-squared is significant in G-BC1 and G-BC2, whereas it is not in G-LS1 and G-LS2, and the opposite case occurs for the coefficient of (lnLlnK). However, substantial differences exist not only in t-values and estimates, but also in signs of estimates between the group-specific models and the other models. The coefficient of labor variable is estimated to be positive in SS, LS, and BC, while it is negative in the G-LSs and the G-BCs. Moreover, a larger number of production function coefficients are significant in the group-specific models than in the other models. There is no substantial difference between the estimates of G-LS1 and G-LS2 as well as between the estimates of G-BC1 and G-BC2. This indicates the production function estimates are not sensitive to how countries are categorized. The production function is also estimated by the Cuesta (2000) model, but the estimates are substantially different from others. As discussed above, the Cuesta model includes N parameters relevant to temporal pattern of technical inefficiency. Therefore, the estimates may be imprecise when the panel data sample has large cross-sectional observations relative to time-series observations as our sample is the case (49 countries over 26 years). Overall, the group-specific models lead to substantially different regression coefficients as compared to SS, LS, and BC, but there is no substantial difference within the group-specific models. To check whether the estimated production functions are consistent with theoretical and observed relations between outputs and inputs, the elasticity of inputs and returns to scale are calculated, as presented in the bottom rows of Table 3. The group-specific models estimate labor elasticity at 0.634–0.667, which is consistent to the measures of labor compensation (Krueger, 1999). The returns to scale estimates (1.013–1.044) of the group-specific models are also consistent with the previous theoretical and empirical findings (Basu and Fernald, 1997). On the other hand, the restricted models provide decreasing returns to scale measures and labor elasticity estimates lower than 0.5. In particular, the SS model estimates the least returns to scale at 0.862 as well as the least labor elasticity at 0.399. The elasticity measures of the Cuesta model are erratic. Again, this reflects implausibility of this model to this panel data set with a large N and small T. Generally, as the hypothesis tests in Table 2 indicate, the group-specific models provide input elasticity estimates which are more consistent to theoretical and observed findings than the restricted models.

513

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517 Table 4 Average technical efficiency comparison.

Group Group Group Group

1 2 3 4

Total

SS

LS

G-LS1

G-LS2

BC

G-BC1

G-BC2

Cuesta

0.613 0.353 0.286 0.269

0.736 0.532 0.366 0.404

0.837 0.726 0.413 0.486

0.847 0.746 0.420 0.497

0.587 0.341 0.262 0.267

0.844 0.703 0.406 0.471

0.858 0.725 0.416 0.485

0.292 0.033 0.079 0.036

0.340

0.482

0.598

0.612

0.330

0.584

0.485

0.072

Fig. 1. Temporal pattern of average technical efficiency.

^ it follows the same method used by Lee (2006): We now turn to the measurement of technical efficiency. The separation of ûit from a

a^ t ¼ max ht ðg^ g Þa^ i ;

ð15Þ

i

^ and the inefficiency term uit is then estimated as ^ i ¼ ½hðg ^ g Þ0 hðg ^ g Þ1 hðg ^ g Þ0 ei ðbÞ, where a

^ t  ht ðg ^i; ^ g Þa ^ it ¼ a u

8i 2 Group g:

ð16Þ

Since the dependent variable is expressed in natural log form, the technical efficiency scores are calculated from Eq. (16) as

^ t  ht ðg ^ i Þ: ^ it Þ ¼ exp½ða ^ g Þa Tb E it ¼ expðu

ð17Þ

Table 4 reports the group average as well as total average of the estimated efficiency levels. For SS, LS, and BC, we obtained a mean efficiency score of approximately 33–48%. On the other hand, the scores are substantially higher for the four group-specific models in the range of 49– 61%. The fact that the average efficiency scores of G-LSs are approximately equal or slightly greater than those of G-BCs is intriguing because completely unrestricted specifications of LS and Lee (2006) have been known to yield unreasonably variable temporal patterns of efficiency. Note that the technical efficiency measure herein is a relative concept, and thus, the higher the variance, the smaller the average efficiency. However, all seven models except Cuesta yield identical rank of average efficiency scores by group: Group 1 (G6 countries) has the highest average efficiency scores and Group 2 (Western countries) has the second highest average among the four groups. Fig. 1 displays the yearly average of the estimated efficiency scores. LS and BC present upward-sloping efficiency curves while the group-specific models have a more or less constant efficiency curve. The results of LS and BC imply that the efficiency gap between the most efficient country and the rest of the countries has been reduced on average over time, while those of the group-specific models imply that the efficiency gap has been more or less constant over time even though year-to-year fluctuations have occurred. In other words, LS and BC draw convergence of efficiency across different countries of different levels of economic maturity while the group-specific models do not.2 This empirical difference may have significant implications for growth studies even though we will not discuss such applications further as we are interested not in growth studies, but rather in comparing different models. Fig. 2 displays the temporal variation in average technical efficiency of the four groups. As expected from the nature of the model, the time-invariant SS model derives a constant pattern of efficiency and both LS and BC yield an identical temporal pattern for the four groups. The average efficiency score is increasing over time for LS and BC. However, the group-specific models yield different temporal patterns of average efficiency scores for each group. According to G-LS1, the average efficiency levels of Group 1 are more or less constant over time, but those of Group 2 are slightly decreasing; thus the gap between the two groups slightly widens as t approaches T. Group 3 (East Asian countries) is the least efficient group in the initial period, but its efficiency level has climbed fast enough to catch and surpass the efficiency level of Group 4 and to close the gap with Group 2. GBC1 yields almost the same results as G-LS1. Table 5 reports the average of Spearman’s rank correlation coefficients between the efficiency levels estimated using the seven alternative models. Since the ranks of efficiency vary over time for the group-specific models, the rank correlations are not the same for t = 1, 2, . . . , T. Therefore the numbers in Table 5 are averages of the yearly rank correlations. SS has high rank correlations over 0.9 with

2

We, hereinafter, will not discuss the estimates of Cuesta (2000) since they are already found unreasonable in Tables 3 and 4.

514

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

Fig. 2. Temporal pattern of group-average technical efficiency.

Table 5 Spearman rank correlation coefficients.

SS LS G-LS1 G-LS2 BC G-BC1 G-BC2 Cuesta

SS

LS

G-LS1

G-LS2

BC

G-BC1

G-BC2

Cuesta

1 0.93 0.812 0.796 0.980 0.836 0.827 0.794

1 0.931 0.922 0.972 0.946 0.943 0.622

1 0.996 0.864 0.993 0.994 0.462

1 0.853 0.986 0.992 0.431

1 0.884 0.878 0.708

1 0.996 0.508

1 0.484

1

LS and BC, but relatively low correlations with the four group-specific models. The rank correlation coefficients of BC with all four groupspecific models are also below 0.9. However, LS, which assumes identical temporal variation in efficiency across different countries as does BC, has all of the rank correlation coefficients over 0.9, indicating the advantage of its flexible specification. The efficiency score of any estimator tends to be correlated highly with the efficiency scores for other estimators in the group-specific models. All of the relevant correlation coefficients are over 0.99. This implies that choosing a specific estimator among the group-specific models is not necessarily a crucial issue for this particular dataset when ranking countries in terms of their efficiency levels. Table 6 presents efficiency levels and ranks over time for some individual countries. The top portion of the table gives the efficiency levels for the countries with median efficiency levels in each time period. Again, due to the nature of the SS, LS, and BC models, the same country has the median efficiency in all time periods. SS and BC estimate Chile and LS selects Morocco as the countries with the median

515

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517 Table 6 Technical efficiency of the median and efficiency rank of selected countries. T

SS

LS

G-LS1

G-LS2

BC

G-BC1

G-BC2

Cuesta

1

Chile (0.311)a Chile (0.311) Chile (0.311) Chile (0.311) Chile (0.311) Chile (0.311)

Morocco (0.431) Morocco (0.436) Morocco (0.464) Morocco (0.499) Morocco (0.551) Morocco (0.582)

Ireland (0.655) Ireland (0.646) Ireland (0.643) Guatemala (0.627) Yugoslavia (0.590) Yugoslavia (0.574)

Chile (0.668) Finland (0.657) Finland (0.650) Guatemala (0.639) Portugal (0.613) Israel (0.617)

Chile (0.238) Chile (0.267) Chile (0.296) Chile (0.327) Chile (0.357) Chile (0.388)

Finland (0.590) Ireland (0.618) Guatemala (0.620) Chile (0.600) Yugoslavia (0.571) Mexico (0.572)

Finland (0.594) Ireland (0.640) Sweden (0.641) Chile (0.617) Yugoslavia (0.599) Israel (0.616)

Thailand (0.059) Chile (0.039) Taiwan (0.026) Austria (0.016) Austria (0.009) Hong Kong (0.006)

Finland 1 6 11 16 21 26

30b 30 30 30 30 30

29 29 29 29 29 29

24 24 24 22 21 20

24 25 25 22 21 20

30 30 30 30 30 30

25 24 23 22 22 23

25 24 22 22 21 21

33 33 33 34 34 35

Taiwan 1 6 11 16 21 26

39 39 39 39 39 39

36 36 36 36 36 36

39 37 37 35 34 23

40 37 37 35 34 24

38 38 38 38 38 38

39 37 35 35 33 22

39 37 36 35 33 24

35 30 25 23 20 20

6 11 16 21 26

a b

The numbers in () are efficiency scores. Rank of efficiency level.

efficiency level in every year. It is not reasonable to assume that every country maintains the same efficiency rank over 26 years. On the other hand, the country with median efficiency is different in every period for the group-specific models. For example, G-LS1 estimates Ireland with median efficiency in the early periods (t = 1, 6, 11), but it estimates Yugoslavia as a median efficiency country in the late LS

SS 1 0.8 0.6 0.4 0.2 0

0.8 0.6 0.4 0.2 0

1 3 5 7 9 11 13 15 17 19 21 23 25 UK

Ireland

Korea

1 3 5 7 9 11 13 15 17 19 21 23 25 UK

Keyna

BC

Ireland

Korea

Keyna

G-LS1

0.8

1 0.8 0.6 0.4 0.2 0

0.6 0.4 0.2 0

1 3 5 7 9 11 13 15 17 19 21 23 25

1 3 5 7 9 11 13 15 17 19 21 23 25 UK

Ireland

Korea

UK

Keyna

Ireland

Korea

Keyna

Cuesta

G-BC1 0.5

1 0.8 0.6 0.4 0.2 0

0.4 0.3 0.2 0.1

1 3 5 7 9 11 13 15 17 19 21 23 25 UK

Ireland

Korea

Keyna

0

1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526 UK

Ireland

Korea

Fig. 3. Technical efficiency of the selected countries from each group.

Keyna

516

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

periods (t = 21, 26). According to G-BC1, the median efficiency country is Finland in t = 1, Ireland in t = 6, and Mexico in t = 26. Specifically, G-BC1 estimates three countries from each of Group 2 and Group 4 in the six different time periods as the countries with median efficiency, while G-LS1 selects five countries from Group 2 and one country from Group 4. The variations in median efficiency country are possible due to the flexibility of the group–specific models. Table 6 also presents the efficiency ranks in each period for selected countries. We chose Finland with the median efficiency in t = 1 in GBC1, the model best fitted to this dataset according to our various test results. The efficiency rank of SS, LS, or BC is either 29 or 30 throughout the sample period, but its ranks fluctuate moderately over time in the group-specific models. G-BC1 estimates that Finland is ranked at 25 in t = 1 based on efficiency level. Its efficiency rank rises continuously to 22 in t = 16 and 21, but falls slightly back to 23 in t = 26. The rest of the group-specific models estimate the rank of Finland slightly differently with respect to temporal variation. LS-1 and BC-2 show the efficiency rank to decline slightly and continuously, while LS-2 yields the rank as rising in the earlier period but as declining continuously from t = 11. Finland belongs to Group 2 (Western countries), which shows moderate efficiency temporal fluctuation. We chose Taiwan from Group 3 (East Asian countries), which has shown dynamic economic growth. SS, LS, and BC estimate the efficiency rank of Taiwan at 39, 36, and 38, respectively. Again, it is not reasonable to assume that the countries that experienced rapid economic growth for a long time period like Taiwan maintained their efficiency ranks constantly over the sample period. However, the four group-specific models yield large temporal variations in efficiency ranks, especially in the later periods. G-BC1 estimates its rank at 39 in t = 1 for SS, LS, and BC, but shows the rank rising consistently to 35 in t = 11, 33 in t = 21, and 22 in t = 26. The rest of the group-specific models yield the same results. Fig. 3 displays the time pattern of estimated technical efficiency for randomly selected countries, one from each of the four groups (the UK from Group 1, Ireland from Group 2, Korea from Group 3, and Kenya from Group 4). The estimated efficiency scores of the four selected countries from SS are constant and those of LS and BC increase over time, but in the same pattern by construction. G-LS1 and G-BC1 yield significantly different temporal patterns of efficiency for the selected countries. The estimated efficiency increases slightly for the UK and stay more or less constant for Ireland and Kenya, and increases continuously for Korea. Even though the overall results from G-LS1 and GBC1 are similar, Fig. 3 displays some meaningful differences. For example, the estimate of G-LS1 implies that the UK experienced a rise of efficiency level in the earlier period, but a fall after t = 3. In addition, its efficiency level experienced up and down fluctuations followed by a quick rise after t = 10, but stayed constant in the latter half of the sample period. However, the estimate of G-BC1 shows that the efficiency score of UK rose rapidly in the earlier period and stayed constant after t = 10. This difference occurs because of the smoothing effect of the exponential parametric function adopted in G-BC1. 5. Concluding remarks In this paper, we considered a stochastic frontier model that allows not only for group-specific temporal patterns of technical efficiency, but also for a parametric function of the temporal pattern. This model is a straightforward extension of the BC model in the way in which it allows for a group-specific parameter in the exponential function. In other words, it is a restriction of Cuesta (2000) in the way in which it assumes that firms from the same group have the same firm-specific parameter. Unlike the Lee (2006) model, which requires a dataset with large N and small T, this model is also useful for a panel dataset with large T. We treat the individual effects as ‘‘fixed”. Following Lee (2006), the concentrated least squares estimator is developed along with its asymptotic properties. We also apply this stochastic frontier model along with other previously introduced models to the measurement of efficiency by using the Penn World data. Our empirical results can be summarized as follows. First, the four group-specific models (G-LS1, G-LS2, G-BC1, and G-BC2) generate similar production function estimates for this panel dataset, but somewhat different estimates from the rest of the models (SS, LS, BC and Cuesta). Nonetheless, our specification tests suggest the G-BC1 model to be the most useful for this sample data. Second, the estimated efficiency scores are substantially different across different models with respect to not only average measures but also individual temporal patterns of efficiency. However, the four group-specific models generate similar estimates of efficiency score, indicating that the efficiency measures are not sensitive to grouping and to the assumption of h (unrestricted parameter or parametric function). This insensitivity to the assumption of h is expected because the parametric function of G-BCs cannot be rejected and the specification of G-LSs nests that of G-BCs. A somewhat unexpected finding is that G-LSs with completely unrestricted specifications of h, which has been known to yield unreasonably variable efficiency measures, present quite stable efficiency scores. Third, our overall empirical results show the group-specific models yield much more variation in temporal patterns of efficiency across countries. This application demonstrates that group-specific stochastic frontier models with a parametric function of temporal pattern can feasibly be applied in a real empirical data analysis. Acknowledgements The author acknowledges the editorial assistance of D. Kim. This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-B00278). Appendix A. The covariance matrix of k = (b, g1, . . . , gG) The probability limit of the matrix of second-order derivatives of SSE is

2 2

AðkÞ ¼ lim E N!1

1 @ SSE N @k@k0

!

6 6 6 6 ¼6 6 6 6 6 6 6

Q X1

QX

0

2



ðhðg1 Þ d1 Þ Q 1 ½d1 d1  hð g Þ0 hðg Þ

   0

0

0

1

0

1

Q X2

...

0

...

3

Q XG

7 7 7 7 0 ... 0 7; 7 7 7 0 7 h i 0 gG Þ0 dG Þ2 7 7 . . . . . . 0 Q G dG dG  ðhð hðg Þ0 hðg Þ 0

G

G

517

Y.H. Lee / European Journal of Operational Research 200 (2010) 508–517

where N 1X X 0i M g X i ; N!1 N i¼1 1X 0 ¼ lim X i M g dg ai N!1 N i2G

Q X ¼ lim Q Xg

g ¼ 1; 2; . . . ; G;

g

Q g ¼ lim

N!1

1X 2 a N i2G i

g ¼ 1; 2; . . . ; G;

g

dg ¼

dhðgg Þ dgg

g ¼ 1; 2; . . . ; G:

The probability limit of the matrix of cross-product of first derivatives of SSE is

2 6 6 6     6 6 1 @SSE @SSE  ¼6 BðkÞ ¼ lim E 6 N!1 N @k @k0 6 6 6 6 6 6

QX   

h

2

Q 1 þ NhðgG1Þr0 hðg 1

1

Q X1 ih i 0 g1 Þ0 d1 Þ2 d1 d1  ðhð Þ hðg Þ0 hðg Þ 1

0

 0

0

1

Q X2

...

0

...

Q XG

3

7 7 7 7 7 0 ... 0 7: 7 7 7 7 0 h ih i7 0 ðhðgG Þ0 dG Þ2 7 GG r2 ...... 0 Q G þ Nhðg Þ0 hðg Þ dG dG  hðg Þ0 hðg Þ 7 G G G G 0

References Ahn, S.C., Lee, Y.H., Schmidt, P., 2001. GMM estimation of linear panel data models with time-varying individual effects. Journal of Econometrics 101, 219–255. Amemiya, T., 1985. Advanced Econometrics. Havard University Press, Cambridge, Mass. Basu, S., Fernald, J.G., 1997. Returns to scale in U.S. production: Estimates and implication. Journal of Political Economy 105, 249–283. Battese, G.E., Coelli, T.J., 1992. Frontier production functions, technical efficiency and panel data with application to paddy farmers in India. Journal of Productivity Analysis 3, 153–169. Chamberlain, G., 1984. Panel data. In: Griliches, Z., Intriligator, M. (Eds.), Handbook of Econometrics, vol. 2. North-Holland. Cornwell, C., Schmidt, P., 1996. Production frontiers and efficiency measurement. In: Matyas, L., Sevestre, P. (Eds.), The Econometrics of Panel Data. Kluwer Academic Publishers., Norwell, MA. Cuesta, R.A., 2000. A production model with firm-specific temporal variation in technical inefficiency: With application to Spanish Dairy Farms. Journal of Productivity Analysis 13, 139–149. Gallant, R., 1985. Nonlinear Statistical Models. John Wiley & Sons, New York. Greene, W.H., 1997. Frontier production functions. In: Pesaran, M.H., Schmidt, P. (Eds.), Handbook of Applied Econometrics, Micro-Economics, vol. 2. Oxford: Blackwell Publishers, Ltd.. Good, D.H., Roller, L.H., Sickles, R.C., 1995. Airline efficiency differences between Europe and the US: Implications for the pace of EC integration and domestic regulation. European Journal of Operational Research 80, 508–518. Han, C., Orea, L., Schmidt, P., 2005. Estimation of a panel data model with parametric temporal variation in individual effects. Journal of Econometrics 126 (2), 241–267. Kim, S.H., Lee, Y.H., 2006. The productivity debate of east asia revisited: A stochastic Frontier approach. Applied Economics 38, 1697–1706. Krueger, A.B., 1999. Measuring labor’s share. American Economic Review 89, 45–51. Kumbhakar, S.C., 1990. Production frontiers, panel data, and time-varying technical inefficiency. Journal of Econometrics 46, 201–211. Kumbhakar, S.C., Lovell, C.A.K., 2000. Stochastic Frontier Analysis. Cambridge University Press, Cambridge. Lee, Y.H., 1991. Panel data models with multiplicative individual and time effects: Application to compensation and frontier production functions. Unpublished Ph.D Dissertation, Michigan State University. Lee, Y.H., 2006. A stochastic production Frontier model with group-specific temporal variation in technical efficiency. European Journal of Operational Research 174, 1616– 1630. Lee, Y.H., Schmidt, P., 1993. A production frontier model with flexible temporal variation in technical inefficiency. In: Fried, H., Lovell, C.A.K., Schmidt, P. (Eds.), The Measurement of Productive Efficiency: Techniques and Applications. Oxford University Press, Oxford. Neyman, J., Scott, E.L., 1948. Consistent estimates based on partially consistent observations. Econometrica 16, 1–32. Schmidt, P., 1986. Frontiers production function. Econometric Reviews 4, 289–328. Schmidt, P., Sickles, R.C., 1984. Production frontiers and panel data. Journal of Business and Economic Statistics 2, 367–374. Summers, R., Heston, A., 1991. The Penn World Table (mark 5): An expanded set of international comparisons, 1950–1988. Quarterly Journal of Economics 106, 1–41.