Efficiency Measurement with the Weibull Stochastic Frontier*

Transcription

1 OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 69, 5 (2007) doi: /j x Efficiency Measurement with the Weibull Stochastic Frontier* Efthymios G. Tsionas Department of Economics, Athens University of Economics and Business, Athens, Greece ( tsionas@aueb.gr) Abstract In this paper we consider the Weibull distribution as a model for technical efficiency. The distribution has a shape and scale parameter like the gamma distribution and can be a reasonable competitor in practice. The techniques are illustrated using artificial data as well as a panel of Spanish dairy farms. I. Introduction Beginning with Aigner, Lovell and Schmidt (1977) and Meeusen and van den Broeck (1977) stochastic frontier models with half-normal or exponential one-sided error terms have become popular in the measurement of production efficiency. Greene (1990) has proposed the gamma distribution as a plausible model, and Stevenson (1980) the truncated normal distribution. In production frontiers one would normally expect to find negative skewness in the composed error term. The standard way to generate negative skewness is to assume a positively skewed inefficiency distribution. However, it is often the case in practice that estimated production frontiers show positive skewness, which certain authors (Green and Mayes, 1991) interpret as evidence in favour of super-efficiency. Carree (2002) suggested an alternative explanation by pointing out that there are in fact negatively skewed distributions for the one-sided error term which are reasonable on a priori grounds and, naturally, generate positively skewed composed errors. One such distribution is the binomial and another is the Weibull. Carree (2002) points out that [i]n contrast to the *Many thanks are due to an anonymous referee for several helpful comments and suggestions on earlier versions of this paper and to Subal Kumbhakar for providing the data. JEL Classification numbers: C13, D Blackwell Publishing Ltd and the Department of Economics, University of Oxford, Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

2 694 Bulletin conclusion of super-efficiency in case of a positive skewness, the example of the binomial distribution shows that a positive skewness suggests a one-sided distribution that has low probabilities for small inefficiencies and high probabilities of large inefficiencies. (...) Hence, (...) a large fraction attains considerable inefficiencies (p. 105). 1 Carree (2002) proposes a simple method of moments estimator when the one-sided distribution is binomial but did not consider estimation under the assumption of a Weibull distribution for inefficiencies because this model is more complex. Considering the Weibull is important because the binomial is a discrete distribution and thus it is difficult to argue that it can be a reasonable model in practice. In this paper, we consider Bayesian analysis of the stochastic frontier model when the one-sided error follows a Weibull distribution. With the exception of Misra (2002) who uses maximum simulated likelihood to estimate the parameters, this distribution has not been analysed before because of computational problems. We show that this is, in fact, not a serious problem because computations can be performed using Markov chain Monte Carlo (MCMC) techniques, and we provide finite sample inferences on all parameters of the model, as well as predictive and actual inefficiencies. The model and inference techniques are presented in section II. An illustration using artificial data is presented in section III. An empirical application to Spanish dairy farms is discussed in section IV. II. The model and inference techniques The model considered in this paper applies to panel data and is given by the following y it = x itβ + v it + J Ψ t u i, (1) v it σ IN(0, σ 2 ), (2) p(u i c, θ) = cθu c 1 i exp( θu c i ), c, θ, u i > 0, (3) t = 1,..., T, i = 1,..., n. Here, T is the number of observations for firm i, x it a k 1 vector of explanatory variables, v it the measurement error, and the u i values independent non-negative random variables independent of v it with a Weibull distribution whose density is given by equation (3). Moreover, Ψ t 0 is a time-specific component that we parameterize as Ψ t = exp(η(t 1)), η a parameter representing the rate of change of technical inefficiency over time (Battese and Coelli, 1992), and J = 1 or +1 depending on whether we have a production or cost frontier respectively. Technical inefficiency 2 is given by Ψ t u i and is allowed to vary in a parametric way over time thus avoiding the 1 This discussion applies to cost frontiers as well. The composed error of cost frontiers must be positively skewed so a positively skewed one-sided distribution is often assumed. The finding of negative skewness in least squares residuals might be taken as evidence that the one-sided distribution is, in fact, negatively skewed. 2 For a detailed discussion of the problems arising in Bayesian cross-sectional stochastic frontiers, see Fernandez, Osiewalski and Steel (1997).

3 Efficiency measurement with the Weibull stochastsic frontier 695 assumption of time-invariance. 3 Furthermore, it is assumed that both v it and u i are independent of x it. (For more details on stochastic frontier analysis, see Kumbhakar and Lovell, 2000.) In equation (3), θ is a scale parameter and c a shape parameter. For c = 1 the distribution reduces to exponential. The distribution has a positive mode for c > 1, otherwise the mode is zero. The important feature of the distribution is that it exhibits negative skewness for c > Let y i = [y i1, y i2,..., y iti ] be the T i 1 vector of observations for the dependent variable of the i the firm, X i the associated T i k matrix of explanatory variables, and y and X denote the data in obvious notation. The joint probability density of y i is given by p(y i X i, β, σ, c, θ, η) = (2πσ 2 ) T/ 2 cθ 0 exp { 1 2σ 2 } T (y it x itβ J Ψ t u i ) 2 u c 1 i exp( θu c i )du i. (4) The integral in equation (4) is not available in closed form (unless c = 1), so the likelihood function n L(β, σ, c, θ, η; y, X ) = p(y i X i, β, σ, c, θ, η) i = 1 cannot be computed in closed form. For that reason, we treat the u i values as parameters, and integrate them out using Monte Carlo methods. The kernel of the posterior distribution augmented by the latent inefficiencies, u = [u 1,..., u n ], is given by { p(β, σ, c, θ, η, u y, X ) σ N (cθ) N exp { n i = 1 1 2σ 2 u c 1 i exp( θu c i ) n t = 1 i = 1 t = 1 } } T (y it x itβ J Ψ t u i ) 2 p(β, σ, c, θ, η) (5) where p(β, σ, c, θ, η) is the prior distribution and N = nt denotes the total number of observations. We will assume that p(β, σ c, θ, η) σ ( N + 1) exp( Q/2σ 2 ) where N and Q are parameters. In this prior we have Q/σ 2 χ 2 N, where χ2 ν denotes the chi-squared distribution with ν degrees of freedom, and p(β σ, c, θ, η) const. 4 For Q = N = 0 we obtain the familiar default prior p(β, σ c, θ, η) σ 1. The proposed prior for σ provides some flexibility relative to the default prior and is the conditionally conjugate prior (see equation 7 below). For the remaining parameters, we specify gamma priors of the form 3 It is possible to set η = 0 and T i = 1 so that we treat the panel as a cross-section. 4 The conditions for existence of the posterior from this model in spite of the improper prior on β are given in Fernandez et al. (1997).

4 696 Bulletin θ G(A, B), c G(C, D), p(η) const., (6) where G(a, b) denotes the gamma distribution with density f (x a, b) = ba Γ(a) xa 1 exp( bx) (a, b, x > 0). It should be noted that our numerical techniques allow us to specify an arbitrary prior distribution for parameter η. Based on the kernel posterior distribution we can use Gibbs sampling 5 to perform the computations. To that end, we extract the following conditional distributions: β σ, c, θ, η, u, y, X N k ( β(u, η), σ 2 (X X ) 1 ) (7) (ỹ(u, η) X β) (ỹ(u, η) X β) σ 2 β, c, θ, η, u, y, X χ 2 N (8) θ β, σ, c, η, u, y, X G p(c β, σ, θ, η, u, y, X ) c n + C 1 exp ( ( θ 2σ 2 Åi n + A, ) n u c i + B i = 1 ( n n )) u c i + c ln u i D i = 1 i = 1 (9) (10) p(u i β, σ, c, θ, η, y, X ) exp { (u i m Å } i ) 2 u c 1 i exp( θu c i ), u i > 0 (11) p(η β, σ, c, θ, u, y, X ) exp { 1 2σ 2 n i = 1 t = 1 T [y it x itβ J exp(η(t 1))u i ] }, 2 (12) where β(u, η) = (X X ) 1 X ỹ(u, η), and ỹ(u, η) is a vector whose elements are ỹ it (u, η) = y it J Ψ t u i. Moreover, m Å i = J T i t = 1 ζt 1 (y it x itβ), σ 2 C = σ2 Åi, i C i T C i = ζ 2(t 1) = 1 ζ2t and ζ = exp(η). 1 ζ 2 t = 1 The distributions in equations (6) (8) are well known, and random number generation from them is straightforward. For the distribution in equation (9) we use acceptance sampling from a gamma distribution, G(n + C, λ) where the scale parameter is given by λ = (n + C 1)/ c, and c solves the nonlinear equation 5 For Bayesian analysis in the context of stochastic frontiers, see van den Broeck et al. (1994), Koop, Osiewalski and Steel (1997), Koop, Steel and Osiewalski (1995), and Osiewalski and Steel (1998). For a detailed review of MCMC applications in econometrics, see Geweke (1999).

5 Efficiency measurement with the Weibull stochastsic frontier 697 θc n i = 1 ( n ) u c i ln u i c ln u i D (n + C 1) = 0. i = 1 This construction leads to the maximum acceptance rate relative to all gamma distributions with shape parameter n + C, and performed very well in practice giving an average acceptance rate close to 95%. The nonlinear equation is solved using bisection. The distribution in equation (11) is not in any known family but random number generation can be implemented as follows. Denote the mean of the distribution by μ i = 0 0 u i p(u i β, σ, c, θ, η, y, X )du i p(u i β, σ, c, θ, η, y, X )du i, for all i = 1,..., n. Both integrals must be computed numerically because they are not available in closed form. The distribution whose kernel density is given by equation (10) is approximated by a Weibull with shape parameter c and scale parameter λ i = (Γ(1 + c 1 )/ μ i ) so that the two distributions have the same mean. The density of the approximating distribution is g(u i c, λ i ) = cλ i ui c 1 exp( λ i ui c ). We use a Metropolis rule 6 to generate a draw as follows. If u (0) i is the current draw and u Å i denotes a candidate draw from the distribution whose density is g(u i c, λ i ), we accept the candidate draw with probability { min 1, p(uå i β, σ, c, θ, η, y, X )/ g(u Å } i c, λ i ), p(u (0) i β, σ, c, θ, η, y, X )/ g(u (0) i c, λ i ) else we set the draw to u (0) i. In practice, we scale the parameters λ i by the same factor, that is, we use Λλ i for some Λ > 0 which is chosen to ensure that the sampler does not accept or reject too often. 7 The distribution in equation (12) is also not in any known family. We use a Metropolis update where, given the current draw η (0), the candidate is drawn from η Å N(η (0), Φσ 2 ) and is accepted with probability { min 1, p(ηå } β, σ, c, θ, u, y, X ). p(η (0) β, σ, c, θ, u, y, X ) The constant Φ is adjusted to provide a reasonable acceptance rate. 8 Efficiency measurement in Bayesian stochastic frontier models follows the principles in Koop, Steel and Osiewalski (1995). Define r it = exp( Ψ t u i ) to be technical 6 For c > 1 the distribution is log-concave so specialized rejection techniques could have been used. These techniques are, unfortunately, computationally expensive when n is very large and this is the reason for not using them in this paper. Finally, for c = 1 we have an exponential distribution and the posterior conditional distribution is truncated normal, see Koop et al. (1995). 7 The constant Λ is chosen so that the acceptance rate is close to 25%. To implement numerical integration a 10-point Simpson s rule is used. 8 An alternative is to use a normal proposal centered at the least squares estimate of η derived from linearizing exp(η(t 1)) with respect to η. It has been found difficult to calibrate the constant Φ so that the acceptance rate is reasonable and this alternative has not been given further consideration.

6 698 Bulletin efficiency. The average firm-specific efficiency can be computed as a Monte Carlo average of the draws for r it values and the same principles can be followed to obtain inferences on time-invariant efficiency defined as exp( u i ). Other moments can be computed in the same way. The predictive efficiency distribution is the prior efficiency distribution given by equation (2) averaged with respect to posterior draws for the parameters c and θ. This distribution can be used to provide inferences about the efficiency level of a typical or as yet unobserved firm. III. Prior elicitation To elicit the parameters we determine the unconditional distribution of u.as p(u θ, c) = θcu c 1 exp( θu c ) and the priors are we have p(θ) = BA Γ(A) θa 1 exp( Bθ), p(u) = p(c) = DC Γ(C) cc 1 exp( Dc) p(u θ, c)p(θ)p(c)dθ dc. 0 0 The parameter θ can be integrated out analytically using properties of the gamma distribution, and we obtain p(u c) = cba Γ(A + 1) u c 1 Γ(A) (u c + B). (13) A + 1 The integral p(u) = 0 p(u c)p(c)dc can be computed using numerical integration, and one can also determine the density of r = exp( u) given as p r (r) = p( ln r)r 1. Using numerical integration 9 we can determine the mean of the distribution, and a point r such that E(r) = r rp r (r)dr, p r (r)dr = Using exponential priors on the parameters, i.e. A = C = 1 we can determine the parameters B and D that minimize the distance (E(r) E) 2 + ( r F) 2 where E and F 9 Numerical integration is implemented using 100-point Gaussian quadrature.

7 Efficiency measurement with the Weibull stochastsic frontier 699 TABLE 1 Prior elicitation of parameters B and D Target E(r) Target r B D Notes: The table provides the required values of parameters B and D in the prior distribution of θ and c defined in equation (6). The other prior parameters, A and C are set equal to one. The parameters B and D are computed for given values of E(r) which is average prior efficiency and r which is the lower 5% point of the prior distribution of technical efficiency. are target values for prior expected efficiency, and prior 5% efficiency. These values are reported in Table 1 for selected values of E and F. Therefore, based on the results of Table 1 we can elicit the parameters B and D by answering the question what is prior mean efficiency E(r) and what is a lower 5% bound r for technical efficiency? IV. Artificial data To make sure that the techniques perform well, a panel data set has been constructed with n = 100 units, T = 5 time periods and k = 2 regressors one of which is the intercept and the other is constructed as a standard normal random variable. The coefficients are β 1 = β 2 = 1, the error standard deviation is σ = 0.1, and the other parameters are θ = 10, η = 0.01 and c = 3.8. For this data set, the inefficiency distribution is negatively skewed so that the composed error term exhibits positive skewness. Moreover, the one-sided error component accounts for nearly 68% of the variability of the composed error term so the signal-to-noise ratio is quite large in this application, mitigating the effects of practical non-identification problems resulting from the fact that the shape parameter exceeds Based on the stated parameter values, average efficiency is about 61%. Regarding the prior, we have N = 1, Q = 10 3, A = 1, B = 0.3, C = D = 1. The prior mean and standard deviation of θ are 3.33, and for c they are equal to 1. These choices imply that average prior efficiency is 70% and r = 0.1 according to Table 1.

8 700 Bulletin Figure 1. Marginal posterior distributions

9 Efficiency measurement with the Weibull stochastsic frontier 701 Figure 2. Posterior predictive efficiency distribution for the artificial data We have performed 150,000 Gibbs iterations and the first 10,000 were discarded. Convergence was assessed using Geweke s (1992) diagnostic. The Metropolis sampler for the u i values has an average acceptance rate (across iterations and observations) close to 2.3%. The average acceptance rate for η was about 8.0%. Marginal posterior densities have been computed by skipping every other 10th draw to mitigate the impact of autocorrelation due to MCMC. The empirical results seem satisfactory, and marginal posterior distributions of the parameters are reported in Figure 1. Figure 2 presents the true efficiency distribution (straight line) and the posterior predictive efficiency distribution (for the first time period, i.e. t = 1). The true efficiency distribution is the distribution of exp( U) when U follows a Weibull distribution with c = 3.8 and θ = 10 while the posterior predictive has been computed by averaging the distributions of exp( U (m) ) where U (m) follows a Weibull distribution with parameters c (m) and θ (m), and m denotes the MCMC draw. The two distributions are quite close, indicating that reasonable inferences can be made based on the posterior predictive efficiency distribution. V. Empirical application Data on 80 Spanish dairy farms observed from 1993 to 1998 were used to illustrate the new techniques. These dairy farms are small family farms operated mostly by the family members. I use litres of milk as output and number of cows, kilograms of concentrates, hectares of land and labour (measured in man-equivalent units) as

10 702 Bulletin TABLE 2 Posterior moments for regression coefficients for the Spanish dairy data Mean SD Cows Conc Land Labour Time Notes: The table reports posterior mean and posterior standard deviations for the slope parameters of the Cobb Douglas stochastic production frontier applied to the Spanish dairy data. conc. stands for concentrate. inputs (see Alvarez, Arias and Kumbhakar, 2003 for details). I also used time as an additional regressor to capture technical change. I proceed with Bayesian computations using Gibbs sampling with 40,000 iterations, the first 20,000 of which were discarded to mitigate the impact of start-up effects. 10 To specify the prior, I assume θ G(1, 0.3), c G(1, 1) and 0.001/ σ 2 χ 2 1. These are the same choices that I made before in connection with the artificial data and imply that technical efficiency is approximately 70% on average and can be lower than 10% with a small probability This is a reasonable choice but I will also provide detailed sensitivity analysis. Posterior moments of parameters and functions of interest are reported in Table 2, and technical efficiency results are presented in Figure 3. Comparisons are made with an exponential and a gamma distribution for technical inefficiency. For the gamma distribution, we have f (u) = θc Γ(c) uc 1 exp( θu). For the exponential distribution we have c = 1 in the previous expression. Technical efficiency is very similar for the gamma and Weibull specifications but different compared with the exponential, so restricting attention to simple distributional assumptions can be dangerous. It seems that the extension to a Weibull or gamma is worthwhile in this application. Posterior results for the regression coefficients are reported in Table 2 and posterior moments for the various shape and scale parameters and technical inefficiency are reported in Table 3. Based on the posterior mean and standard deviation (SD) of η it is likely that technical inefficiency is time-invariant. Turning attention to the other parameters, the posterior mean of c for the Weibull specification is and the 10 Sensitivity analysis with respect to widely differing initial conditions has been performed and posterior distributions of important functions of interest have been computed and compared. These posteriors are indistinguishable suggesting that the Gibbs sampler converges fast. Convergence is diagnosed using Geweke s (1992) diagnostic. Results are available upon request. The average acceptance rate of the Metropolis step was about 27%.

11 Efficiency measurement with the Weibull stochastsic frontier 703 Figure 3. Technical efficiency measures posterior SD is For the gamma distribution, the posterior mean of the shape parameter is with posterior SD When c exceeds 3 the gamma distribution is nearly symmetric. Formal Bayesian model comparison is clearly necessary here because we need to compare alternative models for technical inefficiency. Given the likelihood function L(θ; Y ) where θ denotes the parameters and Y is the data and given a prior distribution p(θ) the marginal likelihood is m(y ) = L(θ; Y ) p(θ)dθ, and is simply the integrating constant of the posterior distribution. This quantity can be obtained using a Laplace approximation as described in Lewis and Raftery (1997). 11 This approximation requires only the posterior mean and posterior covariance matrix of the parameters which are available from the Monte Carlo simulation. Based on values of the log-marginal likelihood it turns out that the Weibull model performs best relative both to the gamma and exponential specifications. The Bayes factor in favour of the Weibull and against the gamma model is about 1.48 while the 11 The likelihood function cannot be obtained in closed form for the gamma and Weibull models. The latent inefficiency variables are integrated out using quadrature and the resulting approximate likelihood is used in conjunction with the Laplace approximation.

12 704 Bulletin TABLE 3 Posterior moments of other parameters for the Spanish dairy data Weibull Gamma Exponential Mean SD Mean SD Mean SD c η / σ θ FSI u LML Notes: The table reports posterior mean and posterior standard deviations of shape and scale parameters of the Weibull stochastic production frontier model applied to the Spanish dairy data. FSI is firm-specific inefficiency and is the average over all draws, firms and time periods, u denotes the time-invariant component of technical inefficiency, and LML is the log-marginal likelihood. TABLE 4 Sensitivity analysis for posterior means c and θ corresponding to 500 different priors E(c data) E(θ data) Minimum Maximum % interval Notes: Results correspond to 500 different priors. E(c data) and E(θ data) denote the posterior means of parameters c and θ. Minimum and maximum denote the minimum and maximum values of these posterior expectations across all 500 different priors. Bayes factor against the exponential and in favour of Weibull is overwhelming. Based on this evidence, the Weibull model seems to be a useful alternative to the gamma specification. Finally, I perform sensitivity analysis with respect to the prior. Recall that the priors are θ G(A, B) and c G(C, D). We will not consider changes in the prior of σ 2 so we have to account for changes in the four hyperparameters A, B, C, and D. 12 Our strategy is as follows. We generate 500 values for the hyperparameters from a uniform distribution in [0, 10] 4. For each of the 500 configurations of the hyperparameters we use the sampling-importance-resampling (SIR) algorithm of Rubin (1988) to transform the existing posterior sample to an approximate sample from the new posterior which results from the different prior. The size of the new sample is set to 10% of the size of the existing posterior sample. For each of the 500 different priors we obtain 500 sets of posterior draws for all parameters of the model. We focus on the 12 In practice it is hard to elicit the hyper-parameters of the prior for σ because this cannot be interpreted in an economic sense. Therefore, it seems reasonable to choose such hyper-parameters so that the resulting prior for σ is rather diffuse.

13 Efficiency measurement with the Weibull stochastsic frontier 705 posterior means of parameters c and θ, and some statistics are reported in Table 4. The changes in posterior means of these parameters across all 500 priors are found to be minor. The 95% interval for the posterior mean of c is (0.97, 2.98) and for parameter θ it is (5.87, 21.12). The posterior means corresponding to the baseline prior fall within these intervals and posterior standard deviations seem to be broadly consistent with the variation of posterior means across different priors. Average posterior predictive efficiency is almost invariant across all priors and is close to 78%. VI. Conclusions The paper presented efficient computational techniques for stochastic frontier models with Weibull-distributed one-sided error terms. These models allow for both positive and negative skewness and, in that sense, they are able to cope with Carree s (2002) criticism that usually employed models impose heavy structure on the data. We have illustrated the techniques in the context of artificial data, as well as an empirical application to Spanish dairy farms. Final Manuscript Received: November 2006 References Aigner, D. J., Lovell, C. A. K. and Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models, Journal of Econometrics, Vol. 6, pp Alvarez, A., Arias, C. and Kumbhakar, S. (2003). Empirical Consequences of Direction Choice in Technical Efficiency Analysis, manuscript, SUNY Binghamton, NY. Battese, G. E. and Coelli, T. J. (1992). Frontier production functions, technical efficiency and panel data: With application to paddy farmers in India, Journal of Productivity Analysis, Vol. 3, pp van den Broeck, J., Koop, G., Osiewalski, J. and Steel, M. F. J. (1994). Stochastic frontier models: a Bayesian perspective, Journal of Econometrics, Vol. 61, pp Carree, M. A. (2002). Technological inefficiency and the skewness of the error component in stochastic frontier analysis, Economics Letters, Vol. 77, pp Fernandez, C., Osiewalski, J. and Steel, M. F. J. (1997). On the use of panel data in stochastic frontier analysis with improper priors, Journal of Econometrics, Vol. 79, pp Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, in Bernardo J., Berger J., Dawid A. P. and Smith A. F. M. (eds), Bayesian Statistics 4, Oxford University Press, Oxford, pp Geweke, J. (1999). Using simulation methods for Bayesian econometric models: inference, development and communication (with discussion and rejoinder), Econometric Reviews, Vol. 18, pp Greene, W. H. (1990). A gamma-distributed stochastic frontier model, Journal of Econometrics, Vol. 46, pp Green, A. and Mayes, D. (1991). Technical inefficiency in manufacturing industries, Economic Journal, Vol. 101, pp Koop, G., Steel, M. F. J. and Osiewalski, J. (1995). Posterior analysis of stochastic frontier models using Gibbs sampling, Computational Statistics, Vol. 10, pp Koop, G., Osiewalski, J. and Steel, M. F. J. (1997). Bayesian efficiency analysis through individual effects: Hospital cost frontiers, Journal of Econometrics, Vol. 76, pp Kumbhakar, S. C. and Lovell, C. A. K. (2000). Stochastic Frontier Analysis, Cambridge University Press, Cambridge.

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