OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 69, 5 (2007) 0305-9049 doi: 10.1111/j.1468-0084.2007.00475.x Efficiency Measurement with the Weibull Stochastic Frontier* Efthymios G. Tsionas Department of Economics, Athens University of Economics and Business, Athens, Greece (e-mail: [email protected]) 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 nor- mally 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 evi- dence in favour of ‘super-efficiency’. Carree (2002) suggested an alternative ex- planation 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, D24. 693 © Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2007. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. 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 distribu- tion 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 assump- tion 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 distribu- tion 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 infer- ences 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 = + v + yit xit it J tui,(1) 2 vit | ∼ IN(0, ), (2) | = c−1 − c p(ui c, ) c ui exp( ui ), c, , ui > 0, (3) t = 1, ..., T, i = 1, ..., n. Here, T is the number of observations for firm i, xit a k ×1 vector of explanatory vari- ables, vit the measurement error, and the ui values independent non-negative random variables independent of vit with a Weibull distribution whose density is given by equation (3). Moreover, t ≥ 0 is a time-specific component that we parameterize = as t exp((t − 1)), 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 inefficiency2 is given by tui and is allowed to vary in a parametric way over time thus avoiding the 1This 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. 2For a detailed discussion of the problems arising in Bayesian cross-sectional stochastic frontiers, see Fernandez, Osiewalski and Steel (1997). © Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2007 Efficiency measurement with the Weibull stochastsic frontier 695 3 assumption of time-invariance. Furthermore, it is assumed that both vit and ui are independent of xit. (For more details on stochastic frontier analysis, see Kumbhakar and Lovell, 2000.) In equation (3), 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 ex- = × hibits negative skewness for c > 3.602. Let yi [yi1, yi2, ..., yiTi ] be the Ti 1 vector of observations for the dependent variable of the i the firm, Xi the associated Ti × k matrix of explanatory variables, and y and X denote the data in obvious notation. The joint probability density of yi is given by ∞ 1 T p(y | X , , , c, , ) = (22)−T/ 2c exp − (y − x − J u )2 i i 2 it it t i 2 = 0 t 1 × c−1 − c ui exp( ui )dui. (4) The integral in equation (4) is not available in closed form (unless c = 1), so the likelihood function n = L(, , c, , ; y, X ) p(yi | Xi, , , c, , ) i = 1 cannot be computed in closed form. For that reason, we treat the ui values as param- eters, and integrate them out using Monte Carlo methods. The kernel of the posterior = distribution augmented by the latent inefficiencies, u [u1, ..., un] , is given by 1 n T p(, , c, , , u | y, X ) ∝−N (c)N exp − (y − x − J u )2 22 it it t i i = 1 t = 1 n × c−1 − c ui exp( ui ) p( , , c, , )(5) i = 1 where p(, , c, , ) is the prior distribution and N = nT denotes the total number of observations. We will assume that ¯ p(, | c, , ) ∝ −(N + 1) exp(−Q/¯ 22) ¯ ¯ ¯ 2 ∼ 2 2 where N and Q are parameters. In this prior we have Q/ χN¯ , where χ denotes the chi-squared distribution with degrees of freedom, and p( | , c, , ) ∝ const.4 For Q¯ = N¯ = 0 we obtain the familiar default prior p(, | c, , )∝−1. The proposed prior for provides some flexibility relative to the default prior and is the condition- ally conjugate prior (see equation 7 below). For the remaining parameters, we specify gamma priors of the form 3 = = It is possible to set 0 and Ti 1 so that we treat the panel as a cross-section. 4The conditions for existence of the posterior from this model in spite of the improper prior on are given in Fernandez et al. (1997). © Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2007 696 Bulletin ∼ G(A, B), c ∼ G(C, D), p() ∝ const., (6) where G(a, b) denotes the gamma distribution with density ba f (x | a, b) = xa−1 exp(−bx)(a, b, x > 0). (a) It should be noted that our numerical techniques allow us to specify an arbitrary prior distribution for parameter . Based on the kernel posterior distribution we can use Gibbs sampling5 to perform the computations. To that end, we extract the following conditional distributions: ˜ 2 −1 | , c, , , u, y, X ∼ Nk ((u, ), (X X ) )(7) (y˜(u, ) − X )(y˜(u, ) − X ) | , c, , , u, y, X ∼ χ2 (8) 2 N n | ∼ + c + , , c, , u, y, X G n A, ui B (9) i = 1 n n | ∝ n + C−1 − c + − p(c , , , , u, y, X ) c exp ui c ln ui D (10) i = 1 i = 1 Å (u − m )2 p(u | , , c, , , y, X ) ∝ exp − i i uc−1 exp(−uc), u > 0(11) i 2 i i i 2 Åi 1 n T p( | , , c, , u, y, X ) ∝ exp − [y − x − J exp((t − 1))u ]2 , (12) 22 it it i i = 1 t = 1 where ˜(u, ) = (X X )−1X y˜(u, ), and y˜(u, ) is a vector whose elements are = y˜it(u, ) yit − J tui. Moreover, Å J Ti t−1(y − x ) 2 = t = 1 it it 2 = mi , Åi , Ci Ci T 1 − 2T C = 2(t−1) = and = exp(). i 1 − 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, ) where the scale parameter is given by = (n + C − 1)/ c¯, and c¯ solves the nonlinear equation 5For 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).