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Bayesian estimation of inefficiency heterogeneity in stochastic frontier models


Estimation of the one sided error component in stochastic frontier models may erroneously attribute firm characteristics to inefficiency if heterogeneity is unaccounted for. However, unobserved inefficiency heterogeneity has been little explored. In this work, we propose to capture it through a random parameter which may affect the location, scale, or both parameters of a truncated normal inefficiency distribution using a Bayesian approach. Our findings using two real data sets, suggest that the inclusion of a random parameter in the inefficiency distribution is able to capture latent heterogeneity and can be used to validate the suitability of observed covariates to distinguish heterogeneity from inefficiency. Relevant effects are also found on separating and shrinking individual posterior efficiency distributions when heterogeneity affects the location and scale parameters of the one-sided error distribution, and consequently affecting the estimated mean efficiency scores and rankings. In particular, including heterogeneity simultaneously in both parameters of the inefficiency distribution in models that satisfy the scaling property leads to a decrease in the uncertainty around the mean scores and less overlapping of the posterior efficiency distributions, which provides both more reliable efficiency scores and rankings.

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  1. 1.

    We also studied these effects using models that follow half-normal and exponential distributions for the inefficiency. These results are available from the authors upon request.

  2. 2.

    In a previous study, Caudill and Ford (1993) also found biased estimates of the frontier parameters.

  3. 3.

    More details on this criterion and an approximate lower bound for the LPS are described in Fernandez et al. (2001).

  4. 4.

    After performing some tests Greene (2004) chose a model that includes Gini and GDP in the inefficiency and the rest of covariates in the production function.

  5. 5.

    Among models with inefficiency heterogeneity, rank correlation is very high (0.99).

  6. 6.

    Similar results were obtained from other scaling-type models following half-normal and exponential distributions but they performed a bit worse in terms of fit and predictive performance. Results are available from authors upon request.

  7. 7.

    The original data set includes 256 observations, ten years of observations for an extra airline company. We excluded this firm since we do not have data for the exogenous variables of this airline.

  8. 8.

    Coelli et al. (1999) evaluate both alternatives for a technical efficiency analysis and conclude statistically in favor of a model including them in the inefficiency term.

  9. 9.

    For all models, monotonicity conditions were found to be not satisfied because of negative signs obtained for prices coefficients. This result was also obtained by Greene (2008). Therefore, we impose regularity conditions by requiring the cost function to have positive elasticities on prices (∂c it /∂p it  > 0). We follow the procedure described in Griffin and Steel (2007) by restricting coefficients β1 to β4 to be positive through truncated normal prior distributions for these parameters.

  10. 10.

    In fact, most of the efficiency studies applied to airlines have treated size and network environment variables as frontier drivers (see Coelli et al. 1999).


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The authors would like to thank Mark Steel and Jim Griffin for their comments and suggestions as well as the participants of the 33rd National Congress on Statistics and Operations Research and the Permanent Seminar on Efficiency and Productivity of Universidad de Oviedo. Financial support from the Spanish Ministry of Education and Science, research projects ECO2012-3401, MTM2010-17323 and SEJ2007-64500 is also gratefully acknowledged.

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Correspondence to Jorge E. Galán.

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Galán, J.E., Veiga, H. & Wiper, M.P. Bayesian estimation of inefficiency heterogeneity in stochastic frontier models. J Prod Anal 42, 85–101 (2014).

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  • Stochastic frontier models
  • Efficiency
  • Unobserved heterogeneity
  • Bayesian inference

JEL Classification

  • C11
  • C23
  • C51
  • D24