Skip to main content

Bayesian estimation of inefficiency heterogeneity in stochastic frontier models

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  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).

References

  1. Aigner D, Lovell C, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6:21–37

    Article  Google Scholar 

  2. Alvarez A, Amsler C, Orea L, Schmidt P (2006) Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics. J Prod Anal 25:201–212

    Article  Google Scholar 

  3. Battese G, Coelli T (1992) Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. J Prod Anal 3:153–169

    Article  Google Scholar 

  4. Battese G, Coelli T (1995) A model for technical inefficiency effects in a stochastic frontier production model for panel data. Empir Econ 20:325–332

    Article  Google Scholar 

  5. Caudill S, Ford J (1993) Biases in frontier estimation due to heteroscedasticity. Econ Lett 41:17–20

    Article  Google Scholar 

  6. Caudill S, Ford J, Gropper D (1995) Frontier estimation and firm-specific inefficiency measures in the presence of heteroskedasticity. J Bus Econom Stat 13:105–111

    Google Scholar 

  7. Celeux G, Forbes F, Robert C, Titterington D (2006) Deviance information criteria for missing data models. Bayesian Anal 4:651–674

    Article  Google Scholar 

  8. Coelli T, Perelman S, Romano E (1999) Accounting for environmental influences in stochastic frontier models: with application to international airlines. J Prod Anal 11:251–273

    Article  Google Scholar 

  9. Evans D, Tandon A, Murray C, Lauer J (2000) The comparative efficiency of national health systems in producing health: an analysis of 191 countries. Discussion paper no. 29, World Health Organization, EIP/GPE/EQC

  10. Fernandez C, Ley E, Steel M (2001) Benchmark priors for Bayesian model averaging. J Econom 100:381–427

    Article  Google Scholar 

  11. Ferreira J, Steel M (2007) Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers. J Econom 137:641–673

    Article  Google Scholar 

  12. Good I (1952) Rational decisions. J R Stat Soc B 14:107–114

    Google Scholar 

  13. Greene W (1990) A gamma-distributed stochastic frontier model. J Econom 46:141–164

    Article  Google Scholar 

  14. Greene W (2004) Distinguishing between heterogeneity and inefficiency: stochastic frontier analysis of the World Health Organization’s panel data on national health care systems. Health Econ 13:959–980

    Article  Google Scholar 

  15. Greene W (2005) Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. J Econom 126:269–303

    Article  Google Scholar 

  16. Greene W (2008) The econometric approach to efficiency analysis. The measurement of productive efficiency and productivity growth, chap 2. Oxford University Press, Inc., New York, pp 959–980

    Google Scholar 

  17. Griffin J, Steel M (2004) Semiparametric Bayesian inference for stochastic frontier models. J Econom 123:121–152

    Article  Google Scholar 

  18. Griffin J, Steel M (2007) Bayesian stochastic frontier analysis using WinBUGS. J Prod Anal 27:163–176

    Article  Google Scholar 

  19. Griffin J, Steel M (2008) Flexible mixture modelling of stochastic frontiers. J Prod Anal 29:33–50

    Article  Google Scholar 

  20. Hadri K (1999) Estimation of a doubly heteroscedastic stochastic frontier cost function. J Prod Anal 17:359–363

    Google Scholar 

  21. Hadri K, Guermat C, Whittaker J (2003a) Estimating farm efficiency in the presence of double heteroscedasticity using panel data. J Appl Econ 2:255–268

    Google Scholar 

  22. Hadri K, Guermat C, Whittaker J (2003b) Estimation of technical inefficiency effects using panel data and doubly heteroscedastic stochastic production frontiers. Empir Econ 28:203–222

    Article  Google Scholar 

  23. Huang H (2004) Estimation of technical inefficiencies with heterogeneous technologies. J Prod Anal 21:277–296

    Article  Google Scholar 

  24. Huang H, Liu J (1994) Estimation of a non-neutral stochastic frontier production function. J Prod Anal 5:171–180

    Article  Google Scholar 

  25. Koop G, Osiewalski J, Steel M (1997) Bayesian efficiency analysis through individual effects: hospital cost frontiers. J Econom 76:77–106

    Article  Google Scholar 

  26. Koop G, Steel M, Osiewalski J (1995) Posterior analysis of stochastic frontier models using Gibbs sampling. Comput Stat 10:353–373

    Google Scholar 

  27. Kumbhakar S, Ghosh S, McGuckin J (1991) A generalized production frontier approach for estimating determinants of inefficiency in US dairy farms. J Bus Econ Stat 9:279–286

    Google Scholar 

  28. Kumbhakar S, Lovell C (2000) Stochastic frontier analysis. Cambridge University Press, New York

    Book  Google Scholar 

  29. Li Y, Zeng T, Yu J (2012) Robust deviance information criterion for latent variable models. Singapore Management University, Research Collection School of Economics (Open Access). Paper 1403

  30. Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb–Douglas production functions with composed errors. Int Econ Rev 8:435–444

    Article  Google Scholar 

  31. Reifschnieder D, Stevenson R (1991) Systematic departures from the frontier: a framework for the analysis of firm inefficiency. Int Econ Rev 32:715–723

    Article  Google Scholar 

  32. Richardson S (2002) Discussion of spiegelhalter et al. J R Stat Soc B 64:626–627

    Google Scholar 

  33. Simar L, Lovell C, van den Eeckaut P (1994). Stochastic frontiers incorporating exogenous influences on efficiency. Discussion paper no. 9403, Institut de Statistique, Universit Catholique de Louvain

  34. Spiegelhalter D, Best N, Carlin B, van der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc 64(4):583–639

    Article  Google Scholar 

  35. Stevenson R (1980) Likelihood functions for generalized stochastic frontier estimation. J Econom 13:57–66

    Article  Google Scholar 

  36. Tsionas E (2002) Stochastic frontier models with random coefficients. J Appl Econom 17:127–147

    Article  Google Scholar 

  37. van den Broeck J, Koop G, Osiewalski J, Steel M (1994) Stochastic frontier models: a bayesian perspective. J Econom 61:273–303

    Article  Google Scholar 

  38. Wang H (2002) Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. J Prod Anal 18:241–253

    Article  Google Scholar 

  39. Wang H, Schmidt P (2002) One step and two step estimation of the effects of exogenous variables on technical efficiency levels. J Prod Anal 18:129–144

    Article  Google Scholar 

  40. WHO (2000) Health systems: improving performance. The World Health report. World Health Organization, Geneva

Download references

Acknowledgments

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jorge E. Galán.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Galán, J.E., Veiga, H. & Wiper, M.P. Bayesian estimation of inefficiency heterogeneity in stochastic frontier models. J Prod Anal 42, 85–101 (2014). https://doi.org/10.1007/s11123-013-0377-4

Download citation

Keywords

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

JEL Classification

  • C11
  • C23
  • C51
  • D24