Journal of Productivity Analysis

, Volume 42, Issue 1, pp 55–65 | Cite as

Understanding prediction intervals for firm specific inefficiency scores from parametric stochastic frontier models

  • Phill Wheat
  • William Greene
  • Andrew Smith


This paper makes two important contributions to the literature on prediction intervals for firm specific inefficiency estimates in cross sectional SFA models. Firstly, the existing intervals in the literature do not correspond to the minimum width intervals and in this paper we discuss how to compute such intervals and how they either include or exclude zero as a lower bound depending on where the probability mass of the distribution of \( u_{i} |\varepsilon_{i} \) resides. This has useful implications for practitioners and policy makers, with greatest reductions in interval width for the most efficient firms. Secondly, we propose an ‘asymptotic’ approach to incorporating parameter uncertainty into prediction intervals for firm specific inefficiency (given that in practice model parameters have to be estimated) as an alternative to the ‘bagging’ procedure suggested in Simar and Wilson (Econom Rev 29(1):62–98, 2010). The approach is computationally much simpler than the bagging approach.


Stochastic frontier Prediction intervals Efficiency 

JEL Classification

C12 L25 L51 L92 


  1. Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6(1):21–37CrossRefGoogle 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 25:201–212CrossRefGoogle Scholar
  3. Amsler C, Leonard M, Schmidt P (2010) Estimation and inference in parametric deterministic frontier models, working paperGoogle Scholar
  4. Battese GE, Coelli TJ (1992) Frontier production functions and the efficiencies of Indian Farms Using Panel data from ICRISAT’s village level studies. J Quant Econ 5:327–348Google Scholar
  5. Bera AK, Sharma SC (1999) Estimating production uncertainty in stochastic frontier production frontier models. J Prod Anal 12:187–210CrossRefGoogle Scholar
  6. Coelli T, Rao DSP, O’Donnell CJ, Battese GE (2005) An introduction to efficiency and productivity analysis, 2nd edn. New York, SpringerGoogle Scholar
  7. Cuesta RA (2000) A production model with firm-specific temporal variation in technical inefficiency: with application to Spanish dairy farms. J Prod Anal 13(2):139–152CrossRefGoogle Scholar
  8. Econometric Software Inc. (2010) LIMDEP, user’s manual., Plainview, NY
  9. Farrell MJ (1957) The measurement of productive efficiency. J R Stat Soc Ser A Gen 120(3):253–290CrossRefGoogle Scholar
  10. Flores-Lagunes A, Horrace WC, Schnier KE (2007) Identifying technically efficient fishing vessels: a non-empty, minimal subset approach. J Appl Econom 22:729–745CrossRefGoogle Scholar
  11. Greene WH (2008) The econometric approach to efficiency analysis. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency growth, 2nd edn. Oxford University Press, New YorkGoogle Scholar
  12. Greene WH (2011) Econometric analysis, 7th edn. Prentice Hall, New YorkGoogle Scholar
  13. Hjalmarsson L, Kumbhakar SC, Heshmati A (1996) DEA, DFA and SFA: a comparison. J Prod Anal 7:303–327CrossRefGoogle Scholar
  14. Horrace WC (2005) On ranking and selection from independent truncated normal distributions. J Econom 126:335–354CrossRefGoogle Scholar
  15. Horrace WC, Schmidt P (1996) Confidence statements for efficiency estimates from stochastic frontier models. J Prod Anal 7(2/3):257–282CrossRefGoogle Scholar
  16. Horrace WC, Schmidt P (2000) Multiple comparisons with the best, with economic applications. J Appl Econom 15:1–26CrossRefGoogle Scholar
  17. Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19:233–238CrossRefGoogle Scholar
  18. Kim Y, Schmidt P (2008) Marginal comparisons with the best and the efficiency measurement problem. J Bus Econ Stat 26(2):253–260CrossRefGoogle Scholar
  19. Krinsky I, Robb A (1986) On approximating the statistical properties of elasticities. Rev Econ Stat 68(4):715–719CrossRefGoogle Scholar
  20. Kumbhakar SC, Löthgren M (1998) A Monte Carlo analysis of technical inefficiency predictors. Working paper series in economics and finance, no. 229, Stockholm School of EconomicsGoogle Scholar
  21. Meeusen W, van Den Broeck J (1977) Efficiency estimation from Cobb-Douglas production functions with composed error. Intern Econ Rev 18(2):435–444CrossRefGoogle Scholar
  22. Pitt M, Lee L (1981) The measurement and sources of technical inefficiency in Indonesian weaving industry. J Dev Econ 9:43–64CrossRefGoogle Scholar
  23. R Development Core Team (2010) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
  24. Simar L, Wilson PW (2010) Inferences from cross-sectional, stochastic frontier models. Econom Rev 29(1):62–98CrossRefGoogle Scholar
  25. Smith A, Wheat P (2012) Evaluating alternative policy responses to franchise failure: Evidence from the passenger rail sector in Britain. J Transp Econ Policy 46(1):25–43Google Scholar
  26. Taube R (1988) Möglichkeiten der Effizienzmess ung von öffentlichen Verwaltungen. Duncker & Humbolt GmbH, BerlinGoogle Scholar
  27. Train KE (2009) Discrete choice methods with simulation, 2nd edn. Cambridge University Press, New YorkCrossRefGoogle Scholar
  28. Waldman DM (1984) Properties of technical efficiency estimators in the stochastic frontier model. J Econom 25:353–364CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of LeedsLeedsUK
  2. 2.Department of Economics, Stern School of BusinessUniversity of New YorkNew York CityUSA

Personalised recommendations