Journal of Productivity Analysis

, Volume 42, Issue 1, pp 45–54 | Cite as

Stochastic frontier models with threshold efficiency

  • Sungwon Lee
  • Young Hoon Lee


This paper proposes a tail-truncated stochastic frontier model that allows for the truncation of technical efficiency from below. The truncation bound implies the inefficiency threshold for survival. Specifically, this paper assumes a uniform distribution of technical inefficiency and derives the likelihood function. Even though this distributional assumption imposes a strong restriction that technical inefficiency has a uniform probability density over [0, θ], where θ is the threshold parameter, this model has two advantages: (1) the reduction in the number of parameters compared with more complicated tail-truncated models allows better performance in numerical optimization; and (2) it is useful for empirical studies of the distribution of efficiency or productivity, particularly the truncation of the distribution. The Monte Carlo simulation results support the argument that this model approximates the distribution of inefficiency precisely, as the data-generating process not only follows the uniform distribution but also the truncated half-normal distribution if the inefficiency threshold is small.


Stochastic frontier Technical efficiency Threshold inefficiency Uniform distribution Productivity distribution 

JEL classification

C13 C21 D24 L11 



Young Hoon Lee would like to acknowledge the supports by the National Research Foundation of Korea Grant funded by Korean Government (NRF-2010-330-B00069) and Sogang University Research Grant.


  1. Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econom 6:21–37CrossRefGoogle Scholar
  2. Almanidis P, Sickles R (2010) The skewness problem in stochastic frontier models : fact or fiction?, forthcoming in Exploring Research Frontiers in Contemporary Statistics and Econometrics: A Festschrift in Honor of Leopold Simar, Ingrid Van Keilegom and Paul Wilson (eds), Springer PublishingGoogle Scholar
  3. Amemiya T (1973) Regression analysis when the dependent variable is truncated normal. Econometrica 41(6):997–1016CrossRefGoogle Scholar
  4. De Loecker J (2009) Product differentiation, multi-product firms and estimating the impact of trade liberalization on productivity, Working Paper, Princeton UniversityGoogle Scholar
  5. Dunne T, Klimek S, Schmitz JA (2008) Does Foreign competition spur productivity? evidence from Post WWII US cement manufacturing, Working Paper, Federal Reserve Bank MinneapolisGoogle Scholar
  6. Feng Q, Horrace WC (2012) Alternative technical efficiency measures: skew, bias and scale. J Appl Econom 27:253–268CrossRefGoogle Scholar
  7. Good DH, Nadiri MI, Roller L-H, Sickles RC (1993) Efficiency and roductivity growth comparison of European and US air carriers: a first look at the data. J Prod Anal 4:115–125CrossRefGoogle Scholar
  8. Greene WH (2008) The econometric approach to efficiency analysis. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency and productivity growth. Oxford University Press, Oxford Google Scholar
  9. Holmes TJ, Schmitz JA (2001) Competition at work: Railroad vs. monopoly in US shipping. Fed Reserve Bank Minneap Q Rev 19(1):3–29Google Scholar
  10. Holmes TJ, Schmitz JA (2010) Competition and productivity: a review of evidence. Annu Rev Econ 2:619–642CrossRefGoogle Scholar
  11. Jondrow J, Lovell C, Materov I, Schmidt P (1982) On the estimation of technical inefficiency in the stochastic frontier production function model. J Econom 19:233–238CrossRefGoogle Scholar
  12. Lee YH (1996) Tail truncated stochastic frontier models. J Econ Theory Econom 2:137–152Google Scholar
  13. Matsa D (2011) Competition and product quality in the supermarket industry. Q J Econ 126(3):1539–1591Google Scholar
  14. Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb-Douglas production functions with composed error. Intern Econ Rev 18:435–555CrossRefGoogle Scholar
  15. Qian J, Sickles R (2008) Stochastic frontiers with bounded inefficiency, mimeo, Rice UniversityGoogle Scholar
  16. Schmidt P, Sickles R (1984) Production frontiers and panel data. J Bus Econ Stat 2(4):367–374Google Scholar
  17. Stevenson RE (1980) Generalized stochastic frontier estimation. J Econom 13:57–66Google Scholar
  18. Syverson C (2004) Market structure and productivity : a concrete example. J Polit Econ 112:1181–1222CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Sogang UniversitySeoulSouth Korea

Personalised recommendations