Journal of Productivity Analysis

, Volume 34, Issue 3, pp 239–255 | Cite as

Fractional regression models for second stage DEA efficiency analyses

  • Esmeralda A. Ramalho
  • Joaquim J. S. Ramalho
  • Pedro D. Henriques


Data envelopment analysis (DEA) is commonly used to measure the relative efficiency of decision-making units. Often, in a second stage, a regression model is estimated to relate DEA efficiency scores to exogenous factors. In this paper, we argue that the traditional linear or tobit approaches to second-stage DEA analysis do not constitute a reasonable data-generating process for DEA scores. Under the assumption that DEA scores can be treated as descriptive measures of the relative performance of units in the sample, we show that using fractional regression models is the most natural way of modeling bounded, proportional response variables such as DEA scores. We also propose generalizations of these models and, given that DEA scores take frequently the value of unity, examine the use of two-part models in this framework. Several tests suitable for assessing the specification of each alternative model are also discussed.


Second-stage DEA Fractional data Specification tests One outcomes Two-part models 

JEL Classification

C12 C13 C25 C51 



The authors thank the editor, an associate editor and the referees for valuable comments that helped to substantially improve the paper. Financial support from Fundação para a Ciência e a Tecnologia is also gratefully acknowledged (grant PTDC/ECO/64693/2006).


  1. Aranda-Ordaz FJ (1981) On two families of transformations to additivity for binary response data. Biometrika 68(2):357-363CrossRefGoogle Scholar
  2. Banker RD, Natarajan R (2008) Evaluating contextual variables affecting productivity using data envelopment analysis. Oper Res 56(1):48–58CrossRefGoogle Scholar
  3. Chilingerian JA (1995) Evaluating physician efficiency in hospitals: a multivariate analysis of best practices. Eur J Oper Res 80:548–574CrossRefGoogle Scholar
  4. Coelli TJ, Rao DS, O’Donnell CJ, Battese GE (2005) An introduction to efficiency and productivity analysis, 2nd edn. Springer, New YorkGoogle Scholar
  5. Czado C (1994) Parametric link modification of both tails in binary regression. Stat Papers 35:189-201CrossRefGoogle Scholar
  6. Daraio C, Simar L (2005) Introducing environmental variables in nonparametric frontier models: a probabilistic approach. J Prod Anal 24:93–121CrossRefGoogle Scholar
  7. Davidson R, MacKinnon JG (1981) Several tests for model specification in the presence of alternative hypotheses. Econometrica 49(3):781–793CrossRefGoogle Scholar
  8. Duan N (1983) Smearing estimate: a nonparametric retransformation method. J Am Stat Assoc 78:605–610CrossRefGoogle Scholar
  9. Gillespie J, Schupp A, Taylor G (1997) New alternative enterprise: the case of ratite industry. J Agric Appl Econ 29(2):409–418Google Scholar
  10. Gouriéroux C, Monfort A (1994) Testing non-nested hypotheses. In: Engle RF, McFadden DL (eds) Handbook of econometrics, volume IV. Elsevier Science, Amsterdam, pp 2585–2637Google Scholar
  11. Gouriéroux C, Monfort A, Trognon A (1984) Pseudo maximum likelihood methods: applications to Poisson models. Econometrica 52(3):701–720CrossRefGoogle Scholar
  12. Grosskopf S (1996) Statistical inference and nonparametric efficiency: a selective survey. J Prod Anal 7:161–176CrossRefGoogle Scholar
  13. Hoff A (2007) Second stage DEA: comparison of approaches for modelling the DEA score. Eur J Oper Res 181:425–435CrossRefGoogle Scholar
  14. Kneip A, Park BU, Simar L (1998) A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econ Theory 14:783–793CrossRefGoogle Scholar
  15. Kravtsova V (2008) Foreign presence and efficiency in transition economies. J Prod Anal 29:91–102CrossRefGoogle Scholar
  16. Latruffe L, Davidova S, Balcombe K (2008) Application of a double bootstrap to investigation of determinants of technical efficiency of farms in Central Europe. J Prod Anal 29:183–191CrossRefGoogle Scholar
  17. McDonald J (2009) Using least squares and tobit in second stage DEA efficiency analyses. Eur J Oper Res 197(2):792–798CrossRefGoogle Scholar
  18. McDonald JF, Moffitt RA (1980) The uses of tobit analysis. Rev Econ Stat 62(2):318–321CrossRefGoogle Scholar
  19. Nagler J (1994) Scobit: an alternative estimator to logit and probit. Am J Polit Sci 38(1):230–255CrossRefGoogle Scholar
  20. Pagan A, Vella F (1989) Diagnostic tests for models based on individual data: a survey. J Appl Econ 4:S29–S59CrossRefGoogle Scholar
  21. Papke LE, Wooldridge JM (1996) Econometric methods for fractional response variables with an application to 401(k) plan participation rates. J Appl Econ 11(6):619-632CrossRefGoogle Scholar
  22. Poirier DJ (1980) A Lagrange multiplier test for skewness in binary models. Econ Lett 5:141–143CrossRefGoogle Scholar
  23. Pregibon D (1980) Goodness of link tests for generalized linear models. Appl Stat 29(1):15–24CrossRefGoogle Scholar
  24. Prentice RL (1976) A generalization of the probit and logit methods for dose response curves. Biometrics 32:761–768CrossRefGoogle Scholar
  25. Ray SC (1991) Resource-use efficiency in public schools: a study of Connecticut data. Manag Sci 37(12):1620–1628CrossRefGoogle Scholar
  26. Ramalho EA, Ramalho JJS, Murteira J (2010) Alternative estimating and testing empirical strategies for fractional regression models. J Econ Surv. doi:10.1111/j.1467-6419.2009.00602.x
  27. Ruggiero J (1998) Non-discretionary inputs in data envelopment analysis. Eur J Oper Res 111:461–469CrossRefGoogle Scholar
  28. Simar L, Wilson P (2007) Estimation and inference in two-stage, semi-parametric models of production processes. J Econ 136:31–64Google Scholar
  29. Smith RJ (1989) On the use of distributional misspecification checks in limited dependents variable models. Econ J 99:178–192CrossRefGoogle Scholar
  30. Stukel TA (1988) Generalized logistic models. J Am Stat Assoc 83(402):426–431CrossRefGoogle Scholar
  31. Taylor JMG (1988) The cost of generalizing logistic regression. J Am Stat Assoc 83(404):1078–1083CrossRefGoogle Scholar
  32. Wang H, Schmidt P (2002) One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. J Prod Anal 18:129–144CrossRefGoogle Scholar
  33. Whitemore AS (1983) Transformations to linearity in binary regression. SIAM J Appl Math 43(4):703–710CrossRefGoogle Scholar
  34. Wooldridge JM (2002) Econometric analysis of cross section and panel data. The MIT Press, CambridgeGoogle Scholar
  35. Zelenyuk V, Zheka V (2006) Corporate governance and firm’s efficiency: the case of a transitional country, Ukraine. J Prod Anal 25:143–157CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Esmeralda A. Ramalho
    • 1
  • Joaquim J. S. Ramalho
    • 1
  • Pedro D. Henriques
    • 1
  1. 1.Department of Economics and CEFAGE-UEUniversidade de ÉvoraÉvoraPortugal

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