Estimating efficiency effects in a panel data stochastic frontier model

  • Satya Paul
  • Sriram ShankarEmail author


This paper proposes a panel data based stochastic frontier model which accommodates time-invariant unobserved heterogeneity along with efficiency effects. The efficiency effects are specified by a standard normal cumulative distribution function of exogenous variables which ensures the efficiency scores to lie in a unit interval. The model is within-transformed and then estimated with non-linear least squares. The finite sample properties of the proposed estimator are investigated through a set of Monte Carlo experiments. The experiments suggest that our estimation procedure generally performs well also in small samples. Finally, an empirical illustration based on widely used panel data on Indian farmers reveals the simplicity and easy applicability of the model.


Fixed effects Stochastic frontier Technical efficiency Standard normal cumulative distribution function Monte Carlo simulations Non-linear least squares 

JEL classification

C51 D24 Q12 



We are grateful to two anonymous referees and Prasada Rao for their useful comments and suggestions on earlier drafts of this paper. We are also thankful to Hung-Jen Wang for providing us the farm level data used for empirical exercise in this paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. Aigner DJ, Lovell CAK, Schmidt P (1977) Formulation and estimation of stochastic frontier production function models. J Econ 6:21–37CrossRefGoogle 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 Product Anal 25:201–212CrossRefGoogle Scholar
  3. Amemiya T (1985) Advanced econometrics. Harvard University Press, Cambridge, MAGoogle Scholar
  4. Barrett CB (1996) On price risk and the inverse farm size–productivity relationship. J Dev Econ 51:193–215CrossRefGoogle Scholar
  5. Battese G, Coelli T, Colby T (1989) Estimation of frontier production functions and the efficiencies of Indian farms using panel data from ICRISTAT’s village level studies. J Quant Econ 5:327–348Google Scholar
  6. Battese GE, Coelli TJ (1988) Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. J Econ 30:387–399CrossRefGoogle Scholar
  7. Battese G, Coelli T (1992) Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. J Product Anal 3:153–169CrossRefGoogle Scholar
  8. Battese GE, Coelli TJ (1995) A model for technical inefficiency effects in a stochastic frontier production function for panel data. Empir Econ 20:325–332CrossRefGoogle Scholar
  9. Belotti F, Ilardi G (2018) Consistent inference in fixed-effects stochastic frontier models. J Econ 202:161–177CrossRefGoogle Scholar
  10. Benjamin D (1995) Can unobserved land quality explain the inverse productivity relationship? J Dev Econ 46:51–84CrossRefGoogle Scholar
  11. Bhalla SS, Roy P (1988) Mis-specification in farm productivity analysis: the role of land quality. Oxf Economic Pap 40:55–73CrossRefGoogle Scholar
  12. Carter M (1984) Identification of the inverse relationship between farm size and productivity: An empirical analysis of peasant agricultural production. Oxf Economic Pap 36:131–146CrossRefGoogle Scholar
  13. Caudill SB, Ford JM (1993) Biases in frontier estimation due to heteroscedasticity. Economic Lett 41:17–20CrossRefGoogle Scholar
  14. Caudill SB, Ford JM, Gropper DM (1995) Frontier estimation and firm-specific inefficiency measures in the presence of heteroscedasticity. J Bus Economic Stat 13:105–111Google Scholar
  15. Chen Y, Schmidt P, Wang H (2014) Consistent estimation of the fixed effects stochastic frontier model. J Econ 181:65–76CrossRefGoogle Scholar
  16. Coelli TJ, Battese GE (1996) Identification of factors which influence the technical inefficiency of Indian farmers. Australian. J Agric Econ 40:103–28Google Scholar
  17. Colombi R, Kumbhakar SC, Martini G, Vittadini G (2014) Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency. J Product Anal 42:123–136CrossRefGoogle Scholar
  18. Deprins D (1989) Estimation de frontieres de production et mesures de l’efficacite technique. CIACO, Louvain-la-Neuve, BelgiumGoogle Scholar
  19. Deprins P, Simar L (1989a) Estimating technical efficiencies with corrections for environmental conditions with an application to railway companies. Ann Public Cooperative Econ 60:81–102CrossRefGoogle Scholar
  20. Deprins P, Simar L (1989b) Estimation de frontieres deterministes avec factuers exogenes d’inefficacite. Annales d’Economie et de Statistiqu 14:117–150CrossRefGoogle Scholar
  21. Eswaran M, Kotwal A (1986) Access to capital and agrarian production organization. Economic J 96:482–498CrossRefGoogle Scholar
  22. Greene WH (2004) Fixed effects and bias due to the incidental parameters problem in the Tobit model. Econom Rev 23:125–147CrossRefGoogle Scholar
  23. Greene WH (2005) Reconsidering heterogeneity in panel data estimators of the stochastic frontier model. J Econ 126:269–303CrossRefGoogle Scholar
  24. Hadri K (1999) Estimation of a doubly heteroskedastic stochastic frontier cost function. J Bus Economic Stat 17:359–363Google Scholar
  25. Hansson H (2008) Are larger farms more efficient? A farm level study of the relationships between efficiency and size on specialized dairy farms in Sweden. Agric Food Sci 17:325–337CrossRefGoogle Scholar
  26. Helfand SM, Levine ES (2004) Farm size and the determinants of productive efficiency in the Brazilian Center-West. Agric Econ 31:241–249CrossRefGoogle Scholar
  27. Heltberg R (1998) Rural market imperfections and the farm-size productivity relationship: Evidence from Pakistan. World Dev 26:1807–1826CrossRefGoogle Scholar
  28. Horrace W, Schmidt P (1996) Confidence statements for efficiency estimates from stochastic frontier models. J Product Anal 7:257–282CrossRefGoogle Scholar
  29. Huang CJ, Liu JT (1994) Estimation of a non-neutral stochastic frontier production function. J Product Anal 5:171–180CrossRefGoogle Scholar
  30. Jondrow J, Lovell CAK, Materov IS, Schmidt P (1982) On the estimation of technical inefficiency in stochastic frontier production function model. J Econ 19:233–238CrossRefGoogle Scholar
  31. Kagin J, Taylor JE, Yúnez-Naude A (2016) Inverse productivity or inverse efficiency? Evidence from Mexico. J Dev Stud 52:396–411CrossRefGoogle Scholar
  32. Kalirajan K (1981) An econometric analysis of yield variability in paddy production. Can J Agric Econ 29:283–294CrossRefGoogle Scholar
  33. Kumbhakar SC (1990) Production frontiers, panel data, and time-varying technical inefficiency. J Econ 46:201–212CrossRefGoogle Scholar
  34. Kumbhakar SC (1991) Estimation of technical inefficiency in panel data models with firm and time-specific effects. Econ Lett 36:43–48CrossRefGoogle Scholar
  35. Kumbhakar SC, Ghosh S, McGuckin JT (1991) A generalized production frontier approach for estimating determinants of inefficiency in US dairy farms. J Bus Economic Stat 9:279–286Google Scholar
  36. Kumbhakar SC, Heshmati A (1995) Efficiency measurement in Swedish dairy farms: an application of rotating panel data, 1976–88. Am J Agric Econ 77:660–674CrossRefGoogle Scholar
  37. Kumbhakar SC, Hjalmarsson L (1993) Technical efficiency and technical progress in Swedish dairy farms. In: Fried HO, Lovell CAK, Schmidt SS (eds) The measurement of productive efficiency-techniques and applications, Oxford University Press, UK, p 256–270Google Scholar
  38. Kumbhakar SC, Hjalmarsson L (1995) Labour-use efficiency in Swedish social insurance offices. J Appl Econ 10:33–47CrossRefGoogle Scholar
  39. Kumbhakar S, Lovell CAK (2000) Stochastic frontier analysis. Cambridge University Press, Cambridge, UKCrossRefGoogle Scholar
  40. Lancaster T (2000) The incidental parameters problem since 1948. J Econ 95:391–414CrossRefGoogle Scholar
  41. Meeusen W, van den Broeck J (1977) Efficiency estimation from Cobb-Douglas production functions with composed error. Int Economic Rev 18:435–444CrossRefGoogle Scholar
  42. Newey W, McFadden D (1994) Large sample estimation and hypothesis testing. In: Engle, RF, McFadden, D (eds) Handbook of Econometrics. vol 4. Elsevier Science B.V., USA, p 2111–2245Google Scholar
  43. Neyman J, Scott E (1948) Consistent estimates based on partially consistent observations. Econometrica 16:1–32CrossRefGoogle Scholar
  44. Parmeter CF, Kumbhakar SC (2014) Efficiency analysis: a primer on recent advances. Foundations and trends(R) in econometrics. Now publishers, Boston - Delft. vol 7. 191–385CrossRefGoogle Scholar
  45. Parmeter CF, Wang HJ, Kumbhakar SC (2017) Nonparametric estimation of the determinants of inefficiency. J Product Anal 47:205–221CrossRefGoogle Scholar
  46. Paul S, Shankar S (2018) On estimating efficiency effects in a stochastic frontier model. Eur J Operational Res 271:769–774CrossRefGoogle Scholar
  47. Pitt MM, Lee MF (1981) The measurement and sources of technical inefficiency in the Indonesian weaving industry. J Dev Econ 9:43–64CrossRefGoogle Scholar
  48. Rao CR (1973) Linear statistical inference and its applications, 2nd edn. John Wiley & Sons, New York, NYCrossRefGoogle Scholar
  49. Robinson PM (1988) Root-N-consistent semiparametric regression. Econometrica 56:931–954CrossRefGoogle Scholar
  50. Schmidt P, Sickles RC (1984) Production frontiers and panel data. J Bus Economic Stat 2:367–374Google Scholar
  51. Sen A (1966) Peasants and dualism with or without surplus labor. J Political Econ 74:425–450CrossRefGoogle Scholar
  52. Simar L, Lovell CAK, van den Eeckaut P (1994) Stochastic frontiers incorporating exogenous influences on efficiency. Discussion Papers No. 9403, Institut de Statistique, Universite de LouvainGoogle Scholar
  53. Simar L, Wilson PW (2007) Estimation and inference in two-stage, production processes. J Econ 136:31–64CrossRefGoogle Scholar
  54. Tsionas EG, Kumbhakar SC (2012) Firm heterogeneity, persistent and transient technical inefficiency: a generalized true random-effects model. J Appl Econ 29:110–132CrossRefGoogle Scholar
  55. Wang HJ (2002) Heteroscedasticity and non-monotonic efficiency effects of a stochastic frontier model. J Product Anal 18:241–253CrossRefGoogle Scholar
  56. Wang HJ, Ho CW (2010) Estimating fixed-effect panel stochastic frontier models by model transformation. J Econ 157:286–296CrossRefGoogle Scholar
  57. Wang HJ, Schmidt P (2002) One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. J Product Anal 18:129–144CrossRefGoogle Scholar
  58. Wikstrom D (2015) Consistent method of moments estimation of the true fixed effects model. Econ Lett 137:62–69CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centre for Social Research and MethodsAustralian National UniversityCanberraAustralia
  2. 2.Centre for Economics and GovernanceAmrita UniversityKollamIndia
  3. 3.Centre for Social Research and Methods & Research School of EconomicsAustralian National UniversityCanberraAustralia

Personalised recommendations