Piecewise Frontier Analysis

  • Christopher J. O’DonnellEmail author


Estimating/predicting levels of efficiency involves estimating production frontiers. A widely-used estimation approach involves enveloping scatterplots of data points as tightly as possible without violating any assumptions that have been made about production technologies. Some of the most common assumptions lead to estimated frontiers that are comprised of multiple linear segments (or pieces). The associated frontiers are known as piecewise frontiers. This chapter explains how to estimate the unknown parameters of so-called piecewise frontier models (PFMs). It then explains how the estimated parameters can be used to analyse efficiency and productivity change. The focus is on data envelopment analysis (DEA) estimators.


  1. Afsharian M, Ahn H (2015) The overall Malmquist index: a new approach for measuring productivity changes over time. Ann Oper Re 226(1):1–27CrossRefGoogle Scholar
  2. Afsharian M, Podinovski V (2018) A linear programming approach to efficiency evaluation in nonconvex metatechnologies. Eur J Oper Res 268(1):268–280CrossRefGoogle Scholar
  3. Allen R, Athanassopoulos A, Dyson R, Thanassoulis E (1997) Weight restrictions and value judgements in data envelopment analysis: evolution, development and future directions. Ann Oper Res 73:13–34CrossRefGoogle Scholar
  4. Anik AR, Rahman S, Sarker J (2017) Agricultural productivity growth and the role of capital in South Asia (1980–2013). Sustainability 9(3):470CrossRefGoogle Scholar
  5. Baležentis T (2015) The sources of the total factor productivity growth in Lithuanian family farms: A Färe-Primont index approach. Prague Econ Papers 24(2):225–241CrossRefGoogle Scholar
  6. Banker R (1993) Maximum likelihood, consistency and data envelopment analysis: a statistical foundation. Manag Sci 39(10):1265–1273CrossRefGoogle Scholar
  7. Banker R (1996) Hypothesis tests using data envelopment analysis. J Prod Anal 7(2–3):139–159CrossRefGoogle Scholar
  8. Banker R, Charnes A, Cooper W (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092CrossRefGoogle Scholar
  9. Banker RD, Charnes A, Cooper WW, Schinnar AP (1981) A bi-extremal principle for frontier estimation and efficiency evaluations. Manag Sci 27(12):1370–1382CrossRefGoogle Scholar
  10. Baráth L, Fertö I (2017) Productivity and convergence in European agriculture. J Agri Econ 68(1):228–248CrossRefGoogle Scholar
  11. Battese G, Rao D, O’Donnell C (2004) A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. J Prod Anal 21(1):91–103CrossRefGoogle Scholar
  12. Beltran-Esteve M, Gomez-Limon J, Picazo-Tadeo A, Reig-Martinez E (2014) A metafrontier directional distance function approach to assessing eco-efficiency. J Prod Anal 41:69–83CrossRefGoogle Scholar
  13. Bogetoft P, Otto L (2015) Benchmarking: benchmark and frontier analysis using DEA and SFA. R package version 2015-7-8Google Scholar
  14. Boles J (1966) Efficiency squared-efficient computation of efficiency indexes. In: Thirty Ninth annual meeting of the Western farm economics association, pp 137–142Google Scholar
  15. Brännlund R, Färe R, Grosskopf S (1995) Environmental regulation and profitability: an application to Swedish pulp and paper mills. Environ Resour Econ 6:23–36CrossRefGoogle Scholar
  16. Brännlund R, Chung Y, Färe R, Grosskopf S (1998) Emissions trading and profitability: the Swedish pulp and paper industry. Environ Resour Econ 12:345–356CrossRefGoogle Scholar
  17. Bressler R (1966) The measurement of productive efficiency. In: Thirty Ninth annual meeting of the Western farm economics association, pp 129–136Google Scholar
  18. Briec W, Kerstens K, Van de Woestyne I (2016) Congestion in production correspondences. J Econ 119(1):65–90Google Scholar
  19. Briec W, Kerstens K, Van de Woestyne I (2018) Hypercongestion in production correspondences: an empirical exploration. Appl Econ 50(27):2938–2956CrossRefGoogle Scholar
  20. Burley H (1980) Productive efficiency in U.S. manufacturing: a linear programming approach. Rev Econ Stat 62(4):619–622CrossRefGoogle Scholar
  21. Carrington R, Coelli T, Rao D (2005) The performance of Australian universities: conceptual issues and preliminary results. Econ Papers 24(2):145–163CrossRefGoogle Scholar
  22. Carrington R, O’Donnell C, Rao D (2016) Australian university productivity growth and public funding revisited. Stud High Educ 1–22:,079.2016.1259,306
  23. Casu B, Ferrari A, Zhao T (2013) Regulatory reform and productivity change in Indian banking. Rev Econ Stat 95(3):1066–1077CrossRefGoogle Scholar
  24. Charnes A, Cooper W (1962) Programming with linear fractional functionals. Naval Res Logistics Q 9(3–4):181–186CrossRefGoogle Scholar
  25. Charnes A, Cooper W, Rhodes E (1978) Measuring the efficiency of decision-making unit. Eur J Oper Res 2(6):429–444 (see also “Corrections” op.cit. 3(4):339)CrossRefGoogle Scholar
  26. Charnes A, Cooper W, Rhodes E (1981) Evaluating program and managerial efficiency: an application of data envelopment analysis to Program follow through. Manag Sci 27(6):668–697CrossRefGoogle Scholar
  27. Chattopadhyay S, Heffley D (1994) Are for-profit nursing homes more efficient? Data envelopment analysis with a case-mix constraint. East Econ J 20(2):171–186Google Scholar
  28. Coelli T, Grifell-Tatje E, Perelman S (2002) Capacity utilisation and profitability: a decomposition of short-run profit efficiency. Int J Prod Econ 79(3):261–278CrossRefGoogle Scholar
  29. Coelli T, Rao D, O’Donnell C, Battese G (2005) An introduction to efficiency and productivity analysis, 2nd edn. Springer, New YorkGoogle Scholar
  30. Cooper W, Seiford L, Zhu J (2000) A unified additive model approach for evaluating inefficiency and congestion with associated measures in DEA. Socio-Econ Plann Sci 34(1):1–25CrossRefGoogle Scholar
  31. Cooper W, Seiford L, Zhu J (2001) Slacks and congestion: response to a comment by R. Fare and S. Grosskopf. Socio-Econ Plann Sci 35(3):205–215Google Scholar
  32. Curi C, Guarda P, Lozano-Vivas A, Zelenyuk V (2013) Is foreign-bank efficiency in financial centers driven by home or host country characteristics? J Prod Anal 40:367–385CrossRefGoogle Scholar
  33. Demchuk P, Zelenyuk V (2009) Testing differences in efficiency of regions within a country: the case of Ukraine. J Prod Anal 32(2):81–102CrossRefGoogle Scholar
  34. Deprins D, Simar L, Tulkens H (1984) Measuring labor efficiency in post offices. In: Marchand M, Pestieau P, Tulkens H (eds) The performance of public enterprises: concepts and measurements. North Holland Publishing Company, AmsterdamGoogle Scholar
  35. Fan Y, Li Q, Weersink A (1996) Semiparametric estimation of stochastic production frontier models. J Bus Econ Stat 14(4):460–468Google Scholar
  36. Färe R, Grosskopf S (1998) Congestion: a note. Socio-Econ Plann Sci 32(1):21–23CrossRefGoogle Scholar
  37. Färe R, Grosskopf S (2000) Slacks and congestion: a comment. Socio-Econ Plann Sci 34(1):27–33CrossRefGoogle Scholar
  38. Färe R, Grosskopf S (2001) When can slacks be used to identify congestion? An answer to W. W. Cooper, L. Seiford and J. Zhu. Socio-Econ Plann Sci 35(3):217–221Google Scholar
  39. Färe R, Grosskopf S, Logan J (1983) The relative efficiency of Illinois electric utilities. Resour Energy 5(4):349–367CrossRefGoogle Scholar
  40. Färe R, Grosskopf S, Lovell C (1985) The measurement of efficiency of production. Kluwer Academic Publishers, BostonCrossRefGoogle Scholar
  41. Färe R, Grosskopf S, Kokkelenberg E (1989) Measuring plant capacity, utilization and technical change: a non-parametric approach. Int Econ Rev 30(3):655–666CrossRefGoogle Scholar
  42. Färe R, Grosskopf S, Lee H (1990) A nonparametric approach to expenditure constrained profit maximization. Am J Agri Econ 72(3):574–581CrossRefGoogle Scholar
  43. Färe R, Grosskopf S, Lovell C (1994) Production frontiers. Cambridge University Press, CambridgeGoogle Scholar
  44. Färe R, Grosskopf S, Norris M, Zhang Z (1994) Productivity growth, technical progress, and efficiency change in industrialized countries. Am Econ Rev 84(1):66–83Google Scholar
  45. Färe R, Grosskopf S, Lee WF (2001) Productivity and technical change: the case of Taiwan. Appl Econ 33(15):1911–1925CrossRefGoogle Scholar
  46. Färe R, Grosskopf S, Hernandez-Sancho F (2004a) Environmental performance: an index number approach. Resour Energy Econ 26(4):343–352CrossRefGoogle Scholar
  47. Färe R, Grosskopf S, Weber W (2004b) The effect of risk-based capital requirements on profit efficiency in banking. Appl Econ 36(15):1–13CrossRefGoogle Scholar
  48. Farrell M (1957) The measurement of productive efficiency. J R Stat Soc Ser A (General) 120(3):253–290CrossRefGoogle Scholar
  49. Farrell M, Fieldhouse M (1962) Estimating efficient production functions under increasing returns to scale. J R Stat Soc Ser A (General) 125(2):252–267CrossRefGoogle Scholar
  50. Forsund F, Sarafoglou N (2002) The origins of data envelopment analysis. J Prod Anal 17(1):23–40CrossRefGoogle Scholar
  51. Gijbels I, Mammen E, Park B, Simar L (1999) On estimation of monotone and concave frontier functions. J Am Stat Assoc 94(445):220–228CrossRefGoogle Scholar
  52. Grosskopf S (1996) Statistical inference and nonparametric efficiency: a selective survey. J Prod Anal 7:161–176CrossRefGoogle Scholar
  53. Hoff A (2007) Second stage DEA: comparison of approaches for modelling the DEA score. Eur J Oper Res 181(1):425–435CrossRefGoogle Scholar
  54. Huang C, Fu TT (1999) An average derivative estimation of stochastic frontier. J Prod Anal 12:45–53CrossRefGoogle Scholar
  55. Huang MY, Juo JC, Fu TT (2015) Metafrontier cost Malmquist productivity index: an application to Taiwanese and Chinese commercial banks. J Prod Anal 44:321–335CrossRefGoogle Scholar
  56. Islam N, Xayavong V, Kingwell R (2014) Broadacre farm productivity and profitability in south-western Australia. Aust J Agri Resour Econ 58(2):147–170CrossRefGoogle Scholar
  57. Johnes J, Izzeldin M, Pappas V (2014) A comparison of performance of Islamic and conventional banks 2004–2009. J Econ Behav Organ 103(Supplement):S93–S107CrossRefGoogle Scholar
  58. Johnson A, Kuosmanen T (2011) One-stage estimation of the effects of operational conditions and practices on productive performance: asymptotically normal and efficient, root-n consistent StoNEZD method. J Prod Anal 36(2):219–230CrossRefGoogle Scholar
  59. Johnson A, Kuosmanen T (2012) One-stage and two-stage DEA estimation of the effects of contextual variables. Eur J Oper Res 220(2):559–570CrossRefGoogle Scholar
  60. Kerstens K, Vanden Eeckaut P (1999) Estimating returns to scale using non-parametric deterministic technologies: a new method based on goodness-of-fit. Eur J Oper Res 113(1):206–214CrossRefGoogle Scholar
  61. Kerstens K, O’Donnell C, Van de Woestyne I (2015) Frontier metatechnologies and convexity: a restatement. Document de travail du LEM 2015-08, Universit’e de LilleGoogle Scholar
  62. Khan F, Salim R, Bloch H (2015) Nonparametric estimates of productivity and efficiency change in Australian broadacre agriculture. Aust J Agri Resour Econ 59(3):393–411CrossRefGoogle Scholar
  63. Kneip A, Park B, Simar L (1998) A note on the convergence of nonparametric DEA estimators for production efficiency scores. Econometric Theory 14(6):783–793CrossRefGoogle Scholar
  64. Kneip A, Simar L, Wilson P (2008) Asymptotics and consistent bootstraps for DEA estimators in nonparametric frontier models. Econometric Theory 24:1663–1697CrossRefGoogle Scholar
  65. Kneip A, Simar L, Wilson P (2011) A computationally efficient, consistent bootstrap for inference with non-parametric DEA estimators. Comput Econ 38(4):483–515CrossRefGoogle Scholar
  66. Kneip A, Simar L, Wilson P (2015) When bias kills the variance: central limit theorems for DEA and FDH efficiency scores. Econometric Theory 31(2):394–422CrossRefGoogle Scholar
  67. Korostelev A, Simar L, Tsybakov A (1995) Efficient estimation of monotone boundaries. Ann Stat 23(2):476–489CrossRefGoogle Scholar
  68. Koutsomanoli-Filippaki A, Margaritis D, Staikouras C (2012) Profit efficiency in the European Union banking industry: a directional technology distance function approach. J Prod Anal 37:277–293CrossRefGoogle Scholar
  69. Kumar S, Russell R (2002) Technological change, technological catch-up, and capital deepening: relative contributions to growth and convergence. Am Econ Rev 92(3):527–548CrossRefGoogle Scholar
  70. Kumbhakar S, Park B, Simar L, Tsionas E (2007) Nonparametric stochastic frontiers: a local maximum likelihood approach. J Econometrics 137(1):1–27CrossRefGoogle Scholar
  71. Kumtong R, Saosaovaphak A, Chaiboonsri C (2017) Measuring the total factor productivity for international seaport in South East Asia (ASEAN). J Adv Res Soc Sci Humanit 2(3):154–167Google Scholar
  72. Kuosmanen T, Kortelainen M (2012) Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints. J Prod Anal 38(1):11–28CrossRefGoogle Scholar
  73. Kuosmanen T, Sipiläinen T (2009) Exact decomposition of the Fisher ideal total factor productivity index. J Prod Anal 31(3):137–150CrossRefGoogle Scholar
  74. Laurenceson J, O’Donnell CJ (2014) New estimates and a decomposition of provincial productivity change in China. China Econ Rev 30:86–97CrossRefGoogle Scholar
  75. Li Q (1996) Nonparametric testing of closeness between two unknown distribution functions. Econometric Rev 15(3):261–274CrossRefGoogle Scholar
  76. Li Q (1999) Nonparametric testing the similarity of two unknown density functions: local power and bootstrap analysis. J Nonparametric Stat 11:189–213CrossRefGoogle Scholar
  77. McDonald J (2009) Using least squares and tobit in second stage DEA efficiency analyses. Eur J Oper Res 197(2):792–798CrossRefGoogle Scholar
  78. Milner C, Weyman-Jones T (2003) Relative national efficiency and country size: evidence for developing countries. Rev Dev Econ 7(1):1–14CrossRefGoogle Scholar
  79. Mohammad A (2015) Analysis of total factor productivity change among Indian commercial banks using Fare-Primont index. Asian J Res Banking Finance 5(2):183–193CrossRefGoogle Scholar
  80. Mugera A, Langemeier M, Ojede A (2016) Contributions of productivity and relative price change to farm-level profitability change. Am J Agri Econ 98(4):1210–1229CrossRefGoogle Scholar
  81. Nguyen PA, Simioni M (2015) Productivity and efficiency of Vietnamese banking system: new evidence using Färe-Primont index analysis. Appl Econ 47(41):4395–4407CrossRefGoogle Scholar
  82. O’Donnell C (2010a) DPIN version 1.0: A program for decomposing productivity index numbers. Centre for efficiency and productivity analysis working papers WP01/2010, University of QueenslandGoogle Scholar
  83. O’Donnell C (2010b) Measuring and decomposing agricultural productivity and profitability change. Aust J Agri Resour Econ 54(4):527–560CrossRefGoogle Scholar
  84. O’Donnell C (2012) Nonparametric estimates of the components of productivity and profitability change in U.S. agriculture. Am J Agri Econ 94(4):873–890CrossRefGoogle Scholar
  85. O’Donnell C, Nguyen K (2013) An econometric approach to estimating support prices and measures of productivity change in public hospitals. J Prod Anal 40(3):323–335CrossRefGoogle Scholar
  86. O’Donnell C, Rao D, Battese G (2008) Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Econ 34(2):231–255CrossRefGoogle Scholar
  87. O’Donnell C, Fallah-Fini S, Triantis K (2017) Measuring and analysing productivity change in a metafrontier framework. J Prod Anal 47(2):117–128CrossRefGoogle Scholar
  88. Park B, Simar L, Weiner C (2000) The FDH estimator for productivity efficiency scores. Econometric Theory 16(6):855–877CrossRefGoogle Scholar
  89. Pastor J, Asmild M, Lovell C (2011) The biennial Malmquist productivity change index. Socio-Econ Plann Sci 45(1):10–15CrossRefGoogle Scholar
  90. Podinovski V (2004a) Production trade-offs and weight restrictions in data envelopment analysis. J Oper Res Soc 55(12):1311–1322CrossRefGoogle Scholar
  91. Podinovski V (2004b) Suitability and redundancy of non-homogeneous weight restrictions for measuring the relative efficiency in DEA. Eur J Oper Res 154:380–395CrossRefGoogle Scholar
  92. Podinovski V (2007) Computation of efficient targets in DEA models with production trade-offs and weight restrictions. Eur J Oper Res 181(2):586–591CrossRefGoogle Scholar
  93. Podinovski V, Bouzdine-Chameeva T (2013) Weight restrictions and free production in data envelopment analysis. Oper Res 61(2):426–437CrossRefGoogle Scholar
  94. Portela M, Thanassoulis E (2005) Profitability of a sample of Portuguese bank branches and its decomposition into technical and allocative components. Eur J Oper Res 162(3):850–866CrossRefGoogle Scholar
  95. Rahman S, Salim R (2013) Six decades of total factor productivity change and sources of growth in Bangladesh agriculture (1948–2008). J Agri Econ 64(2):275–294CrossRefGoogle Scholar
  96. Ray S (1988) Data envelopment analysis: nondiscretionary inputs and efficiency: an alternative interpretation. Socio-Econ Plann Sci 22(4):167–176CrossRefGoogle Scholar
  97. Seiford L, Thrall R (1990) Recent developments in DEA: the mathematical programming approach to frontier analysis. J Econometrics 46(1–2):7–38CrossRefGoogle Scholar
  98. Seitz W (1966) Efficiency measures for steam-electric generating plants. In: Thirty Ninth annual meeting of the Western farm economics association, pp 143–151Google Scholar
  99. Shiu A, Zelenyuk V (2012) Production efficiency versus ownership. In: van Keilegom I, Wilson P (eds) Exploring research frontiers in contemporary statistics and econometrics. Springer, chap 2, pp 23–44Google Scholar
  100. Simar L, Wilson P (1998) Sensitivity analysis of efficiency scores: how to bootstrap in nonparametric frontier models. Manag Sci 44(1):49–61CrossRefGoogle Scholar
  101. Simar L, Wilson P (2000) Statistical inference in nonparametric frontier models: the state of the art. J Prod Anal 13(1):49–78CrossRefGoogle Scholar
  102. Simar L, Wilson P (2007) Estimation and inference in two-stage, semi-parametric models of production processes. J Econometrics 136:31–64CrossRefGoogle Scholar
  103. Simar L, Wilson P (2011a) Inference by the m out of n bootstrap in nonparametric frontier models. J Prod Anal 36(1):33–53CrossRefGoogle Scholar
  104. Simar L, Wilson P (2011b) Two-stage DEA: Caveat emptor. J Prod Anal 36(2):205–218CrossRefGoogle Scholar
  105. Simar L, Wilson P (2013) Estimation and inference in nonparametric frontier models: recent developments and perspectives. Found Trends Econometrics 5(3–4):183–337Google Scholar
  106. Simar L, Wilson P (2015) Statistical approaches for non-parametric frontier models: a guided tour. Int Stat Rev 83(1):77–110CrossRefGoogle Scholar
  107. Simar L, Zelenyuk V (2006) On testing equality of distributions of technical efficiency scores. Econometric Rev 25(4):497–522CrossRefGoogle Scholar
  108. Sitorus B (1966) Productive efficiency and redundant factors of production in traditional agriculture of underdeveloped countries. In: Thirty Ninth annual meeting of the Western farm economics association, pp 153–158Google Scholar
  109. Suhariyanto K, Thirtle C (2001) Asian agricultural productivity and convergence. J Agri Econ 52(3):96–110CrossRefGoogle Scholar
  110. Temoso O, Villano R, Hadley D (2015) Agricultural productivity, efficiency and growth in a semi-arid country: a case study of Botswana. Afr J Agri Resour Econ 10(3):192–206Google Scholar
  111. Tozer P, Villano R (2013) Decomposing productivity and efficiency among Western Australian grain producers. J Agri Resour Econ 38(3):312–326Google Scholar
  112. Tulkens H, Vanden Eeckaut P (1995) Non-parametric efficiency, progress and regress measures for panel data: methodological aspects. Eur J Oper Res 80:474–499CrossRefGoogle Scholar
  113. Zelenyuk V (2009) Power of significance test of dummy variables in two-stage efficiency analysis model. Appl Econ Lett 16(15):1493–1495CrossRefGoogle Scholar
  114. Zhang N, Zhou P, Choi Y (2013) Energy efficiency, CO\(_2\) emission performance and technology gaps in fossil fuel electricity generation in Korea: a meta-frontier non-radial directional distance function analysis. Energy Policy 56(May):653–662CrossRefGoogle Scholar
  115. Zhu J (2009) Quantitative models for performance evaluation and benchmarking: data envelopment analysis with spreadsheets, 2nd edn. SpringerGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.University of QueenslandBrisbaneAustralia

Personalised recommendations