Deterministic Frontier Analysis

  • Christopher J. O’DonnellEmail author


Production frontiers are often represented by distance, revenue, cost and/or profit functions. These functions can sometimes be written in the form of regression models in which the explanatory variables are deterministic (i.e., not random). This chapter explains how to estimate and draw inferences concerning the unknown parameters in so-called deterministic frontier models (DFMs). It then explains how the estimated parameters can be used to predict levels of efficiency and analyse productivity change. The focus is on least squares estimators and predictors.


  1. Afriat S (1972) Efficiency estimation of production functions. Int Econ Rev 13(3):568–598Google Scholar
  2. Aigner D, Chu S (1968) On estimating the industry production function. Am Econ Rev 58(4):826–839Google Scholar
  3. Althin R, Fare R, Grosskopf S (1996) Profitability and productivity changes: an application to Swedish pharmacies. Ann Oper Res 66:219–230Google Scholar
  4. Amsler C, Leonard M, Schmidt P (2013) Estimation and inference in parametric deterministic frontier models. J Prod Anal 40(3):293–305Google Scholar
  5. Baltagi B, Griffin J (1988) A general index of technical change. J Polit Econ 96(1):20–41Google Scholar
  6. Banker R, Conrad R, Strauss R (1986) A comparative application of data envelopment analysis and translog methods: an illustrative study of hospital production. Manage Sci 32(1):30–44Google Scholar
  7. Banker R, Chang H, Cunningham R (2003) The public accounting industry production function. J Acc Econ 35(2):255–281Google Scholar
  8. Bhargava A, Franzini L, Narendranathan W (1982) Serial correlation in the fixed effects model. Rev Econ Stud 49(4):533–549Google Scholar
  9. Bickel P, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1:1071–1095Google Scholar
  10. Breusch T (1978) Testing for autocorrelation in dynamic linear models. Aust Econ Papers 17:334–355Google Scholar
  11. Breusch T, Pagan A (1979) A simple test for heteroskedasticity and random coefficient variation. Econometrica 47:1287–1294Google Scholar
  12. Breusch T, Pagan A (1980) The Lagrange multiplier test and its applications to model specification in econometrics. Rev Econ Stud 47:239–253Google Scholar
  13. Caselli F, Coleman W (2006) The world technology frontier. Am Econ Rev 93(3):499–522Google Scholar
  14. Chaudhary M, Khan M, Naqvi K, Ahmad M (1999) Estimates of farm output supply and input demand elasticities: the translog profit function approach [with comments]. Pak Dev Rev 37(4):1031–1050Google Scholar
  15. Coelli T, Perelman S (1999) A comparison of parametric and non-parametric distance functions: with application to European railways. Eur J Oper Res 117(2):326–339Google Scholar
  16. Coelli T, Estache A, Perelman S, Trujillo L (2003) A primer on efficiency measurement for utilities and transport regulators. WBI Development Studies, The World BankGoogle Scholar
  17. Coelli T, Gautier A, Perelman S, Saplacan-Pop R (2013) Estimating the cost of improving quality in electricity distribution: a parametric distance function approach. Energy Policy 53:287–297Google Scholar
  18. Cragg J, Donald S (1993) Testing identifiability and specification in instrumental variable models. Econometric Theory 9:222–240Google Scholar
  19. Das A, Kumbhakar S (2012) Productivity and efficiency dynamics in Indian banking: an input distance function approach incorporating quality of inputs and outputs. J Appl Econometrics 27(2):205–234Google Scholar
  20. 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
  21. Diewert W (1980) Aggregation problems in the measurement of capital. In: Usher D (ed) The measurement of capital. University of Chicago PressGoogle Scholar
  22. Diewert W, Fox K (2010) Malmquist and Törnqvist productivity indexes: returns to scale and technical progress with imperfect competition. J Econ 101(1):73–95Google Scholar
  23. Durbin J, Watson G (1950) Testing for serial correlation in least squares regression. I. Biometrika 37(3–4):409–428Google Scholar
  24. Durbin J, Watson G (1951) Testing for serial correlation in least squares regression. II. Biometrika 38(1–2):159–179Google Scholar
  25. Enders W (2004) Applied econometric time series, 2nd edn. Wiley, New JerseyGoogle Scholar
  26. Engle R, Granger C (1987) Cointegration and error-correction: representation, estimation and testing. Econometrica 55:251–276Google Scholar
  27. Engle R, Yoo B (1987) Forecasting and testing in co-integrated systems. J Econometrics 35(1):143–159Google Scholar
  28. Fan Y (1994) Testing the goodness of fit of a parametric density function by Kernel method. Econometric Theory 10(2):316–356Google Scholar
  29. Färe R, Grosskopf S, Lovell C, Yaisawarng S (1993) Derivation of shadow prices for undesirable outputs: a distance function approach. Rev Econ Stat 75(2):374–380Google Scholar
  30. Ferrari A (2006) The internal market and hospital efficiency: a stochastic distance function approach. Appl Econ 38(18):2121–2130Google Scholar
  31. Førsund F, Lovell C, Schmidt P (1980) A survey of frontier production functions and of their relationship to efficiency measurement. J Econometrics 13(1):5–25Google Scholar
  32. Fuentes H, Grifell-Tatjé E, Sergio P (2001) A parametric distance function approach for Malmquist productivity index estimation. J Prod Anal 15(2001):79–94Google Scholar
  33. Glesjer H (1969) A new test for heteroskedasticity. J Am Stat Assoc 64:316–323Google Scholar
  34. Godfrey L (1978a) Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica 46(6):1293–1301Google Scholar
  35. Godfrey L (1978b) Testing for multiplicative heteroskedasticity. J Econometrics 8:227–236Google Scholar
  36. Goldfeld S, Quandt R (1972) Nonlinear methods in econometrics. North-Holland, AmsterdamGoogle Scholar
  37. Greene W (1980a) Maximum likelihood estimation of econometric frontier functions. J Econometrics 13(1):27–56Google Scholar
  38. Greene W (1980b) On the estimation of a flexible frontier production model. J Econometrics 13(1):101–115Google Scholar
  39. Greene W (2008) Econometric analysis, 6th edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  40. Grosskopf S, Margaritis D, Valdmanis V (1995) Estimating output substitutability of hospital services: a distance function approach. Eur J Oper Res 80(1995):575–587Google Scholar
  41. Hajargasht G, Coelli T, Rao D (2008) A dual measure of economies of scope. Econ Lett 100(2):185–188Google Scholar
  42. Hill R, Griffiths W, Lim G (2011) Principles of econometrics, 4th edn. Wiley, New JerseyGoogle Scholar
  43. Hsieh CT, Klenow P (2009) Misallocation and manufacturing TFP in China and India. Q J Econ 124(4):1403–1448Google Scholar
  44. Koenker R, Bassett G (1982) Robust tests for heteroscedasticity based on regression quantiles. Econometrica 50(1):43–61Google Scholar
  45. Kopp R, Diewert W (1982) The decomposition of frontier cost function deviations into measures of technical and allocative efficiency. J Econometrics 19(2–3):319–331Google Scholar
  46. Kumbhakar S (1997) Modeling allocative inefficiency in a translog cost function and cost share equations: an exact relationship. J Econometrics 76(1–2):351–356Google Scholar
  47. Kumbhakar S (2001) Estimation of profit functions when profit is not maximum. Am J Agri Econ 83(1):1–19Google Scholar
  48. Kumbhakar S (2006) Specification and estimation of nonstandard profit functions. Empirical Econ 31(1):243–260Google Scholar
  49. Kumbhakar S, Bhattacharyya A (1992) Price distortions and resource-use efficiency in Indian agriculture: a restricted profit function approach. Rev Econ Stat 74(2):231–239Google Scholar
  50. Kumbhakar S, Lovell C (2000) Stochastic frontier analysis. Cambridge University PressGoogle Scholar
  51. Nadiri M, Nandi B (1999) Technical change, markup, divestiture, and productivity growth in the U.S. telecommunications industry. Rev Econ Stat 81(3):488–498Google Scholar
  52. Newey W, Powell J (1987) Asymmetric least squares estimation and testing. Econometrica 55(4):819–847Google Scholar
  53. O’Donnell C (2016) Using information about technologies, markets and firm behaviour to decompose a proper productivity index. J Econometrics 190(2):328–340Google Scholar
  54. O’Donnell C, Coelli T (2005) A Bayesian approach to imposing curvature on distance functions. J Econometrics 126(2):493–523Google Scholar
  55. O’Mahony M, Timmer M (2009) Output, input and productivity measures at the industry level: the EU KLEMS database. Econ J 119(538):F374–F403Google Scholar
  56. Orea L (2002) Parametric decomposition of a generalized Malmquist productivity index. J Prod Anal 18:5–22Google Scholar
  57. Pagan A, Ullah A (1999) Nonparametric econometrics. Cambridge University Press, Cambridge, UKGoogle Scholar
  58. Pesaran H (2004) General diagnostic tests for cross section dependence in panels. CESifo working paper 1229, Ifo Institute, Center for Economic StudiesGoogle Scholar
  59. Ray S (1998) Measuring scale efficiency from a translog production function. J Prod Anal 11(2):183–194Google Scholar
  60. Reig-Martinez E, Picazo-Tadeo A, Hernandez-Sancho F (2001) The calculation of shadow prices for industrial wastes using distance functions: an analysis for Spanish ceramic pavements firms. Int J Prod Econ 69(2001):277–285Google Scholar
  61. Richmond J (1974) Estimating the efficiency of production. Int Econ Rev 15(2):515–521Google Scholar
  62. Rosenblatt M (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann Stat 3:1–14Google Scholar
  63. Schmidt P (1976) On the statistical estimation of parametric frontier production functions. Rev Econ Stat 58(2):238–239Google Scholar
  64. Solow R (1957) Technical change and the aggregate production function. Rev Econ Stat 39(3):312–320Google Scholar
  65. Stern D (2010) Derivation of the Hicks, or direct, elasticity of substitution using the input distance function. Econ Lett 108(3):349–351Google Scholar
  66. Tsekouras K, Pantzios CJ, Karagiannis G (2004) Malmquist productivity index estimation with zero-value variables: the case of Greek prefectural training councils. Int J Prod Econ 89(1):95–106Google Scholar
  67. Vardanyan M, Noh DW (2006) Approximating pollution abatement costs via alternative specifications of a multi-output production technology: a case of the US electric utility industry. J Environ Manage 80(2)Google Scholar
  68. White H (1980) A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 48:817–838Google Scholar
  69. Winsten C (1957) Discussion on Mr. Farrell’s paper. J R Stat Soc Ser A (General) 120(3):282–84Google Scholar
  70. Zhang T, Garvey E (2008) A comparative analysis of multi-output frontier models. J Zhejiang Univ Sci A 9(10):1426–1436Google Scholar
  71. Zheng S, Bloch H (2014) Australia’s mining productivity decline: implications for MFP measurement. J Prod Anal 41(2):201–212Google Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.University of QueenslandBrisbaneAustralia

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