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Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics


  • Peter Schmidt
  • Antonio Alvarez
  • Christine Amsler


In this paper, we are interested in a stochastic frontier model in which observable characteristics of the firms affect their levels of technical inefficiency. Let u ≥ 0 be the one-sided error reflecting technical inefficiency, and let z be a set of variables that affect u. We write u as u(z,δ) to reflect its dependence on z and some parameters δ. Various models in the existing literature specify the distribution of u(z,δ). We are interested in models that satisfy the scaling property, which says that u(z,δ) can be written as a scaling function h(z, δ) times a random variable u* that does not depend on z. This property implies that changes in z affect the scale but not the shape of u(z,δ). This paper reviews the existing literature and identifies models that do and do not have the scaling property. It also discusses practical advantages of the scaling property. The scaling property is argued to be intuitively appealing; it allows estimation by nonlinear least squares; it allows a distribution-free interpretation of the parameters δ that show how z affects inefficiency; and it underlies the model of Battese and Coelli, Journal of Productivity Analysis, 1992, which is currently the only model to allow correlation over time when inefficiency depends on firm characteristics. The paper shows how to test the hypothesis of scaling, and other interesting hypotheses, in the context of the model of Wang, Journal of Productivity Analysis, 2002. Finally, two empirical examples are given.

Suggested Citation

  • Peter Schmidt & Antonio Alvarez & Christine Amsler, 2004. "Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics," Econometric Society 2004 Far Eastern Meetings 520, Econometric Society.
  • Handle: RePEc:ecm:feam04:520

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    References listed on IDEAS

    1. Han, Chirok & Orea, Luis & Schmidt, Peter, 2005. "Estimation of a panel data model with parametric temporal variation in individual effects," Journal of Econometrics, Elsevier, vol. 126(2), pages 241-267, June.
    2. Caudill, Steven B. & Ford, Jon M., 1993. "Biases in frontier estimation due to heteroscedasticity," Economics Letters, Elsevier, vol. 41(1), pages 17-20.
    3. Reifschneider, David & Stevenson, Rodney, 1991. "Systematic Departures from the Frontier: A Framework for the Analysis of Firm Inefficiency," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 32(3), pages 715-723, August.
    4. Cuesta, Rafael A. & Orea, Luis, 2002. "Mergers and technical efficiency in Spanish savings banks: A stochastic distance function approach," Journal of Banking & Finance, Elsevier, vol. 26(12), pages 2231-2247.
    5. Kumbhakar, Subal C., 1990. "Production frontiers, panel data, and time-varying technical inefficiency," Journal of Econometrics, Elsevier, vol. 46(1-2), pages 201-211.
    6. Hung-jen Wang & Peter Schmidt, 2002. "One-Step and Two-Step Estimation of the Effects of Exogenous Variables on Technical Efficiency Levels," Journal of Productivity Analysis, Springer, vol. 18(2), pages 129-144, September.
    7. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-430, March.
    8. Stevenson, Rodney E., 1980. "Likelihood functions for generalized stochastic frontier estimation," Journal of Econometrics, Elsevier, vol. 13(1), pages 57-66, May.
    9. Battese, George E. & Coelli, Tim J., 1988. "Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data," Journal of Econometrics, Elsevier, vol. 38(3), pages 387-399, July.
    10. Kumbhakar, Subal C & Ghosh, Soumendra & McGuckin, J Thomas, 1991. "A Generalized Production Frontier Approach for Estimating Determinants of Inefficiency in U.S. Dairy Farms," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(3), pages 279-286, July.
    11. Wang, Hung-Jen, 2003. "A Stochastic Frontier Analysis of Financing Constraints on Investment: The Case of Financial Liberalization in Taiwan," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(3), pages 406-419, July.
    12. Caudill, Steven B & Ford, Jon M & Gropper, Daniel M, 1995. "Frontier Estimation and Firm-Specific Inefficiency Measures in the Presence of Heteroscedasticity," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(1), pages 105-111, January.
    13. Hung-Jen Wang, 2002. "Heteroscedasticity and Non-Monotonic Efficiency Effects of a Stochastic Frontier Model," Journal of Productivity Analysis, Springer, vol. 18(3), pages 241-253, November.
    14. Battese, G E & Coelli, T J, 1995. "A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data," Empirical Economics, Springer, vol. 20(2), pages 325-332.
    15. Aigner, Dennis & Lovell, C. A. Knox & Schmidt, Peter, 1977. "Formulation and estimation of stochastic frontier production function models," Journal of Econometrics, Elsevier, vol. 6(1), pages 21-37, July.
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    More about this item


    stochastic frontier; scaling property; technical inefficiency;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation


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