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On The Distribution of Estimated Technical Efficiency in Stochastic Frontier Models

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Abstract

We consider a stochastic frontier model with error [epsilon]=v-u, where v is normal and u is half normal. We derive the distribution of the usual estimate of u,E(u[epsilon]). We show that as the variance of v approaches zero, E(u[epsilon])-u converges to zero, while as the variance of v approaches infinity, E(u[epsilon]) converges to E(u). We graph the density of E(u[epsilon]) for intermediate cases. To show that E(u[epsilon]) is a shrinkage of u towards its mean, we derive and graph the distribution of E(u[epsilon]) conditional on u. We also consider the distribution of estimated inefficiency in the fixed-effects panel data setting.
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  • Wei Siang Wang & Peter Schmidt, 2007. "On The Distribution of Estimated Technical Efficiency in Stochastic Frontier Models," CEPA Working Papers Series WP022007, School of Economics, University of Queensland, Australia.
    • Handle: RePEc:qld:uqcepa:25
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    1. Greene, William H., 1980. "Maximum likelihood estimation of econometric frontier functions," Journal of Econometrics, Elsevier, vol. 13(1), pages 27-56, May.
    2. Waldman, Donald M., 1984. "Properties of technical efficiency estimators in the stochastic frontier model," Journal of Econometrics, Elsevier, vol. 25(3), pages 353-364, July.
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    5. William C. Horrace & Peter Schmidt, 2002. "Confidence Statements for Efficiency Estimates from Stochastic Frontier Models," Econometrics 0206006, University Library of Munich, Germany.
    6. Meeusen, Wim & van den Broeck, Julien, 1977. "Efficiency Estimation from Cobb-Douglas Production Functions with Composed Error," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(2), pages 435-444, June.
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    Cited by:

    1. Eduardo Fé & Richard Hofler, 2013. "Count data stochastic frontier models, with an application to the patents–R&D relationship," Journal of Productivity Analysis, Springer, vol. 39(3), pages 271-284, June.
    2. William C. Horrace & Hyunseok Jung & Yi Yang, 2023. "The conditional mode in parametric frontier models," Journal of Productivity Analysis, Springer, vol. 60(3), pages 333-343, December.
    3. Federico Belotti & Giuseppe Ilardi, 2012. "Consistent Estimation of the “True” Fixed-effects Stochastic Frontier Model," CEIS Research Paper 231, Tor Vergata University, CEIS, revised 18 Apr 2012.
    4. Daniel Wikström, 2015. "A finite sample improvement of the fixed effects estimator applied to technical inefficiency," Journal of Productivity Analysis, Springer, vol. 43(1), pages 29-46, February.
    5. Subal C. Kumbhakar & Christopher F. Parmeter & Valentin Zelenyuk, 2022. "Stochastic Frontier Analysis: Foundations and Advances I," Springer Books, in: Subhash C. Ray & Robert G. Chambers & Subal C. Kumbhakar (ed.), Handbook of Production Economics, chapter 8, pages 331-370, Springer.
    6. Jun Cai & Qu Feng & William C. Horrace & Guiying Laura Wu, 2021. "Wrong skewness and finite sample correction in the normal-half normal stochastic frontier model," Empirical Economics, Springer, vol. 60(6), pages 2837-2866, June.
    7. Luis Orea & Jevgenijs Steinbuks, 2018. "Estimating Market Power In Homogenous Product Markets Using A Composed Error Model: Application To The California Electricity Market," Economic Inquiry, Western Economic Association International, vol. 56(2), pages 1296-1321, April.
    8. Orea, Luis, 2019. "The Econometric Measurement of Firms’ Efficiency," Efficiency Series Papers 2019/02, University of Oviedo, Department of Economics, Oviedo Efficiency Group (OEG).
    9. Jun Cai & William C. Horrace & Christopher F. Parmeter, 2021. "Density deconvolution with Laplace errors and unknown variance," Journal of Productivity Analysis, Springer, vol. 56(2), pages 103-113, December.
    10. Yujian Jin & Lihong Yu & Yan Wang, 2022. "Green Total Factor Productivity and Its Saving Effect on the Green Factor in China’s Strategic Minerals Industry from 1998–2017," IJERPH, MDPI, vol. 19(22), pages 1-20, November.
    11. William Horrace & Christopher Parmeter, 2011. "Semiparametric deconvolution with unknown error variance," Journal of Productivity Analysis, Springer, vol. 35(2), pages 129-141, April.
    12. Gómez-Gallego, Juan Cándido & Gómez-García, Juan & Pérez-Cárceles, María Concepción, 2012. "Appropriate Distribution of Cost Inefficiency Estimates as Predictor of Financial Instability /La distribución de la ineficiencia estimada como predictor de inestabilidad financiera," Estudios de Economia Aplicada, Estudios de Economia Aplicada, vol. 30, pages 1071(12.)-1, Diciembre.
    13. Shinji Yane & Sanford Berg, 2013. "Sensitivity analysis of efficiency rankings to distributional assumptions: applications to Japanese water utilities," Applied Economics, Taylor & Francis Journals, vol. 45(17), pages 2337-2348, June.
    14. Anup Bhandari & Pradip Maiti, 2012. "Efficiency of the Indian leather firms: some results obtained using the two conventional methods," Journal of Productivity Analysis, Springer, vol. 37(1), pages 73-93, February.
    15. Qu Feng & William Horrace & Guiying Laura Wu, 2013. "Wrong Skewness and Finite Sample Correction in Parametric Stochastic Frontier Models Abstract: In parametric stochastic frontier models, the composed error is specified as the sum of a two-sided noise," Center for Policy Research Working Papers 154, Center for Policy Research, Maxwell School, Syracuse University.
    16. Thomas Graaff, 2020. "On the estimation of spatial stochastic frontier models: an alternative skew-normal approach," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 64(2), pages 267-285, April.
    17. Zangin Zeebari & Kristofer Månsson & Pär Sjölander & Magnus Söderberg, 2023. "Regularized conditional estimators of unit inefficiency in stochastic frontier analysis, with application to electricity distribution market," Journal of Productivity Analysis, Springer, vol. 59(1), pages 79-97, February.
    18. Federico Belotti & Giuseppe Ilardi & Andrea Piano Mortari, 2019. "Estimation of Stochastic Frontier Panel Data Models with Spatial Inefficiency," CEIS Research Paper 459, Tor Vergata University, CEIS, revised 30 May 2019.
    19. Wei Wang & Christine Amsler & Peter Schmidt, 2011. "Goodness of fit tests in stochastic frontier models," Journal of Productivity Analysis, Springer, vol. 35(2), pages 95-118, April.
    20. Phill Wheat & Alexander D. Stead & William H. Greene, 2019. "Robust stochastic frontier analysis: a Student’s t-half normal model with application to highway maintenance costs in England," Journal of Productivity Analysis, Springer, vol. 51(1), pages 21-38, February.
    21. Alecos Papadopoulos, 2023. "The noise error component in stochastic frontier analysis," Empirical Economics, Springer, vol. 64(6), pages 2795-2829, June.
    22. María Concepción Pérez-Cárceles & Juan Cándido Gómez-Gallego & Juan Gómez-García, 2016. "Distribution of cost inefficiency in stochastic frontier approach: evidence from Spanish banking," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(16), pages 3030-3041, December.
    23. Stead, Alexander D. & Wheat, Phill & Greene, William H., 2023. "Robust maximum likelihood estimation of stochastic frontier models," European Journal of Operational Research, Elsevier, vol. 309(1), pages 188-201.
    24. Bellio, Ruggero & Grassetti, Luca, 2011. "Semiparametric stochastic frontier models for clustered data," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 71-83, January.
    25. Daniel Wikström, 2016. "Modified fixed effects estimation of technical inefficiency," Journal of Productivity Analysis, Springer, vol. 46(1), pages 83-86, August.

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