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Inference in Dynamic, Nonparametric Models of Production: Central Limit Theorems for Malmquist Indices

Author

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  • Kneip, Alois
  • Simar, Léopold

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Wilson, Paul W.

Abstract

The Malmquist index gives a measure of productivity in dynamic settings and has been widely applied in empirical work. The index is typically estimated using envel- opment estimators, particularly data envelopment analysis (DEA) estimators. Until now, inference about productivity change measured by Malmquist indices has been problematic, including both inference regarding productivity change experienced by particular firms as well as mean productivity change. This paper establishes properties of a DEA-type estimator of distance to the conical hull of a variable returns to scale production frontier. In addition, properties of DEA estimators of Malmquist indices for individual producers are derived as well as properties of geometric means of these estimators. The latter requires new central limit theorem results, extending the work of Kneip, Simar, and Wilson (2015, Econometric Theory 31, 394–422). Simulation results are provided to give applied researchers an idea of how well inference may work in practice in finite samples. Our results extend easily to other productivity indices, including the Luenberger and Hicks–Moorsteen indices.

Suggested Citation

  • Kneip, Alois & Simar, Léopold & Wilson, Paul W., 2021. "Inference in Dynamic, Nonparametric Models of Production: Central Limit Theorems for Malmquist Indices," LIDAM Reprints ISBA 2021023, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvar:2021023
    DOI: https://doi.org/10.1017/s0266466620000237
    Note: In: Econometric Theory, Vol. 37, no.3, p. 537-572 (2021)
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    References listed on IDEAS

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    1. Léopold Simar & Paul W. Wilson, 1998. "Sensitivity Analysis of Efficiency Scores: How to Bootstrap in Nonparametric Frontier Models," Management Science, INFORMS, vol. 44(1), pages 49-61, January.
    2. Kneip, Alois & Simar, Léopold & Wilson, Paul W., 2008. "Asymptotics And Consistent Bootstraps For Dea Estimators In Nonparametric Frontier Models," Econometric Theory, Cambridge University Press, vol. 24(6), pages 1663-1697, December.
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    4. Kneip, Alois & Park, Byeong U. & Simar, Léopold, 1998. "A Note On The Convergence Of Nonparametric Dea Estimators For Production Efficiency Scores," Econometric Theory, Cambridge University Press, vol. 14(6), pages 783-793, December.
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    7. Ball, V. Eldon & Lovell, C.A. Knox & Luu, H. & Nehring, Richard F., 2004. "Incorporating Environmental Impacts in the Measurement of Agricultural Productivity Growth," Journal of Agricultural and Resource Economics, Western Agricultural Economics Association, vol. 29(3), pages 1-25, December.
    8. S.‐O. Jeong & B. U. Park, 2006. "Large Sample Approximation of the Distribution for Convex‐Hull Estimators of Boundaries," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 139-151, March.
    9. Park, B.U. & Jeong, S.-O. & Simar, L., 2010. "Asymptotic distribution of conical-hull estimators of directional edges," LIDAM Reprints ISBA 2010025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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    Cited by:

    1. Manh Pham & Léopold Simar & Valentin Zelenyuk, 2024. "Statistical Inference for Aggregation of Malmquist Productivity Indices," Operations Research, INFORMS, vol. 72(4), pages 1615-1629, July.
    2. Léopold Simar & Valentin Zelenyuk & Shirong Zhao, 2023. "Further improvements of finite sample approximation of central limit theorems for envelopment estimators," Journal of Productivity Analysis, Springer, vol. 59(2), pages 189-194, April.
    3. Jin, Qianying & Kerstens, Kristiaan & Van de Woestyne, Ignace, 2020. "Metafrontier productivity indices: Questioning the common convexification strategy," European Journal of Operational Research, Elsevier, vol. 283(2), pages 737-747.
    4. Valentin Zelenyuk, 2023. "Productivity analysis: roots, foundations, trends and perspectives," Journal of Productivity Analysis, Springer, vol. 60(3), pages 229-247, December.
    5. Léopold Simar & Valentin Zelenyuk, 2018. "Improving Finite Sample Approximation by Central Limit Theorems for DEA and FDH efficiency scores," CEPA Working Papers Series WP072018, School of Economics, University of Queensland, Australia.
    6. Paul W. Wilson & Shirong Zhao, 2023. "Investigating the performance of Chinese banks over 2007–2014," Annals of Operations Research, Springer, vol. 321(1), pages 663-692, February.
    7. Arocena, Pablo & Saal, David S. & Urakami, Takuya & Zschille, Michael, 2020. "Measuring and decomposing productivity change in the presence of mergers," European Journal of Operational Research, Elsevier, vol. 282(1), pages 319-333.
    8. Léopold Simar & Paul W. Wilson, 2023. "Another look at productivity growth in industrialized countries," Journal of Productivity Analysis, Springer, vol. 60(3), pages 257-272, December.
    9. Robin C. Sickles, 2023. "Special Symposium on Lifetime Achievements of Rolf Färe and Shawna Grosskopf," Journal of Productivity Analysis, Springer, vol. 60(3), pages 227-228, December.
    10. Léopold Simar & Paul W. Wilson, 2020. "Hypothesis testing in nonparametric models of production using multiple sample splits," Journal of Productivity Analysis, Springer, vol. 53(3), pages 287-303, June.
    11. Simar, Léopold & Zelenyuk, Valentin, 2020. "Improving finite sample approximation by central limit theorems for estimates from Data Envelopment Analysis," European Journal of Operational Research, Elsevier, vol. 284(3), pages 1002-1015.
    12. Simar, Léopold & Zelenyuk, Valentin & Zhao, Shirong, 2024. "Inference for aggregate efficiency: Theory and guidelines for practitioners," European Journal of Operational Research, Elsevier, vol. 316(1), pages 240-254.
    13. Mette Asmild & Dorte Kronborg & Anders Rønn-Nielsen, 2018. "Testing productivity change, frontier shift, and efficiency change," IFRO Working Paper 2018/07, University of Copenhagen, Department of Food and Resource Economics.
    14. Daraio, Cinzia & Di Leo, Simone & Simar, Léopold, 2024. "Conical FDH Estimators of Directional Distances and Luenberger Productivity Indices for General Technologies," LIDAM Discussion Papers ISBA 2024009, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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