Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T22:18:36.462Z Has data issue: false hasContentIssue false
  • English
  • Français

INFERENCE IN DYNAMIC, NONPARAMETRIC MODELS OF PRODUCTION: CENTRAL LIMIT THEOREMS FOR MALMQUIST INDICES

Published online by Cambridge University Press:  15 June 2020

Alois Kneip ,
Léopold Simar  and
Paul W. Wilson
Alois Kneip
Affiliation:
Universität Bonn
Léopold Simar
Affiliation:
Université Catholique de Louvain-la-Neuve
Paul W. Wilson*
Affiliation:
Clemson University
*
Address correspondence to Paul W. Wilson, Department of Economics and School of Computing, Clemson University, Clemson, SC, USA; e-mail: pww@clemson.edu.
Get access Rights & Permissions [Opens in a new window]

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 envelopment 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.

Type
ARTICLES
Information
Econometric Theory , Volume 37 , Issue 3 , June 2021 , pp. 537 - 572
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Technical support from the Cyber Infrastructure Technology Integration group at Clemson University is gratefully acknowledged. Any remaining errors are solely our responsibility.

References

REFERENCES

Bahadur, R.R. & Savage, L.J. (1956) The nonexistence of certain statistical procedures in nonparametric problems. The Annals of Mathematical Statistics 27, 1115–1122.CrossRefGoogle Scholar
Ball, V.E., Lovell, C.A.K., Luu, H., & Nehring, R. (2004) Incorporating environmental impacts in the measurement of agricultural productivity growth. Journal of Agricultural and Resource Economics 29, 436–460.Google Scholar
Casu, B., Ferrari, A., & Zhao, T. (2013) Regulatory reform and productivity change in Indian banking. The Review of Economics and Statistics 95, 1066–1077.CrossRefGoogle Scholar
Caves, D.W., Christensen, L.R., & Diewert, W.E. (1982) The economic theory of index numbers and the measurement of input, output and productivity. Econometrica 50, 1393–1414.CrossRefGoogle Scholar
Delis, M.D., Molyneux, P., & Pasiouras, F. (2011) Regulations and productivity growth in banking: Evidence from transition economies. Journal of Money, Credit and Banking 43, 735–764.CrossRefGoogle Scholar
Egilmez, G. & McAvoy, D. (2013) Benchmarking road safety of U.S. states: A DEA-based Malmquist productivity index approach. Accident Analysis & Prevention 53, 55–64.CrossRefGoogle ScholarPubMed
Färe, R. (1988) Fundamentals of Production Theory. Springer-Verlag.CrossRefGoogle Scholar
Färe, R., Grosskopf, S., Lindgren, B., & Roos, P. (1992) Productivity changes in Swedish pharmacies 1980–1989: A non-parametric Malmquist approach. Journal of Productivity Analysis 3, 85–101.CrossRefGoogle Scholar
Färe, R., Grosskopf, S., & Lovell, C.A.K. (1985) The Measurement of Efficiency of Production. Kluwer-Nijhoff Publishing.CrossRefGoogle Scholar
Färe, R., Grosskopf, S., & Margaritis, D. (2008) Productivity and efficiency: Malmquist and more. In Fried, H., Lovell, C.A.K., & Schmidt, S. (eds.), The Measurement of Productive Efficiency, Chapter 5, 2nd Edition, pp. 522–621. Oxford University Press.CrossRefGoogle Scholar
Farrell, M.J. (1957) The measurement of productive efficiency. Journal of the Royal Statistical Society A 120, 253–281.CrossRefGoogle Scholar
Gray, H.L. & Schucany, R.R. (1972) The Generalized Jackknife Statistic. Marcel Decker, Inc.Google Scholar
Grifell-Tatjé, E. & Lovell, C.A.K. (1995) A note on the Malmquist productivity index. Economic Letters 47, 169–175.CrossRefGoogle Scholar
Jeong, S.O. (2004) Asymptotic distribution of DEA efficiency scores. Journal of the Korean Statistical Society 33, 449–458.Google Scholar
Jeong, S.O. & Park, B.U. (2006) Large sample approximation of the distribution for convex-hull estimators of boundaries. Scandinavian Journal of Statistics 33, 139–151.CrossRefGoogle Scholar
Kneip, A., Park, B., & Simar, L. (1998) A note on the convergence of nonparametric DEA efficiency measures. Econometric Theory 14, 783–793.CrossRefGoogle Scholar
Kneip, A., Simar, L., & Wilson, P.W. (2008) Asymptotics and consistent bootstraps for DEA estimators in non-parametric frontier models. Econometric Theory 24, 1663–1697.CrossRefGoogle Scholar
Kneip, A., Simar, L., & Wilson, P.W. (2015) When bias kills the variance: Central limit theorems for DEA and FDH efficiency scores. Econometric Theory 31, 394–422.CrossRefGoogle Scholar
Kneip, A., Simar, L., & Wilson, P.W. (2016) Testing hypotheses in nonparametric models of production. Journal of Business and Economic Statistics 34, 435–456.CrossRefGoogle Scholar
Malmquist, S. (1953) Index numbers and indifference surfaces. Trabajos de Estatística y de Investigación Operativa 4, 209–242.Google Scholar
Nishimizu, M. & Page, J. (1982) Total factor productivity growth, technological progress and technical efficiency change: Dimensions of productivity change in Yugoslavia. Economic Journal 92, 920–936.CrossRefGoogle Scholar
Park, B.U., Jeong, S.-O., & Simar, L. (2010) Asymptotic distribution of conical-hull estimators of directional edges. Annals of Statistics 38, 1320–1340.CrossRefGoogle Scholar
Ramakrishna, G., Ramulu, M., & Kumar, B.P. (2016) Financial crisis and the performance of commercial banks: Indian experience. Journal of International Economics 7, 35–55.Google Scholar
Shephard, R.W. (1970) Theory of Cost and Production Functions. Princeton University Press.Google Scholar
Simar, L., Vanhems, A., & Wilson, P.W. (2012) Statistical inference for DEA estimators of directional distances. European Journal of Operational Research 220, 853–864.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (1998) Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science 44, 49–61.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (1999) Estimating and bootstrapping Malmquist indices. European Journal of Operational Research 115, 459–471.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (2011) Inference by the m out of n bootstrap in nonparametric frontier models. Journal of Productivity Analysis 36, 33–53.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (2013) Estimation and inference in nonparametric frontier models: Recent developments and perspectives. Foundations and Trends in Econometrics 5, 183–337.CrossRefGoogle Scholar
Simar, L. & Wilson, P.W. (2015) Statistical approaches for non-parametric frontier models: A guided tour. International Statistical Review 83, 77–110.CrossRefGoogle Scholar
Wilson, P.W. (2008) FEAR: A software package for frontier efficiency analysis with R. Socio-Economic Planning Sciences 42, 247–254.CrossRefGoogle Scholar
Wilson, P.W. (2011) Asymptotic properties of some non-parametric hyperbolic efficiency estimators. In Van Keilegom, I. & Wilson, P. W. (eds.), Exploring Research Frontiers in Contemporary Statistics and Econometrics, pp. 115–150. Springer-Verlag.CrossRefGoogle Scholar
Wilson, P.W. (2018) Dimension reduction in nonparametric models of production. European Journal of Operational Research 267, 349–367.CrossRefGoogle Scholar
Supplementary material: PDF

Kneip et al. supplementary material

Kneip et al. supplementary material

Download Kneip et al. supplementary material(PDF)
PDF 424 KB