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Non Parametric Estimation of a Polarization Measure

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  • Gordon Anderson
  • Oliver Linton
  • Yoon-Jae Wang

Abstract

This paper develops a methodology for nonparametric estimation of a polarization measure due to Anderson Ge and Leo (2006) based on kernel estimation techniques. We give the asymptotic theory of our estimator, which in some cases is non standard due to boundary value problems. We also propose a method for conducting inference based on estimation of unknown quantities in the limiting distribution and show that our method yields consistent inference in all cases we consider. We investigate finite sample proerties of our estimator by simulation methods. We give an application to the study of polarization in China in recent years.

Suggested Citation

  • Gordon Anderson & Oliver Linton & Yoon-Jae Wang, 2009. "Non Parametric Estimation of a Polarization Measure," Working Papers tecipa-363, University of Toronto, Department of Economics.
    • Handle: RePEc:tor:tecipa:tecipa-363
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    References listed on IDEAS

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    1. Oliver Linton & Esfandiar Maasoumi & Yoon-Jae Whang, 2005. "Consistent Testing for Stochastic Dominance under General Sampling Schemes," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(3), pages 735-765.
    2. Esteban, Joan & Ray, Debraj, 1994. "On the Measurement of Polarization," Econometrica, Econometric Society, vol. 62(4), pages 819-851, July.
    3. Andrews, Donald W.K., 1995. "Nonparametric Kernel Estimation for Semiparametric Models," Econometric Theory, Cambridge University Press, vol. 11(3), pages 560-586, June.
    4. Victor Chernozhukov & Han Hong & Elie Tamer, 2007. "Estimation and Confidence Regions for Parameter Sets in Econometric Models," Econometrica, Econometric Society, vol. 75(5), pages 1243-1284, September.
    5. Jean-Yves Duclos & Joan Esteban & Debraj Ray, 2004. "Polarization: Concepts, Measurement, Estimation," Econometrica, Econometric Society, vol. 72(6), pages 1737-1772, November.
    6. Clemons, Traci E. & Jr., Edwin L. Bradley, 2000. "A nonparametric measure of the overlapping coefficient," Computational Statistics & Data Analysis, Elsevier, vol. 34(1), pages 51-61, July.
    7. Knight, J. & Shi, L. & Renwei, Z., 1999. "A Spatial Analysis of Wages and Incomes in Urban China: Divergent Means, Convergent Inequality," Economics Series Working Papers 99209, University of Oxford, Department of Economics.
    8. Donald W. K. Andrews, 1999. "Estimation When a Parameter Is on a Boundary," Econometrica, Econometric Society, vol. 67(6), pages 1341-1384, November.
    9. Anderson, Gordon & Ge, Ying, 2005. "The size distribution of Chinese cities," Regional Science and Urban Economics, Elsevier, vol. 35(6), pages 756-776, November.
    10. Anderson, Gordon, 2004. "Toward an empirical analysis of polarization," Journal of Econometrics, Elsevier, vol. 122(1), pages 1-26, September.
    11. Schmid, Friedrich & Schmidt, Axel, 2006. "Nonparametric estimation of the coefficient of overlapping--theory and empirical application," Computational Statistics & Data Analysis, Elsevier, vol. 50(6), pages 1583-1596, March.
    12. Holmström, Lasse & Klemelä, Jussi, 1992. "Asymptotic bounds for the expected L1 error of a multivariate kernel density estimator," Journal of Multivariate Analysis, Elsevier, vol. 42(2), pages 245-266, August.
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    Citations

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    Cited by:

    1. Sokbae (Simon) Lee & Yoon-Jae Whang, 2009. "Nonparametric tests of conditional treatment effects," CeMMAP working papers CWP36/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Gordon Anderson & Oliver Linton & Teng Leo, 2012. "A polarization-cohesion perspective on cross-country convergence," Journal of Economic Growth, Springer, vol. 17(1), pages 49-69, March.
    3. Amal Helu & Hani Samawi & Robert Vogel, 2011. "Nonparametric overlap coefficient estimation using ranked set sampling," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 385-397.
    4. Gordon Anderson, 2010. "Polarization Of The Poor: Multivariate Relative Poverty Measurement Sans Frontiers," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 56(1), pages 84-101, March.
    5. Hani M. Samawi & Amal Helu & Robert Vogel, 2011. "A nonparametric test of symmetry based on the overlapping coefficient," Journal of Applied Statistics, Taylor & Francis Journals, vol. 38(5), pages 885-898, February.
    6. Gordon Anderson & Teng Wah Leo & Oliver Linton, 2010. "Making Inferences About Rich Country - Poor Country Convergence: The Polarization Trapezoid and Overlap measures," Working Papers tecipa-387, University of Toronto, Department of Economics.
    7. Maria Grazia Pittau & Roberto Zelli & Paul A. Johnson, 2010. "Mixture Models, Convergence Clubs, And Polarization," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 56(1), pages 102-122, March.
    8. Gordon Anderson & Kinda Hachem, 2009. "Institutions and Economic Outcomes: A Dominance Based Analysis of Causality and Multivariate Welfare With Discrete and Continuous Variables," Working Papers tecipa-378, University of Toronto, Department of Economics.

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    More about this item

    Keywords

    kernel estimation; Inequality; Overlap Coefficient; Poissonization.;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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