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On the Decomposition by Subgroups of the Gini Index and Zenga's Uniformity and Inequality Indexes

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  • Paolo Radaelli

Abstract

Besides a brief historical outline of the Gini index decomposition proposals, we compare, in a subgroups framework, the decompositions of the Gini index and of the uniformity and inequality indexes recently proposed by Zenga. The two decompositions follow a similar scheme: in both cases the overall index can be at first expressed as a weighted average of convenient “cross” measures, with their own interpretation, and afterwards it is decomposed into a within and a between term by merely distinguishing measures evaluated within the same subgroup from the ones regarding different subgroups. This procedure does not depend on an a priori definition of the within or the between term and allows their contribution to be naturally evaluated preserving the structure of the index itself. In the last section the two decompositions are applied to the Italian net household wealth provided by the 2006 central Bank of Italy sample survey on household budgets. Après une concise revue de la littérature sur la décomposition de l'indice de concentration de Gini en sous‐groupes, cette article confronte les décompositions de la mesure de Gini avec celles d'uniformité et d'inégalité récemment proposées par Zenga. Les deux décompositions suivent un approche similaire: l'indice global d'inégalité s'écrit sous la forme d'une moyenne pondérée des adéquates mesures croisés, de facile interprétation. Après il est décomposé en deux composantes: celle entre les groupes et celle dans les groupes, simplement avec la séparation des comparaisons à l'intérieur de chaque groupe de celles entre les différents groupes. Cette procédure est indépendant d'une définition a priori des éléments entre les groupes et dans les groupes et elle est capable de calculer leurs contributions à l'inégalité totale et de préserver la structure de l'indice. Dans le dernier paragraphe les deux décompositions sont appliquées aux données sur la richesse des ménages italien récoltées par le sondage de 2006 de la Banque Centrale d'Italie.

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  • Paolo Radaelli, 2010. "On the Decomposition by Subgroups of the Gini Index and Zenga's Uniformity and Inequality Indexes," International Statistical Review, International Statistical Institute, vol. 78(1), pages 81-101, April.
    • Handle: RePEc:bla:istatr:v:78:y:2010:i:1:p:81-101
    DOI: 10.1111/j.1751-5823.2010.00100.x
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    2. Francesca Greselin & Simone Pellegrino & Achille Vernizzi, 2021. "The Zenga Equality Curve: A New Approach to Measuring Tax Redistribution and Progressivity," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 67(4), pages 950-976, December.
    3. Greselin Francesca, 2014. "More Equal and Poorer, or Richer but More Unequal?," Stochastics and Quality Control, De Gruyter, vol. 29(2), pages 99-117, December.
    4. Ogwang Tomson, 2014. "A Convenient Method of Decomposing the Gini Index by Population Subgroups," Journal of Official Statistics, Sciendo, vol. 30(1), pages 91-105, March.
    5. Matti Langel & Yves Tillé, 2012. "Inference by linearization for Zenga’s new inequality index: a comparison with the Gini index," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1093-1110, November.
    6. Pasquazzi Leo & Zenga Michele, 2018. "Components of Gini, Bonferroni, and Zenga Inequality Indexes for EU Income Data," Journal of Official Statistics, Sciendo, vol. 34(1), pages 149-180, March.
    7. Luigi Grossi & Mauro Mussini, 2017. "Inequality in Energy Intensity in the EU-28: Evidence from a New Decomposition Method," The Energy Journal, International Association for Energy Economics, vol. 0(Number 4).
    8. Francesca Greselin & Simone Pellegrino & Achille Vernizzi, 2017. "Lorenz versus Zenga Inequality Curves: a New Approach to Measuring Tax Redistribution and Progressivity," Working papers 046, Department of Economics, Social Studies, Applied Mathematics and Statistics (Dipartimento di Scienze Economico-Sociali e Matematico-Statistiche), University of Torino.
    9. D'Errico, Marco & Macchiarelli, Corrado & Serafini, Roberta, 2015. "Differently unequal: Zooming-in on the distributional dimensions of the crisis in euro area countries," Economic Modelling, Elsevier, vol. 48(C), pages 93-115.
    10. Francesco Porro & Michele Zenga, 2020. "Decomposition by subpopulations of the Zenga-84 inequality curve and the related index $$\zeta $$ζ: an application to 2014 Bank of Italy survey," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 187-207, March.
    11. Frank A. Cowell & Philippe Kerm, 2015. "Wealth Inequality: A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 29(4), pages 671-710, September.
    12. Michele Zenga, 2016. "On the decomposition by subpopulations of the point and synthetic Zenga (2007) inequality indexes," METRON, Springer;Sapienza Università di Roma, vol. 74(3), pages 375-405, December.
    13. Francesca Greselin & Simone Pellegrino & Achille Vernizzi, 2020. "The Social Welfare Implications of the Zenga Index," Papers 2006.12623, arXiv.org.
    14. Skedgell, Kyleigh & Kearney, Christopher A., 2018. "Predictors of school absenteeism severity at multiple levels: A classification and regression tree analysis," Children and Youth Services Review, Elsevier, vol. 86(C), pages 236-245.

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