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Modeling inequality and spread in multiple regression


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  • Rolf Aaberge
  • Steinar Bjerve
  • Kjell Doksum


We consider concepts and models for measuring inequality in the distribution of resources with a focus on how inequality varies as a function of covariates. Lorenz introduced a device for measuring inequality in the distribution of income that indicates how much the incomes below the u$^{th}$ quantile fall short of the egalitarian situation where everyone has the same income. Gini introduced a summary measure of inequality that is the average over u of the difference between the Lorenz curve and its values in the egalitarian case. More generally, measures of inequality are useful for other response variables in addition to income, e.g. wealth, sales, dividends, taxes, market share and test scores. In this paper we show that a generalized van Zwet type dispersion ordering for distributions of positive random variables induces an ordering on the Lorenz curve, the Gini coefficient and other measures of inequality. We use this result and distributional orderings based on transformations of distributions to motivate parametric and semiparametric models whose regression coefficients measure effects of covariates on inequality. In particular, we extend a parametric Pareto regression model to a flexible semiparametric regression model and give partial likelihood estimates of the regression coefficients and a baseline distribution that can be used to construct estimates of the various conditional measures of inequality.

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Paper provided by in its series Papers with number math/0610852.

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Date of creation: Oct 2006
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Publication status: Published in IMS Lecture Notes--Monograph Series 2006, Vol. 49, 120-130
Handle: RePEc:arx:papers:math/0610852

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  1. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, Econometric Society, vol. 39(6), pages 1037-39, November.
  2. Aaberge, Rolf, 2008. "Ranking Intersecting Lorenz Curves," IZA Discussion Papers 3852, Institute for the Study of Labor (IZA).
  3. Claudio Zoli, 1999. "Intersecting generalized Lorenz curves and the Gini index," Social Choice and Welfare, Springer, Springer, vol. 16(2), pages 183-196.
  4. Shorrocks, Anthony F & Foster, James E, 1987. "Transfer Sensitive Inequality Measures," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 54(3), pages 485-97, July.
  5. Takemi Yanagimoto & Masaaki Sibuya, 1976. "Isotonic tests for spread and tail," Annals of the Institute of Statistical Mathematics, Springer, Springer, vol. 28(1), pages 329-342, December.
  6. Rojo, Javier & He, Guo Zhong, 1991. "New properties and characterizations of the dispersive ordering," Statistics & Probability Letters, Elsevier, Elsevier, vol. 11(4), pages 365-372, April.
  7. Muliere, Pietro & Scarsini, Marco, 1989. "A note on stochastic dominance and inequality measures," Journal of Economic Theory, Elsevier, Elsevier, vol. 49(2), pages 314-323, December.
  8. Rolf Aaberge, 2000. "Characterizations of Lorenz curves and income distributions," Social Choice and Welfare, Springer, Springer, vol. 17(4), pages 639-653.
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Cited by:
  1. Andreas Behr & Ulrich Pötter, 2010. "What determines wage differentials across the EU?," Journal of Economic Inequality, Springer, Springer, vol. 8(1), pages 101-120, March.


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