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The relationship between the absolute deviation from a quantile and Gini’s mean difference

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  • Shlomo Yitzhaki
  • Peter Lambert

Abstract

We investigate the relationship between Gini’s mean difference (GMD), the mean absolute deviation, the least absolute deviation and the absolute deviation from a quantile. The latter can all be interpreted as equivalents either to the GMD of a transformed distribution or to a between-group GMD measure, according to the particular partition of the data. They all possess properties of the GMD but each omits the intra-group variability—and they give rise to different regression techniques. We argue that the analyst using one of these techniques should justify the omission of the intra-group variability from the analysis. Copyright Sapienza Università di Roma 2013

Suggested Citation

  • Shlomo Yitzhaki & Peter Lambert, 2013. "The relationship between the absolute deviation from a quantile and Gini’s mean difference," METRON, Springer;Sapienza Università di Roma, vol. 71(2), pages 97-104, September.
    • Handle: RePEc:spr:metron:v:71:y:2013:i:2:p:97-104
    DOI: 10.1007/s40300-013-0015-y
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    References listed on IDEAS

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    1. Peter J. Lambert & Andre' Decoster, 2005. "The Gini coefficient reveals more," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 373-400.
    2. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    3. Edna Schechtman & Shlomo Yitzhaki & Taina Pudalov, 2011. "Gini’s multiple regressions: two approaches and their interaction," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(1), pages 67-99.
    4. Pakes, Ariel, 1982. "On the Asymptotic Bias of Wald-Type Estimators of a Straight Line When Both Variables Are Subject to Error," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 23(2), pages 491-497, June.
    5. Hiroshi Konno & Hiroaki Yamazaki, 1991. "Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market," Management Science, INFORMS, vol. 37(5), pages 519-531, May.
    6. repec:bla:econom:v:50:y:1983:i:197:p:3-17 is not listed on IDEAS
    7. Koenker, Roger & Bassett, Gilbert, Jr, 1982. "Robust Tests for Heteroscedasticity Based on Regression Quantiles," Econometrica, Econometric Society, vol. 50(1), pages 43-61, January.
    8. Yitzhaki, Shlomo, 2002. "Do we need a separate poverty measurement?," European Journal of Political Economy, Elsevier, vol. 18(1), pages 61-85, March.
    9. Shlomo Yitzhaki, 2003. "Gini’s Mean difference: a superior measure of variability for non-normal distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 285-316.
    10. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, September.
    11. Liang Peng, 2003. "Least absolute deviations estimation for ARCH and GARCH models," Biometrika, Biometrika Trust, vol. 90(4), pages 967-975, December.
    12. Yusif Simaan, 1997. "Estimation Risk in Portfolio Selection: The Mean Variance Model Versus the Mean Absolute Deviation Model," Management Science, INFORMS, vol. 43(10), pages 1437-1446, October.
    13. Frick, Joachim R. & Goebel, Jan & Schechtman, Edna & Wagner, Gert G. & Yitzhaki, Shlomo, 2006. "Using Analysis of Gini (ANOGI) for Detecting Whether Two Subsamples Represent the Same Universe: The German Socio-Economic Panel Study (SOEP) Experience," EconStor Open Access Articles and Book Chapters, ZBW - Leibniz Information Centre for Economics, vol. 34(4), pages 427-468.
    14. Frick, Joachim R. & Goebel, Jan & Schechtman, Edna & Wagner, Gert G. & Yitzhaki, Shlomo, 2004. "Using Analysis of Gini (ANoGi) for Detecting Whether Two Sub-Samples Represent the Same Universe: The SOEP Experience," IZA Discussion Papers 1049, Institute of Labor Economics (IZA).
    15. Schechtman, E. & Yitzhaki, S., 1999. "On the proper bounds of the Gini correlation," Economics Letters, Elsevier, vol. 63(2), pages 133-138, May.
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    Cited by:

    1. Ndéné Ka & Stéphane Mussard, 2015. "Book Review of The Gini Methodology: A Primer on a Statistical Methodology," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 13(2), pages 321-324, June.
    2. Shlomo Yitzhaki & Peter Lambert, 2014. "Is higher variance necessarily bad for investment?," Review of Quantitative Finance and Accounting, Springer, vol. 43(4), pages 855-860, November.
    3. Téa Ouraga, 2019. "A note on Gini Principal Component Analysis," Economics Bulletin, AccessEcon, vol. 39(2), pages 1076-1083.
    4. Ndéné Ka & Stéphane Mussard, 2016. "ℓ 1 regressions: Gini estimators for fixed effects panel data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(8), pages 1436-1446, June.
    5. Ndene Ka & Stephane Mussard, 2015. "l1 Regressions: Gini Estimators for Fixed Effects Panel Data," Cahiers de recherche 15-02, Departement d'économique de l'École de gestion à l'Université de Sherbrooke.
    6. Iddo Eliazar & Giovanni M. Giorgi, 2020. "From Gini to Bonferroni to Tsallis: an inequality-indices trek," METRON, Springer;Sapienza Università di Roma, vol. 78(2), pages 119-153, August.

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    More about this item

    Keywords

    GMD; LAD; MAD; QUAD; C2; C10; C58;
    All these keywords.

    JEL classification:

    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

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