A new interpretation of the Gini correlation
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A new interpretation of the Gini correlation Tomson Ogwang1
Received: 17 January 2015 / Accepted: 8 September 2015 © Sapienza Università di Roma 2015
Abstract We provide a new way of interpreting the Gini correlation in the income source decomposition of the Gini index as a linear function of the ratio of the concordance pseudoGini (which measures the degree of agreement between the ranking of the incomes within a particular source and the ranking with respect to total income) to the within-source Gini (which measures income disparities within a particular income source). As a by-product, we discuss the modifications of the stochastic approach to the Gini index to take survey weights into account. We also discuss the income source decomposition of the Gini index when the incomes are weighted. Keywords Income source decomposition · Stochastic approach · Gini correlation · Gini index · Pseudo-Gini · Weighted data
1 Introduction Income source decomposition of an inequality measure entails breaking down overall inequality, as measured by, say, the Gini index, into the contributions of the various income components. From a policy perspective, the goal of inequality decompositions by income source is to identify which income sources are equalizing and which sources are disequalizing, or to compute the elasticity of the inequality measure with respect to changes in the incomes from a particular source. An intermediate step with respect to several Gini income source decomposition methods involves determining the degree of association between the rank of each income component and that component’s income or total income. Unfortunately, Pearson’s correlation is not the best measure of association when one variable is a rank transformation and the other variable is discrete or continuous. This is because in this situation Pearson’s correlation lies
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Tomson Ogwang [email protected] Department of Economics, Brock University, 500 Glenridge Avenue, St. Catharines, ON L2S 3A1, Canada
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within a narrower range than the usual [−1, +1], as pointed out by Shih and Huang [14] and Schechtman and Yitzhaki [11], among others. In contrast, the Gini correlation, which was first introduced by Schechtman and Yitzhaki [11], lies within the [−1, +1] range in the same situation, which makes it appealing by providing a fixed goalpost for comparisons. Hence, the Gini correlation which represents a hybrid of Pearson’s correlation (where both variables are discrete or continuous) and Spearman’s rank correlation (where both variables are expressed in ranks) is used in economics to decompose the Gini index by income source (e.g., [6,7]). The correlation is also used in portfolio analysis in finance (e.g., [11,13]). The use of the Gini correlation extends to other fields such as gene analysis in the biological sciences (e.g., [8]). In fact, as pointed out by Shalit [13] and Schechtman and Yitzhaki [11], the Gini correlation is applicable to any situation for which one variable can be expressed as an un-weighted or weighted sum of
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