Decomposition and Normalization of Absolute Differences, When Positive and Negative Values Are Considered:applications to the Gini Coefficient

Abstract

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Income and Expenditure Inequalities among Households in Poland

“…There are some methods to deal with this problem, e.g. to standardise this value in a proper way (see, for instance, Raffinetti, Siletti & Vernizzi 2015;Ostasiewicz & Vernizzi 2017). Here, however, as the decomposition is what is being investigated, there will simply be presented the shares of each of the terms of the decomposition within the overall inequality, and with respect to the shares the non-standardised character of inequality does not matter.…”

Bottom Incomes and the Measurement of Poverty and Inequality

“…We find that the traditional approaches produce non-trivial corrections of up to 2.3 points of the Gini, and 1.5 points of the poverty headcount ratio. The enduring problem with these approaches is that they do not use all information available in surveys, they do not replace unreliable zero or negative incomes with more realistic values, and they produce income distributions that are truncated at the bottom or have discontinuous point-mass at zero incomes (Ostasiewicz and Vernizzi, 2017).…”

“…When the income distribution includes both negative and nonnegative incomes, the Gini coefficient can be obtained using the Lorenz curve from the Gini coefficients among negative incomes () and among nonnegative incomes () by knowing the population share of households with negative incomes () and their share of aggregate net income (, a negative share). Refer to Figure 1 for derivation (also refer to Ostasiewicz and Vernizzi, 2017). Here is computed nonparametrically from data, is observed, is observed or computed in a corrected income distribution, and is estimated nonparametrically or parametrically using the corrected distribution of negative incomes.…”

“…When the income distribution includes both negative and nonnegative incomes, the Gini coefficient can be obtained using the Lorenz curve from the Gini coefficients among negative incomes (G N ) and among nonnegative incomes (G 1−N ) by knowing the population share of households with negative incomes ( N ) and their share of aggregate net income (S N , a negative share). Refer to Figure 1 for derivation (also refer to Ostasiewicz and Vernizzi, 2017).…”

Decomposition of the Gini index in the presence of observations with negative values

“…In general, with such sources the issue arises that in the presence of negative inputs, the Gini index is no longer restricted to the interval 〈0,1〉).Some methods of overcoming this difficulty have been proposed (see e.g. [Berrebi, Silber 1985;Raffinetti et al 2015;Ostasiewicz, Vernizzi 2017]). Podder's methodology allows in a natural way to divide a source with both positive and negative inputs, and to deal with them as the composition of two different sources -separately estimating their contributions to the total inequality.…”