How to deal with negative values in the calculation of the Gini coefficient

There could be several reasons for an income unit to have net incomes that are negative Often, in real surveys, in addition to positive incomes, one can also observe negative incomes, especially when assessing income units and financial assets such as, for instance, capital gains. Two other important factors affecting a positive net income value are that the business of a self-employed individual might have made a loss during the period under consideration, or that the transfers made by an individual to other individuals might have exceeded his total income. Another example can be given by tax systems admitting negative income taxes, which can be originated, for example, by children. The most common practice is to eliminate the observations with negative values. Other researchers suggest converting negative values into zero. In order to avoid Gini coefficients greater than 1, Chen, Tsaur and Rhai (1982), introduce a new normalization. The idea if the authors, was subsequently refined by Berrebi and Silber (1985). Recently Raffinetti, Siletti and Vernizzi (2014), have suggested a further normalization which can be applied provided that the sum of negative values and that of positive values remain unchanged. Chen and al.’s normalization keeps fixed the distribution of negative values, whilst Raffinetti et al. consider the maximum inequality which can arise in presence both of positive and negative values. In this paper we shall go through the approaches here enlisted, tackling them by the absolute difference average approach. By simulations performed on the basis of the bank of Italy SHIW (2012) we shall compare the effects of erasing negative values with the approach suggested by Raffinetti, Siletti and Vernizzi (2014).