Gini and undercoverage at the upper tail: a simple approximation

This paper investigates the impact of top-distribution undercoverage on the Gini coefficient. First, we show that failing to correct for underreporting and nonresponse at the top does not necessarily result in an underestimated Gini coefficient. Then, we establish analytical conditions under which the Gini coefficient of the uncorrected distribution is higher than the Gini coefficient of the true distribution, i.e. the distribution that incorporates underreporting and/or nonresponse. In addition, we propose a Gini approximation based on the Atkinson approximation $$G=G_{1-p}\cdot (1-S_{p})+S_{p}$$ G = G 1 - p · ( 1 - S p ) + S p to correct for underreporting at the top. Under plausible assumptions, the approximation proposed is very close to the real Gini coefficient. We also show that before correcting for underreporting we need to first address the issue of nonresponse as otherwise the approximation may be strongly upper biased. Finally, this work proposes a procedure to estimate the fraction of nonrespondents at the very top under the assumption that underreported incomes belong to individuals with incomes higher than those reported in a household survey. To evaluate the proposed methodologies, this paper uses Chile and Canada as examples, where we include undistributed business profits—an underreported source of income—to measure income inequality.