题目：Measuring inequality for winners and losers: extended Lorenz curves and Gini indices for possiblynegative variables
主讲人：Tianyu He 加拿大麦吉尔大学经济系
Measuring inequality is not straight forward, as we face challenges such as selection of variables, distributional features of interest and reference points. The last problem becomes obvious in the presence of negative observations, as common inequality measurements are not defined. The extended inequality measurements have been proposed to incorporate negative values, which, however, have been used essentially as descriptive devices so far. To make these tools formal, we provide a comprehensive study on estimation and asymptotic inference for these extended inequality measurements, including extended Lorenz curves and extended Gini indices. We establish asymptotic normality of the estimators and build confidence intervals for the extended Lorenz curves at a fixed vector of percentiles. To make inference for the entire curves, we prove that the empirical extended Lorenz curves convergence uniformly to Gaussian processes and construct uniform confidence bands for the entire curves by nonparametric bootstrap based on supreme-type test statistics. We also provide asymptotic results of extended Gini indices by U-statistic theory and delta method. The simulation evidence shows that the proposed methods exhibit well.
We apply our methods to study inequality of household financial incomes in Italy where a non-negligible proportion of financial losses are observed. We observe that the inequality of financial gains is much higher than that of financial losses. Moreover, the traditional inequality metrics, by ignoring negative observations, underestimate the overall inequality level. Through a decomposition of positive Gini index, we find that the inequality of positive financial gains and gross inequality between financial gains and losses are important source of overall inequality. We exploit the usage of extended inequality metrics as dispersion measures by investigating the dispersion of cross-sectional US stock returns where nearly half of observations are negative. We find that dispersion of positive stock returns is larger than that of losses. By decomposing the positive Gini index, we find that more than half of overall dispersion comes from gross dispersion between positive and negative returns. We further find that the dispersion of cross-sectional monthly stock returns is very high from 2000 to 2019 and slightly higher during periods of bear markets.
Tianyu He is a Ph.D. candidate in economics from McGill University, Canada. His research focuses on econometrics, applied econometrics, and labor Economics.