Extreme value estimation for heterogeneous data

We develop a universal econometric formulation of the empirical power laws possibly driven by parameter heterogeneity. Our approach extends classical extreme value theory to specifying the behavior of the empirical distribution of a general data set with possibly heterogeneous marginal distributions and a complex dependence structure. The main assumption is that in the intermediate tail the empirical distribution approaches some heavy-tailed distribution with a positive extreme value index. In this setup the Hill estimator consistently estimates this extreme value index and, on a log-scale, extreme quantiles are consistently estimated. We discuss several model examples that satisfy our conditions and demonstrate in simulations how heterogeneity may generate the dynamics of empirical power laws. We observe a dynamic cross-sectional power law for the new confirmed COVID-19 cases and deaths per million people across countries, and show that this international inequality is largely driven by the heterogeneity of the countries' scale parameters.