Days of Zipf and Covid? Looking for evidence of Zipf’s Law in the infected Brazil

Does the ranking of Covid-19 cases by municipality follow Zipf’s law (i.e. an estimated Pareto exponent of one)? This note tries to answer this question using daily data from Brazil for the Mar 30, 2020 -Oct 22, 2021 period. We used a Poisson Pseudo Maximum Likelihood (PPML) estimator and the result is that the Pareto exponent of the ranking of Covid-19 cases converges to the Pareto exponent obtained for the population ranking. For comparison purposes, we did the same exercise using Italian regions. Contrary to Brazil, the Pareto exponent for the ranking of Covid-19 cases stabilizes in a first moment and may now be converging to the population’s one. We try to advance some rationale for this contrasting behavior between the two countries. In addition we show via Monte Carlo simulation that, in the presence of heteroscedasticity, the PPML estimator for the tail exponent has better properties than the OLS estimator with Gabaix correction