Why you should also use OLS estimation of tail exponents

TL;DR

This paper argues that, with a small-sample correction, OLS estimation of Pareto tail exponents can be unbiased and may outperform the Hill MLE, especially when the data is only approximately Pareto or regularly varying.

Contribution

It demonstrates that OLS, with a small-sample correction, is a viable and potentially superior alternative to Hill MLE for estimating tail exponents in certain conditions.

Findings

OLS estimator is unbiased with small-sample correction

MLE emphasizes smaller observations more heavily

OLS can outperform MLE in practical tail estimation scenarios

Abstract

Even though practitioners often estimate Pareto exponents running OLS rank-size regressions, the usual recommendation is to use the Hill MLE with a small-sample correction instead, due to its unbiasedness and efficiency. In this paper, we advocate that you should also apply OLS in empirical applications. On the one hand, we demonstrate that, with a small-sample correction, the OLS estimator is also unbiased. On the other hand, we show that the MLE assigns significantly greater weight to smaller observations. This suggests that the OLS estimator may outperform the MLE in cases where the distribution is (i) strictly Pareto but only in the upper tail or (ii) regularly varying rather than strictly Pareto. We substantiate our theoretical findings with Monte Carlo simulations and real-world applications, demonstrating the practical relevance of the OLS method in estimating tail exponents.