Rank-1/2: A Simple Way to Improve the OLS Estimation of Tail Exponents

Abstract

Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log (Rank)= c - blog (Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank-1/2, and run log (Rank-1/2) c - blog (Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent \zeta is not the OLS standard error, but is asymptotically (2/n)^{1/2} \zeta. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf's law for the U.S. city size distribution.

Keywords: power law, heavy-tailedness, OLS log-log rank-size regression, bias, standard errors, Zipf's law

JEL Classification: C13, C14, C16