RANK−1/2: A SIMPLE WAY TO IMPROVE THE OLS ESTIMATION OF TAIL EXPONENTS∗ Xavier Gabaix Stern School of Business, New York University, and NBER

Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log (Rank) = a − b log (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) = a − b log (Size). The shift of 1/2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent ζ is not the OLS standard error, but is asymptotically (2/n)ζ. 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.