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

A popular way to estimate a Pareto exponent is to run an OLS regression: log (Rank) = c - blog (Size), and take b as an estimate of the Pareto exponent. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and argue that, if one wants to use an OLS regression, one should use the Rank -1/2, and run log (Rank- 1/2) = c-b log (Size). The shift of 1/2 is optimal, and cancels the bias to a leading order. The standard error on the Pareto exponent is not the OLS standard error, but is asymptotically (2/n)^{1/2}b. To obtain this result, we provide asymptotic expansions for the OLS estimate in such log-log rank-size regression with arbitrary shifts in the ranks. The arguments for the asymptotic expansions rely on strong approximations to martingales with the optimal rate and demonstrate that martingale convergence methods provide a natural and conceptually simple framework for deriving the asymptotics of the tail index estimates using the log-log rank-size regressions.