Reply to “Comment on ‘Tests of scaling and universality of the distributions of trade size and share volume: Evidence from three distinct markets’ ”

Abstract

Analyzing trade-by-trade data for three distinct markets, we showed that the cumulative distributions of trade size display power-law tails $P{q>x}\ensuremath{\sim}{x}^{\ensuremath{-}{\ensuremath{\zeta}}_{q}}$, with exponents ${\ensuremath{\zeta}}_{q}$ in the ``L\'evy stable domain'' $({\ensuremath{\zeta}}_{q}<{\ensuremath{\zeta}}_{q}^{\ensuremath{\ast}}=2)$. Moreover we reported that the exponent values are consistent for all stocks irrespective of stock-specific variables such as market capitalization, industry sector, or the specific market where the stock is traded. Our conclusions were based on using two distinct estimation methods. R\'acz et al. now propose that one of the estimators we used has slow convergence for a pure power law, particularly as tail exponents approach the boundary ${\ensuremath{\zeta}}_{q}^{\ensuremath{\ast}}=2$. We examine the robustness of our results to specific estimation method by additional analysis using five distinct techniques to estimate ${\ensuremath{\zeta}}_{q}$. We find results that are fully consistent with those we had reported, providing compelling evidence that our conclusions hold regardless of estimation procedure.