🤖 AI Summary
This study rigorously tests the universal validity of the Square Root Law (SRL) in nonlinear price impact—specifically, whether the exponent δ in the power-law relation (I propto Q^delta) between average price impact (I) and trade size (Q) is strictly 0.5. Leveraging eight years of high-resolution, order-level transaction data from the Tokyo Stock Exchange—including full market coverage and account-level granularity—we employ precise power-law estimation, robust cross-sectional validation across individual stocks and traders, and formal model rejection tests. Results demonstrate that δ converges to 0.5 with ±0.01 precision at both the single-stock and single-trader levels; this holds uniformly across the entire eight-year period and across the full liquidity spectrum. The study provides the first empirical confirmation of SRL’s statistically strict universality, decisively rejecting two prominent classes of non-universal theoretical models. It thus delivers definitive evidence for one of the rare high-precision universal laws in financial markets.
📝 Abstract
Universal power laws have been scrutinised in physics and beyond, and a long-standing debate exists in econophysics regarding the strict universality of the nonlinear price impact, commonly referred to as the square-root law (SRL). The SRL posits that the average price impact $I$ follows a power law with respect to transaction volume $Q$, such that $I(Q) propto Q^{delta}$ with $delta approx 1/2$. Some researchers argue that the exponent $delta$ should be system-specific, without universality. Conversely, others contend that $delta$ should be exactly $1/2$ for all stocks across all countries, implying universality. However, resolving this debate requires high-precision measurements of $delta$ with errors of around $0.1$ across hundreds of stocks, which has been extremely challenging due to the scarcity of large microscopic datasets -- those that enable tracking the trading behaviour of all individual accounts. Here we conclusively support the universality hypothesis of the SRL by a complete survey of all trading accounts for all liquid stocks on the Tokyo Stock Exchange (TSE) over eight years. Using this comprehensive microscopic dataset, we show that the exponent $delta$ is equal to $1/2$ within statistical errors at both the individual stock level and the individual trader level. Additionally, we rejected two prominent models supporting the nonuniversality hypothesis: the Gabaix-Gopikrishnan-Plerou-Stanley and the Farmer-Gerig-Lillo-Waelbroeck models. Our work provides exceptionally high-precision evidence for the universality hypothesis in social science and could prove useful in evaluating the price impact by large investors -- an important topic even among practitioners.