Strict Universality of the Square-Root Law in Price Impact across Stocks: A Complete Survey of the Tokyo Stock Exchange

Universal power laws have been scrutinized 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)∝Q^{δ} with δ≈1/2. Some researchers argue that the exponent δ should be system specific, without universality. Conversely, others contend that δ should be exactly 1/2 for all stocks across all countries, implying universality. However, resolving this debate requires high-precision measurements of δ 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 behavior 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 over eight years. Using this comprehensive microscopic dataset, we show that the exponent δ 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 [Gabaix et al., Nature (London) 423, 267 (2003).NATUAS0028-083610.1038/nature01624; Q. J. Econ 121, 461 (2006).10.1162/qjec.2006.121.2.461; Farmer et al., Quant. Finance 13, 1743 (2013)NATUAS0028-083610.1038/nature01624]. Our Letter 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.