The "double" square-root law: Evidence for the mechanical origin of market impact using Tokyo Stock Exchange data

Understanding the impact of trades on prices is a crucial question for both academic research and industry practice. It is well established that impact follows a square-root impact as a function of traded volume. However, the microscopic origin of such a law remains elusive: empirical studies are particularly challenging due to the anonymity of orders in public data. Indeed, there is ongoing debate about whether price impact has a mechanical origin or whether it is primarily driven by information, as suggested by many economic theories. In this paper, we revisit this question using a very detailed dataset provided by the Japanese stock exchange, containing the trader IDs for all orders sent to the exchange between 2012 and 2018. Our central result is that such a law has in fact microscopic roots and applies already at the level of single child orders, provided one waits long enough for the market to "digest" them. The mesoscopic impact of metaorders arises from a "double" square-root effect: square-root in volume of individual impact, followed by an inverse square-root decay as a function of time. Since market orders are anonymous, we expect and indeed find that these results apply to any market orders, and the impact of synthetic metaorders, reconstructed by scrambling the identity of the issuers, is described by the very same square-root impact law. We conclude that price impact is essentially mechanical, at odds with theories that emphasize the information content of such trades to explain the square-root impact law.