Inferring Microscopic Financial Information from the Long Memory in Market-Order Flow: A Quantitative Test of the Lillo-Mike-Farmer Model

In financial markets, the market-order sign exhibits strong persistence, widely known as the long-range correlation (LRC) of order flow; specifically, the sign autocorrelation function (ACF) displays long memory with power-law exponent gamma, such that C(tau) proportional to tau-gamma for large time-lag tau. One of the most promising microscopic hypotheses is the order-splitting behavior at the level of individual traders. Indeed, Lillo, Mike, and Farmer (LMF) introduced in 2005 a simple microscopic model of order-splitting behavior, which predicts that the macroscopic sign correlation is quantitatively associated with the microscopic distribution of metaorders. While this hypothesis has been a central issue of debate in econophysics, its direct quantitative validation has been missing because it requires large microscopic datasets with high resolution to observe the order-splitting behavior of all individual traders. Here we present the first quantitative validation of this LMF prediction by analyzing a large microscopic dataset in the Tokyo Stock Exchange market for more than nine years. On classifying all traders as either order-splitting traders or random traders as a statistical clustering, we directly measured the metaorder-length distributions P(L) proportional to L-alpha-1 as the microscopic parameter of the LMF model and examined the theoretical prediction on the macroscopic order correlation gamma approximate to alpha - 1. We discover that the LMF prediction agrees with the actual data even at the quantitative level. We also discuss the estimation of the total number of the order-splitting traders from the ACF prefactor, showing that microscopic financial information can be inferred from the LRC in the ACF. Our Letter provides the first solid support of the microscopic model and solves directly a long-standing problem in the field of econophysics and market microstructure.