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In financial markets, the market order sign exhibits strong persistence,widely known as the long-range correlation (LRC) of order flow; specifically,the sign correlation function displays long memory with power-law exponent$\gamma$, such that $C(\tau) \propto \tau^{-\gamma}$ for large time-lag $\tau$.One of the most promising microscopic hypotheses is the order-splittingbehaviour at the level of individual traders. Indeed, Lillo, Mike, and Farmer(LMF) introduced in 2005 a simple microscopic model of order-splittingbehaviour, which predicts that the macroscopic sign correlation isquantitatively associated with the microscopic distribution of metaorders.While this hypothesis has been a central issue of debate in econophysics, itsdirect quantitative validation has been missing because it requires largemicroscopic datasets with high resolution to observe the order-splittingbehaviour of all individual traders. Here we present the first quantitativevalidation of this LFM prediction by analysing a large microscopic dataset inthe Tokyo Stock Exchange market for more than nine years. On classifying alltraders as either order-splitting traders or random traders as a statisticalclustering, we directly measured the metaorder-length distributions$P(L)\propto L^{-\alpha-1}$ as the microscopic parameter of the LMF model andexamined the theoretical prediction on the macroscopic order correlation:$\gamma \approx \alpha - 1$. We discover that the LMF prediction agrees withthe actual data even at the quantitative level. Our work provides the firstsolid support of the microscopic model and solves directly a long-standingproblem in the field of econophysics and market microstructure.

Can we infer microscopic financial information from the long memory in market-order flow?: a quantitative test of the Lillo-Mike-Farmer model