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 $C(\tau )\propto {\tau }^{-\gamma }$ 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 $P(L)\propto L^{-\alpha -1}$ as the microscopic parameter of the LMF model and examined the theoretical prediction on the macroscopic order correlation $\gamma \approx \alpha -1$. 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.

Figure 1

Schematic of the LRC of the market-order flow and the order-splitting hypothesis (particularly, the LMF model); as a shorthand notation for $+1$ ($-1$), “$+$” (“$-$”) signifies a buy (sell). (a) As a microscopic model, we assume the presence of STs. Also, STs successively submit the child orders with the same sign for $L$ times, where $L$ is called the metaorder length and obeys power law statistics, $P(L)\propto L^{-\alpha -1}$. (b) Consequently, the LRC appears as a macroscopic phenomenon. The LMF theory predicts a quantitative relation $\gamma =\alpha -1$, which we empirically establish in this Letter through data analysis.

Figure 2

Presence of the STs by our strategy clustering. (a) Empirical PDF for the percentage of STs in each market, showing direct evidence of the presence of STs. Typically, 25% of all traders were STs. (b) Empirical PDF for STs contribution to market orders in each market. Typically, 80% of all the market orders were issued by the STs, implying their overwhelming contribution to market orders.

Figure 3

The aggregated CCDFs for STs with our strategy clustering. (a) Empirical CCDF aggregated regarding metaorder length $L$ among STs using data for the Toyota Motor Corporation in 2020 as a typical example. The CCDF obeys the power law $P_{>}(L)\propto L^{-\alpha }$ with $\alpha \approx 1.62$. Likewise, most empirical aggregated CCDF for STs obey similar power laws. (b) Empirical PDF of the power-law exponent $\alpha$ for all the markets. The power-law exponents were evaluated systematically using Clauset’s algorithm [15, 16] across all the markets. The exponent $\alpha$ typically satisfies $1<\alpha <2$, consistent with the standard assumption for the LMF model.

Figure 4

(a),(b) Scattered box plots between $\alpha$ and $\gamma$ with the median, the first and third quartiles for (a) the ACF and (b) the PSD methods, exhibiting excellent agreement with the LMF prediction (2) (black line). $\gamma$ was evaluated using the approximate unbiased estimator ${\gamma }_{\text{unbiased}}$, based on the NLLS estimator and the LMF model. (c),(d) Scattered box plots between the LMF estimator $N_{\mathrm{ST}}^{\mathrm{LMF}}$ and the actual total number of the STs $N_{\mathrm{ST}}$ for (c) the ACF and (d) the PSD methods for the datapoints with $\alpha <2$. The LMF estimator is highly correlated with the true value of $N_{\mathrm{ST}}$ as the classical theory predicts, but systematically underestimates $N_{\mathrm{ST}}$, such that $N_{\mathrm{ST}}^{\mathrm{LMF}}\lesssim N_{\mathrm{ST}}$. This observation is consistent with a generalized LMF theory [18] with heterogeneous intensities $\{{\lambda }^{(i)}{\}}_i$.