**3. Method**

The preceding section showed that the limit order generation processes of the HFTs tended to be in sync with traders' orders or orders in the order book. To clarify how 134 HFTs' buy–sell limit orders react to the other order events, their order processes are modeled by the multivariate Hawkes process. In this section, we introduce the multivariate Hawkes process that we used in this study (see Section 3.1) and explain the parameter estimation method based on maximum likelihood estimation (see Section 3.2). We then describe the validity of the estimation results (see Section 3.3).
