  A Hawkes process model with a time-varying background rate is developed for
analyzing the high-frequency financial data. In our model, the logarithm of the
background rate is modeled by a linear model with a relatively large number of
variable-width basis functions, and the parameters are estimated by a Bayesian
method. Our model can capture not only the slow time-variation, such as in the
intraday seasonality, but also the rapid one, which follows a macroeconomic
news announcement. By analyzing the tick data of the Nikkei 225 mini, we find
that (i) our model is better fitted to the data than the Hawkes models with a
constant background rate or a slowly varying background rate, which have been
commonly used in the field of quantitative finance; (ii) the improvement in the
goodness-of-fit to the data by our model is significant especially for sessions
where considerable fluctuation of the background rate is present; and (iii) our
model is statistically consistent with the data. The branching ratio, which
quantifies the level of the endogeneity of markets, estimated by our model is
0.41, suggesting the relative importance of exogenous factors in the market
dynamics. We also demonstrate that it is critically important to appropriately
model the time-dependent background rate for the branching ratio estimation.
