Before revision:  We also demonstrate that the level of the endogeneity of markets, quantified by the branching ratio of the Hawkes process, is overestimated if the time-variation is not considered .
After revision:  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, quantified by the branching ratio of the Hawkes process, is overestimated if the time-variation is not considered .