The statistical estimate of the branching ratio η of the Hawkes model, when fitted to windows of mid-price changes, has been reported to approach criticality ($\eta =1$) as the fitting window becomes large. In this study – using price changes from the EUR/USD currency pair traded on the Electronic Broking Services (EBS) interbank trading platform and the S&P 500 E-mini futures contract traded at the Chicago Mercantile Exchange (CME) – it is shown that the estimated branching ratio depends little upon window size and is usually far from criticality. This is done by controlling for exogenous non-stationarities/heterogeneities at inter- and intraday scales, accomplished by using information criteria to select the degree of flexibility of the Hawkes immigration intensity, either piecewise constant or adaptive logspline, estimated using an expectation maximization (EM) algorithm. The (positive) bias incurred by keeping the immigration intensity constant is small for time scales up to two hours, but can become as high as 0.3 for windows spanning days. This emphasizes the importance of choosing an appropriate model for the immigration intensity in the application of Hawkes processes to financial data and elsewhere. The branching ratio is also found to have an intraday seasonality, where it appears to be higher during times where market activity is dominated by supposedly reflexive automated decisions and a lack of fundamental news and trading. The insights into the microstructure of the two considered markets derived from our Hawkes process fits suggest that equity futures exhibit more complex non-stationary features, are more endogenous, persistent and traded at higher speed than spot foreign exchange. We complement our point process study with EM-estimates of integer-valued autoregressive (INAR) time series models at even longer scales of months. Transferring our methodologies to the aggregate bin-count setting confirms that, even at these very long scales, criticality can be rejected.

据报道，霍克斯模型的分支比η的统计估计值适合于中等价格变化的窗口时，已接近临界值（$\eta =\mathrm{1个}$）随着拟合窗口变大。在这项研究中-使用在电子经纪服务（EBS）银行间交易平台上交易的欧元/美元货币对的价格变化和在芝加哥商业交易所（CME）交易的S＆P 500 E-mini期货合约进行的交易-表明估计的分支比率几乎不依赖于窗口大小，通常远非临界。这是通过控制日间和日间尺度上的外源非平稳性/异质性来完成的，这是通过使用信息标准来选择霍克斯移民强度的柔性程度来实现的，霍克移民强度是分段常数还是自适应对数线，使用期望最大化（EM）进行估算） 算法。在长达两个小时的时间范围内，保持移民强度恒定所产生的（正）偏见很小，但对于跨天的Windows可能会高达0.3。这强调了在将霍克斯流程应用于金融数据和其他方面时，为移民强度选择适当模型的重要性。还发现分支比率具有盘中季节性，在市场活动主要由据称自反的自动决策以及缺乏基本新闻和交易主导的时期内，分支比率似乎更高。从我们的霍克斯过程拟合中得出的两个考虑市场的微观结构的见解表明，股票期货表现出比即期外汇更复杂的非平稳特征，更内生，持久且交易速度更快。我们使用整数值自回归（INAR）时间序列模型的EM估计值（甚至更长的月份数）对点过程研究进行补充。将我们的方法学转移到总的bin-count设置可以确认，即使在如此长的范围内，关键度也可以被拒绝。