Scale-, time- and asset-dependence of Hawkes process estimates on high frequency price changes

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 (η = 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 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.