Branching-ratio approximation for the self-exciting Hawkes process

We introduce a model-independent approximation for the branching ratio of Hawkes self-exciting point processes. Our estimator requires knowing only the mean and variance of the event count in a sufficiently large time window, statistics that are readily obtained from empirical data. The method we propose greatly simplifies the estimation of the Hawkes branching ratio, recently proposed as a proxy for market endogeneity and formerly estimated using numerical likelihood maximization. We employ our method to support recent theoretical and experimental results indicating that the best fitting Hawkes model to describe S&P futures price changes is in fact critical (now and in the recent past) in light of the long memory of financial market activity.

Figure 1

Applying the mean-variance branching ratio approximation method to a simulated Hawkes processes with an exponential kernel form and scale parameter $\beta =1.0$. $\alpha$ is varied to decide the branching ratio, and $\mu$ is varied to keep the average event rate fixed at $\Lambda =1.0$. The process is simulated for $T=100000s$, and the branching ratio estimate is made using Eq. (7) with a window size $W=20$. The mean and [5%,95%] confidence bands are calculated over an ensemble of 100 processes for each parameter set.

Figure 2

Applying the mean-variance branching ratio approximation method to simulated data. Shaded area represents the [5%,95%] confidence interval. We note that the branching ratio estimate converges on the expected value of $0.75$ as the window size increases. For very large $W$, we lack a sufficient number of event count observations to estimate the variance of $N_W$ with precision and the confidence interval grows considerably.

Figure 3

$1-\tilde{n}$ for simulated near-critical ($n=0.99,\epsilon =0.35$) power-law Hawkes process. The value for $1-\tilde{n}$ that we obtain approximately scales as $\sim W^{-0.35}$ for large $W$. Note that the simulated process is only “near critical” (with a branching ratio $n=1-1\times {10}^{-2}$) so for very large $W$ the curve levels off and converges to $1\times {10}^{-2}$.

Figure 4

Reproduction of the flash-crash branching ratio analysis of Filimonov and Sornette using the mean-variance estimator. Our results compare well with those obtained by maximizing the likelihood of the exponential model (with 1-s randomization). The dashed line is the E-mini S&P price. The points are placed at the end of the 10-min period for which the branching ratio estimate is made. The shaded region is a [10%,90%] confidence interval generated by bootstrap resampling.

Figure 5

$1-\tilde{n}$, the estimate of the branching ratio as a function of window size for E-mini S&P midprice changes in 2010. The mean and variance of $N_W$ are estimated on the full year. The change of power-law behavior occurs around 100 s. Note that we have “stitched” together all 5-min bins of regular trading hours (09:30 to 16:00 EST) that contain at least one event (this solves a problem arising from missing data at half days). We have then detrended the intraday seasonality by dividing each 5-min event count by a normalized average event rate for each 5 min of the trading day calculated on the full year.

Figure 6

Estimates of the branching ratio for 15-min windows using the method of Filimonov and Sornette [3] and estimates using the mean-variance estimator with a window size of $W=30s$. Note that our MLE results differ somewhat from the plot presented in [3]. We attribute this to differences in the data source or identification of midprice changes.

Figure 7

Estimates of the branching ratio on 2-month periods using the mean-variance estimator with a window size that follows Moore's law: $W_t=W_0{e}^{-c(t-t_0)}$ with $c=-ln(1/2)/\left(18\mathrm{months}\right)$ and $t_0=1998$. We again stitch together periods of regular trading hours and detrend the event count by the intraday U shape for each year. When $W$ is appropriately rescaled, the branching ratio estimate is approximately constant through time, for all values of $W_0$, and tends to $n=1.0$ for large $W_0$.