The Critical Branching Ratio Debate (Reflexivity Near One)
Is the estimated branching ratio evidence that market activity is self-sustaining?
Estimate the Hawkes branching ratio under several kernels, baselines and sample windows, then compare fit and residuals. A value near one is model dependent and cannot directly measure all endogenous market activity or settle the criticality debate.
Criticality remains debated and the fitted branching ratio is not a direct measure of all endogeneity.
Evidence map
| Aspect | Finding |
|---|---|
| What it is | Two groups fitted Hawkes processes to the same market — mid-price changes of the E-mini S&P 500 futures at the CME — and reached opposite conclusions about whether endogeneity has been rising. |
| Key result / formula | Filimonov & Sornette (2012), with an exponential kernel fitted on short windows, found the branching ratio n rising from about 0.3 in 1998 to above 0.7 after 2007 — read as reflexivity growing with automated trading, with only a minority of price moves since 2007 attributable to exogenous news. |
| Why it matters for backtesting | The debate fixes the precision to attach to any "endogeneity" or "criticality" statistic: with an exponential kernel it tells a story of recent change, with a power-law kernel a story of permanent criticality, and with an unmodelled intraday seasonality possibly a story about the seasonality itself. |
What it is
The disagreement is not about the data; it is about the kernel, and it is the clearest lesson in the literature that a branching ratio is a model output, not a measurement.
Key result / formula
Hardiman, Bercot & Bouchaud (2013), on 1998-2011, fitted a power-law kernel instead and found it decays with exponent close to −1.15 below about 10³ s and about −1.45 from 10³ to 10⁶ s; its integral is close to one in every period, so on their reading the market was always near criticality (n ≈ 1) and nothing has changed. Filimonov & Sornette (2015) replied with a calibration study: power-law kernels are biased upward by outliers and by edge effects at the ends of the window, the regularisation of the kernel at short times moves the estimate, the likelihood has several local maxima, and — most damaging — a mixture of pure Poisson processes with regime changes in the baseline yields an apparently critical branching ratio when the true value is zero.
Why it matters for backtesting
Before using a branching ratio as a regime indicator or a crash-risk gauge, refit it under both kernel families, on windows of different length, and against a baseline that varies in time; if the conclusion flips, the indicator has no content. Neither side claims the branching ratio predicts direction — a market near criticality is one whose activity feeds on itself, which is a statement about clustering.
Source
Filimonov & Sornette, "Quantifying reflexivity in financial markets: Toward a prediction of flash crashes", Physical Review E 85(5), 2012, 056108; Hardiman, Bercot & Bouchaud, "Critical reflexivity in financial markets: a Hawkes process analysis", European Physical Journal B 86, 2013, 442; Filimonov & Sornette, "Apparent criticality and calibration issues in the Hawkes self-excited point process model: application to high-frequency financial data", Quantitative Finance 15(8), 2015. Primary source