Hurst Exponent: Persistence, Mean-Reversion, or Random Walk
The <strong>Hurst exponent (H)</strong> is a statistical measure of time-series persistence. It sits on a 0-to-1 scale. Values below 0.5 indicate mean-reverting behaviour — the series tends to pull back toward a central value. Values near 0.5 indicate random-walk behaviour — past moves do not predict future direction. Values above 0.5 indicate trending behaviour — past moves tend to continue. For cointegration analysis, rolling Hurst readings tell you whether a pair's spread is currently correcting or drifting.
What it measures
Intuitively, the Hurst exponent asks: does this series "remember" its past? A mean-reverting series (H < 0.5) forgets quickly — a spike today doesn't imply a spike tomorrow. A trending series (H > 0.5) remembers — a move today makes a continuation move more likely. A random walk (H ≈ 0.5) is memoryless in a specific mathematical sense.
Closelook calculates Hurst on rolling windows of pair-spreads. For a pair like SPY-GLD, the spread's Hurst tells you whether the current market is treating the relationship as a mean-reverting equilibrium (healthy) or as a trending divergence (breaking).
How to read it
The interpretation bands we use:
- H < 0.4 — strongly mean-reverting. The spread is correcting. Tradeable mean-reversion setups live here.
- 0.4-0.6 — neutral / random walk. No persistent directional bias. Statistical tests may still show cointegration, but the near-term behaviour is noise.
- H > 0.6 — trending. The spread is moving in one direction with momentum. This is either an opportunity for trend-following or a warning that the cointegrated relationship is breaking down.
- H > 0.8 — strongly trending. In a cointegrated pair, this is a red flag. The historical mean-reversion mechanism has stopped working.
The trajectory matters more than the level. A Hurst falling from 0.55 to 0.35 tells you the pair is shifting from random-walk to mean-reverting — a setup entering tradeable territory. A Hurst rising from 0.45 to 0.75 tells you the opposite: a once-stable relationship is trending away from equilibrium.
Why it matters
Cointegration tests (Engle-Granger, Johansen) give binary answers: the pair either passes or fails. Hurst adds a continuous dimension. Two pairs can both pass the cointegration test yet have very different Hurst readings — one reverting cleanly, the other drifting. Trading the first makes sense; trading the second is trading the statistical residue of a relationship that no longer describes current behaviour.
For macro regime analysis, Hurst on the key cross-asset pairs is a leading indicator. When Hurst across multiple pairs rises together, the market is in a trending regime where historical relationships don't anchor prices — a setup that has historically preceded major regime changes.
In the Lab
The Lab shows both a current Hurst reading per pair and the 52-week Hurst heatmap — pairs × weekly buckets, colour-coded by Hurst band. The heatmap makes regime shifts visible: a pair that was blue (mean-reverting) for a year and flips to amber (trending) deserves attention regardless of whether its spread Z-score is stretched.