Hurst exponent

The rescaled-range estimate

The classic estimator, due to Hurst and popularised by Mandelbrot, works on a window of N returns. Compute the mean of the window, then the cumulative deviations from that mean; the range R is the difference between the highest and lowest cumulative deviation. Divide R by the standard deviation S of the returns in the window, and the exponent is H = ln(R/S) ÷ ln(N). A random walk produces R/S growing like the square root of N, which gives H near 0.5; persistent series grow faster, anti-persistent ones slower.

• Single-scale estimate: one window, one N, no regression across scales.
• Biased upward for small N, which is why longer windows are preferred.
• Undefined when S is zero (constant returns); the honest output is no value.

How to read it

The spread of the estimate is wide on short samples. The guide on this site notes that a pure random walk measured over 128 bars can print anywhere from about 0.32 to 0.72 by chance, so a threshold near 0.5 separates regimes only loosely. Longer windows narrow the spread but respond later.

• H < 0.5: a move up makes the next move down more likely; the series keeps returning to a mean.
• H ≈ 0.5: no memory; the past window gives no information about continuation.
• H > 0.5: moves feed on themselves; the series trends within the window.
• The exponent lags regime changes by design, because it needs new data to re-estimate.
• The same market can read mean-reverting on one timeframe and trending on another.

In Senzoukria

The chart indicator "Hurst Exponent (R/S)" (catalog id hurst-rs) computes the single-scale estimator described above on the last N returns, with N defaulting to 100 rather than 20 because of the small-sample bias. Its catalog entry lists the five steps, uses a population standard deviation, and returns no value during warm-up or when S or R is zero. The indicator label in the desktop reads "Hurst · single-scale R/S" and its note states that short windows are unstable and do not establish predictive edge.

The Hurst exponent also drives the Hurst MR Signal described in the mean reversion guide, where it acts as the regime filter, and it appears in the scripting template "Mean reversion under regime (HMM + Hurst)", which is Python only because it needs numpy.

Common mistakes

• Reading H as a forecast of direction; it describes behaviour, not sign.
• Using a short window and treating a value of 0.45 as a confirmed mean-reverting regime.
• Comparing H across instruments computed with different windows or return definitions.
• Expecting the estimator to signal a regime change on the bar where it happened.