What is the Hurst exponent, and how does it distinguish between mean-reverting, random, and trending time series?

The Hurst exponent (H) quantifies the degree of long-range dependence (or anti-persistence) in a time series. It reveals whether price movements tend to continue in the same direction (trending), reverse direction (mean-reverting), or move randomly.

Interpretation:

Hurst Value	Behavior	Description	Trading Implication
H = 0.5	Random walk	No memory; past returns don't predict future	No edge; passive investing
H > 0.5	Persistent / Trending	Up moves followed by up, down by down	Momentum strategies work
H < 0.5	Anti-persistent / Mean-reverting	Up moves followed by down, and vice versa	Mean-reversion strategies work

Estimation Method (R/S Analysis):

The Rescaled Range (R/S) method works as follows:

1. Divide the time series into sub-periods of length n
2. For each sub-period, compute the range R (max cumulative deviation minus min) and standard deviation S
3. Average R/S across all sub-periods
4. Repeat for different values of n
5. The Hurst exponent is the slope of log(R/S) vs log(n)

E(R/S) ~ c x n^H

Worked Example:

Analyst Rowan at Ferndale Quant examines daily returns of two assets over 1,000 trading days:

Sub-period length (n)	log(n)	log(R/S) Asset A	log(R/S) Asset B
10	1.00	0.65	0.40
25	1.40	1.01	0.53
50	1.70	1.27	0.61
100	2.00	1.52	0.68
250	2.40	1.87	0.78

Slope for Asset A: (1.87 - 0.65) / (2.40 - 1.00) = 1.22 / 1.40 = H = 0.87 (strongly trending) Slope for Asset B: (0.78 - 0.40) / (2.40 - 1.00) = 0.38 / 1.40 = H = 0.27 (strongly mean-reverting)

Trading Strategy Mapping:

For Asset A (H = 0.87): Rowan deploys a trend-following strategy:

• Enter long when 20-day SMA crosses above 50-day SMA
• Hold until the trend reverses
• Expected Sharpe ratio improvement vs. buy-and-hold: +0.3

For Asset B (H = 0.27): Rowan deploys a mean-reversion strategy:

• Buy when price drops 2 standard deviations below 20-day mean
• Sell when it reverts to the mean
• Expected Sharpe ratio improvement: +0.4

Limitations:

1. Non-stationarity: the Hurst exponent can change over time. A market may be trending in one regime and mean-reverting in another.
2. Sample size sensitivity: reliable estimation requires long time series (1,000+ observations)
3. Short-range vs long-range: R/S analysis can conflate short-range autocorrelation with true long-range dependence. DFA (Detrended Fluctuation Analysis) is a more robust alternative.
4. Not a crystal ball: even a high Hurst exponent doesn't guarantee future trending behavior

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