Using the Hurst Exponent to Compare Mean Reversion and Trend

Summary

The article introduces a lag-based estimate of the Hurst exponent and applies it to simulated mean-reverting data and adjusted SPY prices. The method compares the variability of price differences across a range of lags, fits a line to the log-scaled measurements, and uses its slope to estimate H. Values below, near, or above 0.5 are interpreted as evidence consistent with mean reversion, random-walk behavior, or persistence, respectively.

For SPY, the author reports that the estimated behavior changes with the lag range: shorter lags suggest moderate mean reversion, intermediate lags approach random-walk behavior, and longer lags indicate more trending behavior. The author also says these patterns appear across different sample segments, including periods that look strongly upward trending by eye. The key caveat is that lag selection materially affects the estimate; the result is not a fixed label for an asset. The examples do not establish a tradable strategy or account for statistical uncertainty, regime changes, or transaction costs.

Key ideas

• The Hurst estimate is derived from how price-difference variability changes across lags.
• Values around 0.5 are associated with random-walk-like behavior in the article’s interpretation.
• Shorter lag ranges for SPY suggest mean reversion, while longer ranges suggest persistence.
• Estimated behavior can vary substantially with the selected lag range.
• A visual price trend alone may not reveal shorter-horizon statistical behavior.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.