Interpreting the Regression Slope in Hurst Exponent Estimation

Summary

The document describes a discrepancy encountered while implementing a Hurst exponent estimation method based on a power-law relationship and regression. The author reports reproducing a guide’s results within an acceptable error, but finding estimates about 0.5 higher than expected: a geometric Brownian motion simulation produced a value near 1, and a mean-reverting series produced a value in the 0.7 range. The author observes that dividing the regression slope by two brings the estimates into line with expectations.

The central issue is how the slope maps to the Hurst exponent and why one implementation includes that factor of one-half while another apparently does not. The document does not include an answer or derive the scaling relationship, so it does not establish which implementation is correct. Its value is as a methodological question about definitions and normalization in Hurst estimation; results depend on the particular power law, regression setup, and implementation being compared.

Key ideas

• The author compares Hurst estimates from a power-law regression with expected behavior for simulated processes.
• The reported estimates appear roughly 0.5 higher before scaling the regression slope.
• Dividing the slope by two produces estimates closer to the author’s expectations.
• The document asks how the slope relates mathematically to the Hurst exponent but provides no answer.
• Comparisons require checking the exact power-law definition and estimation procedure.

Tags

# Dividing H in the Hurst power law function to get the Hurst exponent?


# Dividing H in the Hurst power law function to get the Hurst exponent?












For my own learning I have been following the guide here. It is highly instructive.

Implementing this in R I was able to reproduce the authors results on the data sets provided within some acceptable amount of error. However I noticed something funny. The data seems adjust up by exactly 0.50.

For a GBM process I got a value of approximately 1. The author of the website linked encounters nearly the same thing evaluating the raw Alcoa price series. This is wrong, obviously, since a GBM should have a value near 0.5.

There also seems to be a similar bias in the other data sets, and I have found generating a mean reverting data set results in a value in the .7x range, again obviously biased up almost 0.5.

This confused me because my math is correct per the website. Digging around I've stumbled on this example website implementing it. In there the author clearly indicates H is found by dividing the slope of the regression results by 2. Magically, my H is corrected using the same technique.

So this leaves me quite confused.

- Why does the author of the bearcave not divide their H by 2?

- What is the purpose of this division? What is the justification for it magically fixing my math?

Thank you!

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