Enter the rescaled range inputs (range of cumulative deviations and standard deviation) and the window size (T, number of observations) into the calculator to estimate the Hurst coefficient.

Hurst Coefficient Calculator

Enter any 3 values to calculate the missing variable (single-scale approximation using H ≈ log(R/S) / log(T)).

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Hurst Coefficient Formula

The rescaled-range (R/S) method is commonly summarized by the scaling relationship R/S ≈ c·TH. If you assume the constant c ≈ 1 and use a single window size T, you get the following single-scale approximation.

H \approx \frac{\log(R/S)}{\log(T)}

Variables:

  • H is the Hurst coefficient (Hurst exponent)
  • R is the range of cumulative deviations (computed over a window)
  • S is the standard deviation (over the same window)
  • T is the window size (number of observations in the window; unitless)

To estimate the Hurst coefficient using rescaled-range analysis, compute R/S for one or more window sizes T. A standard approach uses multiple T values and estimates H as the slope of a line fit to log(R/S) versus log(T). The calculator above uses the single-scale approximation H ≈ log(R/S) / log(T) (effectively assuming c ≈ 1).

What is the Hurst Coefficient?

The Hurst coefficient, also known as the Hurst exponent, is a measure of the long-term memory of time series data. It indicates whether a time series is a random walk, mean-reverting, or trending. A Hurst coefficient value (H) between 0.5 and 1 suggests a trending time series, while a value between 0 and 0.5 indicates a mean-reverting series. A value of exactly 0.5 suggests a completely random series.

How to Calculate Hurst Coefficient?

The following steps outline how to calculate the Hurst coefficient.

  1. Choose one or more window sizes (T), where T is the number of observations per window.
  2. For each window, compute the cumulative deviations from the window mean and take the range of that cumulative series (R).
  3. Compute the standard deviation for the same window (S).
  4. Compute the rescaled range (R/S) for each chosen T.
  5. Estimate H as the slope of log(R/S) vs. log(T) across multiple T values (or use the calculator above for the single-scale approximation H ≈ log(R/S) / log(T)).

Example Problem : 

Use the following variables as an example problem to test your knowledge (single-scale approximation).

Range of cumulative deviations (R) = 10

Standard deviation (S) = 2

Window size (T) = 5 observations