Rescaled range

The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978).

[1] Its purpose is to provide an assessment of how the apparent variability of a series changes with the length of the time-period being considered.

The slope of this line gives the Hurst exponent, H. If the time series is generated by a random walk (or a Brownian motion process) it has the value of H = 1/2.

Many physical phenomena that have a long time series suitable for analysis exhibit a Hurst exponent greater than 1/2.

For example, observations of the height of the Nile River measured annually over many years gives a value of H = 0.77.

Several researchers (including Peters, 1991) have found that the prices of many financial instruments (such as currency exchange rates, stock values, etc.)

According to a model[3] of Fractional Brownian motion this is referred to as long memory of positive linear autocorrelation.

However it has been shown[4] that this measure is correct only for linear evaluation: complex nonlinear processes with memory need additional descriptive parameters.

Several studies using Lo's[5] modified rescaled range statistic have contradicted Peters' results as well.

Using this adjusted rescaled range, he concludes that stock market return time series show no evidence of long-range memory.