Rescaled range

The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst. It is calculated from the dividing the range of the values exhibited in a portion of the time series by the standard deviation of the values over the same portion of the time series. For example, consider a time series {2, 5, 3, 7, 8, 12, 4, 2} which has a range, R, of 12 - 2 = 10. Its standard deviation, s, is 3.69, so the rescaled range is R/s = 2.71. In this example, the number of observations, n, of the time series is 8.

If we consider the same time series, but increase the number of observations of it, the rescaled range will generally also increase. The increase of the rescaled range can be characterized by making a plot of the logarithm of R/s vs. the logarithm of n. The slope of this line gives the Hurst exponent, H.  If the time series is generated by a random walk (or a Gaussian 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.) also have H > 1/2. This means that they have a behavior that is distinct from a random walk, and therefore the time series is not generated by a stochastic process that has the nth value independent of all of the values before this. This is referred to as long term memory. However this result is controversial and several studies using Lo's (Lo, 1991) modified rescaled range statistic have contradicted Peters' results.