Rescaled range (R/S) analysis

To estimate the Hurst exponent, one must first estimate the dependence of the rescaled range on the time span n of observation.^[7] A time series of full length N is divided into a number of nonoverlapping shorter time series of length n, where n takes values N, N/2, N/4, ... (in the convenient case that N is a power of 2). The average rescaled range is then calculated for each value of n.

For each such time series of length ${\displaystyle n}$, ${\displaystyle X=X_{1},X_{2},\dots ,X_{n}\,}$, the rescaled range is calculated as follows:^[6][7]

1. Calculate the mean;
   $${\displaystyle m={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\,.}$$

   
2. Create a mean-adjusted series;
   $${\displaystyle Y_{t}=X_{t}-m\quad {\text{ for }}t=1,2,\dots ,n\,.}$$

   
3. Calculate the cumulative deviate series ${\displaystyle Z}$;
   $${\displaystyle Z_{t}=\sum _{i=1}^{t}Y_{i}\quad {\text{ for }}t=1,2,\dots ,n\,.}$$

   
4. Compute the range ${\displaystyle R}$;
   $${\displaystyle R(n)=\operatorname {max} \left(Z_{1},Z_{2},\dots ,Z_{n}\right)-\operatorname {min} \left(Z_{1},Z_{2},\dots ,Z_{n}\right).}$$

   
5. Compute the standard deviation ${\displaystyle S}$;
   $${\displaystyle S(n)={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}\left(X_{i}-m\right)^{2}}}.}$$

   
6. Calculate the rescaled range ${\displaystyle R(n)/S(n)}$ and average over all the partial time series of length ${\displaystyle n.}$

The Hurst exponent is estimated by fitting the power law ${\displaystyle \mathbb {E} [R(n)/S(n)]=Cn^{H}}$ to the data. This can be done by plotting ${\displaystyle \log[R(n)/S(n)]}$ as a function of ${\displaystyle \log n}$, and fitting a straight line; the slope of the line gives ${\displaystyle H}$. A more principled approach would be to fit the power law in a maximum-likelihood fashion.^[16] Such a graph is called a box plot. However, this approach is known to produce biased estimates of the power-law exponent. For small ${\displaystyle n}$ there is a significant deviation from the 0.5 slope. Anis and Lloyd^[17] estimated the theoretical (i.e., for white noise) values of the R/S statistic to be:

$${\displaystyle \mathbb {E} [R(n)/S(n)]={\begin{cases}{\frac {\Gamma ({\frac {n-1}{2}})}{{\sqrt {\pi }}\Gamma ({\frac {n}{2}})}}\sum \limits _{i=1}^{n-1}{\sqrt {\frac {n-i}{i}}},&{\text{for }}n\leq 340\\{\frac {1}{\sqrt {n{\frac {\pi }{2}}}}}\sum \limits _{i=1}^{n-1}{\sqrt {\frac {n-i}{i}}},&{\text{for }}n>340\end{cases}}}$$

where ${\displaystyle \Gamma }$ is the Euler gamma function. The Anis-Lloyd corrected R/S Hurst exponent is calculated as 0.5 plus the slope of ${\displaystyle R(n)/S(n)-\mathbb {E} [R(n)/S(n)]}$.

Confidence intervals

No asymptotic distribution theory has been derived for most of the Hurst exponent estimators so far. However, Weron^[18] used bootstrapping to obtain approximate functional forms for confidence intervals of the two most popular methods, i.e., for the Anis-Lloyd^[17] corrected R/S analysis:

and for DFA:

Here ${\displaystyle M=\log _{2}N}$ and ${\displaystyle N}$ is the series length. In both cases only subseries of length ${\displaystyle n>50}$ were considered for estimating the Hurst exponent; subseries of smaller length lead to a high variance of the R/S estimates.

Note

In the above definition two separate requirements are mixed together as if they would be one.^[24] Here are the two independent requirements: (i) stationarity of the increments, x(t+T) − x(t) = x(T) − x(0) in distribution. This is the condition that yields longtime autocorrelations. (ii) Self-similarity of the stochastic process then yields variance scaling, but is not needed for longtime memory. E.g., both Markov processes (i.e., memory-free processes) and fractional Brownian motion scale at the level of 1-point densities (simple averages), but neither scales at the level of pair correlations or, correspondingly, the 2-point probability density.

An efficient market requires a martingale condition, and unless the variance is linear in the time this produces nonstationary increments, x(t+T) − x(t) ≠ x(T) − x(0). Martingales are Markovian at the level of pair correlations, meaning that pair correlations cannot be used to beat a martingale market. Stationary increments with nonlinear variance, on the other hand, induce the longtime pair memory of fractional Brownian motion that would make the market beatable at the level of pair correlations. Such a market would necessarily be far from "efficient".

An analysis of economic time series by means of the Hurst exponent using rescaled range and Detrended fluctuation analysis is conducted by econophysicist A.F. Bariviera.^[25] This paper studies the time varying character of Long-range dependency and, thus of informational efficiency.

Hurst exponent has also been applied to the investigation of long-range dependency in DNA,^[26] and photonic band gap materials.^[27]