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H was originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The Hurst exponent is non-deterministic in that it expresses what is actually observed in nature; it is not calculated so much as it is estimated.
The Hurst exponent is used as a measure of the long term memory of time series, i.e. the autocorrelation of the time series. Where a value of 0 < H < 0.5 indicates a time series with negative autocorrelation (e.g. a decrease between values will probably be followed by an increase), and a value of 0.5 < H < 1 indicates a time series with positive autocorrelation (e.g. an increase between values will probably be followed by another increase). A value of H=0.5 indicates a true random walk, where it is equally likely that a decrease or an increase will follow from any particular value (e.g. the time series has no memory of previous values)
DefinitionThe Hurst exponent, H, is defined in terms of the asymptotic behaviour of the rescaled range as a function of the time span of a time series as follows;[2][3]
where;
To calculate the Hurst exponent, one must estimate the dependence of the rescaled range on the time span n of observation[3]. A time series of full length N is divided into a number of shorter time series of length n = N, N/2, N/4, ... The average rescaled range is then calculated for each value of n.
For a (partial) time series of length n, , the rescaled range is calculated as follows:[2][3]
1. Calculate the mean;
2. Create a mean-adjusted series;
3. Calculate the cumulative deviate series Z;
4. Compute the range R;
5. Compute the standard deviation S;
6. Calculate the rescaled range R(n) / S(n) and average over all the partial time series of length n.
The Hurst exponent is estimated by fitting the power law to the data.
EstimatingThere are a variety of techniques that exist for estimating H, however assessing the accuracy of the estimation can be a complicated issue. Mathematically, in one technique, the Hurst exponent can be estimated such that:[4][5]
for a time series
may be defined by the scaling properties of its structure functions Sq(τ):
where q > 0, τ is the time lag and averaging is over the time window
usually the largest time scale of the system.
Practically, in nature, there is no limit to time, and thus H is non-deterministic as it may only be estimated based on the observed data; e.g., the most dramatic daily move upwards ever seen in a stock market index can always be exceeded during some subsequent day.[6]
H is directly related to fractal dimension, D, such that D = 2 - H. The values of the Hurst exponent vary between 0 and 1, with higher values indicating a smoother trend, less volatility, and less roughness.
In the above mathematical estimation technique, the function H(q) contains information about averaged generalized volatilities at scale τ (only q = 1, 2 are used to define the volatility). In particular, the H1 exponent indicates persistent (H1 > ½) or antipersistent (H1 < ½) behavior of the trend.
For the BRW (brown noise, 1/f²) one gets
while for the pink noise (1/f) and white noise we have
For the popular Levy stable processes and truncated Levy processes with parameter α it has been found that
In the above definition two separate requirements are mixed together as if they would be one.[7] Here are the two independent requirements: (i) stationarity of the increments, x(t+T)-x(t)=x(T) in distribution. this is the condition that yields long time autocorrelations. (ii) Self-similarity of the stochastic then yields variance scaling, but is not needed for long time 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). 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 long time pair memory of fBm that would make the market beatable at the level of pair correlations. Such a market would necessarily be far from "efficient".
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