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The Hurst Exponent and Technical Analysis

Introduction

It is reasonable to ask whether a given financial time series is predictable before modelling the data and trying to forecast its development.

The Hurst Exponent is a numerical estimate of the predictability of a time series. It is defined as the relative tendency of a time series to either regress to a longer term mean value or 'cluster' in a direction.

The reason the Hurst Exponent is an estimate and not a definitive measure is because the algorithm operates under the assumption that the time series is a pure fractal, which is not entirely true for most financial time series.

This is however of low importance and what really makes the Hurst Exponent such a valuable asset in technical analysis is that it provides a means of classifying time series in terms of predictability.

 

Interpreting the Hurst Exponent

The values of the Hurst Exponent range between 0 and 1.

• A Hurst Exponent value H close to 0.5 indicates a random walk (a Brownian time series). In a random walk there is no correlation between any element and a future element and there is a 50% probability that future return values will go either up or down. Series of this type are hard to predict.
• A Hurst Exponent value H between 0 and 0.5 exists for time series with "anti-persistent behaviour". This means that an increase will tend to be followed by a decrease (or a decrease will be followed by an increase). This behaviour is sometimes called "mean reversion" which means future values will have a tendency to return to a longer term mean value. The strength of this mean reversion increases as H approaches 0.
• A Hurst Exponent value H between 0.5 and 1 indicates "persistent behavior", that is the time series is trending. If there is an increase from time step [t-1] to [t] there will probably be an increase from [t] to [t+1]. The same is true of decreases, where a decrease will tend to follow a decrease. The larger the H value is, the stronger the trend. Series of this type are easier to predict than series falling in the other two categories.

Note that the Hurst Exponent is a different measure than volatility. An index or a fund has a relative low volatility, but still can have an H close to 0.5. Mature markets often have Hurst Exponents closer to 0.5 than emerging markets indicating that they are more efficient and less predictable.

The Hurst Exponent thus provides a method of classifying time series, which can be beneficial in identifying for instance which stocks have greater short term predictability. We could create a portfolio consisting of stocks with particular Hurst Exponent values and investigate their profit generating characteristics. If a particular stock has its Hurst Exponent drop below a threshold value, all investment positions in this asset could be closed.

In conjunction with technical indicators or neural networks, Hurst Exponent estimation can help determine which assets to forecast and which ones to ignore. This can be particularly useful in neural nets where models can focus more on time series with higher predictability.

 

About the Hurst Exponent

The Hurst Exponent occurs in several areas of applied mathematics, including fractals and chaos theory, long memory processes and spectral analysis. Hurst Exponent estimation has been applied in areas ranging from biophysics to computer networking.

The Hurst Exponent is directly related to the fractal dimension of a process, which gives a measure of the roughness of the process. The fractal dimension has been used to measure the roughness of coastlines, for example. Other applications exist in computer graphics (the simulation of mountains and hills), biology (measurement of the boundary of a mold colony) and medicine (measurement of neuronal growth).

The Hurst Exponent is not so much calculated as estimated. There are a variety of techniques that exist for estimating H, however assessing the accuracy of the estimation can be a complicated issue. The accuracy and fidelity of H are also limited by the Heisenberg Uncertainty Principle, a boundary for all measurements.

In Optimal Trader the Hurst Exponent is estimated by calculating the average rescaled range (R/S) over multiple overlapping regions (of different sizes) of the data. Only regions with lengths larger than 31 are used. This technique is robust and results in small standard deviations, but the estimated value is a bit biased. For a time series with a true Hurst Exponent value of 0.5 this technique on average produces an estimate with value 0.53 and standard deviation 0.06 when the data size is 1024 samples.

 

More information

www.bearcave.com/misl/misl_tech/wavelets/hurst/
Ian Kaplan
A description of the history of the Hurst Exponent, how it can be estimated and where it can be applied.

Hurst Exponent and Financial Market Predictability
Bo Quian and Khaled Rasheed
University of Georgia, USA
Experiments with neural networks showing that time series with large Hurst Exponents can be predicted more accurately than those series with Hurst Exponents close to 0.5.