# Can Hurst exponent be used to characterize nonlinear dependence in time series?

It appears to me that the answer is no, because Hurst exponent measures persistence in terms of autocorrelation, which is a linear measure. So even if a time series of asset returns is driven by nonlinear or chaotic dynamics, the dependence would not be captured by Hurst exponent as long as the ACF isn't significant at any lag.

Is my understanding correct?

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The Hurst exponent can be used to distinguish between an AR(1) process and an ARIFMA process. Dependence is more often used contemporaneously. For instance, a t copula could capture the non-linear dependence between two time series. Presumably you could use some sort of t autocopula to do something similar, but I'm skeptical to what benefits you could get from doing that. – John Sep 10 '12 at 19:48
I am more interested in assessing the predictability of time series in univariate settings. For example, if a time series is driven by a deterministic nonlinear dynamic, it can appear totally random if you look at its ACF, even though in fact it's not random at all. From what I understand, the Hurst exponent cannot help to detect the nonlinear predictability in this case, but I just want to see what people's thoughts are. – ezbentley Sep 10 '12 at 22:06

## 1 Answer

As you suggest, in the case of non-stationary time series, the Hurst exponent is not suitable to measure the time seires persistence for the reasons you cited in the question. Particularly, when $H(q)$ is a non-linear function of q, as in the non-stationary time-series case, the time-series has to be analysed as it is a multi-fractal system (to deal this topic with, look at the 2nd reference below).

As regards the measure of the time series persistence for non-linear dynamics, I suggest to implement a detrended fluctuation analysis, that produce an indicator, highly correlated with the Hurst exponent, that is more appropriate to model non stationary, changing with time, data.

For more references about the topic, look at the Peng's seminal paper:

Peng, C.K. et al. (1994). "Mosaic organization of DNA nucleotides". Phys. Rev. E 49: 1685–1689.

an to the Mandelbrot's book about using fractal distributions in the financial markets:

Mandelbrot, Benoît B., The (Mis)Behavior of Markets, A Fractal View of Risk, Ruin and Reward (Basic Books, 2004), pp. 186-195

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