Hurst Exponents, Nonlinear Dependence, and Detrended Fluctuation Analysis
Summary
The document considers whether the Hurst exponent can identify nonlinear dependence in financial time series. It notes that a Hurst-based persistence measure may fail to reveal nonlinear dynamics when ordinary autocorrelation is insignificant, and cautions against using it for non-stationary series. It also points to multifractal analysis when scaling behavior varies with the order parameter.
As an alternative for non-stationary or time-varying data, the answer recommends detrended fluctuation analysis, which produces an indicator related to the Hurst exponent. The discussion cites foundational work on fluctuation analysis and a book applying fractal ideas to markets, but provides no empirical comparison, implementation details, or validation on asset returns. These pointers are suggestions rather than evidence that the method captures every form of nonlinear dependence; method choice should reflect the series’ properties.
Key ideas
- A Hurst exponent may not reveal nonlinear dependence that is not reflected in autocorrelation.
- The answer cautions that standard Hurst analysis is unsuitable for some non-stationary series.
- Nonlinear scaling behavior may call for a multifractal framework.
- Detrended fluctuation analysis is suggested for persistence in non-stationary, changing data.
- The document offers references but no market-specific empirical validation.
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Full text
# Can Hurst exponent be used to characterize nonlinear dependence in time series? # 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? ## Answer by Quantopik (score 1) https://quant.stackexchange.com/a/17482 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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