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Volatility of Linear and Nonlinear Time Series

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  • Tomer Kalisky
  • Yosef Ashkenazy
  • Shlomo Havlin

Abstract

Previous studies indicate that nonlinear properties of Gaussian time series with long-range correlations, $u_i$, can be detected and quantified by studying the correlations in the magnitude series $|u_i|$, i.e., the ``volatility''. However, the origin for this empirical observation still remains unclear, and the exact relation between the correlations in $u_i$ and the correlations in $|u_i|$ is still unknown. Here we find analytical relations between the scaling exponent of linear series $u_i$ and its magnitude series $|u_i|$. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared to linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series $sgn(u_i)$]. Our results of magnitude series correlations may help to identify linear and nonlinear processes in experimental records.

Suggested Citation

  • Tomer Kalisky & Yosef Ashkenazy & Shlomo Havlin, 2004. "Volatility of Linear and Nonlinear Time Series," Papers cond-mat/0406310, arXiv.org.
  • Handle: RePEc:arx:papers:cond-mat/0406310
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    Cited by:

    1. Wu, Zhenyu & Shang, Pengjian, 2017. "Nonlinear transformation on the transfer entropy of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 392-400.

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