Elsevier

Chaos, Solitons & Fractals

Volume 138, September 2020, 109914
Chaos, Solitons & Fractals

A note on power-law cross-correlated processes

https://doi.org/10.1016/j.chaos.2020.109914Get rights and content

Highlights

  • •

    We introduce co-self-similar processes and extend the concept of cumulative range to the bivariate case.

  • •

    Co-self-similar processes are equivalent to those with co-affine increments.

  • •

    We prove the validity of MFHXA to calculate the bivariate Hurst exponent.

  • •

    We introduce a power-law cross-correlated version of the FD algorithm and test how it performs compared to DCCA and MFHXA.

  • •

    We show that power-law cross-correlation has nothing to do with correlation.

Abstract

In this paper, some mathematical support is provided to properly justify the validity of the so-called multifractal height cross-correlation analysis (MFHXA), first contributed by Kristoufek (2011)[1]. With this aim, we extend several concepts from univariate random functions and their increments to the bivariate case. Specifically, we introduce the bivariate cumulative range as well as the concepts of co-self-similar processes and random processes with co-affine increments of bivariate parameter. They allow us to introduce a new procedure, named multifractal cross-correlated fractal dimension (MFXFD) algorithm. We theoretically prove the validity of such a novel approach to calculate the bivariate Hurst exponent of a pair of processes with co-affine increments. Interestingly, that class of random functions is characterised theoretically in terms of co-self-similar processes. Moreover, we prove that a pair of univariate self-similar processes are co-self-similar and their bivariate Hurst exponent coincides with the mean of their univariate parameters. Finally, we test the behavior of the new algorithm to calculate the bivariate Hurst exponent of a pair of time series. Both DCCA and MFHXA procedures are involved in such an empirical comparison. Our results suggest that the new MFXFD performs (at least) as well as the other approaches with shorter deviations with respect to the mean of the bivariate Hurst exponents.

Section snippets

Introduction: multifractality in financial data

A large amount of empirical studies have shown that stock returns exhibit fat tails and some other features that are difficult to be modeled by simple distributions. Such statistical properties are known as stylized facts and appear to be universal for all kind of financial data. In this way, Cont provides in Cont [2] an interesting review about the statistical properties of financial data.

The first stylized fact regards the fat-tailed distributions. Part of the financial literature has

Multifractal power-law cross-correlation algorithms: a literature review

The goal of all proposed methods of multifractal power-law cross-correlation analysis is to detect possible multifractal long-range cross-correlations between two time series. Applications of such techniques has been extended to different fields. The seminal paper on this topic was due to Meneveau et al., where the authors analyzed the relationship between the dissipation rates of kinetic energy and passive scalar fluctuations in fully developed turbulence [50]. Such a method is known as the

Power-law cross-correlated processes

In [62], it was highlighted that the bivariate Hurst exponent does not provide information about possible relations between a pair of series itself. Thus, a careful comparison involving the bivariate self-similarity exponent and the separate Hurst exponents of the two series becomes necessary.

The detrended cross-correlation analysis (DCCA/DXA) was introduced by Podobnik and Stanley [51] as a natural step from DFA towards power-law cross-correlation analysis. That approach is based on scaling of

Theoretical support for MFHXA

Algorithm MFHXA (see [1]) is based on the assumptions that two processes Xt and Yt satisfy the following equation (c.f. [1, Eq. (2)]):E[(ΔτXtΔτYt)q2]∝τqHx,y(q),

Next, we prove that for a pair of random processes with co-affine increments of parameter H, the previous equation is satisfied and hence the MFHXA algorithm can be used to calculate the generalized bivariate Hurst exponent Hx,y(q)=H.

Theorem 4.1

Let Xt and Yt be two random functions with co-affine increments of bivariate parameter Hx,y. Then MFHXA

Introducing MFXFD

In this section we introduce the power-law cross-correlation version of the FD algorithm to calculate the Hurst exponent.

First, recall that the FD approach (c.f. [77], [78]) is based on the following equation:E[Rτq(Xt)]∝τqHx,y(q)

Now, we introduce the multifractal cross fractal dimension (MFXFD) method, which is valid for processes Xt and Yt that satisfies the following equation:E[Rτq2(Xt,Yt)]∝τqHx,y(q)

As always, in order to calculate Hx,y(q) for some q > 0, one must take logarithms in the

Conclusions

In this paper, we have introduced the concept of co-self-similar processes, which are the generalization of self-similar processes to the bivariate case. Moreover, we have proved that they constitute a natural context where algorithms such as the MFHXA have most sense. In addition, we have introduced a novel procedure (MFXFD) to properly calculate the bivariate parameter of such processes and empirically tested its behavior for this purpose as an alternative to DCCA or MFHXA approaches.

In

CRediT authorship contribution statement

M. Fernández-Martínez: Supervision, Conceptualization, Project administration. M.A. Sánchez-Granero: Writing - original draft, Investigation, Methodology. M.P. Casado Belmonte: Formal analysis, Validation, Software, Methodology. J.E. Trinidad Segovia: Visualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors would like to express their gratitude to the anonymous reviewers whose suggestions, comments, and remarks have allowed them to enhance the quality of this paper.

References (78)

  • E. Bacry et al.

    Singularity spectrum of fractal signals from wavelet analysis: exact results

    J Stat Phys

    (1993)
  • T. Lux

    The Markov-switching multifractal model of asset returns: GMM estimation and linear forecasting of volatility

    J Bus Econ Stat

    (2008)
  • W.C. Jun et al.

    Understanding volatility correlation behavior with a magnitude cross-correlation function

    Phys Rev E

    (2006)
  • F. Wang et al.

    Cross-correlation detection and analysis for California’s electricity market based on analogous multifractal analysis

    Chaos

    (2013)
  • Z. Jiang et al.

    Multifractal cross wavelet analysis

    Fractals

    (2017)
  • Y. Wei et al.

    Multifractal temporally weighted detrended cross-correlation analysis to quantify power-law cross-correlation and its application to stock markets

    Chaos

    (2017)
  • P. Oswiecimka et al.

    Detrended cross-correlation analysis consistently extended to multifractality

    Phys Rev E

    (2014)
  • L. Kristoufek

    Can the bivariate hurst exponent be higher than an average of the separate hurst exponents?

    Phys A

    (2015)
  • L. Kristoufek

    Power-law cross-correlations estimation under heavy tails

    Commun Nonlinear Sci Numer Simul

    (2016)
  • L. Kristoufek

    On the interplay between short and long term memory in the power-law cross-correlations setting

    Phys A

    (2015)
  • L. Kristoufek

    Spectrum-based estimators of the bivariate hurst exponent

    Phys Rev E

    (2014)
  • G.F. Zebende

    DCCA cross-correlation coefficient: quantifying level of cross-correlation

    Phys A

    (2011)
  • L. Kristoufek

    Detrending moving-average cross-correlation coefficient: measuring cross-correlations between non-stationary series

    Phys A

    (2014)
  • L. Kristoufek

    Detrended fluctuation analysis as a regression framework: estimating dependence at different scales

    Phys Rev E

    (2015)
  • L. Kristoufek

    Multifractal height cross-correlation analysis: a new method for analyzing long-range cross-correlations

    EPL (Europhys Lett)

    (2011)
  • R. Cont

    Empirical properties of asset returns: stylized facts and statistical issues

    Quant Finance

    (2001)
  • B.B. Mandelbrot

    The variation of certain speculative prices

    J Bus

    (1964)
  • T. Lux et al.

    Financial power laws: empirical evidence, models, and mechanisms

    Chaos Solitons Fractals

    (2016)
  • P. Carr et al.

    The fine structure of asset returns: an empirical investigation

    J Bus

    (2002)
  • Y.S. Kim et al.

    The modified tempered stable distribution, GARCH models and option pricing

    Probab Math Stat

    (2009)
  • M.C.A.B. Hols et al.

    The limiting distribution of extremal exchange rate returns

    J Appl Econom

    (1991)
  • R. Cont et al.

    Scaling in stock market data: stable laws and beyond

  • P. Gopikrishnan et al.

    Inverse cubic law for the distribution of stock price variations

    Eur Phys J B

    (1998)
  • T. Lux et al.

    The oxford handbook of computational economics and finance, chapter multifractal models in finance: their origin, properties, and applications

    (2018)
  • Mandelbrot B.B., Fisher A.J., Calvet L.E. A multifractal model of asset returns. (September 15, 1997). Cowles...
  • B.B. Mandelbrot

    A multifractal walk down wall street

    Sci Am

    (1999)
  • L. Calvet et al.

    Multifractality in asset returns: theory and evidence

    Rev Econ Stat

    (2002)
  • N. Vandewalle et al.

    Sparseness and roughness of foreign exchange rates

    Int J Modern Phys C

    (1998)
  • F. Schmitt et al.

    Multifractal analysis of foreign exchange data

    Appl Stoch Models Data Anal

    (1999)

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    1

    Partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain and FEDER, Spain, grant PGC2018-097198-B-I00.

    2

    Partially supported by Fundación Séneca of Región de Murcia (Murcia, Spain), grant 20783/PI/18.

    3

    Partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain and FEDER, Spain, grant PGC2018-101555-B-I00.

    4

    Partially supported by UAL/CECEU/FEDER, Spain, grant UAL18-FQM-B038-A.

    5

    Author acknowledges the support of CDTIME.

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