Abstract
We study whether, and if yes then how, a varying auto-correlation structure in different parts of distributions is reflected in the multifractal properties of a dynamic process. Utilizing the quantile autoregressive process with Gaussian copula using three popular estimators of the generalized Hurst exponent, our Monte Carlo simulation study shows that such dynamics translate into multifractal dynamics of the generated series. The tail-dependence of the auto-correlations forms strong enough non-linear dependencies to be reflected in the estimated multifractal spectra and separated from the case of the standard auto-regressive process. With a quick empirical example from financial markets, we argue that the interaction is more important for the asymmetric tail dependence. In addition, we discuss and explain the often reported paradox of higher multifractality of shuffled series compared to the original financial series. In short, the quantile-dependent auto-correlation structures qualify as sources of multifractality and they are worth further theoretical examination.
Highlights
Multifractality as a characteristic of complex dynamic systems was developed in the 1970s and 1980s in two separate branches—in the works of Novikov [1] and Mandelbrot [2,3] studying turbulence in fluid mechanics, and in the works of Grassberger [4,5] and Hentschel and Procaccia [6]generalizing the fractal dimension, information dimension and correlation dimension into a unified framework through Rényi entropy describing strange attractors in nonlinear dynamics
Generalizing the notion of fractality, which in finance mostly translates into long-range dependence and/or power-law distributions, the multifractality of financial time series suggests that the correlation structure is non-linear and complex, usually intertwined with a non-Gaussian distribution of the stochastic process
We studied the effect of quantile-dependent auto-correlation structure on the multifractality of such time series
Summary
Multifractality as a characteristic of complex dynamic systems was developed in the 1970s and 1980s in two separate branches—in the works of Novikov [1] and Mandelbrot [2,3] studying turbulence in fluid mechanics, and in the works of Grassberger [4,5] and Hentschel and Procaccia [6]generalizing the fractal dimension, information dimension and correlation dimension into a unified framework through Rényi entropy describing strange attractors in nonlinear dynamics. Generalizing the notion of fractality, which in finance mostly translates into long-range dependence and/or power-law distributions, the multifractality of financial time series suggests that the correlation structure is non-linear and complex, usually intertwined with a non-Gaussian distribution of the stochastic process. This leads to the standard separation of multifractality into two sources—complex (non-linear) correlations and heavy-tailed distributions [14]. The former goes back to the long-range dependence generalization, as long-range dependent processes are characterized by a slowly decaying auto-correlation function, usually in an asymptotically hyperbolic manner, i.e., following a power-law, which implies a power-law scaling of variance of their integrated process
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