Volatility of Linear and Nonlinear Time Series

TL;DR

This paper analytically explores the relationship between the correlations in linear and nonlinear time series and their magnitudes, proposing a model for generating multifractal series and aiding in identifying underlying processes.

Contribution

It provides analytical relations between the scaling exponents of linear series and their magnitude series, and introduces a simple model for generating multifractal time series with long-range correlations.

Findings

Nonlinear series show stronger or equal correlations in magnitude series compared to linear series.

Analytical relations between the scaling exponents of $u_i$ and $|u_i|$ are derived.

A model for generating multifractal series with long-range magnitude correlations is proposed.

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.