For stationary time series, the cross-covariance and the cross-correlation as functions of time lag $n$ serve to quantify the similarity of two time series. The latter measure is also used to assess whether the cross-correlations are statistically significant. For nonstationary time series, the analogous measures are detrended cross-correlations analysis (DCCA) and the recently proposed detrended cross-correlation coefficient, ${\ensuremath{\rho}}_{\mathrm{DCCA}}(T,n)$, where $T$ is the total length of the time series and $n$ the window size. For ${\ensuremath{\rho}}_{\mathrm{DCCA}}(T,n)$, we numerically calculated the Cauchy inequality $\ensuremath{-}1\ensuremath{\le}{\ensuremath{\rho}}_{\mathrm{DCCA}}(T,n)\ensuremath{\le}1$. Here we derive $\ensuremath{-}1\ensuremath{\le}{\ensuremath{\rho}}_{\mathrm{DCCA}}(T,n)\ensuremath{\le}1$ for a standard variance-covariance approach and for a detrending approach. For overlapping windows, we find the range of ${\ensuremath{\rho}}_{\mathrm{DCCA}}$ within which the cross-correlations become statistically significant. For overlapping windows we numerically determine---and for nonoverlapping windows we derive---that the standard deviation of ${\ensuremath{\rho}}_{\mathrm{DCCA}}(T,n)$ tends with increasing $T$ to $1/T$. Using ${\ensuremath{\rho}}_{\mathrm{DCCA}}(T,n)$ we show that the Chinese financial market's tendency to follow the U.S. market is extremely weak. We also propose an additional statistical test that can be used to quantify the existence of cross-correlations between two power-law correlated time series.

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Statistical tests for power-law cross-correlated processes

In this short report, we investigate the ability of the DCCA coefficient to measure correlation level between non-stationary series. Based on a wide Monte Carlo simulation study, we show that the DCCA coefficient can estimate the correlation coefficient accurately regardless the strength of non-stationarity (measured by the fractional differencing parameter d ). For a comparison, we also report the results for the standard Pearson correlation coefficient. The DCCA coefficient dominates the Pearson coefficient for non-stationary series.

/pdf/measuring-correlations-between-non-stationary-series-with-oxqkvhs302.pdf

Measuring correlations between non-stationary series with DCCA coefficient

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

https://iopscience.iop.org/article/10.1088/1361-6633/ab42fb/pdf

Multifractal analysis of financial markets: a review.

Multifractal analysis of financial markets

Based on the daily price data of the Chinese Yuan (RMB)/US dollar exchange rate and the Shanghai Stock Composite Index, we conducted an empirical analysis of the cross-correlations between the Chinese exchange market and stock market using the multifractal cross-correlation analysis method. The results demonstrate the overall significance of the cross-correlation based on the analysis of a statistic. Multifractality exists in cross-correlations, and the cross-correlated behavior of small fluctuations is more persistent than that of large fluctuations. Moreover, using the rolling windows method, we find that the cross-correlations between the Chinese exchange market and stock market vary with time and are especially sensitive to the reform of the RMB exchange rate regime. The previous reduction in the flexibility of the RMB exchange rate in July 2008 strengthened the persistence of cross-correlations and decreased the degree of multifractality, whereas the enhancement of the flexibility of the RMB exchange rate in June 2010 weakened the persistence of cross-correlations and increased the multifractality. Finally, several relevant discussions are provided to verify the robustness of our empirical analysis.

Multifractal detrended cross-correlations between the Chinese exchange market and stock market

Multifractal detrended cross-correlation analysis (MF-DXA) has been developed to detect the long-range power-law cross-correlation of considered signals in the presence of non-stationarity. However, crossovers arising from extrinsic periodic trends make the scaling behavior difficult to analyze. We introduce a Fourier filtering method to eliminate the trend effects and systematically investigate the multifractal cross-correlation of simulated and real traffic signals. The crossover locations are found approximately corresponding to the periods of underlying trend. Traffic velocity on one road and flows on adjacent roads show strong cross-correlation. They also present weak multifractality after periodic trends are removed. The traffic velocity and flow are cross-correlated in opposite directions which is accordant to their actual evolution.

Gang Chen

Papers

Multifractal Fourier detrended cross-correlation analysis of traffic signals