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Statistical tests for power-law cross-correlated processes.

Boris Podobnik1, Zhi-Qiang Jiang, Wei-Xing Zhou

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Summary
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This study derives the bounds for the detrended cross-correlation coefficient (ρ(DCCA)) in nonstationary time series. It establishes statistical significance ranges and shows weak correlation between Chinese and U.S. financial markets.

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Area of Science:

  • Time Series Analysis
  • Econometrics
  • Statistical Physics

Background:

  • Cross-correlation quantifies time series similarity, but standard methods fail for nonstationary data.
  • Detrended Cross-correlations Analysis (DCCA) and its coefficient ρ(DCCA)(T,n) are recent advancements for nonstationary series.

Purpose of the Study:

  • To derive the theoretical bounds of the detrended cross-correlation coefficient (ρ(DCCA)(T,n)).
  • To determine statistical significance ranges for cross-correlations using overlapping windows.
  • To analyze the relationship between Chinese and U.S. financial markets using the DCCA framework.

Main Methods:

  • Derivation of the Cauchy inequality (-1 ≤ ρ(DCCA)(T,n) ≤ 1) for both standard variance-covariance and detrending approaches.
  • Numerical determination of statistical significance ranges for overlapping windows.
  • Derivation of the standard deviation of ρ(DCCA)(T,n) for nonoverlapping windows, showing it tends to 1/T as T increases.

Main Results:

  • The Cauchy inequality for ρ(DCCA)(T,n) is theoretically derived.
  • Statistical significance of cross-correlations is determined for overlapping windows.
  • The standard deviation of ρ(DCCA)(T,n) is shown to approach 1/T for large time series lengths.
  • An extremely weak following tendency of the Chinese financial market towards the U.S. market is demonstrated.

Conclusions:

  • The derived bounds and significance ranges enhance the applicability of DCCA for nonstationary time series.
  • The analysis reveals a negligible cross-correlation between the Chinese and U.S. financial markets.
  • A new statistical test is proposed for quantifying cross-correlations in power-law correlated time series.