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

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## Abstract

In the paper, we introduce a new measure of correlation between possibly non-stationary series. As the measure is based on the detrending moving-average cross-correlation analysis (DMCA), we label it as the DMCA coefficient $\rho_{DMCA}(\lambda)$ with a moving average window length $\lambda$. We analytically show that the coefficient ranges between -1 and 1 as a standard correlation does. In the simulation study, we show that the values of $\rho_{DMCA}(\lambda)$ very well correspond to the true correlation between the analyzed series regardless the (non-)stationarity level. Dependence of the newly proposed measure on other parameters -- correlation level, moving average window length and time series length -- is discussed as well.

## Suggested Citation

• Ladislav Kristoufek, 2013. "Detrending moving-average cross-correlation coefficient: Measuring cross-correlations between non-stationary series," Papers 1311.0657, arXiv.org.
• Handle: RePEc:arx:papers:1311.0657
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File URL: http://arxiv.org/pdf/1311.0657

## References listed on IDEAS

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1. Kristoufek, Ladislav, 2014. "Measuring correlations between non-stationary series with DCCA coefficient," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 402(C), pages 291-298.
2. Zebende, G.F. & da Silva, M.F. & Machado Filho, A., 2013. "DCCA cross-correlation coefficient differentiation: Theoretical and practical approaches," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(8), pages 1756-1761.
3. Schumann, Aicko Y. & Kantelhardt, Jan W., 2011. "Multifractal moving average analysis and test of multifractal model with tuned correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(14), pages 2637-2654.
4. Kantelhardt, Jan W. & Zschiegner, Stephan A. & Koscielny-Bunde, Eva & Havlin, Shlomo & Bunde, Armin & Stanley, H.Eugene, 2002. "Multifractal detrended fluctuation analysis of nonstationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 87-114.
5. Ladislav Kristoufek, 2012. "Multifractal Height Cross-Correlation Analysis: A New Method for Analyzing Long-Range Cross-Correlations," Papers 1201.3473, arXiv.org, revised Jan 2012.
6. Sergio Arianos & Anna Carbone, 2008. "Cross-correlation of long-range correlated series," Papers 0804.2064, arXiv.org, revised Mar 2009.
7. Gao-Feng Gu & Wei-Xing Zhou, 2010. "Detrending moving average algorithm for multifractals," Papers 1005.0877, arXiv.org, revised Jun 2010.
8. Serinaldi, Francesco, 2010. "Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2770-2781.
9. Zhi-Qiang Jiang & Wei-Xing Zhou, 2011. "Multifractal detrending moving average cross-correlation analysis," Papers 1103.2577, arXiv.org, revised Mar 2011.
10. He, Ling-Yun & Chen, Shu-Peng, 2011. "A new approach to quantify power-law cross-correlation and its application to commodity markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(21), pages 3806-3814.
11. Ying-Hui Shao & Gao Feng Gu & Zhi-Qiang Jiang & Wei-Xing Zhou & Didier Sornette, 2012. "Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series," Papers 1208.4158, arXiv.org.
12. Barunik, Jozef & Kristoufek, Ladislav, 2010. "On Hurst exponent estimation under heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3844-3855.
13. Zebende, G.F., 2011. "DCCA cross-correlation coefficient: Quantifying level of cross-correlation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(4), pages 614-618.
14. Wei-Xing Zhou, 2008. "Multifractal detrended cross-correlation analysis for two nonstationary signals," Papers 0803.2773, arXiv.org.
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## Citations

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Cited by:

1. Kristoufek, Ladislav, 2014. "Leverage effect in energy futures," Energy Economics, Elsevier, vol. 45(C), pages 1-9.
2. Kristoufek, Ladislav, 2015. "Power-law correlations in finance-related Google searches, and their cross-correlations with volatility and traded volume: Evidence from the Dow Jones Industrial components," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 428(C), pages 194-205.

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