A new approach to quantify power-law cross-correlation and its application to commodity markets

We proposed a new method: Detrended Moving-average Cross-correlation Analysis (DMCA) to detect the power-law cross-correlation between two correlated non-stationary time series by combining Detrended Cross-Correlation Analysis (DCCA) and Detrended Moving Average (DMA). In order to compare the performance of DMCA and DCCA in the detection of cross-correlation, and to estimate the influence of periodic trend, we generate two cross-correlated time series x(i) and y(i) by a periodic two-component fractionally autoregressive integrated moving average (ARFIMA) process. Then we apply both methods to quantify the cross-correlations of the generated series, whose theoretical values are already known to us. By comparing the results we obtained, we find that the performance of this new approach is comparable to DCCA with less calculating amounts; our method can also reduce the impact of trends; furthermore, DMCA (for background and forward moving average case) outperforms DCCA in more accurate estimation when the analyzed times series are short in length. To provide an example, we also apply this new method to the time series of the real-world data from Brent and WTI crude oil spot markets, to investigate the complex cross-market correlation between these commodity markets. In all, our method is another practical choice to detect the cross-correlation between two short period non-stationary time series, and has potential application to real world problems.

[1]  Ivo Grosse,et al.  Power-law correlated processes with asymmetric distributions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  H. Stanley,et al.  Effect of trends on detrended fluctuation analysis. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Wei-Xing Zhou,et al.  Multiscaling behavior in the volatility return intervals of Chinese indices , 2008 .

[4]  H. Stanley,et al.  Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series , 2002, physics/0202070.

[5]  H. Stanley,et al.  Quantifying cross-correlations using local and global detrending approaches , 2009 .

[6]  Gabjin Oh,et al.  Understanding volatility correlation behavior with a magnitude cross-correlation function. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Ivo Grosse,et al.  Time-lag cross-correlations in collective phenomena , 2010 .

[8]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[9]  Marcel Ausloos,et al.  CROSSING OF TWO MOBILE AVERAGES: A METHOD FOR MEASURING THE ROUGHNESS EXPONENT , 1998 .

[10]  Ivo Grosse,et al.  Fractionally integrated process with power-law correlations in variables and magnitudes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Wei-Xing Zhou,et al.  Statistical properties of volatility return intervals of Chinese stocks , 2008, 0807.1818.

[12]  R. Palmer,et al.  Time series properties of an artificial stock market , 1999 .

[13]  Ling-Yun He,et al.  Multifractal spectrum analysis of nonlinear dynamical mechanisms in China’s agricultural futures markets , 2010 .

[14]  Boris Podobnik,et al.  Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes , 2007, 0709.0838.

[15]  Wei-Xing Zhou,et al.  Detrending moving average algorithm for multifractals. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Wei‐Xing Zhou.  Multifractal detrended cross-correlation analysis for two nonstationary signals. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Yi-Ming Wei,et al.  Impact of speculator's expectations of returns and time scales of investment on crude oil price behaviors , 2009 .

[18]  Ling-Yun He,et al.  Nonlinear bivariate dependency of price–volume relationships in agricultural commodity futures markets: A perspective from Multifractal Detrended Cross-Correlation Analysis , 2011 .

[19]  J. Richman,et al.  Physiological time-series analysis using approximate entropy and sample entropy. , 2000, American journal of physiology. Heart and circulatory physiology.

[20]  R. Mantegna.  Hierarchical structure in financial markets , 1998, cond-mat/9802256.

[21]  H. Stanley,et al.  Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Wei‐Xing Zhou,et al.  Scaling and memory in the return intervals of realized volatility , 2009, 0904.1107.

[23]  H. Stanley,et al.  Multifractal properties of price fluctuations of stocks and commodities , 2003, cond-mat/0308012.

[24]  H. Stanley,et al.  Cross-correlations between volume change and price change , 2009, Proceedings of the National Academy of Sciences.

[25]  Takayuki Mizuno,et al.  Correlation networks among currencies , 2006 .

[26]  Ling-Yun He,et al.  Are developed and emerging agricultural futures markets multifractal? A comparative perspective , 2010 .

[27]  K Kaski,et al.  Time-dependent cross-correlations between different stock returns: a directed network of influence. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  R. Kálmán,et al.  Irregularity, volatility, risk, and financial market time series. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[29]  Edgar E. Peters.  Fractal Market Analysis: Applying Chaos Theory to Investment and Economics , 1994 .

[30]  H. Stanley,et al.  Power-law autocorrelated stochastic processes with long-range cross-correlations , 2007 .

[31]  M Tumminello,et al.  A tool for filtering information in complex systems. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[32]  Kazuko Yamasaki,et al.  Scaling and memory in volatility return intervals in financial markets. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[33]  The empirical analysis for fractal features and long-run memory mechanism in petroleum pricing systems , 2007 .

[34]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[35]  Sergio Arianos,et al.  Detrending moving average algorithm: A closed-form approximation of the scaling law , 2007 .

[36]  Wei-Xing Zhou,et al.  Recurrence interval analysis of high-frequency financial returns and its application to risk estimation , 2009, 0909.0123.

[37]  R. Coelho,et al.  Sector analysis for a FTSE portfolio of stocks , 2007 .

[38]  V. Plerou,et al.  Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series , 1999, cond-mat/9902283.

[39]  Ling-Yun He,et al.  Are crude oil markets multifractal? Evidence from MF-DFA and MF-SSA perspectives , 2010 .

[40]  H. Stanley,et al.  Detrended cross-correlation analysis: a new method for analyzing two nonstationary time series. , 2007, Physical review letters.

[41]  Clive W. J. Granger,et al.  An introduction to long-memory time series models and fractional differencing , 2001 .

[42]  A. Carbone,et al.  Second-order moving average and scaling of stochastic time series , 2002 .

[43]  Recurrence interval analysis of trading volumes. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  A. Carbone.  Detrending Moving Average algorithm: a brief review , 2009, 2009 IEEE Toronto International Conference Science and Technology for Humanity (TIC-STH).

[45]  P. Gopikrishnan,et al.  Inverse cubic law for the distribution of stock price variations , 1998, cond-mat/9803374.