Statistical tests for power-law cross-correlated processes

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, ${\rho }_{\mathrm{DCCA}}(T,n)$, where $T$ is the total length of the time series and $n$ the window size. For ${\rho }_{\mathrm{DCCA}}(T,n)$, we numerically calculated the Cauchy inequality $-1\le {\rho }_{\mathrm{DCCA}}(T,n)\le 1$. Here we derive $-1\le {\rho }_{\mathrm{DCCA}}(T,n)\le 1$ for a standard variance-covariance approach and for a detrending approach. For overlapping windows, we find the range of ${\rho }_{\mathrm{DCCA}}$ 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 ${\rho }_{\mathrm{DCCA}}(T,n)$ tends with increasing $T$ to $1/T$. Using ${\rho }_{\mathrm{DCCA}}(T,n)$ 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.

Figure 1

PDF of critical points ${\rho }_c(\alpha ,{\alpha }^{\prime },T,n)$ for the statistical test of Eq. (4), where $\alpha ={\alpha }^{\prime }=0.5$.

Figure 2

Scaling in critical points. Critical points ${\rho }_c(T,n)$ for the statistical test of Eq. (4) with the 95$\%$ confidence level. For varying values of series size $T$, we show ${\rho }_c(\alpha ,{\alpha }^{\prime },T,n)$ vs window size $n$.

Figure 3

Normality in data. (a) Linear-linear plot. Standard deviation of the statistical test ${\rho }_c(T,n)$ of Eq. (4) vs series size $T$ for different choice of window size $n$. For each $n$, standard deviation vs $T$ scales with a characteristic exponent for a Gaussian PDF. (b) Log-log plot. Standard deviation of the statistical test ${\rho }_c(T,n)$ of Eq. (4) vs window size $n$ for different choice of series size $T$. For each $T$, standard deviation vs $n$ scales with a characteristic exponent for a Gaussian PDF.

Figure 4

Statistical test $\rho (T,n)$ of Eq. (4) vs window size $n$ calculated for absolute returns series between NYA and a Shanghai index SSEC, NYA, and a German index DAX, and between NYA and NASDAQ Industrial. Shown are also the critical values for the 95$\%$ confidence level under assumption of no cross-correlations. The area below ${\rho }_c(T,n)$ vs $n$ means insignificant cross-correlations. Chinese index and U.S. index follow each other very weakly implying that U.S. financial market does not affect the Chinese financial market very strongly. We also show $\rho (T,n)$ of Eq. (4) vs window size $n$ calculated for returns series between NYA and NASDAQ.

Figure 5

Statistical test $\rho (T,n)$ of Eq. (4) vs window size $n$ between NYA and a Shanghai index SSEC, NYA calculated for two subperiods. We show both returns and absolute values of returns. Chinese index and U.S. index follow each other stronger during the second half of an 11-year period, characterized by a world recession and market crashes. However, the influence between these two indices is still much weaker than that between DAX and NYA (see Fig. 4).