Quantifying and modeling long-range cross correlations in multiple time series with applications to world stock indices

We propose a modified time lag random matrix theory in order to study time-lag cross correlations in multiple time series. We apply the method to 48 world indices, one for each of 48 different countries. We find long-range power-law cross correlations in the absolute values of returns that quantify risk, and find that they decay much more slowly than cross correlations between the returns. The magnitude of the cross correlations constitutes “bad news” for international investment managers who may believe that risk is reduced by diversifying across countries. We find that when a market shock is transmitted around the world, the risk decays very slowly. We explain these time-lag cross correlations by introducing a global factor model (GFM) in which all index returns fluctuate in response to a single global factor. For each pair of individual time series of returns, the cross correlations between returns (or magnitudes) can be modeled with the autocorrelations of the global factor returns (or magnitudes). We estimate the global factor using principal component analysis, which minimizes the variance of the residuals after removing the global trend. Using random matrix theory, a significant fraction of the world index cross correlations can be explained by the global factor, which supports the utility of the GFM. We demonstrate applications of the GFM in forecasting risks at the world level, and in finding uncorrelated individual indices. We find ten indices that are practically uncorrelated with the global factor and with the remainder of the world indices, which is relevant information for world managers in reducing their portfolio risk. Finally, we argue that this general method can be applied to a wide range of phenomena in which time series are measured, ranging from seismology and physiology to atmospheric geophysics.

Figure 1

Cross correlations among the $N=48$ world financial index returns each of size $T=2744.$ (a) The empirical PDF of the coefficients of the cross correlation matrix $\mathsf{C}$ calculated between index returns with increasing $\Delta t$ quickly converges to the Gaussian form. The normal distribution is the distribution of the pairwise cross correlations for finite length uncorrelated time series, which is a normal distribution with mean zero and standard deviation $\frac{1}{\sqrt{T}}$. (b) The empirical PDF of the coefficients of the matrix $\tilde{\text{C}}$ calculated between index volatilities approaches the PDF of the random matrix more slowly than in (a).

Figure 2

Long-range magnitude cross correlations. The largest singular value ${\lambda }_L$ obtained from the spectrum of the matrices $\text{C}$ and $\tilde{\text{C}}$ versus time lag $\Delta t$. With increasing $\Delta t$, the largest singular values obtained for $\text{C}$ of returns decays more quickly than $\tilde{\text{C}}$ calculated for absolute values of returns. The magnitude cross correlations decay as a power-law function with the scaling exponent of $\approx 0.25$.

Figure 3

Short-range autocorrelations of the global factor. (a) Time series of the global factor. (b) The autocorrelation function (ACF) of the global factor. The region between dashed lines is the 95% confidence interval for the no autocorrelation hypothesis. Autocorrelations are smaller than 0.132 except $\Delta t=0$, and become insignificant after time lag $\Delta t=2$, with no more than one significant autocorrelation for every 20 time lags. Therefore, only short-range autocorrelations can be found in the global factor.

Figure 4

Long-range autocorrelations of the magnitude global factor. (a) Time series of magnitudes of the global factor. (b) Autocorrelations of magnitudes of the global factor. The region between dashed lines is the 95% confidence interval for the no autocorrelation hypothesis. Autocorrelations are much larger than the autocorrelations of the global factor itself, as large as 0.359 at $\Delta t=2$, and is still larger than 0.2 until $\Delta t=33$. For every time lag, the autocorrelation is significant even after $\Delta t=100$. Therefore long-range autocorrelations exist in the magnitudes of the global factor.

Figure 5

(a) Conditional volatility of the global factor, showing that the clusters in the conditional volatilities may serve to predict market crashes. In each cluster, the height indicates the size of the market crash, and the width indicates its duration. (b) The 100-day forecasted volatility of the global factor, using the past data ranging from 4 Jan 1999 through 10 July 2009. It will converge to the unconditional volatility asymptotically.

Figure 6

Cross correlation between the global factor $M_t$ and each individual index $R_{i,t}$, $i=1,2,\dots ,48$. The global factor has large correlation with most of the indices. However, there are indices that are not much correlated with the global factor. Of the 48 indices, ten have a correlation smaller than 0.1 between the global factor, corresponding to the indices for Iceland, Malta, Nigeria, Kenya, Israel, Oman, Qatar, Pakistan, Sri Lanka, and Mongolia. Hence, unlike most countries, the economies of these ten countries are more independent of the world economy.