  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 constitute "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 auto-correlations 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 10 indices 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.
