This color plot shows the value of the cross-correlation C(t1, t2) between the S&P500 in the time interval [t1, t1+5 years] and the Nikkei in the time interval [t2, t2+5 years], where t1 and t2 are varied over large time intervals shown in the figure. t1 is the abscissa and t2 is the ordinate. The colored contours plot the value of the cross-correlation coefficient C(t1, t2) as a function of t1 and t2. Regions in red mean large cross-correlations and thus good matching between the two indices in their respective intervals. We can observe a line of crests outlined by the violet straight line which expresses statistically through C(t1, t2) the remarkable matching between the two indices. The violet line is the best linear fit to this line of crests and corresponds mathematically to equation (2). This figure thus confirms the visual matching by a simple and robust statistical analysis and uncovers the novel feature of a significant time contraction of the patterns of the S&P500 compared with those of the Nikkei, as explained above. The implication is clear: it is naive to expect a perfect superposition described by a simple translation.
Discussion of the Nikkei-S&P500 matching patterns: The matching between the Nikkei and the S&P500 time series obtained here by a rigorous quantitative analysis of the cross-correlation of the two shifted time series should not lead to the belief that the S&P500 index is bound to follow blindly this correlation in the future. In contrast with chartism or technical analysis, we try to develop a scientific understanding of these bubble-antibubble phases. The similitude between the Nikkei and US markets are part of the search for "universal" properties, that allow us to establish a theory (in short, a theory is a story of repeatable/reproducible occurrences). Using this theory then allows us to describe idiosyncratic behaviors, that is, deviations from one case to another, or in other words, the parts of the evolutions that are not universal. This is what should give us an hedge for predictions.
Already, as early as September 2002, in our paper [*] based on an analysis carried on the stock market time series available up to Aug. 25, 2002, we wrote that we could see a clear difference between the Nikkei and the SP500. Thus the qualitative analogy is there but, quantitatively, there are serious differences. Technically, after two years and a half after the top in Dec. 31, 1989, we find that the Nikkei has started to shift to another antibubble regime while no such shift is yet detectable after more than three years since the start of the antibubble in the US. In addition, the US markets have been characterized by much stronger crashes and rallies, modelled below by our "zero-phase-Weierstrass" functions. These two facts suggest to us that the herding forces are even stronger in the US and that investors react even more on hair-trigger to any "news".
To sum up, the similarities between the shifted Nikkei and the S&P500 are qualitative: bubble preceding antibubble, strong speculation and herding, similar fear and herding in the anti-bubble regime, some problems with bad loans or bad accounting, strong commitment from the central banks and governments to provide liquidity and cash... But there are differences and these differences can be detected. Thus, we are not proponents of a superposition of the two time series to predict the future evolution of the US stock markets. It is clear to us that their future will be different, according to the forecasts proposed below.
[*] D. Sornette and W.-X. Zhou, The US 2000-2002 Market Descent: How
Much Longer and Deeper? Quantitative Finance 2 (6), 468-481 (2002) http://www.iop.org/EJ/S/1/NCA203394/RCM0rqd2bn5eBW0XZGGwvA/toc/1469-7688/2/6
(http://arXiv.org/abs/cond-mat/0209065)