European stock market dependencies when price changes are unusually large

Abstract

This article studies dependencies between European stock markets when returns are unusually large ‘extreme’, using daily data on stock market indices for Germany, the UK, France, The Netherlands and Italy from 1973 to 2001. Dependency is measured by the conditional probability of an unusually large return in one market given an unusually large return in another and is estimated using an approach from multivariate extreme value theory. It finds the following. First, dependencies between markets in situations of unusually large returns have become closer over time. Second, they are generally higher for large negative returns than for large positive ones. Third, dependencies differ depending on the country pair considered. For example, stock markets in the Netherlands and France are more closely and those in the UK and Italy less closely linked to the German market. Fourth, overall dependencies are quite symmetric, in the sense that the conditional probability for an unusually large change given a large change in the other country is similar irrespective of which of the two countries the probability is conditioned on.

Acknowledgements

The author is grateful to Pedro de Lima, Florian Pelgrin, Catalin Stărică and participants at seminars at the Deutsche Bundesbank and at the Annual Conference of the European Economic Association for helpful discussions and comments. Nonetheless, the author takes full responsibility for the final results.

Notes

¹ These are relatively large and liquid markets, accounting together for more than a fifth of total world stock market capitalization. Another advantage of this choice of markets is that the daily data reflect market conditions at about the same time. Including data for markets from time zones that differ more substantially (e.g. USA or Japan) would raise the problem of non-synchronous data collection which complicates the comparison of daily data for the different markets. While this problem could be reduced by looking at weekly data, such an approach would not allow one to focus on the very short-term, i.e. daily dependencies, and reduce the efficiency of the estimators which require very large data samples.

² The distribution of a one-dimensional variable r _t is said to be regularly varying at ∞ with index α > 0 if the tail probabilities can be written as , for all x > 0, where L is a function converging to a positive constant as x goes to infinity. Intuitively, the index α means that the probability of the random variable r _t larger than a given value x decreases as x ^{− α}, for x large. See Embrechts et al. (1997). For the multivariate case see Stărică (2000).

³ It should be noted that the focus is on return pairs where both returns are either positive or negative. This implies that one loses those pairs where one return is positive and the other negative. However, there are only very few such observations when returns are unusually large. Estimates are made using S-Plus procedures, building on those kindly obtained from Catalin Stărică.

⁴ Each probability estimate is based on a specific choice of k, the number of upper-order statistics. The choices of k are (column wise) as follows: 95, 95, 95, 95, 95, 95 for the UK; 100, 100, 100, 100, 100, 100 for Italy; 45, 45, 50, 45, 45, 35 for France and 35, 30, 50, 35, 45, 30 for Netherlands. As the ks differ for positive, negative and any returns for each country, the three estimated probabilities for each country are not necessarily systematically related each one to another. The estimates are found to be robust with respect to the choice of k. For example, when k is varied, the estimated probability changed only by 1% in many cases and in all but three cases the variation was smaller than 5%. The three exceptions are absolute returns in the case of The Netherlands and negative returns in the case of The Netherlands and France. In the case of the latter, reducing k by 10 lowers the estimated probability and increasing k by 10 raises it, in each case by about 0.09. In the other two cases the maximum variation could attain plus 0.09 (absolute, The Netherlands) and minus 0.15 (negative, The Netherlands). Also, controlling for the stock market crash in 1987 (by excluding data for October and November 1987), the estimated parameters change only very little.