TABLE 5.4 Estimated tail indices of cross rates.

Estimated tail index values and their 95% confidence ranges, for cross FX rates. The tail index values are based on the subsample bootstrap method using the Hill estimator, and the confidence ranges result from the jackknife method. All the cross rates of the lower part were subject to the regulations of the European Monetary System (EMS). The computed cross rates are obtained via the two bilateral rates against the USD (see Equation 2.1). The estimations are performed on samples from January 1988 to June 1994 (6j years). The time intervals are measured in 7?-time (see Chapter 6).

In Section 5.3.1, we have seen that cross rates behaved differently when both exchanged currencies were members of the EMS in the 1990s. A difference is also found when considering the tail behavior, as shown in Table 5.4. The sample was chosen accordingly, during the lifetime of the EMS. The upper block of Table 5.4 has non-EMS cross rates for comparison, the lower block has EMS cross rates. The 1-day column is missing in Table 5.4 as the sample size is smaller than that of Table 5.3. The tail index values of EMS cross rates are around 2.7, distinctly lower than the typical value of 3.5 found for other cross rates in the upper part of Table 5.4 and other FX rates in Table 5.3. The 30-min columns of both tables should mainly be considered, but the values of the other columns confirm the same fact that EMS cross rates have fatter tails.

The cross rates are computed from USD rates according to Equation 2.1. As compared to direct quotes, these computed cross rates have larger spreads and an artificially generated volatility (i.e., noise due to the price uncertainty within the spread and asynchronous fluctuations of the used USD rates). Therefore we have also analyzed direct cross rate quotes in the limited sample (since October 1992) where they are available. These direct quotes have less short-term noise and lead to slightly but systematically lower tail index values than those of computed cross rates. The small tail index of EMS cross rates indicates that the reduced volatility induced by the EMS setup is at the cost of a larger probability of extreme events, which may lead to realignments of the system. This is an argument against the credibility of institutional setups such as the EMS.

Like the drift exponent as a function of time, the tail index reflects the institutional setup and, in a further interpretation, the way different agents on the markets interact. The tail index can therefore be considered as another empirical measure of market regulation and market efficiency. A large tail index indicates free interactions between agents with different time horizons, with a low degree of regulation and thus a smooth adjustment to external shocks. A small tail index indicates the opposite.

With very few exceptions, the estimated tail indices are between 2 and 4. A few confidence ranges extend to values outside this range, but this is due to the limited number of observations mainly for the longer return measurement intervals. Using Equation 5.2, we conclude that the second moment of the return distribution is finite and the fourth moment usually diverges. In Section 5.6.1, this fact leads us to preferring the autocorrelation of absolute returns to that of squared returns, which relies on the finiteness of the fourth moment.

Tables 5.3 and 5.4 indicate that the distribution of FX returns belongs to the class of fat-tailed nonstable distributions that have a finite tail index larger than 2. Furthermore, the very high values of the kurtosis in Tables 5.1 and 5.2 and the growth of these values with increasing sample size provide additional evidence in favor of this hypothesis.12 From Tables 5.3 and 5.4, one can also verify the invariance of the tail index under aggregation, except for the longest intervals, where the small number of observations becomes a problem in getting significant estimates of a. The smaller number of data for large intervals forces the estimation algorithm to use a larger fraction of this data, closer to the center of the distribution. Thus the empirically measured tail properties become distorted by properties of the center of the distribution, which, for a > 2 and under aggregation, approaches the normal distribution (with a = oo) as a consequence of the central limit theorem.

In Table 5.5, we present the results of a estimations of interbank money market cash interest rates for five different currencies and two maturities. Although generally exhibiting lower as, the results are close to those of the FX rates. The message seems to be the same: fat tails, finite second moment,13 and nonconverg-ing fourth moment. . We also find a relative stability of the tail index with time aggregation. The estimations for daily returns give more consistent values than in the case of the FX rates. Yet the estimations are more noisy as one would expect from data of lower frequency, and this is reflected in the high errors displayed in Table 5.5. The tail index estimation is quite consistent but can significantly jump from one time interval to the next as is the case for GBP and CHF 6 months. This market is much less liquid than the FX market. Interest rate markets with higher liquidity can be studied in terms of interest rate derivatives, which are traded in

12 Simulations in Gielens et al. (1996) and McCulloch (1997) show that one cannot univocally distinguish between a fat-tailed nonstable and a thin-tailed distribution only on the basis of low estimated values of the tail index. However, the confidence intervals around the estimated tail index values and the diverging behavior of the kurtosis are strong evidence in favor of the fat-tail hypothesis.

13 In most of the cases, except perhaps the 6-month interest rate for JPY, a is significantly larger than 2. In the JPY case, if the first 2 years are removed, we get back to values for a around 3.

TABLE 5.5 Estimated tail index for cash interest rates. Estimated tail index values of cash interest rates and their 95% confidence ranges, for five different currencies and two maturities. The tail index values are based on the subsample bootstrap method using the Hill estimator, and the confidence ranges result from the jackknife method. The time intervals are measured in #-time (see Chapter 6). The estimations are performed on samples going from January 2, 1979, to June 30, 1996; "m" refers to a month.

futures markets such as London International Financial Futures Exchange (LIFFE) in London or Singapore International Monetary Exchange (SIMEX) in Singapore.

To study the stability of the tail index under aggregation, we observe how the estimates change with varying sample size. We do not know any theoretical tail index for empirical data, but we compare the estimates with the estimation done on the "best" sample we have, 30-min returns from January 1987 to June 1996. To study the tail index of daily returns, we use an extended sample of daily data from July 1, 1978, to June 30, 1996. The small-sample bias can be studied by comparing the results to the averages of the results from two smaller samples: one from June 1, 1978, to June 30, 1987, and another one from July 1, 1987, to June 30, 1996. These two short samples together cover the same period as the large sample. The results are given in Table 5.6.14 When going from the short samples to the long 18-year sample, we see a general decrease of the estimated a toward the values reached for 30-min returns. The small-sample bias is thus reduced, but probably not completely eliminated. We conclude that, at least for the FX rates against the USD, the a estimates from daily data are not accurate enough even if the sample covers up to 18 years. The case of gold and silver is different because the huge fluctuations of the early 1980s have disappeared since then. The picture in this case is blurred by the changing market conditions.

A similar analysis of 30-min returns reinforces the obtained conclusions. In this case, the two shorter samples are from July 1, 1988, to June 30, 1992, and from July 1, 1992, to June 30,1996. The large sample is again the union of the two shorter samples. A certain small-sample bias is found also for the 30-min returns of most rates comparing the two last columns of Table 5.3, but this bias is rather small. This is an expected result because the number of 30-min observations is much larger than that of the daily observations.

For the short samples we do not give the errors because we present only the average of both samples. The errors are larger for the short samples than for the long sample.