Let X], Xi, ... , Xn be a sequence of n observations drawn from a stationary i.i.d. process whose probability distribution function F is unknown. We assume that the distribution is fat-tailed-that is, the tail index a is finite.10 Let us define X(\) > Xci) > • > ( ) as the descending order statistics from Xi, X2, ... , X„.

Extreme value theory states that the extreme value distribution of the ordered data must belong to one of just three possible general families, regardless of the original distribution function F (Leadbetter et al, 1983). Besides, if the original distribution is fat-tailed, there is only one general family it can belong to

G(x) = I a (5.3)

( exp(- x a) x > 0, a > 0

where G(x) is the probability that Xi\) exceeds x. There is only one parameter to estimate, a, which is called the tail index. The stable distributions (excluding the Gaussian distribution), the Student-? model, and the unconditional distribution of the ARCH-process all fall in the domain of attraction of this type of distribution.

To give more intuition to these statements, we plot the logarithm of the order statistics m as a function of the difference between the logarithms of the most extreme observation, lnX(i), and the mh observation in the ordered sequence, lnX(m). Such a plot is shown on Figure 5.7 for the case of a Student-r distribution with 4 degrees of freedom. Because we are in the domain of attraction of exp(-x~a), it is trivial to see that the problem of estimating a becomes the problem of estimating the slope of the tangent at m -> 0 of the curve shown in Figure 5.7. We see that a straight line with a slope equal to 4 is indeed a good tangent to the curve, as it should be because the theoretical tail index of the Student-r distribution is equal to the number of degrees of freedom. Although the behavior of In X(m) is quite regular on Figure 5.7 because we took the average values over 10 Monte Carlo simulations, it is not always so and the problem of how to choose the number of points that are far in the tail is not trivial. One needs a more formal way to estimate the tail index. The estimator we present here is a way of estimating the slope of the tangent shown in Figure 5.7. There are other ways of studying the tail index by directly fitting the distribution of ordered data to some known distribution."

We concentrate our efforts on the estimator first proposed by Hill (1975)

I m - l

Ynm = - Y\ln Xd) ~ In X(m) where m > 1 (5.4)

m - 1 i-J

This estimator was proven to be a consistent estimator of = I/a for fat-tailed distributions in Mason (1982). From Hall (1982) and Goldie and Smith (1987), it follows that (Yn.m -y)m]/2 is asymptotically normally distributed with mean zero

10 A good review of the definitions used in this chapter can be found in Leadbetter et al. (1983). A good reference to learn about these methods is the book by Embrechts el al. (1997).

ln(X(1))-ln(X(m))

FIGURE 5.7 The logarithm of the order statistics m is plotted as a function of the difference between the logarithm of the most extreme observation and the logarithm of the ordered random observations. The data are drawn from a Student-f distribution with 4 degrees of freedom averaged over I0 replications of a Monte-Carlo simulation. The straight line represents the theoretical tangent to this curve.

and variance y2. In fact the Hill estimator is the maximum likelihood estimator of and a = \/y holds for the tail index. For finite samples, however, the expected value of the Hill estimator is biased. As long as this bias is unknown, the practical application of the Hill estimator to empirical samples is difficult. A related problem is that p„,m depends on m, the number of order statistics, and there is no easy way to determine which is the best value of m. Extending a bootstrap estimation method proposed by Hall (1990), Danielsson et al. (1997) solved the problem by means of a subsample bootstrap procedure, which is described and discussed by Pictet et al. (1998). Many independent subsamples (or resamples) are drawn from the full sample and their tail behaviors are statistically analyzed, which leads to the best choice of m. For such a statistical analysis, the subsamples have to be distinctly smaller than the full sample. On the other hand, the subsamples should still be large enough to contain some representative tail observations, so the method greatly benefits from a large sample size to begin with.

Tail index values of some FX rates have been estimated by a subsample bootstrap method and are presented in Table 5.3. The confidence ranges indicated for all values are standard errors times 1.96. Assuming a normally distributed error, this corresponds to 95% confidence. The standard errors have been obtained through the jackknife method, which can be characterized as follows. The data

TABLE 5.3 Estimated tail indices of FX rates Estimated tail index values a and their 95% confidence ranges, for main FX rates against the USD, gold (XAU), and silver (XAG) and some of the main (computed) cross rates against the DEM, Muller et al. (1998). The tail index values are based on the subsample bootstrap method using the Hill estimator, the confidence ranges result from the jackknife method. 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 1, 1987, to June 30, 1996. The time intervals are measured in tf-time (see Chapter 6).

sample is modified in 10 different ways, each time removing one-tenth of the total sample. The tail index is separately computed for each of the 10 modified samples. An analysis of the deviations between the 10 results yields an estimate of the standard error, which is realistic because it is based on the data rather than restrictive theoretical assumptions. The methodology is explained by Pictet etal. (1998).

All the FX rates against the USD as well as the presented cross rates have a tail index between 3.1 and 3.9 (roughly around 3.5). These values are found in the 30-min column of Table 5.3. The other columns are affected by lower numbers of observations and thus wider confidence ranges. The chosen cross rates are computed from USD rates according to Equation 2.1. None of them was part of the European Monetary System (EMS), except GBP-DEM for a period much shorter than the analyzed sample. Gold (XAU-USD) and silver (XAG-USD) have higher tail index values above 4. These markets differ from FX. Their volatilities were very high in the 1980s, followed by a much calmer behavior in the 1990s, a structural change that may have affected the tail statistics.