7.8 The Extremal Index

So far our discussions of extreme values are based on the assumption that the data are iid random variables. However, in reality extremal events tend to occur in clusters because of the serial dependence in the data. For instance, we often observe large returns (both positive and negative) of an asset after some news event. In this section we extend the theory and applications of extreme values to cases in which the data form a strictly stationary time series. The basic concept of the extension is extremal index, which allows one to characterize the relationship between the dependence structure of the data and their extremal behavior. Our discussion will be brief. Interested readers are referred to Beirlant et al. (2004, Chapter 10) and Embrechts et al. (1997).

Let x1, x2, … be a strictly stationary sequence of random variables with marginal distribution function F(x). Consider the case of n observations {xi|i = 1, … , n}. As before, let x(n) be the maximum of the data, that is, x(n) = max{xi}. We seek the limiting distribution of (x(n) − βn)/αn for some suitably chosen normalizing constants αn > 0 and βn. If {xi} were iid, Section 7.5 shows that the only possible nondegenerate limits are the extreme value distributions. What is the limiting distribution when {xi} are serially dependent?

To answer this question, we start with a heuristic argument. Suppose that the serial dependence of the stationary series xi decays quickly so that xi and xi+ℓ are essentially independent when ℓ is sufficiently large. In other words, assume that the long-range dependence of xi vanishes quickly. Now divide the data into disjoint blocks of size k. Specifically, let g = [n/k] be the largest integer less than or equal to n/k. The ith block of the data is then {xj|j = (i − 1) * k + 1, … , i * k}, where it is understood that the (g + 1)th block may contain less than k observations. Let xk, i be the maximum of the ith block, that is, xk, i = max{xj|j = (i − 1) * k + 1, … , i * k}. The collection of block maxima is {xk, i|i = 1, … , g + 1}. From the definitions, it is easy to see that

7.44

That is, the sample maximum is also the maximum of the block maxima. If the block size k is sufficiently large and the block maximum xk, i does not occur near the end of the ith block, then xk, i and xk, i+1 are sufficiently far apart and essentially independent under the assumption of weak long-range dependence in {xi}. Consequently, {xk, i|i = 1, … , g + 1} can be regarded as a sample of iid random variables, and the limiting distribution of its maximum, which is x(n), should be the extreme value distribution. The prior discussion shows that, under some proper condition, the limiting distribution of the maximum of a strictly stationary time series is also the extreme value distribution.

The proper condition needed for the maximum x(n) of a strictly stationary time series to have the extreme value limiting distribution is obtained by Leadbetter (1974) and known as the D(un) condition. Details are given in the next section. The prior heuristic argument also suggests that, even though the limiting distribution of x(n) is also the extreme value distribution, the parameters associated with the limiting distribution, however, will not be the same as those when {xi} are iid random samples because the limiting distribution depends on the marginal distribution of the underlying sequences. For the iid sequences, the marginal distribution is F(x), but for a stationary series the underlying sequences are the block maxima xk, i whose marginal distribution is not F(x). The marginal distribution of xk, i depends on k and the strength of serial dependence in {xi}.

7.8.1 The D(un) Condition

Consider the sample x1, x2, … , xn. To place limits on the long-range dependence of {xi}, let un be a sequence of thresholds increasing at a rate for which the expected number of exceedances of xi over un remains bounded. Mathematically, this says that , where F( · ) is the marginal cumulative distribution function of xi. For any positive integers p and q, suppose that iv (v = 1, … , p) and jt (t = 1, … , q) are arbitrary integers satisfying

where j1 − ip ≥ ℓn, where ℓn is a function of the sample size n such that ℓn/n → 0 as . Let A1 = {i1, i2, … , ip} and A2 = {j1, j2, … , jq} be two sets of time indices. From the prior condition, elements in A1 and A2 are separated by at least ℓn time periods. The condition D(un) is satisfied if

7.45

where as . This condition says that any two events of the form and can become asymptotically independent as the sample size n increases when the index subsets A1 and A2 of {1, 2, … , n} are separated by a distance ℓn which satisfies ℓn/n → 0 as . The D(un) condition looks complicated, but it is relatively weak. For instance, consider Gaussian sequences with autocorrelation ρn for lag n. The D(un) condition is satisfied if ρnln(n) → 0 as ; see Berman (1964).

Leadbetter's Theorem 1

Suppose that {xi|i = 1, … , n} is a strictly stationary time series for which there exist sequences of constants αn > 0 and βn and a nondegenerate distribution function F*( · ) such that

where → d denotes convergence in distribution. If D(un) holds with un = αnx + βn for each x such that F*(x) > 0, then F*(x) is an extreme value distribution function.

The prior theorem shows that the possible limiting distributions for the maxima of strictly stationary time series satisfying the D(un) condition are also the extreme value distributions. As noted before, the dependence can affect the limiting distribution, however. The effect of the dependence appears in the marginal distribution of the block maxima xk, i. To state the effect more precisely, let be a sequence of iid random variables such that the marginal distribution of is the same as that of the stationary time series xi. Let be the maximum of . Leadbetter (1983) establishes the following result.

Leadbetter's Theorem 2

If there exist sequences of constants αn > 0 and βn and a nondegenerate distribution function such that

if the condition D(un) holds with un = αnx + βn for each x such that , and if P[(x(n) − βn)/αn ≤ x] converges for some x, then

for some constant θ ∈ (0, 1].

The constant θ is called the extremal index. It plays an important role in determining the limiting distribution F*(x) for the maximum of a strictly stationary time series. To see this, we provide some simple derivations for the case of ξ ≠ 0. From the result of Eq. (7.16), is the generalized extreme value distribution and assumes the form

where ξ ≠ 0 and 1 + ξ(x − β)/α > 0. In other words, we assume that for the iid sequence , the limiting extreme distribution of has parameters ξ, β and α. Based on Theorem 2 of Leadbetter (1983), we have

7.46

where ξ* = ξ, α* = αθξ, and β* = β − α(1 − θξ)/ξ. Therefore, for a stationary time series {xi} satisfying the D(un) condition, the limiting distribution of the sample maximum is the generalized extreme value distribution with the shape parameter ξ, which is the same as that of the iid sequences. On the other hand, the location and scale parameters are affected by the extremal index θ. Specifically, α* = αθξ and β* = β − α(1 − θξ)/ξ. Results for the case of ξ = 0 can be derived via the same approach and we have α* = α and β* = β + αln(θ).

A formal definition of the extremal index is as follows: Let {xi} be a strictly stationary time series with marginal cumulative distribution function F(x) and θ a nonnegative number. Assume that for every τ > 0 there exists a sequence of thresholds un such that

7.47

7.48

Then θ is called the extremal index of the time series {xi}. See Embrechts et al. (1997). Note that, for the corresponding iid sequence , under the assumption that Eq. (7.47) holds, we have

where we have used the property . Thus, the definition also highlights the role played by the extremal index θ.

7.8.2 Estimation of the Extremal Index

There are several ways to estimate the extremal index θ of a strictly stationary time series {xi}. Each estimation method is associated with an interpretation of the extremal index. In what follows, we discuss some of the estimation methods.

7.8.2.1 The Blocks Method

From the definition of the extremal index θ, we have, for a large n, that

provided that n[1 − F(un)] → τ > 0. Hence

7.49

This limiting relationship suggests a method to estimate θ. The denominator can be estimated by the sample quantile, namely

where I(C) = 1 if the augment C holds and = 0 otherwise, that is, I(C) is the indicator variable for the statement C, and N(un) denotes the number of exceedances of the sample over the threshold un. The numerator P(x(n) ≤ un) is harder to estimate. One possibility is to use the block maxima. Specifically, let k = k(n) be a properly chosen block size that depends on the sample size n and, as before, let g = [n/k] be the integer part of n/k. For simplicity, assume that n = gk. The ith block consists of {xj|j = (i − 1) * k + 1, … , i * k} and let xk, i be the maximum of the ith block. Using Eq. (7.44) and the approximate independence of block maxima, we have

The probability P(xk, i ≤ un) can be estimated from the block maxima, that is,

where G(un) is the number of blocks such that the block maximum exceeds the threshold un. Combining the estimators for numerator and denominator, we obtain

7.50

where the subscript b signifies the blocks method. Note that N(un) is the number of exceedances of the sample {xi} over the threshold un and G(un) is the number of blocks with one or more exceedances. Using approximation based on Taylor expansion of ln(1 − x), we obtain a second estimator:

Based on the results of Hsing et al. (1988), this estimator can also be interpreted as the reciprocal of the mean cluster size of the limiting compound Poisson process N(un).

7.8.2.2 The Runs Method

O'Brien (1987) proved, under certain weak mixing condition, that

where , where s is a function of the sample size n satisfying some growth conditions, including and s/n → 0 as . See Beirlant et al. (2004) and Embrechts et al. (1997) for details. This result has been used to construct an estimator of θ based on runs:

where N(un) is the number of exceedances of the sample {xi} over the threshold un, k is a function of n, and Ai, n = {xi > un, xi+1 ≤ un, … , xi+k ≤ un}. Note that Ai, n denotes the event that an exceedance is followed by a run of k observations below the threshold. Since k/n → 0 as , we can write the runs estimator as

Finally, other estimators of θ are available in the literature. See, for instance, the methods discussed in Beirlant et al. (2004). For demonstration, we consider, again, the negative daily log returns of IBM stock from July 3, 1962, to December 31, 1998. Figure 7.10 shows the estimates of the extremal index for various thresholds when the block size k = 10. We chose k = 10 because the daily log returns have weak serial dependence. The estimates are based on the blocks method, that is, . From the plot, we see that 0.82 for threshold 0.025. Indeed, a simple direct calculation using k = 10 and threshold 0.025 gives . The plot also shows that the estimate of the extremal index might be sensitive to the choices of threshold and block size k.

7.8.3 Value at Risk for a Stationary Time Series

The relationship between F*(x) of the maximum of a stationary time series and of its iid counterpart established in Theorem 2 of Leadbetter (1983) can be used to calculate the VaR of a financial position when the associated log returns form a stationary time series. Specifically, from P(x(n) ≤ un) ≈ [F(x)]nθ, the (1 − p)th quantile of F(x) is the (1 − p)nθth quantile of the limiting extreme value distribution of x(n). Consequently, the VaR of Eq. (7.28) based on the extreme value theory becomes

7.51

where n is the length of the subperiod. From the formula, we risk underestimating the VaR if the extremal index is overlooked.

As an illustration, again consider the negative daily log returns of IBM stock from July 3, 1962, to December 31, 1998. Using = 0.823, the 1% VaR for the long position of $10 millions on the stock for the next trading day becomes 3.2714 for the case of choosing n = 63 days in parameter estimation. As expected, this is higher than the 3.0497 of Example 7.6 when the extremal index is neglected.

7.8.3.1 R Demonstration

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