Extreme Values, Multivariate

A Summary of Dependence Estimation and Visualization in Multivariate Extremes with Applications to Financial Data by Hsing et al. (2005)

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Hsing et al. set out to develop a new function for measuring extremal dependence nonparametrically. They begin by reviewing some of the goals of extreme value analysis. Namely, one of those is to understand the dependence of a random process at extreme levels. If we let \mathbf{X_i} = (X_{i,1}, …, X_{i,m}), i = 1,2, \ldots be an iid sequence of random vectors, then of interest is doing inference on the dependence structure of componentwise maxima M_{n,j} = \vee_{i = 1}^n, 1 \leq j \leq m.

The authors motivate the topic with common examples where understanding of the dependence structure can lead to different behaviors. For example, when designing an investment portfolio is is necessary to understand the dependence of individual losses among the individual assets that make up the portfolio. One can also imagine that it would be critical to understand how extreme rainfall is temporally dependent to design infrastructure that can mitigate possible floods.

They note that some existing methods for measuring extremal dependence assume a parametric form for F, and can be misleading if the model is incorrect. Before describing their method Hsing et al. review how the distribution of a random vector can be equivalently formulated using copulas, and hence also the dependence structure of a random vector is captured by its copula. A copula is a (multivariate) cumulative distribution function with uniform margins defined by

C_F(u_1, …, u_m) = P(F(X_1)\leq u_1, … F(X_m) \leq u_m)

for (u_1, …, u_m) \in [0,1]^m

An extreme copula, which is the focus of their method, is one that satisfies
C^t(u_1, …, u_m) = C(u_1^t, …, u_m^t)

Their approach uses the Pickands representation of a copula, which I will not include here but essentially uses a change of measure to one (the spectral measure \Phi(\theta)) on [0, \pi/2].
Consider the data with m dimensions. And consider \theta_2,…,\theta_m to be angles in [0,\pi/2] after the change in measure. They defined the tail dependence to be calculated as

\rho(\theta_2,…,\theta_m)=\frac{(1+cot\theta_2+…+cot\theta_m)-(\psi(\theta_2,…,\theta_m))}{(1+cot\theta_2+…+cot\theta_m)-(1\vee cot\theta_2\vee…\vee cot \theta_m)}

Where \psi is a function defined using the new measure in [0,\pi/2]. Values of \rho near 0 represent weak dependence and values near 1 represents strong dependence. Since the measure is not known, consequently \psi is not known so it is necessary to estimate it using a nonparametric estimator.

The application of this method was done returns of Annually Compounded Zero Coupon Swap Rates with different maturities and currencies. The tail dependence was estimated for combinations maturities. Some of the results showed strong dependence between 7 days and 30 days, 10 years with 15/20/30 years.

There is moderate dependence between 7 days and 6 months. There is weak dependence between 7 days and 30 years and between 30 days and 1/30 years.

The dependence between 30 days and 60 days varies with \theta. For values of \theta close to \pi/4 the dependence becomes moderate and for values far from \pi/4 the dependence becomes strong.

Besides the bivariate analysis 2 trivariate examples were analyzed. The first one is the dependence of swap rates with 7 days maturity between currencies USD, EUR and GBP. This data showed low tail dependence with values increasing in the direction of the edge of the parameter space, but values still smaller than .5.

The second trivariate shows the dependence between swap rates with maturities 5,6 and 7 years for EUR. Almost all values are close to 1, the dependence decreases a little for values of \theta_2,\theta_3 close to \pi/4, but it is still bigger than 0.8.

Based on the simulation study that they did in the paper the method seems to work fine. During the simulation this estimator was able to retrieve the correct dependence. One missing thing on the paper was moments when this may fail, a problem presented during the analysis of the real data is that the lack of data in some parts of the parameter space does not allows to do estimation. However, no other possible problem or maybe extension was presented.

Reference:

Hsing T, Klüppelberg C, Kuhn G. Dependence Estimation and Visualization in Multivariate Extremes with Applications to Financial Data. Extremes. 2004;7(2):99-121.

Written by:
Greg, Mauricio

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