where ||X(j,m) ||2 denotes the j-th order statistics of ||X (1) ||2 , . . . , ||X (m) ||2 and

2.4 Estimation and empirical results 45

k = k(m) ’ ∞ is chosen in an appropriate way. For a discussion on the right

choice we refer the reader to Embrechts, Kl¨pppelberg and Mikosch (1997),

u

Drees and Kaufmann (1998), and Groeneboom and de Wolf (2002).

The relationship between the tail index ±, the density generator, and the ran-

dom variable ||X||2 is established in Schmidt (2002b). Observe that the latter

TDC estimator is biased, even though the tail-index estimator is unbiased.

The quantlet TailCoeffEstimElliptical estimates the coinciding upper and

lower TDC of a bivariate elliptically distributed data-set utilizing the Hill es-

timator. The inputs of this quantlet are the two-dimensional data-set and the

threshold k. The result of the application is assigned to the variable TDC and

contains the estimated coinciding lower and upper TDC.

TDC = TailCoeffEstimElliptical(data,threshold)

Estimates the upper and lower tail-dependence coe¬cient based

on formulas (2.10) and (2.11).

2.4 Estimation and empirical results

The ¬gures below reveal that tail dependence is indeed often found in ¬nancial

data. Provided are two scatter plots of daily negative log-returns of a tuple of

ˆ (1)

¬nancial securities and the corresponding upper TDC estimate »U for various

k (for notational convenience we drop the index m). Data-set D1 contains

negative daily stock-log-returns of BMW and Deutsche Bank and data-set D2

consists of negative daily exchange-rate log-returns of DEM-USD$ and YEN-

USD$. For modeling reasons we assume that the daily log-returns are iid

observations. Both plots show the presence of tail dependence and the order of

magnitude of the tail-dependence coe¬cient. Moreover, the typical variance-

bias problem for various threshold values k can be observed. In particular,

a small k comes along with a large variance of the TDC estimate, whereas

an increasing k results in a strong bias. In the presence of tail dependence,

ˆ (1)

k is chosen such that the TDC estimate »U lies on a plateau between the

decreasing variance and the increasing bias. Thus for the data-set D1 we take

ˆ (1)

k between 80 and 110 which provides a TDC estimate of »U,D1 = 0.28, whereas

ˆ (1) = 0.14.

for D2 we estimate » U,D2

An application of TDC estimations is given within the Value at Risk (VaR)

46 2 Tail dependence

0.5

0.13

-log returns Dt.Bank

0.3

lambda

0.07

0.1

0.01

-0.01 0.03 0.07 0.11 0 100 200 300 400

-log returns BMW k

Figure 2.4: Scatter plot of BMW versus Dt. Bank negative daily stock log-

returns (2325 data points) and the corresponding TDC estimate

ˆ (1)

»U for various k.

0.35

0.035

-log returns Fr-US$

lambda

0.20

0.020

0.005 0.05

0.00 0.01 0.02 0.03 0.04 0 200 400 600

-log returns DM-US$ k

Figure 2.5: Scatter plot of DEM-USD versus YEN-USD negative daily

exchange-rate log-returns (3126 data points) and the corresponding

ˆ (1)

TDC estimate »U for various k.

framework of asset portfolios. VaR calculations relate to high quantiles of

portfolio-loss distributions and asset return distributions, respectively. In par-

ticular, VaR estimations are highly sensitive towards tail behavior and tail de-

pendence of the portfolio™s asset-return distributions. Fitting the asset-return

random vector towards a multi-dimensional distribution by utilizing a TDC

2.4 Estimation and empirical results 47

estimate leads to more accurate VaR estimates. See Schmidt and Stadtm¨ller

u

(2002) for an extension of the bivariate tail-dependence concept to the multi-

dimensional framework. Observe that upper tail dependence for a bivariate

random vector (X1 , X2 ) is equivalent to

»U = lim P(X2 > V aR1’± (X2 )|X1 > V aR1’± (X1 )) > 0. (2.12)

±’0

In the last part of this chapter we investigate some properties of the above in-

troduced TDC estimators. 1000 independent copies of m = 500, 1000, 2000 iid

random vectors of a bivariate standard t-distribution with θ = 1.5, 2, 3 degrees

of freedom are generated and the upper TDC™s are estimated. The empirical

bias and mean-squared-error (MSE) for all three introduced TDC estimation

methods are derived and presented in Table 2.4, 2.5, 2.6, 2.7, and 2.8, re-

spectively. For the parametric approach we apply the procedure explained in

Section ?? and estimate ρ by a trimmed Pearson-correlation coe¬cient with

trimming portion 0.05% and θ by a Hill-type estimator. Here we choose the op-

timal threshold value k according to Drees and Kaufmann (1998). Observe that