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9.2. Estimation du coefficient de dépendance de queue 313
same tail index. We can also add that, as asserted by Loretan and Phillips (1994) or Longin (1996),
we cannot reject the hypothesis that the tail index remains the same over time. Nevertheless, it
seems that during the first period from July 1962 to December 1979, the tail indexes were sightly
larger than during the second period from January 1980 to december 2000.
3.4 Determination of the coefficient of tail dependence
Using the just established empirical fact that we cannot reject the hypothesis that the assets, the
market and the residues have the same tail index, we can use the theorem of Appendix A and
its second corollary stated in section 2. This allows us to conclude that one cannot reject the
hypothesis of a non-vanishing tail dependence between the assets and the market.
In addition, the coefficient of tail dependence is given by equations (12) and (13). These equations
provide two ways for estimating the coefficient of tail dependence: non-parametric with (12) and
parametric with (13). The first one is more general since it only requires the hypothesis of a regular
variation, while the second one explicitly assumes that the factor and the residues have distributions
with power law tails.
To estimate the tail dependence according equation (12), we need only to determine the constant l
defined (8). Consider N sorted realizations of X and Y denoted by x1,N ≥ x2,N ≥ · · · ≥ xN,N and
y1,N ≥ y2,N ≥ · · · ≥ yN,N, the quantile of F −1
−1
X (u) and of FY (u) are estimated by
−1 −1
FX
ˆ (u) = x[(1−u)·N],N and FY
ˆ (u) = y[(1−u)·N],N, (22)
where [·] denotes the integer part. Thus, the constant l is non-parametrically estimated by
ˆ lk = xk,N
yk,N
as k → 0 or N. (23)
As u goes to zero or one (or k goes to zero or N), the number of observations decreases dramatically.
However, we observe a large interval of small or large k’s such that the ratio of the empirical
quantiles remains remarkably stable and thus allows for an accurate estimation of l. A more
precise estimation could be performed with a kernel-based quantile estimator (see Shealter and
Marron (1990) or Pagan and Ullah (1999) for instance). A non-parametric estimator for λ is then
obtained by replacing l by its estimated value in equation (12)
ˆλNP =
1
max 1, ˆ 1
α =
l
ˆβ max 1, xk,N
α . (24)
ˆβ·yk,N
It can also be advantageous to follow a parametric approach, which generally allows for a more
accurate estimation of (the ratio of) the quantiles, provided that the assumed parametric form of
the distributions is not too far from the true one. For this purpose, we will use formula (13) which
requires the estimation of the scale factors for the different assets. To get the scale factors, we
proceed as follows. Consider a variable X which asymptotically follows a power law distribution
Pr{X > x} ∼ C · x −α . Given a rank ordered sample x1,N ≥ x2,N ≥ · · · ≥ xN,N, the scale factor C
can be consistently estimated from the k largest realizations by
Ĉ = k
N · (xk,N) α . (25)
The estimated value of the scale factor must not depends on the rank k for k large enough, in order
for the parameterization of the distribution in terms of a power law to hold true. Thus, denoting
14