Page 137 - Special Topic Session (STS) - Volume 2

STS474 Hideatsu T.
where Φθ is the distribution function of bivariate normal distribution with mean
0 1
( ) and covariance matrix ( ) , Φ is the distribution function of univariate
0 1
standard normal distribution.

(i)-(iii) are examples of well-known Archimedean copula:
( , ) = ( −1 ( ) + −1 ( )), (2.1)
2
2
1
1
where its generator : ℝ → [0,1] is a convex and decreasing function
+
satisfying (0) = 1 and (∞) = 0.
Considering bivariate case (d = 2), one is often interested in
dependence between two random variables X1 and X2 in the tail of their
distributions.

Definition 2.1 The coefficient of upper tail dependence between 1 and 2 is
defined by
≔ lim Ρ( > 2 −1 () | > 1 −1 ()),

1
2
↑1
assuming that the limit on the right-hand side exists. Similarly, the coefficient
of lower tail dependence between X1 and X2 is defined by
≔ lim Ρ( > 2 −1 () | ≤ 1 −1 ())
2

1
↓0
assuming that the limit on the right-hand side exists. When (0; 1], 1 and
U
X2 are said to be upper tail dependent, and when = 0, they are said to be
U
asymptotically upper tail independent. And similarly for .
L
If and are continuous, these coefficients of tail dependence depend
2
1
only on the copula associated with , and it holds that
̅ (, ) (, )
= lim , = lim ,
U
L
↑1 1 −
↓0
̅
where ( , ) ≔ 1 − − + ( , ) is a survival copula of .
1
2
2
2
1
1
 For Frank and Gauss family, = = 0;

 For Gumbel-Hougaard family, = 2 − 2 1/ , = 0;

 For Clayton family, = 0, = 2 −1/ ( > 0), = 0 ( ≤ 0)).
U
L

3. Methodology
We shall assume that values of dependent variable are observed
repeatedly overtime = 1, . . . , , so that we have a spatial panel structure;
see Anselin et al. (2008), and LaSage and Pace (2009).
= + , = 1, . . . , (3.1)

We consider the following copula families for the distribution of :

(1) Elliptical copula: This is just a copula associated with a nelliptical
distribution, and so it has a symmetric dependence structure.
(2) Skew t-copula: Let ∼ (, ) and ∼ Ig(/2, /2) be

independent (‘Ig’ stands for inverse Gaussian). Then = + √
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