Page 136 - Special Topic Session (STS) - Volume 2

STS474 Hideatsu T.
= +
(1.1)
The term is called a spatial lag. It follows that = ( − ) , where
−1

is the identity matrix of size . Thus, the dependence structure of random
vector is completely determined by = ( − ) . If ~(0, ), as is
2
−1

often assumed, then is also normally distributed, and it cannot possess
neither asymmetric dependence nor tail dependence. It also lacks to
incorporate a nonlinear structure.
We would like to build a model in which the disturbance vector is also
spatially associated but not through a weight matrix, so that we can capture
nonlinear, asymmetric dependence structure in the tail. These kinds of
dependence structures have been empirically observed in the finance and
insurance literature. To this end, one way is to employ the copula approach.
Krupskii et al. (2018) have recently suggested a model based on factor copulas,
but it seems somewhat too ad hoc.

2. Copula
Let = ( , … , ) be a d-dimensional distribution function, and be the
1

th marginal distribution function of . According to Sklar’s theorem, there
d
exists copula (a distribution function on [0; 1] with uniform marginals) such
that
( , … , ) = ( ( ), … , ( )), for all ( , … , )
1

1
1

1

is called the copula associated with , and when is continuous, the copula
associated with is uniquely determined and is given by
( 1 −1 ( ), … , −1 ( )) , 0 ≤ ≤ 1, = 1, … , .

1
Copulas have recently been drawing some attention mainly as a tool to
model various dependence among random variables, including the fields of
financial risk management and multivariate survival analysis; see Nelsen (2006)
and Joe (2015) for an introduction to the topic. Popular bivariate families of
copulas are as follows.
(i) Clayton family
(, ) = ( − + − − 1) −1/ , [−1, ∞) \ {0}

(ii) Gumbel-Hougaard family
1/

(, ) = {− [(− ) + (− ) ] }, ≥ 1

(iii) Frank family
1 ( − 1)( − 1)
(, ) = log (1 + ) , ℝ

− 1

(iv) Gauss family
(, ) = Φ (Φ −1 (), Φ −1 ())

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