Copula-based Multivariate GARCH Model with ... - Economics

Frank copula: The generator for Frank copula is ϕ θ (x) =− ln( e−θx −1). For θ ∈ R\{0}, the
e −θ −1
CDF and PDF for Frank copula are
C Frank (u 1 ,u 2 ; θ) = − 1 ∙
θ ln 1+ (e−θu 1
− 1)(e −θu 2
¸
− 1)
(e −θ , (26)
− 1)
c Frank (u 1 ,u 2 ; θ) =
−θ(e −θ − 1)e −θ(u 1+u 2 )
[(e −θ − 1) + (e −θu 1 − 1)(1 − e −θu 2 − 1)] 2 .
The dependence structure described by Frank copula is symmetric: θ>0 for positive dependence,
θ → 0 for independence, and θ0 (Nelsen 1999, Example 4.21) is:
³
C Clayton
1,··· ,m (u; θ) = u −θ
1 + ···+ u −θ
m − m +1´−1/θ
.
The m-variate Gumbel copula function with θ ≥ 1 (Nelsen 1999, Example 4.23) is:
½ h
C1,··· Gumbel
,m (u; θ) =exp − (− ln u 1 ) θ +(− ln u 2 ) θ + ···+(− ln u n ) θi ¾ 1/θ
.
The m-variate Frank copula function is:
C Frank
1,··· ,m (u; θ) =− 1 θ ln "1+
¡
e
−θu 1
− 1 ¢¡ e −θu 2
− 1 ¢ ···¡e −θun − 1 ¢ #
(e −θ − 1) n−1
. (27)
22