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

models using the three foreign exchange rates. The empirical results from the in-sample and OOS
analysis clearly demonstrate the advantages of the new model.
7 Appendix
We present here some details on copula functions for two widely used copula families — elliptical
copula family and Archimedean copula family. The former includes the Gaussian copula and the
Student’s t copula. The latter includes Gumbel copula, Clayton copula and Frank copula. We also
discuss the survival copulas of Archimedean copulas and m-variate Archimedean copulas.
7.1 Elliptical copulas
Gaussian copula: Let R be the symmetric, positive definite correlation matrix and Φ R (·, ·) be
the standard bivariate normal distribution with correlation matrix R.
bivariate Gaussian copula is:
The density function of
c Gaussian (u 1 ,u 2 )= 1
|R| 1/2 exp(−1 2 η0 (R −1 −I)η), (25)
where η =(Φ −1 (u 1 ) Φ −1 (u 2 )) 0 and Φ −1 (·) is the inverse of the univariate normal CDF. The bivariate
Gaussian copula is:
C Gaussian (u 1 ,u 2 ; R) =Φ R
¡
Φ −1 (u 1 ), Φ −1 (u 2 ) ¢ .
Hu (2003) shows the bivariate Gaussian copula can be approximated by Taylor expansion:
C Gaussian (u 1 ,u 2 ; θ) ≈ u 1 u 2 + θ · φ(Φ −1 (u 1 ))φ(Φ −1 (u 2 )),
where φ is the density function of univariate Gaussian distribution and θ is the correlation coefficient
between η 1 and η 2 . Both the upper tail dependence λ U and the lower tail dependence λ L are zero,
reflecting the asymptotic tail independence of Gaussian copula.
Student’s t copula: Letω c be the degree of freedom, and T R,ωc (·, ·) be the standard bivariate
Student’s t distribution with degree of freedom ω c and correlation matrix R. The density function
of bivariate Student’s t copula is:
c Student’s t (u 1 ,u 2 ; R,ω c )=|R| − 1 Γ( ω c+2
2
2
)Γ( ω c
2
) (1 + η0 R −1 η
Γ( ωc+1
2
) 2 Π 2 i=1 (1 + η2 i
ω c
) − ω c+2
2
ω c
) − ω c+1
2
where η =(t −1
ω c
(u 1 ),t −1
ω c
(u 2 )) 0 ,u 1 = t ω1 (x), u 2 = t ω2 (y), and t ωi (·) is the univariate Student’s t
CDF with degree of freedom ω i . The bivariate Student’s t copula is
C Student’s t (u 1 ,u 2 ; R,ω c )=T R,ωc
¡
t
−1
ω c
(u 1 ),t −1
ω c
(u 2 ) ¢ .
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