Modelling dependence in finance using copulas - Thierry Roncalli's ...

Estimation
The log-likelihood function is
l (u; ρ) = − T 2 ln |ρ| − 1 2
T∑
ςt
⊤
t=1
(
ρ −1 − I ) ς t
and the ML estimate of ρ is also ˆρ ML = 1 T
T∑
t=1
ς ⊤ t ς t.
Two-stage (Joe and Xu [1996]) and omnibus (Genest, Ghoudi and
Rivest [1995]) estimators can then be obtained with
ς t = ( Ψ [F 1 ]
(
x
t
1
)
, . . . , Ψ [FN ]
(
x
t
N
))
:
1. IFM estimate: F n = MLE of the n th marginal distribution.
2. Omnibus estimate : F n = n th empirical distribution.
⇒ The data are mapped to uniforms and transformed with the
inverse gaussian distribution. The correlation parameter ρ of the
Normal copula is then equal to the Pearson product moment of the
transformed data.
Modelling dependence in finance using copulas
Copulas and multivariate financial models 2-7