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

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By Hoeffding’s Lemma and Sklar’s Theorem, the off-diagonal element σ 12,t of the conditional
covariance matrix Σ t between η 1,t and η 2,t at time t, can be expressed as
ZZ
σ 12,t (θ t )= [C (F 1 (η 1 ; θ 1,t ),F 2 (η 2 ; θ 2,t ); θ 3,t ) − F 1 (η 1 ; θ 1,t )F 2 (η 2 ; θ 2,t )] dη 1 dη 2 . (11)
R 2
For simplicity, we assume that the marginal standard normal distribution (for which θ 1 ,θ 2 are
known) and the copula parameter θ 3 is not time-varying: θ t ≡ θ = θ 3 . 3 This makes σ 12,t (θ t ) ≡
σ 12 (θ) and Σ t (θ t ) ≡ Σ(θ).
The log-likelihood function for {η t } n t=1 is:
L η (θ) =
nX
ln f 1 (η 1,t )+lnf 2 (η 2,t )+lnc(F 1 (η 1,t ),F 2 (η 2,t ); θ). (12)
t=1
Because r t = H 1/2
t Σ −1/2 η t , the log-likelihood function for {r t } n t=1 is:
L r (θ, α) =L η (θ)+
nX
¯
ln ¯Σ 1/2 (θ)H −1/2
(α) ¯ , (13)
t=1
¯
where ¯Σ 1/2 H −1/2
t ¯ is the Jacobian of the transformation from η t to r t , and α is the parameter
vector in the MGARCH model for H t (DCC,VC,SBEKK).WemaximizeL r (θ, α) to estimate all
parameters in one step, with the diagonal elements of Σ being normalized (σ ii =1)for identification.
Remark 1: Because e t = Σ −1/2
t η t ,ifΣ −1/2
t = Σ −1/2 ≡ (a ij ),thene 1,t = a 11 η 1,t + a 12 η 2,t and
e 2,t = a 12 η 1,t + a 22 η 2,t would be linear combinations of two dependent random variables η 1,t and
η 2,t . Evenifeachofη 1,t and η 2,t has the margins of standard normal distribution, the marginal
distributions of e 1,t and e 2,t are not normal because η 1,t and η 2,t are not independent. If normal
margins of η t are chosen for non-independent copula, then the marginal distributions of e t are
non-normal. A nice feature of the C-MGARCH model is to allow the non-normal margins of r t
even if we assume the normality of η t . Therefore, the C-MGARCH model not only allows the
non-normal joint distribution of r t but also allow the (implied) non-normal marginal distributions
of the elements of r t . Note that if the copula for η t is the independence copula, then we obtain
the bivariate normality for r t if we use the normal margins for the elements of η t . Therefore, the
better fit of the non-independence copula of η t may be due to the non-normality of the bivariate
joint density of r t and also due to the non-normality of the margins of the elements of r t . ¤
3 θ 3,t may be modelled to be time-varying. For example, for Gumbel copula, θ 3,t =1+exp(a + bθ 3,t−1 + cu 1,t−1 +
du 2,t−1).
t
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