Does Tail Dependence Make A Difference In the ... - Boston College

Assuming twice differentiability of the conditional joint distribution, the copula model as well as the conditional
marginal distribution yields the decomposition for the conditional joint density function
f(ɛ it , ɛ jt ) = f i (ɛ it )f j (ɛ jt )c(u, v; θ)
where u = F i (ɛ it ) and v = F j (ɛ jt )
Assuming the parameters in the marginal and copula densities are independent, we can separately estimate the
parameters of the copula model θ by maximizing the log likelihood of the copula density function.
ˆθ = argmax
θ
T∑
log c(u, v; θ)
t=1
where c(u, v; θ) = ∂2 C(u,v)
∂u∂v
is the density function for the copula model.
4.2.1 Dynamic Copula Models
Patton (2006) and Creal (2008) et al suggested similar observation-driven dynamic copula models for which
the dependence parameter is a parametric function of lagged data and an autoregressive term. In this paper, We
used the ”Generalized Autoregressive Score”(GAS) model (Creal,et al. 2011) to estimate time-varying parameters
for a wide range of copula models in the same dynamic framework. Since the parameters of copula θ t are often
constrained to lie in a particular range, this approach applies a strictly increasing transformation (e.g., log, logistic,
arctan) to the copula parameter, and then model the dynamics of transformed parameters f t without constraints.
Let the copula model be C(U t , V t ; θ t ). The time varying evolution of transformed parameter f t can be modeled
as
f t = h(θ t ) ⇐⇒ θ t = h −1 (f t )
Where f t+1 = ω + βf t + αI −1/2
t s t
s t = ∂ ∂θ logC(U t, V t ; θ t )
I t = E t−1 (s t s ′ t)
where I −1/2
t s t is the standardized score of the copula log-likelihood 24 , which defines a steepest ascent direction
for improving the model local fit in term of likelihood. For the student t copula, the transformation function
ρ t = 1−exp(−ft)
1+exp(−f is used to ensure that the conditional correlation ρ t) t takes an value inside (−1, 1).
Table 2 presents the bivariate copula models studied in this paper. Since different copula models imply
different dependence structures in the tail of the distribution, it is essential to be aware how well the competing
specifications of copula models are capable of accurately estimating the underlying dependence process. A very
simple and reliable way to select the best fitting model is to compare the value of the log likelihood function of
different copula models and choose the ones with the highest likelihood. 25 The metrics that will be used to test
24 The score of the copula log-likelihood can be estimated numerically if there is no closed form solution.
25 Generally speaking, the standard likelihood ratio test can’t be performed when the models are non-nested. However, Rivers and
Vuong (2002) discussed how to construct non-nested likelihood ratio tests. Therefore, non-nested models can still be compared by their
log-likelihood values.
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