݄
௧
ሺݔǡ ݕȁ࣠
௧ିଵ
ሻ ൌ ߲
ଶ
ሾܥ
௧
ሺܨ
௧
ሺݔȁ࣠
௧ିଵ
ሻǡ ܩ
௧
ሺݕȁ࣠
௧ିଵ
ሻȁ࣠
௧ିଵ
ሻሿ ߲ܨ
௧
ሺݔȁ࣠
௧ିଵ
ሻ ߲ܩ
௧
ሺݕȁ࣠
௧ିଵ
ሻ ڄ
߲ܨ
௧
ሺݔȁ࣠
௧ିଵ
ሻ ߲ݔ
ڄ ߲ܩ
௧
ሺݕȁ࣠
௧ିଵ
ሻ ߲ݕ
ൌܿ
௧
ሺܨ
௧
ሺݔȁ࣠
௧ିଵ
ሻǡ ܩ
௧
ሺݕȁ࣠
௧ିଵ
ሻȁ࣠
௧ିଵ
ሻ ڄ݂
௧
ሺݔȁ࣠
௧ିଵ
ሻ ڄ݃
௧
ሺݕȁ࣠
௧ିଵ
ሻ ܿሺݑǡ ݒȁ࣠
௧ିଵ
ሻ ൌ ݄
௧
ሺݔǡ ݕȁ࣠
௧ିଵ
ሻ ݂
௧
ሺݔȁ࣠
௧ିଵ
ሻ ڄ݃
௧
ሺݕȁ࣠
௧ିଵ
ሻ 5
where ݑ ؠ ܨ
௧
ሺݔȁ࣠
௧ିଵ
ሻand ݒ ؠܩ
௧
ሺݕȁ࣠
௧ିଵ
ሻ. Ones may refer to [ 7
] for another class of copulas, which is also known as Archimedean copulas.
3. THE MODEL FOR MARGINS
The marginal distributions that we used to build a joint multivariate distribution are Normal, t-student, Skew-t student. The model for each marginal time series by a general AR1-GJR1,1 model for the
continuously compounded returns is given by ݎ
ǡ௧
ൌ ܿ
ܿ
ଵ
ݎ
ǡ௧ିଵ
݁
ǡ௧
6 ݁
ǡ௧
ൌ ߪ
ǡ௧
ߝ
ǡ௧
ǡ ߝ
ǡ௧
ܵ݇݁ݓ െ ݐ ሺߥ
௧
ǡ ߣ
௧
ሻ 7
ߪ
ǡ௧ ଶ
ൌ ߱
ߙ
݁
ǡ௧ିଵ ଶ
ߚ
ߪ
ǡ௧ିଵ ଶ
ߛ݁
ǡ௧ିଵ ଶ
ǡషభ
ழ
8 where
݁
ǡ௧
and ݁
ǡ௧ିଵ
are the residuals and one lagged residual of the model ݅ and the distributions of ߝ
ǡ௧
are ܰሺߤ
௧
ǡ ߪ
௧
ሻǡ ݐሺߥ
௧
ǡ ߣ
௧
ሻǡ and െ ݐሺߥ
௧
ǡ ߣ
௧
ሻ, where the skewed-t densities is given by ݂൫ߝ
ǡ௧
Ǣ ߥ
௧
ǡ ߣ
௧
൯ ൌ ە
ۖ ۔
ۖ ۓ ܾܿ ቆͳ ͳ
ߥ
௧
െ ʹ ൬ ܾߝ
ǡ௧
ܽ ͳ െ ߣ
௧
൰
ଶ
ቇ
ିሺఔ
ାଵሻȀଶ
ߝ
ǡ௧
൏ െܽȀܾ ܾܿ ቆͳ
ͳ ߥ
௧
െ ʹ ൬ ܾߝ
ǡ௧
ܽ ͳ ߣ
௧
൰
ଶ
ቇ
ିሺఔ
ାଵሻȀଶ
ߝ
ǡ௧
െܽȀܾ 9
with constants ܽǡ ܾ and ܿ defined as
ܽ ൌ Ͷߣܿ ൬ ߥ
௧
െ ʹ ߥ
௧
െ ͳ൰ǡܾ
ଶ
ൌ ͳ ͵ߣ
௧
െ ܽ
ଶ
ǡܿ ൌ Ȟ ቀ
ఔ
ାଵ ଶ
ቁ Ȟ ቀ
ఔ
ଶ
ቁ ඥߨሺߥ
௧
െ ʹሻ where the parameters
ߥ
௧
and ߣ
௧
representing the degrees of freedom and asymmetry, respectively.
4. PARAMETER ESTIMATIONS: INFERENCE FOR MARGIN IFM
Rearranging Equation 5 and putting parameters taken into account to the density function, it gives ݄
௧
ሺݔ
௧
ǡ ݕ
௧
ȁ࣠
௧ିଵ
Ǣ ߠ
ሻ ൌ݂
௧
൫ݔห࣠
௧ିଵ
Ǣ ߠ
൯ ڄ ݃
௧
൫ݕห࣠
௧ିଵ
Ǣ ߠ
൯ ሶ
ڄ ܿ
௧
ሺݑǡ ݒȁ࣠
௧ିଵ
Ǣ ߠ
ሻ 10
where ߠ
ൌ ሾߠ
ǡ ߠ
ǡ ߠ
ሿ is a vector of parameters of the joint density. Equation 10 suggests that the conditional density function
݄can be decomposed into two problems and estimation will be carried out sequentially; firstly to identify the conditional distribution of the margins for
ܺ and ܻ and secondly to establish a functional form for copula
ܥ. Thus, the log-likelihood function of Equation 10 is given by ݄
௧
ሺݔ
௧
ǡ ݕ
௧
ȁ࣠
௧ିଵ
Ǣ ߠ
ሻ
் ௧ୀଵ
ൌ ݂
௧
൫ݔ
௧
ห࣠
௧ିଵ
Ǣ ߠ
൯
் ௧ୀଵ
݃
௧
൫ݕ
௧
ห࣠
௧ିଵ
Ǣ ߠ
൯
் ௧ୀଵ
International Journal of Applied Mathematics and Statistics
88
ܿ
௧
ሺݑ
௧
ǡ ݒ
௧
ȁ࣠
௧ିଵ
Ǣ ߠ
ሻ
் ௧ୀଵ
11 According to the IFM method, the parameters of the marginal distributions are estimated sequentially,
in two steps: 1 Estimating the parameters
ሺߠ
ǡ ߠ
ሻ of the marginal distributions, ܨ
௧
and ܩ
௧
using maximum likelihood Estimation method MLE method:
ߠ
ൌ ݂
௧
൫ݔ
௧
ห࣠
௧ିଵ
Ǣ ߠ
൯
் ௧ୀଵ
12 ߠ
ൌ ݃
௧
൫ݕ
௧
ห࣠
௧ିଵ
Ǣ ߠ
൯
் ௧ୀଵ
13 2 Estimating the copula parameter
ߠ
, given ߠ
and ߠ
ߠ
ൌ ሾܿ
௧
ሺܨ
௧
൫ݔ
௧
ห࣠
௧ିଵ
Ǣ ߠ
൯ǡ ܩ
௧
൫ݕ
௧
ห࣠
௧ିଵ
Ǣ ߠ
൯ሿ
் ௧ୀଵ
14 It is just like the ML estimator method, it verifies the properties of asymptotic normality.
5. TAIL DEPENDENCE