THE MODEL FOR MARGINS PARAMETER ESTIMATIONS: INFERENCE FOR MARGIN IFM

݄ 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 ૚ ௘ ೔ǡ೟షభ 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 ‹ˆߝ ௜ǡ௧ ൏ െܽȀܾ ܾܿ ቆͳ ൅ ͳ ߥ ௧ െ ʹ ൬ ܾߝ ௜ǡ௧ ൅ ܽ ͳ ൅ ߣ ௧ ൰ ଶ ቇ ିሺఔ ೟ ାଵሻȀଶ ‹ˆߝ ௜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 ෍ Ž‘‰ ݄ ௧ ሺݔ ௧ ǡ ݕ ௧ ȁ࣠ ௧ିଵ Ǣ ߠ ௛ ሻ ் ௧ୀଵ ൌ෍ Ž‘‰ ݂ ௧ ൫ݔ ௧ ห࣠ ௧ିଵ Ǣ ߠ ௙ ൯ ் ௧ୀଵ ൅ ෍ Ž‘‰ ݃ ௧ ൫ݕ ௧ ห࣠ ௧ିଵ Ǣ ߠ ௚ ൯ ் ௧ୀଵ International Journal of Applied Mathematics and Statistics 88 ൅ ෍ Ž‘‰ ܿ 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෠ ௙ ൌ ƒ”‰ ƒš ෍ Ž‘‰ ݂ ௧ ൫ݔ ௧ ห࣠ ௧ିଵ Ǣ ߠ ௙ ൯ ் ௧ୀଵ 12 ߠ෠ ௚ ൌ ƒ”‰ ƒš ෍ Ž‘‰ ݃ ௧ ൫ݕ ௧ ห࣠ ௧ିଵ Ǣ ߠ ௚ ൯ ் ௧ୀଵ 13 2 Estimating the copula parameter ߠ ௖ , given ߠ෠ ௙ and ߠ෠ ௚ ߠ෠ ௖ ൌ ƒ”‰ ƒš ෍ Ž‘‰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