Although the univariate GARCH model is that volatility in the capture of assets have a strong advantage, but to estimate the correlation between asset returns not as good as multivariate GARCH model (MGARCH) model. MGARCH model used more often and relatively simple model is Bollerslev (1990) proposed a static conditional correlation model (Constant Conditional Correlation), the model terms and conditions of the variance covariance matrix is defined as:
(3.10)
Which is the conditional variance to take the diagonal entries of the n * n formed by the diagonal matrix, R is the fixed conditional correlation coefficient matrix, Rii = 1, Rij = (ρij).
(3.11)
Compared to other MGARCH models, DCC model is not the greatest conditions while the estimated correlation coefficient is divided into the average GARCH model parameters α, β and unconditional correlation coefficients for the two parts:
(2.12)
Where is the standardized residual sequence Zi, tZj, t structure unconditional correlation coefficient, α, β, respectively MGARCH model pre-standardized residuals squared coefficients and initial conditions of the coefficients of items Heteroscedasticity meet α ≥ 0, β ≥ 0 and α + β <1
DCC model can unity as follows:
(3.13)
Which, on behalf of all time t-1 information set,
Further use of maximum likelihood estimation, if, as the parameters Dt, as the parameters Rt, then the logarithmic likelihood function can be expressed as:
(3.14)
When ignoring the case of constant nlog2π, (3.14) can be expressed as:
(3.15)
To simplify, divided into two steps:
(3.16)
This likelihood function can be divided into two waves (2.18) type and the correlation (2.19) type of two parts:
(3.17)
(3.18)
Therefore
(3.19)
Thus, Dcc model will be divided into two stages to estimate the time-varying conditional correlation coefficient matrix: First, obtain the likelihood function part of the volatility (3.17) where the parameters of maximum; Second, the substitution (3.18) where seeking to maximize the relevance of some parameters.
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