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

and Italian Lira). Section 6 concludes. Section 7 is Appendix on copulas.
2 MGARCH Models
We begin with a brief review of three MGARCH models. Suppose a vector of the m return series
{r t } n t=1 with E(r t|F t−1 ) ≡ µ t =0and E(r t r 0 t|F t−1 ) ≡ H t where F t−1 is the information set (σ-field)
at time t−1. For simplicity, we assume the conditional mean µ t is zero. For H t ,manyspecifications
have been proposed.
Engle and Kroner (1995) propose the BEKK model
H t = Ω + A(r t−1 r 0 t−1)A 0 + BH t−1 B 0 , (1)
With the scalar or diagonal specifications on A and B, we obtain the scalar BEKK (SBEKK) or
the diagonal BEKK. We use the SBEKK in Section 5, which is
H t =(1− a − b)¯Ω + a(r t−1 r 0 t−1)+bH t−1 , (2)
where ¯Ω = n −1 P n
t=1 r tr 0 t isthesamplecovariancematrixofr t .
Instead of modeling H t directly, conditional correlation models decompose H t into D t R t D t ,
where D 2 t ≡ diag(H t ). As the conditional covariance matrix for ε t ≡ D −1
t r t is the conditional
correlation matrix for r t , The DCC model of Engle (2002) models Q t , the covariance matrix of ε t ,
via a variance-targeting scalar BEKK model:
Q t =(1− a − b) ¯Q + a(ε t−1 ε 0 t−1)+bQ t−1 , (3)
where ¯Q is the sample covariance matrix of ˆε t . A transformation R t = diagQ −1
t Q t diagQ −1
t makes
the conditional correlation matrix for r t .
The VC model of Tse and Tsui (2002) uses the following specification
R t =(1− a − b) ¯R + a ˜R t−1 + bR t−1 , (4)
where ¯R is the positive definite unconditional correlation matrix with ones in diagonal, and ˜R t =
P M
i=1 ε P
1,t−iε 2,t−i /
M
i=1 2,t−i´1/2. ε2 1
³ PM
i=1 ε2 1,t−i
1 In Tse and Tsui (2002), a necessary condition to guarantee ˜R t positive definite is M > k. Another necessary
condition for non-singularity of ˜R t, which should be added, is that M should be bigger than the maximum number
of observations of consecutive zeros of ε i,t , i =1,...,k. In the empirical section, we set M =5, which is transaction
days in one week.
2