"Frontmatter". In: Analysis of Financial Time Series

GARCH MODELS FOR BIVARIATE RETURNS 363normal N(0, G t ), and the log likelihood function of the data becomes extremelysimple. Indeed, we have the log probability density of a t as[]l(a t , Σ t ) = l(b t , Ξ t ) =− 1 k∑ln(g ii,t ) + b2 it, (9.15)2g ii,twhere for simplicity the constant term is omitted and g ii,t is the variance of b it .Using the Cholesky decomposition to reparameterize Σ t has several advantages.First, from Eq. (9.14), Σ t is positive definite if g ii,t > 0foralli. Consequently, thepositive definite constraint of Σ t can easily be achieved by modeling ln(g ii,t ) insteadof g ii,t . Second, elements of the parameter vector Ξ t in Eq. (9.13) have nice interpretations.They are the coefficients and residual variances of multiple linear regressionsthat orthogonalize the shocks to the returns. Third, the correlation coefficientbetween a 1t and a 2t isρ 21,t =i=1√σ 21,tσ11,t√ = q 21,t × √ ,σ11,t σ 22,t σ22,twhich is time-varying if q 21,t ̸= ρ √ σ 22,t / √ σ 11,t where ρ is a constant. For example,if q 21,t = c ̸= 0, then ρ 21,t = c √ σ 11,t / √ σ 22,t , which continues to be time-varyingprovided that the variance ratio σ 11,t /σ 22,t is not a constant. This time-varying propertyapplies to other correlation coefficients when the dimension of r t is greater than2 and is a major difference between the two approaches for reparameterizing Σ t .Using Eq. (9.11) and the orthogonality among the transformed shocks b it ,weobtainσ ii,t = Var(a it | F t−1 ) =i∑qiv,t 2 g vv,t,v=1σ ij,t = Cov(a it , a jt | F t−1 ) =i = 1,...,k,j∑q iv,t q jv,t g vv,t , j < i, i = 2,...,k,v=1where q vv,t = 1forv = 1,...,k. These equations show the parameterization of Σ tunder the Cholesky decomposition.9.2 GARCH MODELS FOR BIVARIATE RETURNSSince the same techniques can be used to generalize many univariate volatility modelsto the multivariate case, we focus our discussion on the multivariate GARCHmodel. Other multivariate volatility models can also be used.For a k-dimensional return series r t , a multivariate GARCH model uses “exactequations” to describe the evolution of the k(k + 1)/2-dimensional vector Ξ t overtime. By exact equation, we mean that the equation does not contain any stochastic