"Frontmatter". In: Analysis of Financial Time Series

REPARAMETERIZATION 359To illustrate, for k = 2, we have ϱ t = ρ 21,t andΞ t = (σ 11,t ,σ 22,t ,ρ 21,t ) ′ , (9.4)which is a 3-dimensional vector, and for k = 3, we have ϱ t = (ρ 21,t ,ρ 31,t ,ρ 32,t ) ′andΞ t = (σ 11,t ,σ 22,t ,σ 33,t ,ρ 21,t ,ρ 31,t ,ρ 32,t ) ′ , (9.5)which is a six-dimensional vector.If a t is a bivariate normal random variable, then Ξ t is given in Eq. (9.4) and theconditional density function of a t given F t−1 is[]1f (a 1t , a 2t | Ξ t ) =exp −2π√σ Q(a 1t, a 2t , Ξ t )11,t σ 22,t (1 − ρ21,t 2 ) 2(1 − ρ21,t 2 ) ,whereQ(a 1t , a 2t , Ξ t ) = a2 1tσ 11,t+ a2 2tσ 22,t− 2ρ 21,ta 1t a 2t√ σ11,t σ 22,t.The log probability density function of a t relevant to the maximum likelihood estimationis{l(a 1t , a 2t , Ξ t ) =− 1 ln[σ 11,t σ 22,t (1 − ρ21,t 2 2)]1+1 − ρ21,t2( )}a21t+ a2 2t− 2ρ 21,ta 1t a 2t√ . (9.6)σ 11,t σ 22,t σ11,t σ 22,tThis reparameterization is useful because it models covariances and correlationsdirectly. Yet the approach has several weaknesses. First, the likelihood functionbecomes complicated when k ≥ 3. Second, the approach requires a constrainedmaximization in estimation to ensure the positive definiteness of Σ t . The constraintbecomes complicated when k is large.9.1.2 Cholesky DecompositionThe second reparameterization of Σ t is to use the Cholesky decomposition; seeAppendix A of Chapter 8. This approach has some advantages in estimation as itrequires no parameter constraints for the positive definiteness of Σ t ; see Pourahmadi(1999). In addition, the reparameterization is an orthogonal transformation so thatthe resulting likelihood function is extremely simple. Details of the transformationare given next.