"Frontmatter". In: Analysis of Financial Time Series

360 MULTIVARIATE VOLATILITY MODELSBecause Σ t is positive definite, there exist a lower triangular matrix L t with unitdiagonal elements and a diagonal matrix G t with positive diagonal elements suchthatΣ t = L t G t L ′ t . (9.7)This is the well-known Cholesky decomposition of Σ t . A nice feature of the decompositionis that the lower off-diagonal elements of L t and the diagonal elements of G thave nice interpretations. We demonstrate the decomposition by studying carefullythe bivariate and three-dimensional cases. For the bivariate case, we haveΣ t =[ ]σ11,t σ 21,t, Lσ 21,t σ t =22,twhere g ii,t > 0fori = 1 and 2. Using Eq. (9.7), we haveΣ t =[ ] [ ]1 0g11,t 0, Gq 21,t 1 t =,0 g 22,t[ ] [ ]σ11,t σ 12,t g11,t q 21,t g 11,t=σ 12,t σ 22,t q 21,t g 11,t g 22,t + q21,t 2 g .11,tEquating elements of the previous matrix equation, we obtainσ 11,t = g 11,t , σ 21,t = q 21,t g 11,t , σ 22,t = g 22,t + q 2 21,t g 11,t. (9.8)Solving the prior equations, we haveg 11,t = σ 11,t , q 21,t = σ 21,tσ 11,t, g 22,t = σ 22,t − σ 2 21,tσ 11,t. (9.9)However, consider the simple conditional linear regressiona 2t = βa 1t + b 2t , (9.10)where b 2t denotes the error term. From the well-known least squares theory, we haveβ = Cov(a 1t, a 2t )Var(a 1t )= σ 21,tσ 11,t,Var(b 2t ) = Var(a 2t ) − β 2 Var(a 1t ) = σ 22,t − σ 2 21,tσ 11,t.Furthermore, the error term b 2t is uncorrelated with the regressor a 1t . Consequently,using Eq. (9.9), we obtaing 11,t = σ 11,t , q 21,t = β, g 22,t = Var(b 2t ), b 2t ⊥a 1t ,