"Frontmatter". In: Analysis of Financial Time Series

310 VECTOR TIME SERIESSince G is a diagonal matrix, the components of b t are uncorrelated. MultiplyingL −1 from left to model (8.8), we obtainL −1 r t = L −1 φ 0 + L −1 Φr t−1 + L −1 a t = φ ∗ 0 + Φ∗ r t−1 + b t , (8.9)where φ ∗ 0= L −1 φ 0 is a k-dimensional vector and Φ ∗ = L −1 Φ is a k × kmatrix. Because of the special matrix structure, the kth row of L −1 is in the form(w k1 ,w k2 ,...,w k,k−1 , 1). Consequently, the kth equation of model (8.9) is∑k−1k∑r kt + w ki r it = φk,0 ∗ + ∗ ki r i,t−1 + b kt , (8.10)i=1where φ ∗ k,0 is the kth element of φ∗ 0 and ∗ ki is the (k, i)th element of Φ∗ . Becauseb kt is uncorrelated with b it for 1 ≤ i < k, Eq. (8.10) shows explicitly the concurrentlinear dependence of r kt on r it ,where1≤ i ≤ k − 1. This equation is referred to asa structural equation for r kt in the econometric literature.For any other component r it of r t , we can rearrange the VAR(1) model so thatr it becomes the last component of r t . The prior transformation method can then beapplied to obtain a structural equation for r it . Therefore, the reduced-form model(8.8) is equivalent to the structural form used in the econometric literature. In timeseries analysis, the reduced-form model is commonly used for two reasons. The firstreason is ease in estimation. The second and main reason is that the concurrent correlationscannot be used in forecasting.Example 8.3. To illustrate the transformation from a reduced-form model tostructural equations, consider the bivariate AR(1) model[r1t]=r 2t[ ] 0.2+0.4i=1[ ][ ] [ ]0.2 0.3 r1,t−1 a1t+ , Σ =−0.6 1.1 r 2,t−1 a 2tFor this particular covariance matrix Σ, the lower triangular matrixL −1 =[ ] 1.0 0.0−0.5 1.0[ ] 2 1.1 1provides a Cholesky decomposition (i.e., L −1 Σ(L ′ ) −1 is a diagonal matrix). PremultiplyingL −1 to the previous bivariate AR(1) model, we obtain[ ][ ] [ ] [ 1.0 0.0 r1t 0.2 0.2 0.3= +−0.5 1.0 r 2t 0.3 −0.7 0.95[ ] 2 0G = ,0 0.5][r1,t−1r 2,t−1][ ]b1t+ ,b 2t