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28 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(11) (eq. 2.6)To deal with these problems, Þrst standardize the innovations. Sincethe correlation matrix is given bywhere Λ =⎛⎝1√σ11001√σ22R = ΛΣΛ =⎞Ã1ρρ 1!,⎠ is a matrix with the inverse of the standarddeviations on the diagonal and zeros elsewhere. The error covariancematrix can be decomposed as Σ = Λ −1 RΛ −1 . This means the Woldvector moving-average representation (2.4) can be re-written asq t==⎛∞X⎝j=0⎛∞X⎝j=0C j L j ⎞⎠ Λ −1 (Λ² t )D j L j ⎞⎠ v t . (2.5)where D j ≡ C j Λ −1 ,v t ≡ Λ² t and E(v t v 0 t)=R. The newly deÞnedinnovations v 1t and v 2t both have variance of 1.Now to unambiguously attribute an innovation to q 1t , you mustorthogonalize the innovations by taking the unique upper triangularCholeski matrix à decomposition ! of the correlation matrix R = S 0 S,s11 swhere S =12. Now insert SS −1 into the normalized moving0 s 22average (2.5) to get⎛ ⎞∞Xq t= ⎝ D j L j ⎠ S ³ ∞XS −1 v t´= B j L j η t , (2.6)j=0where B j ≡ D j S = C j Λ −1 S and η t ≡ S −1 v t , is the 2×1 vectorofzeromeanorthogonalized innovations with covariance matrix E(η t η 0 = I). tNote that S −1 is also upper triangular.Now write out the individual equations in (2.6) to getq 1t =q 2t =∞Xj=0∞Xj=0b 11,j η 1,t−j +b 21,j η 1,t−j +∞Xj=0∞Xj=0j=0b 12,j η 2,t−j , (2.7)b 22,j η 2,t−j . (2.8)