TimeSeriesBook.pdf

2 b 21 σ 12 the first solution b 1 12 can be

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2 - b 21 σ 12 . The first solution b (1) 12 can be excluded because it violates the assumption b 12 b 21 6 = 1 which stands in contradiction to the positive-definiteness of the covariance matrix Σ. Inserting the second solution back into the solution for ω 2 d and ω 2 s , we finally get: ω 2 d = σ 2 1 σ 2 2 - σ 2 12 b 2 21 σ 2 1 - 2 b 21 σ 12 + σ 2 2 > 0 ω 2 s = ( σ 2 2 - b 21 σ 12 ) 2 b 2 21 σ 2 1 - 2 b 21 σ 12 + σ 2 2 > 0 . Because Σ is a symmetric positive-definite matrix, σ 2 1 σ 2 2 - σ 2 12 and the de- nominator b 2 21 σ 2 1 - 2 b 21 σ 12 + σ 2 2 are strictly positive. Thus, both solutions yield positive variances and we have found the unique admissible solution. 15.5.2 The General Approach The general case of long-run restrictions has a structure similar to the case of short-run restrictions. Take as a starting point again the structural VAR (15.2) from Section 15.2.2: AX t = Γ 1 X t - 1 + . . . + Γ p X t - p + BV t , V t WN(0 , Ω) , A (L) X t = BV t where { X t } is stationary and causal with respect to { V t } . As before the matrix A is normalized to have ones on its diagonal and is assumed to be invertible, Ω is a diagonal matrix with Ω = diag( ω 2 1 , . . . , ω 2 n ), and B is a matrix with ones on the diagonal. The matrix polynomial A (L) is defined as A (L) = A - Γ 1 L - . . . - Γ p L p . The reduced form is given by Φ(L) X t = Z t , Z t WN(0 , Σ) where AZ t = BV t and A Φ j = Γ j , j = 1 , . . . , p , respectively A Φ(L) = A (L).
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306 CHAPTER 15. INTERPRETATION OF VAR MODELS The long-run variance of { X t } , J , (see equation (11.1) in Chapter 11) can be derived from the reduced as well as from the structural form, which gives the following expressions: J = Φ(1) - 1 Σ Φ(1) - 1 0 = Φ(1) - 1 A - 1 B B 0 A 0- 1 Φ(1) - 1 0 = A (1) - 1 B B 0 A (1) - 1 0 = Ψ(1)Σ Ψ(1) 0 = Ψ(1) A - 1 B B 0 A 0- 1 Ψ(1) 0 where X t = Ψ(L) Z t denotes the causal representation of { X t } . The long-run variance J can be estimated by adapting the methods in Section 4.4 to the multivariate case. Thus, the above equation system has a similar structure as the system (15.5) which underlies the case of short-run restrictions. As before, we get n ( n + 1) / 2 equations with 2 n 2 - n unknowns. The nonlinear equation system is therefore undetermined for n 2. Therefore, 3 n ( n - 1) / 2 additional equations or restrictions are necessary to achieve identification. Hence, conceptually we are in a similar situation as in the case of short-run restrictions. 22 In practice, it is customary to achieve identification through zero restric- tions where some elements of Ψ(1) A - 1 B , respectively Φ(1) - 1 A - 1 B , are set a priori to zero. Setting the ij -th element [Ψ(1) A - 1 B ] ij = [Φ(1) - 1 A - 1 B ] ij equal to zero amounts to set the cumulative effect of the j -th structural dis- turbance V jt on the i -th variable equal to zero. If the i -th variable enters X t in first differences, as was the case for Y t in the previous example, this zero restriction restrains the long-run effect on the level of that variable.
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