TimeSeriesBook.pdf

Proof the proof is a straightforward extension of the

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Proof. The proof is a straightforward extension of the univariate case. As in the univariate case, the coefficient matrices which make up the causal representation can be found by the method of undetermined coeffi- cients, i.e. by equating Φ( z )Ψ( z ) = Θ( z ). In the case of the VAR(1) process,
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236 CHAPTER 12. VARMA PROCESSES the { Ψ j } have to obey the following recursion: 0 : Ψ 0 = I n z : Ψ 1 = ΦΨ 0 = Φ z 2 : Ψ 2 = ΦΨ 1 = Φ 2 . . . z j : Ψ j = ΦΨ j - 1 = Φ j The recursion in the VAR(2) case is: 0 : Ψ 0 = I n z : - Φ 1 + Ψ 1 = 0 Ψ 1 = Φ 1 z 2 : - Φ 2 - Φ 1 Ψ 1 + Ψ 2 = 0 Ψ 2 = Φ 2 + Φ 2 1 z 3 : - Φ 1 Ψ 2 - Φ 2 Ψ 1 + Ψ 3 = 0 Ψ 3 = Φ 3 1 + Φ 1 Φ 2 + Φ 2 Φ 1 . . . Remark 12.1. Consider a VAR(1) process with Φ = 0 φ 0 0 with φ 6 = 0 then the matrices in the causal representation are Ψ j = Φ j = 0 for j > 1 . This means that { X t } has an alternative representation as a VMA(1) pro- cess because X t = Z t + Φ Z t - 1 . This simple example demonstrates that the representation of { X t } as a VARMA process is not unique. It is therefore impossible to always distinguish between VAR and VMA process of higher or- ders without imposing additional assumptions. These additional assumptions are much more complex in the multivariate case and are known as identify- ing assumptions. Thus, a general treatment of this identification problem is outside the scope of this book. See Hannan and Deistler (1988) for a general treatment of this issue. For this reason we will concentrate exclusively on VAR processes where these identification issues do not arise. Example We illustrate the above concept by the following VAR(2) model: X t = 0 . 8 - 0 . 5 0 . 1 - 0 . 5 X t - 1 + - 0 . 3 - 0 . 3 - 0 . 2 0 . 3 X t - 2 + Z t with Z t WN 0 0 , 1 . 0 0 . 4 0 . 4 2 . 0 .
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12.4. COMPUTATION OF COVARIANCE FUNCTION 237 In a first step, we check whether the VAR model admits a causal represen- tation with respect to { Z t } . For this purpose we have to compute the roots of the equation det( I 2 - Φ 1 z - Φ 2 z 2 ) = 0: det 1 - 0 . 8 z + 0 . 3 z 2 0 . 5 z + 0 . 3 z 2 - 0 . 1 z + 0 . 2 z 2 1 + 0 . 5 z - 0 . 3 z 2 = 1 - 0 . 3 z - 0 . 35 z 2 + 0 . 32 z 3 - 0 . 15 z 4 = 0 . The four roots are: - 1 . 1973 , 0 . 8828 ± 1 . 6669 ι, 1 . 5650. As they are all outside the unit circle, there exists a causal representation which can be found from the equation Φ( z )Ψ( z ) = I 2 by the method of undetermined coefficients. Multiplying the equation system out, we get: I 2 - Φ 1 z - Φ 2 z 2 + Ψ 1 z - Φ 1 Ψ 1 z 2 - Φ 2 Ψ 1 z 3 2 z 2 - Φ 1 Ψ 2 z 3 - Φ 2 Ψ 2 z 4 . . . = I 2 . Equating the coefficients corresponding to z j , j = 1 , 2 , . . . : z : Ψ 1 = Φ 1 z 2 : Ψ 2 = Φ 1 Ψ 1 + Φ 2 z 3 : Ψ 3 = Φ 1 Ψ 2 + Φ 2 Ψ 1 . . . . . . z j : Ψ j = Φ 1 Ψ j - 1 + Φ 2 Ψ j - 2 . The last equation shows how to compute the sequence { Ψ j } recursively: Ψ 1 = 0 . 8 - 0 . 5 0 . 1 - 0 . 5 Ψ 2 = 0 . 29 - 0 . 45 - 0 . 17 0 . 50 Ψ 3 = 0 . 047 - 0 . 310 - 0 . 016 - 0 . 345 . . . 12.4 Computation of the Covariance Func- tion of a Causal VAR Process As in the univariate case, it is important to be able to compute the covari- ance and the correlation function of VARMA process (see Section 2.4). As explained in Remark 12.1 we will concentrate on VAR processes. Consider first the case of a causal VAR(1) process: X t = Φ X t - 1 + Z t Z t WN(0 , Σ) .
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238 CHAPTER 12. VARMA PROCESSES Multiplying the above equation first by X 0 t and then successively by X 0 t - h from the left, h = 1 , 2 , . . .
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