TimeSeriesBook.pdf

T one prefers the invertible solution remark 23 if x

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t , one prefers the invertible solution. Remark 2.3. If { X t } is a stationary solution to the stochastic difference equation Φ(L) X t = Θ(L) Z t with Z t WN(0 , σ 2 ) and if in addition Φ( z )Θ( z ) 6 = 0 for | z | ≤ 1 then X t = X j =0 ψ j Z t - j , Z t = X j =0 π j X t - j , where the coefficients { ψ j } and { π j } are determined for | z | ≤ 1 by Ψ( z ) = Θ( z ) Φ( z ) and Π( z ) = Φ( z ) Θ( z ) , respectively. In this case { X t } is causal and invert- ible with respect to { Z t } . Remark 2.4. If { X t } is an ARMA process with Φ(L) X t = Θ(L) Z t such that Φ( z ) 6 = 0 for | z | = 1 then there exists polynomials ˜ Φ( z ) and ˜ Θ( z ) and a white noise process { ˜ Z t } such that { X t } fulfills the stochastic difference equation ˜ Φ(L) X t = ˜ Θ(L) ˜ Z t and is causal with respect to { ˜ Z t } . If in addition Θ( z ) 6 = 0 for | z | = 1 then ˜ Θ(L) can be chosen such that { X t } is also invertible with respect to { ˜ Z t } (see the discussion of the AR(1) process after the definition of causality and Brockwell and Davis (1991, p. 88)). 2.4 Computation of the autocovariance func- tion of an ARMA process Whereas the autocovariance function summarizes the external and directly observable properties of a time series, the coefficients of the ARMA process give information of its internal structure. Although there exists for each ARMA model a corresponding autocovariance function, the converse is not true as we have seen in Section 1.3 where we showed that two MA(1) pro- cesses are compatible with the same autocovariance function. This brings up a fundamental identification problem. In order to shed some light on the relation between autocovariance function and ARMA models it is necessary
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40 CHAPTER 2. ARMA MODELS to be able to compute the autocovariance function for a given ARMA model. In the following, we will discuss three such procedures. Each procedure relies on the assumption that the ARMA process Φ(L) X t = Θ(L) Z t with Z t WN(0 , σ 2 ) is causal with respect to { Z t } . Thus there exists a represen- tation of X t as a weighted sum of current and past Z t ’s: X t = j =0 ψ j Z t - j with j =0 | ψ j | < . 2.4.1 First procedure Starting from the causal representation of { X t } , it is easy to calculate its autocovariance function given that { Z t } is white noise. The exact formula is proved in Theorem (6.4). γ ( h ) = σ 2 X j =0 ψ j ψ j + | h | , where Ψ( z ) = X j =0 ψ j z j = Θ( z ) Φ( z ) for | z | ≤ 1 . The first step consists in determining the coefficients ψ j by the method of undetermined coefficients. This leads to the following system of equations: ψ j - X 0 <k j φ k ψ j - k = θ j , 0 j < max { p, q + 1 } , ψ j - X 0 <k p φ k ψ j - k = 0 , j max { p, q + 1 } . This equation system can be solved recursively (see Section 2.3): ψ 0 = θ 0 = 1 , ψ 1 = θ 1 + ψ 0 φ 1 = θ 1 + φ 1 , ψ 2 = θ 2 + ψ 0 φ 2 + ψ 1 φ 1 = θ 2 + φ 2 + φ 1 θ 1 + φ 2 1 , . . . Alternatively one may view the second part of the equation system as a linear homogeneous difference equation with constant coefficients (see Section 2.3).
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