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# 32 chapter 2 arma models in the case φ 1 the

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32 CHAPTER 2. ARMA MODELS In the case | φ | > 1 the solution (2.2) does not converge. It is, however, possible to iterate the difference equation forward to obtain: X t = φ - 1 X t +1 - φ - 1 Z t +1 = φ - k - 1 X t + k +1 - φ - 1 Z t +1 - φ - 2 Z t +2 - . . . - φ - k - 1 Z t + k +1 . This suggests to take X t = - X j =1 φ - j Z t + j as the solution. Going through similar arguments as before it is possible to show that this is indeed the only stationary solution. This solution is, however, viewed to be inadequate because X t depends on contemporaneous and future shocks Z t + j , j = 0 , 1 , . . . . Note, however, that there exists an AR(1) process with | φ | < 1 which is observationally equivalent, in the sense that it generates the same autocorrelation function, but with new shock or forcing { e Z t } (see next section). In the case | φ | = 1 there exists no stationary solution (see Section 1.4.4) and therefore, according to our definition, no ARMA process. Processes with this property are called random walk, unit root processes or integrated pro- cesses. They play an important role in economics and are treated separately in Chapter 7. 2.3 Causality and invertibility If we view { X t } as the state variable and { Z t } as an impulse or shock, we can ask whether it is possible to represent today’s state X t as the outcome of current and past shocks Z t , Z t - 1 , Z t - 2 , . . . In this case we can view X t as being caused by past shocks and call this a causal representation . Thus, shocks to current Z t will not only influence current X t , but will propagate to affect also future X t ’s. This notion of causality rest on the assumption that the past can cause the future but that the future cannot cause the past. See section 15.1 for an elaboration of the concept of causality and its generalization to the multivariate context. In the case that { X t } is a moving-average process of order q , X t is given as a weighted sum of current and past shocks Z t , Z t - 1 , . . . , Z t - q . Thus, the moving-average representation is already the causal representation. In the case of an AR(1) process, we have seen that this is not always feasible. For | φ | < 1, the solution (2.2) represents X t as a weighted sum of current and past shocks and is thus the corresponding causal representation. For | φ | > 1, no such representation is possible. The following Definition 2.2 makes the

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2.3. CAUSALITY AND INVERTIBILITY 33 notion of a causal representation precise and Theorem 2.1 gives a general condition for its existence. Definition 2.2 (Causality) . An ARMA ( p, q ) process { X t } with Φ(L) X t = Θ(L) Z t is called causal with respect to { Z t } if there exists a sequence { ψ j } with the property j =0 | ψ j | < such that X t = Z t + ψ 1 Z t - 1 + ψ 2 Z t - 2 + . . . = X j =0 ψ j Z t - j with ψ 0 = 1 . The above equation is referred to as the causal representation of { X t } with respect to { Z t } . The coefficients { ψ j } are of great importance because they determine how an impulse or a shock in period t propagates to affect current and future X t + j , j = 0 , 1 , 2 . . . In particular, consider an impulse e t 0 at time t 0 , i.e. a time series which is equal to zero except for the time t 0 where it takes on the values e t 0 . Then, { ψ t -
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