For the integrated process the forecast error can be

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For the integrated process the forecast error can be written as X t + h - e P t X t + h = Z t + h + (1 + ψ 1 ) Z t + h - 1 + . . . + (1 + ψ 1 + ψ 2 + . . . + ψ h - 1 ) Z t +1 . The forecast error variance therefore is E X t + h - e P t X t + h 2 = 1 + (1 + ψ 1 ) 2 + . . . + (1 + ψ 1 + . . . + ψ h - 1 ) 2 σ 2 . This expression increases with the length of the forecast horizon h , but is no longer bounded. It increases linearly in h to infinity. 2 The precision of the forecast therefore not only decreases with the forecasting horizon h as in the case of the trend-stationary model, but converges to zero. In the example above of the ARIMA(0,1,1) process the forecasting error variance is E ( X t + h - P t X t + h ) 2 = 1 + ( h - 1) (1 + θ ) 2 σ 2 . This expression clearly increases linearly with h . 7.1.3 Impulse Response Function The impulse response function (dynamic multiplier) is an important analyt- ical tool as it gives the response of the variable X t to the underlying shocks. In the case of the trend-stationary process the impulse response function is e P t X t + h ∂Z t = ψ h -→ 0 for h → ∞ . 2 Proof : By assumption { ψ j } is absolutely summable so that Ψ(1) converges. Moreover, as Ψ(1) 6 = 0, there exists ε > 0 and an integer m such that h j =0 ψ j > ε for all h > m . The squares are therefore bounded from below by ε 2 > 0 so that their infinite sum diverges to infinity.
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146 CHAPTER 7. INTEGRATED PROCESSES The effect of a shock thus declines with time and dies out. Shocks have there- fore only transitory or temporary effects. In the case of an ARMA process the effect even declines exponentially (see the considerations in Section 2.3). 3 In the case of integrated processes the impulse response function for ∆ X t implies : e P t X t + h ∂Z t = 1 + ψ 1 + ψ 2 + . . . + ψ h . For h going to infinity, this expression converges j =0 ψ j = Ψ(1) 6 = 0. This implies that a shock experienced in period t will have a long-run or permanent effect. This long-run effect is called persistence . If { X t } is an ARMA process then the persistence is given by the expression Ψ(1) = Θ(1) Φ(1) . Thus, for an ARIMA(0,1,1) the persistence is Ψ(1) = Θ(1) = 1 + θ . In the next section we will discuss some examples. 7.1.4 The Beveridge-Nelson Decomposition The Beveridge-Nelson decomposition represents an important tool for the un- derstanding of integrated processes. 4 It shows how an integrated time series of order one can be represented as the sum of a linear trend, a random walk, and a stationary series. It may therefore be used to extract the cyclical com- ponent (business cycle component) of a time series and can thus be viewed as an alternative to the HP-filter (see section 6.5.2) or to more elaborated so-called structural time series models (see sections 17.1 and 17.4.2). Assuming that { X t } is an integrated process of order one, there exists, according to Definition 7.1, a causal representation for { X t } : X t = δ + Ψ(L) Z t with Z t WN ( 0 , σ 2 ) with the property Ψ(1) 6 = 0 and j =0 j | ψ j | < . Before proceeding to the main theorem, we notice the following simple, but extremely useful polyno- 3 The use of the partial derivative is just for convenience. It does not mean that
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