Suppose that x t follows an arma process φl x t c

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Suppose that { X t } follows an ARMA process Φ(L)∆ X t = c + Θ(L) Z t then μ t = δ +Ψ(1) Z t can be identified as the trend component. This means that the trend component can be recursively determined from the observations by applying the formula μ t = Φ(L) Θ(L) Ψ(1) X t . The cyclical component is then simply the residual: ε t = X t - μ t . In the above decomposition both the permanent (trend) component as well as the stationary (cyclical) component are driven by the same shock Z t . A more sophisticated model would, however, allow that the two compo- nents are driven by different shocks. This idea is exploited in the so-called structural time series analysis where the different components (trend, cycle,
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7.2. OLS ESTIMATOR WITH INTEGRATED VARIABLES 149 season, and irregular) are modeled as being driven by separated shocks. As only the series { X t } is observed, not its components, this approach leads to serious identification problems. See the discussion in Harvey (1989), Han- nan and Deistler (1988), or Mills (2003). In Sections 17.1 and 17.4.2 we will provide an overall framework to deal with these issues. Examples Let { X t } be a MA(q) process with ∆ X t = δ + Z t + . . . + θ q Z t - q then the persistence is given simply by the sum of the MA-coefficients: Ψ(1) = 1+ θ 1 + . . . + θ q . Depending on the value of these coefficients. The persistence can be smaller or greater than one. If { X t } is an AR(1) process with ∆ X t = δ + φ X t - 1 + Z t and assuming | φ | < 1 then we get: ∆ X t = δ 1 - φ + j =0 φ j Z t - j . The persistence is then given as Ψ(1) = j =0 φ j = 1 1 - φ . For positive values of φ , the persistence is greater than one. Thus, a shock of one is amplified to have an effect larger than one in the long-run. If { X t } is assumed to be an ARMA(1,1) process with ∆ X t = δ + φ X t - 1 + Z t + θZ t - 1 and | φ | < 1 then ∆ X t = δ 1 - φ + Z t +( φ + θ ) j =0 φ j Z t - j - 1 . The persistence is therefore given by Ψ(1) = 1 + ( φ + θ ) j =0 φ j = 1+ θ 1 - φ . The computation of the persistence for the model estimated for Swiss GDP in Section 5.6 is more complicated because a fourth order difference 1 - L 4 has been used instead of a first order one. As 1 - L 4 = (1 - L)(1+L+ L 2 + L 3 ), it is possible to extend the above computations also to this case. For this purpose we compute the persistence for (1 + L + L 2 + L 3 ) ln BIP t in the usual way. The long-run effect on ln BIP t is therefore given by Ψ(1) / 4 because (1 + L + L 2 + L 3 ) ln BIP t is nothing but four times the moving- average of the last four values. For the AR(2) model we get a persistence of 1.42 whereas for the ARMA(1,3) model the persistence is 1.34. Both values are definitely above one so that the permanent effect of a one-percent shock to Swiss GDP is amplified to be larger than one in the long-run. Campbell and Mankiw (1987) and Cochrane (1988) report similar values for the US.
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