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FIGURE 22.4 ACF and PACF of selected stochastic processes: (a) AR(2): a1 = 0.5, a2 = 0.3; (b) MA(2): p1 = 0.5, p2 = 0.3; (c) ARMA(1, 1): a1 = 0.5, p1 = 0.5.

Remembering that the ACF and PACF shown there are sample quantities, we do not have a nice pattern as suggested in Table 22.1. The autocorrelations decline up to lag 4, then except at lags 8 and 12, the rest of them are statistically not different from zero (the solid lines shown in this figure give the approximate 95% confidence limits). The partial autocorrelations with spikes at lag 1, 8, and 12 seem statistically significant but the rest are not; if the partial correlation coefficient were significant only at lag 1, we could have identified this as an AR(1) model. Let us therefore assume that the process that generated the (first-differenced) GDP is at the most an AR(12) process. Of course, we do not have to include all the AR terms up to 12, for from the partial correlogram we know that only the AR terms at lag 1, 8, and 12 are significant.

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