Applied Time Series Analysis I. Proceedings of the First by D.F. Findley

By D.F. Findley

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Extra resources for Applied Time Series Analysis I. Proceedings of the First Applied Time Series Symposium Held in Tulsa, Oklahoma, May 14–15, 1976

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5) are invertible, then the series generated may have interesting properties. For example, suppose x t is formed by where E t is a Gaussian zero-mean white noise, then and all T '1 0, so that x t will appear to be white noise if just covariances are considered even though it is potentially forecastable from its own past, although in a non-linear fashion. However, it may be shown that this model is never invertible, for any non-zero value of a, and so in practice it cannot be used for forecasting.

Gray et al. 54 THEOREM 7. ) Hn(m+ ) [l;f;+J')' l -- n J ' n+l J ' (m+l) (m) Hn +l [l;f i + j ) and Hn + 2 [1;f i + j ) ~ not ~, then 4 G(m) n+l G (m+l) _ G (m) n n G (m) + n (29) 1 - R (m) n+l where (m+1) r n+l r(m) n+l R(m) n+l (30) In order to calculate the G-transform recursively, it is clear from (29) that a recursive calculation of R~~i also is needed. This is accomplished by the following relations which also were established in [51). [Sns r (m+l) n (m) r n +1 lmHl n (m) -1] , -r n (m+l) s.

Use the algo~ithm o~ Pye and A~chison described above to calculate the G (J), si (J), and r (J) recursively to some i i preassigned maximum values of i and j. Select the proper G. (j) as G-spectral estimator from the "stabling off" p~operty. If Gi (j) does not stable off, inspect the r. (j) and si (j) to determine p and q. 5, and near peaks), it is recommended that the r. J and J. si (j) be considered for possibly a better choice of j and i 60 H. L. Gray et al. in G (j). This procedure is implemented in the program i given in Appendix 2.

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