By Akira Takayama

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If the lhs. 47) are also MA(5)s . 47), a system = = 56 3. ARIMA Models and Signal Extraction of six equations is obtained (one equation for each nonzero covariance) . , and V,, ; a total of eight unknowns. 47) . 46) with nonnegative spectra an admissible decomposition. The structural model will not be identified, in general, because an infinite number of admissible decompositions are possible . 46) . The spectral zero translates into a unit root in 8p (B) and in 8. (B), so that the two components Pt and St become noninvertible.

18) of a stationary process, Zt = 'l1(B)at, the representation is of no help from the point of view of fitting a model because , in general, the polynomial 'l1(B) will contain an infinite number of parameters. Therefore we use a rational approximation of the type 'l1(B)= O(B) ¢J(B) , where O(B) and ¢J(B) are finite polynomials in B of order q and p, respectively . Then we can write ¢J(B)Zt = O(B)at. 1) The model is the autoregressive moving average process of orders p and q; in brief, the ARMA(p,q) model.

The model is highly parsimonious, and the three parameters can be given a structural interpretation. 1, when 81 ~ -1, the trend behavior generated by the model becomes more and more stable and, when 84 ~ -1, the same thing happens to the seasonal component . 99 produces perfectly behaved invertible models. 99, certain (mutually exclusive) things can explain the result: (1) seasonality is practically deterministic; (2) there is no seasonality, and the model is overdifferenced . Determining which of the two is the correct explanation is rather simple by testing for the significance of seasonal dummy variables.