On the evaluation of the information matrix for multiplicative seasonal time-series models

This paper gives a procedure for evaluating the Fisher information matrix for a general multiplicative seasonal autoregressive moving average time-series model. The method is based on the well-known integral specification of Whittle [Ark. Mat. Fys. Astr. (1953) vol. 2. pp. 423-434] and leads to a system of linear equations, which is independent of the seasonal period and has a closed solution. It is shown to be much simpler, in general, than the method of Klein and Melard [Journal of Time Series Analysis (1990) vol. 11, pp. 231-237], which depends on the seasonal period. It is also shown that the nonseasonal method of McLeod [Biometrika (1984) vol. 71, pp. 207-211] has the same basic features as that of Klein and Melard. Explicit solutions are obtained for the simpler nonseasonal and seasonal models in common use, a feature which has not been attempted with the Klein-Melard or the McLeod approaches. Several illustrations of these results are discussed in detail.