A Note on the Information Matrix for Multiplicative Seasonal Autoregressive Moving‐Average Models

Abstract

. It is shown that there is an invariance property for each of the elements of the information matrix of a multiplicative seasonal autoregressive moving-average time-series model, which enables the integral specification of Whittle (1953a,b) to be solved in a straightforward way. The resulting non-iterative closed procedure shares the property possessed by the piecemeal approach of Godolphin and Bane (2006) of being independent of the seasonal period, but our procedure is preferable if one or more orders of the seasonal components of the model are greater than unity. The procedure is therefore simpler, in general, than the iterative method of Klein and Melard (1990) that depends necessarily on the seasonal period. In the strictly non-seasonal case this invariance property prescribes a non-iterative closed procedure for evaluating the information matrix which improves on the methods of Godolphin and Unwin (1983), Friedlander (1984), McLeod (1984) and Klein and Spreij (2003). Three illustrations of the approach are given.