Duality and other properties of multiplicative seasonal autoregressive-moving average models

Model duality is defined between four models referred to as the primal, the dual, the autoregressive adjoint and the moving average adjoint. A duality theorem which generalizes the results of Box & Pierce (1970) and Pierce (1970) is presented. Applications of this duality theorem to autoregressive-moving average models and multiplicative seasonal autoregressive-moving average models are discussed. These applications include: (i) a convenient method for calculating the covariance matrix of the estimated parameters; (ii) convenient formulae for the variances of the residual autocorrelations; (iii) the distribution of the inverse partial autocorrelations. Finally, a useful approximation to the covariance determinant of multiplicative seasonal models is derived.