A unified approach to the study of sums, products, time-aggregation and other functions of ARMA processes

Abstract

Conditions under which sums, products and time-aggregation of ARMA processes follow ARMA models are derived from a single theorem. This characterizes these processes in terms of difference equations satisfied by their autocovariance function. From this we obtain necessary and sufficient conditions for a function of a Gaussian ARMA process and the product of two possibly dependent Gaussian ARMA processes to be ARMA. We show that the sum and product of two ARMA processes related by a Box and Jenkins transfer function model belong to the ARMA family.