A Unified Theory for Arma Models with Varying Coefficients: One Solution Fits All

A new explicit solution representation is provided for ARMA recursions with drift and either deterministically or stochastically varying coefficients. It is expressed in terms of the determinants of banded Hessenberg matrices and, as such, is an explicit function of the coefficients. In addition to computational efficiency, the proposed solution provides a more explicit analysis of the fundamental properties of such processes, including their Wold-Cram´er decomposition, their covariance structure and their asymptotic stability and efficiency. Explicit formulae for optimal linear forecasts, based either on finite or infinite sequences of past observations are provided. The practical significance of the theoretical results in this work is illustrated with an application to U.S. inflation data. The main finding is that inflation persistence increased after 1976, whereas from 1986 onwards the persistence declines and stabilizes to even lower levels than the pre-1976 period.