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For the large family of ARMA models with variable coefficients (TV-ARMA),either deterministic or stochastic, we provide an explicit and computationallytractable representation based on the general solution of the associated lineardifference equation. Analogous representations are established for thefundamental properties of such processes, including the Wold-Cram\'{e}rdecomposition and their covariance structure as well as explicit optimal linearforecasts based on a finite set of past observations. These results aregrounded on the principal determinant, that is a banded Hessenbergianrepresentation of a restriction of the Green function involved in the solutionof the linear difference equation associated with TV-ARMA models, built upsolely of the autoregressive coefficients of the model. The $L_2$ convergenceproperties of the model are a consequence of the absolute summability of theaforementioned Hessenbergian representation, which is in line with theasymptotic stability and efficiency of such processes. The invertibility of themodel is also a consequence of an analogous condition, but now the Greenfunction is built up of the moving average coefficients. The structuralasymmetry between constant and deterministically time-varying coefficientmodels, that is the backward and forward asymptotic efficiency differ in anessential manner, is formally demonstrated. An alternative approach to theHessenbergian solution representation is described by an equivalent procedurefor manipulating time-varying polynomials. The practical significance of thetheoretical results in this work is illustrated with an application to U.S.inflation data. The main finding is that inflation persistence increased after1976, whereas from 1986 onwards the persistence declines and stabilizes to evenlower levels than the pre-1976 period.

A unified theory for ARMA models with varying coefficients: One solution fits all