A Unified Theory for ARMA Models With Varying Coefficients : One Solution Fits All

For the large family of ARMA models with variable coefficients we provide an explicit and computationally tractable solution representation, which yields the fundamental properties of such processes, including the Wold-Cramer decomposition and the covariance structure. These results are founded on a banded Hessenbergian representation of the Green's function, built up solely of the auto-regressive coefficients of the model. A generic condition, also expressed in terms of the Green's function,guarantees the convergence of the above mentioned properties. This condition is in line with the asymptotic stability and efficiency of such processes, while, their invertibility is guaranteed by an analogous condition, but now the Green's function is built up of the moving average coefficients. As a consequence, we illustrate mathematically a structural asymmetry between constant and 'time dependent' coefficient models, that is in the latter ones the backward and forward asymptotic efficiency differ structurally from one another. An alternative approach to the Hessenbergian solution representation is described by a simple procedure for manipulating polynomials with variable coefficients. 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 on wards the persistence declines and stabilizes to even lower levels than the pre-1976 period