Unified Theory for the Large Family of Time Varying Models with Arma Representations: One Solution Fits All

For the large family of ARMA models with variable coeffcients we obtain an explicit and computationally tractable solution that generates all their fundamental properties, including theWold-Cramer decomposition and their covariance structure, thus unifying the invertibility conditions which guarantee both their asymptotic stability and main properties. The one sided Green's function, associated with the homogeneous solution, is expressed as a banded Hessenbergian formulated exclusively in terms of the autoregressive parameters of the model. The proposed methodology allows for a unified treatment of these `time varying' systems. We also illustrate mathematically one of the focal points in Hallin's (1986) analysis. Namely, that in a time varying setting the backward asymptotic effciency is different from the forward one. Equally important it is shown how the linear algebra techniques, used to obtain the general solution, are equivalent to a simple procedure for manipulating polynomials with variable coeffcients. The practical significance of the suggested approach is illustrated with an application to U.S. in ation data. The main finding is that in ation persistence increased after 1976, whereas from 1986 onwards the persistence reduces and stabilizes to even lower levels than the pre-1976 period.