  For the large family of ARMA models with variable coefficients (TV-ARMA),
either deterministic or stochastic, we provide an explicit and computationally
tractable representation based on the general solution of the associated linear
difference equation. Analogous representations are established for the
fundamental properties of such processes, including the Wold-Cram\'{e}r
decomposition and their covariance structure as well as explicit optimal linear
forecasts based on a finite set of past observations. These results are
grounded on the principal determinant, that is a banded Hessenbergian
representation of a restriction of the Green function involved in the solution
of the linear difference equation associated with TV-ARMA models, built up
solely of the autoregressive coefficients of the model. The $L_2$ convergence
properties of the model are a consequence of the absolute summability of the
aforementioned Hessenbergian representation, which is in line with the
asymptotic stability and efficiency of such processes. The invertibility of the
model is also a consequence of an analogous condition, but now the Green
function is built up of the moving average coefficients. The structural
asymmetry between constant and deterministically time-varying coefficient
models, that is the backward and forward asymptotic efficiency differ in an
essential manner, is formally demonstrated. An alternative approach to the
Hessenbergian solution representation is described by an equivalent procedure
for manipulating time-varying polynomials. 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.
