Non-stationary q-dependent processes and time-varying moving-average models: invertibility properties and the forecasting problem

Abstract

The spectral factorization problem was solved in Hallin (1984) for the class of (non-stationary) m-variate MA(q) stochastic processes, i.e. the class of second-order q-dependent processes. It was shown that such a process generally admits an infinite (mq(mq +1)/2-dimensional) family of possible MA(q) representations. The present paper deals with the invertibility properties and asymptotic behaviour of these MA(q) models, in connection with the problem of producing asymptotically efficient forecasts. Invertible and borderline non-invertible models are characterized (Theorems 3.1 and 3.2). A criterion is provided (Theorem 4.1) by which it can be checked whether a given MA model is a Wold–Cramér decomposition or not; and it is shown (Theorem 4.2) that, under mild conditions, almost every MA model is asymptotically identical with some Wold–Cramér decomposition. The forecasting problem is investigated in detail, and it is established that the relevant invertibility concept, with respect to asymptotic forecasting efficiency, is what we define as Granger-Andersen invertibility rather than the classical invertibility concept (Theorem 5.3). The properties of this new invertibility concept are studied and contrasted with those of its classical counterpart (Theorems 5.2 and 5.4). Numerical examples are also treated (Section 6), illustrating the fact that non-invertible models may provide asymptotically efficient forecasts, whereas invertible models, in some cases, may not. The mathematical tools throughout the paper are linear difference equations (Green's matrices, adjoint operators, dominated solutions, etc.), and a matrix generalization of continued fractions.