Spectral Factorization of Nonstationary Moving Average Processes

Abstract

We solve here the general nonstationary multivariate MA spectral factorization problem, i.e. the problem of obtaining all the possible MA models (with time-dependent coefficients) corresponding to a given (time-dependent) autocovariance function. Our result (Theorem 8) relies on a symbolic generalization (Theorem 1) of the classical factorization property of the characteristic polynomial associated with stationary autocovariance functions, and is obtained by means of a matrix extension of ordinary continued fractions. We also give necessary and sufficient conditions for an autocovariance function to be an MA autocovariance function and for a process to be an MA one (Theorems 6 and 7).