Noninvertible Transfer Functions and their Forecasts

Abstract

A transfer function relating a time series $y_t$ to present and past values of a series $x_t$ need not possess an inverse. When $(x_t, y_t)$ is a covariance stationary process, it is shown that noninvertibility in this transfer function has the effect of reducing the error variance of the minimum mean-square-error predictor of $y_t$ one or more steps ahead. In deriving these results a "dual" series to $x_t$ is constructed, which has univariate stochastic structure identical to that of $x_t$ itself, and an associated dual transfer function relating it to $y_t$ which is invertible.