Printed in Great Britain The relevance of a probabilistic form of invertibility

Recent interest in polynomial moving average models has raised the question of their invertibility. A probabilistic notion of invertibility is presented and applied to polynomial moving averages using a stochastic Liapunov criterion. Some key words: Liapunov function; Moving average model; Stochastic stability. The concept of the invertibility of an observed stochastic process Yt, functionally related to an unobservable white noise process e,, has been discussed by Kashyap & Rao (1976) and Granger & Andersen (1978a; 1978b). Invertibility is concerned with estimation of the values of et that gave rise to some observed sequence of Y('B. A model relating the Yt and et processes is said to be invertible if there is an estimation procedure yielding estimates it such that the error et = it — e, tends to zero in some sense as the number of observations on Yt tends to infinity. Some suggested criteria are J£(ef) tends to zero, plim (e,) = 0, or e, tends to zero with probability one. A generalization of the last criterion is obtained by defining a model to be a invertible if e, tends to zero with probability at least a. An example of an a invertible model not known to be invertible will now be presented. Let et be a sequence of mutually independent standard normal variates and suppos