Biquadratic functions: stationary and invertibility in estimated time-series models

Abstract

It is important that the estimates of the parameters of an autoregressive moving-average (ARMA) model should satisfy the conditions of stationarity and invertibility. It can be shown that the unconditional maximum-likelihood estimates are bound to fill these conditions regardless of the size of the sample from which they are derived; and, in some quarters, it has been argued that they should be used in preference to any other estimates when the size of he sample is small. However, the maximum-likelihood estimates are difficult to obtain; and, in practice, estimates are usually derived from a least-squares criterion. In this paper we show that, if an appropriate form of least-squares criterion is adopted, then we can likewise guarantee that the conditions of stationarity and invertibility will be fulfilled. We also re-examine several of the alternative procedures for estimating ARMA models to see whether the criterion functions from which they are derived have the appropriate form.