ARMA processes have maximal entropy among time series with prescribed autocovariances and impulse responses

Abstract

The maximum-entropy approach to the estimation of the spectral density of a time series has become quite popular during the last decade. It is closely related to the fact that an autoregressive process of order p has maximal entropy among all time series sharing the same autocovariances up to lag p. We give a natural generalization of this result by proving that a mixed autoregressive-moving-average process (ARMA process) of order (p, q) has maximal entropy among all time series sharing the same autocovariances up to lag p and the same impulse response coefficients up to lag q. The latter may be estimated from a finite record of the time series, for example by using a method proposed by Bhansali (1976). By the way, we give a result on the existence of ARMA processes with prescribed autocovariances up to lag p and impulse response coefficients up to lag q.