A note on non-negative ARMA processes

Abstract

Recently, there has been much research on developing models suitable for analysing the volatility of a discrete-time process. Since the volatility process, like many others, is necessarily non-negative, there is a need to construct models for stationary processes which are non-negative with probability one. Such models can be obtained by driving autoregressive moving average (ARMA) processes with non-negative kernel by non-negative white noise. This raises the problem of finding simple conditions under which an ARMA process with given coefficients has a non-negative kernel. In this article, we derive a necessary and sufficient condition. This condition is in terms of the generating function of the ARMA kernel which has a simple form. Moreover, we derive some readily verifiable necessary and sufficient conditions for some ARMA processes to be non-negative almost surely. © 2006 Blackwell Publishing Ltd.