ARMA models with bilinear innovations

It is well known that any purely non-deterministic stationary process (X_t) with finite variance can be written as an infinite moving average in terms of its innovation process. This property is widely used in the linear methods of estimation and prediction of time series but these methods may give poor results when the innovations are not independent. This often occurs in practical situations. To improve the quality of fit and forecasts the class of models under consideration has to be enlarged. However it may be desirable to preserve the linear representation property of the process (X_t) for the purposes of interpretation, computation and forecasting. This is generally not the case with global non linear models. In this paper we try to solve the problem by means of ARMA models with bilinear white noise. We consider the probabilistic as well as the statistical properties of this class of models (stationarity, invertibility, moments, prediction, identification, estimation, tests). Finally, applicability is studied via simulations and sunspot numbers.