Probabilistic Properties of Parametric Dual and Inverse Time Series Models Generated by ARMA Models

Abstract

For the class of autoregressive-moving average (ARMA) processes, we examine the relationship between the dual and the inverse processes. It is demonstrated that the inverse process generated by a causal and invertible ARMA (p, q) process is a causal and invertible ARMA (q, p) model. Moreover, it is established that this representation is strong if and only if the generating process is Gaussian. More precisely, it is derived that the linear innovation process of the inverse process is an all-pass model. Some examples and applications to time reversibility are given to illustrate the obtained results.

Keywords:

• All-pass time series models
• Inverse and dual processes
• Inverse and ordinary autocorrelations
• Non linear time series
• Time reversibility
• Weak ARMA models.

Mathematics Subject Classification:

• Primary 62M10; Secondary 60G10