Comparing ARMA Processes with Roots of Modulus 1 and Polynomial Regression
Consider a process defined as where &#x1d539; is the backward operator, Q a polynomial with all zeros of modulus ≥1, and (Y t ) an ergodic stationary process. We show that , even if some zeros (real or complex) have modulus 1. In particular, if (Y t ) is a white noise, it becomes the innovation of (X t ). It follows that the polynomial regression model where P is a polynomial of degree p and (ϵ t ) is a white noise, is a zero-mean IMA(p + 1, p + 1) and that (ϵ t ) is the innovation of the MA part. A practical consequence of this fact is that the ARIMA model remains competitive even if the model is a genuine polynomial regression. Various numerical simulations illustrate that point.