Comparing ARMA Processes with Roots of Modulus 1 and Polynomial Regression

作者：Blanke Delphine; Bosq Denis^*

来源：Communications in Statistics - Theory and Methods, 2012, 41(16-17): 2851-2863.

DOI：10.1080/03610926.2011.617482

• ARMA
• IMA
• Innovation
• Polynomial regression
• Real and complex roots of modulus one
• Seasonality

Consider a process defined as
X-t = Q(B)Y-t, t is an element of Z,
where B is the backward operator, Q a polynomial with all zeros of modulus >= 1, and (Y-t) an ergodic stationary process. We show that Y-t is an element of (sp) over bar {X-s, s <= t}, 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
Z(t) = P(t) + epsilon(t), t is an element of Z,
where P is a polynomial of degree p and (epsilon(t)) is a white noise, is a zero-mean IMA (p + 1, p + 1) and that (epsilon(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.

• 出版日期2012