Partial autocorrelation parameterisation of models with unit roots on the unit circle

TL;DR

This paper introduces a new parametrization method for autoregressive models with unit roots using partial autocorrelation coefficients, revealing algebraic properties and potential for model factorization.

Contribution

It proposes a novel parametrization of ARMA models with unit roots via partial autocorrelations, including algebraic properties and model factorization insights.

Findings

Partial autocorrelation sequences can be split at values of 1 or -1.

Splitting sequences corresponds to factors of the polynomial.

Provides a foundation for estimation procedures.

Abstract

We propose a parametrization of autoregressive unit roots ARMA models (ARUMA) with partial autocorrelation coefficients to specify the autoregressive and integrated part of the model. We obtain the algebraic properties of the partial autocorrelations in the context of unit roots. The main result is that if a partial autocorrelation sequence contains some values equal to 1 or -1, then it can be split at these values into sub-sequences each of which represents the partial autocorrelations of a factor of the overall polynomial on the left-hand side of the model. A separate paper will discuss the details of the estimation procedure and its properties. An implementation is provided by Boshnakov and Halliday (2022) R package sarima, https://cran.r-project.org/package=sarima (function sarima).