Note on AR(1)-characterisation of stationary processes and model fitting

TL;DR

This paper explores the AR(1) characterization of stationary processes, analyzing special cases where the estimation method fails, which are linked to degenerate processes, and provides insights into model fitting limitations.

Contribution

It offers a detailed analysis of the special cases where AR(1) model estimation fails, clarifying their connection to degenerate processes.

Findings

Identification of degenerate processes as failure cases

Conditions under which AR(1) estimation may fail

Insights into limitations of autocovariance-based estimators

Abstract

It was recently proved that any strictly stationary stochastic process can be viewed as an autoregressive process of order one with coloured noise. Furthermore, it was proved that, using this characterisation, one can define closed form estimators for the model parameter based on autocovariance estimators for several different lags. However, this estimation procedure may fail in some special cases. In this article we provide a detailed analysis of these special cases. In particular, we prove that these cases correspond to degenerate processes.