On asymptotic normality of sequential LS-estimates of unstable autoregressive processes

TL;DR

This paper demonstrates that sequential least squares estimates for unstable autoregressive processes are asymptotically normal, even on the boundary of the stability region, using a novel property of the Fisher information matrix.

Contribution

It introduces a new approach for establishing the asymptotic normality of sequential LSEs for unstable AR processes, including boundary cases.

Findings

Sequential LSEs are asymptotically normal inside and on the boundary of the stability region.

A new property of the Fisher information matrix underpins the asymptotic normality.

The distribution of the stopping time is explicitly derived.

Abstract

For estimating the unknown parameters in an unstable autoregressive AR(p), the paper proposes sequential least squares estimates with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix which holds both inside the stability region of AR(p) process and on the part of its boundary. The asymptotic distribution of the stopping time is derived.