Asymptotic Theory for M-Estimates in Unstable AR(p) Processes with Infinite Variance Innovations

TL;DR

This paper derives the asymptotic distribution of M-estimators in non-stationary AR(p) processes with infinite variance innovations, revealing their convergence properties and stochastic integral representations.

Contribution

It introduces the asymptotic theory for M-estimators in unstable AR(p) models with heavy-tailed innovations, extending existing results to infinite variance cases.

Findings

M-estimators have higher convergence rates than least squares in certain non-stationary models.

Asymptotic distributions are expressed as Itô stochastic integrals.

Results apply to innovations in the domain of attraction of stable laws with index 0<α≤2.

Abstract

In this paper, we present the asymptotic distribution of M-estimators for parameters in non-stationary AR(p) processes. The innovations are assumed to be in the domain of attraction of a stable law with index $0<\alpha\le2$. In particular, when the model involves repeated unit roots or conjugate complex unit roots, M-estimators have a higher asymptotic rate of convergence compared to the least square estimators and the asymptotic results can be written as It\^{o} stochastic integrals.