A Note on Bootstrapping M-estimates from Unstable AR(2) Process with Infinite Variance Innovations

TL;DR

This paper investigates the validity of bootstrap methods for M-estimates in non-stationary AR(2) models with infinite variance innovations, demonstrating their effectiveness under certain conditions.

Contribution

It establishes the asymptotic validity of bootstrap procedures for M-estimates in unstable AR(2) processes with heavy-tailed errors, extending inference methods to infinite variance cases.

Findings

Bootstrap with $m=o(n)$ resampling is approximately valid for heavy-tailed AR(2) models.

The results apply to innovations in the domain of attraction of a stable law with index $0<eta extless2$.

Provides theoretical support for bootstrap inference in non-stationary models with infinite variance innovations.

Abstract

The limiting distribution for M-estimates in a non-stationary autoregressive model with heavy-tailed error is computationally intractable. To make inferences based on the M-estimates, the bootstrap procedure can be used to approximate the sampling distribution. In this paper, we show that the bootstrap scheme with $m=o(n)$ resampling sample size when $m/n \to 0$ is approximately valid in a multiple unit roots time series with innovations in the domain of attraction of a stable law with index $0<\alpha\leq2$.