Limit theory for an AR(1) model with intercept and a possible infinite variance

TL;DR

This paper investigates the asymptotic behavior of the least squares estimator in AR(1) models with intercepts under various conditions, including infinite variance errors and different autoregressive parameters.

Contribution

It derives the limit distributions of the estimator considering infinite variance errors and different regimes of the AR(1) coefficient, extending classical results.

Findings

Limit distributions vary with |ρ| < 1, |ρ| > 1, and near unit root cases.

Infinite variance errors significantly affect convergence rates.

Different regimes lead to distinct asymptotic behaviors.

Abstract

In this paper, we derive the limit distribution of the least squares estimator for an AR(1) model with a non-zero intercept and a possible infinite variance. It turns out that the estimator has a quite different limit for the cases of $|\rho| < 1$, $|\rho| > 1$, and $\rho = 1 + \frac{c}{n^\alpha}$ for some constant $c \in R$ and $\alpha \in (0, 1]$, and whether or not the variance of the model errors is infinite also has a great impact on both the convergence rate and the limit distribution of the estimator.