Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations

Abstract

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(0,4) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index α∈(0,4); and no restriction on the rate of ρn is necessary.