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 ε (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 - ρ2n, but no condition on the rate of ρn is required. It is shown that, for the tail index α ε (0, 2), the LSE is inconsistent, for α = 2, log n/(1 - ρ2n)- consistent, and for α ε (2, 4), n1-2/α/(1 - ρ2n)-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 ε (0, 4); and no restriction on the rate of ρn is necessary.