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

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 = ρ n is derived uniformly over stationary values in [ 0 , 1 ) , focusing on ρ n → 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 − ρ n 2 , but no condition on the rate of ρ n is required. It is shown that, for the tail index α ∈ ( 0 , 2 ) , the LSE is inconsistent, for α = 2 , log n / ( 1 − ρ n 2 ) -consistent, and for α ∈ ( 2 , 4 ) , n 1 − 2 / α / ( 1 − ρ n 2 ) -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 ρ n is necessary.