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 rho=rho(n) is derived uniformly over stationary values in [0,1), focusing on rho(n)-> 1 as sample size n tends to infinity. For tail index alpha is an element of(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-rho(2)(n), but no condition on the rate of rho(n) is required. It is shown that, for the tail index alpha is an element of(0,2), the LSE is inconsistent, for alpha=2, logn/(1-rho(2)(n))-consistent, and for alpha is an element of(2,4), n(1-2/alpha)/(1-rho(2)(n))-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 alpha is an element of(0,4); and no restriction on the rate of rho(n) is necessary.