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.

Automated summary

This paper examines stationary autoregressive models with heavy-tailed G-GARCH or augmented GARCH noises, deriving limit theory for the least squares estimator of the autoregression coefficient. The asymptotic distributions of the estimators are established for different tail indices of the G-GARCH innovations, and it is shown that the LSE consistency depends on the tail index. This work extends the existing limit theory by considering errors with conditional heteroscedastic variance and heavy tails, without imposing restrictions on the rate of rho(n).