Nearly nonstationary processes under infinite variance GARCH noises

Let Y_t be an autoregressive process with order one, i.e., Y_t = μ + ϕ_nY_t−1 + ε_t, where [ε_t] is a heavy tailed general GARCH noise with tail index α. Let \({{\hat \phi }_n}\) be the least squares estimator (LSE) of ϕ_n For μ = 0 and α < 2, it is shown by Zhang and Ling (2015) that \({{\hat \phi }_n}\) is inconsistent when Y_t is stationary (i.e., ϕ_n ≡ ϕ < 1), however, Chan and Zhang (2010) showed that \({{\hat \phi }_n}\) is still consistent with convergence rate n when Y_t is a unit-root process (i.e., ϕ_n = 1) and [ε_t] is a GARCH(1, 1) noise. There is a gap between the stationary and nonstationary cases. In this paper, two important issues will be considered: (1) what about the nearly unit root case? (2) When can ϕ be estimated consistently by the LSE? We show that when ϕ_n = 1 − c/n, then \({{\hat \phi }_n}\) converges to a functional of stable process with convergence rate n. Further, we show that if lim_n→∞k_n(1 − ϕ_n) = c for a positive constant c, then \({k_n}({\hat \phi _n} - \phi )\) converges to a functional of two stable variables with tail index α/2, which means that ϕ_n can be estimated consistently only when k_n → ∞.