Nearly nonstationary processes under infinite variance GARCH noises

Let Y-t be an autoregressive process with order one, i.e., Y-t = mu+phi(n) Yt-1 + epsilon(t), where {epsilon(t)} is a heavy tailed general GARCH noise with tail index alpha. Let (phi) over cap (n) be the least squares estimator (LSE) of phi(n). For mu = 0 and alpha < 2, it is shown by Zhang and Ling (2015) that <(phi)over cap>(n) is inconsistent when Y-t is stationary (i.e., (phi) over cap (n) equivalent to phi < 1), however, Chan and Zhang (2010) showed that <(phi)over cap>(n) is still consistent with convergence rate n when Y t is a unit-root process (i.e., phi(n) = 1) and epsilon(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. be estimated consistently by the LSE? We show that when phi(n) = 1 c/n, then (phi) over cap (n) converges to a functional of stable process with convergence rate n. Further, we show that if lim n ->infinity k(n)(1 - phi(n)) = c for a positive constant c, then k(n)((phi) over cap (n) - phi) converges to a functional of two stable variables with tail index alpha/2, which means that phi(n) can be estimated consistently only when k(n) -> infinity.