Nearly nonstationary processes under infinite variance GARCH noises

Abstract

AbstractLet Yt be an autoregressive process with order one, i.e., Yt = μ + ϕnYt−1 + εt, where [εt] is a heavy tailed general GARCH noise with tail index α. Let $${{\hat \phi }_n}$$ ϕ ^ n be the least squares estimator (LSE) of ϕn For μ = 0 and α < 2, it is shown by Zhang and Ling (2015) that $${{\hat \phi }_n}$$ ϕ ^ n is inconsistent when Yt is stationary (i.e., ϕn ≡ ϕ < 1), however, Chan and Zhang (2010) showed that $${{\hat \phi }_n}$$ ϕ ^ n is still consistent with convergence rate n when Yt is a unit-root process (i.e., ϕn = 1) and [εt] 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 ϕn = 1 − c/n, then $${{\hat \phi }_n}$$ ϕ ^ n converges to a functional of stable process with convergence rate n. Further, we show that if limn→∞kn(1 − ϕn) = c for a positive constant c, then $${k_n}({\hat \phi _n} - \phi )$$ k n ( ϕ ^ n − ϕ ) converges to a functional of two stable variables with tail index α/2, which means that ϕn can be estimated consistently only when kn → ∞.