Nearly nonstationary processes under infinite variance GARCH noises

• Published in: Applied Mathematics-A Journal of Chinese Universities Vol. 37; no. 2; pp. 246 - 257
• Main Authors: Zhang, Rong-mao, Liu, Qi-meng, Shi, Jian-hua
• Format: Journal Article
• Language: English
• Published: Singapore Springer Nature Singapore 2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Autoregressive processes; Convergence; Mathematical analysis; Mathematics; Mathematics and Statistics; heavy-tailed; 62M10; GARCH noises; 60F17; 62E20; stable processes and unit-root
• ISSN: 1005-1031; 1993-0445
• DOI: 10.1007/s11766-022-4442-5

Let Y t be an autoregressive process with order one, i.e., Y t = μ + ϕ n Y t −1 + ε t , where [ ε t ] is a heavy tailed general GARCH noise with tail index α . Let ϕ ^ 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 ϕ ^ n is inconsistent when Y t is stationary (i.e., ϕ n ≡ ϕ < 1), however, Chan and Zhang (2010) showed that ϕ ^ 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 ϕ ^ 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 ( ϕ ^ n − ϕ ) 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 → ∞.