Limit Theory for Moderate Deviations from a Unit Root Under Innovations with a Possibly Infinite Variance

Abstract

An asymptotic theory was given by Phillips and Magdalinos (J Econom 136(1):115–130, 2007) for autoregressive time series Y _t  = ρY _t−1 + u _t , t = 1,...,n, with ρ = ρ _n  = 1 + c/k _n , under (2 + δ)-order moment condition for the innovations u _t , where δ > 0 when c < 0 and δ = 0 when c > 0, {u _t } is a sequence of independent and identically distributed random variables, and (k _n ) _{n ∈ ℕ} is a deterministic sequence increasing to infinity at a rate slower than n. In the present paper, we established similar results when the truncated second moment of the innovations $l(x)=\textsf{E} [u_1^2I\{|u_1|\le x\}]$ is a slowly varying function at ∞, which may tend to infinity as x → ∞. More interestingly, we proposed a new pivotal for the coefficient ρ in case c < 0, and formally proved that it has an asymptotically standard normal distribution and is nuisance parameter free. Our numerical simulation results show that the distribution of this pivotal approximates the standard normal distribution well under normal innovations.