Quantile inference for near-integrated autoregressive time series under infinite variance and strong dependence

Consider a near-integrated time series driven by a heavy-tailed and long-memory noise ε t = ∑ j = 0 ∞ c j η t − j , where { η j } is a sequence of i . i . d random variables belonging to the domain of attraction of a stable law with index α . The limit distribution of the quantile estimate and the semi-parametric estimate of the autoregressive parameters with long- and short-range dependent innovations are established in this paper. Under certain regularity conditions, it is shown that when the noise is short-memory, the quantile estimate converges weakly to a mixture of a Gaussian process and a stable Ornstein–Uhlenbeck (O–U) process while the semi-parametric estimate converges weakly to a normal distribution. But when the noise is long-memory, the limit distribution of the quantile estimate becomes substantially different. Depending on the range of the stable index α , the limit distribution is shown to be either a functional of a fractional stable O–U process or a mixture of a stable process and a stable O–U process. These results indicate that although the quantile estimate tends to be more efficient for infinite variance time series, extreme caution should be exercised in the long-memory situation.