INFERENCE FOR UNSTABLE LONG-MEMORY PROCESSES WITH APPLICATIONS TO FRACTIONAL UNIT-ROOT AUTOREGRESSIONS

An autoregressive time series is said to be unstable if all of its cha racteristic roots lie on or outside the unit circle, with at least one on the unit circle. This paper aims at developing asymptotic inferent ial schemes for an unstable autoregressive model generated by long-mem ory innovations. This setting allows both nonstationarity and long-mem ory behavior in the modeling of low-frequency phenomena. In developing these procedures, a novel weak convergence result for a sequence of l ong-memory random variables to a stochastic integral of fractional Bro wnian motions is established. Results of this paper can be used to tes t for unit roots in a fractional AR model.