584

**Inference for Unstable Long-Memory
Processes with Applications to Fractional
Unit Root Autoregressions**

**Ngai Hang Chan and Norma Terrin **

### Abstract:

*An autoregressive time series is said to be unstable if all of its
characteristic roots lie on or outside the unit circle,
with at least one on the unit circle. This paper
aims at developing asymptotic inferential schemes for an unstable
autoregressive model generated by long-memory innovations. This
setting allows both nonstationarity and long-memory behavior in the
modeling of
low frequency phenomena. In developing these procedures, a novel
weak convergence result for a sequence of long-memory random variables
to a stochastic integral of fractional Brownian motions is
established. Results of this paper can be used
to test for unit roots in a fractional AR model.
*
*Keywords and phrases:* Fractional Brownian motion, least
squares, long-range dependence, stochastic integral, unstable.

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