This paper considers nonstationary fractional autoregressive
integrated moving-average (p,d,q) models
with the fractionally differencing parameter d ∈
(− 1/2,1/2) and the autoregression function with roots on or
outside the unit circle. Asymptotic inference is based on the
conditional sum of squares (CSS) estimation. Under some suitable
conditions, it is shown that CSS estimators exist and are consistent.
The asymptotic distributions of CSS estimators are expressed as
functions of stochastic integrals of usual Brownian motions. Unlike
results available in the literature, the limiting distributions
of various unit roots are independent of the parameter d over
the entire range d ∈ (− 1/2,1/2). This allows
the unit roots and d to be estimated and tested separately
without loss of efficiency. Our results are quite different from the
current asymptotic theories on nonstationary long memory time series.
The finite sample properties are examined for two special cases through
simulations.