  We make an observation that facilitates exact likelihood-based inference for
the parameters of the popular ARFIMA model without requiring stationarity by
allowing the upper bound $\bar{d}$ for the memory parameter $d$ to exceed
$0.5$. We observe that estimating the parameters of a single non-stationary
ARFIMA model is equivalent to estimating the parameters of a sequence of
stationary ARFIMA models, which allows for the use of existing methods for
evaluating the likelihood for an invertible and stationary ARFIMA model. This
enables improved inference because many standard methods perform poorly when
estimates are close to the boundary of the parameter space. It also allows us
to leverage the wealth of likelihood approximations that have been introduced
for estimating the parameters of a stationary process. We explore how
estimation of the memory parameter $d$ depends on the upper bound $\bar{d}$ and
introduce adaptive procedures for choosing $\bar{d}$. Via simulations, we
examine the performance of our adaptive procedures for estimating the memory
parameter when the true value is as large as $2.5$. Our adaptive procedures
estimate the memory parameter well, can be used to obtain confidence intervals
for the memory parameter that achieve nominal coverage rates, and perform
favorably relative to existing alternatives.
