Likelihood Inference for Possibly Non-Stationary Processes via Adaptive Overdifferencing

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 $0.5$: estimating the parameters of a single non-stationary ARFIMA model is equivalent to estimating the parameters of a sequence of stationary ARFIMA models. This 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 $\bar{d}$. We show via simulation how our adaptive procedures estimate the memory parameter well, relative to existing alternatives, when the true value is as large as 2.5.