A GPH-Filtered Hannan--Rissanen Information Criterion for ARFIMA Order Selection

We propose a simple two-stage order selector for finite-order ARFIMA models. First, a preliminary log-periodogram estimate of the memory parameter is used to fractionally filter the data. Second, a Hannan--Rissanen residual construction is applied to the filtered series, and the autoregressive and moving-average orders are selected by a generalized information criterion over a growing candidate rectangle. The search bounds are allowed to satisfy \(P_n,Q_n\to\infty\), whereas the true orders remain fixed and finite. The penalty is allowed to be larger than the ordinary BIC penalty so that it dominates the error introduced by preliminary long-memory estimation and by the Hannan--Rissanen residual approximation. We prove a uniform residual-variance approximation over the growing rectangle and combine it with a population separation argument between the true finite ARMA representation and underfitted alternatives. The resulting generalized information-criterion selector is consistent.