Model Selection in Nonstationary Weak FARIMA Models

We study the problem of order selection for Fractional AutoRegressive Integrated Moving Average (FARIMA) models driven by uncorrelated innovations admitting α-mixing dependence. The innovation process is allowed to exhibit general forms of temporal dependence while remaining free of serial correlation. Within this framework, we analyze the asymptotic behavior of the Bayesian Information Criterion (BIC) and a quasi-Bayesian Information Criterion (QBIC) based on quasi-maximum likelihood estimation. We establish the consistency of both criteria for selecting the autoregressive and moving average orders of FARIMA models under mild regularity conditions. These results extend existing consistency properties obtained for FARIMA models with independent or martingale difference innovations and provide, to the best of our knowledge, the first theoretical justification of BIC-type order selection procedures for long-memory models induced by uncorrelated yet generally dependent noise.