Data Driven Order Selection for Projection Estimator of the Spectral Density of Time Series with Long Range Dependence

Fractional exponential (FEXP) models have been introduced by Robinson (1991) and Beran (1993) to model the spectral density of a covariance stationary long-range dependent process. In this class of models, the spectral density f ( x ) of the process is decomposed as f ( x ) = |1 − exp( ix )|^{−2 d} f _*( x ), where f _*( x ) accounts for the short-memory component. In this contribution, FEXP models are used to construct semi-parametric estimates of the fractional differencing coefficient and of the spectral density, by considering an infinite Fourier series expansion of log f _*( x ). A data-driven order selection procedure, adapted from the Mallows' C _p procedure, is proposed to determine the order of truncation. The optimality of the data-driven procedure is established, under mild assumptions on the short-memory component f _*( x ). A limited Monte-Carlo experiment is presented to support our claims.