Model selection for broadband semiparametric estimation of long memory in time series

We study the properties of Mallows' CL criterion for selecting a fractional exponential (FEXP) model for a Gaussian long-memory time series. The aim is to minimize the mean squared error of a corresponding regression estimator (d) over cap (FEXP) of the memory parameter, d. Under conditions which do not require that the data were actually generated bay a FEXP model, it is known that the mean squared error MSE = E[(d) over cap (FEXP) - d](2) can converge to zero as fast as (log n)/n, where n is the sample size, assuming that the number of parameters grows slowly with n in a deterministic fashion. Here, we suppose that the number of parameters in the FEXP model is chosen so as to minimize a local version of CL, restricted to frequencies in a neighborhood of zero. We show that, under appropriate conditions, the expected value of the local CL is asymptotically equivalent to MSE. A combination of theoretical and simulation results give guidance as to the choice of the degree of locality in CL.