Estimation of the long memory parameter by fitting fractionally differenced autoregressive models

We examine the estimation of the memory parameter d of I(d) series, by fitting an auto-regressive AR(k) representation where k approaches infinity simultaneously with the observed series length n. Under some conditions on the growth of k with respect to n, and on the short memory component of the spectral density which admits an infinite autoregressive representation, the estimator is shown to be ?(k/n) consistent and asymptotically normal, where k may be taken to be proportional to log n. The joint asymptotic distribution of the long memory parameter and the estimated autoregressive coefficients (increasing in number) is derived.