Estimation of the memory parameter by fitting fractionally differenced autoregressive models

Estimation of the memory parameter, d, by fitting a fractionally differenced autoregression of order p, where p approaches infinity simultaneously with the observed series length, n, is examined. Under some conditions on growth of p with respect to n and on the short-memory component, which admits an infinite autoregressive representation with coefficients a j , the estimator is shown to be p / n consistent and asymptotically normal, where p may be taken to be proportional to log n . The joint asymptotic distribution of the estimators of d and of the a j is also derived.