On parameter estimation for locally stationary long-memory processes

We consider parameter estimation for time-dependent locally stationary long-memory processes. The asymptotic distribution of an estimator based on the local infinite autoregressive representation is derived, and asymptotic formulas for the mean squared error of the estimator, and the asymptotically optimal bandwidth are obtained. In spite of long memory, the optimal bandwidth turns out to be of the order n−1/5 and inversely proportional to the square of the second derivative of d. In this sense, local estimation of d is comparable to regression smoothing with iid residuals.