Estimation of Long Memory in the Presence of a Smooth Nonparametric Trend

Abstract

We consider semiparametric estimation of the long-memory parameter of a stationary process in the presence of an additive nonparametric mean function. We use a semiparametric Whittle-type estimator, applied to the tapered, differenced series. Because the mean function is not necessarily a polynomial of finite order, no amount of differencing will completely remove the mean. We establish a central limit theorem for the estimator of the memory parameter, assuming that a slowly increasing number of low frequencies are trimmed from the estimator's objective function. We find in simulations that tapering and trimming, applied either separately or together, are essential for the good performance of the estimator in practice. In our simulation study, we also compare the proposed estimator of the long-memory parameter with a direct estimator obtained from the raw data without differencing or tapering, and finally we study the question of feasible inference for the regression function. We find that the proposed estimator of the long-memory parameter is potentially far less biased than the direct estimator, and consequently that the proposed estimator may lead to more accurate inference on the regression function.