EXACT LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION WITH UNKNOWN MEAN AND TIME TREND

Recently, Shimotsu and Phillips (2005, <italic>Annals of Statistics</italic> 33, 1890–1933) developed a new semiparametric estimator, the exact local Whittle (ELW) estimator, of the memory parameter (<italic>d</italic>) in fractionally integrated processes. The ELW estimator has been shown to be consistent, and it has the same <inline-graphic>null</inline-graphic> asymptotic distribution for all values of <italic>d</italic>, if the optimization covers an interval of width less than 9/2 and the mean of the process is known. With the intent to provide a semiparametric estimator suitable for economic data, we extend the ELW estimator so that it accommodates an unknown mean and a polynomial time trend. We show that the two-step ELW estimator, which is based on a modified ELW objective function using a tapered local Whittle estimator in the first stage, has an <inline-graphic>null</inline-graphic> asymptotic distribution for <inline-graphic>null</inline-graphic> (or <inline-graphic>null</inline-graphic> when the data have a polynomial trend). Our simulation study illustrates that the two-step ELW estimator inherits the desirable properties of the ELW estimator.