The distribution of the low frequency periodogram ordinates of fractionally differenced series and their inclusion in two estimators of the differencing parameter

The commonly held belief that the low frequency ordinates of the periodogram of fractionally differenced models are unbiased for the spectral density, independent and exponentially distributed gives rise to some methods of estimating the differencing parameter d, in particular, Geweke and Porter-Hudak (GPH),(1983) and Janacek (1994). Hurvich and Beltrao (1993) and Hurvich and Ray (1995) gave results on the bias and expressions for the distribution under certain conditions which led them to suggest using a tapered periodogram and omitting the first ordinate for the GPH estimator. In this paper we make further empirical investigations into the distribution of both tapered and untapered low frequency periodogram ordinates for values of d in the range [-2.25,2]. We then investigate which range of ordinates should be included in the GPH and Janacek estimators for models with a variety of short term components and values of d, including nonstationary and noninvertible. Surprisingly we find that these estimators are usually better when the initial ordinates are included. We report the mean errors and root mean square errors obtained from these simulations and use these to compare the two estimators and to assess their performance.