Large-sample properties of the periodogram estimator of seasonally persistent processes

Seasonally persistent models were first introduced by Andel (1986) and Gray et al. (1989) to extend autoregressive moving-average and fractionally differenced models and to encompass long-memory quasi-periodic behaviour. These models are, for certain ranges of parameters, stationary, and we prove here that the behaviour of the periodogram and other tapered estimators cannot be simply extended from the work of Kunsch (1986) and Hurvich & Beltrao (1993) on long memory induced by a pole at the origin. We demonstrate that potentially large both positive and negative bias can be found from the same value of the long-memory parameter, and that the new distribution can be easily written down in the case of Gaussian processes. We also consider using both the cosine taper and the sine taper. The extended least squares estimator is also considered in this context.