Frequency Domain Local Bootstrap in long memory time series.

Bootstrap techniques in the frequency domain have been proved to be effective instruments to approximate the distribution of many statistics of weakly dependent (short memory) series. However their validity with long memory has not been analysed yet. This paper proposes a Frequency Domain Local Bootstrap (FDLB) based on resampling a locally studentised version of the periodogram in a neighbourhood of the frequency of interest. A bound of the Mallows distance between the distributions of the original and bootstrap periodograms is offered for stationary and non-stationary long memory series. This result is in turn used to justify the use of FDLB for some statistics such as the average periodogram or the Local Whittle (LW) estimator. Finally, the finite sample behaviour of the FDLB in the LW estimator is analysed in a Monte Carlo, comparing its performance with rival alternatives.