Frequency domain local bootstrap in short and long memory time series

Bootstrap methods in the frequency domain are effective instruments to approximate the distribution of many statistics of weakly dependent (short memory) series but their validity with long memory remains mostly unsolved. This article proposes a frequency domain local bootstrap (FDLB) based on resampling a locally studentized version of the periodogram in a neighborhood of the frequency of interest. A bound of the Mallows distance between the distributions of the periodogram and its FDLB bootstrap counterpart is offered for anti-persistent, weakly dependent, stationary, and non stationary long memory series. This result is in turn used to justify the use of the FDLB for some statistics that are weighted averages of periodogram ordinates. Finally, the validity of the FDLB to estimate the distribution of the local Whittle estimator is proved and its finite sample behavior analyzed in a Monte Carlo and in an empirical application, comparing its performance with rival alternatives.