We review concepts in frequency domain inference for time series data, in particular the periodogram and its statistical properties, persistence, (long-memory) and the Whittle likelihood. We study distributional results for seasonally persistent processes (where the spectral density function (SDF) of the underlying process has a singularity at a frequency away from zero) that facilitate likelihood construction and estimation. Full likelihood procedures based on these results yield estimators for key parameters in the SDF that have identical asymptotic properties to non likelihood-based estimators, but appear to have superior small sample performance. Simulated and real data examples are given.
This is joint work with Sofia Olhede and Emma McCoy, Imperial College London.