Seasonal and cyclical long-memory in time series
This thesis deals with the issue of persistence, focusing on economic time series, and extending the subject to seasonal and cyclical long memory time series. Such processes are defined. In the frequency domain they are characterized by spectral poles/zeros at some frequency ω between 0 and π. First we review some of the work done to date on seasonality and long memory, and we focus on research that try to link both issues. One of the limitations of the existing work is the imposition of asymptotic symmetry in the spectral density around ω. We describe some processes that allow for spectral asymmetry around the frequency ω where the pole/zero occurs. They are naturally described in the frequency domain, and they imply two possibly different persistence parameters describing the behaviour of the spectrum to the right and left of ω. Two semiparametric methods of estimating the persistence parameters in the frequency domain, which have been proposed for the symmetric case ω = 0 and are based on a partial knowledge of the spectral density around ω, are extended to ω ≠ 0 and their asymptotic properties are analysed. These are the log-periodogram regression and the local Whittle or Gaussian semiparametric estimates. Their performance in finite samples is studied via Monte Carlo analysis. Some semiparametric Wald and LM type tests on the symmetry of the spectral density at ↓ and on the equality of persistence parameters at different frequencies are proposed, showing their good asymptotic properties. Their performance in finite samples is analysed through a small Monte Carlo study. All these techniques are applied to a monthly UK inflation series from January 1915 to April 1996, where we test not only the symmetry of the spectral poles but also the equality of persistence parameters across seasonal frequencies. Finally some concluding remarks and possible extensions are suggested.