Semiparametric inference in seasonal and cyclical long memory processes

Several semiparametric estimates of the memory parameter in standard long memory time series are now available. They consider only local behaviour of the spectrum near zero frequency about which the spectrum is symmetric. However long-range dependence can appear as a spectral pole at any Nyqvist frequency (reflecting seasonal or cyclical long-memory), where the spectrum need display no such symmetry We introduce Seasonal/Cyclical Asymmetric Long Memory (SCALM) processes that allow differing rates of increase on either side of such a pole. To estimate the two consequent memory parameters we extend two semiparametric methods that were proposed for the standard case of a spectrum diverging at the origin, namely the log-periodogram and Gaussian or Whittle methods. We also provide three tests of symmetry. Monte Carlo analysis of finite sample behaviour and an empirical application to UK inflation data are included. Our models and methods allow also for the possibility of negative dependence, described by a possibly asymmetric spectral zero.