Semiparametric Inference in Seasonal and Cyclical Long Memory Processes

Abstract

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 longrange dependence can appear as a spectral pole at any Nyqvist frequency (re¯ecting 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 logperiodogram and Gaussian or Whittle methods. We also provide three tests of symmetry. Monte Carlo analysis of ®nite sample behaviour and an empirical application to UK in¯ation data are included. Our models and methods allow also for the possibility of negative dependence, described by a possibly asymmetric spectral zero.

On a class of minimum contrast estimators for Gegenbauer random fields

“…Very detailed discussions about estimation of parameters of seasonal/cyclical long memory time series were given in Arteche and Robinson (2000), Giraitis et al (2001), Ferrara and Guégan (2001), and references therein. It was shown that the case of spectral singularities outside the origin is much more difficult comparing to the situation of a spectral pole at 0.…”

“…Also in some applications the spectral density singularity location can be estimated in advance. Various methods, including semiparametric, wavelet, and pseudo-maximum likelihood techniques, of the estimation of a singularity location were discussed by, for example, Arteche and Robinson (2000), Giraitis et al (2001), and Ferrara and Guégan (2001).…”

Fractional integration and cointegration in US financial time series data

“…Note that we adopt the Type II definition of fractional integration (see Marinucci and Robinson 1999). Finally, one should also note that fractional integration may also occur at some other frequencies away from 0, as in the case of seasonal/cyclical models (see Arteche 2002;Arteche and Robinson 2000;Hassler et al 2009 among others).…”

Asymptotic behavior of functionals of cyclic long-range dependent random fields

“…In particular, models with singularities at non-zero frequencies are of great importance in time series analysis. Many time series show cyclical/seasonal evolutions with peaks in spectral densities whose locations define periods of the cycles, see [1,2]. Among the extensive literature on long-range dependence, relatively few publications are devoted to cyclical long-range dependent processes.…”