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This article is devoted to study the effects of the S-periodical fractionaldifferencing filter $(1-L^S)^{D_t}$. To put this effect in evidence, we havederived the periodic auto-covariance functions of two distinct univariateseasonally fractionally differenced periodic models. A multivariaterepresentation of periodically correlated process is exploited to provide theexact and approximated expression auto-covariance of each models. Thedistinction between the models is clearly obvious through the expression ofperiodic autocovariance function. Besides producing different autocovariancefunctions, the two models differ in their implications. In the first model, theseasons of the multivariate series are separately fractionally integrated. Inthe second model, however, the seasons for the univariate series arefractionally co-integrated. On the simulated sample, for each models, with thesame parameters, the empirical periodic autocovariance are calculated andgraphically represented for illustrating the results and support the comparisonbetween the two models.

Two Distinct Seasonally Fractionally Differenced Periodic Processes