  This article is devoted to study the effects of the S-periodical fractional
differencing filter $(1-L^S)^{D_t}$. To put this effect in evidence, we have
derived the periodic auto-covariance functions of two distinct univariate
seasonally fractionally differenced periodic models. A multivariate
representation of periodically correlated process is exploited to provide the
exact and approximated expression auto-covariance of each models. The
distinction between the models is clearly obvious through the expression of
periodic autocovariance function. Besides producing different autocovariance
functions, the two models differ in their implications. In the first model, the
seasons of the multivariate series are separately fractionally integrated. In
the second model, however, the seasons for the univariate series are
fractionally co-integrated. On the simulated sample, for each models, with the
same parameters, the empirical periodic autocovariance are calculated and
graphically represented for illustrating the results and support the comparison
between the two models.
