Two Distinct Seasonally Fractionally Differenced Periodic Processes

This article is devoted to study the e¤ects of the S-periodical fractional di¤erencing filter (1-L^S)^Dt . To put this e¤ect in evidence, we have derived the periodic auto-covariance functions of two distinct univariate seasonally fractionally di¤erenced 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 auto-covariance function. Besides producing di¤erent autocovariance functions, the two models di¤er 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.