# Two Distinct Seasonally Fractionally Differenced Periodic Processes

## TL;DR

This paper investigates two types of seasonally fractionally differenced periodic processes, deriving their autocovariance functions and comparing their implications through theoretical analysis and simulations.

## Contribution

It introduces and distinguishes two models of seasonally fractionally differenced processes, providing explicit autocovariance functions and empirical comparisons.

## Key findings

- Distinct autocovariance functions for the two models
- Different implications for seasonality and co-integration
- Empirical autocovariance supports theoretical differences

## Abstract

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.