Seasonal fractional long-memory processes. A semiparametric estimation approach

This paper explores seasonal and long-memory time series properties by using the seasonal fractional ARIMA model when the seasonal data has one and two seasonal periods and short-memory counterparts. The stationarity and invertibility parameter conditions are established for the model studied. To estimate the memory parameters, the method given in Reisen, Rodrigues and Palma (2006 a,b) is generalized here to deal with a time series with two seasonal fractional long-memory parameters. The asymptotic properties are established and the accuracy of the method is investigated through Monte Carlo experiments. The good performance of the estimator indicates that it can be an alternative competitive procedure to estimate seasonal long-memory time series data. Artificial and PM10 series were considered as examples of applications of the proposed estimation method.

💡 Research Summary

This paper addresses a notable gap in the analysis of time‑series that exhibit both long‑memory behavior and multiple seasonal cycles. While the conventional ARFIMA framework captures fractional integration, it fails to accommodate more than one seasonal period, a situation common in many environmental, economic, and engineering data sets. The authors therefore propose an extended Seasonal Fractional ARIMA (SARFIMA) model that incorporates two distinct seasonal frequencies, denoted (s_{1}) and (s_{2}). The model can be written as

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