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This paper explores seasonal and long-memory time series properties by usingthe seasonal fractional ARIMA model when the seasonal data has one and twoseasonal periods and short-memory counterparts. The stationarity andinvertibility parameter conditions are established for the model studied. Toestimate 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 seasonalfractional long-memory parameters. The asymptotic properties are establishedand the accuracy of the method is investigated through Monte Carlo experiments.The good performance of the estimator indicates that it can be an alternativecompetitive procedure to estimate seasonal long-memory time series data.Artificial and PM10 series were considered as examples of applications of theproposed estimation method.

Seasonal fractional long-memory processes. A semiparametric estimation approach