In this article, we presented a variety of modelling approaches for the El Niño Southern Oscillation (ENSO) time series data trying to capture two features that are present in the data, in particular the persistence and the seasonal structure. For this purpose, we use long memory models based on fractional integration. The results indicate that the four ENSO series examined (El Niño 1–2; El Niño 3, El Niño 4 and El Niño 3.4) can be described in terms of a seasonal I(d) model with an order of integration higher than 0 and smaller than 1 in all four cases. The lowest degree of persistence is found in the series corresponding to El Niño 1.2, with an order of seasonal integration of 0.39. This seasonal fractional differencing parameter, however, is found to be higher than 0.5 (and thus non-stationary) for the remaining three series.

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