I am a self learner, and I am studying time series analysis.

I came through the fact that ARIMA can be used to model a time series which is not stationary (Integrated ARMA model). The non stationarity taken in to account in an ARIMA model is the nonstationarity that can be removed by differencing.

I am wondering whether a seasonality is the type of nonstationarity that can be removed by differencing?

If yes, so for a non seasonal data, we may use ARIMA modeling, so why we have SARIMA modeling ?

If no, how seasonality can be removed in order to make data stationary?


Differencing in ARIMA is successful at removing trends. For removing seasonality, seasonal differencing may be applied. The differencing in ARIMA is like taking derivative, but in discrete time. If your trend is linear, once differentiated, series become (hopefully) stationary. If the trend is second order, i.e. parabolic, then twice differencing it will stationarize the series. But, none of these capture seasonality well. Consider taking derivative of a cosine; it'll be a sine. You won't get rid of it via differentiating/differencing.

Seasonality removal can be addressed in several ways. One is simply seasonal differencing. Another way is to deal with it via a complete SARIMA model. Or, STL decomposition.

  • $\begingroup$ Thank you very much for the very clear answer. $\endgroup$ – Nizar Jun 15 at 16:36
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    $\begingroup$ Just a warning: differencing a trend-stationary series will produce an integrated moving-average process, which is why deterministic trends are not handled by differencing but rather by including a variable representing the deterministic time trend ($t$, $t^2$ and such). If you want to learn more about the problem, check the keyword overdifferencing. @Nizar $\endgroup$ – Richard Hardy Jun 16 at 7:35

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