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Q&A

In a SARIMA model, does the order in which I difference the time series matter?

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In a SARIMA model $\text{ARIMA}(p,d,q)\times(P,D,Q)_S$, we can write it as $$\Phi_P(B^S)\phi(B)(1-B^S)^D(1-B)^d x_t= \delta +\Theta_Q(B^S)\theta(B)w_t$$ where the capital letters are the parameters for the multiplicative part of the model.

When I try to find the integration order ($d$ and $D$), is there a problem whether I start by differencing first with respect to $B^S$ and only then w.r.t. $B$, or the other way around?

From the polynomial representation above, I would say that we can... however, I would like to be sure.

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