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Can someone could help me write the mathematical/backshift equation for the seasonal ARIMA (0,0,1) x (2,1,2) with 12 periods? I'm confused with how to go about this. I would prefer an equation involving $B$, $Y_{t}$, $e_{t}$, $\phi$, $\Phi$, $\theta$, and $\Theta$.

This is different from other models that have previously been answered and I don’t understand the explanation in FPP2.


marked as duplicate by Ferdi, kjetil b halvorsen, Michael Chernick, Richard Hardy, Frans Rodenburg Oct 31 '18 at 3:32

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


This section and this section of the excellent free online book Forecasting: Principles and Practice (2nd ed.) by Athanasopoulos & Hyndman allow you to translate an $\text{ARIMA}(0,0,1)(2,1,2)_{12}$ model into

$$ (1-\Phi_1B^{12}-\Phi_2B^{24})(1-B^{12})y_t = (1+\theta_1B)(1+\Theta_1B^{12}+\Theta_2B^{24})\epsilon_t, $$


  • $B$ is the backshift operator, $Bx_t=x_{t-1}$
  • the term $(1-\Phi_1B^{12}-\Phi_2B^{24})$ comes from the $\text{AR}(2)_{12}$ component
  • the term $(1-B^{12})$ is the seasonal integration of order 1
  • the term $(1+\theta_1B)$ comes from the $\text{MA}(1)$ component
  • the term $(1+\Theta_1B^{12}+\Theta_2B^{24})$ comes from the $\text{MA}(2)_{12}$ component.

Depending on your software, there may be an additive intercept $c$ on the right hand side. Note also that some packages use "$-$" instead of "$+$" on the right hand ($\text{MA}$) side, so parameter estimates may have flipped signs.

  • $\begingroup$ You have said before that it is easier to write a new answer than to look for a duplicate, but... I think we have more than enough answers on this topic, so we should probably stick to marking new ones as duplicates even if it takes a little work to find the canonical thread(s). Otherwise we will end up with dozens of very similar threads. $\endgroup$ – Richard Hardy Oct 30 '18 at 18:40
  • $\begingroup$ @RichardHardy: I did search for duplicates and was sure I had already answered at least one similar question myself. And I didn't find anything. So I proceeded to answer. Next time, I guess I could wait for a day and see if someone does find a dupe. Perhaps we need a more specific tag like "ARIMA-translation"... $\endgroup$ – Stephan Kolassa Oct 30 '18 at 21:52

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