How do I write a mathematical equation for seasonal ARIMA (0,0,1) x (2,1,2) period 12 [duplicate]

This question already has an answer here:

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 RodenburgOct 31 '18 at 3:32

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

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

where

• $$B$$ 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.

• 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. – Richard Hardy Oct 30 '18 at 18:40
• @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"... – Stephan Kolassa Oct 30 '18 at 21:52