# Solved – How to write a mathematical equation for seasonal ARIMA (0,0,1) x (2,1,2) period 12

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.

Contents

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.

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