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.

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.

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