ARIMA SARIMA model mathematical formula

Question

I need help writing a SARIMA model I have obtained mathematically. My model is

ARIMA(1,0,4)(2,0,2) period 12.

I understand what the different parts actually mean but get very lost trying to write out the mathematical model. I have tried to follow other examples but as the models differ it makes it hard to apply it to what I have.

Answer 1

You can start from Hyndman book at the section about Seasonal ARIMA here. Once you know the mathematical definition of B operator* it will be easy to write the extended formula of your model. Your model ARIMA(1,0,4)(2,0,2) period 12 becomes something like: $y_t = c + a_1 y_{t-1} + e_t + b_1 e_{t-1} + b_2 e_{t-2} + b_3 e_{t-3} + b_4 e_{t-4} + d_1 y_{t-12} + d_2 y_{t-24} + f_1 e_{t-12} + f_2 e_{t-24}$

with $a, b, c, d, f$ the coefficients.

*For definition of B operator, see this

Answer 2

Writing the model using operator polynomials $$ \phi(B)\Phi(B^{12})y_t = \theta_0 + \theta(B)\Theta(B^{12})e_t $$ that is, for an ARIMA$(1,0,4)\times(2,0,2)_{12}$ process, the model is $$ (1-\phi_1B)(1-\Phi_1B^{12}-\Phi_2B^{24})y_t = \theta_0 + (1-\theta_1B-\theta_2B^2-\dots-\theta_4B^4)(1-\Theta_1B^{12}-\Theta_2B^{24})e_t. $$ Expanding terms and rearranging \begin{align} y_t &= \theta_0 + \phi_1 y_{t-1} \\&+ \Phi_1 y_{t-12} - \phi_1\Phi_1 y_{t-13} \\&+\Phi_2 y_{t-24}-\phi_1\Phi_2y_{t-25} \\ &+e_t-\theta_1e_{t-1}-\dots-\theta_4e_{t-4} \\ &-\Theta_1 e_{t-12}+\theta_1\Theta_1e_{t-13}+\dots+\theta_4\Theta_1 e_{t-16} \\ &-\Theta_2 e_{t-24}+\theta_1\Theta_2e_{t-25}+\dots+\theta_4\Theta_2 e_{t-28}. \end{align}