I'm trying to write a seasonal ARIMA model ARIMA(1,0,3)(1,2,0) period 5 mathematically but I don't seem to be able to follow what this resource is saying otexts arima
The example they use is ARIMA(1,1,1)(1,1,1)4 so it makes it very hard for me to follow what's actually going on because all the numbers are the same.
Could someone help me write out my model mathematically and possibly explain how it's done?
The general $SARIMA(p,d,q)(P,D,Q)_m$ process $X_t$ is the solution of the following equation
$$\Phi(B^m)\phi(B)\nabla_m^D\nabla^d X_t=\Theta(B^m)\theta(B)Z_t,$$
where $Z_t$ is the white noise process. $\nabla_mX_t=X_{t}-X_{t-m}$, $\nabla X_t=X_{t}-X_{t-1}$, \begin{align} \Phi(B^m)&=1-\Phi_1B^m -\dots-\Phi_PB^{PM}\\ \phi(B)&=1-\phi_1B-\dots-\phi_p B^{p}\\ \Theta(B^m)&=1-\Theta_1 B^m-\dots-\Theta_Q B^{Qm}\\ \theta(B&)=1-\theta_1B-\dots-\theta_qB^q \end{align} and $B^nX_t=X_{t-n}$.
Now take your example of $ARIMA(1,0,3)(1,2,0)_5$. This means that \begin{align} \Phi(B^m)&=\Phi(B^5)=1-\Phi_1B^5\\ \phi(B)&=1-\phi_1B\\ \Theta(B^m)&=\Theta(B^5)=1\\ \theta(B&)=1-\theta_1B-\theta_2B^2-\theta_3B^3 \end{align} So in your case $X_t$ must satisfy the equation: \begin{align} (1-\Phi_1B^5)(1-\phi_1B)\nabla_5^2X_t=(1-\theta_1B-\theta_2B^2-\theta_3B^3)Z_t \end{align} which we can rewrite as \begin{align} (1-\phi_1 B - \Phi_1 B^5 +\phi_1\Phi_1 B^{6})\nabla_5^2X_t=Z_t-\theta_1Z_{t-1}-\theta_2Z_{t-2}-\theta_3Z_{t-3}. \end{align} Now $\nabla^2_5X_t=\nabla_5(X_t-X_{t-5})=(X_t-2X_{t-5}+X_{t-10})$ and I leave the last step for the reader to complete.
