# How To Write Seasonal ARIMA model mathematically

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?

• You might like to take a look at answers to these two questions: Q1 Q2, which - while they individually have some repeated values - because they repeat different things, between them should give you a clear idea of the general approach. – Glen_b Jan 14 '14 at 1:52

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.