Multiplicative Seasonal ARMA Model

Definition

A Multiplicative Seasonal ARMA Model, denoted as $\text{ARMA}(p,q)\times (P,Q{)}_s$, is a model that combines non-seasonal and seasonal components. It has a seasonal period $s$, AR characteristic polynomials $\varphi (x)$ (non-seasonal) and $\Phi (x)$ (seasonal), and MA characteristic polynomials $\theta (x)$ (non-seasonal) and $\Theta (x)$ (seasonal).

The model is defined by:

$$\Phi (B^s)\varphi (B)Y_t=\Theta (B^s)\theta (B)e_t$$

Interpretation

Sometimes a time series exhibits both short-term (non-seasonal) and long-term (seasonal) autocorrelation. This model captures both by multiplying the operators. It’s more parsimonious than a general ARMA model with many parameters.