Seasonal MA(Q) Model

Definition

Let:

• $Y_t$ : time series
• $e_t$ : white noise
• $s$ : seasonal period

A Seasonal MA(Q) Model is defined as

$$Y_t=e_t-{\Theta }_1{e}_{t-s}-{\Theta }_2{e}_{t-2s}-\cdots -{\Theta }_Q{e}_{t-Qs}$$

with the seasonal MA characteristic polynomial equation:

$$\Theta (x)=1-{\Theta }_1{x}^s-{\Theta }_2{x}^{2s}-\cdots -{\Theta }_Q{x}^{Qs}$$

Interpretation

It is an extension of the standard MA model, but the errors are separated by the seasonal lag $s$ instead of lag 1.

Properties

• Invertibility: This model is invertible if the absolute values of the roots of $\Theta (x)=0$ are all greater than 1.
• Autocorrelation: Like a non-seasonal MA(q) model, a seasonal MA(Q) model has non-zero autocorrelation only at lags $s,2s,\dots ,Qs$, with:

$${\rho }_{ks}\ne 0\text{ for }k=1,2,\dots ,Q$$

and zero otherwise. For a seasonal MA(1) model (where $s=12$), the autocorrelation is non-zero only for lag 12.