Seasonal AR(P) Model

Definition

Let:

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

A Seasonal AR(P) Model (Model Musiman AR(P)) is defined as

$$Y_t={\Phi }_1{Y}_{t-s}+{\Phi }_2{Y}_{t-2s}+\cdots +{\Phi }_P{Y}_{t-Ps}+e_t$$

with the seasonal AR characteristic polynomial equation:

$$\Phi (x)=1-{\Phi }_1{x}^s-{\Phi }_2{x}^{2s}-\cdots -{\Phi }_P{x}^{Ps}$$

Interpretation

Similar to a standard AR model, but the current value depends on past values at seasonal intervals (e.g., this January depends on last January).

Properties

• Stationarity: This model is stationary if the absolute values of the roots of $\Phi (x)=0$ are all greater than 1.
• Autocorrelation (ACF): The ACF at multiples of the seasonal period ($ks$) decays exponentially, while autocorrelation at other lags is zero. For AR(1) seasonal model, ${\rho }_{12k}={\Phi }^k$ for $k=1,2,\dots$
• Partial Autocorrelation (PACF): Cuts off after lag $P$ (which is a multiple of the seasonal period), while other lags are zero.