Seasonal Autoregressive Integrated Moving Average (SARIMA) Model

The SARIMA model is an extension of the ARIMA model, often used when we suspect a model may have a seasonal effect.

By definition, the seasonal auto-regressive integrated moving average - SARIMA(p, d, q)(P, D, Q)s - process is a multiplicative of two ARMA processes of the differenced time series.$$(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}})(1-L)^d (1-L^s)^D x_t = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t$$ $$y_t = (1-L)^d (1-L^s)^D $$

Where:

• $x_t$ is the original non-stationary output at time $t$.
• $y_y$ is the differenced (stationary) output at time $t$.
• $d$ is the non-seasonal integration order of the time series.
• $p$ is the order of the non-seasonal AR component.
• $P$ is the order of the seasonal AR component.
• $q$ is the order of the non-seasonal MA component.
• $Q$ is the order of the seasonal MA component.
• $s$ is the seasonal length.
• $D$ is the seasonal integration order of the time series.
• $a_t$ is the innovation, shock, or error term at time $t$.
• $\{a_t\}$ time series observations are independent and identically distributed (i.e., $i.i.d$) and follow a Gaussian distribution (i.e., $\Phi(0,\sigma^2)$).

Assuming $y_t$ follows a stationary process with a long run mean of $\mu$, then taking the expectation from both sides, we can express $\phi_o$ as follows:$$\phi_o = (1-\phi_1-\phi_2-\cdots-\phi_p)(1-\Phi_1-\Phi_2-\cdots-\Phi_P)$$

Thus, the SARIMA(p, d, q)(P, D, Q)s process can now be expressed as:$$(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}}) (y_t -\mu) = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t$$ $$z_t=y_t-\mu$$ $$(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}}) z_t = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t$$

In sum, $z_t$ is the differenced signal after we subtract its long-run average.

Remarks

1. The variance of the shocks is constant or time-invariant.
2. The order of the seasonal or non-seasonal AR (or MA) component is solely determined by the order of the last lagged variable with a non-zero coefficient. In principle, you can have fewer parameters than the order of the component.
   Consider the following SARIMA(0,1,1) (0,1,1)12 process:$$(1-L)(1-L^{12})x_t-\mu = (1+\theta L)(1+\Theta L^{12})a_t$$

Related Links

• Wikipedia - Autoregressive moving average model.

References

• James Douglas Hamilton; Time Series Analysis, Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896.
• Tsay, Ruey S.; Analysis of Financial Time Series, John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0-471-690740.