The airline model is a special, but often used, case of multiplicative ARIMA model. For a given seasonality length (s), the airline model is defined by four(4) parameters: $\mu$, $\sigma$, $\theta$, and $\Theta$).
- $s$ is the length of seasonality.
- $\mu$ is the model mean.
- $\theta$ is the coefficient of first lagged innovation.
- $\Theta$ is the coefficient of s-lagged innovation.
- $\left [a_t\right ]$ is the innovations time series.
- The AirLine model can be viewed as a "cascade" of two models:
- The first model is non-stationary: $$(1-L^s)(1-L)Y_t = Z_t$$
- The second model is wide-sense stationary: $$Z_t = \mu + (1-\theta L)(1-\Theta L^s)a_t$$
- The stationary component is a special form of the moving average model.
- The airline model of order ($s$) has 4 free parameters: $\mu\,,\sigma\,\,,\theta\,,\Theta$