# Airline Model

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 $\Theta$).

$$(1-L^s)(1-L)Y_t = \mu + (1-\theta L)(1-\Theta L^s)a_t$$ OR $$Z_t = (1-L^s)(1-L)Y_t = \mu + (1-\theta L)(1-\Theta L^s)a_t$$ OR $$Z_t = \mu -\theta \times a_{t-1}-\Theta \times a_{t-s} +\theta\times\Theta \times a_{t-s-1}+ a_t$$

Where:

• $s$ 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.

## Remarks

1. The AirLine model can be viewed as a "cascade" of two models:
1. The first model is non-stationary: $$(1-L^s)(1-L)Y_t = Z_t$$
2. The second model is wide-sense stationary: $$Z_t = \mu + (1-\theta L)(1-\Theta L^s)a_t$$
2. The stationary component is a special form of the moving average model.
3. The airline model of order ($s$) has 4 free parameters: $\mu\,,\sigma\,\,,\theta\,,\Theta$