# Airline Model

The airline model is a special, but often used, case of multiplicative SARIMA model.

1. 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 coefficient of first lagged innovation
• $$\Theta$$ is the coefficient of s-lagged innovation.
• $$\left [a_t\right ]$$ is the innovations time series.
Remarks
1. $$\left[Y_t\right]$$ is not a stationary process, but the differenced time series $$\left[Y_t\right]$$ is.
2. After we difference $$Y_t$$ (i.e. $$Z_t$$, the airline model is simplified to a special MA(s) model
3. The airline model has 5 parameters: $$\mu\,,\sigma\,,s\,,\theta\,,\Theta$$
Requirements