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.

## Notes

- the AirLine model can be viewed as a "cascade" of two models:
- The first model is a 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: $\mu\,,\sigma\,\,,\theta\,,\Theta$

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## References

- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740

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