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. $$ (1-L^s)(1-L)Y_t = \mu + (1-\theta L)(1-\Theta L^s)a_t$$ 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.



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


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