Intervention analysis in Box-Jenkins framework crosspoinds to time-series regression with arma errors if the noise is stationary or arima errors if the noise is non-stationary.
For a seasonal time series data with increasing trend, the noise model can be express as
$$
N_t = \frac{\Theta(B)}{(1-B)(1-B^{12})\Phi(B)} \eta_t
$$
If there is a step $S_t$ (0 before intervention and 1 after intervention) and a pulse $P_t$ (1 at intervention and 0 elsewhere) interventions, the model then can be expressed as
$$
Y_t=\beta_1S_t+\beta_2P_t+\frac{\Theta(B)}{(1-B)(1-B^{12})\Phi(B)} \eta_t
$$
Also because there may different responses to the interventions, say graduate change in level is by $\frac{\omega S_t}{1-\delta B}$ or decayed responses $\frac{\omega P_t}{1-\delta B}$.
$$
Y_t=\frac{\omega S_t}{1-\delta B}+\frac{\omega P_t}{1-\delta B}+\frac{\Theta(B)}{(1-B)(1-B^{12})\Phi(B)} \eta_t
$$
Therefore my question is:
if the data is seasonal time series, then in the practice, does it mean we need to perform difference $(1-B)(1-B^{12})S_t$ and $(1-B)(1-B^{12})P_t$ along with $(1-B)(1-B^{12})Y_t$ anyways when consider those interventions?
Thanks and Regards
Best Answer
The differencing implied by the denominator of your error term must be applied to $Y_t$, $S_t$ and $P_t$. That is, your model is equivalent to $$ \nabla\nabla_{12}Y_t=\frac{\omega \nabla\nabla_{12}S_t}{1-\delta B}+\frac{\omega \nabla\nabla_{12}P_t}{1-\delta B}+\frac{\Theta(B)}{\Phi(B)} \eta_t, $$ where $\nabla\nabla_{12} = (1-B)(1-B^{12})$. This is a transfer function model with ARMA errors which is how it would actually be estimated.
If you intended that the pulse and step apply to the differenced $Y$ series, then you need to doubly integrate $S$ and $P$ in the model (as suggested by @IrishStat). That is $$ Y_t=\frac{\omega S_t}{\nabla\nabla_{12}(1-\delta B)}+\frac{\omega P_t}{\nabla\nabla_{12}(1-\delta B)}+\frac{\Theta(B)}{\nabla\nabla_{12}\Phi(B)} \eta_t. $$