Intervention analysis in time-series regression with seasonal ARIMA errors 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
 A: 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
$$
Y_t=\frac{\omega S_t}{\nabla\nabla_{12}(1-\delta B)}+\frac{\omega P_t}{\nabla\nabla_{12}(1-\delta B)}+\frac{\Theta(B)}{\nabla\nabla_{12}\Phi(B)} \eta_t.
$$
A: If you wish to estimate the model that you specified you would specify regular and seasonal differencing on Y and provide the two doubly integrated intervention series. It appears you are doing intervention modelling and not intervention detection prior to intervention modelling. The differencing operator in the noise component will essentially return your two indicators to the required status.
A: I don't know enough to totally parse your question, but I believe the usual practice is to first do a regression with indicator variables $S_t$ and $P_t$, then do your ARIMA on the residuals.
