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# 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