2 edited title
| link

Intervention analysis in Timetime-Seriesseries regression with seasonal arimaARIMA errors

source | link

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