Gradual Permanent Impact

In Time Series , the gradual permanent impact pattern implies that the increase or decrease due to the intervention is gradual, and that the final permanent impact becomes evident only after some time. This type of intervention can be summarized by the expression:
Impact t = * Impact t-1 +
(for all t time of impact, else = 0).
Note that this impact pattern is defined by the two parameters (delta) and (omega). If is near 0 (zero), then the final permanent amount of impact will be evident after only a few more observations; if is close to 1, then the final permanent amount of impact will only be evident after many more observations. As long as the d parameter is greater than 0 and less than 1 (the bounds of system stability), the impact will be gradual and result in an asymptotic change (shift) in the overall mean by the quantity:
Asymptotic change in level = /( 1- )