Interrupted time series where the impact is not immediate


No cake for spunky
We are analyzing a program change based on various interval dependent variables over time. We are going to use interrupted time series and segmented regression (both) the intervention occurred in different units at different times. We were going to use this as a control, to see if the change (if it occurred) could logically be tied to the intervention. If it occurred in the unit that got the intervention, but not the one that did not, it was more evidence the change was tied to the intervention.

The problem is that the change in the dependent variable is likely to not occur immediately, but gradually over time. So I am not sure how we can tell in these gradual type changes if the change was tied to the intervention or not. I know theoretical ways to do this, in ARIMA for example, but none seem realistic to do in practice.