Correct. They only work when the background trend is extremely stable and you have no other changes at the time of the intervention. This seems quite rare in the real world.
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Exactly, and the step change depends on whether you allow a slope change (and vice versa), if you assume the long-term trend is linear, etc. You can't just look at the data points and see, you have to make assumptions and the results depend strongly on those assumptions
If you had an intervention rolled out across 20 different populations at different times, the degree to which those assumptions are wrong would probably just dilute your effect (still bad but I think you're less likely to find random spurious effects)
If you had the intervention rolled out over 20 populations at different times then you would be best doing a multiple groups and multiple time periods DiD. Like @epiellie.bsky.social did for COVID masks in schools!
You have to model the trend correctly (linear or otherwise) or else your estimate will certainly be wrong. You then ASSUME that trend continues. That is indeed a big assumption.
RE step vs slope changes, again this is a modelling assumption, based on how you believe the effect should be acting.
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RE step vs slope changes, again this is a modelling assumption, based on how you believe the effect should be acting.