Suppose I want to measure the effect of tornadoes on voter turnout at the county level. I suspect that counties that experience tornadoes will have lower voter turnout in the subsequent election than counties that do not, but I also expect the effect will decay the further away from Election Day the tornadoes occur (though I do not have priors regarding the nature of that decay).
Is there a modeling strategy or strategies I should use to assess whether such a decay exists and, if it does, measure it?
Edited for clarification: If I have some counties that experienced tornadoes in June, some in August, and some in October, I wouldn't expect all of those tornadoes to be equally effective at suppressing voter turnout. I would expect the October tornadoes to be most effective, because they occur right before Election Day, and June tornadoes to be least effective, because there are several months between them and Election Day. What I'm looking for is a model that doesn't assume all tornadoes should be equally effective at supressing voter turnout (e.g., a simple turnout~tornadoes regression) but rather accounts for the variation in treatment timing - sort of like a tornadoes*time interaction, but the time variable is undefined for "control" counties that did not experience tornadoes.