Models of costly voting give rise to strategic turnout: in a district in which party A has a big advantage, supporters of party A will have low turnout in equilibrium in order to make the election close. That’s because only when the election is close will voters have an incentive to turnout and vote, which is costly.
Looking at elections data it is hard to identify strategic turnout. Low turnout is perfectly consistent with non-strategic voters who just have high costs of voting.
Redistricting offers an interesting source of variation that could help. Suppose that a state has just undergone redistricting and a town has been moved from a district with a large majority for one party into a more competitive district. Non-strategic voters in that town will not change their behavior.
But strategic voters will have different incentives in the new district. In particular we should see an increase in turnout among voters in the town that is new to the district. And this increase in turnout should be larger than any change in turnout observed for voters who remained in the district before and after redistricting.
There are probably a slew of testable implications that could be derived from models of strategic turnout based on whether the new district is more or less competitive than the old one, whether the stronger party is the same or different from the stronger party in the old district, and whether the town leans toward or against the stronger party in the new district.