The research paper here shows that for every inch of rain above normal levels, the Democratic party's margin in elections decreases by about 1 percentage point. What are some ways that the Republicans and Democrats can exploit this for electoral benefit?
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They can't exploit this because correlation does not imply causation.
When you look at enough different sets of data, you can find the strangest correlations. But that doesn't necessarily mean that one of them is causally linked to the other.
It can also mean that both figures actually depend on the same third figure (so artificially influencing one won't affect the other). But it can also mean that it is just a complete coincidence that the numbers line up until now and there is no point in using knowledge about the development of one to predict future developments of the other.
In this case I expect the latter.