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There's a decent amount of academic work on this. IIRC, a one percentage point increase in a tax rate (sales, property, income, etc) causes about a 3% increase in migration probability. Migration probability is quite low, though, something like 3% per year.
It probably is very messy. States with high taxes are also likely to have onerous regulations and busy body politics.
The article also points out that Alaska has a low tax rate but a lot of emigration. Apparently, it's cold and most people don't like being eaten by a bear when they go for a run.
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It is messy, but there are lots of empirical strategies to deal with it.
One of the simplest is the idea that changes explain changes: i.e. Alaska has always been cold and full of bears, so that can't be the reason people are now moving away.
For the impact of taxes, you'd look at how migration patterns change after a tax change.
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