What is the license for the data shared on Jounral of Political Economics I don't think Journal of Political Economics has suggested a specific license. However, the journal did say that the data policy is adopted from American Economic Review (AER). AER did suggest a license of Creative Commons Attribution International 4.0 (CC-BY). 

That license allows reuse of data without asking permission (I just have to cite it). So my question is, may I also use the data released by JPE without asking permission?
A "suggested" license isn't necessarily imposed on authors. Especially one that is second hand. Ask the authors if you can't otherwise get a definitive answer. 

Some journals might impose a license, but you should be sure, rather than make assumptions.