Wow, the OP or mods deleted my comment critical of the paper. Anyway, I just had a look at the paper itself. You'd think this would be a diff-in-diff paper, comparing trade between two countries as the ratings rise and fall. You'd be wrong. The benchmark regression table includes no country-pair fixed effects however. Neither does the next table or the one after that.
He's picking up a cross-sectional correlation in approval ratings, with millions of observations and very few controls.