The astute Kurt Mitman noted that Greenwood, et.al. issued a later corrective (pdf) to their earlier 2014 AER piece (ungated here), and therefore their estimates in that piece (which I very recently cited) contain no real information. I apologize for having cited the piece, as I was not aware of the later correction.
It would be incorrect to conclude, however, that assortative mating has no connection to income inequality. Some of the other evidence for that connection is cited in my piece, but more specifically Greenwood et. al. have since revised their own revision (pdf), and produced a more integrated model. A gated version of this piece was just published in American Economic Journal: Macroeconomics this last week.
In the integrated model, the focus is on family and labor supply decisions more generally. For instance, let’s say a couple keeps its income up by not divorcing: is that “no divorce” or rather “assortative mating” which is contributing to income inequality? After all, high income, high education, well-paired couples do divorce a lot less, so that may be a semantic distinction from a causal point of view. The authors suggest that their integrated family model explains about a third of the rise in income inequality from 1960-2005 (see for instance p.46, with a summary of their overall approach and results starting on p.47), and assortative mating is very much a part of their bigger-picture story.