They buried the lede again: "ID requirements have no effect on fraud either – actual or perceived."
The web site is disingenously named: "National Bureau of Economic Research". It is not an agency of the US federal government: "NBER is a private, non-profit, non-partisan organization dedicated to conducting economic research and to disseminating research findings among academics, public policy makers, and business professionals". That is, they're under no obligation to be not shade the truth.
This is actually a near-duplicate: https://news.ycombinator.com/item?id=19134345 . The abstract has changed from concluding that election reform money should be spent somewhere other than voter ID laws to "Overall, our results suggest that efforts to reform voter ID laws may not have much impact on elections." That's a far more political conclusion than "spend your money on other things, this one has no effect", which would tend to mitigate against voter ID if it has no real effect one way or the other. Why pay for checking ID if we don't need to?