They referred to this debate as an “academic kerfuffle,” but we believe the debate has been constructive, because it has brought greater clarity over the ideas shaping austerity policies in both the United States and Europe.

The most important insight for anyone following this debate, and one that Ms. Reinhart and Mr. Rogoff acknowledge, is that there is no evidence supporting the claim that countries will consistently experience a sharp decline in economic growth once public debt levels exceed 90 percent of G.D.P. Although the two of them partly backed away from that claim in a 2012 paper in The Journal of Economic Perspectives, they have now done so more definitively, saying the 90 percent figure is not “a magic threshold that transforms outcomes, as conservative politicians have suggested.”

However, Ms. Reinhart and Mr. Rogoff stubbornly maintain that “growth is about 1 percentage point lower when debt is 90 percent or more of gross domestic product,” a core finding of their 2010 paper.

There are serious problems with this claim. The most obvious is that the median growth figures they reported in the 2010 paper are distorted by the same coding error and partial exclusion of data from Australia, Canada and New Zealand that tainted their average growth figures. When we corrected for these errors, the difference in median economic growth rates was only 0.4 percentage points between countries whose public-debt-to-G.D.P. ratio was between 60 percent and 90 percent, and those where the ratio was over 90 percent (2.9 percent median growth, versus 2.5 percent). The difference between 0.4 percent and 1 percent is quite substantial when we’re talking about national economic growth.

(Ms. Reinhart and Mr. Rogoff have substantial disagreements with us about the proper selection and weighting of data. They elaborated on these points in their Op-Ed appendix. We have presented all our data, calculations and methodological arguments on the Web site of the Political Economy Research Institute at the University of Massachusetts, Amherst, where we teach.)