|KL Divergence using Gaussian distributions from Wikipedia.
Noah Smith once again deleted my comment on his blog, so I'll just have to preserve it (well, the gist of it) here.
He discussed an argument against rational expectations he'd never considered before. Since counterfactual universes are never realized, one can never explore the entire state space to learn the fundamental probability distribution from which macro observable are drawn. Let's call this probability distribution A. The best we can get is some approximation B.
If this sounds familiar, it's exactly the way one would approach this with the information equilibrium model as I discussed several months ago.
In that post, I showed that the KL divergence measures information loss in the macroeconomy based on the difference between the distributions A and B.
D(A||B) = ΔI
That was the content of my comment on Noah's post. I go a bit further at the link and say that this information loss is measured by the difference between the price level and how much NGDP changes when the monetary base changes
ΔI ~ P - dN/dM = dN*/dM - dN/dM
Which to me seems intuitive: it compares how much the economy should grow from an expansion of the money supply (ideally) to how much it actually does grow.
Just the aggregate ΔI is measured, however. Two different distributions B, B' and B'' can have the same KL divergence so this doesn't give us a way to estimate A better.
Now rational expectations are clearly wrong at some given level of accuracy, but then so are Newton's laws. The question of whether you can apply rational expectations depends on the size of ΔI. Since ΔI is roughly proportional to nominal shocks (the difference between where the economy is and where it should be based on growth of M alone ) and these nominal shocks are basically the size of the business cycle, it means rational expectations are not a useful approximation to make when analyzing the business cycle.
As far as I know, this is the first macroeconomic estimate of the limits of the rational expectations assumption that doesn't compare it to a different model of expectations (e.g. bounded rationality, adaptive expectations). (There are lots of estimates for micro.)
 In case anyone was curious, this also illustrates the apparent inconsistency between e.g. this post where nominal shocks are negative and e.g. this post where they are positive. It depends on whether you apply them before including the effect of inflation or after. Specifically
0 = dN*/dM - (dN/dM + σ) = (dN*/dM - σ) - dN/dM