In the New Keynesian model, demand shocks induce deviations from flexible-price allocations, which in turn drive inflation. It follow that our main business-cycle shock is consistent with the New Keynesian model only if the cycles it induces are characterized by a strong comovement between the real quantities and inflation.
We instead find that the main business-cycle shock is nearly orthogonal to inflation at all frequencies. For instance, the shock that targets unemployment accounts for almost 70% of the business-cycle variation in that variable and only for 10% of the business-cycle variation in inflation. And conversely, the shock that targets inflation explains 80% of thee business-cycle variation in inflation and only 9% of thee business-cycle variation in unemployment.
A similar disconnect is present between inflation and the labor share, an often-used proxy of the real marginal cost in the New Keynesian literature.
That is from a new working paper by George-Marios Angeletos, Fabrice Collard, and Harris Dellas. The authors conclude that the data most likely suggest a demand-driven story, but with a lesser role for nominal rigidities and Phillips curves.