Ekaterina V. Peneva and Jeremy B. Rudd have a new paper with the Fed (pdf):
We use a time-varying parameter/stochastic volatility VAR framework to assess how the passthrough of labor costs to price inflation has evolved over time in U.S. data. We find little evidence that changes in labor costs have had a material effect on price inflation in recent years, even for compensation measures where some degree of passthrough to prices still appears to be present. Our results cast doubt on explanations of recent inflation behavior that appeal to such mechanisms as downward nominal wage rigidity or a differential contribution of long-term and short-term unemployed workers to wage and price pressures.
Economists knew or figured that to be the case a while ago, I am glad to see it being relearned.