In an NBER working paper, James Stock and Mark Watson argue that the correlation between cyclically sensitive inflation (CSI) and bandpass filtered activity measures is high and has not declined over the last decades, contrary to standard measures of the slope of the Phillips curve.
… we construct a new price index designed to maximize the cyclical variation in the price index. This index, which we call Cyclically Sensitive Inflation (CSI), estimates the weights on the component prices to maximize the correlation of the CSI with our bandpass measure of aggregate cyclical variation. … this index places low weights on tradeable goods, such as energy, motor vehicles & parts, and durable household equipment. The index also places low weight on the least well-measured sectors, such as clothing & footwear and final consumption of nonprofit institutions serving households (NPISH). The sectors that receive the greatest weight – housing excluding gas & electric utilities, followed by food & beverages for off-premises consumption, and recreational services – tend to be both locally determined (nontradeable) and relatively well-measured.