We investigate the flattening Phillips relation by making two departures from standard specifications. First, we measure slack using real activity variables that are bandpass filtered or year-over-year changes in activity (these are similar), instead of gaps. Second, we study the components of inflation instead of the standard aggregates. We find that some inflation components have strong and stable correlations with the cyclical component of real activity; these components tend to be relatively well-measured and domestically determined. Other components, typically prices that are poorly measured or internationally determined, have weak and/or unstable correlations with cyclical activity. We construct a new inflation index, cyclically sensitive inflation, that weights the components by their joint cyclical covariation with real activity. The index has strong and stable correlations with cyclical activity and provides a real-time measure of cyclical movements in inflation.
Stock, James H., and Mark W. Watson. "Slack and Cyclically Sensitive Inflation." Journal of Money, Credit and Banking 52.S2 (December 2020): 393-428.