Does excluding food and energy prices from the Consumer Price Index (CPI) produce a measure that better captures permanent price changes? To examine this question, we decompose CPI inflation and “core” inflation into their permanent and transitory components, using a correlated unobserved-components model. The stationarity of inflation may be time-varying, so we examine the performance of the core measure of inflation separately for periods in which inflation is I(1) and I(0). For a period in which inflation appears to be I(1), we find that core inflation and the permanent component of overall inflation are closely related, but there are some caveats. For a period in which inflation appears to be I(0), we decompose the core and overall price levels and find that the permanent component of the core CPI is much more volatile than the actual core series and that the core excludes volatile permanent shocks to the overall price level.