Does excluding food and energy prices from the Consumer Price Index (CPI) produce a measure that 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. One of the key aspects of the correlated unobserved components model is that it allows shocks to the permanent component to potentially be more variable than shocks to the series itself, due to offsetting transitory shocks correlated with the permanent shocks. The stationarity of inflation may be time-varying, so we examine the performance of the core measure of inflation for periods during which it appears that inflation is I(1) and for periods during which it appears that inflation is 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, although core inflation does have some drawbacks as a measure of permanent inflation. 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 core CPI is much more volatile than the actual core series and that core excludes volatile permanent shocks to the overall price level.