This is frustrating me. People (e.g. the Atlanta Fed Macroblog, the St Louis Fed Economic Synopses (pdf)) still aren't getting it. What can I do to attract attention to my simple point? Think up some totally insulting inflammatory blog post title? Nope, that's not really me. I'm just going to try again. And use bold.
You can't test whether core inflation is a useful indicator for a central bank to look at just by seeing whether core inflation forecasts future total inflation (or whatever the bank is targeting). You can't test whether anything is a useful indicator for central banks to look at that way. Everything ought to look useless by that test, if the bank is doing it right. What you are testing is whether the bank is doing it right.
If a central bank is targeting (say) 2% total inflation at a (say) 2-year horizon, and if it's doing it right, then deviations of total inflation from 2% ought to be uncorrelated with anything that the bank knew 2 years ago. This means that nothing (i.e. nothing in the bank's information set) should forecast 2-year ahead total inflation, if the bank is reacting correctly to those indicators. Core should fail to forecast future total inflation. Total inflation should fail to forecast future total inflation. Trimmed mean inflation should fail to forecast future total inflation. Unemployment should fail to forecast future inflation. Everything should fail to forecast future total inflation.
This is an immediate implication of rational expectations (on the part of the central bank). The bank sets monetary policy, looking at the indicators in its information set, so that the bank's forecast of 2-year ahead total inflation is equal to the 2% target. Deviations of actual inflation from 2% are therefore forecast errors. Under rational expectations (on the part of the bank) forecast errors should be unforecastable from anything in the bank's information set. They wouldn't be rational if you could forecast errors.
When you look at correlations between core inflation and future total inflation, you are not testing whether core inflation is a useful indicator of future headline inflation that the bank should pay attention to. You are doing something quite different. If you do find a positive correlation between core inflation and future total inflation, it means one of three things: 1. the bank is not targeting total inflation; 2. The bank's target for total inflation has changed over time; 3. the bank is targeting total inflation, and the target is not changing over time, but the bank is getting it wrong, and isn't responding as strongly to core inflation as it should, and so is violating rational expectations.
Now, if you want to learn how to do it properly, go and read my old post.