The new core PCE inflation number for March comes out May 1st. In preparation for that, I was looking at the dynamic equilibrium model for PCE inflation and adding more shocks to see how well the data could fit. In the process, I noticed something odd/interesting:
These are all positive shocks to PCE inflation, but notice anything about the dates? Let me add NBER recessions on this picture:
Each recession is associated with a positive shock to PCE inflation that precedes it. The only exceptions are the early 2000s recession (for which there is a debate on whether or not it is a recession) and the early 1960s recession (where there isn't data). Actually, it is not entirely out of the question to add one for the early 2000s . Since these shocks precede the recessions, they'll precede the shocks to unemployment (adding the dynamic equilibrium model of unemployment from e.g. here)
This reproduces a "Phillips curve"-like behavior. Inflation rises when unemployment has been falling for awhile after an unemployment shock. Just after a positive inflation shock, we get a shock to unemployment. Therefore inflation will tend to fall (since the shock is over) while unemployment is rising. These fluctuations are likely happening on top of the demographic transition of the 1960s and 70s.
If we are headed into another recession (per here), this might explain the higher inflation of the past year or so (core PCE inflation was over 3% in Jan of 2016 and 2017, having not been above 3% since 2012):
This is interesting as it means rising inflation is a sign of an upcoming recession (the center of the inflation shock precedes the center of the unemployment shock by about 1.3 years on average). However, this could be a just-so story. Inflation rises because unemployment gets low. But as recessions are random with roughly a mean time between them of about 8 years, it just appears we get recessions after unemployment has been falling for awhile (and we get a rise in inflation).
Update 1 May 2017
I had forgotten about the low CPI number earlier in April which should have prepared us for the very low March 2017 number: -1.7% (continuously compounded annual rate of change).
 I don't necessarily think it's useful, but here it is: