David Beckworth has an excellent post discussing the Fed's confused relationship with the Phillips Curve. Recall that the Fed has the Keynesian interpretation of the Phillips Curve---which basically says that rising inflation is caused by an overheating economy. Or more simply; inflation is procyclical.
In fact, whether inflation is procyclical or countercyclical is up to the Fed---it depends on the monetary policy regime. David shows that the Bank of Israel makes inflation countercyclical by producing a nice smooth path of NGDP growth. Rather than being some sort of fundamental law of nature, the Phillips Curve is actually the outcome of policy choices. Check out David's very interesting graphs of Israeli inflation, RGDP growth and NGDP growth.
David also has a tweet that directed me to the FOMC minutes:
Almost all participants who commented agreed that a Phillips curve-type of inflation framework remained useful as one of their tools for understanding inflation dynamics and informing their decisions on monetary policy. Policymakers pointed to a number of possible reasons for the difficulty in estimating the link between resource utilization and inflation in recent years. These reasons included an extended period of low and stable inflation in the United States and other advanced economies during which the effects of resource utilization on inflation became harder to identify, the shortcomings of commonly used measures of resource gaps, the effects of transitory changes in relative prices, and structural factors that had made business pricing more competitive or prices more flexible over time. It was noted that research focusing on inflation across U.S. states or metropolitan areas continued to find a significant relationship between price or wage inflation and measures of resource gaps.
This reminds me of (highly flawed) cross sectional studies of the relationship between fiscal policy and output. I can't emphasize enough that you cannot draw macro conclusions from regional studies. Even if (as I claim) there is no reliable Phillips curve relationship at the macro level---that any observed relationship reflects bad monetary policy---you would expect this sort of correlation at the regional level. You'd expect higher inflation in booming regions than in depressed regions.
PS. I will be interviewed on monetary policy by Charlie Deist for 1 hour tomorrow morning (8-9am Sunday, Pacific time, or 11-12am, EST) Here is the link.