It’s instead a very specific critique of the idea that they think they know the level of u*. That’s not an asterisk — it’s a star; this variable is called “u-star” and it’s a very big deal in Fed-ville. It is the lowest unemployment rate that Fed economists believe to be consistent with stable inflation. In other words, it’s the unemployment rate at full employment.

The Fed thinks that over the long run, u* is 4.8 percent. That is, as the economy moves toward full resource utilization and the labor market closes in on full employment, the unemployment rate will settle in at this level. The fact that the current rate — 4.6 percent; let’s call it u — is already below their long-term u* is not a problem; short-term fluctuations around u* are acceptable to them. But that’s why they’re raising rates: They believe that meeting their dual mandate of stable prices and full employment requires tapping the economic brakes when u is less than u*.

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There’s a very big problem with this reasoning: Nobody knows what u* is.

It is an unobservable variable that can be inferred only through various techniques, most of which involve the correlation between unemployment and inflation. That’s far from crazy, by the way, and there has, at times, been a clear, negative correlation between labor market tautness and price pressures.

But this is not one of those times. Take a close look at the figure below, which plots the unemployment rate and the Fed’s preferred inflation measure (PCE core, which leaves out volatile food and oil prices; the latter is, of course, especially important, but that price is set in world markets). The theory in play here is that as the economy closes in on full employment, inflation should begin to accelerate.

That relationship seems very clear in the 1960s. After that, there are a bunch of nasty bips and bops for a while. Then, in the late 1980s, you see the jobless rate hitting low levels and inflation accelerating, again, in line with the theory. If you squint, you can see a trace of that same pattern toward the end of the 1990s expansion, but given that the jobless rate was a very low 4 percent in 2000, that bit of acceleration could legitimately be viewed as a small price to pay for the prodigious benefits to low- and middle-wage workers that full employment was spinning off.

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Since then, there’s just not a lot to see here. True, when the economy tanked in 2008, the rate of inflation fell sharply, but — and this is key — as the jobless rate has since fallen fast, allegedly hitting u* in recent months, prices have hardly accelerated at all.

As you might have guessed, we have much trickier ways of teasing out this relationship, by trying to account for inflationary expectations, the inclusion of other variables that influence inflation and geographical variation (some recent work has found a negative correlation between prices and unemployment at the metro/city level). But, if we’re being honest, I think we have to admit that recent work by President Obama’s Council of Economic Advisers (CEA) is pretty dispositive.

The CEA found that our ability to accurately identify u* (called the NAIRU in their figure, shown below) has crashed and burned in recent years. The statistical confidence interval — a measure of certainty of the estimate and the shared area in the graph — around u* ranges from 0 to 6 percent.

This is unsurprising given both the dynamics in the first figure above and earlier work by influential macro-statisticians who, looking into this very problem 20 years ago, found such “imprecision” to “raise serious questions about the role that estimates of [u*] should play in discussions of monetary policy.” And remember, that finding came well before the unemployment/inflation relationship became even more tenuous.

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There are good questions as to why that uncertainty band explodes in the CEA’s figure. Higher inequality and reduced worker bargaining clout has reduced the extent to which lower unemployment lifts wages (also, in recent work, I showed the wage-to-price pass-through to be much diminished). Expanding global supply chains have loosened supply constraints that in the past created price pressures. And the Fed itself has been very effective at “anchoring” inflation expectations, so even if inflation spikes a bit, producers don’t overreact — and, for example, raise prices further — as they expect inflation to settle back down.

In this regard, I certainly understand their policy goal here: ensure inflationary expectations remain anchored. But, with the exception of a few months in early 2012, core inflation has undershot the Fed’s 2 percent target for eight years running!

So, here’s what I conclude from all this:

Low- and middle-income workers have little to show from this expansion. It is essential that they have a chance to reap the benefits of running a truly tight labor market.

Itty-bitty, fully expected rate hikes are not a huge deal, but it’s still the case that such preemptive actions by the Fed, as they are intended to slow the pace of tightening the job market, could reduce the likelihood that these wage and income gains will occur.

Thus, if their rationale is that u is less than u*, then they have way too much faith in their knowledge of u*, and their unwarranted certainty could be costly to working families. We’d all be better off if they used inflation itself (actual and expected), not u*, as their guidepost.