This is a bit technical at times, but it makes an important point that is a bit buried in the discussion that I'd like to highlight.
The first step in the discussion is to explain what a non-linear Phillips curve is, and why a non-linear specification is needed:
Compensation Growth and Slack in the Current Economic Environment, by M. Henry Linder, Richard Peach, and Robert Rich, NY Fed: ...Our analysis is based on the estimation of a nonlinear wage-inflation Phillips curve that draws upon the modeling approach outlined in a Boston Fed paper by Fuhrer, Olivei, and Tootell. The key feature of the nonlinear Phillips curve is that the impact of a change in slack depends on the level of slack. These features are illustrated in the chart below, where the slope of the Phillips curve becomes steeper as the unemployment rate moves further below the natural rate of unemployment (higher resource utilization), while the slope becomes flatter as the unemployment rate moves further above it (lower resource utilization).
Why might the Phillips curve flatten out as the unemployment rate rises further above the natural rate of unemployment? As a reminder, what matters for labor market decisions is the real wage rate—the nominal wage adjusted for the price level (or cost of living). One explanation for the flattening of the Phillips curve is downward real wage rigidity—that is, a more sluggish response of real wages when the unemployment rate is high (see the Boston Fed paper by Holden and Wulfsberg for a more detailed discussion of theories of real wage resistance during an economic downturn). In a situation of high unemployment, wage growth becomes relatively stable around the recent level of underlying inflation, so that real wages don’t fall sufficiently to clear the labor market.
Here's how they estimate the non-linear Phillips curve and an explanation of why the sample begins in 1997:
Our Phillips curve model relates four-quarter growth in nominal compensation per hour (for the nonfarm business sector) to economic slack, controlling for movements in trend productivity growth and expected inflation. Our measure of slack is the Congressional Budget Office (CBO) estimate of the unemployment gap—the percentage point deviation between the actual unemployment rate and the CBO estimate of the natural rate of unemployment. For trend productivity growth, we use an average of the (annualized) quarterly growth rate of productivity. For expected inflation, we construct a ten-year personal consumption expenditure (PCE) survey measure by adjusting the Survey of Professional Forecasters’ ten-year expected CPI inflation series to account for the average differential between CPI and PCE inflation. As the chart below shows, expected inflation has been extremely stable during the post-1997 period. To provide additional observations for estimation and to conduct the analysis in a low-inflation environment with well-anchored expectations, we use data that cover the period from 1997 through the present.
Our model relates economic slack to an adjusted compensation measure, where we subtract the values of trend productivity growth and expected inflation from the compensation growth series. This adjustment imposes the standard restriction that increases in the real wage rate equal increases in labor productivity in the long run. The chart below provides a scatter plot of the adjusted compensation growth series and the unemployment gap. Negative (positive) values of the unemployment gap represent conditions in which unemployment is below (above) the natural rate of unemployment.
The estimated Phillips curve should have the shape predicted above, and it does:
An examination of the scatter plot shows that the general shape of the data points bears a close resemblance to the chart of the nonlinear Phillips curve, and estimation of the model provides evidence of a statistically significant nonlinear relationship between (adjusted) compensation growth and slack. ...
We also consider two additional criteria to evaluate the nonlinear Phillips curve model—within-sample fit and out-of-sample forecast performance. The within-sample fit is based on estimation of the model using data from the full sample to compare the predicted and actual values of growth in compensation per hour. The out-of-sample forecast performance is based on estimation of the model only using data through 2007:Q4 on the unemployment gap, trend productivity growth, and expected inflation. With the resulting estimated model, we input the actual values of the unemployment gap, trend productivity growth, and expected inflation during the post-2007:Q4 period to generate forecasts of compensation growth. The first forecast corresponds to compensation growth from 2008:Q1 to 2009:Q1.
The next chart plots the four-quarter change in compensation growth, the within-sample predictions, and the post-2007:Q4 out-of-sample forecasts. While the within-sample predictions fail to track some short-run movements in compensation growth, they do capture the general movements in the series. Moreover, both the within-sample predictions and out-of-sample forecasts capture the magnitude of the decline in compensation growth since 2008 as well as its subsequent stability.
Our analysis suggests that a nonlinear wage-inflation Phillips curve fits the data well during the post-1997 episode and complements the results of Fuhrer, Olivei, and Tootell, who find evidence of a nonlinear relationship between price inflation and activity gap measures.
Here's what I want to highlight:
An important conclusion from our analysis is that recent stability in the growth rate of labor compensation measures may not be informative about the extent of slack or its change. That is, stability in labor compensation measures doesn’t imply that the output gap has closed, while changes in the output gap may only have a modest impact on compensation growth.
They also note that this implies real rather than nominal wage rigidity:
In an inflation environment where actual and expected price changes are low, someone might interpret the earlier scatter plot as reflective of downward nominal wage rigidity—the idea that workers and firms have incentives to avoid reductions in nominal wages. However, the nonlinearity between wage growth and slack appears to be evident in other episodes in which large fluctuations in real activity were accompanied by high inflation and high compensation growth (this point is also discussed by Fuhrer, Olivei, and Tootell). Thus, the mild trade-off between compensation/wage growth and resource slack when slack is sizable isn’t unique to recent experience. Moreover, the source of the nonlinearity must stem from downward real wage rigidity, as downward nominal wage rigidity can generate this feature only in a low-inflation environment.
They conclude with:
We recognize that our analysis comes with important caveats... Nevertheless, if the nonlinear relationship between slack and wage/price inflation is an important feature of the data, then it will be critical for policymakers to identify other indicators that may be more responsive to slack and provide a quick and more reliable read on its movements.
It is not surprising at all that wage movements would be uninformative about labor market conditions when wages adjust sluggishly to economic conditions, but the prevalence of claims about the condition of the labor market based upon measures of compensation is a signal that people have missed this point. There can be both considerable slack in the economy (so let's do something about it), and relatively stable wages.