Understanding U.S. Wage Dynamics

Yasser Abdih; Stephan Danninger

doi:10.5089/9781484362082.001.A001

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Understanding U.S. Wage Dynamics

Author: Mr. Yasser Abdih and Mr. Stephan Danninger

Contributor Notes

Author’s E-Mail Address: yabdih@imf.org; sdanninger@imf.org.

Publication Date:: 15 Jun 2018

eISBN:: 9781484362082

Language:: English

Keywords:: WP; Phillips curve; wage growth; nominal wage; growth model; growth rate; growth depression

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In this paper, we undertake empirical analysis to understand U.S. wage behavior since the beginning of the new millennium. At the macroeconomic level, we find that a productivity-augmented Phillips curve model explains the data fairly well. The model reveals that the upward pressure on wage growth from recent tightening in the labor market has been dampened by a persistent decline in trend labor productivity growth and the share of income that accrues to labor. These themes are reinforced and complemented at the micro-economic level. Lower regional unemployment puts an upward pressure on wages of individuals, although this effect has become weaker since 2008. But there is downward pressure on wages for individuals with occupations that are exposed to automation and offshoring, and in industries with a higher concentration of large firms. All these factors appear to play a role illustrating why it is difficult to single out any one culprit for the observed wage growth moderation.

Abstract

I. Introduction

Job growth in the U.S. has been robust since the end of the great recession. It has now converged to rates seen at end-2006 (Figure 1). At the same time, unemployment has fallen steadily and is now near levels not seen since the late 1960s.

Yet wage growth has remained subdued (Figure 2). U.S. hourly earnings have grown by 3.2 percent year-on-year, on average, in the new millennium through end-2007, the dawn of the great recession (GR). This is 1.5 times the average growth to date since the end of the recession. The employment cost index for worker compensation, a broader wage measure, points to even larger differences—with a ratio of pre-to-post GR wage growth of almost 2.

Many observers have argued that this apparent puzzle suggests that the wage Phillips curve has flattened (see Leduc and Wilson (2017) for a recent example). Others explored whether lackluster wage growth means that there is more slack than implied by the headline unemployment rate (see, for example, Smith (2014), and IMF (2017)). And some papers pointed to the possibility that wages are rising slowly now, but could accelerate once the unemployment gap turns significantly negative, pointing to potential nonlinearities in the wage Phillips curve (Fisher and Koenig (2014), Kumar and Orrenius (2014), Donayre and Panovska (2016), and Nalewaik (2016)).

In this paper, we take a fresh look at the data and attempt to explain the lack of a meaningful pick-up in wage growth. We do so using a battery of empirical exercises covering both macro-econometric models with aggregate data, and micro-econometric wage models with data at the level of individuals/workers.

Our analysis draws on three strands of the literature: The wage Phillips curve literature (following the seminal paper of Phillips (1958)); the wage curve micro literature (that builds on Blanchflower and Oswald (1994)), and the human capital literature of wage determination in the tradition of Mincer (1974). We build on this literature and make the following main contributions: At the macroeconomic level, our Phillips curve model is estimated within a general-to-specific methodology that includes all potential determinants of wage growth highlighted in the literature together with a rich dynamic structure. In that way, the model nests competing explanations of subdued wage growth within one unified framework, tests the relevance of each, and sheds light, quantitatively, on the relative importance of the key wage drivers. At the micro level, we introduce a series of variables related to different explanations into one well-specified empirical model. By performing a horse race in which variable fits best, we provide evidence of the significance of all three key explanations: technology, globalization, and market concentration.

Our key findings are as follows. Estimates of a wage Phillips curve for the U.S. suggest that weak wage growth is mostly a product of low labor productivity growth. The model fits the data well, its parameters show no sign of structural instability over time, and the null hypothesis of linearity cannot be rejected. The model allows a decomposition of the drivers behind compensation growth and shows that, throughout the current recovery, there has been upward pressure on compensation growth as labor market slack has diminished. However, this has been offset by a contemporaneous decline in labor productivity growth. Additional factors that have kept the growth rate of real compensation below that of labor productivity include technological progress—linked to the automation of routine tasks—falling unionization rates, and trade globalization. The results are robust to using alternative measures of wages and labor market slack.

These themes are broadly reinforced by results from a micro-econometric model of wage determination, that uses earnings observations from the Current Population Survey. Controlling for various individual specific characteristics (including education, gender, and employment characteristics), an augmented Mincer-type model reveals that a lower local/regional rate of unemployment puts upward pressure on wage growth. But, in contrast to the macro results, this effect has become weaker than in the pre-2008 period—implying a flattening of the wage-unemployment relationship even after structural changes have been taken into account. Further, there is a downward pressure on wages for individuals with occupations that are exposed to automation and offshoring, and in industries with a higher concentration of large firms. This latter finding is in line with Paul Krugman’s (2018) recent emphasis of monopsony power in labor markets as one key factor behind weak wage growth. Finally, the model suggests that college-educated workers in routinizable jobs and in more concentrated industries face further downward pressure on their wages while less-educated workers face a wage discount insofar as they are employed in offshorable tasks. All these factors appear to play a role, illustrating the difficulty in identifying a clear-cut explanation for the observed weakening of wage growth.

The rest of the paper is organized as follows. Section II carries out the macro-econometric analysis of U.S. wages and conducts various robustness checks. Section III undertakes a battery of micro-econometric exercises at the level of individuals. The last section summarizes the key messages and offers some concluding remarks.

II. Macro analysis

We begin by looking at the long run and short run determinants of wage growth at the macroeconomic level. We then use our estimated wage models to address the following questions: has the relationship between labor market slack and wage growth weakened in the aftermath of the global financial crisis? Is the economy close to full employment now or is subdued wage growth an indication that there is more slack than that implied by the headline unemployment rate? Does slower labor productivity growth play a role in depressing wage growth? How important is the effect? Can the model’s estimated coefficients and the time path of the models’ wage growth determinants adequately track recent wage dynamics? By gauging the effects of the macro drivers, we can assess the size of the “missing” wage growth and areas where one needs to possibly dig deeper into micro data.

A. Determinants of Wage Growth

The Long Run

What determines nominal wage growth in the long run? Standard economic theory suggests labor productivity growth plus price inflation. Intuitively, an increase in productivity will make the value to the firm of hiring additional workers exceed the cost. As a result, firms expand employment to maximize their profits. Greater demand for labor boosts real wages, as firms compete for workers.

Statistically, cointegration techniques are suitable for the analysis of long-run relationships between integrated variables, which we carry out here. Specifically, we use Johansen’s (1988, 1991) procedure of maximum likelihood for a finite order VAR to test for cointegration among the growth rates of: (1) The Employment Cost Index (ECI) for compensation of workers (wages, salaries, and employer costs for employee benefits) in the nonfarm business sector; (2) the price index for personal consumption expenditures (PCE); and (3) real output per hour in nonfarm business. By including the growth rates of the variables in the VAR, we are implicitly treating them as individually non-stationary and integrated of order one, I(1). As we will show below through extensive formal testing, this treatment is strongly supported by the data. What we are looking for then is whether a linear combination of the growth rates is stationary, and if so, whether such combination can be simplified further into one where nominal wage growth equals the sum of the growth rates of labor productivity and inflation.

We estimate the VAR over the period 1990Q1-2017Q3. In this VAR and in all other macro regressions, we compute growth rates as annualized log differences of the quarterly level data. Table 1 reports the standard statistics of Johansen’s procedure applied to the estimated VAR. The trace eigenvalue statistic strongly rejects the null hypothesis of no cointegration in favor of at least one cointegrating relationship. At the same time, it cannot reject, at conventional significance levels, the null hypothesis of at most one relationship in favor of more than one. Hence, the evidence points to a unique cointegrating vector, the estimates of which are also shown in the table. The coefficients on inflation (Δp_t) and labor productivity growth (Δy_t) have the right signs and are statistically significant. Numerically, they are each close to minus unity, a hypothesis that we formally test in the right block of Table1. Indeed, likelihood ratio tests cannot reject the null hypothesis that the coefficients are equal to minus one, both individually and jointly. Imposing these restrictions yields the final model which we also report in the table, and reproduce here in standard equation format, where everything but ECI growth (Δw_t) is on the right-hand side of the equation:

Table 1.

Cointegration Analysis of the Baseline VAR Model

Notes: The VAR, estimated over 1990Q1-2017Q3, includes a constant and five lags on each variable (Δw_t, Δp_t, Δy_t,).

\begin{array}{l} (1) & Δ w_{t} = - 0.9 + Δ p_{t} + Δ y_{t} \end{array}

Equation 1 states that, in the long run, wage inflation increases with price inflation and labor productivity growth. But it has fallen short of their sum, as indicated by the negative (and statistically significant) constant term. The constant has an economic interpretation: it roughly captures/mirrors the growth rate of the labor share of income, which has been negative on average, over our period of analysis, 1990-2017. In recent work, we examined the sources of the secular decline in the labor share using data broken down by industry and state (Abdih and Danninger (2017)). We found that the fall was largest, on average, in industries that saw: a high initial intensity of “routinizable” occupations; steep declines in unionization; and a high level of penetration of and competition from imports. Quantitatively, the bulk of the effect came from changes in technology that are linked to the automation of routine tasks, followed by trade globalization.

The bottom right of Table 1 shows Johansen’s (1995) multivariate statistic for testing the null hypothesis of stationarity. For a given variable, the statistic tests the restriction that the cointegrating vector contains a unity for that variable and a zero for all others. For example, the null hypothesis that ECI growth (Δw_t) is stationary implies that the cointegrating vector β′= (1 0 0 0). As Table 1 shows, the tests strongly reject the stationarity of ECI growth, inflation, and productivity growth, with p values that are effectively zero. Because they are multivariate and so involve a larger information set, these tests typically have more power than their univariate counterparts. These results validate our treatment of the growth rates of variable as nonstationary I(1) series. They also preclude the presence of a stationary linear combination of the variables in levels. This can also be seen graphically in Figure 3 which plots w_t – p_t – y_t, where small case letters denote the variables in natural logarithms. This series trends downwards and appears nonstationary. Johansen’s multivariate test confirms this and strongly rejects the null hypothesis that the series is stationary around a determinist trend— the test statistic is Chi^2(2) = 14.5 with a p value of 0.0007. In contrast and as demonstrated above, the variables in growth rates do cointegrate into the stationary linear combination of equation 1. Indeed, it is clear from Figure 3 that Δw_t + 0.9 – Δp_t – Δy_t exhibits mean reversion.

What about labor market slack? Does it play a role in the long run? Conceptually, it should not. To test that, we augment the VAR with the Congressional Budget Office’s (CBO) measure of the unemployment gap. We then estimate the model and test for cointegration. Again, we find evidence of one and only one vector. We also find that the coefficient on the unemployment gap in the cointegrating vector is numerically very small and, statistically, the restriction that it is zero cannot be rejected: Chi^2(1) = 0.77 [0.381]. The cointegrating relationship in equation 1 continues to hold, and for that matter all the other results discussed in the section.

The Short Run

While we find that slack does not affect wage growth in the long run, it can play a key role in the short run. In other words, one would expect shocks hitting the economy to cause deviations of wage inflation from its long run trend and the extent of this deviation could depend on the degree of resource utilization in the economy.

To quantify the impact, we proceed in two steps. First, we generate an estimate of trend wage growth utilizing the cointegrating relationship in equation 1. Second, we model the deviation of actual wage growth from this trend as a function of the unemployment gap in an autoregressive distributed lag model (ADL).

We estimate trend wage growth by plugging into equation 1 a measure of trend inflation and a measure of trend productivity. We measure trend inflation $(Δ p_{t}^{t r e n d})$ by expected long run (10 yr.) PCE inflation reported in the Survey of Professional Forecasters (SPF); we construct trend labor productivity growth $(Δ y_{t}^{t r e n d})$ by passing the actual data through the Baxter and King (1999) filter. The filter width and cutoffs are set equal to the values used in Staiger, Stock, and Watson (2001), which is standard in the literature (see, for example, Yellen (2017)).

We then estimate an initial ADL model where the wage growth gap—i.e., deviation of actual ECI growth from its estimated trend constructed as per above—is regressed on its own 8 lags, and the current value and 8 lags of the unemployment gap (Ugap_t), again the latter measured as actual unemployment minus the CBO estimate of the natural rate. None of the lags on the wage growth gap were statistically significant. Also lags on the unemployment gap beyond the first did not matter. Jointly, all these zero coefficient restrictions were statistically valid: F(15,93) = 1.076 [0.389]. Imposing these valid restrictions yield the following final model:

\begin{array}{l} 2. & Δ w_{t} - Δ p_{t}^{t r e n d} - Δ y_{t}^{t r e n d} = θ + \sum_{i = 0}^{1} λ_{i} (U_{t - i} - {\bar{U}}_{t - i}) + ε_{t} \end{array}

Or equivalently:

\begin{array}{l} 3. & Δ w_{t} = \underset{T r e n d}{\underset{︸}{θ + Δ p_{t}^{t r e n d} + Δ y_{t}^{t r e n d}}} + \underset{C y c l e}{\underset{︸}{\sum_{i = 0}^{1} λ_{i} U g a p_{t - i}}} \end{array} + \underset{N o i s e}{\underset{︸}{ε_{t}}}

where:

B. An Alternative Approach—Same Results

We will come back shortly to the model in equation 3 and discuss its economic and statistical properties, as well as its implications. Before doing that, it may be useful to step back and analyze how this model relates to the literature on the wage Phillips curve. Also, as a check on our above reasoning and analysis, we ask the following: if we start with an unrestricted wage Phillips curve model, rich enough in its dynamics, and general enough to encompass key variables highlighted in the literature, would it be statistically and economically acceptable to reduce it to model 3 above? The short answer is yes. Below we elaborate.

We begin with the following general unrestricted dynamic model:

\begin{array}{l} 4. & Δ w_{t} = θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} + \sum_{i = 1}^{q} ρ_{i} Δ w_{t - i} + \sum_{i = 1}^{p} u_{i} Δ p_{t - i} + \end{array} \sum_{i = 0}^{g} λ_{i} U g a p_{t - i} + α {(w - p - y)}_{t - 1} + ε_{t}

In equation 4, ECI growth depends on its own lags (potentially displaying inertia), long run inflation expectations (capturing forward looking wage setting behavior), past inflation (allowing for potential backward indexation), and labor market slack. We augment this wage Phillips curve with trend labor productivity growth and a “potential” error correction term, the latter allowing ECI growth to react to/correct for last period’s deviation of real wages from labor productivity. The error term is assumed to be white noise.

The above specification encompasses, and nests as special cases, well-known empirical wage Phillips curve models that are rooted in micro-theory, for example those of Blanchard and Katz (1997, 1999), Ball and Moffitt (2001), Gali (2011), and IMF (2017), among others.

Looking at all the above determinants jointly and assessing their relative importance is one contribution of this paper.

We first estimate equation 4 with 8 lags on wage growth, inflation, and the unemployment gap. Extensive testing reveals that this general model (also labeled EQ(a) in Table 2) can be simplified to a more parsimonious model (EQ(g) in Table 2):

The F-statistic testing the null hypothesis that the coefficients on lags 5 through 8 on all variables are jointly zero does not reject—its realized value is 0.65 with a p-value of 0.793. Hence, as Table 2 shows, the reduction from EQ(a) to the one with 4 lags on each variable (EQ(b)) is statistically valid (see notes to Table 2 for a description of model reduction tests).
The restriction that the coefficients on the 4 remaining lags of ECI growth (ρ_is) are jointly zero cannot be rejected (individually, they were also imprecise).
The restriction that the coefficients on the 4 remaining lags of price inflation (μ_is) are jointly zero also cannot be rejected (they were imprecise individually as well).
The restriction that the coefficient on the error correction term (α) is zero cannot be rejected. This is consistent with the multivariate analysis in Section A, which verified that this term is non-stationary.
Lags beyond the first on the unemployment gap (λ_is) were not statistically significant.

Table 2.

ECI Growth: General-to-Specific Modeling

Notes: Equations (a) is equation 4 in the main text, and equations (b) through g are further simplifications involving exclusion restrictions. SC is the Schwarz information criterion; HQ and AIC are the Hanna-Quin and Akaike information criteria. Under tests of model reduction, the table reports the approximate F statistic for testing the null hypothesis (indicated by the model to the right of the arrow) against the maintained hypothesis (indicated by the model to the left of the arrow), and the tail probability associated with that value of the F statistic (in square brackets). The numbers in parentheses after the F statistic are the degrees of freedom. The sample is 1990Q1-2017Q3.

Table 2.

ECI Growth: General-to-Specific Modeling

EQ (a)	EQ (b)	EQ (c)	EQ (d)	EQ (e)	EQ (f)	EQ (g)
$\sum_{i = 1}^{8} ρ_{i} Δ w_{t - i}$	$\sum_{i = 1}^{4} ρ_{i} Δ w_{t - i}$	……	$\sum_{i = 1}^{4} ρ_{i} Δ w_{t - i}$	$\sum_{i = 1}^{4} ρ_{i} Δ w_{t - i}$	……	……
$\sum_{i = 1}^{8} μ_{i} Δ p_{t - i}$	$\sum_{i = 1}^{4} μ_{i} Δ p_{t - i}$	$\sum_{i = 1}^{4} μ_{i} Δ p_{t - i}$	……	$\sum_{i = 1}^{4} μ_{i} Δ p_{t - i}$	……	……
$\sum_{i = 0}^{8} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$
θ	θ	θ	θ	θ	θ	θ
$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$
$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$
α (w – p – y)_t–1	α (w – p – y)_t–1	α (w – p – y)_t–1	α (w – p – y)_t–1	…	…	…
SC = 3.334	2.915	2.787	2.767	2.885	2.596	2.521
HQ = 2.913	2.669	2.598	2.578	2.653	2.480	2.449
AIC= 2.626	2.500	2.469	2.449	2.494	2.401	2.399

Tests of model Reduction
EQ( a) -->	EQ( b):	F(12,82) =	0.650	[0.7934]	EQ( b) -->	EQ( c):	F(4,94) =	0.979	[0.4228]
EQ( a) -->	EQ( c):	F(16,82) =	0.721	[0.7649]	EQ( b) -->	EQ( d):	F(4,94) =	0.498	[0.7370]
EQ( a) -->	EQ( d):	F(16,82) =	0.606	[0.8702]	EQ( b) -->	EQ( e):	F(1,94) =	1.109	[0.2950]
EQ( a) -->	EQ( e):	F(13,82) =	0.681	[0.7758]	EQ( b) -->	EQ( f):	F(9,94) =	0.672	[0.7322]
EQ( a) -->	EQ( f):	F(21,82) =	0.646	[0.8709]	EQ( b) -->	EQ( g):	F(12,94) =	0.955	[0.4971]
EQ( a) -->	EQ( g):	F(24,82) =	0.781	[0.7491]
EQ( c) -->	EQ( f):	F(5,98) =	0.427	[0.8290]	EQ( d) -->	EQ( f):	F(5,98) =	0.828	[0.5328]
EQ( c) -->	EQ( g):	F(8,98) =	0.944	[0.4847]	EQ( d) -->	EQ( g):	F(8,98) =	1.208	[0.3024]
EQ( e) -->	EQ( f):	F(8,95) =	0.617	[0.7619]	EQ( f) -->	EQ( g):	F(3,103) =	1.857	[0.1416]
EQ( e) -->	EQ( g):	F(11,95) =	0.940	[0.5064]

Table 2.

ECI Growth: General-to-Specific Modeling

EQ (a)	EQ (b)	EQ (c)	EQ (d)	EQ (e)	EQ (f)	EQ (g)
$\sum_{i = 1}^{8} ρ_{i} Δ w_{t - i}$	$\sum_{i = 1}^{4} ρ_{i} Δ w_{t - i}$	……	$\sum_{i = 1}^{4} ρ_{i} Δ w_{t - i}$	$\sum_{i = 1}^{4} ρ_{i} Δ w_{t - i}$	……	……
$\sum_{i = 1}^{8} μ_{i} Δ p_{t - i}$	$\sum_{i = 1}^{4} μ_{i} Δ p_{t - i}$	$\sum_{i = 1}^{4} μ_{i} Δ p_{t - i}$	……	$\sum_{i = 1}^{4} μ_{i} Δ p_{t - i}$	……	……
$\sum_{i = 0}^{8} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$	$\sum_{i = 0}^{4} λ_{i} U g a p_{t - i}$
θ	θ	θ	θ	θ	θ	θ
$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$	$β_{1} Δ p_{t}^{t r e n d}$
$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$	$β_{2} Δ y_{t}^{t r e n d}$
α (w – p – y)_t–1	α (w – p – y)_t–1	α (w – p – y)_t–1	α (w – p – y)_t–1	…	…	…
SC = 3.334	2.915	2.787	2.767	2.885	2.596	2.521
HQ = 2.913	2.669	2.598	2.578	2.653	2.480	2.449
AIC= 2.626	2.500	2.469	2.449	2.494	2.401	2.399

Tests of model Reduction
EQ( a) -->	EQ( b):	F(12,82) =	0.650	[0.7934]	EQ( b) -->	EQ( c):	F(4,94) =	0.979	[0.4228]
EQ( a) -->	EQ( c):	F(16,82) =	0.721	[0.7649]	EQ( b) -->	EQ( d):	F(4,94) =	0.498	[0.7370]
EQ( a) -->	EQ( d):	F(16,82) =	0.606	[0.8702]	EQ( b) -->	EQ( e):	F(1,94) =	1.109	[0.2950]
EQ( a) -->	EQ( e):	F(13,82) =	0.681	[0.7758]	EQ( b) -->	EQ( f):	F(9,94) =	0.672	[0.7322]
EQ( a) -->	EQ( f):	F(21,82) =	0.646	[0.8709]	EQ( b) -->	EQ( g):	F(12,94) =	0.955	[0.4971]
EQ( a) -->	EQ( g):	F(24,82) =	0.781	[0.7491]
EQ( c) -->	EQ( f):	F(5,98) =	0.427	[0.8290]	EQ( d) -->	EQ( f):	F(5,98) =	0.828	[0.5328]
EQ( c) -->	EQ( g):	F(8,98) =	0.944	[0.4847]	EQ( d) -->	EQ( g):	F(8,98) =	1.208	[0.3024]
EQ( e) -->	EQ( f):	F(8,95) =	0.617	[0.7619]	EQ( f) -->	EQ( g):	F(3,103) =	1.857	[0.1416]
EQ( e) -->	EQ( g):	F(11,95) =	0.940	[0.5064]

Imposing all the above restrictions jointly cannot be rejected. This yields EQ(g), which is fleshed out in the left block of Table 3. The coefficient estimates on long-term inflation expectations (β₁) and on trend productivity growth (β₂) are numerically very close to one. Statistically, the joint restriction that they are equal to one cannot be rejected: Chi^2(2) = 0.021 [0.9897]. This gives the model in the right block of Table 3, which is the exact same model in equation 3 above.

Table 3.

ECI growth: Final Model

Notes: The estimation sample is 1990Q1-2017Q3.

C. Properties and Key Messages of the Baseline Model

Our baseline model (right block in Table 3 and equation 3 in the text) is economically reasonable. It displays headwinds to wage growth reflected by the negative constant term; the latter is highly statistically significant, and at about one percent, is similar in magnitude to the constant derived from the multivariate analysis of section A. As discussed there, the headwinds capture the secular decline in the labor share of income and the underlying main drivers of technological progress and trade globalization. The slope of the wage Phillips curve is given by the sum of the coefficients on the current and lagged unemployment gap. It is negative, highly statistically significant, and implies, on average, a 0.311 percentage point increase in wage growth in response to a one percentage point reduction in the unemployment gap. This is very close to Staiger, Stock and Watson’s (2001) estimate of −0.280 and Yellen’s (2017) estimate of −0.333—both for the U.S. economy—and IMF’s (2017) estimate of −0.338 for a sample of advanced economies.

The baseline model is statistically well-behaved. Looking through some volatility, it has a reasonably good fit and pseudo out-of-sample forecasts track the realized data fairly well (see Figure 4a). The residuals look stationary (Figure 4b), and formal standard tests do not reject the null hypotheses that they are serially uncorrelated, homoscedastic, and normally distributed. Further, there is no evidence that they follow an ARCH process (Autoregressive Conditional Heteroscedasticity).

When we estimate the model recursively, the parameters display remarkable stability over time, as depicted in Figure 5. This is validated statistically with a battery of recursively estimated chow tests, shown in the bottom half of Figure 5. These statistics test the null hypothesis of parameter constancy and are normalized by the one percent critical value, so that the horizontal line at 1 gives the critical value (for technical details on the chow tests, see Doornik and Hendry (2001)). As can be seen, parameter stability cannot be rejected, providing evidence against the hypothesis that the global financial crisis made the wage Phillips curve flatter. This is consistent with Abdih et. al. (2016) and Laseen et. al. (2016) who reached the same conclusion for both the aggregate and bottom-up price Phillips curves.

The estimated model allows us to decompose ECI growth into contributions from the various determinants. Figure 6 shows the computations since 2000. Clearly, a high level of labor market slack during the great recession and moderating productivity growth have weighed on wage growth. Since then, as the recovery proceeded, rising labor utilization has resulted in an increasing upward pressure on compensation growth. But that has been offset by continued declining contribution from productivity growth. Other headwinds—reflected in the constant term—have also been a drag. These results indicate that once we augment the wage Phillips curve with labor productively growth, “missing” wage growth disappears. In other words, wages are where they should be and the apparent wage puzzle is resolved.

D. Robustness

We now conduct robustness checks along several dimensions. These include using different concepts of slack, testing for nonlinearity in the wage Phillips curve; using an alternative estimate to CBO’s natural rate of unemployment utilizing the Kalman filter; and using different measures of wages. It is fair to say that the results from the baseline model are broadly robust, and the conclusions from the above analysis continue to hold.

Alternative measures of slack

So far, we have used headline unemployment (or U3) to carry out the empirical analysis. Here we consider two alternatives: U6, and the short-term unemployment term (UST). U6 is the broadest measure of unemployment. In addition to the unemployed, it includes people who are marginally attached to the labor force (for example, discouraged workers) as well as people who are working part time but would prefer full time jobs—that is, working part time for economic reasons. UST looks at those that have been unemployed for 26 weeks or less.

To compute slack measures for U6 and UST, we need estimates of their natural rates. We estimate those through a linear transformation of CBO’s measure of the natural rate of headline unemployment (U3). Specifically, and starting with U6, we do the following: First, we formulate an ADL model and regress U6 on its own lags and the current and lagged values of U3. Then, we solve for the long run relationship, which links U6 to U3 after all the dynamics of the model have played out—the coefficient on U3 captures the cumulative long run impact. Finally, we feed the CBO series of the natural rate of U3 into this long run equation to obtain an estimate of the path of the natural rate for U6. We follow the same procedure to estimate the natural rate of short term unemployment, basically replacing U6 with UST in all the above steps.

Figure 7 plots slack or gap measures for U6, UST, and headline unemployment. It is interesting that by any of these measures, the economy is now either at or beyond full employment. This provides some evidence against the hypothesis that subdued wage growth can be squared with or implies that there is still much more slack in the economy than what headline unemployment conveys.

In Table 4, we report the estimated models with unemployment gaps based on U6 and UST. We once again started with general models as in equation 4 and simplified them using statistically acceptable exclusion restrictions into more parsimonious models, and these are the ones we show in Table 4. The block on the left reproduces the estimates of our baseline model for ease of comparison.

Table 4.

ECI growth: Alternative Slack Measures

Notes: The estimation sample is 1990Q1-2017Q3; for U6, data are available starting in 1996Q1.

Several interesting results are worth highlighting. Like in our baseline model, in the U6 model, ECI growth depends on inflation expectations and trend productivity growth with a coefficient of unity on each. In the UST model, because the lagged dependent variable was significant, unity coefficients hold in the long run. Across all models, headwinds captured by the constant term are of roughly the same magnitude. Also, the sum of the coefficients on the unemployment gap—again the slope of the wage Phillips curve—is negative and highly statistically significant. The magnitude in absolute value is largest for the UST model (0.661)—note that for this model, the lagged unemployment gap was not significant and dropped; also, the presence of the lagged dependent variable gives rise to a somewhat larger cumulative impact of the unemployment gap of about 0.82. The second largest magnitude is given by our baseline model (0.311) followed by the U6 model (0.181). This ranking makes sense. Intuitively, one can think of the estimated coefficients as capturing the covariance between the unemployment gap and wage growth divided by the variance of the unemployment gap. The UST gap has the smallest sample variance. As we start broadening the concept of slack to U3gap and then U6gap, the sample variance increases. While the absolute value of the covariance with wage growth also increases with the broadening of the slack concept, it does so at a slower rate than the increase in the variance. The result is that the ratio falls.

This result also implies that there should be no substantial changes in the contribution of slack to ECI growth as we broaden the concept of slack. This is because broader measures also get smaller weights. Indeed, Figures 8a and 8b show decompositions of ECI growth for the U6gap and USTgap models. The charts are broadly similar to each other and to that of the baseline model in Figure 6.

Nonlinearities, and natural unemployment rate estimates from the Kalman filter

Table 5 allows for potential nonlinearities in the augmented wage Phillips curve through a quadratic formulation (middle panel). The coefficient on the square of the headline unemployment gap is imprecisely estimated, and hence the null hypothesis of linearity cannot be rejected. Statistically, this model reduces to the baseline model which we again reproduce in the left panel.

Table 5.

ECI growth: Non-Linearity and the Kalman Filter

Notes: The estimation sample is 1990Q1-2017Q3.

Table 5.

ECI growth: Non-Linearity and the Kalman Filter

Baseline Model					Non-linear Model					Kalman Filter Model
$\begin{array}{l} Δ w_{t} & = & θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} \\ + \sum_{i = 0}^{1} λ_{i} U g a p_{t - i} + ε_{t} \end{array}$					$\begin{array}{l} Δ w_{t} = θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} + ϕ U g a p_{t}^{2} \\ + \sum_{i = 0}^{1} λ_{i} U g a p_{t - i} + ε_{t} \end{array}$					$\begin{array}{l} Δ w_{t} = θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} \\ + \sum_{i = 0}^{1} λ_{i} U g a p_{t}^{K F} + ε_{t} \end{array}$
β₁=1 & β₂=1		Chi^2(2)= 0.021		[0.9897]	β₁=1 & β₂=1		Chi^2(2)= 0.415		0.8125	β₁=1 & β₂=1		Chi^2(2)= 0.0091		[0.9955]
Parameter	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob
θ	−1.100	0.086	−12.80	0.000	θ	−1.099	0.085	−12.86	0.000	θ	−1.104	0.087	−12.75	0.000
λ₀	−0.802	0.261	−3.07	0.003	λ₀	−0.964	0.281	−3.43	0.001	λ₀	−0.818	0.263	−3.11	0.002
λ₁	0.491	0.261	1.88	0.063	λ₁	0.497	0.260	1.91	0.058	λ₁	0.523	0.263	1.99	0.050
					ϕ	0.045	0.030	1.51	0.134
Memo: Cummulative Impact Estimates					Memo: Cummulative Impact Estimates					Memo: Cummulative Impact Estimates
	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob
(λ₀+λ₁)	−0.311	0.050	−6.27	0.000	(λ₀+λ₁)	−0.466	0.114	−4.09	0.000	(λ₀+λ₁)	−0.295	0.047	−6.24	0.000
R-squared		0.496			R-squared		0.507			R-squared		0.488
S.E. of regression		0.778			S.E. of regression		0.774			S.E. of regression		0.777

Notes: The estimation sample is 1990Q1-2017Q3.

Table 5.

ECI growth: Non-Linearity and the Kalman Filter

Baseline Model					Non-linear Model					Kalman Filter Model
$\begin{array}{l} Δ w_{t} & = & θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} \\ + \sum_{i = 0}^{1} λ_{i} U g a p_{t - i} + ε_{t} \end{array}$					$\begin{array}{l} Δ w_{t} = θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} + ϕ U g a p_{t}^{2} \\ + \sum_{i = 0}^{1} λ_{i} U g a p_{t - i} + ε_{t} \end{array}$					$\begin{array}{l} Δ w_{t} = θ + β_{1} Δ p_{t}^{t r e n d} + β_{2} Δ y_{t}^{t r e n d} \\ + \sum_{i = 0}^{1} λ_{i} U g a p_{t}^{K F} + ε_{t} \end{array}$
β₁=1 & β₂=1		Chi^2(2)= 0.021		[0.9897]	β₁=1 & β₂=1		Chi^2(2)= 0.415		0.8125	β₁=1 & β₂=1		Chi^2(2)= 0.0091		[0.9955]
Parameter	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob
θ	−1.100	0.086	−12.80	0.000	θ	−1.099	0.085	−12.86	0.000	θ	−1.104	0.087	−12.75	0.000
λ₀	−0.802	0.261	−3.07	0.003	λ₀	−0.964	0.281	−3.43	0.001	λ₀	−0.818	0.263	−3.11	0.002
λ₁	0.491	0.261	1.88	0.063	λ₁	0.497	0.260	1.91	0.058	λ₁	0.523	0.263	1.99	0.050
					ϕ	0.045	0.030	1.51	0.134
Memo: Cummulative Impact Estimates					Memo: Cummulative Impact Estimates					Memo: Cummulative Impact Estimates
	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob	Parameter	Est.	s.e.	t-value	t-prob
(λ₀+λ₁)	−0.311	0.050	−6.27	0.000	(λ₀+λ₁)	−0.466	0.114	−4.09	0.000	(λ₀+λ₁)	−0.295	0.047	−6.24	0.000
R-squared		0.496			R-squared		0.507			R-squared		0.488
S.E. of regression		0.778			S.E. of regression		0.774			S.E. of regression		0.777

Notes: The estimation sample is 1990Q1-2017Q3.

Next, we generate a new estimate of the path of the natural rate of headline unemployment using the Kalman filter. We assume the natural unemployment rate follows a random walk and jointly estimate its path with the augmented Phillips curve model. Figures 9a shows the natural rate series from the Kaman filter and compares it to the CBO series. It is interesting to note that the two natural rate series display very similar behavior. The CBO series is noticeably higher over the period 2009–14. But even then, the maximum difference is only about 0.3 percentage points. Recent readings from both series put the natural rate below 4.8 percent. Figure 9b shows the corresponding gap measures, which do comove very closely. It is not surprising then that the coefficient estimates of the Phillips curve with the natural rate generated jointly through Kalman filtering are very similar in size, sign, and statistical significance to those of our baseline model. This can be seen in Table 5, if one compares the Kalman filter model (depicted in the right panel) to the baseline model (left panel).

Alternative measures of wages

Here, we carry out our analysis using different measures of wages. These are: the ECI for wages and salaries (no employer costs for employee benefits) of nonfarm business sector workers; average hourly earnings of production and nonsupervisory employees on private nonfarm payrolls; compensation per hour in the nonfarm business sector, and the Atlanta Fed wage tracker.

Table 6a summarizes the long run properties of the models. For all models, tests reveal one and only one cointegrating relationship where nominal wage growth depends on inflation, and productivity growth, with a coefficient of unity on each. This result is identical to our baseline model which used ECI for compensation.

Table 6a.

Johansen Cointegration Analysis for VARs with Alternative Wage Measures