Will the true labor share stand up.
Date: 2015-06-25
Institutions: European Central Bank
Subjects: data collection, economic analysis, economic fluctuation, macroeconomics, pay, wage determination

Working Paper Series 

Jakub Mućk, Peter McAdam and  
Jakub Growiec  

Will the true labor share  
stand up? 

No 1806 / June 2015 

Note: This Working Paper should not be reported as representing the views of the European Central Bank (ECB). 
The views expressed are those of the authors and do not necessarily reflect those of the ECB 

 
 
 
 
Abstract

We document the consequences of ambiguity in the empirical deﬁnition of the
macroeconomic labor share. Depending on its deﬁnition, the properties of short-
run ﬂuctuations, medium-run swings, and long-run stochastic trends of the la-
bor share may vary substantially. Based on a range of historical US time se-
ries, we carry out a systematic exploration of discrepancies between the alterna-
tive labor share deﬁnitions in terms of the observed stochastic trends, shares of
short-, medium- and long-run variation in total volatility of the series, degree of
persistence, mean-reversion properties, and susceptibility to structural breaks.
We conclude that while short-run properties of the labor shares (represented by
cyclical variation below 8 years) are relatively consistent across all deﬁnitions,
their medium-run swings (8-50 years) and long-run trends (≥ 50 years) diverge
substantially. As important applications, we document the implications of our
ﬁndings for growth accounting, the identiﬁcation of short-run responses of the
labor share to technology shocks and for estimating inﬂation.

Keywords: labor share, spectral analysis, persistence, mean reversion, structural
breaks

JEL Codes: C82, E25, E32

ECB Working Paper 1806, June 20151NON TECHNICAL SUMMARY

We document the consequences of ambiguity in the empirical deﬁnition of the macroeconomic
labor share. Based on annual US labor share series spanning 1929–2012 and a quarterly series
beginning in 1947, we carry out a systematic exploration of discrepancies between the alternative
labor share deﬁnitions in terms of the observed stochastic trends, shares of short-, medium- and
long-run variation in total volatility of the series, degree of persistence, mean-reversion properties,
and evidence for structural breaks.

Our results suggest that while short-run properties of the labor shares (represented by cyclical
variation below 8 years) are relatively consistent across all deﬁnitions, their medium- run swings
(8-50 years) and long-run trends (above 50 years) substantially diverge. Having considered the al-
ternatives, we argue that the US series on the share of employees’ compensation in GDP, adjusted
for proprietors’ income is not only most sound theoretically, but also has intuitive, economically
interpretable proprerties. This measure suggests that the US labor share has not only declined
after 1970, but also substantially increased before that, exhibiting a hump-shaped pattern over the
last 84 years. It corroborates the idea that instead of concentrating on the decline in the labor share
since 1970, one could also embrace the larger time span of available data and discuss the (possibly
technology-driven) long cycle in this variable.

We then illustrate the possible range of discrepancies in conclusions when certain macroeco-
nomic studies are based upon different measures of the labor share. We concentrate on two inter-
esting applications. The ﬁrst of them is growth accounting. We conclude that the intertemporal
variability of factor shares has a negligible impact on the resulting TFP measures, but differences
in average levels of factor shares across various measurements can cumulate over years to about
12% of TFP.

Second, we deal with the question of empirical identiﬁcation of the impact of technology
shocks on the labor share. We ﬁnd that the identiﬁed impulse responses vary largely across labor
share deﬁnitions. Finally, we examine the sensitivity of the estimation of inﬂation New Keynesian
Phillips curves to different labor share series.

ECB Working Paper 1806, June 201521

INTRODUCTION

There has recently been a revival of debates surrounding the sources and consequences of shifts
in labor’s share of GDP. This revival is probably due to at least two reasons. First, it has been
argued that the labor share has exhibited a protracted decline since 1970s (Arpaia, Pérez, and
Pichelmann, 2009; Elsby, Hobijn, and Sahin, 2013). Second, it has also been observed that the
labor share is subject to substantial countercyclical short-run volatility (Young, 2004; McAdam
and Willman, 2013).

A key problem in this debate is the fact that there is no consensus as to how the labor share
should be deﬁned. The ambiguity arises from the fact that although total compensation of em-
ployees as well as companies’ aggregated operational surplus are observable, the labor share is
not, because a sizable share of the total value added is generated by the self-employed. This mixed
income cannot be unambiguously understood as either the remuneration of capital or labor. In
consequence, the measured labor share necessarily depends on the assumptions made in relation
to the division of mixed income. Another caveat is related to the treatment of taxes on capital and
labor incomes which may or may not be included in the computation of factor remuneration and
total output (Gollin, 2002).

Although assigning ambiguous income to capital or labor is ultimately a matter of choice, it
is rarely appreciated that this conceptual ambiguity has empirical consequences. These conse-
quences still seem not to have been sufﬁciently researched thus far, further deepening the con-
fusion. For example, it is customary in the business-cycle literature to adjust the labor share by
proprietors’ income (Young, 2004), whereas the structural analysis econometrics literature prefers
to adjust by the fraction of self-employed in total employment (Arpaia, Pérez, and Pichelmann,
2009; Klump, McAdam, and Willman, 2007; Raurich, Sala, and Sorolla, 2012), and neither of these
literatures confronts the role of the assumed deﬁnition. Thus whilst many papers (for example
Gollin’s (2002) seminal contribution) promote discussion on how labor shares could be measured,
there is none which systematically examines and tries to understand those differences. This paper
ﬁlls that important gap.

Our contribution then is to provide a systematic exploration of the dynamic properties of a
range of alternative labor share measures. Our investigation is based on annual US labor share
series spanning 1929–2012 and quarterly series from 1947. We document that these measures are
not only divergent in terms of the implied time trends, which are visible to a naked eye, but also
differ in terms of their other dynamic properties, such as the shares of short-, medium- and long-
run variation in total volatility of the series, degree of persistence, mean-reversion properties, and
susceptibility to structural breaks.

Our results point to the general conclusion that while short-run properties of the labor shares
(represented by cyclical variation below 8 years) are relatively consistent across all deﬁnitions,
their medium-run swings (8-50 years) and long-run trends (≥ 50 years) diverge substantially.
Therefore, it is indeed important to “get factor shares right”, especially if one is interested in the
medium and long run.

Having considered the alternatives, we argue that the US series on the share of employees’
compensation in GDP, adjusted for proprietors’ income following Cooley and Prescott (1995) and
Gomme and Rupert (2007) procedure (which we call PI2-GDP) is probably the most sound the-
oretically, and also has intuitive, economically interpretable empirical properties. It provides the
relatively most consistent message across a range of diverse exercises and applications (discussed
below) while remaining in agreement with known “stylized facts” formulated elsewhere in the lit-
erature (e.g., it is mean-reverting but highly persistent, countercyclical over the short run, and has
recorded a secular decline since 1970). This measure suggests, however, that the US labor share
has not only declined after 1970, but also substantially increased before that, exhibiting a hump-
shaped pattern over the last 84 years. Hence, instead of mostly concentrating the decline in the
labor share since the 1970s, one could also embrace the larger time span of available data; the

ECB Working Paper 1806, June 20153proﬁle as a whole is suggestive of a long cycle of activity (reminiscent of the work of Kondratieff
and Schumpeter).

Having identiﬁed the key differences between the respective labor share deﬁnitions, we con-
centrate on three interesting and well-motivated applications. The ﬁrst is growth accounting,
namely the decomposition of output growth into factor accumulation and technical progress. We
conclude that the intertemporal variability of factor shares has a relatively small impact on the
resulting TFP measures, but differences in average levels of factor shares across various measure-
ments can cumulate over years to about 12% of TFP (which is the difference between PI2-GDP
and the “naive” measure, the payroll share, over 1929–2012), or even 40% in the most extreme
case.

Second, we deal with the question of empirical identiﬁcation of the short-run impact of tech-
nology shocks (Fernald, 2012) on the labor share. Based on an autoregressive distributed lag spec-
iﬁcation and a bi-variate VAR model for quarterly data, we ﬁnd that the identiﬁed impulse re-
sponses vary largely across labor share deﬁnitions. We conclude that although business cycle
properties of all these series are relatively consistent when discussed in isolation, their relation-
ships with other macroeconomic variables react sensitively to changes in measurement.

Finally, we re-examine empirical estimates of the popular New Keynesian Phillips curves
(NKPC) following Galí and Gertler (1999) and the large subsequent literature (e.g., Mavroeidis,
Plagborg-Møller, and Stock, 2014). We ﬁnd large differences in ﬁxed-price durations and the sen-
sitivity (or “slope”) of inﬂation when using different labor share measures. The properties of the
labor share series used can help shed light on the empirical properties of the associated NKPCs.

The remainder of the paper is structured as follows. Section 2 provides some simple theory
In Section 3 we construct the time series of the US labor share un-
background to our study.
der a range of its alternative empirical deﬁnitions. In Section 4 we discuss their basic dynamic
properties, including their degree of persistence and mean reversion properties. In Section 5 we
document the evidence for structural breaks. In Section 6 we carry out a spectral decomposition
of these series into their short-, medium- and long-run components. The implications of the iden-
tiﬁed differences between alternate US labor share series are then discussed in three applications.
Section 7.1 presents an application of our results to growth accounting. Section 7.2 presents the
analysis of labor share responses to technological shocks along the business cycle. Section 7.3 anal-
yses New Keynesian Phillips curves estimation under different labor share deﬁnitions. Section 8
concludes.

2

SOME SIMPLE THEORY BACKGROUND

Understanding the sources of observed variation in the labor share is a complex matter. Alternate
empirical measures of this variable may diverge for many reasons: in the short run, they may
be affected by business cycles, factor utilization rates, ﬁscal and monetary policy changes, etc.,
whereas in the long run there are technological developments, skill patterns, changes in sectoral
composition, economic regulations, changes in tax progressivity, etc. Nonetheless, we can appeal
to simple production-function theory to begin our discussion.

Consider the standard CES production function:

(cid:34)

Yt =

α

(cid:16)

ΓK
t

Kt

(cid:17) ζ−1

ζ + (1 − α)

(cid:17) ζ−1

ζ

(cid:16)

ΓL
t

Lt

(cid:35) ζ
ζ−1

(1)

where Yt represents real output and Kt = κt × Kt, Lt = (cid:96)t × Lt where Kt is the capital stock, Lt
is the labor input, κ and (cid:96) are measures of factor utilization and ζ ≥ 0 is the elasticity of factor
substitution. Function (1) reduces to Cobb Douglas in the limiting case ζ → 1; the ﬁxed factor
proportions Leontief function when ζ → 0; and a linear function (i.e., perfect factor substitution)

ECB Working Paper 1806, June 20154when ζ → ∞. Terms ΓK
tively. Technical progress can then be Hicks-neutral (ΓK
, ΓN > 0), Solow-neutral (ΓK

t and ΓL

t capture capital and labor-augmenting technical progress, respec-
t = 0

t > 0), Harrod-neutral (ΓK

t = ΓL

t = 0) or factor-augmenting (ΓK
Given this, the relative capital-to-labor income share, given competitive factor markets and

t > 0 (cid:54)= ΓL

t > 0, ΓL

t > 0).1

proﬁt maximization, can be expressed as

Θt =

wt Lt
rt Kt

=

1 − α
α

(cid:18) ΓK
t κtKt
ΓL
t (cid:96)tLt

(cid:19) 1−ζ

ζ

,

(2)

where rt and wt denote the user cost (or marginal productivity) of capital and the real wage,
respectively.

For factor incomes shares to be constant, dΘt = 0, requires:

1. ζ = 1. If production is Cobb–Douglas then any trend in the capital–labor ratio, or biased

technical change is completely offset such as to maintain stable factor shares.2

2. If the bias in technical change exactly offsets accumulation of capital per worker (in growth

terms) then factor income shares are stable for any value of ζ.

Otherwise, factor income shares are changed by movements in capital per worker or biases in
technical change or relative movements in factor utilization. The direction of the effect, however,
depends on the value of the substitution elasticity, and, in the case of technology shocks, on their
source (i.e., whether they augment capital or labor):

∂Θt
t /ΓK

t

∂ (cid:0)ΓL

∂Θt
∂ (Kt/Lt)

(cid:1) ,

< 0 for ζ < 1

= 0 for ζ = 1

> 0 for ζ > 1

(3)

For instance, it is often suggested that the immediate post-war period was (in many developed
countries) a period of rising labor share. If we assume that factors are gross complements, ζ < 1
(as in Chirinko (2008); Klump, McAdam, and Willman (2012)), then the conditions for such a
rise would have to involve either increases in capital per worker and/or capital-saving technical
progress:

1.

˙K/K > ˙L/L

2. ΓK > ΓN

By contrast, if factors are gross substitutes these two above margins work in the reverse way.

Either of these margins might explain the rise in the labor share, say, up until the 1970s. But
whether the labor share continues to rise, falls or stabilizes depends on the evolution of these two
inequalities. Rewriting condition (2), the factor ratio (i.e., part 1. of the above) can be shown to be
a function of relative factor prices, the direction of technical bias and relative factor utilization:

log

(cid:19)

(cid:18) Kt
Lt

= φ + ζ log

(cid:19)

(cid:18) wt
rt

+ (1 − ζ) log

(cid:19)

(cid:18) ΓL
t (cid:96)t
ΓK
t κt

(4)

1Neutrality concepts associate innovations to related movements in marginal products and factor ratios. An inno-
vation is Harrod-neutral if relative input shares remain unchanged for a given capital-output ratio. This is also called
labor-augmenting since technical progress raises production equivalent to an increase in the labor supply.

2Although, strictly speaking, technical change (biased or otherwise) cannot be identiﬁed in a Cobb–Douglas setting.

ECB Working Paper 1806, June 20155where φ is a composite constant. Accumulation of capital per worker, thus, occurs if real wages
rise relative to the user cost of capital (i.e., factor prices favor capital accumulation) and, assuming
gross complements, if technical change is net labor-augmenting, or labor is used more intensively
than capital. Under Cobb–Douglas, note, accumulation of capital per worker is driven solely by
relative factor prices, one-to-one.

Biased technical change (part 2. above) is difﬁcult to measure given its latent status. While in
theoretical models of directed technical change, it depends upon ﬁrms’ proﬁt incentives and inno-
vation possibilities (Acemoglu, 2002; Leon-Ledesma and Satchi, 2015), in empirical work, factor-
augmenting technical progress it is often proxied by ﬂexible functional forms (see e.g., Klump,
McAdam, and Willman, 2007).

Keeping the above discussion in mind, from the next section onwards we shall depart from
the above stylized economy with CES production and competitive factor markets, which allowed
us to ﬁnd a direct link between factor shares, factor endowments, and technological progress. In-
stead, our objective will be to investigate the empirical implications of a range of alternative factor
share deﬁnitions applied to US data. Nonetheless, the key insight from the above theory is that the
magnitude of elasticity of substitution between capital and labor, ζ, is key to understanding the
ﬁndings on factor share cyclicality as well as the results of our three empirical applications (dis-
cussed in Section 7) because it affects virtually all relationships between factor shares and other
economic variables (Chirinko, 2002; Irmen, 2011; Cantore, León-Ledesma, McAdam, and Willman,
2014).

3 ALTERNATIVE MEASURES OF THE LABOR SHARE

Formally, the aggregate labor share is deﬁned as the proportion of total remuneration of the labor
force (wtLt) in aggregate output of the economy (GDP or total value added, Yt):

LSt =

wtLt
Yt

.

While such a deﬁnition appears theoretically unambiguous, both the numerator and the denomi-
nator of the above ratio can be measured empirically in various ways (Gollin, 2002), with poten-
tially diverging implications.

3.1 EMPIRICAL MEASUREMENT

The simplest, “naive” way to construct an empirical series of the labor share based on this deﬁ-
nition is to use total compensation of employees (CEt) for the numerator of equation (3). According
to the System of National Accounts, compensation of employees contains the sum of both wages
and other payments to employees. Thus, to derive the labor share, nominal CEt can be simply
divided by nominal output Yt. Thus one computes a measure which we label “Naive GDP”:

Naive-GDP : LSt =

CEt
Yt

,

(5)

where Yt is a generic measure of output. Typically it is GDP; however, in sectoral studies gross
value added (GVA) is often used (see Bentolila and Saint-Paul, 2003; Young, 2010, 2013).

Although straightforward to compute and easily interpretable, this method (the “payroll share”,

cf. Elsby, Hobijn, and Sahin (2013)) has a few crucial empirical disadvantages – the most important
of which is that compensation of employees CEt does not include mixed income, i.e., the ambiguous
income earned by the self-employed, which cannot be directly ascribed to capital or labor. Since
at least part of mixed income remunerates proprietors’ labor, this leads to a systematic underes-
timation of the labor share at the aggregate level. There are at least three ways to deal with this

ECB Working Paper 1806, June 20156issue: (1) assuming that the self-employed (proprietors) face identical average wage as the non-
self-employed; (2) assuming identical labor shares in both groups, and (3) assuming an arbitrary
rule of thumb to divide proprietors’ income. We elaborate on these options below.

The ﬁrst approach to include the ambiguous income in the labor share is to use data on the
number of self-employed (SEt). The key assumption used in this adjustment is that labor com-
pensation is equal on average for both employees (Et) and self-employed workers (SEt). Then the
“naive” labor share is increased by the imputed compensation of the self-employed, as in:

SE-GDP : LSt =

CEt
Yt

(cid:18)

1 +

(cid:19)

.

SEt
Et

(6)

The second way to adjust the labor share refers directly to the concept of mixed income. Pro-
prietors’ income (PIt), as deﬁned in the System of National Accounts, is the ambiguous part of
output which cannot be treated as pure labor or capital income. Labor share estimates can then be
adjusted by the means of a simple reduction of the output by PIt:

PI-GDP : LSt =

CEt
Yt − PIt

.

(7)

This approach is equivalent to assuming that mixed income is split between labor and capital
income in the same proportion as in the rest of the economy.

The third, similar in spirit but more comprehensive approach to dealing with mixed income
has been proposed by Cooley and Prescott (1995), and developed by Gomme and Rupert (2007)
and Ríos-Rull and Santaeulàlia-Llopis (2010). Its starting point is a decomposition of total income
into two components: ambiguous (AIt) and unambiguous (U It) income. Ambiguous income AIt
is the sum of proprietors’ income, taxes on production less subsidies, business current transfer
payments and statistical discrepancies:

AIt = PIt + (Taxt − Subt) + BCTPt + SDISt.

Neither of these amounts is directly attributable to capital or labor.

Unambiguous income U It, on the other hand, is straightforwardly separated into unambigu-

ous labor and capital income components:

U It = ULIt
(cid:124) (cid:123)(cid:122) (cid:125)
CEt

+UKIt,

where the latter consists of consisting of rental income, net interests, current surplus of govern-
ment enterprises, and corporate proﬁts:

The share of capital in unambiguous income (KSU

t ) is obtained as:

UKIt = RIt + N It + GEt + CPt.

KSU

t = 1 − LSU

t =

UKIt + DEPt
U It

=

RIt + N It + GEt + CPt + DEPt
RIt + N It + GEt + CPt + CEt

where DEP is the consumption of ﬁxed capital (Table 1.7.5 of NIPA-BEA).

The key assumption underlying the current adjustment method is that the shares of capital
t AIt. Then,

and labor in ambiguous income are the same as in unambiguous income, AKIt = KSU
the labor share is computed as follows:

PI2-GDP : LSt = (1 − KSt) = 1 −

UKIt + DEPt + AKIt
Yt

.

(8)

The theoretical arguments why PI2-GDP is likely to be a relatively accurate representation
of the “true” labor share are as follows. First, it covers the entire economy and carefully con-
siders many distinct economic quantities, reported in NIPA, including the ambiguous income.

ECB Working Paper 1806, June 20157Hence, from the macroeconomic perspective it should be more robust to structural changes, such
as changes in the sectoral or private vs. public composition of value added, than e.g. the corporate
labor share (Karabarbounis and Neiman, 2014). Second, its core assumption, that the ambiguous
income is split between labor and capital income in the same proportion as in the rest of the
economy, makes this measure much more accurate in the case of long-dated series when com-
pared to series assuming that labor compensation is equal on average for both employees and
self-employed workers (e.g., SE-GVA, see Elsby, Hobijn, and Sahin (2013)): in the early twentieth
century in the US, just like in less developed countries today, most self-employed workers were
farmers who earned much less than the contemporaneous average wage in industry and services.
Finally, Gollin (2002) proposes also an adjustment where the entire proprietors’ income is
treated as compensation of labor. Such an approach likely leads to a sharp overestimation of
the labor share. Accordingly, Johnson (1954) uses an equally simple rule of thumb: two-thirds of
proprietors’ income to labor.

Another issue in constructing the labor share is whether aggregate output Yt in the denomina-
tor is identiﬁed with GDP or gross value added (GVA). It turns out that empirically factor shares
in value added differ systematically from factor shares in GDP (Valentinyi and Herrendorf, 2008).
This argument ought to be borne in mind particularly when GVA is employed in more aggregated
frameworks. For instance, Karabarbounis and Neiman (2014) document a global decline in the la-
bor share, using data on corporate gross value added, which accounts for 60% of overall GVA,
instead of GDP.3

3.2 SOURCES OF DISCREPANCY

There are clear-cut theoretical indications under which assumptions the aforementioned labor
share measures are equivalent. Failure to meet these conditions is then the reason for their dis-
crepancy. We make four points in that regard.

1.

The naive-GDP measure could equal any other measure only in the counterfactual case
where there were no proprietors’ income in the economy. Hence it is always downward biased. The
difference between the payroll share and adjusted labor share measures is the larger, the greater
is the actual share of mixed income in total output.

2.

SE-GDP coincides with PI-GDP if and only if the share of the self-employed in the total

labor force is equal to the share of proprietors’ income in GDP:

SEt
Et + SEt

=

PIt
Yt

.

(9)

Otherwise, the SE-GDP labor share measure exceeds PI-GDP if and only if, on average, employ-
ees obtain a proportionally larger share of output than the self-employed:

.

SEt
Et+SEt

> PIt
Yt

Figure 1 illustrates that after the peak in the self-employed share during the Great Depression,
and a following period of its sharp decline in the 1930s, both sides of equation (9) declined in a
roughly parallel way between World War II and the 1970s. The high share of self employed in the
pre-war period reﬂected the importance of Agriculture and the substitution to self-employment
during the Great Depression. Thereafter, both as a share of output and employment, Agriculture
declined reﬂecting the rise of Manufacturing with its large scale economies and of the public sector
which attracted and absorbed resources from Agriculture.

3Measuring the labor share is not limited to aggregate or sectoral data only. Highly disaggregated data are some-
times also used to estimate aggregate factor shares. For example, Young (1995) used census and survey data to match
the self-employed and other unpaid workers with employees, cross-tabulated by gender, sector, age and other relevant
characteristics. He then imputed implicit labor compensation for the individuals belonging to the labor force groups
which are listed as “unpaid” in ofﬁcial statistics. So imputed labor incomes constitute a micro-founded way of ad-
justing the naive labor share measure. Moreover, the labor share might also gathered from ﬁrm-level data (see e.g.,
Growiec, 2012).

ECB Working Paper 1806, June 20158Figure 1: The Ratio of the Self-Employed to Total Employment and the Share of Mixed Income in GDP.

Notes: The blue line denotes the share of the proprietors’ income in GDP (PIt/GDPt). The green
line represents for the ratio of the self-employed to total employment (SEt/(Et + SEt)).

From the 1970s onwards, that rapid decline in self employed comes to a halt. This reﬂected
factors such as technological changes which helped reduce operating costs and the importance
of scale in favor of smaller-scale enterprises, a greater use of contracting out, demographic shifts,
and so on.

The share of proprietors’ income has a similar overall dynamic to that of the self-employed
share. After 1980, however, the share of proprietors’ income in GDP began to rise despite bottom-
ing out of the share of the self-employed in the labor force. Both lines crossed in late 1990s. Now
it is the self-employed who earn a proportionally larger share of the GDP than employees (which
can be partly due to statistical error, Elsby, Hobijn, and Sahin, 2013) and thus the PI-GDP exceeds
the SE-GDP labor share.

3.

PI2-GDP coincides with PI-GDP as long as ambiguous income which is not directly pro-
prietors’ income (i.e., taxes on production Taxt and business current transfer payments, BCTPt)
is positive and attributed fully to capital. If factually this income is also partly generated by la-
bor, however, then PI2-GDP should be relatively higher while PI-GDP (and, by the same token,
SE-GDP) should be an unambiguously downward-biased measure of the true labor share.

4.

The discrepancy between labor share measures based on GDP and GVA follows from
the difference between both denominators, driven by taxes on production and imports, minus
subsidies.

4 PROPERTIES OF THE US LABOR SHARE SERIES

To construct long historical labor share series for the US economy employing all the aforemen-
tioned measurement methods, we used annual data from National Income and Product Accounts
(NIPA) tables of the Bureau of Economic Analysis (BEA), and quarterly data from the Bureau of
Labor Statistics (BLS). This choice of data sources stems from our wish to construct as long series
as possible; the annual and quarterly series span 1929–2012 and 1947q1–2013q1 respectively. A
detailed description of the constructed series is included in appendix Table A.1.

19401960198020000.050.100.150.200.250.30ECB Working Paper 1806, June 20159To recall, the series computed with the naive method, following equation (5), is denoted as
Naive-GDP (the payroll share of GDP). The next series, SE-GDP, includes adjustment for the
share of the self-employed (equation (6)). Adjustment by proprietors’ income, in the form of a
reduction in the considered measure of output (equation (7)), has been used to construct the series
PI-GDP. A more sophisticated adjustment by proprietors’ income (equation (8)) is employed in
the series PI2-GDP. Keeping measures of the labor’s share of GDP as our benchmark, we also
calculate the “naive” annual labor share in gross value added (GVA) in the private sector (Naive-
GVA, the payroll share of GVA), and in the non-farm private sector (Naive-GVA-NF). The next
two variants are constructed by adjusting the above series by the number of the self-employed in
the corresponding sectors (denoted as SE-GVA and SE-GVA-NF, respectively).4 The last version
of the labor share used in the current study is taken from the BLS. The BLS labor share series is a
quarterly index, whose initial level is not determined.

4.1 GRAPHICAL ANALYSIS

Figures 2 and 3 show the annual and quarterly labor income share time series, respectively. Note
ﬁrst the level differences between the series. For instance, PI2-GDP exceeds Naive-GDP by 11
pp. on average (i.e., almost 1/5th of its level). Systematic differences are substantial also for other
pairs of measures. Such differences, as discussed below in Section 7.1, will for instance have im-
plications for growth-accounting exercises and the retrieval of TFP. Eyeballing the historical series
suggests that over the long run, differences between the variants often systematically diverge.
Thus, the factors which drive a wedge between the series – number of self employed, proprietors’
income, taxes and subsidies – are time-varying. This applies in particular to the comparisons be-
tween adjusted series and their “naive” counterparts. In Appendix D, we address this issue more
systematically by checking stationarity of differences between all possible pairs of alternative labor
share measures. Our results indicate that stationarity of differences is typically rejected.

Figure 2: Annual US Labor Share

Notes:
Left panel: Naive-GDP, PI-GDP, PI2-GDP, SE-GDP
Right panel: Naive-GVA, Naive-GVA-NF, SE-GVA-NF, SE-GVA. NBER recession periods are over-
laid at the appropriate frequency.

Visible discrepancies, however, also relate to dynamics. Most importantly, the “naive” series
as well as series adjusted by mixed income exhibit hump-shaped trajectories, whereas the labor
share modiﬁed by the share of the self-employed records a consistent, strong downward tendency

4Note that SE-GVA is also the “headline measure” of the US labor share in Elsby, Hobijn, and Sahin (2013).

1930194019501960197019801990200020100.500.550.600.650.701930194019501960197019801990200020100.500.550.600.650.700.750.80ECB Working Paper 1806, June 201510Figure 3: Quarterly US Labor Share

Notes:
Left panel: Naive-GDP, PI-GDP, PI2-GDP, SE-GDP
Right panel: Naive-GDP, PI2-GDP, BLS. For comparison all series set at 2000=100.

throughout the period. This particular behavior is likely driven by (a) the sharp fall in the share
of the self-employed in total employment until around 1970 (recall Figure 1), and (b) an overes-
timation of incomes among the self-employed in the immediately following period, identiﬁed by
Elsby, Hobijn, and Sahin (2013).

Moreover, even the much heralded labor income decline since the 1970s is not universal. Series
based on value added have been apparently stable since the 1940s, as has the PI-GDP variant
(annual and quarterly). All series, though, do share a steep fall since the 2001 recession.

Notwithstanding, all of these series are meant to measure the same thing: namely the share of
US national income that goes to labor. All of them have been widely used in various literatures.
Yet, we have little understanding of the properties of these different series: for instance, in line
with Kaldor’s stylized facts, can we view the shares as stable or quasi-stable (e.g., correcting for
structural breaks); how persistent and volatile are they? What, in any given exercise, is the conse-
quence of using one labor share measure rather than another? For instance, if income shares are
not stable how would growth accounting exercises (which retrieve TFP) change? And would it
alter the importance of TFP in accounting for economic growth? If labor’s share of income cannot
be uniquely measured, how would that change debates about income inequality? These are issues
we explore.

4.2

SUMMARY STATISTICS

Summary statistics (and short-run characteristics) are presented in Tables 1 and 2 (where (cid:101)x is the
logged then HP ﬁltered series).5

We computed the cumulative changes over two subperiods with a breakpoint in 1970, and for
most annual series (6 out of 8), the decrease in the labor share after 1970 was smaller than the
strong rise from 1929 to 1970. This applies in particular to PI2-GDP. The series are also mostly

5We applied a smoothing parameter equal to 100 and 1600 to annual and quarterly data, respectively. We also
examined one-sided HP ﬁltered series as well as used the Ravn and Uhlig (2002) adjustment to the smoothing parameter
applied to the annual series, with minimal qualitative differences.

19501960197019801990200020100.550.600.650.7019501960197019801990200020109095100105ECB Working Paper 1806, June 201511characterized by negative skewness (i.e., by a long tail to the left indicating a few very low val-
ues) but with no particular common features in kurtosis (“peakedness”). Likewise the null of
Normality is mostly rejected for annual and quarterly series.

The volatility, relative to output, ranges from 23% − 36% for the annual series (the SE-GVA
series being the most volatile), and 36% − 48% (quarterly) with the BLS series more volatile than
the rest. The auto-correlation varies between 0.43 − 0.64 (annual) to 0.63 − 0.74 (quarterly). The
labor share is generally counter-cyclical (especially in quarterly data and since 1947).6 However
this counter-cyclicality is not especially strong (around -0.2,-0.4 for the quarterly series) and there
is in some cases acylicality (PI-GDP). It is further interesting to note that PI-GDP and PI2-GDP,
though intended to measure the same aspect (namely labor share corrected for proprietors’ in-
come) have such distinct properties: in annual terms the former is apparently pro-cyclical and the
latter a-cyclical; whereas in quarterly data, the former is acyclical and the latter counter-cyclical.

Table 1: Annual Labor Share: Summary Statistics

P
D
G
-
e
v
i
a
N

P
D
G

-
I
P

P
D
G

-
2
I
P

P
D
G
-
E
S

A
V
G
-
e
v
i
a
N

-

F
N
A
V
G
-
E
S

A
V
G
-
E
S

mean
median
st.d.
min
occurrence
max
occurrence
1970-1929
2011-1970
Skewness
Kurtosis
Normality

0.639
0.636
0.023
0.591
2011
0.725
1932
0.033

0.674
0.674
0.017
0.581
1929
0.711
1970
0.115

0.614
0.616
0.016
0.571
1929
0.644
1943
0.098

0.557
0.560
0.023
0.555
1929
0.594
1970
0.062

0.671
0.666
0.026
0.637
2011
0.795
1932
0.089
−0.030 −0.063 −0.066 −0.040 −0.044 −0.068 −0.063 −0.063
−0.621 −0.423 −0.362
2.498
−0.488 −0.242
7.772
0.854
[0.000]
[0.080]
[0.259]
[0.044]

1.034 −0.664 −0.804 −0.514
2.378 −0.845
1.241
[0.007]
[0.014]
[0.000]

0.570
0.583
0.032
0.501
1936
0.620
2001
0.101

0.646
0.647
0.018
0.597
1941
0.698
1932
0.086

0.046
[0.009]

-

F
N
A
V
G
-
e
v
i
a
N
0.588
0.593
0.022
0.531
1941
0.623
2001
0.017

SHORT-RUN CHARACTERISTICS
0.015
0.268

0.013
0.227

0.013
0.226

0.013
0.229

σ
0.015
(cid:101)ls
σ
(cid:101)ls/σ
0.258
(cid:101)y
corr( (cid:101)lst, (cid:101)lst−1) 0.530∗∗∗ 0.528∗∗∗ 0.637∗∗∗ 0.518∗∗∗ 0.434∗∗∗ 0.398∗∗∗ 0.441∗∗∗ 0.540∗∗∗
0.444∗∗∗−0.065 −0.465∗∗∗−0.262∗∗∗−0.196∗∗∗−0.533∗∗∗−0.652∗∗∗
corr( (cid:101)lst, (cid:101)yt)
Note: Normality test is Jarque-Bera. Superscripts ∗∗∗, ∗∗ and ∗ stands for rejection of the null about in-
signiﬁcant auto- or cross -correlation at the 1%, 5% and 10% signiﬁcance level, respectively.

0.015
0.257

0.020
0.355

0.018
0.316

0.174∗∗

4.3 PERSISTENCE

A key property of time series is its persistence. If a time series is subject to a shock, the level of
persistence tells us if, and how soon, the series will revert to its mean: the higher the persistence
the slower the reversion.

6The cyclical co-movement of the labor share with output differs signiﬁcantly among the constructed variants,
In particular, as opposed to other labor share measures, the short-run components of annual Naive-GDP
though.
and PI-GDP measures are signiﬁcantly positively correlated with output. This is driven primarily by the strongly pro-
cyclical behavior of these series in the beginning of the sample, before World War II. We conﬁrm that these series
are again countercyclical when considered for the post-war sub-period only, consistent with their quarterly post-war
counterparts.

ECB Working Paper 1806, June 201512Table 2: Quarterly Labor Share: Summary Statistics

mean
median
st.d.
min
occurrence
max
occurrence
1970-1947
2011-1970
Skewness
Kurtosis
Normality

Naive-GDP

0.566
0.567
0.016
0.522
1948q2
0.598
1970q1
0.062
−0.047
−0.454
−0.143
[0.009]

BLS

2012q3

111.0

104.7
104.9
3.2
94.4

PI-GDP
PI2-GDP
SE-GDP
0.618
0.674
0.632
0.617
0.674
0.634
0.012
0.015
0.019
0.587
0.633
0.581
2012q3
2011q4
2012q3
0.648
0.714
0.664
1970q1
1970q1
1960q4
−0.001
0.022
0.026
−0.050
−0.074
−0.066
−0.105
−0.496
0.171
−0.094
−0.432
0.333
[0.002]
[0.385]
[0.505]
SHORT-RUN CHARACTERISTICS
0.008
0.008
0.008
0.011
0.359
0.357
0.374
0.482
0.658∗∗∗
0.633∗∗∗
0.736∗∗∗
0.637∗∗∗
−0.232∗∗∗ −0.385∗∗∗ −0.275∗∗∗

1960q4
−0.487
−12.841
−0.912
0.881
[0.000]

σ
(cid:101)ls
σ
(cid:101)ls/σ
(cid:101)y
corr( (cid:101)lst, (cid:101)lst−1)
corr( (cid:101)lst, (cid:101)yt)
Note: Changes have been calculated for annual means. See also notes to Table 1.

0.008
0.370
0.674∗∗∗
−0.200∗∗∗

−0.092

Assume the labor share is generated by an AR(1) process:7

LSt = µ + ρLSt−1 + β1t + β2t2,

(10)

This nests three models: (1) only with a constant (β1 = β2 = 0); (2) with a linear trend (β2 = 0); (3)
with a quadratic trend; and model (1+) where (1) is re-estimated using the logged and HP ﬁltered
labor share series, (cid:101)ls, instead of its level.

Models (1) and (1+) are consistent with the usual interpretation of the labor share as being
stable around its long-run mean, µ
1−ρ . In the next two models, the “mean” itself is allowed to shift,
reﬂecting secular trends, long-lasting cycles, structural changes in the economy, and so on. Clearly
the ρ value that emerges from models (2) and (3) captures persistence at the high frequency end
since some of the long-run variation is removed by the included trends.

Point estimates of ρ, although generally high, exhibit substantial heterogeneity (see Tables 3,
4). For example, for annual and quarterly Model 1, ρ ∈ [0.751, 0.944] and ρ ∈ [0.929, 0.982], respec-
tively, which imply half lives of 2.5 − 12 and 2.5 − 9.5 years. For the annual series, the GVA series
are far less persistent. Interestingly, the addition of a linear trend reduces ρ estimates signiﬁcantly
only for the series with GVA as output or adjusted by the self-employed. Models 2 and 3 – as well
as the ﬁltered case 1+ – necessarily contract the persistence and half lives. Extending the autore-
gressive model by a linear trend limits substantially the persistence only for the SE-GDP and BLS
series, for which the linear trend is statistically signiﬁcant. The quadratic of Model 3 naturally ﬁts
the naive series well given the strong hump-shape in its proﬁle.

4.4 PERSISTENCE AND STOCHASTIC VOLATILITY (SV)

A large and growing literature – starting in ﬁnance (Shephard and Andersen, 2008) but recently
evolving into macroeconomics – has emerged to analyze the possibility of time-varying stochastic
volatility in time series. As Fernández-Villaverde and Rubio-Ramírez (2013) discuss, this appears
to be important for aggregate data: periods of high volatility are followed low-volatility periods.

7We also tried an AR(2) speciﬁcation. Despite the fact that, as opposed to the AR(1) model, such a speciﬁcation is
able to capture hump-shaped dynamics with a stationary stochastic process, our results are very similar. The sum of
both autoregressive coefﬁcients is generally close to but signiﬁcantly less than unity. Adding a quadratic trend sub-
stantially reduces the estimated persistence. The differences across various labor share speciﬁcations are of comparable
magnitude. Details are available from the authors upon request.

ECB Working Paper 1806, June 201513Table 3: AR(1) Persistence: Annual Labor Share

Naive-GDP

PI-GDP

PI2-GDP

SE-GDP Naive-GVA Naive-GVA-NF SE-GVA-NF

SE-GVA

ρ

ρ
β1 · 104

ρ
β1 · 104
β2 · 104

0.908∗∗∗

0.840∗∗∗

0.840∗∗∗

0.909∗∗∗
-0.002

0.852∗∗∗
-0.029

0.814∗∗∗
-0.101∗∗∗

0.728∗∗∗
0.727∗∗∗
-0.007∗∗∗

0.759∗∗∗
0.353∗
-0.004∗

0.739∗∗∗
0.197
-0.004∗

Model 1

0.939∗∗∗

0.944∗∗∗

0.726∗∗∗
-0.257∗∗∗

Model 2

0.869∗∗∗
0.118
Model 3

0.716∗∗∗
-0.440∗∗
0.002

0.779∗∗∗
0.655∗∗∗
-0.005∗∗

Model 1+

0.881∗∗∗

0.751∗∗∗

0.872∗∗∗

0.780∗∗∗
0.122∗

0.740∗∗∗
0.030

0.812∗∗∗
−0.130∗

0.700∗∗∗
0.595∗∗
-0.005∗∗

0.738∗∗∗
0.164
-0.002

0.774∗∗∗
-0.504∗
0.004

0.485∗∗∗
ρ
Note: ∗∗∗, ∗∗ and ∗ denote the rejection of null of insigniﬁcance at the 1%, 5% and 10% signiﬁcance level, respectively
(bootstrapped standard errors used). The estimated constants are omitted for brevity.

0.511∗∗∗

0.528∗∗∗

0.388∗∗∗

0.420∗∗∗

0.620∗∗∗

0.454∗∗∗

0.417∗∗∗

Table 4: AR(1) Persistence: Quarterly Labor Share

Naive-GDP

PI-GDP

ρ

0.968∗∗∗

0.929∗∗∗

ρ
β1 · 104

ρ
β1 · 104
β2 · 104

0.971∗∗∗
-0.002

0.884∗∗∗
0.083∗∗∗
-0.003∗∗∗

0.920∗∗∗
-0.004

0.890∗∗∗
0.032∗∗
-0.001∗∗

ρ

0.670∗∗∗

0.627∗∗∗

Note: See Table 3.

PI2-GDP
Model 1
0.975∗∗∗
Model 2
0.958∗∗∗
-0.006
Model 3
0.920∗∗∗
0.033∗∗
-0.002∗∗∗
Model 1+
0.722∗∗∗

SE-GDP

BLS

0.982∗∗∗

0.964∗∗∗

0.864∗∗∗
-0.033∗∗∗

0.823∗∗∗
-0.004
-0.001∗∗

0.913∗∗∗
-2.906∗∗∗

0.843∗∗∗
8.095∗∗
-0.496∗∗∗

0.633∗∗∗

0.655∗∗∗

For robustness, therefore, we additionally estimate the SV-AR(1) process:8

(cid:101)lst = ρ (cid:101)ls (cid:101)lst−1 + eσt υ1,t,

σt = (1 − ρσ) σ + ρσσt−1 + ησυ2,t,

(10a)

(10b)

where, as before, (cid:101)ls is the HP-ﬁltered series of the logged series,9 and υ1,t, υ2,t ∼ N (0, 1). Param-
eters ρ (cid:101)ls and ρσ represent the persistence of the level and volatility equation, respectively; σ is the
unconditional mean of the volatility of the process, σt; and η captures the standard deviation of
the volatility shocks.

Table 5 shows that whilst the quarterly point estimates do not differ signiﬁcantly from the
previous (1+) case (reproduced in the ﬁrst row), there is an efﬁciency gain.10 The data support
moderate time-varying volatility, with persistence similar to that of the labor share series itself. It
is estimated that a one standard deviation volatility shock increases the standard deviation of the
labor share by around e(ησ−1) = 35%.

Figure 4 retrieves the implied stochastic volatility process. Though differences in the level
of stochastic volatility detected, a similar story emerges: namely the increasing volatility built
up over 1970s and early 1980s followed by the “great moderation”. This is followed by a peak

8We thank Benjamin Born for supplying Matlab code to implement the stochastic volatility estimation from Born and
Pfeifer (2014). The Bayesian method used to retrieve these shocks is, for given priors, to evaluate the likelihood using
the sequential importance resampling particle ﬁlter and Randomized Block Metropolis-Hastings algorithm to maximize
the posterior. After ﬁltering, the historical distribution of the volatilities is obtained by a backward-smoothing routine.

9As before, we repeated the exercise with a one-sided HP ﬁlter with minimal qualitative differences.
10We concentrate on quarterly data since that is a frequency often associated with stochastic volatility.

ECB Working Paper 1806, June 201514Table 5: AR(1) and SV-AR(1) Models, Quarterly Labor Share Series

ρ

ρ (cid:101)ls

ρσ

¯σ

ησ

naiveGDP
0.670
{0.581 : 0.759}
0.705
{0.626 : 0.781}
0.717
{0.513 : 0.885}
-5.220
{-5.368 : -5.075}
0.287
{0.206 : 0.376}

PI-GDP
0.627
{0.528 : 0.726}
0.654
{0.578 : 0.732}
0.693
{0.473 : 0.886}
-5.152
{-5.292 : -5.003}
0.293
{0.211 : 0.385}

PI2-GDP
0.722
{0.631 : 0.814}
0.746
{0.673 : 0.820}
0.806
{0.637 : 0.967}
-5.312
{-5.504 : -5.054}
0.259
{0.179 : 0.349}

SE-GDP
0.633
{0.539 : 0.726}
0.685
{0.602 : 0.765}
0.733
{0.550 : 0.889}
-5.167
{-5.322 : -5.003}
0.282
{0.200 : 0.367}

BLS
0.655
{0.562 : 0.749}
0.690
{0.614 : 0.766}
0.737
{0.553 :0.903}
-4.916
{-5.066 : -4.754}
0.281
{0.202 :0.370}

Note: The 95% AR bootstrapped conﬁdence bands used. For the SV-AR(1)
the 95% conﬁdence intervals are given below the median estimates.

of volatility around the 2001 recession which rises again thereafter. Interestingly, the PI2-GDP
series projects a far longer moderation period in terms of stochastic volatility and only spikes (and
even then only temporarily) around the 2007q4 recession. Accordingly, if the researcher were to
examine that series in isolation, she would derive a view of the stability properties of the labor
share quite distinct from the alternatives.

Figure 4: Stochastic Volatility of Labor Share Series

Notes: The plots show eσt. Sample 1946q1 to 2013q1.
Naive-GDP, PI-GDP, PI2-GDP, SE-GDP , BLS (dot-dashed) .

5

STRUCTURAL BREAKS

Ultimately the importance of persistence is to gauge whether a series is or is not stationary (around
a constant or a linear trend). When persistence takes the form of a unit root, the effect of an
innovation is permanent. Since income shares are deﬁned within the unit interval and have not
exhibited corner solutions in history, one’s prior might be that labor share does not contain a unit
root; Tables 3 and 4 suggested as much.

Testing more formally for a unit root, however, is largely inconclusive. We implemented sev-
eral tests (ADF, PP, ADF-GLS, symmetric and asymmetric ADF-ESTAR, and fractional). Results
(see appendix Tables C.1, C.2) vary substantially across the series, reﬂecting the existence of a
clear downward trend in some of them (e.g., SE-GDP), hump-shaped trends in some others (e.g.,
PI2-GDP), and their varying degrees of persistence. Note that some of the aforementioned facts
could be a consequence of changes in the sectoral structure of the US economy (see, e.g., Elsby,
Hobijn, and Sahin, 2013). Since this goes beyond our remit, an indicative discussion of the role of

1950196019701980199020002010468101214x 10−3  ECB Working Paper 1806, June 201515the sectoral makeup of the aggregate labor share has been relegated to Appendix B.

Importantly, there is also no systematic evidence for stationarity when a structural break is
allowed for.11 This outcome may have been caused either by complicated dynamics of the con-
sidered time series – driven by their large persistence and presence of nonlinear trends – or by
the existence of more than one breakpoint in the labor share. Accordingly, we complement our anal-
ysis by applying a multiple breaks detection procedure proposed by Bai and Perron (2003). As
in previous exercises, we consider three assumptions about the deterministic component of the
time series: only constant, linear trend, and quadratic trend. For each case we report the optimal
number of breaks in the data-generating process with corresponding breakpoints. The optimal
number of structural changes is chosen with the BIC criterion, restricted to be at most 5.

Results are presented in appendix Tables C.3-C.4.12 This testing procedure allows for changes
in the mean and/or slope. Which case one relies upon is largely a matter of judgement. For
simplicity and in line with usual interpretations, Figures 5-6 plot the mean breaks detected.

These results indicate strong evidence in favor of multiple structural breaks. However, the
timing of breakpoints varies among different labor share variants. Typically, two to ﬁve structural
changes might be identiﬁed: early 1940s, late 1950s, late 1960s–early 1970s, ﬁrst half of 1980s, and
late 1990s–early 2000s. The ﬁrst (in annual data only, given the sample), third and fourth of these
breaks appear most robust across speciﬁcations, and can be identiﬁed with World War II, the oil
crisis, and the early 1980s recession. Alternatively, the latter two dates might be perceived as a
mark of the beginning of the spread of ICT technologies across the US.

To sum up, for each labor share series we ﬁnd evidence of multiple structural breaks, which
explains why Zivot–Andrews tests of stationarity subject to a structural break might have had
low power. A caveat is some heterogeneity in the dating. For instance, in the case of the PI2-GDP
series, tests suggest either no structural break at all, or three of them: during World War II, in late
1960s, and in early 1980s.

11Our analysis suggests that if there were structural breaks, there have been more than one. This pertains to all
the considered series. First, we ﬁnd that the F statistic of the Chow single breakpoint test, based on a simple data-
generating process including only deterministic components, is below its critical value at any possible breakpoint.
Therefore this test does not allow us to reject the null of no structural break against the alternative of a single break.

12Notice the fall in the labor share from 2001 is not always picked up by the tests reﬂecting the inﬂuence of “trim-

ming” at the end of the sample, as well as the fact that we limit the number of break to at most ﬁve.

ECB Working Paper 1806, June 201516Figure 5: Annual series

naiveGDP

naiveGVA

naiveGVA-NF

1
7

SE-GDP

SE-GVA

SE-GVA-NF

PI-GDP

PI2-GDP

Year1930194019501960197019801990200020100.500.55Year1930194019501960197019801990200020100.500.550.60Year1930194019501960197019801990200020100.550.60Year1930194019501960197019801990200020100.600.650.70Year1930194019501960197019801990200020100.650.700.750.80Year1930194019501960197019801990200020100.600.650.70Year1930194019501960197019801990200020100.60Year1930194019501960197019801990200020100.650.70ECB Working Paper 1806, June 201517Figure 6: Quarterly series

naiveGDP

PI-GDP

PI2-GDP

1
8

SE-GDP

BLS

Year19501960197019801990200020100.550.60Year19501960197019801990200020100.600.65Year19501960197019801990200020100.650.70Year19501960197019801990200020100.600.65Year195019601970198019902000201095.197.6100.1102.6105.1107.6110.1ECB Working Paper 1806, June 2015186

SPECTRAL ANALYSIS

6.1 SPECTRAL DECOMPOSITION OF US LABOR SHARE SERIES

Ambiguity over stationarity, the presence of structural breaks, and the apparent lack of conver-
gence between the labor share series suggests that it is low-frequency aspects that are most impor-
tant to understand when comparing these alternate labor share series. We shall investigate this
issue using spectral techniques. Our motivation in performing spectral analysis is to assess the
importance of ﬂuctuations with given periodicity for the total observed variance of the respective
series and to justify whether oscillations of speciﬁc frequencies systematically co-move between
various deﬁnitions of the labor share.

In our exercise we distinguish between the low-, medium- and high- frequency range. High-
frequency ﬂuctuations are deﬁned as all oscillations with periodicity below 8 years, interpreted as
business-cycle ﬂuctuations (cf. Young, 2004). The second range, the medium-term business cycles,
as formulated by Comin and Gertler (2006), includes all ﬂuctuations with periodicity between 8
and 50 years.13 The longest swings with periodicity higher than 50 years are mapped into the
low-frequency component, interpreted as a stochastic trend.

For spectral techniques, the data should not have a unit root.14 Given the ambiguity in formal
unit root testing, we apply three approaches to excluding the deterministic component: removing
the mean, linear and quadratic trend from log-levels. Of course a demeaned non-stationary series
remains non-stationary, but in the familiar context where the labor share is seen as ﬂuctuating
around a constant mean, it provides a natural benchmark.

The estimated shares of speciﬁc frequencies in the overall variance are reported in Tables 6
and 7. Apart from the SE-GDP and SE-GVA variants,15 the role of the low-frequency component
is substantial. For the demeaned series, long cycles beyond 50 years (variations in the stochastic
trend) are responsible for from 1/4 to almost 2/3 of the overall variance. The contribution of
the low-frequency component is signiﬁcant even if a linear trend is included in data-generating
process. For both transformations, the medium-run component is more important than the short-
run one in the case of all annual series and 4 out of 5 quarterly series.

Table 6: Shares of Speciﬁc Frequencies in Total Variance (%) – Annual Series

PERIODICITY
(IN YEARS)
Naive-GDP
PI-GDP
PI2-GDP
SE-GDP
Naive-GVA
Naive-GVA-NF
SE-GVA-NF
SE-GVA

DEMEANED

EXCL. LINEAR

≥ 50

8 − 50 ≤ 8

≥ 50

8 − 50 ≤ 8

EXCL. QUADRATIC
8 − 50 ≤ 8
≥ 50

73.6
25.8
31.5
18.0
62.4
54.3
30.1
4.1

20.3
58.3
48.3
56.1
33.5
36.8
42.3
53.1

6.1
15.9
20.2
25.9
4.0
8.9
27.6
42.8

72.8
28.0
28.3
15.5
61.0
46.1
15.4
9.4

17.4
56.5
48.9
46.1
21.0
29.5
44.0
49.8

9.9
15.5
22.7
38.4
17.9
24.4
40.5
40.8

2.8
3.5
0.2
15.9
0.9
0.3
1.6
7.3

65.7
73.8
68.1
45.7
57.6
56.1
51.9
49.4

31.5
22.7
31.7
38.4
41.5
43.6
46.5
43.3

Note: The shares have been calculated based on periodogram estimates. Bold indicates the maxi-
mum of frequency share over each respective “stationarizing” processes.

Particularly interesting ﬁndings arise when analyzing the series excluding a quadratic trend.

13To be more precise, our medium frequency component is equivalent to the Comin and Gertler (2006) “low-

frequency subcomponent” of medium-term business cycles.

14The shares of given frequency domains in the total variance of a time series have been computed by cumulating
raw periodogram values over each desired frequency (low, medium and high), and then dividing by total variance.
Such an estimator is only asymptotically consistent, though (for a general overview see Hamilton (1994, chapter 6)),
which provides the caveat that its efﬁciency can be low if the series is short.

15The annual SE-GDP variant should be treated with caution. Robustness of the long cycle to subtracting a quadratic
trend are in the case of this series likely driven by a structural break in the NIPA data on self-employment. To a smaller
extent, this break also inﬂuences the properties of SE-GVA.

ECB Working Paper 1806, June 201519Table 7: Shares of Speciﬁc Frequencies in Total Variance (%) – Quarterly Series

PERIODICITY ≥ 50
(IN YEARS)
Naive-GDP
PI-GDP
PI2-GDP
SE-GDP
BLS

64.8
24.2
42.8
7.0
36.5

DEMEANED

EXCL. LINEAR

8 − 50 ≤ 8

≥ 50

8 − 50 ≤ 8 ≥ 50

EXCL. QUADRATIC
8 − 50 ≤ 8

27.9
51.0
37.9
23.5
36.8

7.4
24.8
19.3
69.5
26.7

70.4
16.1
29.8
6.9
31.7

21.3
60.1
52.4
23.4
35.0

8.3
23.8
17.8
69.7
33.3

3.8
0.8
0.4
3.1
3.3

64.7
69.5
72.6
20.3
44.6

31.6
29.7
27.0
76.6
52.2

Naturally, extraction of a quadratic trend from the labor share data series limits the importance
of the low-frequency component whose contribution to the overall variance falls below 4%. Sec-
ondly, we see that for the “naive” series and for the series adjusted by proprietors’ income, the
share of the medium-term component is almost two times higher than of the high-frequency com-
ponent.

On the other hand, the quarterly SE-GDP series (since 1947) seems to be characterized by
quite distinct spectral characteristics. Most of its variance is concentrated in short-run frequen-
cies, irrespectively of the data transformation.16 PI2-GDP, in contrast, provides a consistent mes-
sage for both the annual and quarterly frequency: around 80% of its total variability is generated
by medium-run cycles and a long-run hump-shaped swing, which can be very well ﬁtted by a
quadratic trend.

6.2 COHERENCE

Next, we perform cross-spectral analysis by computing coherence. This addresses the question
whether the pairs of the different labor share variants systematically co-move within speciﬁc fre-
quency ranges.

Tables 8 and 9 present coherence estimates.17 Keeping in mind that the annual numbers may
be somewhat less reliable due to fewer observations, we ﬁnd that coherence is always signiﬁcant
in the high-frequency domain. This result corroborates the previously formulated conclusion that
labor share series tend to be rather consistent in the short run. Coherence estimates are more am-
biguous in the lower frequencies, though. In the medium- and low-frequency domain we identify
subgroups for which coherence is very high, reﬂecting their deﬁnitional similarity: annual and
quarterly series adjusted by proprietors’ income (PI-GDP and PI2-GDP), all “naive” annual la-
bor share series, all annual series adjusted for self-employment, and a pair consisting of quarterly
series SE-GDP and BLS. Otherwise, the coherence is rather low.

16High-frequency ﬂuctuations are also the most important part of the frequency domain for BLS series but only when

the quadratic trend is extracted from data.

17The coherence statistic for a pair of time series (xt, yt) can be understood as the R2 from xt regressed on yt as a
function of the frequency. Complementarily, one can also compute dynamic correlation coefﬁcients, to control the sign of
the relationship in each given pair. For all the pairs, dynamic correlation is in line with the general intuition, though: it
is positive and signiﬁcant whenever the coherence for a given pair is signiﬁcant. In the case of insigniﬁcant coherence,
the dynamic correlation is not signiﬁcantly different from zero.

ECB Working Paper 1806, June 201520Table 8: Average Coherence Among the Labor Share Series – Annual Data

Periodicity

≥ 50

PI-GDP
8-50 ≤ 8

PI2-GDP
8-50 ≤ 8

≥ 50

SE-GDP
8-50 ≤ 8

Naive-GVA

≥ 50

8-50 ≤ 8

Naive-GVA-NF
8-50 ≤ 8

≥ 50

≥ 50

SE-GVA-NF

≥ 50

8-50 ≤ 8

≥ 50

SE-GVA
8-50 ≤ 8

DEMEANED SERIES

0.52∗∗∗ 0.67∗∗∗ 0.73∗∗∗ 0.09

0.42∗∗ 0.72∗∗∗ 0.47
0.53∗∗∗ 0.63∗∗∗ 0.63∗∗∗ 0.23
0.05

0.19
0.10
0.20

Naive-GDP
PI-GDP
PI2-GDP
SE-GDP
Naive-GVA
Naive-GVA-NF
SE-GVA-NF

0.31∗
0.14
0.65∗∗∗ 0.26

0.34∗
0.43∗∗ 0.86∗∗∗ 0.65∗∗∗ 0.77∗∗∗ 0.89∗∗∗ 0.63∗∗∗ 0.54∗∗∗ 0.42∗∗ 0.15
0.31∗
0.48∗∗∗ 0.17
0.43∗∗
0.48∗∗∗ 0.26
0.56∗∗∗ 0.61∗∗∗ 0.22
0.22
0.37∗∗
0.53∗∗∗ 0.01
0.27∗
0.05
0.07
0.07
0.33∗∗ 0.52∗∗∗ 0.70∗∗∗ 0.78∗∗∗ 0.85∗∗∗
0.46∗∗∗
0.55∗∗∗
0.61∗∗∗ 0.74∗∗∗

0.98∗∗∗ 0.88∗∗∗ 0.81∗∗∗ 0.58∗∗∗ 0.38∗∗ 0.58∗∗∗ 0.37∗∗ 0.20
0.64∗∗∗ 0.62∗∗∗ 0.87∗∗∗ 0.34∗∗ 0.29

0.62∗∗∗ 0.33∗∗ 0.43∗∗ 0.71∗∗∗ 0.04
0.36∗∗ 0.03
0.46∗∗∗ 0.01
0.19
0.41∗∗ 0.11
0.35∗∗ 0.57∗∗∗ 0.26

0.02

DE-TRENDED SERIES

0.00

0.42∗∗ 0.68∗∗∗ 0.16

0.60∗∗∗ 0.60∗∗∗ 0.46∗∗∗ 0.31∗
0.01

0.49∗∗∗ 0.56∗∗∗ 0.46∗∗∗ 0.54∗∗∗ 0.78∗∗∗ 0.37∗∗ 0.46∗∗∗ 0.66∗∗∗ 0.41∗∗ 0.51∗∗∗ 0.50∗∗∗ 0.19
0.66∗∗∗ 0.29∗
0.13
0.21

0.66∗∗∗
Naive-GDP
0.53∗∗∗
0.24
PI-GDP
0.52∗∗∗
0.12
PI2-GDP
0.66∗∗∗ 0.44∗∗ 0.49∗∗∗ 0.65∗∗∗ 0.50∗∗∗ 0.70∗∗∗ 0.81∗∗∗ 0.91∗∗∗
SE-GDP
0.91∗∗∗ 0.91∗∗∗ 0.83∗∗∗ 0.79∗∗∗ 0.84∗∗∗ 0.78∗∗∗ 0.68∗∗∗ 0.81∗∗∗ 0.72∗∗∗
Naive-GVA
0.79∗∗∗ 0.88∗∗∗ 0.94∗∗∗ 0.55∗∗∗ 0.77∗∗∗ 0.60∗∗∗
Naive-GVA-NF
0.86∗∗∗ 0.91∗∗∗ 0.71∗∗∗
SE-GVA-NF
Note: ∗∗∗, ∗∗ and ∗ denote rejection of the null of coherence insigniﬁcance at 1%, 5% and 10% signiﬁcance level, respectively. The spectra for a given pair have been estimated
using the Parzen kernel.

0.54∗∗∗ 0.13
0.56∗∗∗ 0.04
0.01
0.04
0.66
0.25
0.33∗∗ 0.69∗∗∗ 0.56∗∗∗ 0.30∗

0.42∗∗ 0.20
0.28
0.61∗∗∗ 0.49∗∗∗ 0.16
0.11
0.15
0.25

0.26
0.21
0.05

0.11
0.26

0.11
0.11

2
1

Table 9: Average Coherence Among the Labor Share Series – Quarterly Data

PI-GDP
8-50 ≤ 8

PI2-GDP
8-50 ≤ 8

≥ 50

SE-GDP
8-50 ≤ 8

≥ 50

≥ 50

BLS

≥ 50

8-50 ≤ 8

DEMEANED SERIES

0.33∗∗∗ 0.39∗∗∗ 0.68∗∗∗ 0.21∗∗ 0.26∗∗ 0.38∗∗∗ 0.01

0.43∗∗∗
0.01
0.82∗∗∗ 0.81∗∗∗ 0.51∗∗∗ 0.30∗∗∗ 0.32∗∗∗ 0.69∗∗∗ 0.44∗∗∗ 0.48∗∗∗ 0.81∗∗∗
0.55∗∗∗ 0.54∗∗∗ 0.50∗∗∗ 0.64∗∗∗ 0.62∗∗∗ 0.52∗∗∗
0.89∗∗∗ 0.88∗∗∗ 0.89∗∗∗

0.38∗∗∗ 0.04

0.08

Periodicity

Naive-GDP
PI-GDP
PI2-GDP
SE-GDP

DE-TRENDED SERIES

0.78∗∗∗ 0.79∗∗∗ 0.85∗∗∗ 0.64∗∗∗ 0.65∗∗∗ 0.44∗∗∗ 0.18∗

0.26∗∗ 0.77∗∗∗
Naive-GDP
0.83∗∗∗ 0.80∗∗∗ 0.46∗∗∗ 0.24∗∗ 0.41∗∗∗ 0.82∗∗∗ 0.22∗∗ 0.36∗∗∗ 0.87∗∗∗
PI-GDP
0.17∗
0.40∗∗∗
PI2-GDP
0.89∗∗∗ 0.91∗∗∗ 0.85∗∗∗
SE-GDP
Note: ∗∗∗, ∗∗ and ∗ denote rejection of the null of coherence insigniﬁcance at 1%, 5% and 10% signiﬁcance level, respec-
tively. The spectra for a given pair have been estimated using the Parzen kernel.

0.25∗∗ 0.41∗∗∗ 0.08

0.38∗∗∗ 0.89∗∗∗ 0.11

0.12

ECB Working Paper 1806, June 201521The spectral analysis highlights the discrepancies between various labor share series. From
the frequency domain perspective it appears that the most outlying are the series adjusted for
self-employment, which are characterized by substantially different variance decompositions and
insigniﬁcant coherence with other variants in the low- and medium-run frequency. For example,
(our favored series) PI2-GDP is generally incoherent with the SE-GVA measure, considered the
“headline measure” (and subsequently criticized) by Elsby, Hobijn, and Sahin (2013).

Our results obtained so far point to the general conclusion that while short-run properties
of the labor shares are relatively consistent across the considered alternative deﬁnitions, their
medium-run swings and long-run trends diverge substantially. At the same time, ambiguity in
unit root test results is likely due to the high persistence and complicated dynamics of the con-
sidered series. We can also argue that the PI2-GDP measure is probably not only the most sound
theoretically, but also has intuitive empirical properties: (1) it is mean-reverting but highly persis-
tent, with about 80% of its total variance observed in the medium-to-long run frequency range, (2)
it is countercyclical over the short run, (3) it has recorded a secular decline since 1970, and (4) it
can be understood as featuring three economically interpretable structural breaks: during World
War II, in late 1960s, and in early 1980s.

7 THE IMPORTANCE OF THE LABOR SHARE IN ECONOMIC APPLICA-

TIONS

Let us now discuss the potential consequences of using these diverse series interchangeably in
empirical applications. We consider three applications: (1) growth accounting, (2) technology-
labor share VAR analysis, and (3) estimation of New Keynesian Phillips curves. These three were
chosen not only to reﬂect their broad popular application in various literatures, but also because
they help reveal and substantiate some of the differences in the series discussed earlier.

7.1 APPLICATION I: GROWTH ACCOUNTING

We begin with growth accounting, which is a widely-used exercise in macroeconomics, develop-
ment and business-cycle analysis. This exercise decomposes economic growth into that due to
factor accumulation, and technical progress (which is derived residually). The standard growth
accounting equation can be written as:

∆tfpt = ∆yt − ˜αt∆kt − (1 − ˜αt)∆lt,

(11)

where all variables are in logs, and where ˜αt and tfpt denote the (potentially time-varying) capital
share and log total factor productivity, respectively.

We already know that labor shares are time-varying and have different properties across vari-
ants. Accordingly, this should be reﬂected in how we implement growth accounting. With this in
mind, the extraction of TFP can then be done in the following ways:

1. Common Input Factors and Outputs

: GDP in constant USD [NIPA Table 1.6];

Derive TFP across different labor share measures based on common input factors and common
inputs:
Y1
t
K1
t : Chain-Type Quantity Index for the Net Stock of Fixed Assets [FAT Table 1.2];
L1
t
Tables 6.5 and 6.7].

: Full-Time Equivalent Employees plus Self-Employed in all domestic industries [NIPA

2. Common Input Factors But Deﬁnitionally Consistent Outputs

ECB Working Paper 1806, June 201522: GDP in constant USD [NIPA Table 1.6]

Derive TFP across different labor share measures based on common input factors but with
output measures related to the speciﬁc labor-share measure:
Y1
t
Y2
t
Y3
t
t and L1
K1

: Real Gross Value Added in the non-farm business sector [NIPA Table 1.3.6];

: Real Gross Value Added in the private sector [NIPA Table 1.3.6];

t as above.

3. Deﬁnitionally Consistent Factors and Outputs

t , L1

t , K1

t : as above;

: Chain-Type Quantity Index for the Net Stock of Fixed Assets in the private sector [FAT

Derive TFP across different labor share measures based on input factors and output mea-
sures related to the speciﬁc labor-share measure:
t − Y3
Y1
K2
t
Table 1.2]
K3
sector [also FAT Table 1.2];
L2
t
6.5 and 6.7].
L3
t
[NIPA Tables 6.5 and 6.7].

t : Chain-Type Quantity Index for the Net Stock of Fixed Assets in the non-farm business

: Full-Time Equivalent Employees plus Self-Employed in the private sector [NIPA Tables

: Full-Time Equivalent Employees plus Self-Employed in the non-farm business sector

Observe that for naiveGDP, PI-GDP, PI2-GDP and SE-GDP, all these approaches boil down to
the same growth accounting scenario because our reference measures of GDP, capital, and labor
are then also deﬁnitionally consistent.

Within each of these three cases, we assume that factor shares are time-varying following a

Törnquist index:

˜αj,t =

αj,t + αj,t−1
2

,

where j denotes the particular labor share variant used (e.g., naiveGDP, PI-GDP, etc.).

Our purpose therefore is to examine the scope for mis-measurement of TFP (growth and levels)
when factor income shares vary and when differences in shares are compounded with those of
output and the factors. Figure 7 shows the cumulated TFP levels for time-varying income shares,
where the factors are assumed constant across labor share deﬁnitions (the ﬁrst row), where the
output deﬁnitions are additionally allowed to change (second row), and where both the inputs
and outputs are allowed to change consistent with the underlying labor share deﬁnition (ﬁnal
row).

In terms of shape, all series broadly conform to what is commonly understood to be the story
behind US TFP (e.g., Fernald (2007), Shackleton (2013)): exceptionally high TFP growth in the
mid 1930-1940s, the consolidation of those gains in the decades after WW II, followed by a period
of slower residual productivity growth (often dated to the early 1970s), and the acceleration in
productivity towards the end of the sample.

ECB Working Paper 1806, June 201523Figure 7: Cumulative TFP based on Time-Varying Factor Shares (1929=1)
Common Input Factors and Outputs

Common Input Factors, Deﬁnitionally Consistent Outputs

Deﬁnitionally Consistent Input Factors and Outputs

Notes:
Left panel: Naive-GDP, PI-GDP, PI2-GDP, SE-GDP ; Right panel: Naive-GVA, Naive-GVA-NF, SE-GVA-NF, SE-GVA
A log-scale for the level of TFP is used in these graphs for legibility.

1.01.52.02.53.03.54.01930194019501960197019801990200020101.01.52.02.53.03.54.01930194019501960197019801990200020101.01.52.02.53.03.54.01930194019501960197019801990200020101.01.52.02.53.03.54.01930194019501960197019801990200020101.01.52.02.53.03.54.01930194019501960197019801990200020101.01.52.02.53.03.54.0193019401950196019701980199020002010ECB Working Paper 1806, June 201524Table 10: Cumulative Change of TFP, Based on Time-Varying Factor Shares (In %)

naive-GDP

PI-GDP

PI2-GDP

SE-GDP

naive-GVA naive-GVA-NF

SE-GVA-NF

SE-GVA

1929-2011
1929-1945
1945-1970
1970-2011

1929-2011
1929-1945
1945-1970
1970-2011

1929-2011
1929-1945
1945-1970
1970-2011

108
40
27
41

108
40
27
41

108
40
27
41

114
40
31
43

114
40
31
43

114
40
31
43

Common Input Factors and Outputs

120
40
34
46

116
40
32
44

110
40
27
43

112
40
28
43

Common Input Factors, Deﬁnitionally Consistent Outputs

126
29
41
56

116
40
32
44

120
40
34
46

135
33
46
56
Deﬁnitionally Consistent Input Factors and Outputs
136
57
30
49

120
40
34
46

116
40
32
44

139
57
31
51

118
41
32
45

141
33
50
59

140
56
32
51

120
40
34
46

136
29
48
59

148
56
38
54

There are, though, certain level differences between the TFPs generated with the use of the
alternate factor share series. Some speciﬁc discrepancies in dynamics are worth noting too. For
example, productivity and TFP growth are often considered to have exhibited a broken trend in
the early 1970s (e.g., Fernald, 2007). Whilst this is clearly visible for most of the series, it is less
apparent for the GVA series.18

Table 10 shows the cumulative change of TFP based on time-varying factor income shares.
There are indeed substantial cumulative discrepancies. To illustrate, whilst naiveGDP grew by
108% over the whole sample, SE-GVA-NF grew by 140% (in the last accounting scenario).

Interestingly, we also ﬁnd that the TFP deviations for all labor share speciﬁcations against
NaiveGDP have been gradually increasing since World War II, see Figure 7 and Table 11.19 This
is driven by the fact that the post-war period was characterized by rapid physical capital accu-
mulation, and hence the underestimation of the labor share (equivalently, overestimation of the
capital share) in the naive-GDP case has systematically led to an overstating of capital’s contri-
bution to GDP growth, at the cost of understating the role of TFP. By this logic, it should not be
surprising that the relatively highest labor share PI2-GDP implies also the relatively strongest TFP
growth.

7.2 APPLICATION II: TECHNOLOGY SHOCKS AND THE LABOR SHARE

Let us now pass to the short-run question of examining the impact of exogenous technology
shocks on the labor share. The motivation for undertaking such an exercise is the following. First,
it is worthwhile to verify if the apparently consistent short-run properties of all considered labor
share measures carry forward to applied econometric studies of the business cycle. The ques-
tion of the impact of technology shocks seems a reasonable ﬁrst step in this direction. Second, as
argued above the labor share switches from being countercyclical in the short run to being pro-
cyclical in the medium run. Ríos-Rull and Santaeulàlia-Llopis (2010) have found an overshooting

18The absence of a slight hump in the TFP level in the mid 1940s (see middle panel, rhs graph) is caused by the fact
that real GDP/GVA grew at slightly different rates. For example, the most spectacular difference was in 1946 when
GDP fell by 10% while GVA by less than 1%. Note that this does not reappear in the bottom row rhs graph since then
we adjust the inputs consistently with the labor share deﬁnition. Note also that it is the period after WWII so there was
a substantial shift between sectors (government vs private).

19The observed divergence between the GDP-based and GVA-based series in the initial period 1929–1945 in the last

two accounting scenarios is due to real GDP/GVA growing at different rates in the 1940s.

ECB Working Paper 1806, June 201525Table 11: Cumulative Deviation from TFP Based on naiveGDP (In %)

naive-GDP

PI-GDP

PI2-GDP

SE-GDP

naive-GVA naive-GVA-NF

SE-GVA-NF

SE-GVA

1929-2011
1929-1945
1945-1970
1970-2011

1929-2011
1929-1945
1945-1970
1970-2011

1929-2011
1929-1945
1945-1970
1970-2011

0
0
0
0

0
0
0
0

0
0
0
0

6
0
4
2

6
0
4
2

6
0
4
2

Common Input Factors and Outputs

12
0
8
5

8
0
5
2

2
0
0
1

4
0
2
2

Common Input Factors, Deﬁnitionally Consistent Outputs

8
0
5
2

12
0
8
5

18
-11
15
15

27
-7
19
15
Deﬁnitionally Consistent Input Factors and Outputs
28
17
3
8

12
0
8
5

31
17
5
9

8
0
5
2

10
1
5
4

33
-7
23
17

32
16
6
10

12
0
7
4

28
-11
21
18

40
16
11
13

response of the labor share to technology shocks, consistently with its short-run countercyclicality
and a positive correlation of output with lagged labor shares. We shall verify if this property holds
for various labor share deﬁnitions.

The current analysis is based on quarterly data spanning 1948q1-2013q1. Our technological
shock variable is TFP growth, taken from Fernald (2012). Fernald’s TFP measures are superior to
the ones derived in the previous section because they distinguish between heterogenous physical
capital and labor types, whose unit productivities are inferred from data on relative prices.20 An-
other advantage of using Fernald’s TFP time series is that they also include TFP adjusted for capac-
ity utilization, which constitutes an important wedge between the available inputs and currently
produced output. Capacity utilization (varying machine hours, labor hoarding, etc.) can indeed
partially absorb technological shocks before they are transmitted to changes in factor shares.

Let us ﬁrst, though, elaborate on how, in terms of the earlier theory discussion, we view an
“aggregate” TFP shock. To examine this further, we can re-express the CES production function
(1) in per-capita log form and apply a Taylor-series expansion around the point ζ = 1, we derive
(Kmenta, 1967; Klump, McAdam, and Willman, 2012):

yt = αkt + Λk2
t
(cid:20)
2Λ
α

+ α

1 +

(cid:124)

(cid:21)

kt

γK,t + (1 − α)

(cid:20)

1 −

2Λ
(1 − α)

(cid:21)

kt

γN,t + Λ [γK,t − γN,t]2
(cid:125)

(12)

(cid:123)(cid:122)
Φ=log(TFP)

where yt = log[ Yt
Lt

], kt = log[ Kt
Lt

], Λ = (ζ−1)α(1−α)

2ζ

and γK,t = log ΓK,t, γN,t = log ΓN,t.

If ζ → 1 then yt = αγK,t + (1 − α) γN,t + αkt.21 Otherwise the log of TFP is an average
of capital and labor augmenting technologies (with the weights determined by the capital-labor
ratio and the income shares). In the absence of careful estimation, we do not observe Φ in the
factor-augmenting case, for an econometric discussion see León-Ledesma, McAdam, and Willman

20In his TFP computations, Fernald (2012) has interpolated annual data on factor shares from the BLS multifactor
productivity database, claiming that his “results were little affected in experiments with other reasonable choices, such
as using national accounting data”. At this, it is reassuring that our ﬁndings corroborate the robustness of TFP calcula-
tions to changes in factor share deﬁnitions, especially in the short run. See also the work of Basu, Fernald, and Kimball
(2006).

21In other words, the under-identiﬁed Cobb–Douglas form.

ECB Working Paper 1806, June 201526(2010). But its structure suggests we might think of general TFP shocks as driven mostly by labor-
augmenting components.22 As we know from condition (3), if the elasticity of substitution ζ < 1,
that shock will reduce the labor share.

7.2.1 ARDL MODEL

To assess the impact of exogenous technology shocks on the labor share, we ﬁrst estimated a range
of simple autoregressive distributed lag models:

lst = µ + ρlst−1 +

k
∑
i=0

βk

∆tfpt−k + εt

(13)

where lst = log(LSt) and ∆tfpt−k is TFP growth (difference in log TFP levels) lagged k quarters.
Table 12 shows the results.

First, we ﬁnd a negative contemporaneous correlation between the labor share and technolog-
ical shocks. The correlation with lagged TFP growth is positive, though. This is suggestive of a
non-monotonic, overshooting dynamics of the labor share following a TFP shock.23

Second, we ﬁnd that the effect of the technological shock is highest for the BLS and SE-GDP

series, and lowest for PI2-GDP, regardless of whether TFP shocks are capacity-adjusted or not.

Third, we extended model (13) to allow for the presence of asymmetric effects of technological
shocks on the labor share. Thus we have checked whether the labor share reacts differently to
positive and negative TFP shocks. Such results were obtained by splitting ∆tfpt−1 into

∆tfp+

t−1 = ∆tfpt−1I(∆tfpt−1 > 0)

∆tfp−

t−1 = ∆tfpt−1I(∆tfpt−1 < 0)

where I is the indicator function. Under this speciﬁcation, it is still estimated that β0 < 0, i.e., the
immediate effect of technology shocks is still to diminish the labor share. On the other hand, we
also ﬁnd that lagged (non-capacity-adjusted) technological shocks are positively correlated with
the labor share if they are negative, and essentially uncorrelated if they are positive. This means
that negative technological shocks tend to increase the labor share only temporarily (majority of
the immediate negative effect disappears after one period), whereas positive technological shocks
depress the labor share permanently (or at least for a longer time), and the overshooting dynamic
is absent.

This result is largely driven by capacity adjustment, however. We do not ﬁnd any evidence for
asymmetric correlation between TFP shocks adjusted by capacity utilization and the labor share.
Hence, it can be concluded that negative TFP shocks may appear as temporary only because they
induce substantial declines in capacity utilization. If these declines are properly accounted for,
negative TFP shocks tend to increase the labor share permanently as well. The numbers are fairly
symmetric across all labor share speciﬁcations, albeit again, they are somewhat larger for the BLS
and SE-GDP series, and smaller for PI2-GDP.

7.2.2 VAR ANALYSIS

A more sophisticated approach to assessing the impact of exogenous technology shocks on the
labor share requires the researcher to allow for mutual impact of both variables. In its simplest

22For instance ∂Φ/∂γL
∂Φ/∂γK

> 1 if α + 2 (1 + γK − γL) Λ < 1

for α = 0.33, ζ = 0.6 and γK ≈ γL.

2 . For a wide range of values this will typically hold, example

23Using rolling window estimation of equation (13) and its counterpart for HP-ﬁltered series we observe a slight

decrease in the contemporaneous correlation between the labor share and technological shocks over time.

ECB Working Paper 1806, June 201527Table 12: ARDL Model with TFP

Naive-GDP

PI-GDP

PI2-GDP

SE-GDP

BLS

0.955∗∗∗
−0.388∗∗∗
0.109∗∗

NO CAPACITY ADJUSTMENT
0.943∗∗∗
−0.357∗∗∗
0.152∗∗∗

0.987∗∗∗
−0.331∗∗∗
0.0704∗

CAPACITY ADJUSTMENT

0.995∗∗∗
−0.431∗∗∗
0.098∗∗

0.981∗∗∗
−0.528∗∗∗
0.141∗∗

0.987∗∗∗
−0.196∗∗∗
0.140∗∗∗

0.949∗∗∗
−0.233∗∗∗
0.188∗∗∗

0.966∗∗∗
−0.252∗∗∗
0.155∗∗∗
NO CAPACITY ADJUSTMENT + ASYMMETRIC LAG STRUCTURE
0.953∗∗∗
−0.391∗∗∗
0.324∗∗∗
0.006

0.947∗∗∗
−0.359∗∗∗
0.307∗∗∗
0.0809

1.00∗∗∗
−0.435∗∗∗
0.267∗∗
0.019

0.991∗∗∗
−0.289∗∗∗
0.172∗∗∗

0.993∗∗∗
−0.335∗∗∗
0.284∗∗∗

−0.028

0.982∗∗∗
−0.351∗∗∗
0.215∗∗∗

0.985∗∗∗
−0.530∗∗∗
0.265∗
0.084

ˆρ
ˆβ0
ˆβ1

ˆρ
ˆβ0
ˆβ1

ˆρ
ˆβ0
ˆβ−
1
ˆβ+
1

CAPACITY ADJUSTMENT + ASYMMETRIC LAG STRUCTURE

0.960∗∗∗
−0.240∗∗∗
−0.079
0.035

0.985∗∗∗
ˆρ
−0.342∗∗∗
ˆβ0
ˆβ1,−
−0.182
ˆβ1,+
0.056
Note: subscripts ∗∗∗, ∗∗ and ∗ denote the rejection of null about parameter’s insigniﬁcance
at 1%, 5% and 10% signiﬁcance level, respectively. The constant estimated is suppressed for
brevity.

0.956∗∗∗
−0.212∗∗∗
−0.066
0.105∗

0.995∗∗∗
−0.285∗∗∗
−0.116
0.007

0.989∗∗∗
−0.190∗∗∗
−0.094
0.007

form, such relationships can be analyzed by the means of a bivariate VAR model:

zt = a0 +

p
∑
i=1

Φ

izt−i + ut

(14)

where zt =

(cid:21)

(cid:20) ∆tfpt
lst

is the vector of jointly determined dependent variables and ut is a 2 × 1

vector of disturbances. Lag length p shall be selected according to the BIC criterion.

To analyze the dynamic response of the labor share to a technological shock we use orthogonal
impulse response functions. Since residuals from equations in VAR models can be correlated, the
standard IRF analysis does not include such information and, as a result, cannot generate the true
trajectories. Therefore, covariance between residuals is taken into consideration via a Cholesky
decomposition. Our ordering of variables corresponds to the ARDL approach and, correspond-
ingly, ∆tfpt is set as the ﬁrst variable in the system.

We ﬁnd that in both sets of IRFs (Figures 8–9), the effect of a temporary TFP shock is to reduce
the labor share. These results are in line with our earlier theoretical reasoning which points out
that TFP shocks are typically relatively more labor- than capital-augmenting, and that capital and
labor are gross complements, ζ < 1. As with the ARDL case, however, we might speculate that
some fraction of any technological improvement partly complements the existing capital stock or
labor input, and partly raises utilization rates. This latter possibility disguise some of the iden-
tiﬁcation of the technological shock’s effect on factor shares and explains why the response to
utilization-adjusted TFP shocks is generally much smaller and, at the 95% conﬁdence level, only
signiﬁcant in the ﬁrst period.

There are also marked differences in the speed of reversion, with GDPPI and (largely speak-
ing) naiveGDP having returned to their base within a 10 year horizon. For the other series, the
effect is highly protracted and stretches into the domain of medium-term business cycles. Clearly
in general equilibrium models where the labor share plays a non-trivial role (as for example in
labor bargaining models) this differential speed of reversion of income shares from technology
shocks will be very important.

ECB Working Paper 1806, June 201528Figure 8: Response of the Labor Share to a Technology Shock, no Capacity Adjustment

Naive-GDP

PI-GDP

PI2-GDP

SE-GDP

BLS

Notes: Bootstrapped 95% conﬁdence bands shown in blue dotted lines.

Figure 9: Response of the Labor Share to a Technology Shock Adjusted for Capacity Utilization

Naive-GDP

PI-GDP

PI2-GDP

SE-GDP

BLS

Notes: Bootstrapped 95% (68%) conﬁdence bands shown in blue (green) dotted lines.

7.3 APPLICATION III: NEW KEYNESIAN PHILLIPS CURVES

Our ﬁnal application is in the ﬁeld of inﬂation modelling. As in Galí and Gertler (1999) and sub-
sequent literature, the New-Keynesian Phillips Curve literature assumes staggered price setting
under imperfect competition, where a fraction θ of ﬁrms do not change their prices in any given
period. The remaining ﬁrms set prices optimally as a ﬁxed mark-up, µ, on discounted expected

0510152025303540×10-3-6-5-4-3-2-1010510152025303540×10-3-6-5-4-3-2-1010510152025303540×10-3-6-5-4-3-2-1010510152025303540×10-3-6-5-4-3-2-1010510152025303540×10-3-6-5-4-3-2-1010510152025303540×10-3-4-3.5-3-2.5-2-1.5-1-0.500.510510152025303540×10-3-4-3.5-3-2.5-2-1.5-1-0.500.510510152025303540×10-3-4-3.5-3-2.5-2-1.5-1-0.500.510510152025303540×10-3-4-3.5-3-2.5-2-1.5-1-0.500.510510152025303540×10-3-4-3.5-3-2.5-2-1.5-1-0.500.51ECB Working Paper 1806, June 201529(17)

.

marginal costs. When resetting, ﬁrms also take into account that the price may be ﬁxed for many
future periods, yielding the optimal reset price p∗
t ,

p∗
t = (1 − θβ) Et

∞
∑
k=0

(θβ)k (cid:2)mcn

t+k + µ(cid:3)

(15)

where mcn is (the log of) nominal marginal costs, β is a discount factor, and Et is the expectation
operator. The overall price level is then a weighted average of lagged and reset prices, pt =
θ pt−1 + (1 − θ) p∗
t − pt, and constant marginal costs across ﬁrms, the familiar
“New Keynesian Phillips Curve” (NKPC) emerges,

t . Given mcr

t ≡ mcn

where πt = pt − pt−1 is inﬂation and λ = (1−θ)(1−θβ)

θ

represents the reduced-form “slope”.

πt = βEtπt+1 + λ (mcr

t + µ)

(16)

Additionally, it is often assumed that of the 1 − θ price-re-setting ﬁrms a fraction, ω, set their
price according to lagged inﬂation. This implies a NKPC with an intrinsic expectations compo-
nent:

πt = γ f

Etπt+1 + γbπt−1 + λ (mcr

t + µ)

where φ = θ + ω [1 − θ (1 − β)], γ f = θβ

φ , γb = ω

φ , and λ = (1−ω)(1−θ)(1−θβ)

φ

Real marginal costs, mcr, are difﬁcult to measure, though. An early approach was to proxy
them by using the (stationary) deviation of output from a linear/quadratic trend, or a HP-ﬁltered
series. Alternatively, Galí and Gertler (1999) and others argued in favor of proxying real marginal
costs by average real unit labor costs. Under the special case of a (unitary substitution elasticity)
Cobb–Douglas production function, real marginal costs reduce to the labor share; this has tended
to be a common (if not the default) choice in the literature. If the elasticity of substitution between
capital and labor is not unitary, however, such a proxy can lead to biased estimates.

In the following application, we estimate both NKPC forms (speciﬁcations (16) and (17)) over
1960q1-2012q4; the start of the sample is chosen for comparisons with the Gali-Gertler study. Note
that the driving variable, i.e., the λ (·) term, whether it contains the output gap or the labor share,
should, as beﬁts a (price) gap term, be stationary. Stationarity in this context is simply another
way of saying that there is co-integration between the optimal and actual price: p∗
t − pt. In the
case of a typical non-structural output gap measure that stationarity is assured. As we know,
this is less clear for the labor share measures. For instance, revisiting Figure 3, we see (from the
1960s onwards) that SE-GDP and PI2-GDP have exhibited a clear downward trend. The other
three series are only borderline stationary in this period. This has a bearing on the success of the
resulting estimates.

Outwardly, though, the NKPC estimations work relatively well across labor share types: pa-
rameters are correctly signed and tend to be signiﬁcant (Table 13). For example, ˆβ tends to be
around the benchmark region of unity. However, estimates of the duration of price ﬁxedness vary
from 8.5 − 13.8 quarters. Although these durations are high (compared, say, to micro price-setting
evidence) they are by no means untypical in the literature (see the excellent survey by Mavroeidis,
Plagborg-Møller, and Stock (2014)).24

The slope parameters are of more interest here. To repeat, even though the driving variable
should be stationary, at best our labor share series are borderline stationary. Accordingly, the
minimization in the estimation algorithm places unusually low weights on the driving variable
(λ ∈ [0.005, 0.016]). As predicted earlier, the PI2-GDP and SE-GDP variants fare particularly
poorly in that regard: the former never supports a statistically signiﬁcant slope parameter, the
latter supports a signiﬁcant but quantitatively small one. Moreover, both of these speciﬁcations
produce the most unreasonable price setting durations. The naiveGDP and PI-GDP variants,

24For example, Galí, Gertler, and Lopez-Salido (2001), Gagnon and Khan (2005) and Smets and Wouters (2003) for

the euro area.

ECB Working Paper 1806, June 201530Table 13: New Keynesian Phillips Curve Estimates

Naive-GDP

0.891∗∗∗
0.980∗∗∗
0.015∗∗∗
9.2

0.104∗∗
0.883∗∗∗
0.961∗∗∗
0.106∗∗
0.863∗∗∗
0.016∗∗∗
8.5

θ
β
λ
D

ω
θ
β
γb
γ f
λ
D

PI-GDP

PI2-GDP
SPECIFICATION (16)
0.928∗∗∗
1.009∗∗∗
0.005
13.8

0.907∗∗∗
1.004∗∗∗
0.009∗∗
10.7

SPECIFICATION (17)

0.086∗∗
0.901∗∗∗
0.996∗∗∗
0.087∗∗
0.909∗∗∗
0.009∗∗
10.1

0.089
0.924∗∗∗
1.002∗∗∗
0.088
0.914∗∗∗
0.005
13.1

SE-GDP

BLS

0.915∗∗∗
1.009∗∗∗
0.007∗∗
11.8

0.065
0.912∗∗∗
1.002∗∗∗
0.066
0.936∗∗∗
0.007∗∗
11.4

0.909∗∗∗
1.008∗∗∗
0.008∗∗∗
11.0

0.035
0.910∗∗∗
1.004∗∗∗
0.037
0.967∗∗∗
0.008∗∗∗
11.1

Note: The covariance matrix was estimated with a 12 lags Newey-West estimator. The list of in-
struments is the same as in Galí and Gertler (1999): four lags of inﬂation, the labor share, the output
gap, the long-short interest rate spread, wage and commodity price inﬂation.

by contrast, have the lowest durations, signiﬁcant slopes and signiﬁcant parameters across both
NKPC forms.

NKPCs are not, naturally, a fool-proof way of gauging inﬂation movements; there are other
modelling approaches. That is not the main issue, though: our main point was that the NKPC
literature gave a central explanatory role to the labor share of income. However, arguably this
is not what most NKPC paper discuss. Much of the literature has instead become concerned
with estimation and identiﬁcation of dynamics (how much forward and backward-looking price
setting there is), which are the best instruments to use, etc. The question of whether results are
sensitive to which labor share measure we use has received little attention. In our case, though,
we have highlighted that we can tie the success of NKPC estimation to the relative properties of
the available labor share variants.

8 CONCLUSION

We provided a systematic exploration of the dynamic properties of a range of alternative US labor
share measures. We documented that these measures are not only divergent in terms of the im-
plied time trends, which are visible to a naked eye, but also differ in terms of their other dynamic
properties, such as the shares of short-, medium- and long-run variation in total volatility of the
series, degree of persistence, mean-reversion properties, and evidence for structural breaks.

Our results point to the general conclusion that while short-run properties of the labor shares
(represented by cyclical variation below 8 years) are relatively consistent across all deﬁnitions,
their medium-run swings (8-50 years) and long-run trends (≥ 50 years) substantially diverge.

While we generally recommend caution when designing the empirical labor share measure
suited to the given application at hand, we argue that the US series on the share of employees’
compensation in GDP, adjusted for proprietors’ income (which we call PI2-GDP) has intuitive,
economically interpretable properties, covers the entire economy, and thus might be perceived
as the “headline” measure of the US labor share since 1929. This measure, compared to its al-
ternatives, turns out to provide the relatively most consistent message across a range of diverse
exercises and applications discussed in this paper while providing implications which remain in
accordance with known “stylized facts” formulated in the earlier literature.

This measure suggests that the US labor share has not only declined after 1970, but also sub-
stantially increased before that, exhibiting a hump-shaped pattern over the last 84 years. It corrob-
orates the idea that instead of concentrating the decline in the labor share since 1970, one could

ECB Working Paper 1806, June 201531also embrace the larger time span of available data and discuss the (possibly technology-driven)
long cycle in this variable, see (Growiec, McAdam, and Mu´ck, 2015).

We have also presented three interesting applications of our results. We have documented
that while the measurement of factor shares has a modest impact on growth accounting, it can be
a central issue for some more sensitive estimations, such as identifying the short-run impact of
technology shocks on the labor share, and for gauging inﬂation properties.

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ECB Working Paper 1806, June 201534APPENDICES

A DATA DESCRIPTION

Abbreviation
Naive-GDP

PI-GDP

PI2-GDP

SE-GDP

Naive-GVA

Table A.1: Detailed Description of Data Construction

Description
Naive method, where CEt: Compensation of Employees [Table
1.12 NIPA] and Yt: GDP [Table 1.1.5 NIPA]
Adjustment by proprietor’s income, where CEt: Compensation
of Employees [Table 1.12 NIPA] and Yt: GDP [Table 1.1.5 NIPA]
and PIt: Proprietors’ income with IVA and CCAdj [Table 1.12
NIPA].
Extended adjustment by proprietor’s income (see Gomme and
Rupert (2007)). Most of the time series were taken from [Table
1.12 NIPA], apart from the GDP [Table 1.1.5 NIPA] and Consump-
tion of ﬁxed capital [Table 1.7.5 NIPA]
adjustment by self-employed, where Yt is GDP [Table 1.1.5 NIPA],
SEt: self-employment in private economy [Table 6.7 NIPA] and
TEt
is the sum of self-employment excluded and Full-Time
Equivalent Employee [Table 6.5 NIPA]. In order to construct the
quarterly labor share we use the date from BLS: Total Employ-
ment (sum of private [BLS CES0500000001 Series] and govern-
ment [BLS CES9000000001 Series]) and Self-employment (sum of
non-agriculture [BLS LNS12032192 Series] and agriculture self-
employment [BLS LNS12032185 Series]).
Naive method calculated for private sector, where CEt: Compen-
sation of Employees in private sector [Table 1.12 NIPA] and Yt:
GVA in private sector [Table 1.3.5 NIPA]

SE-GVA-NF

Naive-GVA-NF Naive method calculated for non-farm private sector, where CEt:
Compensation of Employees in private sector [Table 1.12 NIPA]
reduced by the CE for farms [Table 1.12 NIPA] and Yt: GVA in
private sector deduced by farms [Table 1.3.5 NIPA]
Adjustment by self-employed, where Yt is GVA for private econ-
omy [Table 1.3.5 NIPA] reduced by GVA in farm sector [Table
1.3.5 NIPA], SEt: self-employment in private economy reduced
by farms [Table 6.7 NIPA] and TEt is the sum of self-employment
excluded by farms sector and Full-Time Equivalent Employee in
private sector [Table 6.5 NIPA]
Adjustment by self-employed , where Yt is GVA for private econ-
omy [Table 1.3.5 NIPA], SEt: self-employment in private economy
[Table 6.7 NIPA] and TEt is the sum of self-employment and Full-
Time Equivalent Employee in private sector [Table 6.5 NIPA]
Labor Share in non-farm business
2005=100

[PRS85006173],

SE-GVA

sector

BLS

Equation Frequency

5

7

8

6

5

5

6

6

–

A&Q

A&Q

A&Q

A&Q

A

A

A

A

Q

Note: All the variables except self-employed data are expressed in current USD. “A” = Annual, “Q” = quarterly
frequencies.

ECB Working Paper 1806, June 201535B INDICATIVE SECTORAL ANALYSIS

One of the hypothesized explanations for the labor share decline since 1970s pertains to changes
in the sectoral structure of the US economy. As argued, e.g., by Elsby, Hobijn, and Sahin (2013),
sectors are subject to various degrees of cross-border integration, and recent decades have wit-
nessed an enormous surge of globalization and offshoring. And when labor-intensive produc-
tion moves to countries with lower labor costs, one could expect the aggregate labor share to
go down. On the other hand, the simultaneous rise of the service sector, and ﬁnancial services
in particular, could have worked in the opposite direction.

However, as presented in Table B.1 and Figure B.1 based on World KLEMS data, factor
shares have been far from constant at the sectoral level as well. For example, in Mining and
quarrying [C] as well as various branches of manufacturing, the labor share has been system-
atically falling throughout the period 1947–2010, whereas in numerous other branches, and
especially non-market service sectors such as Public administration, defence and compulsory
social security [L], Education [M], and Health and social work [N], it has been systematically
rising.

Hence, results of shift-share analyses – i.e., contributions of respective sectors to the total
change in the aggregate labor share – are going to be driven both by the within- and between-
sector component. Figure B.2 illustrates this point. The shares of labor remuneration in man-
ufacturing, as well as agriculture and mining, in total labor remuneration have been systemat-
ically falling throughout the entire period 1947–2010, driven both by declining labor shares in
these sectors and their declining share of total value added. Market services provide a mirror
image of this result. However, the ﬁnancial sector, whose rise was hypothesized to be one of
the drivers of labor share declines, provides a particularly interesting result here. In fact, its
share in total labor remuneration has increased in the recent years, as the increase in its share
of value added has outrun the labor share decline in this sector.

ECB Working Paper 1806, June 201536Table B.1: Share of US Sectors in Gross Value Added, Labor Share in Sectoral GVA, and Unit Root
Tests

share in value added (wi,t)
wi,2010 − wi,1947

labor share (lsi,t)

ADF test

PP test

const

const

trend

trend

−2.27
−3.16∗

lsi,2010 − lsi,1947

¯wi
−0.11
3.24
AtB
−0.10
2.16
15t16
0.01
1.43
17t19
−0.11
0.83
20
−0.02
2.02
21t22
−0.43
0.55
23
−0.10
1.99
24
−0.05
0.70
25
0.01
0.68
26
−0.06
3.06
27t28
−0.01
1.83
29
−0.27
2.32
30t33
0.03
2.58
34t35
−0.35
0.56
36t37
−0.09
1.19
50
0.09
5.37
51
−0.03
5.05
52
60t63
0.09
4.01
−0.34
2.82
64
0.01
10.75
70
71t74
0.33
7.94
−0.23
1.96
C
−0.11
2.56
E
−0.02
4.76
F
−0.03
2.29
H
−0.13
5.13
J
L
0.58
5.42
0.59
4.24
M
0.60
8.53
N
−0.03
3.69
O
0.00
P
0.31
TOT 100.00
0.04
Note: ADF and PP stand for Augmented Dickey-Fuller and Phillips-Perron test, respectively. Superscripts ∗∗∗, ∗∗
and ∗ denote the rejection of null about unit root at 1%, 5% and 10% signiﬁcance level, respectively.

−1.72
−2.22
−1.70
−3.25∗
−1.28
−1.19
−5.26∗∗∗ −2.98∗∗ −5.79∗∗∗
−2.25
−5.56∗∗∗
−5.32∗∗∗ −2.53
−2.46
−2.99∗∗ −3.81∗∗
−2.85
−2.09
−3.26∗
−3.29∗
−0.79
−0.60
−2.58
−1.77
−2.57
−1.76
−4.01∗∗∗ −3.99∗
−2.11
−2.14
−2.60∗
−2.67∗
−2.72
−2.77
−2.58
−2.19
−2.76
−2.31
−1.26
−1.39
−1.30
−1.56
−0.15
0.07
0.98
0.73
−3.13∗∗ −3.33∗
−2.67∗
−2.83
−3.87∗∗
−3.16∗
−0.41
−0.01
−4.80∗∗∗
−5.72∗∗∗ −1.41
−1.28
−3.47∗∗∗ −3.70∗∗ −3.46∗∗∗ −3.77∗∗
−4.75∗∗∗
−5.56∗∗∗ −2.21
−2.01
−2.60∗
−2.77
−2.10
−2.43
−3.33∗
−2.02
−2.72
−1.44
−1.12
−0.91
−1.78
−1.16
−4.37∗∗∗ −3.70∗
−3.81∗∗∗ −3.24∗
−2.02
−2.72∗
−2.00
−1.41
−0.85
−3.14∗∗ −3.55∗∗ −4.81∗∗∗ −5.96∗∗∗
−4.72∗∗∗ −5.59∗∗∗
−2.99∗∗ −3.08
−4.33∗∗∗ −6.24∗∗∗
−2.65∗
−2.74
−3.06∗∗ −3.00
−2.47
−2.41
−7.06∗∗∗ −6.98∗∗∗ −7.77∗∗∗ −7.70∗∗∗
−4.29∗∗∗ −3.74∗∗
−3.03∗∗∗ −2.33

−8.54
−2.21
−3.32
−0.86
−1.17
0.51
−0.08
−0.25
−0.61
−2.98
−1.05
0.27
−1.24
0.00
−0.29
−1.11
−2.61
−3.37
0.93
3.00
11.56
−0.95
0.22
−0.36
0.11
6.52
−3.83
3.88
8.27
0.36
−0.79

¯lsi
0.57
0.60
0.80
0.82
0.71
0.45
0.47
0.65
0.67
0.73
0.62
0.76
0.74
0.78
0.81
0.67
0.83
0.70
0.59
0.05
0.63
0.40
0.30
0.90
0.80
0.62
0.64
0.66
0.68
0.73
1.00
0.61

−3.61∗∗ −2.63∗
−2.44
−2.14
−3.29∗
−2.16

−3.58∗∗∗ −3.19∗
−2.59∗
−2.39
−3.18∗
−1.71
−1.80
−0.70

−4.24∗∗∗

Sectors: Total economy [TOT]; Agriculture, hunting, forestry and ﬁshing [AtB]; Mining and quarrying [C]; Manufacture of food
products, beverages and tobacco products [15t16]; Manufacture of textiles, wearing apparel, dressing and dyeing of fur, luggage,
handbags, saddlery, harness and footwear and tanning and dressing of leather [17t19]; Manufacture of wood and of products of
wood and cork, except furniture; manufacture of articles of straw and plaiting materials [20]; Manufacture of paper and paper
products, publishing, printing and reproduction of recorded media [21t22]; Manufacture of coke, reﬁned petroleum products
and nuclear fuel [23]; Manufacture of chemicals and chemical products [24]; Manufacture of rubber and plastics products [25];
Manufacture of other non-metallic mineral products [26]; Manufacture of basic metals, fabricated metal products, except machin-
ery and equipment [27t28]; Manufacture of machinery and equipment n.e.c. [29]; Manufacture of ofﬁce, accounting, computing
machinery, electrical machinery, apparatus, radio, television, communication equipment, medical, precision and optical instru-
ments, watches and clocks [30t33]; Manufacture of motor vehicles, trailers, semi-trailers and other transport equipment [34t35];
Manufacture of furniture, manufacturing n.e.c. and recycling [35t37]; Electricity, gas and water supply [E]; Construction [F]; Sale,
maintenance and repair of motor vehicles and motorcycles; retail sale of automotive fuel [50]; Wholesale trade and commission
trade, except of motor vehicles and motorcycles [51]; Retail trade, except of motor vehicles and motorcycles; repair of personal
and household goods [52]; Hotels and restaurants [H]; Land transport, transport via pipelines, water transport, air transport,
supporting, auxiliary transport activities and activities of travel agencies [60t63]; Post and telecommunications [64]; Financial in-
termediation [J]; Real estate activities [70]; Renting of machinery and equipment without operator and of personal and household
goods, computer and related activities, research and development and other business activities [71t74]; Public administration,
defence and compulsory social security [L]; Education [M]; Health and social work [N]; Other community, social and personal
service activities [O]; Private households with employed persons [P].

ECB Working Paper 1806, June 201537Figure B.1: Payroll share in the US sectors

AtB

C

15t16

17t19

20

25

21t22

23

26

27t28

30t33

34t35

36t37

F

50

51

24

29

E

52

19501960197019801990200020100.450.550.6519501960197019801990200020100.250.350.4519501960197019801990200020100.450.550.650.7519501960197019801990200020100.700.750.800.8519501960197019801990200020100.750.800.850.9019501960197019801990200020100.640.680.720.7619501960197019801990200020100.10.30.50.719501960197019801990200020100.400.450.500.5519501960197019801990200020100.550.600.650.7019501960197019801990200020100.600.650.700.750.8019501960197019801990200020100.650.700.750.8019501960197019801990200020100.500.600.700.8019501960197019801990200020100.50.60.70.819501960197019801990200020100.650.700.750.800.8519501960197019801990200020100.550.650.750.8519501960197019801990200020100.250.300.350.4019501960197019801990200020100.820.860.900.9419501960197019801990200020100.650.750.8519501960197019801990200020100.600.640.680.7219501960197019801990200020100.650.750.85ECB Working Paper 1806, June 201538H

70

N

60t63

64

71t74

O

L

P

J

M

TOT

Note: solid and dashed lines stand for the payroll share in given sector and its long-run tendency, respectively.

Sectors: Total economy [TOT]; Agriculture, hunting, forestry and ﬁshing [AtB]; Mining and quarrying [C]; Manufacture of food prod-
ucts, beverages and tobacco products [15t16]; Manufacture of textiles, wearing apparel, dressing and dyeing of fur, luggage, handbags,
saddlery, harness and footwear and tanning and dressing of leather [17t19]; Manufacture of wood and of products of wood and cork,
except furniture; manufacture of articles of straw and plaiting materials [20]; Manufacture of paper and paper products, publishing, print-
ing and reproduction of recorded media [21t22]; Manufacture of coke, reﬁned petroleum products and nuclear fuel [23]; Manufacture
of chemicals and chemical products [24]; Manufacture of rubber and plastics products [25]; Manufacture of other non-metallic mineral
products [26]; Manufacture of basic metals, fabricated metal products, except machinery and equipment [27t28]; Manufacture of machin-
ery and equipment n.e.c. [29]; Manufacture of ofﬁce, accounting, computing machinery, electrical machinery, apparatus, radio, television,
communication equipment, medical, precision and optical instruments, watches and clocks [30t33]; Manufacture of motor vehicles, trail-
ers, semi-trailers and other transport equipment [34t35]; Manufacture of furniture, manufacturing n.e.c. and recycling [35t37]; Electricity,
gas and water supply [E]; Construction [F]; Sale, maintenance and repair of motor vehicles and motorcycles; retail sale of automotive
fuel [50]; Wholesale trade and commission trade, except of motor vehicles and motorcycles [51]; Retail trade, except of motor vehicles
and motorcycles; repair of personal and household goods [52]; Hotels and restaurants [H]; Land transport, transport via pipelines, water
transport, air transport, supporting, auxiliary transport activities and activities of travel agencies [60t63]; Post and telecommunications
[64]; Financial intermediation [J]; Real estate activities [70]; Renting of machinery and equipment without operator and of personal and
household goods, computer and related activities, research and development and other business activities [71t74]; Public administration,
defence and compulsory social security [L]; Education [M]; Health and social work [N]; Other community, social and personal service
activities [O]; Private households with employed persons [P].

19501960197019801990200020100.750.800.8519501960197019801990200020100.600.650.700.7519501960197019801990200020100.50.60.70.819501960197019801990200020100.500.550.600.6519501960197019801990200020100.030.040.050.060.0719501960197019801990200020100.40.50.60.719501960197019801990200020100.10.30.50.719501960197019801990200020100.20.40.60.819501960197019801990200020100.20.40.60.819501960197019801990200020100.680.700.720.740.7619501960197019801990200020100.99981.00001.000219501960197019801990200020100.560.580.600.620.64ECB Working Paper 1806, June 201539Figure B.2: Sectoral Decomposition of the Annual US Labor Share

0.00.20.40.61950196019701980199020002010Agriculture, natural resources, miningManufacturing            Construction             Trade, transportation, hotelsFinance                  Public administrationEducation and health servicesProfessional business servicesOther services, private householdsECB Working Paper 1806, June 201540C UNIT ROOT TESTS AND STRUCTURAL BREAKS

Table C.1: Unit Roots Tests: Annual

P
D
G

-
I
P

P
D
G

-
2
I
P

P
D
G
-
E
S

A
V
G
-
e
v
i
a
N

ADF

P
D
G
-
e
v
i
a
N

(1) −2.10
(2) −1.69

(1) −2.82∗ −3.13∗∗ −3.42∗∗ −1.68
(2) −2.08

−2.80

−2.77∗∗ −3.83∗∗∗ −2.25
−2.53

−1.45
−4.36∗∗∗ −4.35∗∗∗ −2.37

PP
−1.83
−3.79∗∗ −4.39∗∗∗ −2.26
ADF-GLS
−0.56
−1.02
−2.83∗ −2.09
H0 : ESTAR

−1.23
−1.61

(1) −0.58
(2) −1.07

−1.18
−1.69

-

F
N
A
V
G
-
e
v
i
a
N

-

F
N
A
V
G
-
E
S

A
V
G
-
E
S

−2.11
−3.01

−2.35
−3.03

−3.73∗∗∗ −3.72∗∗∗
−3.93∗∗ −4.19∗∗∗

−3.55∗∗∗ −2.65∗
−3.56∗∗ −3.39∗

−3.31∗∗∗ −2.37∗∗
−1.52
−3.32∗∗ −3.86∗∗∗ −3.56∗∗

(1) −2.687∗ −2.874∗ −4.007∗∗∗−2.798∗ −1.896 −2.695∗ −3.576∗∗∗−3.423∗∗
(2) −2.013 −2.984 −4.256∗∗∗−3.634∗∗ −3.307∗ −3.426∗∗ −3.329∗ −3.804∗∗

(1)
(2)

4.450∗
2.516

5.248∗∗
4.476

8.038∗∗∗ 4.275∗
2.334
9.190∗∗∗ 10.379∗∗∗ 6.232∗

4.634∗
6.928∗∗

6.439∗∗
6.032∗

6.435∗∗
7.361∗∗

H0 : asymmetric ESTAR

Fractional

0.639
[0.015]
[0.091]

ˆd
0.791
ˆd = 0 [0.072]
ˆd = 1 [0.328]
Note: ∗∗∗, ∗∗ and ∗ denote rejection of the null hypothesis of a unit root for all tests at the 1%, 5% and 10%
signiﬁcance level, respectively. Models (1) and (2) incorporate only a constant and a constant and deter-
ministic trend, respectively. The ADF ESTAR and asymmetric ADF ESTAR follow (Kapetanios, Shin, and
Snell, 2003) and (Sollis, 2009), respectively.

0.884
[0.006]
[0.588]

0.836
[0.047]
[0.448]

0.649
[0.011]
[0.101]

0.577
[0.114]
[0.048]

0.696
[0.003]
[0.154]

0.421
[0.560]
[0.007]

Table C.2: Unit Roots Tests: Quarterly

Naive-GDP

PI-GDP

(1)
(2)

(1)
(2)

(1)
(2)

(1)
(2)

(1)
(2)

ˆd
ˆd = 0
ˆd = 1

−2.10
−1.69

−2.16
−1.77

−0.91
−1.14

−2.237
−2.532

3.981
3.227

1.079
[0.000]
[0.620]

PI2-GDP
ADF
−1.32
−2.12
PP
−1.73
−2.38
ADF-GLS
−0.51
−2.24

−2.46
−2.78

−2.98∗∗
−3.21∗

−1.44
−3.14∗∗

H1 : ESTAR

−3.161∗∗ −1.100
−3.876∗∗ −2.130

SE-GDP

BLS

−0.79
−3.82∗∗

−1.36
−2.74

1.01
−4.19∗∗∗

−1.83
−3.27∗

0.21
−3.51∗∗∗

−0.11
−3.31∗∗

−0.952
−4.681∗∗∗ −3.535∗∗

−1.038

1.051
10.951∗∗∗

4.997∗∗
8.058∗∗

H1 : asymmetric ESTAR
0.854
3.300
Fractional
0.858
[0.000]
[0.378]

0.866
[0.001]
[0.406]

0.784
[0.007]
[0.180]

1.350
7.804∗∗

0.854
[0.001]
[0.364]

ECB Working Paper 1806, June 201541Table C.3: Number of Breaks with Corresponding Breakpoints – Annual Series

P
D
G
-
e
v
i
a
N

1942
(1940, 1944)
1955
(1954, 1959)
1967
(1966, 1968)
1982
(1980, 1984)
2000
(1987, 2004)

1968
(1967, 1969)

1999
(1997, 2000)

P
D
G

-
I
P

P
D
G

-
2
I
P

P
D
G
-
E
S

A
V
G
-
e
v
i
a
N

-

F
N
A
V
G
-
e
v
i
a
N

-

F
N
A
V
G
-
E
S

A
V
G
-
E
S

1941
(1940, 1942)

1942
(1938, 1955)

1940
(1939, 1945)

1945
(1943, 1949)

1945
(1942, 1987)

1940
(1939, 1946)

WITH MEAN

1967
(1964, 1976)
1980
(1977, 1981)

1967
(1963, 1974)
1980
(1978, 1981)

1992
(1990, 1993)

1952
(1951 , 1954)
1967
(1965, 1968)

1967
(1964, 1970)

1942
(1939, 1943)
1955
(1954, 1968)
1968
(1967, 1969)
1982
(1977, 1983)
1999
(1998, 2000)

WITH LINEAR TREND

1946
(1945, 1956)

1940
(1938, 1950)

1942
(1941 , 1945)

1941
(1940, 1946)

1945
(1944, 1951)

1945
(1944, 1952)

1968
(1967, 1969)
1985
(1984, 1987)
1999
(1996, 2000)

1968
(1962, 1969)

1974
(1973, 1975)

1968
(1966, 1969)

1999
(1996, 2000)

1997
(1994, 1998)

1999
(1997, 2000)

1940-1950s

1960-1970s

1980-1990s

2000s

1940-1950s

1960-1970s

1980-1990s

2000s
Note: The breakpoints are calculated in two steps. In the ﬁrst step, we estimate all the possible models with a number of struc-
tural breaks varying from 1 to 5. In the second step, we choose one with the lowest BIC criterion. The years in parentheses are
95% conﬁdence intervals.

ECB Working Paper 1806, June 201542Table C.4: Number of Breaks with Corresponding Breakpoints – Quarterly Series

1940-1950s

1960-1970s

1980-1990s

2000s

1940-1950s

1960-1970s

1980-1990s

1940-1950s
1960-1970s

1980-1990s

2000s

Naive-GDP

PI-GDP

PI2-GDP
WITH MEAN

SE-GDP

BLS

1956q3
(1956q2, 1957q3)
1967q3
(1967q2, 1967q4)
1983q1
(1982q2, 1983q4)

1968q2
(1967q3, 1968q4)
1980q4
(1979q4, 1981q2)

1967q4
(1967q1, 1968q2)
1980q3
(1980q2, 1981q1)

2003q2
( 2002q3, 2003q4)

2003q2
(2000q2, 2005q4)

2003q2
(2002q1, 2003q4)

1962q2
(1961q3, 1962q4)
1983q2
(1983q1, 1984q2)
1993q3
(1992q2, 1993q4)
2003q2
(2002q3, 2003q4)

1983q1
(1982q3, 1985q4)
1992q4
(1989q4, 1993q3)
2003q2
(2002q3, 2003q3)

WITH LINEAR TREND

1958q1
(1957q4, 1959q4 )
1968q2
(1967q3, 1968q3 )

1968q2
(1967q3, 1968q3 )

1969q1
(1968q3, 1969q2)

1957q3
(1957q1, 1958q1)
1969q1
(1968q4, 1969q3)
1978q4
(1978q1, 1979q4 )

1956q3
(1955q2, 1957q1)
1969q1
(1968q4, 1969q3)
1978q4
(1978q2, 1980q1)

1983q1
(1982q3, 1985q4 )
1999q4
(1998q4, 2000q1)

1983q1
(1981q3, 1983q2)
1999q4
(1999q3, 2000q1)

1986q1
(1985q4, 1986q3)
1999q4
(1998q4, 2000q1)

1999q4
(1998q4, 200q1)

1999q4
(1998q4, 200q1)

WITH QUADRATIC TREND

1960q3
(1960q2, 1960q4 )
1970q2
(197q1, 1970q3)
1983q1
(1982q4, 1983q2)
1997q4
(1997q3, 1998q1)

1960q3
(1960q2, 1960q4)
1970q2
(197q1, 1970q3)
1986q3
(1986q2, 1986q4)
1999q4
(1999q3, 200q1)

1960q3
(1960q2, 1960q4)
1970q2
(197q1, 1970q3)
1983q1
(1982q4, 1983q2)
1997q4
(1997q3, 1998q1)

1960q3
(1960q2, 1960q4 )
1970q2
(197q1, 1970q3)
1983q1
(1982q4, 1983q2)
1997q4
(1997q3, 1998q1)

1960q1
(1959q4, 1960q2)
1969q4
(1969q3, 1970q1)
1981q2
(1980q4, 1981q4)
1992q1
(1991q4, 1992q2)
2001q4
(2001q3, 2002q1)

Note: The breakpoints are calculated in two steps. In the ﬁrst step, we estimate all the possible models with a number of structural
breaks varying from 1 to 5. In the second step, we choose one with the lowest BIC criterion. The years in parentheses are conﬁdence
intervals at 95% signiﬁcance level.

ECB Working Paper 1806, June 201543Table C.5: Zivot and Andrews (1992) Test for a Unit Root Subject to a Structural Break – Annual Series

intercept

trend

intercept and trend

τ
−3.31
Naive-GDP
−4.59∗
PI-GDP
−4.44
PI2-GDP
−6.33∗∗∗
SE-GDP
−3.87
Naive-GVA
Naive-GVA-NF −3.98
SE-GVA-NF
SE-GVA
Note: τ and B denote the test statistic in Zivot-Andrews procedure and its breakpoint, respectively. Asterisks
∗∗∗, ∗∗ and ∗ denote rejection of the null about unit root at 1%, 5% and 10% signiﬁcance level, respectively.

τ
−5.12∗∗
−4.48
−4.63
−6.09∗∗∗
−3.86
−4.61
−6.50∗∗∗
−6.85∗∗∗

τ
−4.44∗∗
−3.62
−4.35∗
−6.30∗∗∗
−3.36
−3.93
−4.73∗∗
−6.73∗∗∗

B
1952
1942
2004
1935
1952
1952
1934
1934

B
1975
1944
2002
1941
1971
1974
1936
1941

B
1967
1942
1967
1944
1952
1942
1946
1945

−5.08∗∗∗
−7.27∗∗∗

Table C.6: Zivot and Andrews (1992) Test for a Unit Root Subject to a Structural Break – Quarterly Series

intercept

trend

intercept and trend

τ

Naive-GDP −3.74
−4.32
PI-GDP
−3.74
PI2-GDP
−5.28∗∗∗
SE-GDP
−4.79∗∗
BLS
Note: As in table C.5.

B
1966q2
1966q2
1966q2
1967q3
2009q1

τ
−4.58∗∗
−4.30∗
−3.81
−5.26∗∗∗
−5.05∗∗∗

B
1972q1
1970q1
1970q2
2000q2
2000q4

τ
−5.03∗
−4.53
−4.13
−5.47∗∗
−5.11∗∗

B
1968q3
1974q4
1966q2
1999q1
2000q1

D STATIONARITY OF LOG-DIFFERENCES BETWEEN ALTERNATE LABOR

SHARE VARIANTS

One of the indications that two time series diverge in terms of their dynamics is that their dif-
ference is non-stationary. We have checked if this possibility appears among various labor share
measures by running a series of unit root tests. Formally, we deﬁne the following relationship
between two different empirical measures of the (log) labor share:

log LSi,t = log LSj,t + ηi,j,t

(A.1)

For example, if LSi,t is calculated with thenaive method then ηi,j,t stands for a multiplicative adjust-
ment component, augmenting CEt/Yt in the calculation of LSj,t (according to some other method).
Naturally, when ηi,j,t is stationary then we might expect the ratio (and the difference) between both
labor share series to be broadly stable over time. Apart from the hypothesis about stability of ηi,j,t,
we check also whether that difference might be trend-stationary.

Tables D.7 and D.8 present the results of augmented Dickey-Fuller (ADF) tests for the afore-
mentioned differences. Given that all variables were supposed to measure the same phenomenon,
stationarity of the residual component ηi,j,t at the 5% signiﬁcance level is found surprisingly rarely.
The only clear evidence is provided for the stability of the difference between two different ver-
sions of the labor share adjusted by proprietors’ income (in both annual and quarterly data). There
is also somewhat more consistency in the case of annual series, especially the ones obtained with
GVA instead of GDP, and if one allows for time trends in the differences.

The key message from the unit root tests for the log-differences between the labor share vari-
ants is that most of the proposed adjustments to the naive calculation are not constant over time.
Please recall that there are two key tendencies which are responsible for that facts (cf. Figure 1):
(i) a downward trend in the ratio of the self-employed to employees, and (ii) a systematic decrease
in the share of ambiguous income in total output.

ECB Working Paper 1806, June 201544Table D.7: Unit Root Test for Differences Between Labor Share Variants – Annual Series

P
D
G

-
I
P

P
D
G

-
2
I
P

P
D
G
-
E
S

A
V
G
-
e
v
i
a
N

-

F
N
A
V
G
-
e
v
i
a
N

-

F
N
A
V
G
-
E
S

A
V
G
-
E
S

−2.13
−4.32∗∗∗ −3.55∗∗ −3.12

−3.70∗∗∗ −3.15∗∗ −2.75∗
−2.88

−4.14∗∗∗ −3.63∗∗
−1.36
−3.60∗∗ −3.96∗∗ −4.14∗∗∗ −3.65∗∗

−2.39

−3.58∗∗∗ −0.79
−3.42∗∗ −3.96∗∗∗
−1.53
−3.25∗ −4.53∗∗∗ −5.04∗∗∗ −4.34∗∗∗ −3.59∗∗

−0.82
−2.86

−1.39
−1.79

−2.06
−2.35

−2.77∗ −2.56
−4.63∗∗∗ −3.03

Naive-GDP
(1)
(2)
PI-GDP
(1)
(2)
PI2-GDP
(1)
(2)
SE-GDP
(1)
(2)
Naive-GVA
(1)
(2)
Naive-GVA-NF
(1)
(2)
SE-GVA-NF
(1)
(2)
Note: (1): ADF with intercept. (2): ADF with intercept and a linear trend.

−0.79
−2.01

−0.83
−2.50

−1.25
−2.02

−2.69∗
−1.12
−3.84∗∗ −3.73∗∗

−1.49
−1.68

−1.72
−2.06

−2.51
−1.78
−3.58∗∗ −2.21

−1.56
−2.14

Table D.8: Unit Root Test for Differences Between Labor Share Variants – Quarterly Series

PI-GDP

PI2-GDP

SE-GDP

BLS

−2.75∗
−1.64

Naive-GDP
(1)
(2)
PI-GDP
(1)
(2)
PI2-GDP
(1)
(2)
SE-GDP
(1)
(2)
Notes: See note to table D.7.

−2.56
−2.77

−3.19∗∗
−4.27∗∗∗

−3.40∗∗ −2.13
−2.46
−2.15

−0.79
−2.42

−2.10
−2.45

−0.67
−1.67

−2.30
−2.79

−2.42
−2.66

ECB Working Paper 1806, June 201545Acknowledgements 

We gratefully acknowledge financial support from the Polish National Science Center (Narodowe Centrum Nauki) under the grant Opus 
3 No. 2012/05/B/HS4/02236. The views expressed here belong to the authors alone.  

Jakub Mućk 

Warsaw School of Economics and Narodowy Bank Polski;  
e-mail: jakub.muck@gmail.com  

Peter McAdam 

European Central Bank and University of Surrey;  
e-mail: peter.mcadam@ecb.europa.eu  

Jakub Growiec 

Warsaw School of Economics and Narodowy Bank Polski; 
e-mail: jakub.growiec@gmail.com  

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ISSN 
ISBN 
DOI 
EU catalogue number 

1725-2806 (online) 
978-92-899-1619-6 
10.2866/208728 
QB-AR-15-046-EN-N