# The Gender Pay Gap Revisited with Big Data:
Do Methodological Choices Matter?††thanks: We acknowledge helpful comments by
Philipp Bach, Marina Bonaccolto-Töpfer, Christina Felfe, Martin Huber, Pat
Kline, Michael Knaus, Matthias Krapf, and Michael Lechner, seminar
participants at the University of Basel, University of Linz, and LISER,
Luxemburg, as well as conference participants at EALE/SOLE 2020, Verein für
Socialpolitik 2020, and the IAB 2020 workshop on “Machine Learning in Labor,
Education, and Health Economics”. Anthony Strittmatter gratefully acknowledges
financial support from the Swiss National Science Foundation (Spark Project
190422) and the French National Research Agency (LabEx
Ecodec/ANR-11-LABX-0047). The authors are solely responsible for the analysis
and the interpretation thereof.

Anthony Strittmatter111CREST-ENSAE, Institut Polytechnique Paris, France;
CESifo, Munich, Germany; email<EMAIL_ADDRESS>Conny
Wunsch222Faculty of Business and Economics, University of Basel; University
St. Gallen, Switzerland; email<EMAIL_ADDRESS>
(September 3, 2024)

###### Abstract

The vast majority of existing studies that estimate the average unexplained
gender pay gap use unnecessarily restrictive linear versions of the Blinder-
Oaxaca decomposition. Using a notably rich and large data set of 1.7 million
employees in Switzerland, we investigate how the methodological improvements
made possible by such big data affect estimates of the unexplained gender pay
gap. We study the sensitivity of the estimates with regard to i) the
availability of observationally comparable men and women, ii) model
flexibility when controlling for wage determinants, and iii) the choice of
different parametric and semi-parametric estimators, including variants that
make use of machine learning methods. We find that these three factors matter
greatly. Blinder-Oaxaca estimates of the unexplained gender pay gap decline by
up to 39% when we enforce comparability between men and women and use a more
flexible specification of the wage equation. Semi-parametric matching yields
estimates that when compared with the Blinder-Oaxaca estimates, are up to 50%
smaller and also less sensitive to the way wage determinants are included.

_Keywords:_ Gender Inequality, Gender Pay Gap, Common Support, Model
Specification, Matching Estimator, Machine Learning.

_JEL classification:_ J31, C21

##  1 Introduction

Achieving gender equality, especially equality in pay, is among the top
priorities for governments in many countries. Measuring unexplained
inequalities in pay between women and men, the so-called unexplained gender
pay gap, has been the subject of an extensive literature for more than half a
century (see Blau and Kahn, 2000, 2017; Goldin and Mitchell, 2017; Kunze,
2018; Olivetti and Petrongolo, 2016, for comprehensive reviews). A large
literature focuses on understanding the sources of inequality in pay by
studying the sensitivity of the gender pay gap with regard to the inclusion of
important wage determinants. A much smaller literature focuses on the impact
of methodological choices. We contribute to the latter literature by
investigating the sensitivity of the estimated gender pay gap with regard to
three dimensions: enforcement of overlap in wage determinants across gender,
flexibility of model specification, and estimation method. Our results suggest
that these methodological choices can reduce the estimated unexplained gender
pay gap by as much as 50%, even when keeping relevant wage determinants fixed
across estimates. This suggests that methodological choices are at least as
relevant for gender pay gap estimations as considerations about wage
determinants.

Weichselbaumer and Winter-Ebmer (2005) and Van der Velde et al. (2015)
document that the vast majority of existing studies use a linear version of
the Blinder-Oaxaca decomposition (BO, Blinder, 1973; Oaxaca, 1973) to estimate
the mean unexplained gender pay gap (see Fortin et al., 2011, for a
comprehensive review of BO and other decomposition methods). Thus, BO
estimates serve as the key input for policy makers who aim to achieve equality
in pay. However, most applications of BO impose a number of restrictions that
may not be realistic. Firstly, they typically use relatively inflexible
functional forms for the wage equation. For example, the returns to education
are often assumed to be the same across occupations, age and experience, which
contradicts both theory and empirical evidence (see, e.g., Lemieux, 2014).
Secondly, standard applications of BO do not account for common support
violations, i.e., the lack of observationally comparable men for every woman.
BO extrapolates into regions without support based on the assumed functional
form. If the functional form is misspecified and there is lack of support,
then this extrapolation may lead to bias. Thirdly, BO restricts heterogeneity
in gender pay gaps to variable-specific differences in coefficients. Any
heterogeneity that is not captured by this may bias estimates of the mean
gender pay gap. This is particularly relevant, because heterogeneity in gender
pay gaps is widely acknowledged in the literature (see, e.g., Bach et al.,
2018; Barón and Cobb-Clark, 2010; Bonjour and Gerfin, 2001; Chernozhukov et
al., 2013, 2018a; Goldin, 2014).

Of course, these restrictions can easily be relaxed when working with large
data sets. Researchers can model the inclusion of control variables in more
flexible ways. They can check and enforce common support ex ante, even for
parametric estimators like BO. Furthermore, they can choose more flexible
estimators that unlike BO do not restrict pay gap heterogeneity. Surprisingly
though, these adjustments are rarely implemented in applied work. Therefore,
an important question we explore in this study is whether these adjustments
really matter in practice. To answer this question, we exploit a very large
data set that covers about 37,000 establishments with individual data on more
than 1.7 million employees in Switzerland, which covers almost one third of
all Swiss employees. The data allow us to take full advantage of the
methodological improvements that are possible with existing methods. In
particular, they allow us to go as far as exact matching on all elements of
the rich set of observed wage determinants with resulting cells that are still
large enough for meaningful analysis after enforcing full support.

We start by analyzing common support and the gender pay gap obtained from
exact matching applying the technique of Ñopo (2008). Thereafter, we estimate
the mean unexplained gender pay gap with different parametric and semi-
parametric methods and for various samples that differ in how strictly we
impose common support. As estimators, we consider a linear regression model
(LRM) with a dummy for women, BO, inverse probability weighting (IPW),
augmented IPW (AIPW) as a doubly robust mixture between BO and IPW, propensity
score matching (PSM) and a combination of exact matching and PSM (EXPSM). For
each estimator, we consider three model specifications that differ in how
flexibly we include the observed wage determinants: (i) the baseline model
contains dummy variables for all categories and quadratic terms for the
continuous variables (up to 117 control variables), (ii) the full model
additionally contains higher-order polynomials as well as a large number of
interactions between wage determinants (up to 615 control variables), and
(iii) the machine learning model employs LASSO estimation techniques
(Tibshirani, 1996) to select the relevant wage determinants in a data-driven
way (see Hastie et al., 2009, for an introduction to LASSO). Specifically, we
implement the post-double-selection procedure (Belloni et al., 2013), double-
machine-learning (Chernozhukov et al., 2018b), and T-learner (Künzel et al.,
2019). Furthermore, we study the private and public sectors separately. The
two sectors are subject to different degrees of labor market regulation and
they attract different types of workers, both which influence the size of
observed gender pay gaps (see, e.g., Barón and Cobb-Clark, 2010).

With this study, we contribute to the existing literature in several important
ways. Firstly, we conduct the first comprehensive analysis of common support
in the context of the gender pay gap. Secondly, we are the first to study how
functional form restrictions regarding the inclusion of wage determinants
affect pay gap estimates both within and across different estimators. By
including machine learning techniques for data-driven model specification for
all estimators, we also employ methods that have never before been considered
in the context of the gender pay gap. Thirdly, we study a much more
comprehensive set of estimators than any previous study. Finally, we are the
first to consider all of these dimensions collectively and to vary them
systematically. In total, we estimate the unexplained gender pay gap for five
definitions of common support, six estimators, three model specifications, and
two sectors, resulting in a total of 180 different estimates. This allows us
to isolate the impact of each dimension and to understand their interactions.

The literature that investigates the sensitivity of the gender pay gap with
regard to common support and estimation methods is small. Ñopo (2008) studies
support violations and the resulting unexplained wage gaps with respect to
four exemplary wage determinants with data from Peru. Moreover, he compares
exact matching (EXM) with various linear specifications of BO that differ in
how flexibly three exemplary wage determinants are included, and whether or
not common support is enforced with respect to these variables. Black et al.
(2008) and Goraus et al. (2017) compare BO and EXM estimates of the gender pay
gap. Djurdjevic and Radyakin (2007), Frölich (2007), and Meara et al. (2020)
compare BO estimates of the gender pay gap with different propensity score
matching estimators. Relatedly, Barsky et al. (2002) and Graham et al. (2016)
study the sensitivity of black-white gaps with regard to different parametric
and semi-parametric estimators. Bach et al. (2018) and Briel and Töpfer (2020)
apply machine learning methods for model specification using the post-double-
selection procedure. However, none of these studies provides a comparable
large-scale systematic analysis of gender pay gap estimates with regard to our
considered methodological choices.

Our paper is related to other strands of the literature that investigate the
sensitivity of the gender pay gap with regard to other dimensions. One strand
of literature focuses on the assumptions required for the identification of
gender discrimination. For example, there are studies that investigate biases
due to gender-specific selection into employment (e.g., Chandrasekhar et al.,
2019; Chernozhukov et al., 2020; Maasoumi and Wang, 2019; Machado, 2017;
Neuman and Oaxaca, 2004; Olivetti and Petrongolo, 2008) and potential
endogeneity of the observed wage determinants (e.g., Kunze, 2008; Huber, 2015;
Huber and Solovyeva, 2020; Yamaguchi, 2015). We maintain the same identifying
assumptions for all our estimates. Thus, differences in our estimates result
from the empirical methods and not from the underlying assumptions. To be
precise, we measure the unexplained gender pay gap as the expected relative
wage difference of employed women in the sample with support, compared to
employed men with the same observed wage determinants. This parameter is
informative about equal pay for equal work taking individual choices as a
given. It does not necessarily measure gender discrimination in pay because
gender discrimination might influence eventual pay earlier in life through,
for instance, educational and career choices. Moreover, there may be
unobserved factors that help explain observed gender differences in pay. A
large strand of literature identifies wage determinants that contribute to
explaining the gender pay gap and analyzes how including these wage
determinants changes estimates of the unexplained gender pay gap. We discuss
this literature in Section 2.2 when we describe the variables we observe in
our data. In our study, we keep the wage determinants for which we control
constant across all estimates and only vary how flexibly we include them in
the estimations.

We find that all the methodological choices we consider significantly impact
the size of the estimated unexplained gender pay gap. The estimates decline by
up to 50% when stricter support is enforced, functional forms are relaxed, and
less restrictive estimators are used. The lack of comparable men for each
woman is a particularly serious issue. For 89% and 70% of employed women in
the private and public sector, respectively, there is no support with regard
to at least one observed wage determinant. For the public sector, we find that
lack of support directly explains a large part of the raw gender pay gap.
Moreover, with estimates that are 6-50% lower, enforcing support strongly
affects the estimates of the unexplained gaps for all estimators. Thus,
checking support in applications ex ante and deciding on how strictly it
should be enforced is crucial to applied work.

With estimates that are around 5-19% lower, the flexible inclusion of wage
determinants is very important for the parametric estimators LRM and BO as
well as the hybrid AIPW. In contrast, model specification has little impact on
the results from semi-parametric matching estimators that do not model the
wage equation. These estimators do not restrict heterogeneity in unexplained
wage gaps, which we find to be important as well. Compared to the most
flexible version of BO and with a reasonable choice of common support,
flexible EXPSM reduces the estimated unexplained wage gap by another 14% in
the private sector and 35% in the public sector. The results for semi-
parametric IPW are ambiguous, which might be related to propensity score
values close to one (e.g., Khan and Tamer, 2010; Busso et al., 2014).

Based on our findings, we recommend enforcing common support with respect to
the most important wage determinants ex ante. To estimate the unexplained pay
gap, we recommend combining exact matching on some important wage determinants
with radius matching on a flexibly specified propensity score. This minimizes
the risk of functional form misspecification and offers a reasonable balance
of comparability, precision of the estimate, and representativeness of the
study sample. For the private and public sector, respectively, implementing
these recommendations with our data explains 16% and 43% more of the raw wage
gap than standard BO estimates and results in estimated unexplained pay gaps
that are 23% and 50% lower. An important takeaway for policy makers is that
the commonly reported BO estimates of the gender pay gap can be misleading.

The paper proceeds as follows. The next section describes the data we use. In
Section 3, we explain our empirical model and estimation methods. Section 4
contains the results, starting with the analysis of common support, then
followed by the results for the mean unexplained gender pay gap as a function
of the three dimensions of methodological choices we consider. In Section 5,
we discuss the generalizability of our results and provide recommendations for
applied work. The last section concludes. Online appendices A-D contain
supplementary material.

##  2 Data

###  2.1 Sample selection

We use the 2016 wave of the Swiss Earnings Structure Survey (ESS). The ESS is
a bi-annual survey of approximately 37,000 private and public establishments
with individual data on more than 1.7 million employees, representing almost
one third of all employees in Switzerland. The survey covers salaried jobs in
the secondary and tertiary sectors in establishments with at least three
employees. Sampling is random within strata defined by establishment size,
industry, and geographic location. Participation in the survey is compulsory
for the establishments. The gross response rate is higher than 80%. All
results we report take into account the sampling weights provided by the ESS
that correct for both stratification, and non-response. Typically,
establishments report the required information directly from their
remuneration systems. Thus, the survey effectively includes administrative
data from establishments.

We restrict the analysis to the working population aged between 20 and 59
years (dropping 127,298 employees). We exclude employees for whom we observe
very little support between men and women ex ante. This excludes 70,052
employees with less than 20 percent part-time employment, 2,706 members of the
armed forces or agricultural and forestry occupations,333Based on 2-digit
ISCO-08 codes 01, 02, 03, 61, and 62. and 3,025 employees from the
agricultural, forestry, mining, and tobacco sectors.444Based on 2-digit NOGA
2008 codes 01, 02, 05, and 12. We analyze the gender pay gap separately for
the private and public sector. The public sector offers more regulated wages
and attracts a different selection of employees. For example, women are over-
represented in the public at 56%, but constitute only 43% of the private
sector. Moreover, the public sector is much more homogeneous in terms of
industries and occupations. The baseline sample contains 1,132,042 employees
in the private sector and 405,448 employees in the public sector, after
dropping an additional 54 employees in industries with very few observations
in the public sector.555Based on 2-digit NOGA 2008 codes 10, 31, 45, 47, 55,
and 58.

###  2.2 Data description

The main variable of interest is a standardized wage measure that is provided
by the Federal Statistical Office as part of the ESS. It measures the monthly
full-time-equivalent gross wage including extra payments. The latter comprise
add-ons for shift, Sunday, and night work, other non-standard working
conditions, and irregular payments such as bonuses and Christmas or holiday
salaries, but they exclude overtime premia. Wages are standardized to a 100%
full-time equivalent without overtime hours.

Table 1: Means and standardized differences of selected variables

| Private Sector |  | Public Sector
---|---|---|---
| Mean | Std. |  | Mean | Std.
| Women | Men | Diff. |  | Women | Men | Diff.
| (1) | (2) | (3) |  | (4) | (5) | (6)
Remuneration characteristics
Standardized monthly wage (in CHF) | 6,266 | 7,793 | 31.9 |  | 7,731 | 8,985 | 42.2
Irregular payments | .330 | .411 | 16.9 |  | .138 | .227 | 23.3
Demographic characteristics
Age |  |  |  |  |  |  |
20-29 years | .206 | .186 | 5.1 |  | .159 | .116 | 12.5
30-39 years | .270 | .285 | 3.4 |  | .266 | .250 | 3.7
40-49 years | .275 | .282 | 1.5 |  | .282 | .300 | 3.9
50-59 years | .249 | .248 | .4 |  | .293 | .335 | 8.9
Education |  |  |  |  |  |  |
Higher | .284 | .320 | 7.9 |  | .539 | .570 | 6.2
Vocational | .474 | .466 | 1.7 |  | .279 | .288 | 2.1
No vocational | .187 | .168 | 4.9 |  | .074 | .048 | 11.0
Employment characteristics
Tenure |  |  |  |  |  |  |
$<$ 2 years | .292 | .262 | 6.8 |  | .191 | .161 | 8.0
2-4 years | .250 | .233 | 4.0 |  | .218 | .197 | 5.2
5-7 years | .143 | .138 | 1.6 |  | .193 | .178 | 3.9
8-15 years | .201 | .209 | 2.0 |  | .234 | .262 | 6.6
16-45 years | .113 | .158 | 13.2 |  | .164 | .202 | 9.8
Management position |  |  |  |  |  |  |
Top | .031 | .073 | 18.8 |  | .011 | .040 | 18.7
Upper | .047 | .076 | 12.0 |  | .075 | .109 | 11.6
Middle | .079 | .101 | 7.7 |  | .067 | .109 | 14.8
Lower | .071 | .083 | 4.3 |  | .057 | .077 | 8.2
None | .771 | .667 | 23.2 |  | .736 | .614 | 26.2
Share of women in occupation |  |  |  |  |  |  |
$<$ 25% | .044 | .297 | 71.4 |  | .028 | .107 | 32.0
25-50% | .381 | .518 | 27.7 |  | .296 | .440 | 30.3
50-75% | .346 | .158 | 44.3 |  | .395 | .278 | 24.9
$>$ 75% | .229 | .027 | 63.2 |  | .214 | .131 | 22.2
Part-time and full-time work |  |  |  |  |  |  |
20-49% | .195 | .038 | 50.6 |  | .187 | .059 | 39.8
50-79% | .235 | .046 | 56.4 |  | .303 | .086 | 57.1
80-99% | .164 | .066 | 31.0 |  | .221 | .124 | 25.9
100% | .406 | .850 | 103.4 |  | .290 | .732 | 98.6
Establishment size |  |  |  |  |  |  |
$<$ 20 Employees | .255 | .215 | 9.5 |  | .015 | .017 | 1.4
20-49 Employees | .131 | .162 | 8.9 |  | .029 | .027 | 1.1
50-249 Employees | .237 | .258 | 5.0 |  | .123 | .091 | 10.3
250-999 Employees | .142 | .157 | 4.4 |  | .117 | .108 | 3.0
$>$ 999 Employees | .236 | .207 | 6.9 |  | .716 | .756 | 9.4
Observations | 491,007 | 641,035 |  |  | 227,617 | 177,831 |

Notes: Table is based on the baseline sample before imposing sample
restriction on support. The monthly regular wage is reported by employers. It
is standardized to 100% full-time equivalent wages without overtime hours.
Table A.1 in the Online Appendix A provides a detailed description of all
observed variables. Rosenbaum and Rubin (1983) classify absolute standardized
difference (std. diff.) of more than 20 as “large”.

The data contain a rich set of wage determinants that are considered important
in the gender pay gap literature such as age, education, occupation, industry,
establishment characteristics, wage bargaining, management position, marital
status, and part-time work.666See Bayard et al. (2003), Beaudry and Lewis
(2014), Bertrand et al. (2019), Bertrand et al. (2010); Bertrand and Hallock
(2001), Brenzel et al. (2014), Bruns (2019), Buffington et al. (2016), Card et
al. (2016), Fernández and Wong (2014), Gobillon et al. (2015), Goldin et al.
(2017), Heinze and Wolf (2010), Liu (2016), Manning and Petrongolo (2008),
Oberfichtner et al. (2020), Sin et al. (2020), and Winter-Ebmer and Zweimüller
(1997) among others. Table 1 documents the means and standardized differences
of selected variables by sector and gender (see Table A.1 in Online Appendix A
for a full list of all observed characteristics). In the private sector, the
average standardized monthly wage is 6,266 CHF for women and 7,793 CHF for
men.777The conversion rate of CHF to US dollars was approximately 1:1 in 2016.
In the public sector, the average wage is much higher at 7,731 CHF for women
and 8,985 CHF for men. In both sectors, the share of women receiving irregular
payments such as bonuses is notably smaller than that of men. This aligns with
a similar difference in the share of employees holding a management position.

The distributions of age and education are relatively similar across gender in
both sectors, although employed women tend to have slightly less education
than men. Women are somewhat less likely to have long tenure and significantly
less likely to work full-time than men. Labor market segregation by
occupations is very strong. We illustrate this by showing the employment
shares for occupations grouped by their shares of women. Unsurprisingly, more
women work in female-dominated occupations than men and vice versa. Female-
dominated occupations are more frequent in the public sector. Less than 3% of
men in the private sector, but more than 13% of men in the public sector work
in female-dominated occupations. Gender differences according to the
establishments size show no strong systematic pattern.

While the data are quite rich with regard to available wage determinants,
there are several important variables that we do not observe. One example is
actual work experience (e.g., Cook et al., 2020; Gayle and Golan, 2012).
Potential experience is captured by age and education, and we also observe
tenure. Hence, we capture some aspects of experience, but not all of it. Since
experience is positively related to wages, and women have on average less work
experience, we over-estimate potential violations of equal pay for equal work
in Switzerland in this respect. However, other information that the literature
emphasizes is missing as well. This includes, for example, competitiveness,
children, environment during childhood, gender norms, and non-cognitive
factors.888See, e.g., Flory et al. (2015) and Gneezy et al. (2009, 2003) for
competitiveness, Adda et al. (2017), Anderson et al. (2002), Angelov et al.
(2016), Bailey et al. (2012), Bütikofer et al. (2018), Ejrnæs and Kunze
(2013), Fitzenberger et al. (2016), Kleven et al. (2019a, b), Krapf et al.
(2020), Lundborg et al. (2017), and Waldfogel (1998) for children, Autor et
al. (2019), Bertrand and Pan (2013), and Brenøe and Lundberg (2018) for
environment during childhood, Bertrand et al. (2015) and Roth and Slotwinski
(2018) for gender norms, and Fortin (2008) for non-cognitive factors.
Moreover, we account for neither selection into employment based on unobserved
characteristics, nor for potential endogenity of control variables. Hence, our
estimates are informative about equal pay for equal work when individual
choices are taken as a given, subject to any omitted variable bias that may
result from unobserved factors.

##  3 Econometric model and estimators

###  3.1 Notation and parameter of interest

We denote the gender dummy by $G_{i}$, with $G_{i}=1$ for employed women and
$G_{i}=0$ for employed men. We use the logarithm of the standardized monthly
wage as the outcome variable, which we denote by $Y_{i}$. The raw gender pay
gap is

$\Delta=E[Y_{i}|G_{i}=1]-E[Y_{i}|G_{i}=0].$ (1)

The raw gender pay gap can be decomposed into an explained and unexplained
part. The vector $X_{i}$ contains observed demographic and labor market
characteristics of the employees as well as observed characteristics of their
employers. The predicted wage of employed men, would they have the same
observed characteristics as employed women is $E_{X|G=1}[\mu_{0}(x)]$, with
$\mu_{0}(x)=E[Y_{i}|G_{i}=0,X_{i}=x]$. Adding and subtracting
$E_{X|G=1}[\mu_{0}(x)]$ in (1) gives

$\Delta=\underbrace{E[Y_{i}|G_{i}=1]-E_{X|G=1}[\mu_{0}(x)]}_{\mbox{unexplained
$\delta$}}+\underbrace{E_{X|G=1}[\mu_{0}(x)]-E[Y_{i}|G_{i}=0]}_{\mbox{explained
$\eta$}}.$ (2)

The second difference on the right side of (2) is the part of the raw gender
pay gap that can be explained by gender differences in the observed wage
determinants $X_{i}$. The first difference on the right side of (2) is the
gender pay gap for employed women that cannot be explained by gender
differences in the observed wage determinants. It is the expected difference
in pay of employed women compared to observationally identical employed men,
which we denote by

$\delta=E[Y_{i}|G_{i}=1]-E_{X|G=1}[\mu_{0}(x)].$ (3)

This is the parameter of interest in the majority of studies on gender wage
inequality.

###  3.2 Common support

The unexplained pay gap (3) is only identified if, for all females, there
exist men that are observationally identical with respect to the wage
determinants. Now assume that there is lack of support for some men or women.
Let $S_{i}=1$ for individuals with support and $S_{i}=0$ for individuals
without support. Ñopo (2008) shows that

$\Delta=\underbrace{E[Y_{i}|G_{i}=1,S_{i}=1]-E[Y_{i}|G_{i}=0,S_{i}=1]}_{=\Delta_{S=1}}+\Delta^{s}_{G=1}-\Delta^{s}_{G=0}$

where $\Delta^{s}_{G=g}\equiv
Pr(S_{i}=0|G_{i}=g)[E[Y_{i}|G_{i}=g,S_{i}=0]-E[Y_{i}|G_{i}=g,S_{i}=1]]$
measures wage differences across individuals of gender $g$ (for
$g\in\\{0,1\\}$) in and out of support. The parameter $\Delta_{S=1}$ is the
raw wage gap within support, which we can decompose as

$\begin{array}[]{rl}\Delta_{S=1}=&\underbrace{E[Y_{i}|G_{i}=1,S_{i}=1]-E_{X|G=1,S=1}[\mu_{0}(x)|S_{i}=1]}_{\mbox{$\delta_{S=1}$}}\\\
&+\underbrace{E_{X|G=1,S=1}[\mu_{0}(x)|S_{i}=1]-E[Y_{i}|G_{i}=0,S_{i}=1]}_{\mbox{$\eta_{S=1}$}}.\end{array}$
(4)

The first right-hand term, $\delta_{S=1}$, is the unexplained gender pay gap
for employed women with support. The second right-hand term, $\eta_{S=1}$, is
the part of the raw gender pay gap with support that can be explained by
gender differences in the wage determinants.

We use the following procedure to analyze the influence of common support
violations on gender pay gap estimates. First, we sequentially increase the
number of wage determinants for which we impose common support and then study
the share of females without support. Imposing common support based on a large
number of wage determinants increases ex-ante comparability of women and men.
Second, we estimate the elements of (4) for each sequential step using an
exact matching estimator (see Section 3.3.4 for more details). Third, based on
the sequential analysis, we pick five exemplary definitions of support and
estimate the unexplained gender pay gap on support, $\delta_{S=1}$.
Specifically, we restrict the sample to the employees on support and implement
different estimators for the unexplained gender pay gap, which we explain in
the next section.

###  3.3 Estimators

There exists a very large set of possible parametric and semi-parametric
estimators, doubly-robust mixtures between the two types of estimators, and
non-parametric estimators. We focus on common estimators that are easy to
implement with the objective of showing the main trade-offs in estimator
choice. We apply each estimator with five different support conditions where
we restrict the sample to the observations satisfying the respective support
definition.

####  3.3.1 Dummy regression (LRM)

A simple estimation approach for the unexplained gender pay gap employs a
linear regression model (LRM) with a dummy for women,

$Y_{i}=\alpha+G_{i}{\delta}_{LRM}+X_{i}{\beta}+{\varepsilon}_{i},$ (5)

where $\alpha$ is a constant, the vector ${\beta}$ describes the association
between the wage determinants $X_{i}$ and the wages, and ${\varepsilon}_{i}$
is an error term. The parameter ${\delta}_{LRM}$ can be interpreted as the
unexplained gender pay gap.

The LRM imposes the restriction that the coefficients of the wage determinants
as well as the unexplained gender pay gap are homogeneous across all
individuals. This includes the assumption that the unexplained gender pay gap
is equal for women and men. This modelling assumption is unnecessarily
restrictive, because it can be relaxed by including interaction terms between
gender and the wage determinants, which we discuss in the next section in the
context of the BO decomposition.

####  3.3.2 Blinder-Oaxaca (BO) decomposition

The BO decomposition is a two-step estimator that allows for heterogeneous
unexplained gender pay gaps (Blinder, 1973; Oaxaca, 1973). In the first step,
we estimate in the subsample of men the linear wage model

$Y_{i}=\alpha_{0}+X_{i}{\beta}_{0}+{u}_{i},$ (6)

where $\alpha_{0}$ is an intercept and ${u}_{i}$ is an error term. The
coefficients ${\beta}_{0}$ describe the association between the wage
determinants $X_{i}$ and the wages of men. In the second step, we use the
estimated coefficients from this regression to predict the counterfactual male
wage $\hat{\mu}_{0}(X_{i})\equiv\hat{\alpha}_{0}+X_{i}\hat{\beta}_{0}$ for
each woman in the sample and estimate the mean unexplained gender pay gap for
women using

$\hat{\delta}_{BO}=\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}(Y_{i}-\hat{\mu}_{0}(X_{i}))$
(7)

with $N_{1}=\sum_{i=1}^{N}G_{i}$ (hats indicate estimated parameters). Note
that this estimation procedure is numerically identical to using a linear
prediction for female wages, $Y_{i}=\alpha_{1}+X_{i}{\beta}_{1}+{v}_{i}$, in
the sample of women and calculating the unexplained part of the pay gap as

$\hat{\delta}_{BO}=(\hat{\alpha}_{1}-\hat{\alpha}_{0})+\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}X_{i}(\hat{\beta}_{1}-\hat{\beta}_{0}),$

since
$\displaystyle\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}Y_{i}=\hat{\alpha}_{1}+\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}X_{i}\hat{\beta}_{1}$.

In contrast to the LRM, this version of the BO decomposition allows for gender
differences in the impact of characteristics $X_{i}$ on wages as well as for
heterogeneity in the unexplained gender pay gap that is driven by these
differences.

The BO model corresponds to the LRM augmented with interaction terms between
gender and all observable wage determinants:

$Y_{i}=\alpha_{0}+G_{i}\underbrace{(\alpha_{1}-\alpha_{0})}_{=\alpha_{BO}}+X_{i}{\beta_{0}}+G_{i}X_{i}\underbrace{(\beta_{1}-\beta_{0})}_{=\beta_{BO}}+{\epsilon}_{i}.$
(8)

Using this fully interacted LRM, we could estimate the BO unexplained wage gap
by

$\hat{\delta}_{BO}=\hat{\alpha}_{BO}+\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}X_{i}\hat{\beta}_{BO}.$

This estimation procedure is numerically identical to (7) when we control for
the same characteristics $X_{i}$ as in (6).

####  3.3.3 Inverse probability weighting (IPW) estimators

Inverse probability weighting (IPW) estimators are also two-step estimators
(Hirano et al., 2003; Horvitz and Thompson, 1952). In the first step, we
estimate the conditional probability of being a women, the so-called
propensity score, based on the model

${p}(X_{i})={Pr}(G_{i}=1|X_{i})=F(X_{i}{\gamma}),$

where $F(\cdot)$ is a binary link function (e.g., the logistic distribution).
In the second step, we predict $\hat{p}(X_{i})$ for all observations and
estimate the re-weighted sample average

$\hat{\delta}_{IPW}=\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}Y_{i}-\sum_{i=1}^{N}\hat{W}_{i}^{0}Y_{i},$

with the IPW weights

$\hat{W}_{i}^{0}=\displaystyle\frac{(1-G_{i})\hat{p}(X_{i})}{1-\hat{p}(X_{i})}\Big{/}\displaystyle\sum_{i=1}^{N}\frac{(1-G_{i})\hat{p}(X_{i})}{1-\hat{p}(X_{i})}$
(9)

The denominator in (9) guarantees that the IPW weights add up to one in finite
samples (see, e.g., Busso et al., 2014). In contrast to LRM and BO, IPW does
not impose any specific functional form on the relationship between $X_{i}$
and the wage, and it does not restrict heterogeneity in unexplained gender pay
gaps. A potential disadvantage of IPW is its instability when the conditional
probability of being a women is close to one (e.g., Khan and Tamer, 2010),
which may results in high variance of $\hat{\delta}_{IPW}$. We impose trimming
rules to avoid propensity score values that are too extreme (see the
discussion in, e.g., Lechner and Strittmatter, 2019). In particular, we omit
males with large weights $\hat{p}(X_{i})/(1-\hat{p}(X_{i}))$ above the 99.5%
quantile.999We apply the normalization described in (9) after the trimming. We
document the number of trimmed observations in Table B.1 of Online Appendix B.

Kline (2011) argues that BO estimators are equivalent to propensity score re-
weighting estimators, but in contrast to IPW they model the propensity score
linearly.101010The usual practice of IPW estimators is to use a Logit or
Probit estimator for the propensity score. As a result, the implicit weights
of BO can become negative, which is not possible for the IPW estimator.
However, in the limit to a fully saturated BO model, even the linear
specification of the propensity score would be well behaved and negative
weights would be unlikely.

Augmented inverse probability weighting (AIPW) dates back to Robins et al.
(1994) and has received significant attention since Chernozhukov et al. (2017)
proposed this estimation procedure in the context of machine learning. AIPW is
a doubly-robust mixture between the BO and IPW approach,

$\hat{\delta}_{AIPW}=\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}(Y_{i}-\hat{\mu}_{0}(X_{i}))-\sum_{i=1}^{N}\hat{W}_{i}^{0}(Y_{i}-\hat{\mu}_{0}(X_{i})),$
(10)

with $\hat{\mu}_{0}(X_{i})$ as in BO. The first right-hand term in (10) is
equivalent to the BO in (7). The second right-hand term in (10) has an
expected value of zero, but makes a finite sample adjustment by re-weighting
the observable bias of $\hat{\mu}_{0}(X_{i})$ in the sample of men with the
IPW weights.

AIPW is more robust to misspecification than BO or IPW. In particular,
$\hat{\delta}_{AIPW}$ is consistent even when either $\hat{\mu}_{0}(x)$ or
$\hat{p}(x)$ is misspecified. Moreover, the theoretical properties of AIPW are
well established for generic estimators of the nuisance parameters
$\hat{\mu}_{0}(x)$ and $\hat{p}(x)$. In particular, $\sqrt[4]{N}$-consistency
of $\hat{\mu}_{0}(x)$ and $\hat{p}(x)$ is sufficient to achieve
$\sqrt{N}$-consistency of $\hat{\delta}_{AIPW}$ (in combination with the
cross-fitting procedure described below). This permits the application of
flexible machine learning methods to estimate $\hat{\mu}_{0}(x)$ and
$\hat{p}(x)$, which often have a convergence rate below $\sqrt{N}$. AIPW
combined with machine learning is often called double-machine-learning (see
Knaus, 2020, for a review).

####  3.3.4 Matching estimators

Exact matching (EXM) as used, for example by Ñopo (2008), is a fully non-
parametric estimation approach. We stratify the sample into $K\ll N$ mutually
exclusive groups $W_{i}\in\\{1,...,K\\}$, which are defined based on the
characteristics $X_{i}$. The EXM estimator is

$\hat{\delta}_{EXM}=\frac{1}{N_{1}}\sum_{i=1}^{N}G_{i}\left(Y_{i}-\sum_{j=1}^{N}\frac{\displaystyle
1\\{W_{i}=W_{j}\\}(1-G_{j})Y_{j}}{\displaystyle\sum_{j=1}^{N}1\\{W_{i}=W_{j}\\}(1-G_{j})}\right).$

EXM is more flexible than LRM, BO, IPW and AIPW. However, it suffers from the
curse of dimensionality. When we consider many strata $K$, the risk of lacking
support (i.e., finding no men matching specific strata of women) increases.
Unlike the previous estimators, EXM cannot extrapolate into regions without
support. Therefore, the estimation breaks down in the presence of support
violations. Furthermore, even when there is support, there may be strata with
very few men, which might lead to high variance of $\hat{\delta}_{EXM}$.
Despite these potential disadvantages, our large data set with more than one
million observations remains suitable for EXM. However, we have to define
relatively coarse groups for the two continuous variables of age and tenure to
avoid empty strata. For the same reason, we have to combine some industries
and occupations with only few observations into more loosely similar groups.
We implement EXM as in Ñopo (2008); i.e., we exactly match the stratified
variables that define common support. This implies that the versions with less
restrictive support definitions also match on fewer variables, thus removing
fewer differences in observed wage determinants. In the private [public]
sector, we have 140 [142] strata for the least restrictive support definition
and 844,100 [208,860] strata for the most restrictive one.

We consider two additional semi-parametric matching estimators to account for
potential concerns regarding the curse of dimensionality and unmatched wage
determinants within strata. First, we use propensity score radius matching
(PSM) (see e.g. Frölich, 2007; Lechner and Wunsch, 2009; Lechner et al.,
2011). The propensity score $p(X_{i})$ is the same as for IPW. PSM is a one-
dimensional matching approach (as opposed to the multi-dimensional matching of
EXM). To each woman, we match the men who have a propensity score value within
a certain absolute difference (radius) of the woman’s propensity score value.
We define the radius as the 99% quantile of the distribution of the closest
absolute distances for all women, then omit women without a match within the
radius. Table B.2 in Online Appendix B documents the number of females we use
for the different matching estimators. We lose only a relatively few women who
lack matching men within the defined radius.

Second, we consider mixtures between exact and propensity score radius
matching (EXPSM). Here, we exactly match on the wage determinants that define
support, as in EXM, and apply propensity score radius matching within each
stratum using all wage determinants, as in PSM. In contrast to EXM, the
propensity score corrects semi-parametrically for remaining within-strata
observable differences between women and men. In contrast to PSM, we enforce
exact comparability of women and men with respect to some wage
determinants.111111It should be noted that we implement PSM after matching
exactly on the three variables that define the least restrictive Support 1,
because this improves matching without lack of support. As a result, PSM and
EXPSM are identical for Support 1 but not for the other support definitions.

###  3.4 Flexibility of inclusion of wage determinants

To estimate the so-called nuisance parameters, i.e., the wage equation
$\hat{\mu}_{0}(x)$ and the propensity score $\hat{p}(x)$, we consider three
different models. We call them the baseline, full, and machine learning (ML)
models. The baseline and full models include the same wage determinants, but
we vary the flexibility in terms of non-linear (e.g., polynomials, categorical
dummies) and interaction terms of the wage determinants. The ML model can
select the relevant wage determinants and the model flexibility in a data-
driven way, using the full model specification as input. In practice, it is
often unclear which non-linear and interaction terms are relevant. Using a too
parsimonious model could bias ${\delta}$, due to model misspecification.
Allowing for too much flexibility could lead to imprecise estimates of
${\delta}$ with high variance, especially in smaller samples. Accordingly, we
face a bias-variance trade-off.

####  3.4.1 Manual model specification

In the baseline model, we control for all observed variables in a standard
way. Specifically, we include age (linear and squared), tenure (linear and
squared), vocational education (9 categories), citizen status (6 categories),
marital status (3 categories), occupation (39 [20] categories in the private
[public] sector), industry sector (36 [12] categories in the private [public]
sector), management position (5 categories [and a missing dummy for public
sector]), region (7 categories), establishment size (5 categories), and
employment share as a fraction of full time (4 categories). Furthermore, we
control for the dummy variables of temporary employment, employment contract
with hourly wages, collective wage agreement, overtime hours payment, bonus
payments (e.g., from profit sharing), supplementary wages (e.g., for shift
work), and extra salary (e.g. Christmas and holiday salaries). Overall, the
baseline model includes 117 control variables in the specification for the
private sector and 74 control variables in the specification for the public
sector.

In the full model, we add to the baseline model all non-linear and interaction
terms that we think could potentially be relevant. The non-linear terms are
the polynomials of age and tenure up to order seven, as well as four age and
five tenure categories. Furthermore, we include interaction terms between
control variables to allow for heterogeneous returns to important wage
determinants.121212We partially coarsen the categorical variables to improve
support of the interaction terms. Specifically, we interact occupation and
industry groups with the categorical variables for age, tenure, education,
foreigner status, management position, and establishment size. We also include
interaction terms between establishment size categories and the categorical
variables for age, tenure, vocational education, migration, and management
position. Finally, we interact vocational education and management position
categories with age and tenure categories. Overall, the full model includes
615 control variables in the specification for the private sector and 396
control variables in the specification for the public sector.

####  3.4.2 Data-driven model specification

In contrast to manual model specification, machine learning estimators have
the potential to balance the bias-variance trade-off in a data-driven way. We
use the LASSO to specify our ML models. The LASSO is a penalized regression
method (see Hastie et al., 2009, for a detailed description of the LASSO). It
can balance the bias-variance trade-off by shrinking some coefficients towards
zero. Eventually, some covariates are excluded from the model when the
coefficients are exactly zero. Accordingly, the LASSO is a model selection
device. To obtain the predicted wages of men $\hat{\mu}_{0}(x)$ in the
baseline and full models for BO, we estimate in the sample of men the OLS
model

$(\hat{\alpha}_{0},\hat{\beta}_{0})=\arg\min_{a,b}\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-a-X_{i}b)^{2},$

and predict the conditional male wages by
$\hat{\mu}_{0}(x)=\hat{\alpha}_{0}+x\hat{\beta}_{0}$. In contrast, the LASSO
objective function is

$(\hat{\alpha}_{0},\hat{\beta}_{0})=\arg\min_{a,b}\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-a-X_{i}b)^{2}+\lambda\|b\|_{1},$
(11)

with the penalty term $\lambda\|b\|_{1}=\lambda\sum_{p=1}^{P}|b|$. The tuning
parameter $\lambda\geq 0$ specifies the degree of penalization of the absolute
coefficient size of $\hat{\beta}_{0}$. If $\lambda=0$, the ML model is
equivalent to the full model. If $\lambda>0$, some coefficients
$\hat{\beta}_{0}$ shrink towards zero and eventually approach zero exactly.
Shrinking coefficients to zero is equivalent to excluding the corresponding
control variable in $X_{i}$ from the wage equation. In the extreme case, when
$\lambda\rightarrow\infty$, all coefficients in $\hat{\beta}_{0}$ are exactly
zero and only the intercept $\hat{\alpha}_{0}$ is non-zero, because it is not
penalized. We determine the tuning parameter $\lambda$ with a 5-fold cross-
validation procedure (Chetverikov et al., 2017).131313Alternatively, the
choice of the tuning parameter $\lambda$ can be based on information criteria
(Zou et al., 2007) or data-driven procedures (Belloni et al., 2012). The
cross-validation procedure selects the $\lambda$ that minimises the mean-
squared-error (MSE). To smooth the ML model, we apply the one-standard-error
rule to the minimum $\lambda$ (see, e.g., Hastie et al., 2009).

We use similar models to estimate the propensity score $\hat{p}(x)$. However,
instead of using OLS, we use the Logit estimator for the baseline and full
model. Likewise, we use Logit-LASSO instead of the standard LASSO approach for
the ML model (see Hastie et al., 2016, for an introduction to Logit-
LASSO).141414The only exception is the post-double-selection procedure, for
which we use a linear model to estimate the propensity score, as proposed in
Belloni et al. (2013). In Figures C.1-C.8 in Online Appendix C, we report the
density of the estimated propensity scores by gender.

Post-double-selection (PDS) procedure. The LASSO version of the LRM
corresponds to the PDS procedure of Belloni et al. (2013, 2014). The
conventional LASSO has the purpose of predicting $Y_{i}$, but it is not
designed to obtain an unbiased estimate of the structural parameter
${\delta}_{LRM}$. To illustrate this, the application of LASSO to (5) without
penalizing the gender dummy would most likely lead to the omission of those
wage determinants that are highly correlated with gender, such that an omitted
variable bias is inevitable. The PDS procedure enables LASSO to select the
relevant wage determinants in a data-driven way, without the threat of
omitting important variables. The PDS procedure has three steps. First, the
LASSO selects the relevant control variables in the linear wage equation
$Y_{i}=\alpha_{Y}+X_{i}{\beta_{Y}}+{\nu}_{i}$, which does not contain the the
gender dummy $G_{i}$.151515In contrast to (11), we use the sample of men and
women for the wage equation of the PDS procedure. In practice, the LASSO can
set some coefficients in the vector ${\beta_{Y}}$ to zero, which is equivalent
to excluding the corresponding control variables in $X_{i}$ from the wage
equation. Second, the LASSO selects the relevant dimensions of $X_{i}$ in the
linear gender model $G_{i}=\alpha_{G}+X_{i}{\beta_{G}}+{\eta}_{i}$. As before,
the LASSO estimator can set some coefficients in the vector $\beta_{G}$ to
zero, such that only those control variables with a strongly gender
correlation remain in the model. We denote the union of all wage determinants
with non-zero coefficients in either $\beta_{Y}$ or $\beta_{G}$ by
$\tilde{X}_{i}$. Finally, we estimate the linear OLS model (called ’post-
LASSO’)

$Y_{i}=\alpha_{PDS}+G_{i}{\delta}_{PDS}+\tilde{X}_{i}{\beta}_{PDS}+{\tilde{\varepsilon}}_{i}.$

Belloni et al. (2013) show that the estimator of ${\delta}_{PDS}$ is
consistent and asymptotically normal. The PDS procedure allows for the
inclusion of high-dimensional characteristics $X_{i}$ (i.e., more
characteristics than observations), but requires $\tilde{X}_{i}$ to be sparse.
Some approximation error in the selection of the $\tilde{X}_{i}$ variables is
permitted. The PDS procedure is doubly robust to misspecification of either
the earnings equation or the gender model.

For the same reason as for the LRM, applying LASSO in a naive way to the
interacted BO in (8) could lead to biased estimates of $\alpha_{BO}$ and
$\beta_{BO}$, which could also bias $\hat{\delta}_{BO}$. However, we can use
the LASSO to estimate $\hat{\mu}_{0}(x)$, since $\hat{\mu}_{0}(x)$ is a pure
prediction model for $Y_{i}$, and then use $\hat{\mu}_{0}(x)$ as a plug-in
estimator in (7). Accordingly, (7) provides a way to combine the Blinder-
Oaxaca decomposition with standard machine learning methods. This procedure is
called T-learner in the denomination of Künzel et al. (2019). A possible
disadvantage of this approach is that we do not obtain estimates of the
parameters $\alpha_{BO}$ and $\beta_{BO}$, since they are not required to
estimate $\delta_{BO}$. Alternatively, Bach et al. (2018) propose a procedure
that is based on post-double-selection, but is more flexible and allows us to
obtain estimates of selected parameters in $\alpha_{BO}$ and $\beta_{BO}$.

Cross-fitting. To avoid over-fitting and obtain $\sqrt{N}$ convergence for
AIPW, it is necessary to estimate the ML models of $\hat{\mu}_{0}(x)$ and
$\hat{p}(x)$ in a different sample than $\hat{\delta}_{AIPW}$ (see
Chernozhukov et al., 2018b). We achieve this by using cross-fitting. We
partition the sample into two parts of equal size. We estimate
$\hat{\mu}_{0}(x)$ and $\hat{p}(x)$ with the first partition, extrapolate
their fitted values to the second partition and use these values to estimate
$\hat{\delta}_{AIPW}$. Thereafter, we switch the first and second partition
and repeat the procedure, such that the entire data set is used efficiently.
We report the average $\hat{\delta}_{AIPW}$ across both partitions.

Selected variables. We allow the LASSO to select among all control variables
of the full model. Table C.1 in Online Appendix C documents the number of
selected control variables for all ML models we consider. The final
specifications vary by support and estimation procedure. For the wage
equation, the ML model considers between 371 and 513 control variables in the
different specifications for the private sector and between 215 and 306
control variables for the public sector. For the propensity score model, the
ML model considers between 141 and 428 control variables for the private
sector and between 126 and 230 control variables for the public sector.

Performance. In Tables C.2 and C.3 in Online Appendix C, we report the out-of-
sample prediction power of the different nuisance parameter models. To obtain
the out-of-sample prediction power, we use a cross-fitting procedure.161616We
partition the data in two equally sized samples, using one partition as a
training sample and the other as a test sample. Thereafter, we switch the two
partitions and report the average prediction power across the two samples. The
prediction power of the baseline, full, and ML models does not differ
strongly. Accordingly, the baseline model already explains a significant
amount of the variation in the data. The prediction power of the full model is
systematically better than the prediction power of the baseline model, but the
additional gain is moderate. The ML model cannot systematically outperform the
full model. The prediction power of the full and ML models is fairly similar,
even though the ML model controls for many fewer variables than the full
model. This suggests that the full model is too flexible. However, because of
our large data set, the prediction power of the full model does not
deteriorate compared to the ML model. The degrees of freedom loss of the full
compared to the ML model appear to be of minor importance. However, we expect
that the ML model would outperform the full model in smaller samples.

##  4 Results

###  4.1 Analysis of common support

Our first step is to analyze common support using the approach of Ñopo (2008).
We sequentially increase the number of wage determinants for which we impose
common support. To determine the order in which we add variables, we run a
simple wage regression in the sample of men, i.e. we estimate $\mu_{0}(x)$ in
order to determine the importance of the different variables for explaining
men’s wages. We measure importance as the change in adjusted $R^{2}$ when we
omit one block of variables, e.g. all occupation dummies, but keep all other
wage determinants. We do this separately for the private and public sectors
and sort all variable blocks in decreasing order according to the average
$R^{2}$ changes across both sectors. Thus, we start with the most important
variable block for explaining male wages and finish with the least important.
We report the resulting order of variables and the sector-specific changes in
the adjusted $R^{2}$ in Tables B.3 and B.4 of Online Appendix B for the
private and public sector, respectively.

Our choice of ordering based on importance for explaining male wages is
motivated by the curse of dimensionality. The larger the number of variables
with respect to which we enforce common support, the more likely it is that
there will be empty cells. As a result, we expect the share of the original
sample for which there is support to decrease as we add more and more
variables. This implies that there is some difficulty in achieving a
reasonable balance of comparability, sample size and representativeness of the
remaining sample. Therefore, we want to impose support with respect to
variables that significantly influence the counterfactual male wage, but not
necessarily with respect to variables of minor importance.

In Figure 1, we report how the share of women with support falls as we
increase the number of wage determinants with respect to which we enforce
support. There is full support with respect to the three most important
determinants – management position, education and age – in the private sector,
and only 2 women are off support in the public sector. The loss of
observations due to lack of support is relatively small when we add industry
and occupation, which are both highly relevant for predicting wages. It
becomes larger but remains below 20% if we add establishment size and a dummy
for irregular payments. Thereafter, however, support deteriorates quickly.
Once we enforce support with respect to the 10 variables that are most
important for explaining male wages, 55% of women in the private sector and
32% of women in the public sector have no comparable men with respect to these
variables. The loss of women is stronger in the private sector than in the
public sector because the latter is more homogeneous in terms of occupations
and industries. Subsequently adding the less important wage determinants
further reduces support, resulting in lack of support for 89% of women in the
private sector and 70% of women in the public sector when we enforce full
support with respect to all observed wage determinants.

Figure 1: Analysis of common support by sector

(a) Private sector

(b) Public sector

Notes: The indicated wage determinants are added sequentially from left to
right. The raw gap is the difference between the average log wages of women
and men in the sample with full support with respect to the indicated
variables. The unexplained gap is based on the counterfactual male wage
obtained from exact matching on all wage determinants from the left to the
respective determinant within the sample that has full support with respect to
these variables. The numbers in parentheses indicate the five support
definitions we use for the estimations.

Of course, different orders of variables lead to different evolutions of
support. Table B.5 in Online Appendix B shows how support changes when we add
these variables in three alternative orders. The first one uses sector-
specific $R^{2}$ changes rather than the average over both sectors to order
variables. For the main analysis, we want to keep the results comparable
across sectors, which is why we want the same order for both sectors. The
order resulting from sector-specific $R^{2}$ changes differs across sectors
for some variables, but the overall differences are moderate. Accordingly, the
evolution of support when adding variables is also similar. The second
alternative is a random order. Here support remains higher for a larger number
of added variables, mainly because critical variables such as industry and
tenure are, by chance, added relatively late. This would change with a
different random order in which the are added earlier. The last alternative
uses an increasing order according to the sector-average $R^{2}$ changes as
the other extreme to a decreasing order. Here, support breaks down rather
late. The reasons for this are twofold. First, the ordering prioritizes wage
determinants in which women and men do not greatly differ. Second, many of
these wage determinants are dummies while more important variables (such as
education, occupation and industry) have many different categories. This
mechanically splits the sample in more cells than a dummy variable such that
cell size is reduced.

Besides studying the evolution of support, we also estimate the elements of
(4). The raw gender pay gaps on support, $\Delta_{S=1}$ , can be obtained from
the average gender difference in wages in the sample with support. To estimate
the unexplained gender pay gap for the women with support, $\delta_{S=1}$, we
follow Ñopo (2008) and calculate the pay gap that remains after matching
exactly on all variables for which we enforce support. Thus, when adding
support-relevant wage determinants, we also increase the number of variables
on which we exactly match. The explained gender pay gap on support,
$\eta_{S=1}$, can be calculated as the difference between the raw and
unexplained gender pay gap within support. We report the main insights from
this analysis in Figure 1 and we document the full set of results in Tables
B.3 and B.4 in Online Appendix B.

The raw gender pay gap is relatively insensitive to the support definition in
the private sector. It decreases only slightly, from 18.6% under the weakest
support definition to 17.3% under the strongest one. In contrast, the EXM
estimates of the unexplained and explained gender pay gaps are very sensitive
to the support definition. The unexplained gender pay gap on support falls
from 13.4% under the weakest support definition to 4.2% under the strongest
one. Correspondingly, the explained gender pay gap increases from 5.1% to
13.1% as we exactly match on more wage determinants. Interestingly, the
results differ substantially for the public sector. Here, a substantial part
of the raw gender pay gap is due to women without support. The raw gender pay
gap on support decreases from 13.9% under the weakest support definition to
only 4.5% under the strongest one. The unexplained gender pay gap on support
decreases quickly from 9.7% to to 3.5% when we impose support with respect to,
and exactly match on, the eight most important wage determinants. Thereafter,
imposing stronger support and matching on more variables leaves the
unexplained gender pay gap almost unchanged although the raw gender pay gap
decreases. This finding is in line with strongly regulated wage setting of the
Swiss public sector, which leaves little room for gender pay differences.

For the analysis of the unexplained gender pay gap with different estimation
methods and model flexibility, we pick five definitions of support that
illustrate the trade-offs between comparability, sample size and
representativeness. We indicate the five versions with the respective number
in parentheses in Figure 1. Here it is important to note that for a given
choice of variables it does not matter in which order support is enforced. All
possible orders will lead to exactly the same sample. Support 1 does not
change the sample except for excluding two women in the public sector. Thus,
it effectively resembles the case without support enforcement. Support 2
enforces support with respect to classical and very important wage
determinants. Support 3 covers all quantitatively important wage determinants.
Support 4 adds tenure as the only proxy for actual experience, but it does not
significantly explain wages. Support 5 is the most extreme case, which
enforces full support with respect to all variables. Table B.6 in Online
Appendix B shows how imposing stricter support changes the average
characteristics of the remaining women with support. The changes are largest
with respect to occupation, management position, establishment size, and part-
time employment status. This is important for the interpretation of our
results. In the presence of heterogeneous pay gaps, we cannot expect estimates
to be the same when the sample changes due to stricter enforcement of support.

###  4.2 Analysis of the unexplained gender pay gap

Figure 2: Mean unexplained gender pay gap in the private sector

(a) Baseline model: estimates

(b) Baseline model: difference to benchmark

(c) Full model: estimates

(d) Full model: difference to benchmark

(e) ML model: estimates

(f) ML model: difference to benchmark

Notes: We report all estimates and their standard errors in Table D.1 in
Online Appendix D. Capped lines indicate the 95% confidence intervals of the
estimated unexplained gender pay gap. The black horizontal line in the left
panels shows the BO estimate of Support 1 as a benchmark with the baseline
specification and its 95% confidence band in black dashed lines. The right
panels show the difference from this benchmark in percent, i.e., the
difference from the BO estimate using Support 1 with baseline model
specification. PDS is the abbreviation for post-double-selection procedure.

Figure 3: Mean unexplained gender pay gap in the public sector

(a) Baseline model: estimates

(b) Baseline model: difference to benchmark

(c) Full model: estimates

(d) Full model: difference to benchmark

(e) ML model: estimates

(f) ML model: difference to benchmark

Notes: We report all estimates and their standard errors in Table D.2 in
Online Appendix D. Capped lines indicate the 95% confidence intervals of the
estimated unexplained gender pay gap. The black horizontal line in the left
panels shows the BO estimate of Support 1 as a benchmark with the baseline
specification and its 95% confidence band in black dashed lines. The right
panels show the difference from this benchmark in percent, i.e., the
difference from the BO estimate using Support 1 with baseline model
specification. PDS is the abbreviation for post-double-selection procedure.

In panels (a), (c) and (e) of Figures 2 and 3, we show the estimates of the
average unexplained gender pay gap for the private and public sector,
respectively. The bars show the unexplained gender pay gap estimates of the
different methodological choices we consider and the capped lines show the 95%
confidence interval for each estimate.171717We do not include exact matching
(EXM) as EXM does not control for all wage determinants in Supports (1)-(4),
but EXM has already been discussed in the context of Figure 1. Tables D.1 and
D.2 in Online Appendix D report all estimates including EXM with their
standard error for the private and public sector, respectively. As a
benchmark, the black horizontal line shows the BO estimate in Support 1 (full
sample except for two women in the public sector) with the baseline
specification together with its 95% confidence interval (black dashed lines).
Furthermore, panels (b), (d) and (f) of both figures show the percentage
differences of each estimate from this benchmark.

We find that all of the methodological choices we consider matter greatly.
Compared to the BO benchmark, using a less flexible estimator like the LRM
increases the estimated unexplained gender pay gap by up to 29%. In contrast,
increasing flexibility by including wage determinants more flexibly or by
using a more flexible estimator together with enforcing common support reduces
the estimated pay gap by up to 50%. These differences are substantial. In the
following section, we investigate the role of each methodological choices
separately: enforcement of common support, flexibility in the inclusion of
wage determinants, and estimator choice.

####  4.2.1 Role of stricter support enforcement

Figure 4 shows the differences in the estimated unexplained gender pay gaps
for each support definition relative to Support 1. We report separate results
for each estimator, but pool the results for the different model
specifications (baseline, full, ML) for a better overview.

Figure 4: Difference in estimated unexplained gender pay gap relative to
Support 1

Notes: Difference in percent to the estimate for Support 1 using the
respective estimator. The results for the model specifications (baseline,
full, ML) are pooled.

The estimated pay gaps shrink substantially and, with few unsystematic
exceptions, monotonically with stricter support enforcement. For the BO
estimates, enforcing support with respect to all quantitatively important wage
determinants (Support 3) reduces the estimated pay gap by around 6% in both
sectors, while enforcing full support with respect to all variables (Support
5) reduces estimated gaps by around 26%. With 13-35% for Support 3 and 5-50%
for Support 5, the differences are considerably larger for the semi-parametric
estimators IPW, PSM and EXPSM. Overall, enforcing common support has a very
strong impact on the size of the estimated unexplained gender pay gaps.

There are two possible explanations for this finding. First, stricter support
enforcement makes women and men more comparable. Hence, differences in
observed wage determinants are likely to explain a larger share of the raw
gender pay gap. This is particularly true for the public sector, where a large
part of the raw gender pay gap can be explained by lack of support directly
(see Figure 1). Second, heterogeneity in unexplained gender pay gaps can
explain the differences across support samples that differ increasingly in
their composition (see Table B.6 in Online Appendix B). For example, stricter
support enforcement increasingly removes women in lower-paying jobs. As a
result, the unexplained gender pay gap could either increase or decrease,
depending on whether the unexplained gender pay gap is larger for women in
higher- or lower-paying jobs. However, the direction cannot be inferred from
data, because we cannot estimate the unexplained gender pay gap for the
omitted women who lack comparable men.

The results are consistent with the findings of Ñopo (2008), who shows that
11% of Peru’s gender pay gap estimates can be explained by lack of common
support with regard to age, education, marital status, and migration
condition. We document that support violations can be much more severe when
accounting for a richer set of wage determinants. Furthermore, we show that
all estimators are affected by support violations, including the parametric
estimators that rely on extrapolation.

####  4.2.2 Role of model flexibility

Controlling for wage determinants more flexibly is also important, especially
for the parametric estimators. Figure 5 shows the percentage differences in
the estimated unexplained gender pay gaps between the baseline model of the
respective estimator and the full and ML model specifications. We report
separate results for each estimator, but pool the results of the different
support definitions for a better overview.

Figure 5: Difference in estimated unexplained gender pay gap relative to
baseline model

Notes: Difference in percent of the baseline model estimate with the
respective estimator. The results for the different support definitions are
pooled.

We find that the most flexible full model specification reduces the BO
estimate of the pay gap by 6% in the private sector and 19% in the public
sector. The reductions are only slightly smaller for the ML specification,
which selects variables from the full model in a data-driven way to increase
efficiency.

Model flexibility matters most for the parametric estimators LRM and BO; as
well as for AIPW, which can be viewed as an extension of BO with semi-
parametric small-sample bias adjustment. All three estimators model the wage
equation to estimate the unexplained gender pay gap. In contrast, the semi-
parametric estimators IPW, PSM and EXPSM are much less sensitive to how wage
determinants are included. All three estimators only use the propensity score
as an input factor and do not impose any functional form on the wage equation.
For those estimators, the estimated pay gaps differ by less than 8% from the
baseline model. The most flexible one, EXPSM, exhibits only very small
differences across model specifications: only 0.4% in the private sector and
3% in the public. For all estimators, the difference between the full and ML
models are not very large. A noticeable exception is the IPW estimator, where
the unexplained gender pay gap of the ML specification exceeds even the
baseline specification. Knaus et al. (2020) document low finite sample
performance of IPW in combination with ML, which may explain this result.

Kline (2011) provides a theoretical justification for the sensitivity of the
BO estimator with regard to model flexibility. He argues that BO is more
vulnerable to model misspecification than IPW, because the implicit weighting
scheme of BO allows for negative weights (which is not permitted for IPW).
When the BO model is specified more flexibly, though, negative implicit
weights become less likely.

We draw three conclusions from these results. First, controlling for wage
determinants in a flexible way is important. Second, the more flexibly they
are included, the better; misspecification of functional forms is less likely
and the loss of degrees of freedom is not costly in our very large data set
(but this might be different in smaller data sets). Third, flexible inclusion
of wage determinants is more important for the estimators that incorporate the
wage equation than for estimators that only incorporate the propensity score.
With our very large database, we also find that applying machine learning
methods for variable selection from a very rich set of non-linear and
interaction terms has only a small impact on the estimated pay gaps. This may
be different, though, in smaller samples where efficiency is more of a
concern.

####  4.2.3 Role of estimator choice

The last dimension we vary is estimator choice. Figure 6 shows the differences
in the estimated unexplained gender pay gaps between BO and the respective
estimator for each support. As the flexibility of including wage determinants
affects the parametric and semi-parametric estimators differently, we focus on
the results from the full model with maximum flexibility for all estimators.
This implies that the differences that we observe across estimators mainly
result from differences in the way the estimators restrict possible
heterogeneity in unexplained gender pay gaps. The LRM imposes homogeneous
gender pay gaps. BO allows for heterogeneous gender pay gaps that are driven
by gender differences in the returns to the included wage determinants. In
contrast, the semi-parametric estimators IPW, PSM and EXPSM do not restrict
heterogeneity in the pay gaps at all. AIPW is a mixture between the parametric
BO and a small sample bias adjustment based on the the semi-parametric IPW.
With our very large samples, small-sample bias should be negligible and BO and
AIPW should yield very similar results.

Figure 6: Difference in estimated unexplained gender pay gap relative to BO in
respective support

Notes: Difference in percent of the BO estimate in the respective support
sample. All estimates are for the full model, which is the most flexible
specification.

We find that for each support, LRM estimates of pay gaps are 8-18% higher than
the BO estimate. The differences between the estimated BO and AIPW pay gaps
are small as expected with our large samples. This may be different in smaller
samples, though. For the matching estimators PSM and EXPSM we estimate pay
gaps that are systematically and substantially smaller than the corresponding
BO estimates on the same support by up to 20% in the private and up to 34% in
the public sector. This is in line with substantial heterogeneity in estimated
pay gaps that is widely acknowledged in the literature (e.g. Bach et al.,
2018; Chernozhukov et al., 2018a; Goldin, 2014).

However, we find no systematic pattern for IPW. In the private sector, we find
estimates smaller than with BO and similar to PSM and EXPSM for supports 1-4,
but much larger estimates for support 5. In the public sector, we obtain
estimates that are 5-24% larger than the BO estimates for supports 1-4 and
much smaller for full support 5. The results for IPW show that this estimator
is very sensitive to the population studied.

We conclude that it is important not to restrict pay gap heterogeneity by
using a semi-parametric estimator. Moreover, among the semi-parametric
estimators, PSM and EXPSM are much more robust and precise. The differences
between PSM and EXPSM are small in most cases. With a sufficient numbers of
observations, EXPSM is preferable because it ensures comparability of women
and men in the best way among all considered estimators. Moreover, EXPSM is
the least sensitive to the way we include the observed wage determinants
because exact matching on the variables that define support fully removes all
differences in these variables without imposing any restrictions.

These findings are consistent with the results of Frölich (2007), who
documents that the PSM estimates of the unexplained gender pay gap in the UK
are up to 29% smaller than the BO estimates. Black et al. (2008) also find
strong differences between BO and EXM estimates of the unexplained gender pay
gap in the United States. In particular, the estimates decline by 18% for
white women, 92% for Hispanic women, and 83% for Asian women. However, for
black women they find no strong difference between the BO and EXM estimates.
Likewise, Ñopo (2008) and Goraus et al. (2017) find no strong differences in
the BO and EXM estimates of the unexplained gender pay gap in Peru and Poland,
respectively, but they only account for a very small set of wage determinants.
In Figure 1, we show that the selection of control variables is crucial for
EXM.

##  5 Discussion

###  5.1 Relevance of the Swiss case

A legitimate question is whether the results we obtain with the Swiss data are
relevant for other applications. In terms of data, ours are very similar to
those of the European Union Structure of Earnings Survey (SES). The SES
provides harmonized data on earnings in EU member states, candidate countries,
and EFTA countries. It includes all of the wage determinants that have been
used to define Supports 1-4 (but not 5), which covers all quantitatively
important wage determinants. Moreover, like the Swiss data, actual work
experience is not included. Studies for the US typically use survey data
collected from individuals such as the Current Population Survey (CPS), the US
Census, or the American Community Survey (ACS). Such surveys contain much
richer information on individuals but potentially suffer from measurement
error in self-reported wages. However, the key wage determinants we observe
are included as well, and most studies use a similar set of variables. In
terms of sample size, all of the data sets are also very large, containing
several hundred thousand observations.

With respect to labor market institutions, Switzerland is a particularly
interesting case. On the one hand, it has generous social insurance systems
like many other European countries. On the other hand, it has a very flexible
labor market that is much less regulated than that of other European
countries, which makes it more comparable to countries like the US. Thus,
Switzerland shares important features of both typically European labor markets
and flexible US-type labor markets. The differences we find between the
private and the public sectors in Switzerland are also likely to be relevant
for other countries. Public sectors in most countries share the features that
explain the differences in the Swiss case. They are are typically more
homogeneous in terms of covered occupations with strong concentration in
certain service sectors, they typically attract a higher share of women than
the private sector, they exhibit higher shares of high-skilled workers, and
they are typically more regulated, including their wage setting, than private
sectors (see Brindusa et al., 2012).

In summary, the data and labor market institutions in Switzerland are in many
ways comparable to those in other countries. Therefore, we expect that our
qualitative results extend to other settings as well. Of course, the
magnitudes of gender pay gaps differ greatly across countries (see e.g.
Weichselbaumer and Winter-Ebmer, 2005; Van der Velde et al., 2015). Hence, we
also expect the quantitative impact of methodological choices to differ.
However, given that most countries share key features of female employment and
gender segregation in the labor market, we are confident that the
methodological choices we study are relevant beyond the Swiss context.

###  5.2 Implications for applied research

Our results suggest that deciding how to enforce common support is the first
important choice in applied research about the gender pay gap. Researchers
face a trade-off between stricter support to improve ex-ante comparability,
and sample size. The latter affects both the precision of the estimate and the
representativeness of the study sample. We recommend starting with an analysis
of common support using the approach of Ñopo (2008), adding variables in
decreasing order of their impact on counterfactual male wages. Based on this
analysis, there are two possible criteria for choosing the set of variables
for enforcing support. One option is to use all quantitatively important wage
determinants. In our application, this would result in support definition 3
with 39% of employed women lost in the private sector and 20% lost in the
public sector, which is quite substantial. Alternatively, one could base the
decision on the importance of wage determinants for explaining the raw gender
pay gap rather than the counterfactual male wage. In this case, the researcher
could choose the point at which the unexplained gender pay gap obtained from
exact matching stabilizes. In our data, this is the case somewhere between
Supports 2 and 3 (see Figure 1). As sample size drops considerably after
enforcement of Support 2 plus irregular payments, this would be a natural
choice of the set of variables for enforcing support. This would exclude 19%
of employed women in the private sector and 10% in the public sector with the
composition of the study sample being quite similar to that of the full
sample.

The next important decision is which estimator to use. The results indicate
that restricting how observed wage determinants and gender may affect wages
can have a large impact on the estimated unexplained gender pay gap. When
using BO, including wage determinants in a flexible way is crucial. However,
with a sufficiently large sample, a flexible matching estimator is even
better, as matching does not restrict gender pay gap heterogeneity in any way.
The results suggest that combining exact matching on few very important wage
determinants that also define support with radius matching on a flexibly
specified propensity score works particularly well. It minimizes the risk of
functional form misspecification and, with support defined as described in the
last paragraph, offers a good balance between ensuring comparability,
precision and representativeness of the study sample.

Implementing these recommendations with our data would affect the estimated
unexplained gender pay gap in the following way. Consider Support 3 as the
more conservative of the two alternatives discussed above. In the private
sector, enforcing Support 3 reduces the raw gender pay gap of 18.6% by only
1%. Standard BO with the baseline specification for including wage
determinants that ignores support explains 58% of the raw gap in the full
sample and results in an estimate of the unexplained pay gap of 7.7%.
Enforcing Support 3 reduces this estimate by 5%, using the most flexible
instead of the baseline specification for BO by another 5%, and using semi-
parametric EXPSM with the same flexible specification instead by another 13%.
In total, the methodological choices decline the unexplained gender pay gap
estimates by 23%. Flexible EXPSM yields an estimated unexplained gap of 6% and
explains 68% of the raw wage.

The differences are even larger for the public sector. Enforcing Support 3
already reduces the raw gender pay gap of 13.9% by 14%. Standard BO without
support enforcement explains 54% of the raw gap in the full sample and yields
an estimate of the unexplained pay gap of 6.4%. Enforcing Support 3 reduces
this estimate by 4%. Using the most flexible instead of the baseline
specification for BO has a strong impact, as it reduces the estimate by
another 19%. Using semi-parametric EXPSM with the same flexible specification
instead leads to another even more substantial reduction of 26%. In total, the
methodological choices decline the estimates of the unexplained gender pay gap
by 50%. Flexible EXPSM yields an estimated unexplained gap of only 3.2% and
explains 77% of the raw wage. This once again illustrates the importance of
these methodological choices.

##  6 Conclusion

We study the sensitivity of estimates of the unexplained gender pay gap to
three types of methodological choices: enforcement of comparability of men and
women ex ante, flexibility in the inclusion of wage determinants, and choice
of estimator. We find that all of these choices matter greatly. Implementing
the choices we recommended based on our results using data for the Swiss
private sector, explains 16% more of the raw wage gap than standard BO
estimates and results in estimated unexplained pay gaps that are 23% lower.
For the public sector, the preferred set of choices explains 42% more of the
raw wage gap than standard BO estimates and results in estimated unexplained
pay gaps that are 50% lower.

An important takeaway from our study is that the commonly reported Blinder-
Oaxaca estimates of the unexplained gender pay gap can be misleading for
several reasons. There is a high risk that they compare women to non-
comparable men to an extent that is quantitatively important. Moreover, they
hide gender inequalities that result from this lack of comparability. For
females with lack of support, we know nothing about possible inequalities in
pay. Moreover, lack of support shows the extent of labor market segregation as
an additional source of gender inequality. This is ignored if researchers do
not check for common support. More generally, the restrictions standard BO
imposes bear a high risk of yielding biased estimates of the unexplained
gender pay gap. In particular, they underestimate the full extent of
heterogeneity in pay gaps that is important for targeting affirmative action
campaigns. Future work should study this heterogeneity in more detail.

As a final remark, we would like to emphasize that our findings provide
guidance on how to estimate unexplained gender pay gaps with a given set of
observed wage determinants. The results from such estimations are informative
about equal pay for equal work taking individual choices as a given, subject
to any omitted variable bias that may result from unobserved factors. Actual
experience, having children, and personality traits are examples of such
unobserved factors in the data we have used. Uncovering the extent and sources
of possible gender discrimination in the labor market requires much more than
accounting for unobserved wage determinants, though, such as dealing with
selection into employment and endogenous control variables.

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