Causal Eﬀects of Time-constant Variables withCounterfactual Mediation Modeling:
Evidence from the Gender Wage Gap in Japan∗
Shoki OKUBO(Osaka University)
Abstract
The decomposition of inequality among time-constant variables (e.g., gender) into explained and unexplained components has been a major research agenda in social inequality
studies. Explained components of the gender wage gap in the decomposition method correspond to the indirect wage eﬀect of gender that operates through these mediators. Furthermore,
unexplained components correspond to the direct wage eﬀect of gender on wage that operates
through unobserved mediators (e.g., discrimination).This study links together conventional
decomposition methods, such as Blinder-Oaxaca (BO) and DiNardo-Fortin-Lemieux (DFL),
based on propensity-score weighting and counterfactual mediation modeling together by introducing the sequential ignorability assumption discussed in Imai et al. (2010). The reason
for linking these two methods is that conventional decomposition methods typically control
for post-birth variables that lie on the causal pathway from gender or race (which are basicallyrandomly assigned at birth) to wage but neglect the potential endogeneity that may arise fromthis approach. Moreover, we never directly test the assumptions of conventional decomposition
methods and mediation modeling. Therefore, based on the more recent literature on counterfactual mediation modeling, this study presents more reasonable identifying assumptions andthe sensitivity of the results to diﬀerent sets of assumptions.In addition, this study aims to
re-focus on time-constant variables in statistical causal analysis of social inequality studies.
The analysis focuses on the decomposition of the gender wage gap in Japan. Empirical results
indicate that explained components with four mediators (education, occupation, employment
status, and post) account for nearly 2%-20% of the gender wage gap in hourly wages.

∗ This research was supported by JSPS KAKENHI Grant Number JP25000001. Updated version of this paper isavailable via https://osf.io/sx47e/?view_only=3caefba079ec49d798609ab2c8bd0b5b―139―

Keywords: Gender Wage Gap, Counterfactual Modeling, Sequential Ignorability Assumption,
Sensitivity Analysis

1.

Introduction

This paper links conventional decomposition methods such as DiNardo-FortinLemieux (DFL) based on propensity-score weighting and counterfactual mediation modeling together by introducing sequential ignorability assumption
discussed in Imai et al. (2010). The reason for linking these two methods isthat conventional decomposition methods typically control for post-birth variables that lie on the causal pathway from gender or race which are basicallyrandomly assigned at birth to wage but neglect the potential endogeneity that
may arise from this approach. Moreover, we never directly test the assumptions that lie in conventional decomposition methods and mediation modeling.
Based on the newer literature on counterfactual mediation modeling, this paper
therefore shows more attractive identifying assumptions and the sensitivity of
the results to diﬀerent sets of assumptions.
The decomposition of inequality among time-constant variables (e.g., gender) into explained and unexplained components has been a major research
agenda in social inequality studies. The classical linear decomposition forthe diﬀerence in mean outcomes for two groups A and B assumes following
equations (Blinder 1973 and Oaxaca 1973).
YA = β′AX A + ϵ(1)

YB = β′B X B + ϵ(2)This decomposition assumes that the outcome variable Y is linearly related
to the covariates, X, and that the error term is conditionally independent of X.
Subtracting equation (1) from (2), we getȲB − ȲA = [ β′B (X̄ B − X̄ A ) ] + [ (β′B − β′A ) X̄ A ]explained component(3)

unexplained component

Equation (3) shows that explained component is the group diﬀerence between
A and B in the mean of Y , which would be eliminated if B had A’s covariate
distribution. Unexplained component is the group diﬀerence between A and―140―

B in the mean of Y , which would be eliminated if A had B’s covariate eﬀects.
Many scholars has proposed the weaker-assumption based nonparametric decomposition methods (DiNardo et al. 1996; Frölich 2007). Let ϕ denotes
an unspeciﬁed function and θ A and θ B are covariate eﬀects parameters. We
express
YA = ϕ(X A, θ A ) + ϵ(4)

YB = ϕ(X B, θ B ) + ϵ(5)

Subtracting equation (5) from (4), we getȲB − ȲA = [ϕ̄(X B, θ B ) − ϕ̄(X A, θ B )] + [ϕ̄(X A, θ B ) − ϕ̄(X A, θ A )]���� ����explained component(6)

unexplained component

We obtain ϕ̄(X B, θ B ) and ϕ̄(X A, θ A ) simply by calculating the sample
means. ϕ̄(X A, θ B ) is obtainable from following integration (Fortin et al. 2010).
ϕ̄(X A, θ B ) ≡
=
=

∫

∫V

ϕ(V , θ B ) f (x|T = 0)dx

∫X
x

E(YB |x) f (x|T = 0)dx

ω(x)E(YB |x) f (x|T = 1)dx

= Eω (YB )(7)

Suppose the treatment T = 1 is Group A and T = 0 is Group B. Eω is theweighted mean with weights ω(x)f (x|T = 0)f (x|T = 1)
p(T = 0|x) f (x)/p(T = 0)=p(T = 1|x) f (x)/p(T = 1)
p(T = 1)p(T = 0|x)=p(T = 0)p(T = 1|x)

ω(x) ≡(8)

Scholars have also dealt with decompositions at quantiles in the outcome distribution (Firpo et al. 2009). As Fortin et al. (2011) pointed out, however, most―141―

papers in the decomposition literature jump directly to the estimation issues
without addressing the identiﬁcation strategy. In terms of Rubin Causal Model(RCM) using potential outcome framework, Yamaguchi (2015) also indicatethat the conventional decomposition methods fails to handle endogenous covariates. Figure 1 shows why the conventional methods do not handle potential
endogeneity.

Diagram (a)Diagram (b)Figure1: Causal diagrams with endogenous covariates

In Figure 1, X are endogenous covariates because there is an unobserved
confounder U that aﬀects both X and Y . In diagram (a), covariates X are
confounders of treatment variable T and outcome variable Y , and T ⊥ U |X
holds. This conditional independence is the important key to estimate causal
eﬀects of treatment variables. On the other hand, in diagram (b), covariates
X are mediator variables, and even though T is assumed to be independentof U, T ⊥ U |X does not hold by controlling for covariates X, which arecommon causal descendants of T and U (Pearl 2009; Morgan and Winship2015; Yamaguchi 2015). As Yamaguchi (2015) point out, diagram (a) is thesituation that the RCM assumes for causal analysis with cross-sectional data,
and the condition T ⊥ U |X is equivalent to the ignorability assumption. Since
the ignorability assumption doesn’t hold in diagram (b), we need to reformulate
decomposition analysis for handling endogenous covariates. This paper tackles
this problem by introducing counterfactual mediation modeling based on RCM.

2.

Identification Strategy

2.1

Set upWe ﬁrst refer to the relationship between decomposition analysis and mediation
analysis. Let’s take an example of the gender wage gap. Because the sex is―142―basically randomly assigned at birth and one’s characteristics including humancapital and social capital will be determined after birth, explained componentsof the gender wage gap in the decomposition method correspond to the indirect
wage eﬀect of gender that operates through these mediators. Meanwhile,
unexplained components correspond to the direct wage eﬀect of gender on wagethat operates through unobserved mediators (e.g., discrimination). Hereafter,
we reformulate the conventional decomposition methods from the point of viewof counterfactual mediation modeling.
The popular mediation analysis by Baron and Kenny (1986) shows indirect
eﬀects (IE) as follows.
Mi = α2 + β2Ti + ξ2 Xi + ϵ i2

(9)Yi = α3 + β3Ti + γ Mi + ξ3 Xi + ϵ i3(10)I E = β2 γ

(11)However, recent literatures have pointed out that IE β2 γ in equation (11)would be average causal mediation eﬀects (ACME) only if sequential ignorability assumption holds. From the point of view of causal inference, theidentiﬁcation of a causal mechanism requires the speciﬁcation of an intermediate variable or a mediator that lies on the causal pathway between the treatment
and outcome variables (Imai et al. 2011). In RCM, average causal eﬀects(ACE) can be expressed as following.
E[Yi (Ti = 1) − Yi (Ti = 0)](12)where treatment Ti = 1 or the control Ti = 0 condition. Many of causal
inference literatures mainly focus on whether one treatment variable causally
aﬀects outcome variable and fail to explain how such a causal eﬀect arises.
Counterfactual mediation analysis instead aims to quantify the eﬀect of atreatment variable that operates through a causal mechanism. Let Mi (t) denotethe potential value of a mediator of interest for unit i with the treatment condition
Ti = t. In the same way, let Yi ( t, m) denote the potential outcome if thetreatment and mediating variables equal t and m. In reality, we observe only
one of the potential outcomes, and thus the observed outcome Yi is Yi (Ti, Mi (Ti )).
Observed outcome is dependent upon both the treatment and mediator status.
―143―

2.2

Quantities of interest

Our goal is to clarify how much of the treatment variable is transmitted by the
mediator. Following Robins and Greenland (1992), Pearl (2001) and Imai etal. (2010) let ACME deﬁne as
δi (t) ≡ E[Yi (t, Mi (1)) − Yi (t, Mi (0))](13)

for each unit i. ACME is the diﬀerence between the potential outcome thatwould result under treatment status t, and the potential outcome that would
occur if the treatment status is the same and yet the mediator takes a value thatwould result under the other treatment status Mi (1) and Mi (0). Similarly, wedeﬁne the average causal direct eﬀect (ACDE) asζi (t) ≡ E[Yi (1, Mi (t)) − Yi (0, Mi (t))](14)

for each unit i and each treatment status t = 0, 1. Since we observe only one
mediator conditions in reality, we require additional assumption for identifying
equation (13) and (14).
2.3

Sequential ignorability

Sequential ignorability (SI) assumption discussed in Imai et al. (2010) identiﬁes
ACME and ACDE in equation (13) and (14). SI consist of two parts{Yi (t ′, m), Mi′ (t)} ⊥ Ti |Xi = x

Yi (t ′, m) ⊥ Mi (t)|Ti = t, Xi = x

(15)
(16)

Equation (15) means the treatment variable Ti is assumed to be ignorable
given the pre-treatment covariates Xi . This assumption, held in randomized
experiments, is known as unconfoundedness or no omitted variable bias. Equation (16) assumes the observed mediator is ignorable given the actual treatment
status Ti and pretreatment covariates Xi . This assumption is not standard ignorability assumptions because randomizing both the treatment and mediator
does not identify the ACME (Imai et al. 2011).

―144―

2.4

Sensitivity Analysis

SI can never be tested directly so sensitivity analysis is used. The idea of
violation of SI leads to a correlation between εi2 and εi3 . Let this correlation
denote ρ. Because ACME can be expressed as a function of ρ, the sensitivity
analysis investigates how robust the estimation results are to the violation of
the SI. Mathematical details are discussed in Imai et al. (2010).
2.5

Estimation procedure

We link the method in Yamaguchi (2015) to counterfactual mediation modeling.
Let W denote potential counfounder of the treatment variable T, mediatorvariable M and covariates X. We follow Imai et al. (2010) in counterfactual
mediation analysis from procedure 2 to 6.
1. Obtain a consistent estimate of p(T = 1|W ) and conduct a diagnosis
for an appropriate construction of propensity scores to create a weighted
sample for which statistical independence between T and W hold.
2. Fit models for the observed outcome and mediator variables with propensity score weighting1 created in procedure 1.
3. Simulate model parameters from their sampling distribution.
4. Simulate the potential values of the mediator.
5. Simulate the PO given the simulated values of the mediator.
6. Compute quantities of interest (ACME, ACDE and TE).

3.

Empirical Application

The next step is to conduct simulation studies for investigating the ﬁnite-sample
performance of the estimators, however, we skip the simulation studies in thispaper2 . In this section, we apply these methods using Social Stratiﬁcation andMobility (SSM) 2015 survey data.

1 Huber (2015) clariﬁes using propensity score weighting induces less biased estimates.
2 See Okubo (2018) that simulates and compares ﬁve estimators:

Blinder-Oaxaca decomposition, DiNardoFortin-Lemieux decomposition, Yamaguchi (2015) decomposition, ACME (in other words, average natural
mediation eﬀects) and average controlled mediation eﬀects.

―145―

3.1

Data

SSM survey have been implemented every 10 years since 1955. The SSM
2015 survey is its seventh time, especially focusing on verification of thechange of population structure represented by rapid declining birthrate andaging population in terms of social stratification. Respondents of this survey ismale and female with Japanese nationality of 20 to 79 years old living in Japan
at the end of December 2014. Sampling design followed the basic policy since
the SSM survey in 1995 and stratified two-stage cluster sample design was used
for sampling respondents. The survey was conducted from January to July in2015. The sample size and response rates were 7817 and 50.1% for each.
3.2

Variables

The outcome variable Y is the log hourly wage of the 20-64 years old survey
participants. Treatment variables T is the sex which denotes 1 for female
and 0 for male. Binary mediators M are education (non-colledge graduate
for 1), occupation (non-manager for 1), employment status (non-regular workfor 1), post (non-post for 1). We also consider controlling for pre-treatment
variables W that reflect family background and could potentially confound thetreatment variable and the mediators. Specifically, W contains mother’s andfather’s levels of education which denote 1 for colledge graduate. Similarly tothe standard literature, covariates X characterizing the explained component
include age, employment duration in labor market, employment duration in
non-regular work ever. We restrict the sample to observed hourly wage andexclude self-employed and family workers. Table 1 provides summary statistics
on these variables, namely the mean and standard deviation values for male andfemale.
3.3

Results

We analysed two estimation model: (1) without propensity score weighting
using the potential confounder W , (2) with propensity score weighting using
the potential confounder W . The calculation of statistical uncertainty estimatesis based on the quasi-Bayesian Monte Carlo approximation (King et al. 2000)
and we run 1000 times simulations for the quasi-Bayesian approximation of―146―

Table1: Summary statistics by treatment status T.
Female (T = 1)

Male (T = 0)

Variables

Mean

S.D.

Mean

S. D.

Diﬀerence

Log Hourly Wage（1,000 YEN）

7.022

0.628

7.582

0.600

0.560

Non-colledge graduate

0.775

0.418

0.558

0.498

0.217

Non-manager

0.996

0.065

0.939

0.240

0.057

Non-regular work

0.521

0.500

0.144

0.351

0.377

Non-post

0.866

0.341

0.548

0.498

0.318

Age

43.06911.11044.081

11.3481.012

Regular employment duration

11.0109.941

20.59912.2309.589

Non-regular employment duration

7.197

7.887

1.746

4.576

5.451

Father’s education

12.0223.512

11.799

3.547

0.223

Mother’s education

11.6262.795

11.514

2.960

0.112

Y
M

X

W

n

1421

1408

NOTE: On the right hand side column Diﬀerence calculates the absolute mean diﬀerence between
female’s and male’s mean.

parameter uncertainty. Results in table 2 show that estimates vary across twomodels especially in Panel A. Model 1 without PSW overestimates ACME.
This is because mediator Non-colledge graduate is confounded by W , parent’s
education. Though we are able to use only two variables for W in SSM
2015 data, obtaining more W , pre-treatment variable, would induce diﬀerent
estimates. Focusing on the percentages of TE mediated, equivalent to the
explained component in conventional decomposition analysis, mediator Nonpost best explains the gender wage gap followed by Non-colledge. Employment
status in non-regular work mediate TE around 12 percent and only 2 percent
for non-manager.
The estimates are identiﬁed if the SI holds. We now move on the sensitivity
analysis for investigating how robust these estimation results are to the violationof the SI because the assumption SI can never be directly tested. Figure 3
shows the results of the sensitivity analysis for each counterfactual mediation
―147―

.2
ACME
0
-.2
-.4

-.4

-.2

ACME
0

.2

.4

ACME(ρ).4

ACME(ρ)-1-.50
Sensitivity parameter: ρ

.5

1

-1

(i) non-colledge-.50
Sensitivity parameter: ρ

.5

1

.5

1

(ii) non-manager

.2
ACME
0
-.2
-.4

-.4

-.2

ACME
0

.2

.4

ACME(ρ).4

ACME(ρ)-1-.50
Sensitivity parameter: ρ.51(iii) non-regular work

-1

-.5

0
Sensitivity parameter: ρ(iv) non-post

Figure2: Sensitivity analysis for ACME(ρ): The dashed line represents the estimated ACMEof the sensitivity parameter ρ. The gray area represents the 95% confidence interval.

estimation. The analysis indicates that the results about the direction of the
ACME under SI would be maintained unless ρ is less than -0.150 for (i) noncolledge, -0.098 for (ii) non-manager, -0.207 for (iii) non-regular work and-0.220 for (iv) non-post. This means that the estimation results are plausible
except for (ii) non-manager3 , given even large departures from the ignorability
assumption of the mediator M .

4.

Conclusion

This paper aimed to clarify the identification strategy using counterfactual
mediation modeling instead of conventional decompositions methods such as
Blinder (1973), Oaxaca (1973) and DiNardo et al. (1996). Literatures have
3 As shown in Table 1, the percentage of manager is less than 7% both for female and male. This small percentsindicate there is a limitation for controlling potential confounder of T and M , which induces the small absolutevalue of ρ.

―148―

Table2: Counterfactual Mediation Modeling Estimates(1) Without PSW(2) With PSW

Panel AMediator: Non-colledge
ACME-.0409[-.0514, -.0302]-.0435[-.0561, -.0318]

ADME-.1978[-.2424, -.1547]-.1987[-.2414, -.1527]

TE-.2387[-.2820, -.1948]-.2422[-.2873, -.1967]% of TE mediated.1717[.1736, .2466].1791[.1513, .2210]

ACME-.0048[-.0097, -.0012]-.0044[-.0082, .-0015]

ADME-.1978[-.2424, -.1547]-.1976[-.2419, -.1549]

TE-.2026[-.2455, -.1592]-.2020[-.2446, -.1589]% of TE mediated.0238[.0196, .0303].0217[.0179, .0276]

ACME-.0265[-.0369, -.0175]-.0285[-.0400, .-0184]

ADME-.1978[-.2424, -.1547]-.1976[-.2419, -.1549]

TE-.2243[-.2668, -.1806]-.2262[-.2682, -.1825]% of TE mediated.1186[.0995, .1470].1266[.1065, .1564]

ACME-.0503[-.0626, -.0386]-.0494[-.0615, -.0375]

ADME-.1978[-.2424, -.1547]-.1976[-.2419, -.1549]

TE-.2481[-.2900, -.2041]-.2471[-.2877, -.2041]% of TE mediated.2039[.1736, .2466].2008[.1718, .2422]

Panel BMediator: Non-manager

Panel CMediator: Non-regular work

Panel DMediator: Non-post

NOTE: Sample size n = 2829. Parenthesis [ ] shows 95% confidence interval. TE = ACME +ADME.

clarified conventional decompositions do not control for confounders of thetime-constant treatment variable and/or mediator variables. This is the standard case in estimating the parameters of time-constant variables, as gender or
ethnicity are determined at or prior to birth and therefore precede mediators likeeducation or profession (Yamaguchi 2015; Huber 2015). We proposed the linking of counterfactual mediation modeling that accounts for causal mechanisms,―149―

discussed in Imai et al. (2010), and propensity score weighting decompositionin Yamaguchi (2015).
We conducted the sensitivity analysis for inspecting how robust the estimation
results are to the violation of the SI. The sensitivity analysis implies that theconclusion about the direction of the ACME in gender wage gap under SI would
be maintained. It is the fate of causal inference to examine the uncertainty
and robustness because the assumption in causal inference mostly can not bedirectly verified. Considering the situation sociologists tries to answer thequestion asking how robust empirical results are to sensible changes in model
specification (Western 1996; Young 2009; Young and Holsteen 2017), it isessential procedure to conduct some diagnosis for investigating the uncertainty
and robustness.
This paper also aimed to bring time-constant variables back to the centerof statistical causal analysis in social inequality studies. We agree with thestatement in Yamaguchi (2015: 426).
... the use of panel survey data for causal analysis seems to havediminished the importance of gender or race in statistical causal analyses
because of the lack of a methodological framework to handle such timeconstant exogenous variables as the treatment variable in causal analysis.
I believe that the discussion and the method presented in this paper willlead to a reconsideration of such trends, and it will be complementary to
the experimental audit method, because gender and racial inequality area major substantive research topic in sociology.
Though the counterfactual mediation analysis without randomizing the treatment variables and mediators might not sharply identifies the quantities ofinterest, comparing to the natural experiment such as sharp regression discontinuity, it’s much better than not doing it.

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