Direct and Indirect Effects based on Changes-in-Changes***We have benefited
from comments by Giuseppe Germinario as well as conference/seminar
participants at the Universities of Neuchâtel, Melbourne, Sydney, Hamburg, and
Lisbon, the Luxembourg Institute of Socio-Economic Research, the 2019 meeting
of the Austro-Swiss Region of the International Biometric Society in Lausanne,
and the 2019 meeting of the International Association for Applied Econometrics
in Nicosia. Addresses for correspondence: Martin Huber, Chair of Applied
Econometrics - Evaluation of Public Policies, University of Fribourg, Bd. de
Pérolles 90, 1700 Fribourg, Switzerland<EMAIL_ADDRESS>Mark Schelker,
Chair of Public Economics, University of Fribourg, Bd. de Pérolles 90, 1700
Fribourg, Switzerland<EMAIL_ADDRESS>Anthony Strittmatter, Swiss
Institute for Empirical Economic Research (SEW), University of St.Gallen,
Varnbüelstr. 14, 9000 St.Gallen, Switzerland<EMAIL_ADDRESS>www.anthonystrittmatter.com.

Martin Huber†, Mark Schelker†, Anthony Strittmatter‡

†University of Fribourg, Dept. of Economics
‡University of St. Gallen, Swiss Institute for Empirical Economic Research

Abstract: We propose a novel approach for causal mediation analysis based on
changes-in-changes assumptions restricting unobserved heterogeneity over time.
This allows disentangling the causal effect of a binary treatment on a
continuous outcome into an indirect effect operating through a binary
intermediate variable (called mediator) and a direct effect running via other
causal mechanisms. We identify average and quantile direct and indirect
effects for various subgroups under the condition that the outcome is
monotonic in the unobserved heterogeneity and that the distribution of the
latter does not change over time conditional on the treatment and the
mediator. We also provide a simulation study and an empirical application to
the Jobs II programme.

Keywords: Direct effects, indirect effects, mediation analysis, changes-in-
changes, causal mechanisms, treatment effects.
JEL classification: C21.

##  1 Introduction

Causal mediation analysis aims at disentangling a total treatment effect into
an indirect effect operating through an intermediate variable – commonly
referred to as mediator – as well as the direct effect. The latter includes
any causal mechanisms not operating through the mediator of interest. Even
when the treatment is random, direct and indirect effects are generally not
identified by simply controlling for the mediator without accounting for its
potential endogeneity, as this likely introduces selection bias, see Robins
and Greenland (1992).

This paper suggests a novel identification strategy for causal mediation
analysis based on changes-in-changes (CiC) as suggested by Athey and Imbens
(2006) for evaluating (total) average and quantile treatment effects. We adapt
the approach to the identification of the direct effect and the indirect
effect running through a binary mediator. The outcome variable must be
continuous and is assumed to be observed both prior to and after treatment and
mediator assignment as it is the case in repeated cross sections or panel
data. The key identifying assumptions imply that the continuous outcome is
strictly monotonic in unobserved heterogeneity and that the distribution of
unobserved heterogeneity does not change over time conditional on the
treatment and the mediator (the latter assumption is also known as
stationarity). Given appropriate common support conditions, this permits
identifying direct effects on subpopulations conditional on the treatment and
the mediator states, even if both treatment and mediator assignment are
endogenous.

Augmenting the assumptions by random treatment assignment and weak
monotonicity of the mediator in the treatment allows for causal mediation
analysis in subpopulations defined upon whether and how the mediator reacts to
the treatment. Specifically, we show the identification of direct effects
among those whose mediator is always one (always-takers in the denomination of
Angrist, Imbens, and Rubin, 1996) and never one (never-takers) irrespective of
treatment assignment, respectively. Furthermore, we identify the total,
direct, and indirect treatment effects on those whose mediator value complies
with treatment assignment (compliers). For any set of assumptions, we discuss
the identification of both average and quantile direct and indirect effects.
We note that if appropriately weighted, the respective average effects among
compliers, always-takers, and never-takers add up to the average direct and
indirect effects in the population.

Identification in the earlier mediation literature typically relied on linear
models for the mediator and outcome equations and often neglected endogeneity
issues, see for instance Cochran (1957), Judd and Kenny (1981), and Baron and
Kenny (1986). More recent contributions use more general identification
approaches based on the potential outcome framework and take endogeneity
issues explicitly into consideration. Examples include Robins and Greenland
(1992), Pearl (2001), Robins (2003), Petersen, Sinisi, and van der Laan
(2006), VanderWeele (2009), Imai, Keele, and Yamamoto (2010), Hong (2010),
Albert and Nelson (2011), Imai and Yamamoto (2013), Tchetgen Tchetgen and
Shpitser (2012), Vansteelandt, Bekaert, and Lange (2012), and Huber (2014).
The vast majority of the literature assumes that the covariates observed in
the data are sufficiently rich to control for treatment and mediator
endogeneity. Also in empirical economics, there has been an increase in the
application of such selection on observables approaches, see for instance
Simonsen and Skipper (2006), Flores and Flores-Lagunes (2009), Heckman, Pinto,
and Savelyev (2013), Huber (2015), Keele, Tingley, and Yamamoto (2015), Conti,
Heckman, and Pinto (2016), Huber, Lechner, and Mellace (2017), Bijwaard and
Jones (2019), Bellani and Bia (2018), Huber, Lechner, and Strittmatter (2018),
and Doerr and Strittmatter (2019). Comparably few studies in economics develop
or apply instrumental variable approaches for disentangling direct and
indirect effects, see for instance Frölich and Huber (2017), Powdthavee,
Lekfuangfu, and Wooden (2013), Brunello, Fort, Schneeweis, and Winter-Ebmer
(2016) and Chen, Chen, and Liu (2017). Our paper provides another, CiC-based
identification strategy that neither rests on selection on observables
assumptions nor on instrumental variables for the treatment or the mediator.

While most studies aim at evaluating direct and indirect effects in the total
population, a smaller strand of the literature uses the principal
stratification framework of Frangakis and Rubin (2002) to investigate effects
in subpopulations (or principal strata) defined upon whether and how the
mediator reacts to the treatment, see Rubin (2004). This approach has been
criticized for typically focussing on direct effects on populations whose
mediator is constant (i.e. always- and never-takers) rather than decomposing
direct and indirect effects on compliers and for considering subpopulations
rather than the total population, see VanderWeele (2008) and VanderWeele
(2012). Deuchert, Huber, and Schelker (2017) suggest a difference-in-
differences (DiD) strategy that alleviates such criticisms. Identification
relies on a randomized treatment, monotonicity of the (binary) mediator in the
treatment, and particular common trend assumptions on mean potential outcomes
across principal strata. The latter imply that mean potential outcomes under
specific treatment and mediator states change by the same amount over time
across specific subpopulations. Depending on the strength of common trend and
effect homogeneity assumptions across principal strata, direct and indirect
effects are identified for different subpopulations and under the strongest
set of assumptions even for the total population.

Our paper contributes to this literature on principal strata effects, but
relies on different identifying assumptions than Deuchert, Huber, and Schelker
(2017). While differential time trends across subpopulations are permitted,
our approach restricts the conditional distribution of unobserved
heterogeneity over time. The two sets of assumptions are not nested and their
appropriateness is to be judged in the empirical context at hand. However,
both approaches could be used simultaneously for testing the joint validity of
the identifying assumptions of either method, in which case both CiC and DiD
converge to the same, true average direct and indirect effects. As a further
distinction to Deuchert, Huber, and Schelker (2017), our method also permits
assessing quantile treatment effects (QTEs) rather than average effects only.

In independent work, Sawada (2019) proposes a CiC strategy to tackle non-
compliance in randomized experiments when the exclusion restriction of random
assignment is violated. While there is an overlap in some identification
results of his study and ours (e.g. concerning the direct effect on never-
takers), there are also important differences. First, Sawada (2019)
predominantly focusses on the average treatment effect on the treated under
one-sided non-compliance (ruling out always-takers), which then corresponds to
the total effect on compliers. Our paper in addition disentangles the total
complier effect into direct and indirect components. Second, under two-sided
non-compliance (i.e. the existence of both never- and always-takers), Sawada
(2019) identifies the total complier effect by assuming homogeneity of the
direct effect, while we extend the CiC assumptions to the always-takers for
identifying (direct, indirect, and total) complier effects as well as the
direct effect among always-takers. Third and in contrast to Sawada (2019), we
also provide identification results in the absence of randomization and
monotonicity of the mediator in the treatment. On the other hand, Sawada
(2019) in contrast to our study demonstrates that the CiC strategy does not
necessarily require pre-treatment outcomes, but may exploit any pre-treatment
variable that has similar rank orders (as a function of unobserved
heterogeneity) like the outcome of interest.

We provide a simulation study in which we compare the CiC to the DiD approach
to illustrate our identification results. We also consider an empirical
application to the Jobs II programme previously analysed by Vinokur, Price,
and Schul (1995), a randomized job training intervention designed to analyse
the impact of job training on labour market and mental health outcomes. We
investigate the direct effect of the randomized offer of treatment on a
depression index, as well as its indirect effect through actual participation
in the programme as mediator. The reason for investigating the direct effect
is that treatment assignment could have a motivation or discouragement effect
on those randomly offered or not offered the training. We, however, find the
direct effect estimates to be close to zero and statistically insignificant
and therefore no indication for the violation of the exclusion restriction
when using treatment assignment as instrumental variable for actual
participation. In contrast, the moderately negative total and indirect effects
on those induced to participate by assignment are statistically significant at
least at the 10% level in all but one case and very much in line with the
estimate obtained by instrumental variable regression.

The remainder of this study is organized as follows. Section 2 introduces the
notation and defines the direct and indirect effects of interest. Section 3
presents the assumptions underlying our CiC approach as well as the
identification results. Section 4 provides a simulation study. Section 5
provides an application to Jobs II. Section 6 concludes.

##  2 Notation and effects

###  2.1 Average effects

Let $D$ denote a binary treatment (e.g., receiving the offer to participate in
a training programme) and $M$ a binary intermediate variable or mediator that
may be a function of $D$ (e.g., the actual participation in a training
programme). Furthermore, let $T$ indicate a particular time period: $T=0$
denotes the baseline period prior to the realisation of $D$ and $M$, $T=1$ the
follow up period after measuring $D$ and $M$ in which the effect of the
outcome is evaluated. Finally, let $Y_{t}$ denote the outcome of interest
(e.g., health measures) in period $T=t$. Indexing the outcome by the time
period $t\in\\{0,1\\}$ implies that it is measured both in the baseline period
and after the realisation of $D$ and $M$. To define the parameters of
interest, we make use of the potential outcome notation, see for instance
Rubin (1974), and denote by $Y_{t}(d,m)$ the potential outcome for treatment
state $D=d$ and mediator state $M=m$ in time $T=t$, with $d,m,t,\in\\{0,1\\}$.
Furthermore, let $M(d)$ denote the potential mediator as a function of the
treatment state $d\in\\{0,1\\}$. For notational ease, we will not use any time
index for $D$ and $M$, because either is assumed to be measured at a single
point in time between $T=0$ and $T=1$, albeit not necessarily at the same
point, as $D$ causally precedes $M$. Therefore, $D$ and $M$ correspond to the
actual treatment and mediator status in $T=1$, while it is assumed that no
treatment or mediation takes place in $T=0$.

Using this notation, the average treatment effect (ATE) in the ex-post period
is defined as $\Delta_{1}=E[Y_{1}(1,M(1))-Y_{1}(0,M(0))]$. That is, the ATE
corresponds to the effect of $D$ on the outcome that either affects the latter
directly (net of any effect on the mediator) or indirectly through an effect
on $M$. Indeed, the total ATE can be disentangled into the direct and indirect
effects, denoted by $\theta_{1}(d)=E[Y_{1}(1,M(d))-Y_{1}(0,M(d))]$ and
$\delta_{1}(d)=E[Y_{1}(d,M(1))-Y_{1}(d,M(0))]$, by adding and subtracting
$Y_{1}(1,M(0))$ or $Y_{1}(0,M(1))$, respectively:

$\displaystyle\Delta_{1}$ $\displaystyle=E[Y_{1}(1,M(1))-Y_{1}(0,M(0))],$
$\displaystyle=\underbrace{E[Y_{1}(1,M(1))-Y_{1}(1,M(0))]}_{=\delta_{1}(1)}+\underbrace{E[Y_{1}(1,M(0))-Y_{1}(0,M(0))]}_{=\theta_{1}(0)},$
$\displaystyle=\underbrace{E[Y_{1}(1,M(1))-Y_{1}(0,M(1))]}_{=\theta_{1}(1)}+\underbrace{E[Y_{1}(0,M(1))-Y_{1}(0,M(0))]}_{=\delta_{1}(0)}.$

Distinguishing between $\theta_{1}(1)$ and $\theta_{1}(0)$ or $\delta_{1}(1)$
and $\delta_{1}(0)$, respectively, implies the possibility of interaction
effects between $D$ and $M$ such that the direct and indirect effects could be
heterogeneous across values $d=1$ and $d=0$.

In our approach, we consider the concepts of direct and indirect effects
within specific subpopulations. The latter are either defined conditional on
the treatment and mediator values or conditional on potential mediator values
under either treatment states, which matches the so-called principal stratum
framework of Frangakis and Rubin (2002). As outlined in Angrist, Imbens, and
Rubin (1996) in the context of instrumental variable-based identification, any
individual $i$ in the population belongs to one of four strata, henceforth
denoted by $\tau$, according to their potential mediator status under either
treatment state: always-takers ($a$: $M(1)=M(0)=1$) whose mediator is always
one, compliers ($c$: $M(1)=1$, $M(0)=0$) whose mediator corresponds to the
treatment value, defiers ($de$: $M(1)=0$, $M(0)=1$) whose mediator opposes the
treatment value, and never-takers ($n$: $M(1)=M(0)=0$) whose mediator is never
one. Note that $\tau$ cannot be pinned down for any individual, because either
$M(1)$ or $M(0)$ is observed, but never both.

Let $\Delta_{1}^{\tau}=E[Y_{1}(1,M(1))-Y_{1}(0,M(0))|\tau]$ denote the ATE
conditional on $\tau\in\\{a,c,de,n\\}$; $\theta_{1}^{\tau}(d)$ and
$\delta_{1}^{\tau}(d)$ denote the corresponding direct and indirect effects.
Because $M(1)=M(0)=0$ for any never-taker, the indirect effect for this group
is by definition zero $(\delta_{1}^{n}(d)=E[Y_{1}(d,0)-Y_{1}(d,0)|\tau=n]=0)$
and
$\Delta_{1}^{n}=E[Y_{1}(1,0)-Y_{1}(0,0)|\tau=n]=\theta_{1}^{n}(1)=\theta_{1}^{n}(0)=\theta_{1}^{n}$
equals the direct effect for never-takers. Correspondingly, because
$M(1)=M(0)=1$ for any always-taker, the indirect effect for this group is by
definition zero $(\delta_{1}^{a}(d)=E[Y_{1}(d,1)-Y_{1}(d,1)|\tau=a]=0)$ and
$\Delta_{1}^{a}=E[Y_{1}(1,1)-Y_{1}(0,1)|\tau=a]=\theta_{1}^{a}(1)=\theta_{1}^{a}(0)=\theta_{1}^{a}$
equals the direct effect for always-takers. For the compliers, both direct and
indirect effects may exist. Note that $M(d)=d$ due to the definition of
compliers. Accordingly, $\theta_{1}^{c}(d)=E[Y_{1}(1,d)-Y_{1}(0,d)|\tau=c]$
equals the direct effect for compliers,
$\delta_{1}^{c}(d)=E[Y_{1}(d,1)-Y_{1}(d,0)|\tau=c]$ equals the indirect effect
for compliers, and $\Delta_{1}^{c}=E[Y_{1}(1,1)-Y_{1}(0,0)|\tau=c]$ equals the
total effect for compliers. In the absence of any direct effect, the indirect
effects on the compliers are homogeneous,
$\delta_{1}^{c}(1)=\delta_{1}^{c}(0)=\delta_{1}^{c}$, and correspond to the
local average treatment effect (LATE, e.g., Angrist, Imbens, and Rubin, 1996).
Analogous results hold for the defiers.

As already mentioned, we will also consider direct effects conditional on
specific values $D=d$ and mediator states $M=M(d)=m$, which are denoted by
$\theta_{1}^{d,m}(d)=E[Y_{1}(1,m)-Y_{1}(0,m)|D=d,M(d)=m]$. These parameters
are identified under weaker assumptions than strata-specific effects, but are
also less straightforward to interpret, as they refer to mixtures of two
strata. For instance,
$\theta_{1}^{1,0}(1)=E[Y_{1}(1,0)-Y_{1}(0,0)|D=1,M(1)=0]$ is the effect on a
mixture of never-takers and defiers, as these two groups satisfy $M(1)=0$.
Likewise, $\theta_{1}^{0,0}(0)$ refers to never-takers and compliers
satisfying $M(0)=0$, $\theta_{1}^{0,1}(0)$ to always-takers and defiers
satisfying $M(0)=1$, and $\theta_{1}^{1,1}(1)$ to always-takers and compliers
satisfying $M(1)=1$.

###  2.2 Quantile effects

We denote by $F_{Y_{t}(d,m)}(y)=\Pr(Y_{t}(d,m)\leq y)$ the cumulative
distribution function of $Y_{t}(d,m)$ at outcome level $y$. Its inverse,
$F_{Y_{t}(d,m)}^{-1}(q)=\inf\\{y:F_{Y_{t}(d,m)}(y)\geq q\\}$, is the quantile
function of $Y_{t}(d,m)$ at rank $q$. The total QTE are denoted by
$\Delta_{1}(q)=F_{Y_{1}(1,M(1))}^{-1}(q)-F_{Y_{1}(0,M(0))}^{-1}(q)$. The QTE
can be disentangled into the direct quantile effects, denoted by
$\theta_{1}(q,d)=F_{Y_{1}(1,M(d))}^{-1}(q)-F_{Y_{1}(0,M(d))}^{-1}(q)$, and the
indirect quantile effects, denoted by
$\delta_{1}(q,d)=F_{Y_{1}(d,M(1))}^{-1}(q)-F_{Y_{1}(d,M(0))}^{-1}(q)$.

The conditional distribution function in stratum $\tau$ is
$F_{Y_{t}(d,m)|\tau}(y)=\Pr(Y_{t}(d,m)\leq y|\tau)$ and the corresponding
conditional quantile function is
$F_{Y_{t}(d,m)|\tau}^{-1}(q)=\inf\\{y:F_{Y_{t}(d,m)|\tau}(y)\geq q\\}$ for
$\tau\in\\{a,c,d,n\\}$. Using the previously described stratification
framework, we define the QTE conditional on $\tau\in\\{a,c,de,n\\}$:
$\Delta_{1}^{\tau}(q)=F_{Y_{1}(1,M(1))|\tau}^{-1}(q)-F_{Y_{1}(0,M(0))|\tau}^{-1}(q)$.
The direct quantile treatment effect among never-takers equals
$\Delta_{1}^{n}(q)=F_{Y_{1}(1,0)|n}^{-1}(q)-F_{Y_{1}(0,0)|n}^{-1}(q)=\theta_{1}^{n}(q)$.
The direct quantile effect among always-takers equals
$\Delta_{1}^{a}(q)=F_{Y_{1}(1,1)|a}^{-1}(q)-F_{Y_{1}(0,1)|a}^{-1}(q)=\theta_{1}^{a}(q)$.
The total QTE among compliers equals
$\Delta_{1}^{c}(q)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$, the
direct quantile effect among compliers equals
$\theta_{1}^{c}(q,d)=F_{Y_{1}(1,d)|c}^{-1}(q)-F_{Y_{1}(0,d)|c}^{-1}(q)$, and
the indirect quantile effect among compliers equals
$\delta_{1}^{c}(q,d)=F_{Y_{1}(d,1)|c}^{-1}(q)-F_{Y_{1}(d,0)|c}^{-1}(q)$.
Finally, we define the direct quantile treatment effects conditional on
specific values $D=d$ and mediator states $M=M(d)=m$,

$\displaystyle\theta_{1}^{d,m}(q,1)$
$\displaystyle=F_{Y_{1}(1,m)|D=d,M(1)=m}^{-1}(q)-F_{Y_{1}(0,m)|D=d,M(1)=m}^{-1}(q)\mbox{
and}$ $\displaystyle\theta_{1}^{d,m}(q,0)$
$\displaystyle=F_{Y_{1}(1,m)|D=d,M(0)=m}^{-1}(q)-F_{Y_{1}(0,m)|D=d,M(0)=m}^{-1}(q),$

with the quantile function
$F_{Y_{t}(d,m)|D=d,M(d)=m}^{-1}(q)=\inf\\{y:F_{Y_{t}(d,m)|D=d,M(d)=m}(y)\geq
q\\}$ and the distribution function
$F_{Y_{t}(d,m)|D=d,M(d)=m}(y)=\Pr(Y_{t}(d,m)\leq y|D=d,M(d)=m)$.

###  2.3 Observed distribution and quantile transformations

We subsequently define various functions of the observed data required for the
identification results. The conditional distribution function of the observed
outcome $Y_{t}$ conditional on treatment value $d$ and mediator state $m$, is
given by $F_{Y_{t}|D=d,M=m}(y)=\Pr(Y_{t}\leq y|D=d,M=m)$ for
$d,m\in\\{0,1\\}$. The corresponding conditional quantile function is
$F_{Y_{t}|D=d,M=m}^{-1}(q)=\inf\\{y:F_{Y_{t}|D=d,M=m}(y)\geq q\\}$.
Furthermore,

$Q_{dm}(y):=F_{Y_{1}|D=d,M=m}^{-1}\circ
F_{Y_{0}|D=d,M=m}(y)=F_{Y_{1}|D=d,M=m}^{-1}(F_{Y_{0}|D=d,M=m}(y))$

is the quantile-quantile transform of the conditional outcome from period 0 to
1 given treatment $d$ and mediator status $m$. This transform maps $y$ at rank
$q$ in period 0 ($q=F_{Y_{0}|D=d,M=m}(y)$) into the corresponding $y^{\prime}$
at rank $q$ in period 1 ($y^{\prime}=F_{Y_{1}|D=d,M=m}^{-1}(q)$).

##  3 Identification and Estimation

###  3.1 Identification

This sections discusses the identifying assumptions along with the
identification results for the various direct and indirect effects. We note
that our assumptions could be adjusted to only hold conditional on a vector of
observed covariates. In this case, the identification results would hold
within cells defined upon covariate values. In our main discussion, however,
covariates are not considered for the sake of ease of notation. For notational
convenience, we maintain throughout that $\Pr(T=t,D=d,M=m)>0$ for $t,d,m$
$\in\\{1,0\\}$, implying that all possible treatment-mediator combinations
exist in the population in both time periods. Our first assumption implies
that potential outcomes are characterized by a continuous nonparametric
function, denoted by $h$, that is strictly monotonic in a scalar $U$ that
reflects unobserved heterogeneity.
Assumption 1: Strict monotonicity of continuous potential outcomes in
unobserved heterogeneity.
The potential outcomes satisfy the following model: $Y_{t}(d,m)=h(d,m,t,U)$,
with the general function $h$ being continuous and strictly increasing in the
scalar unobservable $U\in\mathbb{R}$ for all $d,m,t\in\\{0,1\\}$.
Assumption 1 requires the potential outcomes to be continuous implying that
there is a one-to-one correspondence between a potential outcome’s
distribution and quantile functions, which is a condition for point
identification. For discrete potential outcomes, only bounds on the effects
could be identified, in analogy to the discussion in Athey and Imbens (2006)
for total (rather than direct and indirect) effects. Assumption 1 also implies
that individuals with identical unobserved characteristics $U$ have the same
potential outcomes $Y_{t}(d,m)$, while higher values of $U$ correspond to
strictly higher potential outcomes $Y_{t}(d,m)$. Strict monotonicity is
automatically satisfied in additively separable models, but Assumption 1 also
allows for more flexible non-additive structures that arise in nonparametric
models.

The next assumption rules out anticipation effects of the treatment or the
mediator on the outcome in the baseline period. This assumption is plausible
if assignment to the treatment or the mediator cannot be foreseen in the
baseline period, such that behavioral changes affecting the pre-treatment
outcome are ruled out.
Assumption 2: No anticipation effect of $M$ and $D$ in the baseline period.
$Y_{0}(d,m)-Y_{0}(d^{\prime},m^{\prime})=0\mbox{, for
}d,d^{\prime},m,m^{\prime}\\{1,0\\}.$
Similarly, Athey and Imbens (2006) and Chaisemartin and D’Haultfeuille (2018)
assume the assignment to the treatment group does not affect the potential
outcomes as long as the treatment is not yet realized.

Furthermore, we assume conditional independence between unobserved
heterogeneity and time periods given the treatment and no mediation.
Assumption 3: Conditional independence of $U$ and $T$ given $D=1,M=0$ or
$D=0,M=0$.
(a) $U\perp\\!\\!\\!\perp T|D=1,M=0$,
(b) $U\perp\\!\\!\\!\perp T|D=0,M=0$.
Under Assumption 3a, the distribution of $U$ is allowed to vary across groups
defined upon treatment and mediator state, but not over time within the group
with $D=1,M=0$. Assumption 3b imposes the same restriction conditional on
$D=0,M=0$. Assumption 3 thus imposes stationarity of $U$ within groups defined
on $D$ and $M$. This assumption is weaker than (and thus implied by) requiring
that $U$ is constant across $T$ for each individual $i$. For example,
Assumption 3 is satisfied in the fixed effect model $U=\eta+v_{t}$, with
$\eta$ being a time-invariant individual-specific unobservable (fixed effect)
and $v_{t}$ an idiosyncratic time-varying unobservable with the same
distribution in both time periods.

Athey and Imbens (2006) and Chaisemartin and D’Haultfeuille (2018) impose time
invariance conditional on the treatment status, $U\perp\\!\\!\\!\perp T|D=d$,
to identify the average treatment effect on the treated,
$\varphi_{1}=E[Y_{1}(1,M(1))-Y_{1}(0,M(0))|D=1]$ or local average treatment
effect, $\varphi_{1}=E[Y_{1}(1,M(1))-Y_{1}(0,M(0))|\tau=c]$, respectively. We
additionally condition on the mediator status to identify direct and indirect
effects.

For our next assumption, we introduce some further notation. Let
$F_{U|d,m}(u))=\Pr(U\leq u|D=d,M=m)$ be the conditional distribution of $U$
with support $\mathbb{U}_{dm}$.
Assumption 4: Common support given $M=0$.
(a) $\mathbb{U}_{10}\subseteq\mathbb{U}_{00}$,
(b) $\mathbb{U}_{00}\subseteq\mathbb{U}_{10}$.
Assumption 4a is a common support assumption, implying that any possible value
of $U$ in the population with $D=1,M=0$ is also contained in the population
with $D=0,M=0$. Assumption 4b imposes that any value of $U$ conditional on
$D=0,M=0$ also exists conditional on $D=1,M=0$. Both assumptions together
imply that the support of $U$ is the same in both populations, albeit the
distributions may generally differ.

Assumptions 1 to 3 permit identifying direct effects on mixed populations of
never-takers and defiers as well as never-takers and compliers, respectively,
as formally stated in Theorem 1.
Theorem 1: Under Assumptions 1–3,

  * (a)

and Assumption 4a, the average and quantile direct effects under $d=1$
conditional on $D=1$ and $M(1)=0$ are identified:

$\displaystyle\theta_{1}^{1,0}(1)$
$\displaystyle=E[Y_{1}-Q_{00}(Y_{0})|D=1,M=0],$
$\displaystyle\theta_{1}^{1,0}(q,1)$
$\displaystyle=F_{Y_{1}|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).$

  * (b)

and Assumption 4b, the average and quantile direct effects under $d=0$
conditional on $D=0$ and $M(0)=0$ are identified:

$\displaystyle\theta_{1}^{0,0}(0)$
$\displaystyle=E[Q_{10}(Y_{0})-Y_{1}|D=0,M=0],$
$\displaystyle\theta_{1}^{0,0}(q,0)$
$\displaystyle=F_{Q_{10}(Y_{0})|D=0,M=0}^{-1}(q)-F_{Y_{1}|D=0,M=0}^{-1}(q).$

Proof. See Appendix A.

To identify direct effects on further populations, we invoke a conditional
independence assumption that is in the spirit of Assumption 3, but refers to
different combinations of the treatment and the mediator.
Assumption 5: Conditional independence of $U$ and $T$ given $D=0,M=1$ or
$D=1,M=1$.
(a) $U\perp\\!\\!\\!\perp T|D=0,M=1$,
(b) $U\perp\\!\\!\\!\perp T|D=1,M=1$.
Under Assumption 5a, the distribution of $U$ is allowed to vary by treatment
and mediator group, but not over time conditional on $D=0,M=1$. Assumption 5b
imposes the same restriction conditional on $D=1,M=1$.

Assumption 6 is similar to Assumption 4, but imposes common support
conditional on $M=1$ rather than $M=0$.
Assumption 6: Common support given $M=1$.
(a) $\mathbb{U}_{01}\subseteq\mathbb{U}_{11}$,
(b) $\mathbb{U}_{11}\subseteq\mathbb{U}_{01}$.
Assumptions 6a implies that any possible value of $U$ in the population with
$D=0,M=1$ is also contained in the population with $D=1,M=1$. Assumptions 6b
states that any value of $U$ conditional on $D=1,M=1$ exists conditional on
$D=0,M=1$.

Theorem 2 shows the identification of the direct effects on mixed populations
of always-takers and defiers as well as always-takers and compliers.
Theorem 2: Under Assumptions 1-2, 5,

  * (a)

and Assumption 6a, the average and quantile direct effects under $d=1$
conditional on $D=0$ and $M(0)=1$ are identified:

$\displaystyle\theta_{1}^{0,1}(0)$
$\displaystyle=E[Q_{11}(Y_{0})-Y_{1}|D=0,M=1],$
$\displaystyle\theta_{1}^{0,1}(q,0)$
$\displaystyle=F_{Q_{11}(Y_{0})|D=0,M=1}^{-1}(q)-F_{Y_{1}|D=0,M=1}^{-1}(q).$

  * (b)

and Assumption 6b, the average and quantile direct effects under $d=1$ is
identified conditional on $D=1$ and $M(1)=1$ are identified:

$\displaystyle\theta_{1}^{1,1}(1)$
$\displaystyle=E[Y_{1}-Q_{01}(Y_{0})|D=1,M=1],$
$\displaystyle\theta_{1}^{1,1}(q,1)$
$\displaystyle=F_{Y_{1}|D=1,M=1}^{-1}(q)-F_{Q_{01}(Y_{0})|D=1,M=1}^{-1}(q).$

Proof. See Appendix B.

In the instrumental variable framework, any direct effects of the instrument
are typically ruled out by imposing the exclusion restriction, in order to
identify the causal effect of an endogenous regressor on the outcome, see for
instance Imbens and Angrist (1994). By considering $D$ as instrument and $M$
as endogenous regressor,
$\theta_{1}^{1,0}(1)=\theta_{1}^{0,0}(0)=\theta_{1}^{0,1}(0)=\theta_{1}^{1,1}(1)=0$
yield testable implications of the exclusion restriction under Assumptions
1-6.

So far, we did not impose exogeneity of the treatment or mediator. In the
following, we assume treatment exogeneity by invoking independence between the
treatment and the potential post-treatment variables.
Assumption 7: Independence of the treatment and potential mediators/outcomes.
$\\{Y_{t}(d,m),M(d)\\}\perp\\!\\!\\!\perp D\mbox{, for all
}d,m,t,\in\\{0,1\\}.$
Assumption 7 implies that there are no confounders jointly affecting the
treatment on the one hand and the mediator and/or outcome on the other hand.
It is satisfied under treatment randomization as in successfully conducted
experiments. This allows identifying the ATE:
$\Delta_{1}=E[Y_{1}|D=1]-E[Y_{1}|D=0]$.

Furthermore, we assume the mediator to be weakly monotonic in the treatment.
Assumption 8: Weak monotonicity of the mediator in the treatment.
$\Pr(M(1)\geq M(0))=1.$
Assumption 8 is standard in the instrumental variable literature on local
average treatment effects when denoting by $D$ the instrument and by $M$ the
endogenous regressor, see Imbens and Angrist (1994) and Angrist, Imbens, and
Rubin (1996). It rules out the existence of defiers.

As discussed in the Appendix C, the total ATE
$\Delta_{1}=E[Y_{1}|D=1]-E[Y_{1}|D=0]$ and QTE
$\Delta_{1}(q)=F_{Y_{1}|D=1}^{-1}(q)-F_{Y_{1}|D=0}^{-1}(q)$ for the entire
population are identified under Assumption 7. Furthermore, Assumptions 7 and 8
yield the strata proportions, denoted by $p_{\tau}=\Pr(\tau)$, as functions of
the conditional mediator probabilities given the treatment, which we denote by
$p_{(m|d)}=\Pr(M=m|D=d)$ for $d,m\in\\{0,1\\}$ (see Appendix C):

$p_{a}=p_{1|0},p_{c}=p_{1|1}-p_{1|0}=p_{0|0}-p_{0|1},p_{n}=p_{0|1}.$ (1)

Furthermore, Assumptions 2, 7, and 8 imply that (see Appendix C)

$\Delta_{0,c}=E[Y_{0}(1,1)-Y_{0}(0,0)|c]=\frac{E[Y_{0}|D=1]-E[Y_{0}|D=0]}{p_{1|1}-p_{1|0}}=0.$
(2)

Therefore, a rejection of the testable implication
$E[Y_{0}|D=1]-E[Y_{0}|D=0]=0$ in the data would point to a violation of these
assumptions.

Assumptions 7 and 8 permit identifying additional parameters, namely the
total, direct, and indirect effects on compliers, and the direct effects on
never- and always-takers, as shown in Theorems 3 to 5. This follows from the
fact that defiers are ruled out and that the proportions and potential outcome
distributions of the various principal strata are not selective w.r.t. the
treatment.
Theorem 3: Under Assumptions 1–3, 7-8,

  * a)

and Assumption 4a, the average and quantile direct effects on never-takers are
identified:

$\theta_{1}^{n}=\theta_{1}^{1,0}(1)\mbox{ and
}\theta_{1}^{n}(q)=\theta_{1}^{1,0}(q,1).$

  * b)

and Assumption 4, the average direct effect under $d=0$ on compliers is
identified:

$\displaystyle\displaystyle\theta_{1}^{c}(0)=$
$\displaystyle\frac{p_{0|0}}{p_{0|0}-p_{0|1}}\theta_{1}^{0,0}(0)-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}\theta_{1}^{1,0}(1).$

Furthermore, the potential outcome distributions under $d=0$ on compliers are
identified:

$\begin{array}[]{rl}\displaystyle
F_{Y_{1}(1,0)|\tau=c}(y)=&\displaystyle\frac{p_{0|0}}{p_{0|0}-p_{0|1}}F_{Q_{10}(Y_{0})|D=0,M=0}(y)\\\
&\displaystyle\qquad-\frac{p_{0|1}}{p_{0|0}-p_{0|1}c}F_{Y_{1}|D=1,M=0}(y),\end{array}$
(3) $\begin{array}[]{rl}\displaystyle
F_{Y_{1}(0,0)|\tau=c}(y)=&\displaystyle\frac{p_{0|0}}{p_{0|0}-p_{0|1}}F_{Y_{1}|D=0,M=0}(y)\\\
&\displaystyle\qquad-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y).\end{array}$
(4)

Therefore, the direct quantile effect under $d=0$ on compliers,
$\theta_{1}^{c}(q,0)=F_{Y_{1}(1,0)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$, is
identified.

Proof. See Appendix D.

Theorem 4: Under Assumptions 1–2, 5, 7-8,

  * a)

and Assumption 6a, the average and quantile direct effects on always-takers
are identified:

$\theta_{1}^{a}=\theta_{1}^{0,1}(0)\mbox{ and
}\theta_{1}^{a}(q)=\theta_{1}^{0,1}(q,0).$

  * b)

and Assumption 6, the average direct effect under $d=1$ on compliers is
identified:

$\displaystyle\theta_{1}^{c}(1)=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}\theta_{1}^{1,1}(1)-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}\theta_{1}^{0,1}(0).$

Furthermore, the potential outcome distributions under $d=1$ for compliers are
identified:

$\begin{array}[]{rl}\displaystyle
F_{Y_{1}(1,1)|\tau=c}(y)=&\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}F_{Y_{1}|D=1,M=1}(y)\\\
&\displaystyle\qquad-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}F_{Q_{11}(Y_{0})|D=0,M=1}(y),\end{array}$
(5) $\begin{array}[]{rl}\displaystyle
F_{Y_{1}(0,1)|\tau=c}(y)=&\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}F_{Q_{01}(Y_{0})|D=1,M=1}(y)\\\
&\displaystyle\qquad-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}F_{Y_{1}|D=0,M=1}(y).\end{array}$
(6)

Therefore, the direct quantile effect under $d=1$ on compliers
$\theta_{1}^{c}(q,1)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,1)|c}^{-1}(q)$ is
identified.

Proof. See Appendix E.

Theorem 5: Under Assumptions 1-3, 5, 7-8,

  * a)

and Assumptions 4a, 6a, the total average treatment effect on compliers is
identified:

$\displaystyle\Delta_{1}^{c}=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=1,M=1]-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_{0})|D=0,M=1]$
$\displaystyle−\frac{p_{0|0}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=0,M=0]+\frac{p_{0|1}}{p_{1|1}-p_{1|0}}E[Q_{00}(Y_{0})|D=1,M=0].$

Furthermore, the total quantile treatment effect on compliers
$\Delta_{1}^{c}(q)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is
identified using the inverse of (5) and (4).

  * b)

and Assumptions 4a, 6b, the average indirect effect under $d=0$ on compliers
is identified:

$\displaystyle\delta_{1}^{c}(0)=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{01}(Y_{0})|D=1,M=1]-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=0,M=1]$
$\displaystyle-\frac{p_{0|0}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=0,M=0]+\frac{p_{0|1}}{p_{1|1}-p_{1|0}}E[Q_{00}(Y_{0})|D=1,M=0].$

Furthermore, the quantile indirect effect under $d=0$ on compliers
$\delta_{1}^{c}(q,0)=F_{Y_{1}(0,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is
identified using the inverse of (6) and (4).

  * c)

and Assumptions 4b, 6a, the average indirect effect under $d=1$ on compliers
is identified:

$\displaystyle\delta_{1}^{c}(1)=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=1,M=1]-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_{0})|D=0,M=1]$
$\displaystyle-\frac{p_{0|0}}{p_{1|1}-p_{1|0}}E[Q_{10}(Y_{0})|D=0,M=0]+\frac{p_{0|1}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=1,M=0].$

Furthermore, the quantile indirect effect under $d=1$ on compliers
$\delta_{1}^{c}(q,1)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(1,0)|c}^{-1}(q)$ is
identified using the inverse of (5) and (3).

Proof. See Appendix F.

###  3.2 Estimation

As in Assumption 5.1 of Athey and Imbens (2006), we assume standard regularity
conditions, namely that conditional on $T=t$, $D=d$, and $M=m$, $Y$ is a
random draw from that subpopulation defined in terms of $t,d,m$ $\in$
$\\{1,0\\}$. Furthermore, the outcome in the subpopulations required for the
identification results of interest must have compact support and a density
that is bounded from above and below as well as continuously differentiable.
Denote by $N$ the total sample size across both periods and all treatment-
mediator combinations and by $i$ $\in$ $\\{1,...,N\\}$ an index for the
sampled subject, such that $(Y_{i},D_{i},M_{i},T_{i}$) correspond to sample
realizations of the random variables $(Y,D,M,T$).

The total, direct, and indirect effects may be estimated using the sample
analogy principle, which replaces population moments with sample moments (e.g.
Manski, 1988). For instance, any conditional mediator probability given the
treatment, $\Pr(M=m|D=d)$, is to be replaced by an estimate thereof in the
sample,
$\frac{\sum_{i=1}^{N}I\\{M_{i}=m,D_{i}=d\\}}{\sum_{i=1}^{N}I\\{D_{i}=d\\}}$. A
crucial step is the estimation of the quantile-quantile transforms. The
application of such quantile transformations dates at least back to Juhn,
Murphy, and Pierce (1991), see also Chaisemartin and D’Haultfeuille (2018),
Wüthrich (2019), and Strittmatter (2019) for recent applications. First, it
requires estimating the conditional outcome distribution,
$F_{Y_{t}|D=d,M=m}(y)$, by the conditional empirical distribution
$\hat{F}_{Y_{t}|D=d,M=m}(y)=\frac{1}{\sum_{i=1}^{n}I\\{D_{i}=d,M_{i}=m,T_{i}=t\\}}\sum_{i:D_{i}=d,M_{i}=m,T_{i}=t}I\\{Y_{i}\leq
y\\}$. Second, inverting the latter yields the empirical quantile function
$\hat{F}_{Y_{t}|D=d,M=m}^{-1}(q)$. The empirical quantile-quantile transform
is then obtained by

$\displaystyle\hat{Q}_{dm}(y)=\hat{F}_{Y_{1}|D=d,M=m}^{-1}(\hat{F}_{Y_{0}|D=d,M=m}(y)).$

This permits estimating the average and quantile effects of interest. Average
effects are estimated by replacing any (conditional) expectations with the
corresponding sample averages in which the estimated quantile-quantile
transforms enter as plug-in estimates. Taking $\theta_{1}^{1,0}$ (see Theorem
1) as an example, an estimate thereof is

$\displaystyle\hat{\theta}_{1}^{1,0}(1)$ $\displaystyle=$
$\displaystyle\frac{1}{\sum_{i=1}^{n}I\\{D_{i}=1,M_{i}=0,T_{i}=1\\}}\sum_{i:D_{i}=1,M_{i}=0,T_{i}=1}Y_{i}$
$\displaystyle-$
$\displaystyle\frac{1}{\sum_{i=1}^{n}I\\{D_{i}=1,M_{i}=0,T_{i}=0\\}}\sum_{i:D_{i}=1,M_{i}=0,T_{i}=0}\hat{Q}_{00}(Y_{i}).$

Likewhise, quantile effects are estimated based on the empirical quantiles.

For the estimation of total ATE and QTE, Athey and Imbens (2006) show that the
resulting estimators are $\sqrt{N}$-consistent and asymptotically normal, see
their Theorems 5.1 and 5.3. These properties also apply to our context when
splitting the sample into subgroups based on the values of a binary treatment
and mediator (rather than the treatment only). For instance, the implications
of Theorem 1 in Athey and Imbens (2006) when considering subsamples with $D=1$
and $D=0$ carry over to considering subsamples with $D=1,M=0$ and $D=0,M=0$
for estimating the average direct effect on never-takers. In contrast to Athey
and Imbens (2006), however, some of our identification results include the
conditional mediator probabilities $\Pr(M=m|D=d)$. As the latter are estimated
with $\sqrt{N}$-consistency, too, it follows that the resulting effect
estimators are again $\sqrt{N}$-consistent and asymptotically normal. We use a
non-parametric bootstrap approach to calculate the standard errors.
Chaisemartin and D’Haultfeuille (2018) show the validity of the bootstrap
approach for such kind of estimators, which follows from their asymptotic
normality.

For the case that identifying assumptions to only hold conditional on observed
covariates, denoted by $X$, estimation must be adapted to allow for control
variables. Following a suggestion by Athey and Imbens (2006) in their Section
5.1, basing estimation on outcome residuals in which the association of $X$
and $Y$ has been purged by means of a regression is consistent under the
additional assumption that the effects of $D$ and $M$ are homogeneous across
covariates. As an alternative, Melly and Santangelo (2015) propose a flexible
semiparametric estimator that does not impose such a homogeneity-in-covariates
assumption and show $\sqrt{N}$-consistency and asymptotic normality.

##  4 Simulations

To shape the intuition for our identification results, this section presents a
brief simulation based on the following data generating process (DGP):

$T\sim Binom(0.5),\textrm{ }D\sim Binom(0.5),\textrm{ }U\sim
Unif(-1,1),\textrm{ }V\sim N(0,1)$

independent of each other, and

$M=I\\{D+U+V>0\\},\qquad Y_{T}=\Lambda((1+D+M+D\cdot M)\cdot T+U).$

Treatment $D$ as well as the observed time period $T$ are randomized, while
the mediator-outcome association is confounded due to the unobserved time
constant heterogeneity $U$. The potential outcome in period $1$ is given by
$Y_{1}(d,M(d^{\prime}))=\Lambda((1+d+M(d^{\prime})+d\cdot M(d^{\prime}))+U)$,
where $\Lambda$ denotes a link function. If the latter corresponds to the
identity function, our model is linear and implies a homogeneous time trend
$T$ equal to 1. If $\Lambda$ is nonlinear, the time trend is heterogeneous,
which invalidates the common trend assumption of difference-in-differences
models. $M$ is not only a function of $D$ and $U$, but also of the unobserved
random term $V$, which guarantees common support w.r.t. $U$, see Assumptions 4
and 6. Compliers, always-takers, and never-takers satisfy, respectively:
$c=I\\{U+V\leq 0,1+U+V>0\\}$, $a=I\\{U+V>0\\}$, and $n=I\\{1+U+V\leq 0\\}$.

In the simulations with 1,000 replications, we consider two sample sizes
($N=1,000,4,000$) and investigate the behaviour of our change-in-changes
methods as well as the difference-in-differences approach of Deuchert, Huber,
and Schelker (2017) in both a linear ($\Lambda$ equal to identity function)
and nonlinear outcome model where $\Lambda$ equals the exponential function.
To implement the change-in-changes estimators in the simulations as well as
the application in Section 5, we make use of the ‘cic’ command in the qte
R-package by Callaway (2016) with its default values.

Table 1:  Linear model with random treatment

| $\hat{\theta}_{1}^{n}$ | $\hat{\theta}_{1}^{a}$ | $\hat{\Delta}_{c}$ | $\hat{\theta}_{1}^{c}(1)$ | $\hat{\theta}_{1}^{c}(0)$ | $\hat{\delta}_{1}^{c}(1)$ | $\hat{\delta}_{1}^{c}(0)$
---|---|---|---|---|---|---|---
A. Changes-in-Changes
|  $N$=1,000
bias | 0.00 | -0.00 | -0.01 | -0.01 | -0.01 | -0.00 | -0.01
sd | 0.11 | 0.08 | 0.23 | 0.10 | 0.13 | 0.27 | 0.27
rmse | 0.11 | 0.08 | 0.23 | 0.10 | 0.13 | 0.27 | 0.27
true | 1.00 | 2.00 | 3.00 | 2.00 | 1.00 | 2.00 | 1.00
relr | 0.11 | 0.04 | 0.08 | 0.05 | 0.13 | 0.14 | 0.27
|  $N$=4,000
bias | -0.00 | -0.00 | 0.00 | -0.00 | -0.01 | 0.01 | 0.01
sd | 0.06 | 0.04 | 0.12 | 0.05 | 0.07 | 0.14 | 0.14
rmse | 0.06 | 0.04 | 0.12 | 0.05 | 0.07 | 0.14 | 0.14
true | 1.00 | 2.00 | 3.00 | 2.00 | 1.00 | 2.00 | 1.00
relr | 0.06 | 0.02 | 0.04 | 0.02 | 0.07 | 0.07 | 0.14
B. Difference-in-Differences
|  $N$=1,000
bias | 0.01 | -0.00 | -0.01 | -0.01 | 0.00 | -0.02 | 0.00
sd | 0.11 | 0.09 | 0.14 | 0.14 | 0.12 | 0.19 | 0.10
rmse | 0.11 | 0.09 | 0.14 | 0.14 | 0.12 | 0.19 | 0.10
true | 1.00 | 2.00 | 3.00 | 2.00 | 1.00 | 2.00 | 1.00
relr | 0.11 | 0.04 | 0.05 | 0.07 | 0.12 | 0.10 | 0.10
|  $N$=4,000
bias | -0.00 | -0.00 | 0.00 | -0.00 | -0.00 | 0.00 | 0.00
sd | 0.06 | 0.04 | 0.07 | 0.07 | 0.06 | 0.10 | 0.05
rmse | 0.06 | 0.04 | 0.07 | 0.07 | 0.06 | 0.10 | 0.05
true | 1.00 | 2.00 | 3.00 | 2.00 | 1.00 | 2.00 | 1.00
relr | 0.06 | 0.02 | 0.02 | 0.04 | 0.06 | 0.05 | 0.05

Note: ‘bias’, ‘sd’, and ‘rmse’ provide the bias, standard deviation, and root
mean squared error of the respective estimator. ‘true’ and ‘relr’ are the
respective true effect as well as the root mean squared error relative to the
true effect.

Table 1 reports the bias, standard deviation (‘sd’), root mean squared error
(‘rmse’), true effect (‘true’), and the relative root mean squared error in
percent of the true effect (‘relr’) of the respective estimators of
$\theta_{1}^{n}$, $\theta_{1}^{a}$, $\Delta_{c}$, $\theta_{1}^{c}(1)$,
$\theta_{1}^{c}(0)$, $\delta_{1}^{c}(1)$, and $\delta_{1}^{c}(0)$ for the
linear model. In this case, the identifying assumptions underlying both the
change-in-changes (Panel A.) and difference-in-differences (Panel B.)
estimators are satisfied. Specifically, the homogeneous time trend on the
individual level satisfies any of the common trend assumptions in Deuchert,
Huber, and Schelker (2017), while the monotonicity of $Y$ in $U$ and the
independence of $T$ and $U$ satisfies the key assumptions of this paper. For
this reason any of the estimates in Table 1 are close to being unbiased and
appear to converge to the true effect at the parametric rate when comparing
the results for the two different sample sizes.

Table 2:  Nonlinear model with random treatment

| $\hat{\theta}_{1}^{n}$ | $\hat{\theta}_{1}^{a}$ | $\hat{\Delta}_{c}$ | $\hat{\theta}_{1}^{c}(1)$ | $\hat{\theta}_{1}^{c}(0)$ | $\hat{\delta}_{1}^{c}(1)$ | $\hat{\delta}_{1}^{c}(0)$
---|---|---|---|---|---|---|---
A. Change-in-Changes
|  $N$=1,000
bias | 0.01 | -0.14 | -0.48 | -0.35 | -0.11 | -0.37 | -0.13
sd | 0.48 | 5.08 | 8.47 | 6.20 | 1.16 | 8.64 | 4.23
rmse | 0.48 | 5.08 | 8.48 | 6.21 | 1.17 | 8.65 | 4.23
true | 3.49 | 68.09 | 52.42 | 47.70 | 4.72 | 47.70 | 4.72
relr | 0.14 | 0.07 | 0.16 | 0.13 | 0.25 | 0.18 | 0.90
|  $N$=4,000
bias | -0.01 | 0.01 | -0.00 | -0.11 | -0.07 | 0.07 | 0.11
sd | 0.25 | 2.63 | 4.37 | 3.20 | 0.66 | 4.44 | 2.04
rmse | 0.25 | 2.63 | 4.37 | 3.20 | 0.66 | 4.44 | 2.04
true | 3.49 | 68.09 | 52.45 | 47.73 | 4.72 | 47.73 | 4.72
relr | 0.07 | 0.04 | 0.08 | 0.07 | 0.14 | 0.09 | 0.43
B. Difference-in-Differences
|  $N$=1,000
bias | -0.27 | -8.91 | 14.42 | 11.46 | -1.49 | 15.91 | 2.96
sd | 0.46 | 2.62 | 2.58 | 2.62 | 0.47 | 2.61 | 0.47
rmse | 0.53 | 9.29 | 14.65 | 11.76 | 1.56 | 16.12 | 2.99
true | 3.49 | 68.09 | 52.42 | 47.70 | 4.72 | 47.70 | 4.72
relr | 0.15 | 0.14 | 0.28 | 0.25 | 0.33 | 0.34 | 0.63
|  $N$=4,000
bias | -0.28 | -8.79 | 14.51 | 11.57 | -1.51 | 16.02 | 2.94
sd | 0.24 | 1.28 | 1.26 | 1.28 | 0.25 | 1.27 | 0.23
rmse | 0.37 | 8.88 | 14.57 | 11.64 | 1.53 | 16.07 | 2.95
true | 3.49 | 68.09 | 52.45 | 47.73 | 4.72 | 47.73 | 4.72
relr | 0.11 | 0.13 | 0.28 | 0.24 | 0.32 | 0.34 | 0.62

Note: ‘bias’, ‘sd’, and ‘rmse’ provide the bias, standard deviation, and root
mean squared error of the respective estimator. ‘true’ and ‘relr’ are the
respective true effect as well as the root mean squared error relative to the
true effect.

Table 2 provides the results for the exponential outcome model, in which the
time trend is heterogeneous and interacts with $U$ through the nonlinear link
function. While the change-in-changes assumptions hold (Panel A.), average
time trends are heterogeneous across complier types such that the difference-
in-differences approach (Panel B.) of Deuchert, Huber, and Schelker (2017) is
inconsistent. Accordingly, the biases of the change-in-changes estimates
generally approach zero as the sample size increases, while this is not the
case for the difference-in-differences estimates. Change-in-changes yields a
lower root mean squared error than the respective difference-in-differences
estimator in all but one case (namely $\hat{\delta}_{1}^{c}(0)$ with
$N=1,000$) and its relative attractiveness increases in the sample size due to
its lower bias.

Table 3:  Nonlinear model with non-random treatment

| $\hat{\theta}_{1}^{0,1}$ | $\hat{\theta}_{1}^{1,0}$ | $\hat{\Delta}_{c}$ | $\hat{\theta}_{1}^{c}(1)$ | $\hat{\theta}_{1}^{c}(0)$ | $\hat{\delta}_{1}^{c}(1)$ | $\hat{\delta}_{1}^{c}(0)$
---|---|---|---|---|---|---|---
A. Change-in-Changes
|  $N$=1,000
bias | 0.02 | 0.13 | 47.21 | 40.19 | -1.44 | 48.64 | 7.02
sd | 0.71 | 4.56 | 5.45 | 4.11 | 0.75 | 5.53 | 2.92
rmse | 0.71 | 4.56 | 47.52 | 40.40 | 1.62 | 48.96 | 7.60
true | 4.41 | 54.19 | 52.42 | 47.70 | 4.72 | 47.70 | 4.72
relr | 0.16 | 0.08 | 0.91 | 0.85 | 0.34 | 1.03 | 1.61
|  $N$=4,000
bias | -0.00 | 0.06 | 47.38 | 40.13 | -1.53 | 48.91 | 7.25
sd | 0.38 | 2.35 | 2.84 | 2.04 | 0.38 | 2.86 | 1.51
rmse | 0.38 | 2.35 | 47.47 | 40.18 | 1.57 | 48.99 | 7.40
true | 4.40 | 54.18 | 52.45 | 47.73 | 4.72 | 47.73 | 4.72
relr | 0.09 | 0.04 | 0.90 | 0.84 | 0.33 | 1.03 | 1.57
B. Difference-in-Differences
|  $N$=1,000
bias | 0.35 | 19.98 | 29.00 | 27.65 | 0.04 | 28.96 | 1.35
sd | 0.67 | 2.48 | 2.46 | 2.48 | 0.67 | 2.51 | 0.45
rmse | 0.75 | 20.14 | 29.11 | 27.76 | 0.67 | 29.07 | 1.43
true | 4.41 | 54.19 | 52.42 | 47.70 | 4.72 | 47.70 | 4.72
relr | 0.17 | 0.37 | 0.56 | 0.58 | 0.14 | 0.61 | 0.30
|  $N$=4,000
bias | 0.34 | 20.02 | 28.98 | 27.65 | 0.02 | 28.96 | 1.33
sd | 0.35 | 1.22 | 1.19 | 1.22 | 0.35 | 1.24 | 0.23
rmse | 0.49 | 20.06 | 29.01 | 27.68 | 0.35 | 28.99 | 1.35
true | 4.40 | 54.18 | 52.45 | 47.73 | 4.72 | 47.73 | 4.72
relr | 0.11 | 0.37 | 0.55 | 0.58 | 0.07 | 0.61 | 0.29

Note: ‘bias’, ‘sd’, and ‘rmse’ provide the bias, standard deviation, and root
mean squared error of the respective estimator. ‘true’ and ‘relr’ are the
respective true effect as well as the root mean squared error relative to the
true effect.

In our final simulation design, we maintain the exponential outcome model but
assume $D$ to be selective w.r.t. $U$ rather than random. To this end, the
treatment model in (4) is replaced by $D=I\\{U+Q>0\\}$, with the independent
variable $Q\sim N(0,1)$ being an unobserved term. Under this violation of
Assumption 7, complier shares and effects are no longer identified, which is
confirmed by the simulation results presented in Table 3. The bias in the
change-in-changes based total, direct, and indirect effects on compliers do
not vanish as the sample size increases. Furthermore, under non-random
assignment of $D$ (while maintaining monotonicity of $M$ in $D$), the never-
takers’ and always-takers’ respective distributions of $U$ differ across
treatment. Therefore, average direct effects among the total of never or
always-takers, respectively, are not identified. Yet, $\theta_{1}^{1,0}$,
which is still identified by the same estimator as before, yields the direct
effect among never-takers with $D=1$ (as defiers do not exist). Likewise,
$\theta_{1}^{0,1}$ corresponds to the direct effect on always-takers with
$D=0$. Indeed, the results in Table 3 suggest that both parameters are
consistently estimated with the change-in-changes model (Panel A.).

##  5 Application

Our empirical application is based on the JOBS II data by Vinokur and Price
(1999). JOBS II was a randomized job training intervention in the US, designed
to analyse the impact of job training on labour market and mental health
outcomes, see Vinokur, Price, and Schul (1995). It was a modified version of
the earlier JOBS programme, which had been found to improve labour market
outcomes such as job satisfaction, motivation, earnings, and job stability,
see Caplan, Vinokur, Price, and van Ryn (1989) and Vinokur, van Ryn, Gramlich,
and Price (1991), as well as mental health, see Vinokur, Price, and Caplan
(1991). According to the results of Vinokur, Price, and Schul (1995), the JOBS
II programme increased reemployment rates and improved mental health outcomes,
especially for participants having an elevated risk of depression. The JOBS
interventions had an important impact in the academic literature (see e.g.
Wanberg, 2012, Liu, Huang, and Wang, 2014) and the methodology was implemented
in field experiments in Finland (Vuori, Silvonen, Vinokur, and Price, 2002,
Vuori and Silvonen, 2005) and the Netherlands (Brenninkmeijer and Blonk,
2011), suggesting positive effects on labour market integration in either
case.

The JOBS II intervention was conducted in south-eastern Michigan, where 2,464
job seekers were eligible to participate in a randomized field experiment, see
Vinokur and Price (1999). In a baseline period prior to programme assignment,
individuals responded to a screening questionnaire that collected pre-
treatment information on mental health. Based on the latter, individuals were
classified as having either a high or low depression risk and those with a
high risk were oversampled before the training was randomly assigned.111In the
JOBS II intervention, randomization was followed by yet another questionnaire
sent out two weeks before the actual job training, see Vinokur, Price, and
Schul (1995), which also provided information on whether an individual had
been assigned the training. Consequently, the data collected in that
questionnaire must be considered post-treatment as they could be affected by
learning the assignment. Therefore, we rely on the earlier screening data as
the relevant pre-treatment period prior to random programme assignment. The
job training consisted of five 4-hours seminars conducted in morning sessions
during one week between March 1 and August 7, 1991. Members of the treatment
group who participated in at least four of the five sessions received USD 20.
Each of the standardized training sessions consisted, among other aspects, of
the learning and practicing of job search and problem-solving skills.222When
compared to the earlier JOBS programme (Caplan, Vinokur, Price, and van Ryn,
1989), the job training sessions of JOBS II focused more strongly on building
a sense of mastery, personal control and self- efficacy in job search.
Previous research had suggested that an increase in this sense of mastery,
control and self-efficacy improved observed effort in job search behaviour
(Eden and Aviram, 1993). Results in Marshall and Lang (1990) suggest that
mastery is a strong predictor of depression symptoms among women. For a
detailed discussion of the literature, the exact sampling process, the
training programme, and further aspects, see Vinokur, Price, and Schul (1995).
The control group received a booklet with information on job search methods
(Vinokur, Price, and Schul, 1995, p. 44-49).

We analyse the impact of job training on mental health, namely symptoms of
depression 6 months after training participation. The health outcome ($Y$) is
based on a 11-items index of depression symptoms of the Hopkins Symptom
Checklist. For example, respondents were asked how much they were bothered by
symptoms such as crying easily, feeling lonely, feeling blue, feeling
hopeless, having thoughts of ending their lives, or experiencing a loss of
sexual interest. The questions were coded on a 5-point scale, going from ‘not
at all’ (1) to ‘extremely’ (5), and summarized in a depression variable that
consists of the average across all questions.

One-sided non-compliance with the random assignment is a major issue in JOBS
II. While the study design rules out always-takers because members of the
control group did not have access to the job training programme, 45% of those
assigned to training in our data did not participate and are therefore never-
takers, the remaining 55% are compliers. In order to avoid selection bias
w.r.t actual participation, the original JOBS II study by Vinokur, Price, and
Schul (1995) analysed the total effect of the policy (i.e. the intention-to-
treat effect), including those who, despite receiving an offer to participate,
did not take part in the job training. In contrast, we use our methodology to
separate the direct effect of mere training assignment, which is our treatment
$D$, from the indirect effect operating through actual training participation,
which is our mediator $M$, among compliers.333Imai, Keele, and Tingley (2010)
analyse Jobs II in a mediation context as well, but consider a different
mediator, namely job search self-efficacy, and a different identification
strategy based on selection on observables. We also consider the direct effect
on never-takers, which likely differs from that on the compliers. While being
offered (or not offered) the job training might affect compliers’ mental
health by inducing motivation/enthusiasm (or discouragement), it may not have
the same effect among never-takers, who do not attend such seminars
whatsoever.

More concisely, we base identification on Theorem 3a with Assumption 4a for
the average direct effect on never-takers, $\theta_{1}^{n}$, on Theorem 3b
with Assumption 4 for the direct effect on compliers under $d=0$,
$\theta_{1}^{c}(0)$, and Theorem 5 with Assumptions 4b and 6a for the indirect
effect on compliers under $d=1$, $\delta_{1}^{c}(1)$. None of these approaches
requires the presence of always-takers in the sample. We also note that if
random assignment operated through other mechanisms than actual participation
in any of the subpopulations as it may appear reasonable in the context of
mental health outcomes, this would violate the exclusion restriction when
using assignment as instrumental variable for actual participation in a two
stage least squares regression. Given that our identifying assumptions hold,
our approach can therefore be used to statistically test the exclusion
restriction.

Table 4: Descriptive statistics on outcomes in pre- and post-mediator periods

| pre-treatment ($T=0$) | post-mediator ($T=1$)
---|---|---
| sample size | mean | sample size | mean
overall | 1,796 | 1.86 | 1,564 | 1.73
|  | (0.58) |  | (0.67)
$D=0$ | 551 | 1.87 | 486 | 1.78
|  | (0.59) |  | (0.70)
$D=1$ | 1,245 | 1.86 | 1,078 | 1.70
|  | (0.57) |  | (0.66)
mean diff |  | 0.01 |  | 0.08
pval |  | 0.74 |  | 0.03
SD |  | 1.63 |  | 11.79

Note: Standard deviations are in parentheses. ‘mean diff’, ‘pval’, and ‘SD’
are the mean difference, its p-value, and the standardized difference,
respectively.

Our evaluation sample consists of a total of 3,360 observations in the pre-
treatment and post-mediator periods with non-missing information for $D$, $M$,
and $Y$. It is an unbalanced panel due to attrition of roughly 13% of the
initial respondents between the two periods. Table 4 provides summary
statistics for the outcome in the total sample as well as by treatment group
over time. We verify whether randomization was successful by comparing the
outcome means of the treatment and control groups in the pre-treatment period
($T=0$) just prior to the randomization of $D$. The small difference of 0.01
is not statistically significant according to a two sample t-test.
Furthermore, the standardized difference test suggested by Rosenbaum and Rubin
(1985) yields a value of just 1.68 and is thus far below 20, a threshold
frequently chosen for indicating problematic imbalances across treatment
groups. To test for potential attrition bias we also consider these statistics
in the pre-treatment period exclusively among the panel cases that remain in
the sample in the post-mediator period (not reported in Table 4). The p-value
of the t-test amounts to 0.52 and the standardized difference of 3.5 is low
such that attrition bias does not appear to be a concern. We therefore do not
find statistical evidence for a violation of the random assignment of $D$ in
our sample. Table 4 also reports the mean difference in outcomes in the post-
mediator period ($T=1$) 6 months after participation, which is an estimate for
the total (or intention-to-treat) effect of $D$. The difference of 0.08 is
statistically significant at the 5% level.

Table 5:  Empirical results for Jobs II

| Changes-in-Changes | Difference-in-Differences | Type shares
---|---|---|---
| $\hat{\theta}_{1}^{n}$ | $\hat{\Delta}_{c}$ | $\hat{\theta}_{1}^{c}(0)$ | $\hat{\delta}_{1}^{c}(1)$ | $\hat{\theta}_{1}^{n}$ | $\hat{\Delta}_{c}$ | $\hat{\theta}_{1}^{c}(0)$ | $\hat{\delta}_{1}^{c}(1)$ | $\hat{p}(n)$ | $\hat{p}(c)$
est | -0.04 | -0.11 | 0.06 | -0.17 | -0.03 | -0.12 | -0.06 | -0.06 | 0.45 | 0.55
se | 0.05 | 0.06 | 0.05 | 0.08 | 0.05 | 0.06 | 0.05 | 0.07 | 0.01 | 0.01
pval | 0.40 | 0.06 | 0.26 | 0.04 | 0.52 | 0.03 | 0.21 | 0.43 | 0.00 | 0.00

Note: ‘est’, ‘se’, and ‘pval’ provide the effect estimate, standard error, and
p-value of the respective estimator. $\hat{p}(n)$ and $\hat{p}(c)$ are the
estimated never-taker and complier shares. Standard errors are based on
cluster bootstrapping the effects 1999 times where clustering is on the
respondent level.

Table 5 presents the estimation results based on our CiC approach and the DiD
strategy of Deuchert, Huber, and Schelker (2017) when (linearly) controlling
for the gender of respondents in either case. Standard errors rely on cluster
bootstrapping the direct and indirect effects 1999 times, where clustering is
on the respondent level. The CiC and DiD estimates of the direct effects on
never-takers, $\hat{\theta}_{1}^{n}(0)$, as well as on compliers,
$\hat{\theta}_{1}^{c}(0)$, are not statistically significant at conventional
levels. Hence, we do not find statistical evidence for a direct effect of the
mere assignment into the training programme on the depression outcome, which
would point to a violation of the exclusion restriction when using assignment
as instrument for participation. In contrast, we find for both CiC and DiD
negative total effects among compliers $\hat{\Delta}_{c}$ that are
statistically significant at least at the 10% level. In the case of CiC, also
the negative indirect effect among compliers, $\hat{\delta}_{1}^{c}(1)$, is
significant at the 5% level, while this is not in the case for DiD. By and
large, our results point to a moderately negative treatment effect on
depressive symptoms through actual programme participation, rather than
through other (i.e. direct) mechanisms. The CiC estimates
$\hat{\delta}_{1}^{c}(1)$ and $\hat{\Delta}_{c}$ are in fact rather similar to
the result of a two stage least squares regression relying on the exclusion
restriction by using $D$ as instrument for $M$. The latter approach yields a
local average treatment effect on compliers in the post-mediator period of
-0.14 with a heteroskedasticity-robust standard error of 0.07 (significant at
the 5% level).

##  6 Conclusion

We proposed a novel identification strategy for causal mediation analysis with
repeated cross sections or panel data based on changes-in-changes (CiC)
assumptions that are related but yet different to Athey and Imbens (2006)
considering total treatment effects. Strict monotonicity of outcomes in
unobserved heterogeneity and distributional time invariance of the latter
within groups defined on treatment and mediator states are key assumptions for
identifying direct effects within these groups. Additionally assuming random
treatment assignment and weak monotonicity of the mediator in the treatment
permits identifying direct effects on never-takers and always-takers as well
as total, direct, and indirect effects on compliers. We also provided a brief
simulation study and an empirical application to the Jobs II programme.

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Appendices

##  A Proof of Theorem 1

###  A.1 Average direct effect under $\mathbf{d=1}$ conditional on
$\mathbf{D=1}$ and $\mathbf{M(1)=0}$

In the following, we prove that
$\theta_{1}^{1,0}(1)=E[Y_{1}(1,0)-Y_{1}(0,0)|D=1,M_{i}(1)=0]=E[Y_{1}-Q_{00}(Y_{0})|D=1,M=0]$.
Using the observational rule, we obtain
$E[Y_{1}(1,0)|D=1,M(1)=0]=E[Y_{1}|D=1,M=0]$. Accordingly, we have to show that
$E[Y_{1}(0,0)|D=1,M(1)=0]=E[Q_{00}(Y_{0})|D=1,M=0]$ to finish the proof.

Denote the inverse of $h(d,m,t,u)$ by $h^{-1}(d,m,t;y)$, which exists because
of the strict monotonicity required in Assumption 1. Under Assumptions 1 and
3a, the conditional potential outcome distribution function equals

$\begin{array}[]{rl}F_{Y_{t}(d,0)|D=1,M=0}(y)&\stackrel{{\scriptstyle
A1}}{{=}}\Pr(h(d,m,t,U)\leq y|D=1,M=0,T=t),\\\ &=\Pr(U\leq
h^{-1}(d,m,t;y)|D=1,M=0,T=t),\\\ &\stackrel{{\scriptstyle A3a}}{{=}}\Pr(U\leq
h^{-1}(d,m,t;y)|D=1,M=0),\\\ &=F_{U|10}(h^{-1}(d,m,t;y)),\end{array}$ (A.1)

for $d,d^{\prime}\in\\{0,1\\}$. We use these quantities in the following.

First, evaluating $F_{Y_{1}(0,0)|D=1,M=0}(y)$ at $h(0,0,1,u)$ gives

$F_{Y_{1}(0,0)|D=1,M=0}(h(0,0,1,u))=F_{U|10}(h^{-1}(0,0,1;h(0,0,1,u)))=F_{U|10}(u).$

Applying $F_{Y_{1}(0,0)|D=1,M=0}^{-1}(q)$ to both sides, we have

$h(0,0,1,u)=F_{Y_{1}(0,0)|D=1,M=0}^{-1}(F_{U|10}(u)).$ (A.2)

Second, for $F_{Y_{0}(0,0)|D=1,M=0}(y)$ we have

$F_{U|D=1,M=0}^{-1}(F_{Y_{0}(0,0)|D=1,M=0}(y))=h^{-1}(0,0,0;y).$ (A.3)

Combining (A.2) and (A.3) yields,

$h(0,0,1,h^{-1}(0,0,0;y))=F_{Y_{1}(0,0)|D=1,M=0}^{-1}\circ
F_{Y_{0}(0,0)|D=1,M=0}(y).$ (A.4)

Note that $h(0,0,1,h^{-1}(0,0,0;y))$ maps the period 1 (potential) outcome of
an individual with the outcome $y$ in period 0 under non-treatment without the
mediator. Accordingly, $E[F_{Y_{1}(0,0)|D=1,M=0}^{-1}\circ
F_{Y_{0}(0,0)|D=1,M=0}(Y_{0})|D=1,M=0]=E[Y_{1}(0,0)|D=1,M=0]$. We can identify
$F_{Y_{0}(0,0)|D=1,M=0}(y)$ under Assumption 2, but we cannot identify
$F_{Y_{1}(0,0)|D=1,M=0}(y)$. However, we show in the following that we can
identify the overall quantile-quantile transform
$F_{Y_{1}(0,0)|D=1,M=0}^{-1}\circ F_{Y_{0}(0,0)|D=1,M=0}(y)$ under the
additional Assumption 3b.

Under Assumptions 1 and 3b, the conditional potential outcome distribution
function equals

$\begin{array}[]{rl}F_{Y_{t}(d,0)|D=0,M=0}(y)&\stackrel{{\scriptstyle
A1}}{{=}}\Pr(h(d,m,t,U)\leq y|D=0,M=0,T=t),\\\ &=\Pr(U\leq
h^{-1}(d,m,t;y)|D=0,M=0,T=t),\\\ &\stackrel{{\scriptstyle A3b}}{{=}}\Pr(U\leq
h^{-1}(d,m,t;y)|D=0,M=0),\\\ &=F_{U|00}(h^{-1}(d,m,t;y)),\end{array}$ (A.5)

for $d,d^{\prime}\in\\{0,1\\}$. We repeat similar steps as above. First,
evaluating $F_{Y_{1}(0,0)|D=0,M=0}(y)$ at $h(0,0,1,u)$ gives

$F_{Y_{1}(0,0)D=0,M=0}(h(0,0,1,u))=F_{U|00}(h^{-1}(0,0,1;h(0,0,1,u)))=F_{U|00}(u).$

Applying $F_{Y_{1}(0,0)|D=0,M=0}^{-1}(q)$ to both sides, we have

$h(0,0,1,u)=F_{Y_{1}(0,0)|D=0,M=0}^{-1}(F_{U|00}(u)).$ (A.6)

Second, for $F_{Y_{0}(0,0)|D=0,M=0}(y)$ we have

$F_{U|00}^{-1}(F_{Y_{0}(0,0)|D=0,M=0}(y))=h^{-1}(0,0,0;y).$ (A.7)

Combining (A.6) and (A.7) yields,

$h(0,0,1,h^{-1}(0,0,0;y))=F_{Y_{1}(0,0)|D=0,M=0}^{-1}\circ
F_{Y_{0}(0,0)|D=0,M=0}(y).$ (A.8)

The left sides of (A.4) and (A.8) are equal. In contrast to (A.4), (A.8)
contains only distributions that can be identified from observable data. In
particular, $F_{Y_{t}(0,0)|D=0,M=0}(y)=\Pr(Y_{t}(0,0)\leq
y|D=0,M=0)=\Pr(Y_{t}\leq y|D=0,M=0)$. Accordingly, we can identify
$F_{Y_{1}(0,0)|D=1,M=0}^{-1}\circ F_{Y_{0}(0,0)|D=1,M=0}(y)$ by
$Q_{00}(y)\equiv F_{Y_{1}|D=0,M=0}^{-1}\circ F_{Y_{0}|D=0,M=0}(y)$.

Parsing $Y_{0}$ through $Q_{00}(\cdot)$ in the treated group without mediator
gives

$\begin{array}[]{rl}&E[Q_{00}(Y_{0})|D=1,M=0]\\\
&\qquad=E[F_{Y_{1}|D=0,M=0}^{-1}\circ F_{Y_{0}|D=0,M=0}(Y_{0})|D=1,M=0],\\\
&\qquad=E[F_{Y_{1}(0,0)|D=0,M=0}^{-1}\circ
F_{Y_{0}(0,0)|D=0,M=0}(Y_{0}(1,0))|D=1,M=0],\\\ &\qquad\stackrel{{\scriptstyle
A1,A3b}}{{=}}E[h(0,0,1,h^{-1}(0,0,0;Y_{0}(1,0)))|D=1,M=0],\\\
&\qquad\stackrel{{\scriptstyle
A2}}{{=}}E[h(0,0,1,h^{-1}(0,0,0;Y_{0}(0,0)))|D=1,M=0],\\\
&\qquad\stackrel{{\scriptstyle A1,A3a}}{{=}}E[F_{Y_{1}(0,0)|D=1,M=0}^{-1}\circ
F_{Y_{0}(0,0)|D=1,M=0}(Y_{0}(0,0))|D=1,M=0],\\\
&\qquad=E[Y_{1}(0,0)|D=1,M=0]=E[Y_{1}(0,0)|D=1,M(1)=0],\end{array}$ (A.9)

which has data support because of Assumption 4a.

###  A.2 Quantile direct effect under $\mathbf{d=1}$ conditional on
$\mathbf{D=1}$ and $\mathbf{M(1)=0}$

In the following, we prove that

$\displaystyle\theta_{1}^{1,0}(q,1)$
$\displaystyle=F_{Y_{1}(1,0)|D=1,M(1)=0}^{-1}(q)-F_{Y_{1}(0,0)|D=1,M(1)=0}^{-1}(q),$
$\displaystyle=F_{Y_{1}|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).$

For this purpose, we have to show that

$\displaystyle F_{Y_{1}(1,0)|D=1,M(1)=0}(y)$
$\displaystyle=F_{Y_{1}|D=1,M=0}(y)\mbox{ and}$ (A.10) $\displaystyle
F_{Y_{1}(0,0)|D=1,M(1)=0}(y)$ $\displaystyle=F_{Q_{00}(Y_{0})|D=1,M=0}(y),$
(A.11)

which is sufficient to show that the quantiles are also identified. We can
show (A.10) using the observational rule
$F_{Y_{1}(1,0)|D=1,M(1)=0}(y)=F_{Y_{1}|D=1,M=0}(y)=E[1\\{Y_{1}\leq
y\\}|D=1,M=0]$, with $1\\{\cdot\\}$ being the indicator function.

Using (A.9), we obtain

$\begin{array}[]{rl}&F_{Q_{00}(Y_{0})|D=1,M=0}(y)\\\
&\qquad=E[1\\{Q_{00}(Y_{0})\leq y\\}|D=1,M=0],\\\
&\qquad=E[1\\{F_{Y_{1}|D=0,M=0}^{-1}\circ F_{Y_{0}|D=0,M=0}(Y_{0})\leq
y\\}|D=1,M=0],\\\ &\qquad=E[1\\{Y_{1}(0,0)\leq y\\}|D=1,M=0],\\\
&\qquad=F_{Y_{1}(0,0)|D=1,M(1)=0}(y),\end{array}$ (A.12)

which proves (A.11).

###  A.3 Average direct effect under $\mathbf{d=0}$ conditional on
$\mathbf{D=0}$ and $\mathbf{M(0)=0}$

In the following, we show that
$\theta_{1}^{0,0}(0)=E[Y_{1}(1,0)-Y_{1}(0,0)|D=0,M(0)=0]=E[Q_{10}(Y_{0})-Y_{1}|D=0,M=0]$.
Using the observational rule, we obtain
$E[Y_{1}(0,0)|D=0,M(0)=0]=E[Y_{1}|D=0,M=0]$. Accordingly, we have to show that
$E[Y_{1}(1,0)|D=0,M(0)=0]=E[Q_{10}(Y_{0})|D=0,M=0]$ to finish the proof.

First, we use (A.5) to evaluate $F_{Y_{1}(1,0)|D=0,M=0}(y)$ at $h(1,0,1,u)$

$F_{Y_{1}(1,0)|D=0,M=0}(h(1,0,1,u))=F_{U|10}(h^{-1}(1,0,1;h(1,0,1,u)))=F_{U|10}(u).$

Applying $F_{Y_{1}(1,0)|D=0,M=0}^{-1}(q)$ to both sides, we have

$h(1,0,1,u)=F_{Y_{1}(1,0)|D=0,M=0}^{-1}(F_{U|10}(u)).$ (A.13)

Second, for $F_{Y_{0}(1,0)|D=0,M=0}(y)$ we have

$F_{U|10}^{-1}(F_{Y_{0}(1,0)|D=0,M=0}(y))=h^{-1}(1,0,0;y),$ (A.14)

using (A.5). Combining (A.13) and (A.14) yields,

$h(1,0,1,h^{-1}(1,0,0;y))=F_{Y_{1}(1,0)|D=0,M=0}^{-1}\circ
F_{Y_{0}(1,0)|D=0,M=0}(y).$ (A.15)

Note that $h(1,0,1,h^{-1}(1,0,0;y))$ maps the period 1 (potential) outcome of
an individual with the outcome $y$ in period 0 under treatment without the
mediator. Accordingly, $E[F_{Y_{1}(1,0)|D=0,M=0}^{-1}\circ
F_{Y_{0}(1,0)|D=0,M=0}(Y_{0})|D=0,M=0]=E[Y_{1}(1,0)|D=1,M=0]$. We can identify
$F_{Y_{0}(1,0)|D=0,M=0}(y)$ under Assumption 2, but we cannot identify
$F_{Y_{1}(1,0)|D=0,M=0}(y)$. However, we show in the following that we can
identify the overall quantile-quantile transform
$F_{Y_{1}(1,0)|D=0,M=0}^{-1}\circ F_{Y_{0}(1,0)|D=0,M=0}(y)$ under the
additional Assumption 3a.

First, we use (A.1) to evaluate $F_{Y_{1}(1,0)|D=1,M=0}(y)$ at $h(1,0,1,u)$

$F_{Y_{1}(1,0)|D=10,M=0}(h(1,0,1,u))=F_{U|10}(h^{-1}(1,0,1;h(1,0,1,u)))=F_{U|10}(u).$

Applying $F_{Y_{1}(1,0)|D=1,M=0}^{-1}(q)$ to both sides, we have

$h(1,0,1,u)=F_{Y_{1}(1,0)|D=1,M=0}^{-1}(F_{U|10}(u)).$ (A.16)

Second, for $F_{Y_{0}(1,0)|D=0,M=0}(y)$ we have

$F_{U|10}^{-1}(F_{Y_{0}(1,0)|D=1,M=0}(y))=h^{-1}(1,0,0;y),$ (A.17)

using (A.1). Combining (A.16) and (A.17) yields,

$h(1,0,1,h^{-1}(1,0,0;y))=F_{Y_{1}(1,0)|D=1,M=0}^{-1}\circ
F_{Y_{0}(1,0)|D=1,M=0}(y).$ (A.18)

The left sides of (A.15) and (A.18) are equal. In contrast to (A.15), (A.18)
contains only distributions that can be identified from observable data. In
particular, $F_{Y_{t}(1,0)|D=1,M=0}(y)=\Pr(Y_{t}(1,0)\leq
y|D=1,M=0)=\Pr(Y_{t}\leq y|D=1,M=0)$. Accordingly, we can identify
$F_{Y_{1}(1,0)|D=0,M=0}^{-1}\circ F_{Y_{0}(1,0)|D=0,M=0}(y)$ by
$Q_{10}(y)\equiv F_{Y_{1}|D=1,M=0}^{-1}\circ F_{Y_{0}|D=1,M=0}(y)$.

Parsing $Y_{0}$ through $Q_{10}(\cdot)$ in the non-treated group without
mediator gives

$\begin{array}[]{rl}&E[Q_{10}(Y_{0})|D=0,M=0]\\\
&\qquad=E[F_{Y_{1}|D=1,M=0}^{-1}\circ F_{Y_{0}|D=1,M=0}(Y_{0})|D=0,M=0],\\\
&\qquad=E[F_{Y_{1}(1,0)|D=1,M=0}^{-1}\circ
F_{Y_{0}(1,0)|D=1,M=0}(Y_{0}(0,0))|D=0,M=0],\\\ &\qquad\stackrel{{\scriptstyle
A1,A3a}}{{=}}E[h(1,0,1,h^{-1}(1,0,0;Y_{0}(0,0)))|D=0,M=0],\\\
&\qquad\stackrel{{\scriptstyle
A2}}{{=}}E[h(1,0,1,h^{-1}(1,0,0;Y_{0}(1,0)))|D=1,M=0],\\\
&\qquad\stackrel{{\scriptstyle A1,A3b}}{{=}}E[F_{Y_{1}(1,0)|D=0,M=0}^{-1}\circ
F_{Y_{0}(1,0)|D=0,M=0}(Y_{0}(1,0))|D=0,M=0],\\\
&\qquad=E[Y_{1}(1,0)|D=0,M=0]=E[Y_{1}(1,0)|D=0,M(0)=0],\end{array}$ (A.19)

which has data support because of Assumption 4b.

###  A.4 Quantile direct effect under $\mathbf{d=0}$ conditional on
$\mathbf{D=0}$ and $\mathbf{M(0)=0}$

In the following, we prove that

$\displaystyle\theta_{1}^{0,0}(q,0)$
$\displaystyle=F_{Y_{1}(1,0)|D=0,M(0)=0}^{-1}(q)-F_{Y_{1}(0,0)|D=0,M(0)=0}^{-1}(q),$
$\displaystyle=F_{Q_{10}(Y_{0})|D=0,M=0}^{-1}(q)-F_{Y_{1}|D=0,M=0}^{-1}(q).$

For this purpose, we have to show that

$\displaystyle F_{Y_{1}(1,0)|D=0,M(0)=0}(y)$
$\displaystyle=F_{Q_{10}(Y_{0})|D=0,M=0}(y)\mbox{ and}$ (A.20) $\displaystyle
F_{Y_{1}(0,0)|D=0,M(0)=0}(y)$ $\displaystyle=F_{Y_{1}|D=0,M=0}(y),$ (A.21)

which is sufficient to show that the quantiles are also identified. We can
show (A.21) using the observational rule
$F_{Y_{1}(0,0)|D=0,M(0)=0}(y)=F_{Y_{1}|D=0,M=0}(y)=E[1\\{Y_{1}\leq
y\\}|D=0,M=0]$.

Using (A.19), we obtain

$\begin{array}[]{rl}&F_{Q_{10}(Y_{0})|D=0,M=0}(y)\\\
&\qquad=E[1\\{Q_{10}(Y_{0})\leq y\\}|D=0,M=0],\\\
&\qquad=E[1\\{F_{Y_{1}|D=1,M=0}^{-1}\circ F_{Y_{0}|D=1,M=0}(Y_{0})\leq
y\\}|D=0,M=0],\\\ &\qquad=E[1\\{Y_{1}(1,0)\leq y\\}|D=0,M=0],\\\
&\qquad=F_{Y_{1}(1,0)|D=0,M(0)=0}(y),\end{array}$

which proves (A.20).

##  B Proof of Theorem 2

###  B.1 Average direct effect under $\mathbf{d=0}$ conditional on
$\mathbf{D=0}$ and $\mathbf{M(0)=1}$

In the following, we show that
$\theta_{1}^{0,1}(0)=E[Y_{1}(1,1)-Y_{1}(0,1)|D=0,M(0)=1]=E[Q_{11}(Y_{0})-Y_{1}|D=0,M=1]$.
Using the observational rule, we obtain
$E[Y_{1}(0,1)|D=0,M(0)=1]=E[Y_{1}|D=0,M=1]$. Accordingly, we have to show that
$E[Y_{1}(1,1)|D=0,M(0)=1]=E[Q_{11}(Y_{0})|D=0,M=1]$ to finish the proof.

Under Assumptions 1 and 5a, the conditional potential outcome distribution
function equals

$\begin{array}[]{rl}F_{Y_{t}(d,0)|D=1,M=0}(y)&\stackrel{{\scriptstyle
A1}}{{=}}\Pr(h(d,m,t,U)\leq y|D=0,M=1,T=t),\\\ &=\Pr(U\leq
h^{-1}(d,m,t;y)|D=0,M=1,T=t),\\\ &\stackrel{{\scriptstyle A5a}}{{=}}\Pr(U\leq
h^{-1}(d,m,t;y)|D=0,M=1),\\\ &=F_{U|01}(h^{-1}(d,m,t;y)),\end{array}$ (B.1)

for $d,d^{\prime}\in\\{0,1\\}$. We use these quantities in the following.

First, evaluating $F_{Y_{1}(1,1)|D=0,M=1}(y)$ at $h(1,1,1,u)$ gives

$F_{Y_{1}(1,1)|D=0,M=1}(h(1,1,1,u))=F_{U|01}(h^{-1}(1,1,1;h(1,1,1,u)))=F_{U|01}(u).$

Applying $F_{Y_{1}(1,1)|D=0,M=1}^{-1}(q)$ to both sides, we have

$h(1,1,1,u)=F_{Y_{1}(1,1)|D=0,M=1}^{-1}(F_{U|01}(u)).$ (B.2)

Second, for $F_{Y_{0}(1,1)|D=0,M=1}(y)$ we have

$F_{U|01}^{-1}(F_{Y_{0}(1,1)|D=0,M=1}(y))=h^{-1}(1,1,0;y).$ (B.3)

Combining (B.2) and (B.3) yields,

$h(1,1,1,h^{-1}(1,1,0;y))=F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ
F_{Y_{0}(1,1)|D=0,M=1}(y).$ (B.4)

Note that $h(1,1,1,h^{-1}(1,1,0;y))$ maps the period 1 (potential) outcome of
an individual with the outcome $y$ in period 0 under treatment with the
mediator. Accordingly, $E[F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ
F_{Y_{0}(1,1)|D=0,M=1}(Y_{0})|D=0,M=1]=E[Y_{1}(1,1)|D=0,M=1]$. We can identify
$F_{Y_{0}(1,1)|D=0,M=1}(y)=F_{Y_{0}|D=0,M=1}(y)$ under Assumption 2, but we
cannot identify $F_{Y_{1}(1,1)|D=0,M=1}(y)$. However, we show in the following
that we can identify the overall quantile-quantile transform
$F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ F_{Y_{0}(1,1)|D=0,M=1}(y)$ under the
additional Assumption 5b.

Under Assumptions 1 and 5b, the conditional potential outcome distribution
function equals

$\begin{array}[]{rl}F_{Y_{t}(d,1)|D=1,M=1}(y)&\stackrel{{\scriptstyle
A1}}{{=}}\Pr(h(d,m,t,U)\leq y|D=1,M=1,T=t),\\\ &=\Pr(U\leq
h^{-1}(d,m,t;y)|D=1,M=1,T=t),\\\ &\stackrel{{\scriptstyle A5b}}{{=}}\Pr(U\leq
h^{-1}(d,m,t;y)|D=1,M=1),\\\ &=F_{U|11}(h^{-1}(d,m,t;y)),\end{array}$ (B.5)

for $d,d^{\prime}\in\\{0,1\\}$. We repeat similar steps as above. First,
evaluating $F_{Y_{1}(1,1)|D=1,M=1}(y)$ at $h(1,1,1,u)$ gives

$F_{Y_{1}(1,1)|D=1,M=1}(h(1,1,1,u))=F_{U|11}(h^{-1}(1,1,1;h(1,1,1,u)))=F_{U|11}(u).$

Applying $F_{Y_{1}(1,1)|D=1,M=1}^{-1}(q)$ to both sides, we have

$h(1,1,1,u)=F_{Y_{1}(1,1)|D=1,M=1}^{-1}(F_{U|11}(u)).$ (B.6)

Second, for $F_{Y_{0}(1,1)|D=1,M=1}(y)$ we have

$F_{U|11}^{-1}(F_{Y_{0}(1,1)|D=1,M=1}(y))=h^{-1}(1,1,1;y).$ (B.7)

Combining (B.6) and (B.7) yields,

$h(1,1,1,h^{-1}(1,1,0;y))=F_{Y_{1}(1,1)|D=1,M=1}^{-1}\circ
F_{Y_{0}(1,1)|D=1,M=1}(y).$ (B.8)

The left sides of (B.4) and (B.8) are equal. In contrast to (B.4), (B.8)
contains only distributions that can be identified from observable data. In
particular, $F_{Y_{t}(1,1)|D=1,M=1}(y)=\Pr(Y_{t}(1,1)\leq
y|D=1,M=1)=\Pr(Y_{t}\leq y|D=1,M=1)$. Accordingly, we can identify
$F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ F_{Y_{0}(1,1)|D=0,M=1}(y)$ by
$Q_{11}(y)\equiv F_{Y_{1}|D=1,M=1}^{-1}\circ F_{Y_{0}|D=1,M=1}(y)$.

Parsing $Y_{0}$ through $Q_{11}(\cdot)$ in the non-treated group with mediator
gives

$\begin{array}[]{rl}&E[Q_{11}(Y_{0})|D=0,M=1]\\\
&\qquad=E[F_{Y_{1}|D=1,M=1}^{-1}\circ F_{Y_{0}|D=1,M=1}(Y_{0})|D=0,M=1],\\\
&\qquad=E[F_{Y_{1}(1,1)|D=1,M=1}^{-1}\circ
F_{Y_{0}(1,1)|D=1,M=1}(Y_{0}(0,1))|D=0,M=1],\\\ &\qquad\stackrel{{\scriptstyle
A1,A5b}}{{=}}E[h(1,1,1,h^{-1}(1,1,0;Y_{0}(0,1)))|D=0,M=1],\\\
&\qquad\stackrel{{\scriptstyle
A2}}{{=}}E[h(1,1,1,h^{-1}(1,1,0;Y_{0}(0,0)))|D=0,M=1],\\\
&\qquad\stackrel{{\scriptstyle A1,A5a}}{{=}}E[F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ
F_{Y_{0}(1,1)|D=0,M=1}(Y_{0}(0,0))|D=0,M=1],\\\
&\qquad=E[Y_{1}(1,1)|D=0,M=1]=E[Y_{1}(1,1)|D=0,M(0)=1],\end{array}$ (B.9)

which has data support because of Assumption 6a.

###  B.2 Quantile direct effect under $\mathbf{d=0}$ conditional on
$\mathbf{D=0}$ and $\mathbf{M(0)=1}$

In the following, we show that

$\displaystyle\theta_{1}^{0,1}(q,0)$
$\displaystyle=F_{Y_{1}(1,1)|D=0,M(0)=1}^{-1}(q)-F_{Y_{1}(0,1)|D=0,M(0)=1}^{-1}(q),$
$\displaystyle=F_{Q_{11}(Y_{0})|D=0,M=1}^{-1}(q)-F_{Y_{1}|D=0,M=1}^{-1}(q).$

For this purpose, we have to prove that

$\displaystyle F_{Y_{1}(1,1)|D=0,M(0)=1}(y)$
$\displaystyle=F_{Q_{11}(Y_{0})|D=0,M=1}(y)\mbox{ and}$ (B.10) $\displaystyle
F_{Y_{1}(0,1)|D=0,M(0)=1}(y)$ $\displaystyle=F_{Y_{1}|D=0,M=1}(y),$ (B.11)

which is sufficient to show that the quantiles are also identified. We can
show (B.11) using the observational rule
$F_{Y_{1}(0,1)|D=0,M(0)=1}(y)=F_{Y_{1}|D=0,M=1}(y)=E[1\\{Y_{1}\leq
y\\}|D=0,M=1]$.

Using (B.9), we obtain

$\begin{array}[]{rl}&F_{Q_{11}(Y_{0})|D=0,M=1}(y)\\\
&\qquad=E[1\\{Q_{11}(Y_{0})\leq y\\}|D=0,M=1],\\\
&\qquad=E[1\\{F_{Y_{1}|D=1,M=1}^{-1}\circ F_{Y_{0}|D=1,M=1}(Y_{0})\leq
y\\}|D=0,M=1],\\\ &\qquad=E[1\\{Y_{1}(1,1)\leq y\\}|D=0,M=0],\\\
&\qquad=F_{Y_{1}(1,1)|D=0,M(0)=1}(y),\end{array}$ (B.12)

which proves (B.10).

###  B.3 Average direct effect under $\mathbf{d=1}$ conditional on
$\mathbf{D=1}$ and $\mathbf{M(1)=1}$

In the following, we show that
$\theta_{1}^{1,1}(1)=E[Y_{1}(1,1)-Y_{1}(0,1)|D=1,M(1)=1]=E[Y_{1}-Q_{01}(Y_{0})|D=1,M=1]$.
Using the observational rule, we obtain
$E[Y_{1}(1,1)|D=1,M(1)=1]=E[Y_{1}|D=1,M=1]$. Accordingly, we have to show that
$E[Y_{1}(0,1)|D=1,M(1)=1]=E[Q_{01}(Y_{0})|D=1,M=1]$ to finish the proof.

First, using (B.5) to evaluate $F_{Y_{1}(0,1)|D=1,M=1}(y)$ at $h(0,1,1,u)$
gives

$F_{Y_{1}(0,1)|D=1,M=1}(h(0,1,1,u))=F_{U|11}(h^{-1}(0,1,1;h(0,1,1,u)))=F_{U|11}(u).$

Applying $F_{Y_{1}(0,1)|D=1,M=1}^{-1}(q)$ to both sides, we have

$h(0,1,1,u)=F_{Y_{1}(0,1)|D=1,M=1}^{-1}(F_{U|11}(u)).$ (B.13)

Second, for $F_{Y_{0}(0,1)|D=0,M=1}(y)$ we obtain

$F_{U|11}^{-1}(F_{Y_{0}(0,1)|D=1,M=1}(y))=h^{-1}(0,1,0;y),$ (B.14)

using (B.5). Combining (B.13) and (B.14) yields,

$h(0,1,1,h^{-1}(0,1,0;y))=F_{Y_{1}(0,1)|D=1,M=1}^{-1}\circ
F_{Y_{0}(0,1)|D=1,M=1}(y).$ (B.15)

Note that $h(0,1,1,h^{-1}(0,1,0;y))$ maps the period 1 (potential) outcome of
an individual with the outcome $y$ in period 0 under non-treatment with the
mediator. Accordingly, $E[F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ
F_{Y_{0}(1,1)|D=0,M=1}(Y_{0})|D=0,M=1]=E[Y_{1}(1,1)|D=0,M=1]$. We can identify
$F_{Y_{0}(1,1)|D=0,M=1}(y)=F_{Y_{0}|D=0,M=1}(y)$ under Assumption 2, but we
cannot identify $F_{Y_{1}(1,1)|D=0,M=1}(y)$. However, we show in the following
that we can identify the overall quantile-quantile transform
$F_{Y_{1}(1,1)|D=0,M=1}^{-1}\circ F_{Y_{0}(1,1)|D=0,M=1}(y)$ under the
additional Assumption 5a.

First, using (B.1) to evaluate $F_{Y_{1}(0,1)|D=0,M=1}(y)$ at $h(0,1,1,u)$
gives

$F_{Y_{1}(0,1)|D=0,M=1}(h(0,1,1,u))=F_{U|01}(h^{-1}(0,1,1;h(0,1,1,u)))=F_{U|01}(u).$

Applying $F_{Y_{1}(0,1)|D=0,M=1}^{-1}(q)$ to both sides, we have

$h(0,1,1,u)=F_{Y_{1}(0,1)|D=0,M=1}^{-1}(F_{U|01}(u)).$ (B.16)

Second, for $F_{Y_{0}(0,1)|D=0,M=1}(y)$ we obtain

$F_{U|01}^{-1}(F_{Y_{0}(0,1)|D=0,M=1}(y))=h^{-1}(0,1,1;y),$ (B.17)

using (B.1). Combining (B.16) and (B.17) yields,

$h(0,1,1,h^{-1}(0,1,0;y))=F_{Y_{1}(0,1)|D=0,M=1}^{-1}\circ
F_{Y_{0}(0,1)|D=0,M=1}(y).$ (B.18)

The left sides of (B.15) and (B.18) are equal. In contrast to (B.15), (B.18)
contains only distributions that can be identified from observable data. In
particular, $F_{Y_{t}(0,1)|D=0,M=1}(y)=\Pr(Y_{t}(0,1)\leq
y|D=0,M=1)=\Pr(Y_{t}\leq y|D=0,M=1)$. Accordingly, we can identify
$F_{Y_{1}(0,1)|D=1,M=1}^{-1}\circ F_{Y_{0}(0,1)|D=1,M=1}(y)$ by
$Q_{01}(y)\equiv F_{Y_{1}|D=0,M=1}^{-1}\circ F_{Y_{0}|D=0,M=1}(y)$.

Parsing $Y_{0}$ through $Q_{01}(\cdot)$ in the treated group with mediator
gives

$\begin{array}[]{rl}&E[Q_{01}(Y_{0})|D=1,M=1]\\\
&\qquad=E[F_{Y_{1}|D=0,M=1}^{-1}\circ F_{Y_{0}|D=0,M=1}(Y_{0})|D=1,M=1],\\\
&\qquad=E[F_{Y_{1}(0,1)|D=0,M=1}^{-1}\circ
F_{Y_{0}(0,1)|D=0,M=1}(Y_{0}(1,1))|D=1,M=1],\\\ &\qquad\stackrel{{\scriptstyle
A1,A5a}}{{=}}E[h(0,1,1,h^{-1}(0,1,0;Y_{0}(1,1)))|D=1,M=1],\\\
&\qquad\stackrel{{\scriptstyle
A2}}{{=}}E[h(0,1,1,h^{-1}(0,1,0;Y_{0}(0,1)))|D=1,M=1],\\\
&\qquad\stackrel{{\scriptstyle A1,A5b}}{{=}}E[F_{Y_{1}(0,1)|D=1,M=1}^{-1}\circ
F_{Y_{0}(0,1)|D=1,M=1}(Y_{0}(0,1))|D=1,M=1],\\\
&\qquad=E[Y_{1}(0,1)|D=1,M=1]=E[Y_{1}(0,1)|D=1,M(1)=1],\end{array}$ (B.19)

which has data support under Assumption 6b.

###  B.4 Quantile direct effect under $\mathbf{d=1}$ conditional on
$\mathbf{D=1}$ and $\mathbf{M(1)=1}$

In the following, we show that

$\displaystyle\theta_{1}^{1,1}(q,1)$
$\displaystyle=F_{Y_{1}(1,1)|D=1,M(1)=1}^{-1}(q)-F_{Y_{1}(0,1)|D=1,M(1)=1}^{-1}(q),$
$\displaystyle=F_{Y_{1}|D=1,M=1}^{-1}(q)-F_{Q_{01}(Y_{0})|D=1,M=1}^{-1}(q).$

For this purpose, we have to prove that

$\displaystyle F_{Y_{1}(1,1)|D=1,M(1)=1}(y)$
$\displaystyle=F_{Y_{1}|D=1,M=1}(y)\mbox{ and}$ (B.20) $\displaystyle
F_{Y_{1}(0,1)|D=1,M(1)=1}(y)$ $\displaystyle=F_{Q_{01}(Y_{0})|D=1,M=1}(y),$
(B.21)

which is sufficient to show that the quantiles are also identified. We can
show (B.20) using the observational rule
$F_{Y_{1}(1,1)|D=1,M(1)=1}(y)=F_{Y_{1}|D=1,M=1}(y)=E[1\\{Y_{1}\leq
y\\}|D=1,M=1]$.

Using (B.19), we obtain

$\begin{array}[]{rl}&F_{Q_{01}(Y_{0})|D=1,M=1}(y)\\\
&\qquad=E[1\\{Q_{01}(Y_{0})\leq y\\}|D=1,M=1],\\\
&\qquad=E[1\\{F_{Y_{1}|D=0,M=1}^{-1}\circ F_{Y_{0}|D=0,M=1}(Y_{0})\leq
y\\}|D=1,M=1],\\\ &\qquad=E[1\\{Y_{1}(0,1)\leq y\\}|D=1,M=0],\\\
&\qquad=F_{Y_{1}(0,1)|D=1,M(1)=1}(y),\end{array}$

which proves (B.21).

##  C Proof of equations (1) and (2)

$\Delta_{1}=E[Y_{1}|D=1]-E[Y_{1}|D=0]$ and quantile treatment effect
$\Delta_{1}(q)=F_{Y_{1}|D=1}^{-1}(q)-F_{Y_{1}|D=0}^{-1}(q)$

The average total effect for the entire population is identified by,

$\displaystyle\Delta_{1}$ $\displaystyle=E[Y_{1}(1,M(1))]-E[Y_{1}(0,M(0))],$
$\displaystyle\stackrel{{\scriptstyle
A7}}{{=}}E[Y_{1}(1,M(1))|D=1]-E[Y_{1}(0,M(0))|D=0],$
$\displaystyle=E[Y_{1}|D=1]-E[Y_{1}|D=0],$

where the first equality is the definition of $\Delta_{1}$, the second
equality hold by Assumption 7, and the last equality holds by the
observational rule.

We define the conditional distribution $F_{Y_{1}|D=d}(y)=\Pr(Y_{1}\leq y|D=d)$
and $F_{Y_{1}|D=d}^{-1}(q)=\inf\\{y:F_{Y_{1}|D=d}(y)\geq q\\}$. We can show
the identification of the total QTE for the entire population
$\Delta_{1}(q)=F_{Y_{1}|D=1}^{-1}(q)-F_{Y_{1}|D=0}^{-1}(q)$ when we show that
$F_{Y_{1}(1,M(1))}(y)=F_{Y_{1}|D=1}(y)$ and
$F_{Y_{1}(0,M(0))}(y)=F_{Y_{1}|D=0}(y)$. Using Assumption 7 and the
observational rule gives,

$\displaystyle F_{Y_{1}(1,M(1))}(y)$ $\displaystyle=\Pr(Y_{1}(1,M(1))\leq y),$
$\displaystyle\stackrel{{\scriptstyle A7}}{{=}}\Pr(Y_{1}(1,M(1))\leq y|D=1),$
$\displaystyle=\Pr(Y_{1}\leq y|D=1)=F_{Y_{1}|D=1}(y),$

and

$\displaystyle F_{Y_{1}(0,M(0))}(y)$ $\displaystyle=\Pr(Y_{1}(0,M(0))\leq y),$
$\displaystyle\stackrel{{\scriptstyle A7}}{{=}}\Pr(Y_{1}(0,M(0))\leq y|D=0),$
$\displaystyle=\Pr(Y_{1}\leq y|D=0)=F_{Y_{1}|D=0}(y),$

which finishes the proof.

By Assumption 7, the share of a type $\tau$ conditional on $D$ corresponds to
$p_{\tau}$ (in the population), as $D$ is randomly assigned. This implies that
$p_{1|1}=p_{a}+p_{c}$, $p_{1|0}=p_{a}+p_{de}$, $p_{0|1}=p_{n}+p_{de}$, and
$p_{0|0}=p_{n}+p_{c}$. Under Assumption 8, $p_{de}=0$, which finishes the
proof of (1).

Furthermore,
$E[Y_{t}(d,m)|\tau,D=1]=E[Y_{t}(d,m)|\tau,D=0]=E[Y_{t}(d,m)|\tau]$ due to the
independence of $D$ and the potential outcomes as well as the types $\tau$
(which are a deterministic function of $M(d)$) under Assumption 7. It follows
that conditioning on $D$ is not required on the right hand side of the
following equation, which expresses the mean outcome conditional $D=0$ and
$M=0$ as weighted average of the mean potential outcomes of compliers and
never-takers:

$\begin{array}[]{rl}&E[Y_{t}|D=0,M=0]\\\
&\displaystyle\qquad=\frac{p_{n}}{p_{n}+p_{c}}E[Y_{t}(0,0)|\tau=n]+\frac{p_{c}}{p_{n}+p_{c}}E[Y_{t}(0,0)|\tau=c].\end{array}$
(C.1)

Only compliers and never-takers satisfy $M(0)=0$ and thus make up the group
with $D=0$ and $M=0$. After some rearrangements we obtain

$\begin{array}[]{rl}&E[Y_{t}(0,0)|\tau=n]-E[Y_{t}(0,0)|\tau=c]\\\
&\displaystyle\qquad=\frac{p_{n}+p_{c}}{p_{c}}\left\\{E[Y_{t}(0,0)|\tau=n]-E[Y_{t}|D=0,M=0]\right\\}.\end{array}$
(C.2)

Next, we consider observations with $D=1$ and $M=0$, which might consist of
both never-takers and defiers, as $M(1)=0$ for both types. However, by
Assumption 8, defiers are ruled out, such that the mean outcome given
$D_{1}=1$ and $M_{1}=0$ is determined by never-takers only:

$E[Y_{t}|D=1,M=0]\stackrel{{\scriptstyle A7,A8}}{{=}}E[Y_{t}(1,0)|\tau=n].$
(C.3)

Furthermore, by Assumption 2,

$E[Y_{0}(0,0)|\tau=n]\stackrel{{\scriptstyle
A2}}{{=}}E[Y_{0}(1,0)|\tau=n]\stackrel{{\scriptstyle
A7,A8}}{{=}}E[Y_{0}|D=1,M=0].$

Similarly to (C.1) for the never-takers and compliers, consider the mean
outcome given $Z=1$ and $D=1$, which is made up by always-takers and compliers
(the types with $M(1)=1$)

$\begin{array}[]{rl}&E[Y_{t}|D=1,M=1]\\\
&\displaystyle\qquad=\frac{p_{a}}{p_{a}+p_{c}}E[Y_{t}(1,1)|\tau=a]+\frac{p_{c}}{p_{a}+p_{c}}E[Y_{t}(1,1)|\tau=c].\end{array}$
(C.4)

After some rearrangements we obtain

$\begin{array}[]{rl}&E[Y_{t}(1,1)|\tau=a]-E[Y_{t}(1,1)|\tau=c]\\\
&\displaystyle\qquad=\frac{p_{a}+p_{c}}{p_{c}}\left\\{E[Y_{t}(1,1)|\tau=a]-E[Y_{t}|D=1,M=1]\right\\}.\end{array}$
(C.5)

By Assumptions 7 and 8,

$E[Y_{t}|D=0,M=1]=E[Y_{t}(0,1)|\tau=a].$ (C.6)

Now consider (C.5) for period $T=0$, and note that by Assumption 2,
$E[Y_{0}(1,1)|\tau=a]=E[Y_{0}(0,0)|\tau=a]=E[Y_{0}(0,1)|\tau=a]$ and
$E[Y_{0}(1,1)|\tau=c]=E[Y_{0}(0,0)|\tau=c]$.

Combining (C.4), (C.6), and the law of iterative expectations (LIE) gives

$\displaystyle E[Y_{0}|D=1]$ $\displaystyle\qquad\stackrel{{\scriptstyle
LIE}}{{=}}E[Y_{0}|D=1,M=1]\cdot p_{1|1}+E[Y_{0}|D=1,M=0]\cdot p_{0|1},$
$\displaystyle\qquad=E[Y_{0}(1,1)|\tau=c]\cdot p_{c}+E[Y_{0}(1,1)|\tau=a]\cdot
p_{a}+E[Y_{0}(1,0)|\tau=n]\cdot p_{n},$
$\displaystyle\qquad\stackrel{{\scriptstyle A2}}{{=}}E[Y_{0}(1,1)|\tau=c]\cdot
p_{c}+E[Y_{0}(1,1)|\tau=a]\cdot p_{a}+E[Y_{0}(0,0)|\tau=n]\cdot p_{n}.$

Likewise, combining (C.1) and (C.3) gives

$\displaystyle E[Y_{0}|D=0]$ $\displaystyle\qquad\stackrel{{\scriptstyle
LIE}}{{=}}E[Y_{0}|D=0,M=1]\cdot p_{1|0}+E[Y_{0}|D_{0}=1,M=0]\cdot p_{0|0},$
$\displaystyle\qquad=E[Y_{0}(0,1)|\tau=a]\cdot p_{a}+E[Y_{0}(0,0)|\tau=c]\cdot
p_{c}+E[Y_{0}(0,0)|\tau=n]\cdot p_{n},$
$\displaystyle\qquad\stackrel{{\scriptstyle A2}}{{=}}E[Y_{0}(1,1)|\tau=a]\cdot
p_{a}+E[Y_{0}(0,0)|\tau=c]\cdot p_{c}+E[Y_{0}(0,0)|\tau=n]\cdot p_{n}.$

Accordingly,

$\frac{E[Y_{0}|D=1]-E[Y_{0}|D=0]}{p_{1|1}-p_{1|0}}=E[Y_{0}(1,1)|\tau=c]-E[Y_{0}(0,0)|\tau=c]\stackrel{{\scriptstyle
A2}}{{=}}0,$

which proves (2). Accordingly, $E[Y_{0}|D=1]-E[Y_{0}|D=0]=0$ is a testable
implication of Assumption 2, 7, and 8.

##  D Proof of Theorem 3

###  D.1 Average direct effect on the never-takers

In the following, we show that
$\theta_{1}^{n}=E[Y_{1}(1,0)-Y_{1}(0,0)|\tau=n]=E[Y_{1}-Q_{00}(Y_{0})|D=1,M=0]$.
From (C.3), we obtain the first ingredient
$E[Y_{1}(1,0)|\tau=n]=E[Y_{1}|D=1,M=0]$. Furthermore, from (A.9) we have
$E[Q_{00}(Y_{0})|D=1,M=0]=E[Y_{1}(0,0)|D=1,M(1)=0]$. Under Assumption 7 and 8,

$\begin{array}[]{rl}E[Y_{1}(0,0)|D=1,M(1)=0]&\stackrel{{\scriptstyle
A7}}{{=}}E[Y_{1}(0,0)|D=1,\tau=n]\\\ &\stackrel{{\scriptstyle
A8}}{{=}}E[Y_{1}(0,0)|\tau=n].\end{array}$ (D.1)

###  D.2 Quantile direct effect on the never-takers

We prove that

$\displaystyle\theta_{1}^{n}(q)$
$\displaystyle=F_{Y_{1}(1,0)|\tau=n}^{-1}(q)-F_{Y_{1}(0,0)|\tau=n}^{-1}(q),$
$\displaystyle=F_{Y_{1}|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).$

This requires showing that

$\displaystyle F_{Y_{1}(1,0)|\tau=n}(y)$
$\displaystyle=F_{Y_{1}|D=1,M=0}(y)\mbox{ and}$ (D.2) $\displaystyle
F_{Y_{1}(0,0)|\tau=n}(y)$ $\displaystyle=F_{Q_{00}(Y_{0})|D=1,M=0}(y).$ (D.3)

Under Assumptions 7 and 8,

$\begin{array}[]{rl}F_{Y_{t}|D=1,M=0}(y)&=E[1\\{Y_{t}\leq y\\}|D=1,M=0]\\\
&\stackrel{{\scriptstyle A7,A8}}{{=}}E[1\\{Y_{t}(1,0)\leq y\\}|\tau=n]\\\
&=F_{Y_{t}(1,0)|\tau=n}(y),\end{array}$ (D.4)

which proves (D.2). From (A.12), we have

$F_{Q_{00}(Y_{0})|D=1,M=0}(y)=F_{Y_{1}(0,0)|D=1,M(1)=0}(y)=E[1\\{Y_{1}(0,0)\leq
y\\}|D=1,M(1)=0].$

Under Assumption 7 and 8,

$\begin{array}[]{rl}E[1\\{Y_{1}(0,0)\leq
y\\}|D=1,M(1)=0]&\stackrel{{\scriptstyle A7,A8}}{{=}}E[1\\{Y_{1}(0,0)\leq
y\\}|\tau=n]\\\ &=F_{Y_{1}(0,0)|\tau=n}(y),\end{array}$ (D.5)

which proves (D.3).

###  D.3 Average direct effect under $\mathbf{d=0}$ on compliers

In the following, we show that

$\displaystyle\displaystyle\theta_{1}^{c}(0)=$ $\displaystyle
E[Y_{1}(1,0)-Y_{1}(0,0)|\tau=c],$ $\displaystyle=$
$\displaystyle\frac{p_{0|0}}{p_{0|0}-p_{0|1}}E[Q_{10}(Y_{0})-Y_{1}|D=0,M=0]$
$\displaystyle-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}E[Y_{1}-Q_{00}(Y_{0})|D=1,M=0].$

Plugging (D.1) in (C.1) under $T=1$, we obtain

$\begin{array}[]{rl}E[Y_{1}|D=0,M=0]=&\displaystyle\frac{p_{n}}{p_{n}+p_{c}}E[Q_{00}(Y_{0})|D=1,M=0]\\\
&\displaystyle\quad+\frac{p_{c}}{p_{n}+p_{c}}E[Y_{1}(0,0)|\tau=c].\end{array}$

This allows identifying

$\begin{array}[]{rl}E[Y_{1}(0,0)|\tau=c]=&\displaystyle\frac{p_{0|0}}{p_{0|0}-p_{0|1}}E[Y_{1}|D=0,M=0]\\\
&\displaystyle\quad-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}E[Q_{00}(Y_{0})|D=1,M=0].\end{array}$
(D.6)

Accordingly, we have to show the identification of $E[Y_{1}(1,0)|c]$ to finish
the proof. From (A.19) we have
$E[Y_{1}(1,0)|D=0,M=0]=E[Q_{10}(Y_{0})|D=0,M=0]$. Applying the law of
iterative expectations, gives

$\displaystyle E[Y_{1}(1,0)|D=0,M=0]=$
$\displaystyle\frac{p_{n}}{p_{n}+p_{c}}E[Y_{1}(1,0)|D=0,M=0,\tau=n]$
$\displaystyle+\frac{p_{c}}{p_{n}+p_{c}}E[Y_{1}(1,0)|D=0,M=0,\tau=c],$
$\displaystyle\stackrel{{\scriptstyle A7}}{{=}}$
$\displaystyle\frac{p_{n}}{p_{n}+p_{c}}E[Y_{1}(1,0)|\tau=n]+\frac{p_{c}}{p_{n}+p_{c}}E[Y_{1}(1,0)|\tau=c].$

After some rearrangements and using (C.3), we obtain

$E[Y_{1}(1,0)|\tau=c]=\frac{p_{n}+p_{c}}{p_{c}}E[Q_{10}(Y_{0})|D=0,M=0]-\frac{p_{n}}{p_{c}}E[Y_{1}|D=1,M=0].$

This gives

$\begin{array}[]{rl}E[Y_{1}(1,0)|\tau=c]=&\displaystyle\frac{p_{0|0}}{p_{0|0}-p_{0|1}}E[Q_{10}(Y_{0})|D=0,M=0]\\\
&\displaystyle\quad-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}E[Y_{1}|D=1,M=0],\end{array}$
(D.7)

using $p_{n}=p_{0|1}$, and $p_{c}+p_{n}=p_{0|0}$.

###  D.4 Quantile direct effect under $\mathbf{d=0}$ on compliers

We show that

$\displaystyle F_{Y_{1}(1,0)|\tau=c}(y)$
$\displaystyle=\frac{p_{0|0}}{p_{0|0}-p_{0|1}}F_{Q_{10}(Y_{0})|D=0,M=0}(y)-\frac{p_{0|1}}{p_{0|0}-p_{0|1}c}F_{Y_{1}|D=1,M=0}(y)\mbox{
and}$ $\displaystyle F_{Y_{1}(0,0)|\tau=c}(y)$
$\displaystyle=\frac{p_{0|0}}{p_{0|0}-p_{0|1}}F_{Y_{1}|D=0,M=0}(y)-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y),$

which proves that
$\theta_{1}^{c}(q,0)=F_{Y_{1}(1,0)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is
identified.

From (A.20), we have
$F_{Y_{1}(1,0)|D=0,M(0)=0}(y)=F_{Q_{10}(Y_{0})|D=0,M=0}(y)$. Applying the law
of iterative expectations gives

$\displaystyle F_{Y_{1}(1,0)|D=0,M(0)=0}(y)=$
$\displaystyle\frac{p_{n}}{p_{n}+p_{c}}F_{Y_{1}(1,0)|D=0,M(0)=0,\tau=n}(y)$
$\displaystyle+\frac{p_{c}}{p_{n}+p_{c}}F_{Y_{1}(1,0)|D=0,M(0)=0,\tau=c}(y),$
$\displaystyle\stackrel{{\scriptstyle A7}}{{=}}$
$\displaystyle\frac{p_{n}}{p_{n}+p_{c}}F_{Y_{1}(1,0)|\tau=n}(y)+\frac{p_{c}}{p_{n}+p_{c}}F_{Y_{1}(1,0)|\tau=c}(y).$

Using (D.2) and rearranging the equation gives,

$F_{Y_{1}(1,0)|\tau=c}(y)=\frac{p_{0|0}}{p_{0|0}-p_{0|1}}F_{Q_{10}(Y_{0})|D=0,M=0}(y)-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}F_{Y_{1}|D=1,M=0}(y).$
(D.8)

In analogy to (C.1), the outcome distribution under $D=0$ and $M=0$ equals:

$F_{Y_{1}|D=0,M=0}(y)=\frac{p_{n}}{p_{n}+p_{c}}F_{Y_{1}(0,0)|\tau=n}(y)+\frac{p_{c}}{p_{n}+p_{c}}F_{Y_{1}(0,0)|\tau=c}(y).$

Using (D.3) and rearranging the equation gives

$F_{Y_{1}(0,0)|\tau=c}(y)=\frac{p_{0|0}}{p_{0|0}-p_{0|1}}F_{Y_{1}|D=0,M=0}(y)-\frac{p_{0|1}}{p_{0|0}-p_{0|1}}F_{Q_{00}(Y_{0})|D=1,M=0}(y).$
(D.9)

##  E Proof of Theorem 4

###  E.1 Average direct effect on the always-takers

In the following, we show that
$\theta_{1}^{a}=E[Y_{1}(1,1)-Y_{1}(0,1)|\tau=a]=E[Q_{11}(Y_{0})-Y_{1}|D=0,M=1]$.
From (C.6), we obtain the first ingredient $E[Y_{1}(0,1)|a]=E[Y_{1}|D=0,M=1]$.
Furthermore, from (B.9) we have
$E[Q_{11}(Y_{0})|D=0,M=1]=E[Y_{1}(1,1)|D=0,M(0)=1]$. Under Assumption 7 and 8,

$\begin{array}[]{rl}E[Y_{1}(1,1)|D=0,M(0)=1]&\stackrel{{\scriptstyle
A7}}{{=}}E[Y_{1}(1,1)|D=0,\tau=a]\\\ &\stackrel{{\scriptstyle
A8}}{{=}}E[Y_{1}(1,1)|\tau=a].\end{array}$ (E.1)

###  E.2 Quantile direct effect on the always-takers

We prove that

$\displaystyle\theta_{1}^{a}(q)$
$\displaystyle=F_{Y_{1}(1,1)|\tau=a}^{-1}(q)-F_{Y_{1}(0,1)|\tau=a}^{-1}(q),$
$\displaystyle=F_{Q_{11}(Y_{0})|D=0,M=1}^{-1}(q)-F_{Y_{1}|D=0,M=1}^{-1}(q).$

This requires showing that

$\displaystyle F_{Y_{1}(1,1)|\tau=a}(y)$
$\displaystyle=F_{Q_{11}(Y_{0})|D=0,M=1}(y)\mbox{ and}$ (E.2) $\displaystyle
F_{Y_{1}(0,1)|\tau=a}(y)$ $\displaystyle=F_{Y_{1}|D=0,M=1}(y).$ (E.3)

Under Assumptions 7 and 8,

$\begin{array}[]{rl}F_{Y_{t}|D=0,M=1}(y)&=E[1\\{Y_{t}\leq y\\}|D=0,M=1]\\\
&\stackrel{{\scriptstyle A7,A8}}{{=}}E[1\\{Y_{t}(0,1)\leq y\\}|\tau=a]\\\
&=F_{Y_{t}(0,1)|\tau=a},(y).\end{array}$ (E.4)

which proves (E.3). From (B.12), we have

$F_{Q_{11}(Y_{0})|D=0,M=1}(y)=F_{Y_{1}(1,1)|D=0,M(0)=1}(y)=E[1\\{Y_{1}(1,1)\leq
y\\}|D=0,M(0)=1].$

Under Assumption 7 and 8,

$\begin{array}[]{rl}E[1\\{Y_{1}(1,1)\leq
y\\}|D=0,M(0)=1]&\stackrel{{\scriptstyle A7,A8}}{{=}}E[1\\{Y_{1}(1,1)\leq
y\\}|\tau=a]\\\ &=F_{Y_{1}(1,1)|\tau=a}(y),\end{array}$ (E.5)

which proves (E.2).

###  E.3 Average direct effect under $\mathbf{d=1}$ on compliers

In the following, we show that

$\displaystyle\theta_{1}^{c}(1)=$ $\displaystyle
E[Y_{1}(1,1)-Y_{1}(0,1)|\tau=c],$ $\displaystyle=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Y_{1}-Q_{01}(Y_{0})|D=1,M=1]$
$\displaystyle-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_{0})-Y_{1}|D=0,M=1].$

Plugging (E.1) in (C.4), we obtain

$\begin{array}[]{rl}E[Y_{1}|D=1,M=1]=&\displaystyle\frac{p_{a}}{p_{a}+p_{c}}E[Q_{11}(Y_{0})|D=0,M=1]\\\
&\displaystyle\quad+\frac{p_{c}}{p_{a}+p_{c}}E[Y_{1}(1,1)|\tau=c].\end{array}$

This allows identifying

$\begin{array}[]{rl}E[Y_{1}(1,1)|\tau=c]=&\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=1,M=1]\\\
&\displaystyle\quad-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_{0})|D=0,M=1].\end{array}$
(E.6)

From (B.19) we have $E[Y_{1}(0,1)|D=1,M=1]=E[Q_{01}(Y_{0})|D=1,M=1]$. Applying
the law of iterative expectations, gives

$\displaystyle E[Y_{1}(0,1)|D=1,M=1]=$
$\displaystyle\frac{p_{a}}{p_{a}+p_{c}}E[Y_{1}(0,1)|D=1,M=1,\tau=a]$
$\displaystyle+\frac{p_{c}}{p_{a}+p_{c}}E[Y_{1}(0,1)|D=1,M=1,\tau=c],$
$\displaystyle\stackrel{{\scriptstyle A7}}{{=}}$
$\displaystyle\frac{p_{a}}{p_{a}+p_{c}}E[Y_{1}(0,1)|\tau=a]+\frac{p_{c}}{p_{a}+p_{c}}E[Y_{1}(0,1)|\tau=c].$

After some rearrangements and using (C.6), we obtain

$E[Y_{1}(0,1)|\tau=c]=\frac{p_{a}+p_{c}}{p_{c}}E[Q_{01}(Y_{0})|D=1,M=1]-\frac{p_{a}}{p_{c}}E[Y_{1}|D=0,M=1].$

This gives

$\begin{array}[]{rl}E[Y_{1}(0,1)|\tau=c]=&\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{01}(Y_{0})|D=1,M=1]\\\
&\displaystyle\quad-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=0,M=1],\end{array}$
(E.7)

with $p_{a}=p_{1|0}$, and $p_{c}+p_{a}=p_{1|1}$.

###  E.4 Quantile direct effect under $\mathbf{d=1}$ on compliers

We show that

$\displaystyle F_{Y_{1}(1,1)|\tau=c}(y)$
$\displaystyle=\frac{p_{1|1}}{p_{1|1}-p_{1|0}}F_{Y_{1}|D=1,M=1}(y)-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}F_{Q_{11}(Y_{0})|D=0,M=1}(y)\mbox{
and}$ $\displaystyle F_{Y_{1}(0,1)|\tau=c}(y)$
$\displaystyle=\frac{p_{1|1}}{p_{1|1}-p_{1|0}}F_{Q_{01}(Y_{0})|D=1,M=1}(y)-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}F_{Y_{1}|D=0,M=1}(y),$

which proves that
$\theta_{1}^{c}(q,1)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,1)|c}^{-1}(q)$ is
identified.

In analogy to (C.4), the outcome distribution under $D=0$ and $M=0$ equals:

$F_{Y_{1}|D=1,M=1}(y)=\frac{p_{a}}{p_{a}+p_{c}}F_{Y_{1}(1,1)|\tau=a}(y)+\frac{p_{c}}{p_{a}+p_{c}}F_{Y_{1}(1,1)|\tau=c}(y).$

Using (E.2) and rearranging the equation gives

$F_{Y_{1}(1,1)|\tau=c}(y)=\frac{p_{1|1}}{p_{1|1}-p_{1|0}}F_{Y_{1}|D=1,M=1}(y)-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}F_{Q_{11}(Y_{0})|D=0,M=1}(y).$
(E.8)

From (B.21), we have
$F_{Y_{1}(0,1)|D=1,M(1)=1}(y)=F_{Q_{01}(Y_{0})|D=1,M=1}(y)$. Applying the law
of iterative expectations gives

$\displaystyle F_{Y_{1}(0,1)|D=1,M(1)=1}(y)=$
$\displaystyle\frac{p_{a}}{p_{a}+p_{c}}F_{Y_{1}(0,1)|D=1,M(1)=1,\tau=a}(y)$
$\displaystyle+\frac{p_{c}}{p_{a}+p_{c}}F_{Y_{1}(0,1)|D=1,M(1)=1,\tau=c}(y),$
$\displaystyle\stackrel{{\scriptstyle A7}}{{=}}$
$\displaystyle\frac{p_{a}}{p_{a}+p_{c}}F_{Y_{1}(0,1)|\tau=a}(y)+\frac{p_{c}}{p_{a}+p_{c}}F_{Y_{1}(0,1)|\tau=c}(y).$

Using (E.3) and rearranging the equation gives,

$F_{Y_{1}(0,1)|\tau=c}(y)=\frac{p_{1|1}}{p_{1|1}-p_{1|0}}F_{Q_{01}(Y_{0})|D=1,M=1}(y)-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}F_{Y_{1}|D=0,M=1}(y).$
(E.9)

##  F Proof of Theorem 5

###  F.1 Average treatment effect on the compliers

In (E.6) and (D.6), we show that

$\displaystyle\theta_{1}^{c}=$ $\displaystyle
E[Y_{1}(1,1)-Y_{1}(0,0)|\tau=c],$ $\displaystyle=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=1,M=1]-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_{0})|D=0,M=1]$
$\displaystyle−\frac{p_{0|0}}{p_{0|0}-p_{0|1}}E[Y_{1}|D=0,M=0]+\frac{p_{0|1}}{p_{0|0}-p_{0|1}}E[Q_{00}(Y_{0})|D=1,M=0].$

###  F.2 Quantile treatment effect on the compliers

In (E.8) and (D.9), we show that $F_{Y_{1}(1,1)|c}(y)$ and
$F_{Y_{1}(0,0)|c}(y)$ are identified. Accordingly,
$\Delta_{1}^{c}(q)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is
identified.

###  F.3 Average indirect effect under $d=0$ on compliers

In (E.7) and (D.6), we show that

$\displaystyle\delta_{1}^{c}(0)=$ $\displaystyle
E[Y_{1}(0,1)-Y_{1}(0,0)|\tau=c],$ $\displaystyle=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{01}(Y_{0})|D=1,M=1]-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=0,M=1]$
$\displaystyle-\frac{p_{0|0}}{p_{0|0}-p_{0|1}}E[Y_{1}|D=0,M=0]+\frac{p_{0|1}}{p_{0|0}-p_{0|1}}E[Q_{00}(Y_{0})|D=1,M=0].$

###  F.4 Quantile indirect effect under $d=0$ on compliers

In (E.9) and (D.9), we show that $F_{Y_{1}(0,1)|c}(y)$ and
$F_{Y_{1}(0,0)|c}(y)$ are identified. Accordingly,
$\delta_{1}^{c}(q,0)=F_{Y_{1}(0,1)|c}^{-1}(q)-F_{Y_{1}(0,0)|c}^{-1}(q)$ is
identified.

###  F.5 Average indirect effect under $d=1$ on compliers

In (E.6) and (D.7), we show that

$\displaystyle\delta_{1}^{c}(1)=$ $\displaystyle
E[Y_{1}(1,1)-Y_{1}(1,0)|\tau=c],$ $\displaystyle=$
$\displaystyle\frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Y_{1}|D=1,M=1]-\frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_{0})|D=0,M=1]$
$\displaystyle-\frac{p_{0|0}}{p_{0|0}-p_{0|1}}E[Q_{10}(Y_{0})|D=0,M=0]+\frac{p_{0|1}}{p_{0|0}-p_{0|1}}E[Y_{1}|D=1,M=0].$

###  F.6 Quantile indirect effect under $d=1$ on compliers

In (E.8) and (D.8), we show that $F_{Y_{1}(1,1)|c}(y)$ and
$F_{Y_{1}(1,0)|c}(y)$ are identified. Accordingly,
$\delta_{1}^{c}(q,1)=F_{Y_{1}(1,1)|c}^{-1}(q)-F_{Y_{1}(1,0)|c}^{-1}(q)$ is
identified.