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These will turn out to be the same, so we just work with the former. Thus we have that
\begin{eqnarray}
\bar \psi_A = \psi_A^\dagger \gamma_7
\end{eqnarray}
If we choose $\gamma_7$ such that
\begin{eqnarray}
(\gamma_7)^\dagger = - \gamma_7
\end{eqnarray}
we can express the Hermitian conjugates of our gamma matrices as
\begin{eqnarray}
\gamma_\mu^\dagger = \eta\, G^{-1} \gamma_\mu G
\end{eqnarray}
Importantly, with $G=G_1$ in , we have
\begin{eqnarray}
\eta = -1
\end{eqnarray}
This will be important in Appendix  when the consistency of the symplectic Majorana condition is analyzed.
For now, we just recall that the symplectic Majorana condition must take the form
\begin{eqnarray}
\bar \psi_A = \epsilon^{AB} \psi_B^T\, {\cal C}
\end{eqnarray}
where
\begin{eqnarray}
{\cal C}^2 = 1 \hspace{0.7 in} {\cal C}^T = {\cal C} \hspace{0.7in} \gamma_\mu^T = - {\cal C}^{-1} \gamma_\mu {\cal C}
\end{eqnarray}
We now want to reduce from $d=7$ to $d=6$. In particular, we reduce on the time-like direction $x_7$. This entails finding a Euclidean induced metric on the six-dimensional surface . From the point of view of the Clifford algebra, we must remove the matrix $\gamma_7$ to get a six-dimensional Clifford algebra. However, the properties of the matrix $\gamma^7$ remain the same. In fact, we may choose
\begin{eqnarray}
\gamma_7 = \gamma_0\gamma_1\gamma_2\gamma_3\gamma_4\gamma_5
\end{eqnarray}
which satisfies all of the properties ,.
\section{Free differential algebra}
\setcounter{equation}{0}
In this Appendix, we will construct the free differential algebra (FDA) of a supergravity theory with $\mathbb{H}_6$ background in order to motivate the form of the supersymmetry variations given in .
The first step of constructing the FDA is to write down the Maurer-Cartan equations (MCEs), which may be thought of as the geometrization of the (anti-)commutation relations of the superalgebra. In short, instead of defining the algebra via the (anti-)commutators of its generators, the MCEs encode the algebraic structure in integrability conditions. In
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 the supergravity context, a nice introduction to the MCEs, as well as to the free differential algebras to be introduced shortly, may be found in . In the current case, the MCEs are
\begin{eqnarray}
0 ={\cal D} V^a + {1\over 2} \bar \psi_A \gamma^a \gamma^7 \psi^A
\nonumber\\
0= R^{ab} - 4 m^2 V^a V^b + m \bar \psi_A \gamma^{a b} \psi^A
\nonumber\\
0= d A^r - {1\over 2} g \epsilon^{rst} A_s A_t - i \bar \psi_A \psi_B \sigma^{r\,\,AB}
\nonumber\\
0= D \psi_a + m \gamma_a \gamma_7 \psi_A V^a
\end{eqnarray}
Here $a = 1,\dots,6$ and $V^a$ are the six-dimensional frame fields, given in terms of the seven-dimensional spin-connection as $V^a= {1 \over 2m}\omega^{a7}$. These may be compared to the analogous expressions in the dS/AdS cases of .
As a simple check, the second equation of  tells us that when $\psi^A =0$,
\begin{eqnarray}
R_{\mu \nu} = - 20 m^2 g_{\mu \nu}
\end{eqnarray}
which is precisely as expected for an $\mathbb{H}_6$ background.
The next step is to enlarge the MCEs to a free differential algebra (FDA) by adding the following equations for the additional vector and 2-form fields of the full $d=6$ $F(4)$ supergravity theory,
\begin{eqnarray}
dA - m B + \alpha \bar \psi_A \gamma_7 \psi^A = 0 \hspace{1 in} d B + \beta \bar \psi_A \gamma_a \psi^A V^a = 0
\end{eqnarray}
Above, $\alpha$ and $\beta$ are two coefficients, which can be shown  to satisfy
\begin{eqnarray}
\beta = - 2 \alpha
\end{eqnarray}
for our metric conventions. For the ambient space signature $(t,s) = (1,6)$, it is furthermore found that $\beta = 2i$, and thus we have $\alpha = -i$.
We would now like to compare the FDA above to the results of . To do so, we must first shift our notations by shifting
\begin{eqnarray}
\gamma^a \rightarrow \gamma^7 \gamma^a \hspace{0.7 in} \gamma_a \rightarrow - \gamma_7 \gamma_a
\end{eqnarray}
This preserves the square of the gamma matrices, and hence the signature of the metric. The definition of the Dirac conjugate spinor  remains the same under this change. So the FDA for the $\mathbb{H}_6$ theory in these conventions is,
\begin{eqnarray}
0 ={\cal D} V^a + {1\over 2} \bar \psi_A \gamma^a \psi^A
\nonumber\\
0= R^{ab} - 4 m^2 V^a V^b + m \bar \psi_A \gamma^{a b} \psi^A
\nonumber\\
0= d A^r - {1\over 2} g \epsilon^{rst} A_s A_t - i \bar \psi_A \psi_B \sigma^{r\,\,AB}
\nonumber\\
0= D \psi_a - m \gamma_a \psi_A V^a
\nonumber\\
0=dA - m B - i \bar \psi_A \gamma_7 \psi^A
\nonumber\\
0=d B -2 i \bar \psi_A \gamma_7 \gamma_a \psi^A V^a
\end{eqnarray}
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We may now compare the FDA written above to that obtained in the AdS$_6$ case, which for convenience we reproduce below,
\begin{eqnarray}
0 ={\cal D} V^a - {i \over 2} \bar \psi_A \gamma^a \psi^A
\nonumber\\
0= R^{ab} + 4 m^2 V^a V^b + m \bar \psi_A \gamma^{a b} \psi^A
\nonumber\\
0= d A^r - {1\over 2} g \epsilon^{rst} A_s A_t - i \bar \psi_A \psi_B \sigma^{r\,\,AB}
\nonumber\\
0= D \psi_a - i m \gamma_a \psi_A V^a
\nonumber\\
0=dA - m B - i \bar \psi_A \gamma_7 \psi^A
\nonumber\\
0=d B +2  \bar \psi_A \gamma_7 \gamma_a \psi^A V^a
\end{eqnarray}
We see that formally, we may obtain the $\mathbb{H}_6$ FDA from the AdS$_6$ FDA by exchanging
\begin{align}
m \rightarrow - i m \qquad \psi_A \rightarrow \psi_A \qquad \bar \psi_A \rightarrow i \bar \psi_A \qquad A^r \rightarrow i A^r \qquad g \rightarrow - i g \qquad B \rightarrow - B \qquad A \rightarrow i A\nonumber
\end{align}
These exchanges are compatible with the relation $g = 3m$.
Finally, we will check that the $\mathbb{H}_6$ FDA is compatible with the symplectic Majorana condition. This is a statement about the fourth equation of . We begin by defining
\begin{eqnarray}
\nabla \psi_A \equiv D \psi_A - q \gamma_a \psi_A V^a
\end{eqnarray}
where $q = m$ for $\mathbb{H}_6$ and $q=im$ for AdS$_6$. We then find that
\begin{eqnarray}
\overline{\nabla \psi_A} = D \psi_A^\dagger G^{-1} - q^* \psi_A^\dagger G^{-1} G \gamma_a^\dagger G^{-1} V^a = D \bar \psi_A - q^* \eta \,\bar \psi_A \gamma_a V^a
\nonumber\\
\epsilon^{AB} \nabla \psi_B^T {\cal C} = \epsilon^{AB} D\psi_B^T {\cal C} - q \epsilon^{AB} \psi_B^T {\cal C} {\cal C}^{-1} \gamma_a^T {\cal C} V^a = D \bar \psi_A+q \bar \psi_A \gamma_a V^a
\end{eqnarray}
where $\eta$ is defined implicitly in . We thus find that the symplectic Majorana condition is consistent only when
\begin{eqnarray}
-q^* \eta = q
\end{eqnarray}
For $\mathbb{H}_6$, the consistency of the symplectic Majorana condition thus requires $\eta=-1$, which we have already seen to be the case in . On the other hand, in the AdS$_6$ case, one would instead have required $\eta=1$. Checking the results of  confirms that this was so.
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HS theory can also be considered in the sense of AdS/CFT correspondence, where the HS theory of massless HS fields corresponds to a limiting case of the string theory for the string tension going to zero. Large N superconformal field theories were studied as holographic duals for higher spin gauge theories in perturbative expansion around AdS spacetime .
Klebanov and Polyakov  have proposed a duality  between the singlet sector of the critical 3-d O(N) vector model with $(\phi^a\phi^a)^2$ interaction and minimal bosonic theory in AdS4 which contains massless gauge fields with even spin. The analog of AdS4 conjecture by Klebanov and Polyakov appeared in AdS3  conjecturing a duality between a complex scalar coupled to higher-spin fields in Vasiliev's gravity in 3 dimensions and WN minimal model CFT in t'Hooft limit denoted by coset representation
 \begin{equation}
\frac{SU(N)_k\oplus SU(N)_1}{SU(N)_{k+1}},
\end{equation}
where we define the t'Hooft limit with $N,k\rightarrow\infty$ for $\lambda\equiv\frac{N}{k+N}$.
This duality has been verified by the number of studies, correspondence of global symmetries in bulk and at the boundary , correspondence of the 1-loop partition function in the bulk and at the large NCFT  , and partition function of the HS black hole at high temperature in the bulk and at the boundary CFT, as well as for the 3-point functions when $\lambda=\frac{1}{2}$ and $s=2,3,4$ for the scalar-scalar-HS field (00s) correlator in the t'Hooft limit.
Via three-point functions, tests of the conjecture have been done in  (for 00s correlator with general $\lambda$), and in .
In this work, we extract the coupling of the 00s  three-point correlator  by considering the linearised Vasiliev equations of motion, and we verify it by choosing the spin to be three and comparing with result in .
The result corresponds to coupling of the three-point correlation function up to selected normalisation.
While we consider general $\lambda$, the similar work has been done for the fixed $\lambda$ in .
The structure of the work is as follows: In the section two we consider linearised equations of motion in the Vasiliev's theory, in the section three we consider the higher spin field in the metric formulation, and in section four we conclude.
  \section{Linearized equations of motion}
Let us first consider the coefficient coming from the Vasiliev linearised equations. Vasiliev's theory contains five equations for the master fields W which is spacetime 1-form, B and $S_{\alpha}$ which are spacetime 0-forms. The generating functions are dependent on the coordinates of the spacetime, auxiliary bosonic twistor variables (referred to as "oscillators") and Clifford element pairs, where in definitions we follow conventions from .
The oscillators  and various other ingredients are used to define the "deformed" oscillator star-commutation relations which give rise to $hs[\lambda]$ higher spin algebra.
Two of the above mentioned equations that will be of the interest here are
\begin{align}
dW&=W\wedge\star W \\
dB&=W\star B-B\star W
\end{align}
We can rewrite W with projector operators
\begin{equation}
\mathcal{P}_{\pm}=\frac{1\pm\psi}{2}
\end{equation}
for $\psi$ elements of the Clifford pairs such that $W=-\mathcal{P}_{+}A-\mathcal{P}_{-}\overline{A}$
for
\begin{align}
\mathcal{P}_{\pm}\psi_{1}&=\psi_1\mathcal{P}_{\pm}=\pm\mathcal{P}_{\pm} &\mathcal{P}_{\pm}\psi_2=\psi_2\mathcal{P}_{\mp}
\end{align}
where A are Chern-Simons gauge fields which take value in the Lie algebra hs$[\lambda]$. In this formulation the equation
\begin{equation}
dW=W\wedge\star W
 \end{equation}
gives
\begin{align}
dA+A\wedge\star A =0 \\
D\overline{A}+\overline{A}\wedge \star \overline{A}=0
\end{align}
where A and $\overline{A}$ are positive polynomials of the positive degree in products of deformed oscillators.
() and () are in that case equal to field equations $hs[\lambda]\otimes hs[\lambda]$  Chern-Simons theory. \\
The generators of hs$[\lambda]$ are defined with spin index \textit{s} and mode index \textit{m}  as
$V^s_{m}$ for $s\geq2 $
while $|m|<s$ and obey
 star product
\begin{equation}
V_m^s\star V_n^t=\sum_{u=1,2,3}^{s+t-|s-t|-1}g_u^{st}(m,n;\lambda)V^{s+t-u}_{m-n}
\end{equation}
where
\begin{equation}g_u^{st}(m,n;\lambda)=(-1)^{u+1}g_u^{ts}(m,n;\lambda)\end{equation}
are specific coefficients dependent on $\lambda$ and defined according to conventions .
The equations describe interaction of arbitrary higher spin background with lienarized scalars.
The coupling that we are interested in can be extracted from rewriting the master field B as a linearized fluctuation around vacuum value $\mathcal{\nu}$
\begin{align}
B=\mathcal{\nu}+\mathcal{P}_{+}\psi_2C(x,\tilde{y}_{\alpha})+\mathcal{P}_-\psi_2\tilde{C}(x,\tilde{y}_{\alpha})
\end{align}
and expanding the master field $C$ in the deformed oscillators $\tilde{y}_{\alpha}$ in the equation
\begin{equation}
dC+A\star C- C \star \overline{A}=0.
\end{equation}
 That allows us determining the generalised Klein-Gordon (KG) equation in the background of HS fields.
While the expansion of the master field C in formalism of bosonic Vasiliev's theory is given by
 \begin{equation}
 C=C_0^1+C^{\alpha\beta}\tilde{y}_{\alpha}\tilde{y}_{\beta}+C^{\alpha\beta\sigma\lambda}\tilde{y}_{\alpha}\tilde{y}_{\beta}\tilde{y}_{\sigma}\tilde{y}_{\lambda}+...
 \end{equation}
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 with implied star product and symmetric C components.
  Master field components are now separated in physical scalar field $C_0^1$  and higher ones, related to it on-shell by derivatives.
The expansion of C is given by
\begin{equation}
C=\sum_{s=1}^{\infty}\sum_{|m|<s}C_m^sV_m^s
\end{equation}
for $C_m^s\sim C^{\alpha_1\alpha_2..\alpha_{2s-1}}$
where m and number of oscillators $\tilde{y}_{1}$ versus $\tilde{y}_2$ are related with $2m=N_1-N_2$ and $C_m^s$ are functions of spacetime coordinates. Auxiliary tensors are absorbed within a field.
The fields $A$ and $\bar{A}$ are expanded analogously
\begin{align}
A&=\sum_{s=2}^{\infty}\sum_{|m|<s}A_m^sV_m^s & \overline{A}=\sum_{s=2}^{\infty}\sum_{|m|<s}\overline{A}_m^sV_m^s.
\end{align}
The standard procedure of finding the generalised KG equation consists of inserting the expressions for $A$, $\bar{A}$ and $C$ in  () and determining the smallest possible set of equations needed to find the scalar equation in arbitrary background.
Standard procedure can be described considering equation () in AdS background since it is a foundation for the following computations. The vacuum $C_0^1$ equation without AdS fields is ordinary KG  equation while one can determine the higher components in the terms of $C_0^1$.
The AdS connection consists of the spin-2 generators that form SL(2), subalgebra of $hs[\lambda]$
\begin{align}
A&=e^{\rho}V_1^2dz+V_0^2d\rho\\
\bar{A}&=e^{\rho}V_{-1}^2d\bar{z}-V_0^2d\rho
\end{align}
with AdS metric
\begin{equation}
ds^2=d\rho^2+e^{2\rho}dzd\bar{z}.
\end{equation}
The higher spins fields vanish, and we are working in Euclidean metric and  Fefferman-Graham gauge. The general form of the $C$ equation () in the AdS background is
\begin{align}
&\partial_{\rho}C_m^s+2C_m^{s+1}+C_m^{s+1}g_3^{(s+1)2}(m,0)=0  \end{align}
\begin{align}
 \partial C_m^s+e^{\rho}(C_{m-1}^{s-1}+\frac{1}{2}g_2^{2s}(1,m-1)C^s_m\\ \nonumber+\frac{1}{2}g_3^{2(s+1)}(1,m-1)C_{m-1}^{s+1})=0 \end{align}
\begin{align}
\overline{\partial}C^s_m-e^{\rho}( C_{m+1}^{s-1}-\frac{1}{2}g_2^{2s}(-1,m+1)C^s_{m+1}\\+\frac{1}{2}g_3^{2(s+1)}(-1,m+1)C^{s+1}_{m+1})=0 \nonumber
\end{align}
for $|m|<s$, $\partial=\partial_z,\overline{\partial}=\partial_{\overline{z}}$ and the $\lambda$-dependence in the structure constants suppressed.
In the simplest case choosing $s=1,s=2$ one can solve for the higher components in C and obtain the Klein-Gordon KG equation
\begin{equation}
\left[ \partial_{\rho}^2+2\partial_{\rho}+4e^{-2\rho}\partial\bar{\partial}-(\lambda^2-1) \right]C_0^1=0.
\end{equation}
 Consistency condition on equations is that all the components of C have smooth solution when expressed using $C_0^1$. The strategy for determining the minimal set of equations for $C_0^1$ is to select components of C that are of the form $C_{\pm m}^{m+1}$ and therefore the smallest spin for fixed m (e.g. $C_0^1, C_{\pm1}^2,..$). That are minimal components.
 One needs $V_{m,\rho}^s$ equations for fixed m, solve for non-minimal components in terms of minimal ones and $\rho$ derivatives, for $A_{\rho}=-\overline{A}_{\rho}=V_{0}^2$. After solving for minimal ones, one needs to solve $V_{m,z}^s$ and $V_{m,\overline{z}}^s$ equations in terms of $C_0^1$ and its derivatives.
Once that we have expressed the higher components of C in terms of $C_0^1$ we can determine the part that defines the KG equation and the
generalised part that appears due to the HS background.
To obtain the equation of motion for the scalar field up to linear order we consider the variation of the gauge field and apply the KG equation on it. This and the standard procedure for obtaining the linearised equation of motion for the scalar field described above
should be equal once the gauge parameter is chosen conveniently.
That approach can be written in the following way.
First we express the higher components of the C field in terms of the combination of the derivatives on $C_0^1$ in the background AdS.
Focusing on the master field $C$, the equation () is invariant under the $hs[\lambda]\oplus hs[\lambda]$  gauge invariance when
\begin{align}
C\rightarrow C+C\star \bar{\Lambda}-\Lambda\star C
\end{align}
for
\begin{equation}
\Lambda(\rho,z,\bar{z})=\sum_{n=1}^{2s-1}\frac{1}{(n-1)!}(-\partial)^{n-1}\lambda^{(s)}(z,\bar{z})e^{(s-n)\rho}V^{s}_{s-n}.
\end{equation}
Where we take $\Lambda$ to be chiral, so $\bar{\Lambda}=0$.
The field in the higher spin background is obtained by transformation
\begin{align}
\tilde{C}_m^s=C_m^s-(\Lambda\star C)^s_m.
\end{align}
The field $C_m^s$ we express in terms of the $C_0^1$.
To do that we focus on the set of equations (),(20),().
The product of the C field with $\Lambda$ gives combination of higher components of C in AdS background which can, as we will show, be expressed in terms of $C_0^1$. On the field $C_0^1$ we can use the transformation () and obtain
\begin{align}
\tilde{C}_0^1=C_0^1-(\Lambda\star C)^1_0.
\end{align}
Since we are at the linear order, once we have $C_0^1$ we can rewrite it as $\tilde{C}_0^1$ which is defined on the higher spin background.
From the expression for the gauge field $\Lambda$ () and the relation for the star product () we can determine the variation of the scalar field $C^1_0$
\begin{align}
(\delta C)^1_0 &=-\sum_{n=1}^{2s-1}\frac{1}{(n-1)!}(-\partial)^{n-1}\Lambda^{(s)} \\ &\times\frac{1}{2}g_{2s-1}^{ss}(s-n,n-s)C^s_{-(s-n)}e^{(s-n)\rho},
\end{align}
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for $C^s_{-(s-n)}$ an arbitrary component of the master field $C$.
Taking $m\rightarrow -m$ in the set of equations (), (20), () and $s=m+1$ in (), we can iteratively determine the dependence of the $C^{m+1}_m$ on the $C_0^1$. From the equation () we obtain
\begin{equation}
\partial_z C_{-m}^{m+1}+\frac{e^{\rho}}{2}g_3^{2(m+2)}(1,-m-1)C^{m+2}_{-m-1}=0
\end{equation}
taking into consideration that for certain components $C_n^s$ it is required $|n|\leq s-1$ this iteratively leads to relation of $C_m^{m+1}$ and $C_0^1$, and from
the () analogously for $C^{m+1}_{-m}$ and $C_0^1$.
The general form of the $C^s_{\pm}$ is
then given in terms of $C^{m+1}_{\pm m}$ and coefficients $g_u^{ts}(m,n)$.
Knowing $C^s_{\pm}$ and $C^{m+1}_{\pm m}$ allows to obtain
\begin{align}
(\delta C)^1_0&=\sum_{n=1}^sf_{\pm}^{s,n}(\lambda)\partial_z^{n-1}\Lambda^{(s)}\partial_z^{s-n}\phi
\end{align}
for $\phi\equiv C_0^1$ and $f^{s,n}_{\pm}(\lambda)$ expressed in terms  of coefficients $g_u^{st}(m,n)$
Using the replacement $\partial_{\rho}\rightarrow-(1\pm\lambda)$
and
 writing explicitly first few n values for $f_{\pm}^{s,n}(\lambda)$, allows to determine its general expression
\begin{align}
f_{\pm}^{s,n}(\lambda)&=(-1)^s\frac{\Gamma(s+\lambda)}{\Gamma(s-n+1\pm\lambda)}\frac{1}{2^{n-1}(2(\frac{n}{2}-1))!!\left(\frac{n-1}{2}\right)!}\nonumber\\
&\times\prod_{j=1}^{\frac{n-1}{2}}\frac{s+1-n}{2s-2j-1} .
\end{align}
\noindent Substituting () in () one obtains the variation of the scalar field
\small
\begin{align}
\displaystyle{(\delta C)_0^1}&\displaystyle{=\sum_{n=1}^{s}(-1)^s\frac{\Gamma(s\pm\lambda)}{\Gamma(s-n+1\pm\lambda)}\frac{1}{2^{n-1}\left(2\left(\frac{n}{2}\right)-1\right)!!\left(\frac{n-1}{2}\right)!} \nonumber}\\ & \times\prod_{j=1}^{\left( \frac{n-1}{2} \right)}\frac{s+j-n}{2s-2j-1}\partial_z^{n-1}\Lambda^{(s)}\partial_z^{s-n}C_0^1.
\end{align}
\normalsize
To consider the coefficient in front, we focus on the term with the lowest number of $\partial_z$ derivatives on the gauge field $\Lambda^{(s)}$,  obtained for n=1. Then, () becomes
\begin{equation}
(\delta C)_0^1|_{n=1}=(-1)^s\Lambda^{(s)}\partial^{s-1}C_0^1 .
\end{equation}
To obtain the linearised equation of motion for the scalar field we act on () with KG operator ().
This can be written as
\begin{equation}
\Box_{KG}\tilde{C}^1_0=\Box_{KG}C_0^1+\Box_{KG}\delta C_0^1 .
\end{equation}
Taking   $\partial_{\rho}\rightarrow(1\pm\lambda)$ in $f_{\pm}^{s,n}(\lambda)$ we have taken and considering the term with highest number of derivatives on $C_0^1$  leads to
\begin{align}
&\Box_{KG}|_{\text{highest number of derivatives}}(\delta C)_0^1=\\& = (-1)^s4e^{-2\rho}\partial(\bar{\partial}\Lambda^{(s)}\partial^{(s-1)}C_0^1)\\&=(-1)^s4e^{-2\rho}[ \partial\bar{\partial}\bar{\Lambda}^{(s)}\partial^{(s-1)}C_0^1+\bar{\partial}\Lambda^{(s)}\partial^{s}C_0^1\nonumber\\&+\partial\Lambda^{(s)}\bar{\partial}\partial^{(s-1)}C_0^1+\Lambda^{(s)}\bar{\partial}\partial^sC_0^1 ].
\end{align}
The term in () that is of further interest is the one multiplying $4e^{-2\rho}\partial\bar{\partial}$ acting on $\delta C_0^1$ which is convenient to compute in the metric formulation.
\section{Metric formulation}
In the metric formulation we can express the higher spin field of arbitrary spin $s$ with
\begin{equation}
\phi_{\mu_1.....\mu_s}=tr\left( \tilde{e}_{(\mu_1}...\tilde{e}_{\mu_{s-1}}\tilde{E}_{\mu_s)} \right)
\end{equation}
where $\tilde{E}_{\mu s}=\tilde{A}_{\mu}-\tilde{\bar{A}}_{\mu}$ and $\tilde{A}_{\mu}$ and $\tilde{\bar{A}}_{\mu}$ we define below.
The  dreibein is determined from the background AdS metric ()
\begin{align}
e_{z}&=\frac{1}{2}e^{\rho}(L_1+L_{-1})=\frac{1}{2}e^{\rho}(V_1^2+V_{-1}^{2}) \\
e_{\bar{z}}&=\frac{1}{2}e^{\rho}(L_1-L_{-1})=\frac{1}{2}e^{\rho}(V_1^2-V_{-1}^2) \\
e_{\rho}&=L_0=V_0^2.
\end{align}
The invariance of the equation () under the gauge transformation for $hs[\lambda]\oplus hs[\lambda]$ for the fields A means
\begin{align}
A&\rightarrow A+d \Lambda +\left[A,\Lambda \right]_{\star}\equiv \tilde{A}\\
\bar{A}&\rightarrow \bar{A}+d \bar{\Lambda} +\left[\bar{A},\bar{\Lambda} \right]_{\star}\equiv \tilde{\bar{A}}.
\end{align}
Since $\Lambda$ parameter is chiral it means $\bar{\Lambda}=0$ and the field $\tilde{\bar{A}}$ is essentially unchanged. The field $\tilde{A}_{\mu}$ is then
\begin{equation}
\tilde{A}=A_{AdS}+d\Lambda+\left[A_{AdS},\Lambda\right]_{\star}.
\end{equation}
 $d\Lambda$  reads
\small
\begin{align}
d\Lambda&=\sum_{n=1}^{2s-1}\frac{1}{(n-1)!}V_{s-n}^se^{(s-n)\rho}[ (-\partial)^{n-1}\partial\Lambda^{(s)}(z,\bar{z}) dz\\&+(-\partial)^{n-1}\bar{\partial}\Lambda^{(s)}(z,\bar{z})d\bar{z}+(-\partial)^{n-1}\Lambda^{(s)}(z,\bar{z})(s-n)d\rho ]
\end{align}
\normalsize
and
\begin{align}
\left[A_{AdS},\Lambda\right]_{\star}&= [ e^{\rho}V_1^2dz+V_0^2d\rho,\nonumber\\ &\sum_{n=1}^{2s-1}\frac{1}{(n-1)!}(-\partial)^{n-1}\Lambda^{(s)}(z,\bar{z})e^{(s-n)\rho}V^s_{s-n} ]
\end{align}
To read out the coupling we focus on $\bar{z}....\bar{z}$ component of the field $C_0^1$ with lowest number of derivatives on gauge field $\Lambda^{(s)}$. The $\star$ multiplication of the dreibeins in () in that case contributes only with first $g_{u}^{st}(m,n;\lambda)$  coefficient with the each following dreibein that is being multiplied. More explicitly
\begin{align}
e_{\bar{z}}\star e_{\bar{z}}&=\frac{1}{2^2}e^{2\rho}\left(V_1^2-V_{-1}^2\right)\star(V_{1}^2-V_{-1}^2)
\end{align}
From () we notice that the lowest number of derivatives on $\Lambda$ will appear for lowest n, i.e.  for $n=1$ in summation (). Knowing the relation for the trace of higher spin generators, the required generator $V^s_{s-n}$ will than
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 be of the form $V^s_{s-1}$, as we see below, which means that multiplication of HS generators we have to  consider is
\begin{equation}
V_{-1}^2\star V_{-1}^2\star....\star V_{-1}^2.
\end{equation}
Then
\begin{align}
V_{-1}^2\star V_{-1}^2 &=\frac{1}{2}( g_1^{22}(-1,-1)V_{-2}^3+g_{2}^{22}(-1,-1)V_{-2}^2\nonumber\\&+g_{3}^{22}(-1,-1)V_{-2}^1 )
\end{align}
where the $g_2^{22}(-1,-1)=g_3^{22}(-1,-1)=0$. Multiplying with following $V_{-1}^2$, etc. on e can conclude
\begin{equation}
\underbrace{V_{-1}^{2}\star V_{-1}^2\star ....\star V_{-1}^2}_{s-1}=\frac{1}{2^{s-1}}g_1^{2(s-1)}(-1,-(s-2))V_{-(s-1)}^s
\end{equation}
while
\begin{equation}
g_{2}^{2(s-1)}(-1,-(s-2))=g_{3}^{2(s-1)}(-1,-(s-2))=0.
\end{equation}
That  means we have found the contribution to the $\bar{z}...\bar{z}$ component multiplied with lowest derivative on $\Lambda^{(s)}$ due to definition of trace for generators $V_n^s$
\small
\begin{equation}
tr\left( V_m^s V_n^t \right)=N_s\frac{(-1)^{s-m-1}}{(2s-2)!}\Gamma(s+m)\Gamma(s-m)\delta^{st}\delta_{m,-n}.
\end{equation}
\normalsize
for
\begin{equation}
N_s\equiv \frac{3\cdot 4^{s-3}\sqrt{\pi}q^{2s-4}\Gamma(s)}{(\lambda^2-1)\Gamma(s+\frac{1}{2})}(1-\lambda)_{s-1}(1+\lambda)_{s-1}
\end{equation}
and $(a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}$ ascending Pochhammer symbol. The overall constant is set to \begin{equation}tr(V_1^2V_{-1}^2)=-1.
\end{equation}
Let us go back to $\phi_{\bar{z}....\bar{z}}$ component. The star product $e_{\bar{z}}\star...\star e_{\bar{z}}$ will contribute with $\frac{1}{2^{s-1}}e^{(s-1)\rho}V^{s}_{-(s-1)}$ if we consider as explained above the lowest derivative on $\Lambda^{(s)}$. We can denote this as
\begin{align}
e_{\bar{z}}\star....\star e_{\bar{z}}(V_{-1}^2\star ....\star V_{-1}^2)=\frac{1}{2^{s-1}}e^{(s-1)\rho}V^{s}_{-(s-1)}.\end{align} The $\tilde{E}_{\bar{z}_s}=\tilde{A}_{\bar{z}_s}-\tilde{\bar{A}}_{\bar{z}_s}$ needs to be able to satisfy the conditions of the trace () in star multiplication with $e_{\bar{z}}\star...\star e_{\bar{z}}$, the only HS generator that  contributes is $V^s_{s-1}$ generator. When we gauge the field $A_{\bar{\mu}_s}$, $d\bar{z}$ component appears in $d\Lambda$ while $A_{AdS}$ and $[A_{AdS},\Lambda]_{\star}$ do not have $d\bar{z}$ component. The $\tilde{\bar{A}}_{\bar{z}_s}$ has $d\bar{z}$ component that comes from $\bar{A}_{AdS}$ part and it is $e^{\rho}V_{-1}^2d\bar{z}$. This however will not appear with the right number of derivatives on $\Lambda$. Since we have chosen $\Lambda$ to be chiral and $\bar{\Lambda}=0$, that was the only contribution from $\tilde{\bar{A}}_{\bar{z}}$.
Altogether, we can write $\phi_{\bar{z}...\bar{z}}$ component for the $\bar{\partial}\Lambda^{(s)}$ derivative as
\small
\begin{align}
\phi_{\bar{z}....\bar{z}}|_{\bar{\partial}\Lambda^{(s)}}&=tr\left[\frac{1}{2^{s-1}}e^{(s-1)}V^s_{-(s-1)}\star V^s_{s-1}e^{(s-1)\rho}\bar{\partial}\Lambda^{(s)}(z,\bar{z})\right]\\
&=\frac{1}{2^{s-1}}e^{2(s-1)\rho}\bar{\partial}\Lambda^{(s)} N_s.
\end{align}
\normalsize
Inserting the normalisation $N_s$ we obtain
\begin{align}\nonumber
\phi_{\bar{z}...\bar{z}}|_{\bar{\partial}\Lambda^{(s)}}&=\frac{1}{2^{s-1}}e^{2(s-1)\rho}\bar{\partial}\Lambda^{(s)}\\&\times
\frac{3\cdot 4 \sqrt{\pi}4^{4-2s}\Gamma(s)\Gamma(s+\lambda)\Gamma(s-\lambda)}{(\lambda^2-1)\Gamma(s+\frac{1}{2})\Gamma(1-\lambda)\Gamma(1+\lambda)}.
\end{align}
The expression $\phi_{\bar{z}...\bar{z}}$ we want to compare with expression () for highest derivative on $C_0^1$ and $\bar{\partial}\Lambda^{(s)}$.
In the computation of the vertex this would be a term
\begin{equation}
\phi^{z...z}\mathcal{\phi}\nabla_z...\nabla_{z}\mathcal{\phi}
\end{equation}
for $\phi^{z...z}$ higher spin field with $s$ indices and $\mathcal{\phi}$ scalar field. Raising indices
 contributes with a factor $2^se^{-2s\rho}$, so that the field $\phi^{z...z}$ becomes
\begin{align}
\phi^{z...z}&=\frac{1}{2}e^{-2\rho}\bar{\partial}{\Lambda}^{(s)}3\cdot4^{4-2s}\nonumber \\ &\times
\frac{\Gamma(s)\Gamma(s+\lambda)\Gamma(s-\lambda)}{(\lambda^2-1)\Gamma(s+\frac{1}{2})\Gamma(1-\lambda)\Gamma(1+\lambda)}.
\end{align}
When  we take the ratio with $\Box_{KG}|_{\text{highest number of derivatives}(\delta C_0^1)|_{\bar{\partial}\Lambda}}=(-1)^s4e^{-s\rho}\bar{\partial}\Lambda^{(s)}\partial^sC_0^1$ we get (schematically written)
\begin{align}
&\frac{\phi^{z...z}|_{\bar{\partial}\Lambda^{(s)}}}{\Box_{KG}|_{\text{highest number of derivatives}(\delta C)_0^1|_{\bar{\partial}\Lambda^{(s)}}}}\nonumber\\&=(-1)^s\frac{1}{2}3\sqrt{\pi}\frac{4^{4-2s}\Gamma(s)\Gamma(s+\lambda)\Gamma(s-\lambda)}{(\lambda^2-1)\Gamma(s+\frac{1}{2})\Gamma(1-\lambda)\Gamma(1+\lambda)}.
\end{align}
which taking into account the normalisation gives the coupling for the 00s three point function.
\section{Conclusion and Outline}
We have considered the three-point coupling using metric-like formation to express the higher spin field and using the linearised Vasiliev's equations of motion.  The obtained result can also be verified using the alternative methods, for example following the procedure by .
The generalisation of the result to higher point functions would be non-trivial since in order to compute higher order vertices, one would have to consider perturbations around the background AdS field with higher spin fields up to that required higher order.
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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. 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 $M(1)=1$.
\subsection{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
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$\Delta_1(q) = F_{Y_1(1,M(1))}^{-1}(q) -F_{Y_1(0,M(0))}^{-1}(q)$.
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 $M=M(d)=m$,
\begin{align*}
\theta_1^{d,m}(q,1)=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} \\
\theta_1^{d,m}(q,0)=F_{Y_1(1,m)|D=d,M(0)=m}^{-1}(q)-F_{Y_1(0,m)|D=d,M(0)=m}^{-1}(q),
\end{align*}
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 $F_{Y_{t}(d,m)|D=d,M(d)=m}(y) = \Pr(Y_t(d,m) \leq y |D=d,M(d)=m)$.
\subsection{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
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function is $F_{Y_{t}|D=d,M=m}^{-1}(q) = \inf \{y : F_{Y_t|D=d,M=m}(y) \geq q \}$. Furthermore,
\begin{equation*}
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))
\end{equation*}
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 $y'$ at rank $q$ in period 1 ($y'= F_{Y_{1}|D=d,M=m}^{-1}(q)$).
\section{Identification and Estimation}
\subsection{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.\vspace{5 pt}\\
\textbf{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 $d,m,t \in \{0,1\}$.\vspace{5 pt}\\
Assumption 1 requires the potential outcomes to be continuous implying that there
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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  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.\vspace{5 pt}\\
\textbf{Assumption 2:} No anticipation effect of $M$ and $D$ in the baseline period.\\
$Y_0(d,m) - Y_0(d',m') = 0\mbox{, for } d, d', m, m' \{1,0\}.$
\vspace{5 pt}\\
Similarly,  and  assume the assignment to the treatment group does not affect the potential outcomes as long as the treatment is not yet realized. %This is implied by Assumption 1 and 2.
Furthermore, we assume conditional independence between unobserved heterogeneity and time periods given the treatment and no mediation. \vspace{5 pt}\\
\textbf{Assumption 3:} Conditional independence of $U$ and  $T$ given $D=1,M=0$ or $D=0,M=0$.\\
(a) $U T|D=1,M=0$,\\
(b) $U T|D=0,M=0$.\vspace{5 pt}\\
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
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$U$ 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.
 and  impose time invariance conditional on the treatment status, $U T|D=d$, 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. %Furthermore, we can identify the ATE and indirect effects under additional assumptions about the assignment of $D$, which we discuss in Section .
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 $\mathbb{U}_{dm}$. \\ %\vspace{5 pt}\\
\textbf{Assumption 4:} Common support given $M=0$.\\
(a) $\mathbb{U}_{10}\subseteq \mathbb{U}_{00}$,\\
(b) $\mathbb{U}_{00}\subseteq \mathbb{U}_{10}$.\vspace{5 pt}\\
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.\\
\noindent \textbf{Theorem 1:} Under Assumptions 1–3,
\begin{itemize}
\item[(a)] and Assumption 4a, the average and quantile direct effects under $d=1$ conditional on $D=1$ and $M(1)=0$ are identified:
\begin{align*}
\theta_1^{1,0}(1)= E[Y_1-Q_{00}(Y_0)|D=1,M=0], \\
\theta_1^{1,0}(q,1)= F_{Y_1|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_0)|D=1,M=0}^{-1}(q).
\end{align*}
\item[(b)] and Assumption 4b, the average and quantile direct effects under $d=0$
\end{itemize}
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\begin{align*}
\theta_1^{0,0}(0)= E[Q_{10}(Y_0)-Y_1|D=0,M=0], \\
\theta_1^{0,0}(q,0)= F_{Q_{10}(Y_0)|D=0,M=0}^{-1}(q)-F_{Y_1|D=0,M=0}^{-1}(q).
\end{align*}
\begin{itemize}
\item[(a)] and Assumption 6a, the average and quantile direct effects under $d=0$ conditional on $D=0$ and $M(0)=1$ are identified:
\begin{align*}
\theta_1^{0,1}(0)= E[Y_1-Q_{11}(Y_0)|D=0,M=1],\\
\theta_1^{0,1}(q,0)= F_{Y_1|D=0,M=1}^{-1}(q)-F_{Q_{11}(Y_0)|D=0,M=1}^{-1}(q).
\end{align*}
\item[(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:
\begin{align*}
\theta_1^{1,1}(1)= E[Q_{01}(Y_0)-Y_1|D=1,M=1],\\
\theta_1^{1,1}(q,1)= F_{Q_{01}(Y_0)|D=1,M=1}^{-1}(q)-F_{Y_1|D=1,M=1}^{-1}(q).
\end{align*}
\end{itemize}
\textbf{Proof.} See Appendix .
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 . 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.\vspace{5 pt}\\
\textbf{Assumption 7:} Independence of the treatment and potential mediators/outcomes.\\
$ \{Y_t(d,m),M(d)\} D \mbox{, for all } d,m,t, \in \{0,1\}.$\vspace{5 pt}\\
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
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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.\vspace{5 pt}\\
\textbf{Assumption 8:} Weak monotonicity of the mediator in the treatment.\\
$\Pr(M(1) \geq M(0)) =1.$\vspace{5 pt}\\
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  and . It rules out the existence of defiers.
As discussed in the Appendix , 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 ):
\begin{equation}
p_a =p_{1|0}, p_c = p_{1|1}-p_{1|0} = p_{0|0}-p_{0|1}, p_n = p_{0|1}.
\end{equation}
Furthermore, Assumptions 2, 7, and 8 imply that (see Appendix )
\begin{equation}
\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.
\end{equation}
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.\\
\noindent \textbf{Theorem 3:} Under Assumptions 1–3, 7-8,
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\begin{itemize}
\item[a)] and Assumption 4a, the average and quantile direct effects on never-takers are identified:
\begin{equation*}
\theta_1^n= \theta_1^{1,0}(1) \mbox{ and } \theta_1^n(q)= \theta_1^{1,0}(q,1).
\end{equation*}
\item[b)] and Assumption 4, the average direct effect under $d = 0$ on compliers is identified:
\begin{align*}
\displaystyle \theta_1^{c}(0) = \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) .
\end{align*}
Furthermore, the potential outcome distributions under $d = 0$ on compliers are identified:
\begin{equation} \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}
 \end{equation}
 \begin{equation}
 \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}
\end{equation}
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.
\end{itemize}
\textbf{Proof.} See Appendix .
\noindent \textbf{Theorem 4:} Under Assumptions 1–2, 5, 7-8,
\begin{itemize}
\item[a)] and Assumption 6a, the average and quantile direct effects on
always-takers are identified:
\begin{equation*}
\theta_1^a= \theta_1^{0,1}(0) \mbox{ and } \theta_1^a(q)= \theta_1^{0,1}(q,0).
\end{equation*}
\item[b)] and Assumption 6, the average direct effect under $d = 1$ on compliers is identified:
\begin{align*}
\theta_{1}^{c}(1) = \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).
\end{align*}
Furthermore, the potential outcome distributions under $d = 1$ for compliers
\end{itemize}
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are identified:
\begin{equation} \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}
 \end{equation}
 \begin{equation}
 \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}
\end{equation}
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.
\textbf{Proof.} See Appendix .
\noindent \textbf{Theorem 4:} Under Assumptions 1–2, 5, 7-8,
\begin{itemize}
\item[a)] and Assumption 6a, the average and quantile direct effects on
always-takers are identified:
\begin{equation*}
\theta_1^a= \theta_1^{0,1}(0) \mbox{ and } \theta_1^a(q)= \theta_1^{0,1}(q,0).
\end{equation*}
\item[b)] and Assumption 6, the average direct effect under $d = 1$ on compliers is identified:
\begin{align*}
\theta_{1}^{c}(1) = \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).
\end{align*}
Furthermore, the potential outcome distributions under $d = 1$ for compliers are identified:
\begin{equation} \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}
 \end{equation}
 \begin{equation}
 \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}
\end{equation}
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.
\end{itemize}
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\textbf{Proof.} See Appendix .
\noindent \textbf{Theorem 5:} Under Assumptions 1-3, 5, 7-8,
\begin{itemize}
\item[a)] and Assumptions 4a, 6a, the total average treatment effect on compliers is identified:
\begin{align*}
\Delta_1^c= \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] \\
 − \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].
\end{align*}
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 () and ().
\item[b)] and Assumptions 4a, 6b, the average indirect effect
under $d = 0$ on compliers is identified:
\begin{align*}
\delta_1^c(0) = \frac{p_{1|1}}{p_{1|1}-p_{1|0}}E[Q_{11}(Y_0)|D=1,M=1] - \frac{p_{1|0}}{p_{1|1}-p_{1|0}}E[Y_1|D=0,M=1]\\
\hspace{-0.15cm}- \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].
\end{align*}
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 () and ().
\item[c)] and Assumptions 4b, 6a, the average indirect effect
under $d = 1$ on compliers is identified:
\begin{align*}
\delta_1^{c}(1) = \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]\\
\hspace{-0.15cm}- \frac{p_{0|0}}{p_{1|1} - p_{1|0}}E[Q_{00}(Y_0)|D=0,M=0] + \frac{p_{0|1}}{p_{1|1} - p_{1|0}}E[Y_1|D=1,M=0].
\end{align*}
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 () and ().
\end{itemize}
\textbf{Proof.} See Appendix .
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\subsection{Estimation}
 As in Assumption 5.1 of , 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. 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 , see also , , and  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
  \begin{eqnarray*}
 \hat{Q}_{dm}(y) = \hat{F}_{Y_{1}|D=d,M=m}^{-1}(\hat{F}_{Y_{0}|D=d,M=m}(y)).
  \end{eqnarray*}
 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-
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\subsection{Estimation}
-in estimates. Taking $\theta_1^{1,0}$ (see Theorem 1) as an example, an estimate thereof is
 \begin{eqnarray*}
\hat{\theta}_1^{1,0}(1)=\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\\ -\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).
 \end{eqnarray*}
Likewhise, quantile effects are estimated based on the empirical quantiles.
For the estimation of total ATE and QTE,  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  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 , 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. %which can be shown in a two step GMM framework in the spirit of .
We use a non-parametric bootstrap approach to calculate the standard errors.  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  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,  propose a flexible semiparametric estimator that does not impose such a homogeneity-in-covariates assumption and show $\sqrt{N}$-consistency and asymptotic normality.
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\section{Simulations}
To shape the intuition for our identification results, this section presents a brief simulation based on the following data generating process (DGP):
\begin{equation*}
T \sim Binom(0.5),\textrm{ }D \sim Binom(0.5),\textrm{ }U\sim Unif(-1,1),\textrm{ } V\sim N(0,1)
\end{equation*}
independent of each other, and
\begin{equation*}
M=I\{D+U+V>0\},\qquad Y_T=\Lambda((1+D+M+D\cdot M)\cdot T+U).
\end{equation*}
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 $Y_1(d,M(d'))=\Lambda((1+d+M(d')+d\cdot M(d'))+U)$, 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 $n=I\{1+U+V\leq0\}$.
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  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 , we make use of the `cic' command in the \texttt{qte} R-package by  with its default values.
Table  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')
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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 , 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  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  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  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.
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 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 . 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
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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  suggest that both parameters are consistently estimated with the change-in-changes model (Panel A.).
\numberwithin{equation}{section}
\section{Proof of Theorem 1 }
\subsection{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{equation}
\begin{array}{rl}
 F_{Y_t(d,0)|D=1,M=0}(y)  \stackrel{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{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}
\end{equation}
for $d,d' \in \{0,1\}$. 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
\begin{equation*}
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).
\end{equation*}
Applying $F_{Y_1(0,0)|D=1,M=0}^{-1}(q)$ to both sides, we have
\begin{equation}
h(0,0,1,u)  =F_{Y_1(0,0)|D=1,M=0}^{-1}(F_{U|10} ( u)).
\end{equation}
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Second, for $F_{Y_0(0,0)|D=1,M=0}(y)$ we have
\begin{equation}
F_{U|D=1,M=0}^{-1} ( F_{Y_0(0,0)|D=1,M=0}(y)) =   h^{-1}(0,0,0;y).
\end{equation}
Combining () and () yields,
\begin{equation}
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) .
\end{equation}
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{equation}
\begin{array}{rl}
 F_{Y_t(d,0)|D=0,M=0}(y)  \stackrel{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{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}
\end{equation}
for $d,d' \in \{0,1\}$. 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
\begin{equation*}
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).
\end{equation*}
Applying $F_{Y_1(0,0)|D=0,M=0}^{-1}(q)$ to both sides, we have
\begin{equation}
h(0,0,1,u)  =F_{Y_1(0,0)|D=0,M=0}^{-1}(F_{U|00} ( u)).
\end{equation}
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\usepackage{float}
\usepackage{amsmath}
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\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
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Second, for $F_{Y_0(0,0)|D=0,M=0}(y)$ we have
\begin{equation}
F_{U|00}^{-1} ( F_{Y_0(0,0)|D=0,M=0}(y)) =   h^{-1}(0,0,0;y).
\end{equation}
Combining () and () yields,
\begin{equation}
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) .
\end{equation}
The left sides of () and () are equal. In contrast to (), () 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{equation} \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{A1,A3b}{=}  E[h(0,0,1,h^{-1}(0,0,0; Y_0(1,0)))|D=1,M=0], \\
\qquad \stackrel{A2}{=} E[h(0,0,1,h^{-1}(0,0,0; Y_0(0,0)))|D=1,M=0], \\
\qquad \stackrel{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}
\end{equation}
which has data support because of Assumption 4a.
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\subsection{Quantile direct effect under $\mathbf{d=1}$ conditional on $\mathbf{D=1}$ and $\mathbf{M(1)=0}$}
In the following, we prove that
\begin{align*}\theta_1^{1,0}(q,1) = F_{Y_{1}(1,0)|D=1,M(1)=0}^{-1}(q)-F_{Y_{1}(0,0)|D=1,M(1)=0}^{-1}(q),\\= F_{Y_{1}|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).
\end{align*} For this purpose, we have to show that
\begin{align}
F_{Y_{1}(1,0)|D=1,M(1)=0}(y)   = F_{Y_{1}|D=1,M=0}(y) \mbox{ and}  \\
F_{Y_{1}(0,0)|D=1,M(1)=0} (y)  =F_{Q_{00}(Y_{0})|D=1,M=0}(y) ,
\end{align}
which is sufficient to show that the quantiles are also identified. We can show () 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 (), we obtain
\begin{equation}  \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}
\end{equation}
which proves ().
\subsection{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
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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 () to evaluate $F_{Y_1(1,0)|D=0,M=0}(y)$ at $h(1,0,1,u)$
\begin{equation*}
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).
\end{equation*}
Applying $F_{Y_1(1,0)|D=0,M=0}^{-1}(q)$ to both sides, we have
\begin{equation}
h(1,0,1,u)  =F_{Y_1(1,0)|D=0,M=0}^{-1}(F_{U|10} ( u)).
\end{equation}
Second, for $F_{Y_0(1,0)|D=0,M=0}(y)$ we have
\begin{equation}
F_{U|10}^{-1} ( F_{Y_0(1,0)|D=0,M=0}(y)) =   h^{-1}(1,0,0;y),
\end{equation}
using (). Combining () and () yields,
\begin{equation}
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) .
\end{equation}
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 () to evaluate $F_{Y_1(1,0)|D=1,M=0}(y)$ at $h(1,0,1,u)$
\begin{equation*}
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).
\end{equation*}
Applying $F_{Y_1(1,0)|D=1,M=0}^{-1}(q)$ to both sides, we have
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\begin{equation}
h(1,0,1,u)  =F_{Y_1(1,0)|D=1,M=0}^{-1}(F_{U|10} ( u)).
\end{equation}
Second, for $F_{Y_0(1,0)|D=0,M=0}(y)$ we have
\begin{equation}
F_{U|10}^{-1} ( F_{Y_0(1,0)|D=1,M=0}(y)) =   h^{-1}(1,0,0;y),
\end{equation}
using (). Combining () and () yields,
\begin{equation}
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) .
\end{equation}
The left sides of () and () are equal. In contrast to (), () 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{equation}  \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{A1,A3a}{=}  E[h(1,0,1,h^{-1}(1,0,0; Y_0(0,0)))|D=0,M=0], \\
\qquad \stackrel{A2}{=} E[h(1,0,1,h^{-1}(1,0,0; Y_0(1,0)))|D=1,M=0] ,\\
\qquad \stackrel{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} \end{equation}
which has data support because of Assumption 4b.
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\subsection{Quantile direct effect under $\mathbf{d=0}$ conditional on $\mathbf{D=0}$ and $\mathbf{M(0)=0}$}
In the following, we prove that
\begin{align*}
\theta_1^{0,0}(q,0) = F_{Y_{1}(1,0)|D=0,M(0)=0}^{-1}(q)-F_{Y_{1}(0,0)|D=0,M(0)=0}^{-1}(q),\\= F_{Q_{10}(Y_{0})|D=0,M=0}^{-1}(q)-F_{Y_{1}|D=0,M=0}^{-1}(q).
\end{align*}
For this purpose, we have to show that
\begin{align}
F_{Y_{1}(1,0)|D=0,M(0)=0}(y)   = F_{Q_{10}(Y_{0})|D=0,M=0}(y) \mbox{ and}  \\
F_{Y_{1}(0,0)|D=0,M(0)=0}(y)  =F_{Y_{1}|D=0,M=0}(y) ,
\end{align}
which is sufficient to show that the quantiles are also identified. We can show () 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 (), we obtain
\begin{equation*} \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}
\end{equation*}
which proves ().
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\section{Proof of Theorem 2 }
\subsection{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{equation}
\begin{array}{rl}
 F_{Y_t(d,0)|D=1,M=0}(y)  \stackrel{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{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}
\end{equation}
for $d,d' \in \{0,1\}$. 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
\begin{equation*}
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).
\end{equation*}
Applying $F_{Y_1(1,1)|D=0,M=1}^{-1}(q)$ to both sides, we have
\begin{equation}
h(1,1,1,u)  =F_{Y_1(1,1)|D=0,M=1}^{-1}(F_{U|01} ( u)).
\end{equation}
Second, for $F_{Y_0(1,1)|D=0,M=1}(y)$ we have
\begin{equation}
F_{U|01}^{-1} ( F_{Y_0(1,1)|D=0,M=1}(y)) =   h^{-1}(1,1,0;y).
\end{equation}
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\begin{document} \doublespacing \pagestyle{plain}
Combining () and () yields,
\begin{equation}
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) .
\end{equation}
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{equation}
\begin{array}{rl}
 F_{Y_t(d,1)|D=1,M=1}(y)  \stackrel{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{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}
\end{equation}
for $d,d' \in \{0,1\}$. 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
\begin{equation*}
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).
\end{equation*}
Applying $F_{Y_1(1,1)|D=1,M=1}^{-1}(q)$ to both sides, we have
\begin{equation}
h(1,1,1,u)  =F_{Y_1(1,1)|D=1,M=1}^{-1}(F_{U|11} ( u)).
\end{equation}
Second, for $F_{Y_0(1,1)|D=1,M=1}(y)$ we have
\begin{equation}
F_{U|11}^{-1} ( F_{Y_0(1,1)|D=1,M=1}(y)) =   h^{-1}(1,1,1;y).
\end{equation}
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\begin{document} \doublespacing \pagestyle{plain}
\section{Proof of Theorem 3 }
\subsection{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 (), we obtain the first ingredient $E[Y_1(1,0)|\tau=n]=E[Y_1|D=1,M=0]$. Furthermore, from () 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{equation}  \begin{array}{rl} E[Y_1(0,0)|D=1,M(1)=0]\stackrel{A7}{=}E[Y_1(0,0)|D=1,\tau=n]\\\stackrel{A8}{=}E[Y_1(0,0)|\tau=n].
\end{array} \end{equation}
\subsection{Quantile direct effect on the never-takers}
We prove that
\begin{align*}
\theta_1^n (q)= F_{Y_{1}(1,0)|\tau=n}^{-1}(q)- F_{Y_{1}(0,0)|\tau=n}^{-1}(q), \\
= F_{Y_1|D=1,M=0}^{-1}(q)-F_{Q_{00}(Y_{0})|D=1,M=0}^{-1}(q).
\end{align*}
 This requires showing that
\begin{align}
F_{Y_{1}(1,0)|\tau=n}(y) =F_{Y_1|D=1,M=0}(y) \mbox{ and} \\
F_{Y_{1}(0,0)|\tau=n}(y) = F_{Q_{00}(Y_{0})|D=1,M=0}(y).
\end{align}
Under Assumptions 7 and 8,
\begin{equation}
\begin{array}{rl}
F_{Y_t|D=1,M=0} (y) = E[1\{Y_t\leq y\}|D=1,M=0] \\ \stackrel{A7,A8}{=}E[1\{Y_t(1,0)\leq y\}|\tau=n] \\ = F_{Y_{t}(1,0)|\tau=n} (y), \end{array}
\end{equation}
which proves (). From (), we have
\begin{equation*}
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].
\end{equation*}
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\begin{document} \doublespacing \pagestyle{plain}
Under Assumption 7 and 8,
\begin{equation}  \begin{array}{rl} E[1\{Y_1(0,0) \leq y\}|D=1,M(1)=0]\stackrel{A7,A8}{=}E[1\{Y_1(0,0) \leq y\}|\tau=n]\\  = F_{Y_{1}(0,0)|\tau=n}(y),
\end{array} \end{equation}
which proves.
\subsection{Average direct effect under $\mathbf{d = 0}$ on compliers}
In the following, we show that
\begin{align*}
\displaystyle \theta_1^{c}(0) = E[Y_1(1,0)-Y_1(0,0)|\tau=c], \\ =
\frac{p_{0|0}}{p_{0|0} - p_{0|1}}E[Q_{10}(Y_0)- Y_1|D=0,M=0] \\
  - \frac{p_{0|1}}{p_{0|0} - p_{0|1}}E[Y_1 - Q_{00}(Y_0)|D=1,M=0].
\end{align*}
Plugging () in () under $T=1$, we obtain
\begin{equation*}
\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}
\end{equation*}
This allows identifying
\begin{equation}
\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}
\end{equation}
Accordingly, we have to show the identification of $E[Y_1(1,0)|c]$ to finish the proof. From () we have $E[Y_1(1,0)|D=0,M=0] = E[Q_{10}(Y_0)|D=0,M=0]$. Applying
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\begin{document} \doublespacing \pagestyle{plain}
the law of iterative expectations, gives
\begin{align*}
E[Y_1(1,0)|D=0,M=0] = \frac{p_n}{p_n+p_c} E[Y_1(1,0)|D=0,M=0, \tau=n] \\+ \frac{p_c}{p_n+p_c}E[Y_1(1,0)|D=0,M=0,\tau= c], \\
\stackrel{A7}{=} \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].
\end{align*}
After some rearrangements and using (), we obtain
\begin{equation*}
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].
\end{equation*}
This gives
\begin{equation}
\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}
\end{equation}
using $p_n = p_{0|1}$, and $p_c +p_n = p_{0|0}$.
\subsection{Quantile direct effect under $\mathbf{d = 0}$ on compliers}
We show that
\begin{align*}
 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}c}  F_{Y_1|D=1,M=0}(y) \mbox{ and} \\
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)  ,
\end{align*}
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 (), 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 gives
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
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\begin{document} \doublespacing \pagestyle{plain}expectations gives
\begin{align*}
F_{Y_{1}(1,0)|D=0,M(0)=0}(y)   = \frac{p_n}{p_n + p_c} F_{Y_{1}(1,0)|D=0,M(0)=0,\tau=n}(y)\\  +\frac{p_c}{p_n + p_c} F_{Y_{1}(1,0)|D=0,M(0)=0,\tau=c}(y),\\
 \stackrel{A7}{=} \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).
\end{align*}
Using () and rearranging the equation gives,
\begin{equation}
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).
\end{equation}
In analogy to (), the outcome distribution under $D=0$ and $M=0$ equals:
\begin{equation*}
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).
\end{equation*}
Using () and rearranging the equation gives
\begin{equation}
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) .
\end{equation}
\section{Proof of Theorem 4 }
 \subsection{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 (), we obtain the first ingredient $E[Y_1(0,1)|a]=E[Y_1|D=0,M=1]$. Furthermore, from () 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,
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\begin{document} \doublespacing \pagestyle{plain}
 \subsection{Quantile direct effect on the
always-takers}
We prove that
\begin{align*}
\theta_1^a (q) = F_{Y_1(1,1)|\tau=a}^{-1}(q)- F_{Y_1(0,1)|\tau=a}^{-1}(q), \\
= F_{Q_{11}(Y_0)|D=0,M=1}^{-1}(q)-F_{Y_1|D=0,M=1}^{-1}(q).
\end{align*}
This requires showing that
\begin{align}
F_{Y_1(1,1)|\tau=a}(y) =F_{Q_{11}(Y_0)|D=0,M=1}(y) \mbox{ and} \\
F_{Y_1(0,1)|\tau=a}(y)  = F_{Y_1|D=0,M=1}(y).
\end{align}
Under Assumptions 7 and 8,
\begin{equation}
\begin{array}{rl}
F_{Y_t|D=0,M=1} (y) = E[1\{Y_t\leq y\}|D=0,M=1] \\ \stackrel{A7,A8}{=}E[1\{Y_t(0,1)\leq y\}|\tau=a] \\ = F_{Y_{t}(0,1)|\tau=a}, (y). \end{array}
\end{equation}
which proves (). From (), we have
\begin{equation*}
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].
\end{equation*}
Under Assumption 7 and 8,
\begin{equation}  \begin{array}{rl} E[1\{Y_1(1,1) \leq y\}|D=0,M(0)=1]\stackrel{A7,A8}{=}E[1\{Y_1(1,1) \leq y\}|\tau=a]\\  = F_{Y_1(1,1)|\tau=a}(y),
\end{array} \end{equation}
which proves.
\end{document}


\documentclass[a4paper,12pt]{article}
\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[flushmargin]{footmisc}
\usepackage{setspace}
\usepackage[comma]{natbib}
\usepackage{float}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{ae}
\usepackage{caption}
\usepackage[a4paper,top=3cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
\usepackage{graphicx}
\usepackage[colorinlistoftodos]{todonotes}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\begin{document} \doublespacing \pagestyle{plain}
\subsection{Average direct effect
under $\mathbf{d = 1}$ on compliers}
In the following, we show that
\begin{align*}
\theta_{1}^{c}(1) = E[Y_1(1,1)-Y_1(0,1)|\tau=c], \\ =
\frac{p_{1|1}}{ p_{1|1} - p_{1|0}} E[Y_1-Q_{01}(Y_0) |D=1,M=1] \\ -\frac{p_{1|0}}{p_{1|1} - p_{1|0}}E[Q_{11}(Y_0)-Y_1|D=0,M=1].
\end{align*}
Plugging () in (), we obtain
\begin{equation*}
\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}
\end{equation*}
This allows identifying
\begin{equation}
\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}
\end{equation}
From () 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
\begin{align*}
E[Y_1(0,1)|D=1,M=1] = \frac{p_a}{p_a+p_c} E[Y_1(0,1)|D=1,M=1, \tau=a] \\+ \frac{p_c}{p_a+p_c}E[Y_1(0,1)|D=1,M=1,\tau= c], \\
\stackrel{A7}{=} \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].
\end{align*}
After some rearrangements and using (), we obtain
\begin{equation*}
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].
\end{equation*}
\end{document}