${\boldsymbol H}$ stands for unmeasured confounders.

Theorem 3.1'.
Under SCM ${\cal M}_{IV}'$ and Assumptions 3.1 and 3.2, CAPCE $\mathbb{E}[\partial_x Y_{x}|{\boldsymbol w}]$ is identifiable from distributions $\mathbb{P}(X|Z,{\boldsymbol W})$ and $\mathbb{P}(Y|Z,{\boldsymbol W})$ via the integral equation:

\begin{eqnarray}
\label{d1}
\mu(z,{\boldsymbol w})=\int_{\Omega_X} k(z,x,{\boldsymbol w})\mathbb{E}[\partial_x Y_{x}|{\boldsymbol w}] dx,
\end{eqnarray}

where $\mu(z,{\boldsymbol w})=\mathbb{E}[Y|Z=z_0,{\boldsymbol W}={\boldsymbol w}]-\mathbb{E}[Y|Z=z,{\boldsymbol W}={\boldsymbol w}], k(z,x,{\boldsymbol w})=\mathfrak{p}(X\leq x|Z=z,{\boldsymbol W}={\boldsymbol w})-\mathfrak{p}(X\leq x|Z=z_0,{\boldsymbol W}={\boldsymbol w})$, and $z_0$ is a fixed value.

Eq. (<ref>) is guaranteed by Eq. (<ref>), which appears in the proof of Theorem 3.1.

Theorem 3.1' assumes the separabilty as Assumption 3.1 and 3.2. The disadvantage of Theorem 3.1' is that we have to learn $\mathbb{E}[\partial_x Y_{x}|{\boldsymbol w}]$ by functions on $x$ for each ${\boldsymbol w} \in \Omega_{\boldsymbol W}$ respectively. The advantage of Theorem 3.1 is that we can learn $\mathbb{E}[\partial_x Y_{x}|{\boldsymbol w}]$ by one function on $x$ and ${\boldsymbol w}$.

§.§ Derivation of RKHS CAPCE estimator

We show the detailed steps of deriving the RKHS CAPCE estimator.

RKHS estimator.
The RKHS CAPCE estimator is given as $\hat{\mathbb{E}}[\partial_x{Y}_{x}|{\boldsymbol w}]=\hat{\boldsymbol \alpha}^T{\bf K}_{(X,{\boldsymbol W})^{(1)}(x,{\boldsymbol w})}$ where

\begin{eqnarray}
&&\hat{\boldsymbol \alpha}=(\hat{\bf O}\hat{\bf O}^T+N_2\xi {\bf K}_{(X,{\boldsymbol W})^{(1)}(X,{\boldsymbol W})^{(1)}}+N_2\lambda_3 {\bf I}_{N_2})^{-1}\hat{\bf O}\nonumber\\
&&\hspace{2cm}\times\{{\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2\lambda_2 {\bf I}_{N_2})^{-1}({\bf K}_{Z^{(2)}Z^{(2)}}-{\bf K}_{Z^{(2)}z_0})\}\nonumber\\
&&\hat{\bf O}={\bf K}_{(X,{\boldsymbol W})^{(1)}(X,{\boldsymbol W})^{(1)}}({\bf K}_{Z^{(1)}Z^{(1)}}+N_1\lambda_1 {\bf I}_{N_1})^{-1}({\bf K}_{Z^{(1)}Z^{(2)}}-{\bf K}_{Z^{(1)}z_0}),
\end{eqnarray}

$(\lambda_1,\lambda_2,\lambda_3,\xi)$ are regularization parameters, and ${\bf I}_N$ is a $N \times N$ identity matrix.

There are three optimization problems in RKHS estimator, Stage 1 (A) learning linear operator $G_1$, Stage 1 (B) learning linear operator $G_2$, and Stage 2 learning linear operator $H$.  We explain them respectively.

Stage 1 (A).
We denote the feature map be $\psi(z)$ and $\pi(x,{\boldsymbol w})$, where $\pi(x,{\boldsymbol w})=-\int_{-\infty}^x \eta(x',{\boldsymbol w}) dx'$ for some feature function $\eta(x',{\boldsymbol w})$.
The optimization problem in Stage 1 (A)  becomes

\begin{eqnarray}
\min_{G_1 \in {\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})} {N_1}^{-1}\sum\nolimits_{i=1}^{N_1}\left\|\pi(x_i^{(1)},{\boldsymbol w}_i^{(1)})-G_1(\psi(z_i^{(1)}))\right\|^2_{{\cal H}_{X,{\boldsymbol W}}}+\lambda_1\|G_1\|^2_{{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})}.
\end{eqnarray}

using ${\cal D}^{(1)}$. Then, the estimator $\hat{G_1}$ becomes

\begin{eqnarray}
\hat{G_1}(\cdot)=\left<{\pi_{X^{(1)},{\boldsymbol W}^{(1)}}}({\bf K}_{Z^{(1)}Z^{(1)}}+N_1 \lambda_1 {\bf I})^{-1}\psi_Z^{(1)T},\cdot\right>
\end{eqnarray}

where ${\bf K}_{Z^{(1)}Z^{(1)}}$ and ${\bf K}_{X^{(1)}X^{(1)}}$ are the empirical kernel matrices, the $i$-th column of ${\pi_{X^{(1)},{\boldsymbol W}^{(1)}}}$ is $-\int_{-\infty}^{x_i^{(1)}}\eta(x,{\boldsymbol w})dx$, and the $i$-th column of ${\psi_X}^{(1)}$ is $\psi(z_i^{(1)})$.
The prediction values are

\begin{eqnarray}
d_{0}(z)={\pi_{X^{(1)},{\boldsymbol W}^{(1)}}}({\bf K}_{ZZ}+N_1 \lambda_1 I)^{-1}\psi_Z^{(1)T}\psi(z)=-\sum_{i=1}^{N_1}\gamma_i(z)\int_{-\infty}^{x_i^{(1)}}\eta(x,{\boldsymbol w})dx
\end{eqnarray}

where $\gamma(z)=({\bf K}_{Z^{(1)}Z^{(1)}}+N_1 \lambda_1 {\bf I})^{-1}\psi_Z^{(1)T}\psi(z)=({\bf K}_{Z^{(1)}Z^{(1)}}+N_1 \lambda_1 {\bf I})^{-1}{\bf K}_{Z^{(1)}z}$.
Furthermore, the difference in the predication values are

\begin{eqnarray}
d(z)=d_{0}(z)-d_{0}(z_0)=-\sum_{i=1}^{N_1}\{\gamma_i(z)-\gamma_i(z_0)\}\int_{-\infty}^{x_i^{(1)}}\eta(x,{\boldsymbol w})dx
\end{eqnarray}

and $\gamma(z)-\gamma_i(z_0)=({\bf K}_{Z^{(1)}Z^{(1)}}+N_1 \lambda_1 {\bf I})^{-1}\psi_Z^{(1)T}\{\psi(z)-\psi(z_0)\}=({\bf K}_{Z^{(1)}Z^{(1)}}+N_1 \lambda_1 {\bf I})^{-1}({\bf K}_{Z^{(1)}z}-{\bf K}_{Z^{(1)}z_0})$ holds.
Letting $\hat{G_1}=\sum_{j=1}^{N_1}\alpha_j \eta(x_j^{(1)},{\boldsymbol w}_j^{(1)})$ since the optimal $\hat{G_1}$ exists in ${\text{span}}(\{\eta(x_j^{(1)},{\boldsymbol w}_j^{(1)})\}_{j=1}^{N_1})$ from the representer theorem [47].
Then the functional form of $d(z)$ is restricted by

\begin{eqnarray}
&=&-\left<\sum_{j=1}^{N_1}\alpha_j \int_{-\infty}^{x_i^{(1)}}\eta(x,{\boldsymbol w}_i^{(1)})dx,-\sum_{i=1}^{N_1}\{\gamma_i(z)-\gamma_i(z_0)\}\int_{-\infty}^{x_j^{(1)}}\eta(x,{\boldsymbol w}_j^{(1)})dx\right>\\
&=&\sum_{i=1}^{N_1}\sum_{j=1}^{N_1}\alpha_j\{\gamma_i(z)-\gamma_i(z_0)\}\left< -\int_{-\infty}^{x_i^{(1)}}\eta(x,{\boldsymbol w}_i^{(1)})dx,-\int_{-\infty}^{x_j^{(1)}}\eta(x,{\boldsymbol w}_j^{(1)})dx\right>\\
&=&\sum_{i=1}^{N_1}\sum_{j=1}^{N_1}\alpha_j\{\gamma_i(z)-\gamma_i(z_0)\}\left< \pi(x_i^{(1)},{\boldsymbol w}_i^{(1)}),\pi(x_j^{(1)},{\boldsymbol w}_j^{(1)})\right>.
\end{eqnarray}

From the kernel trick, it becomes

\begin{eqnarray}
&=&\sum_{i=1}^{N_1}\sum_{j=1}^{N_1}\alpha_j\{\gamma_i(z)-\gamma_i(z_0)\}k((x_i^{(1)},{\boldsymbol w}_i^{(1)}),(x_j^{(1)},{\boldsymbol w}_j^{(1)}))\\
&=& {\boldsymbol \alpha}^T w(z)
\end{eqnarray}

where $w(z)={\bf K}_{(X,{\boldsymbol W})^{(1)}(X,{\boldsymbol W})^{(1)}}({\bf K}_{ZZ}+N_1\lambda_1 {\bf I})^{-1}({\bf K}_{Z^{(1)}z}-{\bf K}_{Z^{(1)}z_0})$.
Note that the ${\boldsymbol \alpha}$ will be estimated in Stage 2.

Stage 1 (B).
The optimization problem in Stage 1 (B) is

\begin{eqnarray}
\min_{G_2 \in {\cal L}_2({\cal H}_{{Z}},\Omega_Y)} {N_2}^{-1}\sum\nolimits_{i=1}^{N_2}\left\|y_i^{(2)}-G_2(\psi(z_i^{(2)}))\right\|^2+\lambda_2\|G_2\|^2_{{\cal L}_2({\cal H}_{{Z}},\Omega_Y)}
\end{eqnarray}

using ${\cal D}^{(2)}$. As Stage 1 (A), the estimator of $G_2$, $\hat{G_2}$, become

\begin{eqnarray}
\hat{G_2}(\cdot)=\left<{\boldsymbol y}^{(2)}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2 \lambda_2 {\bf I})^{-1}\psi_Z^{(2)T},\cdot\right>
\end{eqnarray}

where ${\bf K}_{Z^{(2)}Z^{(2)}}$ are the gram matrices, the $i$-th column of ${\boldsymbol y}^{(2)}$ is $y_i^{(2)}$.

\begin{eqnarray}
u_{0}(z)={\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2 \lambda_2 {\bf I})^{-1}\psi_Z^{(2)T}\psi(z)=\sum_{i=1}^{N_2}\gamma_i(z)\psi(z)
\end{eqnarray}

where $\gamma(z)={\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2 \lambda_2 I)^{-1}\psi_Z^{(2)T}$. Then,

\begin{eqnarray}
u_{0}(z_0)={\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2\lambda_2 {\bf I})^{-1}{\bf K}_{Z^{(2)}z_0}
\end{eqnarray}

and, the difference of the predication values are

\begin{eqnarray}
u(z)=u_0(z)-u_0(z_0)={\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2\lambda_2 {\bf I})^{-1}({\bf K}_{Z^{(2)}z}-{\bf K}_{Z^{(2)}z_0}).
\end{eqnarray}

This is the estimator of $\mathbb{E}[Y|Z=z]-\mathbb{E}[Y|Z=z_0]$.

Stage 2.
The optimization problem in Stage 2 using ${\cal D}^{(2)}$ is

\begin{eqnarray}
&&\min_{H \in {\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)} {N_2}^{-1}\sum\nolimits_{i=1}^{N_2}\left\|\hat{G}_2(\psi(z_i^{(2)})-\psi(z_0))-H(\hat{G}_1( \psi(z_i^{(2)})-\psi(z_0)))\right\|^2\nonumber\\
&&\hspace{2cm}+\xi\|H\|^2_{{\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)}+\lambda_3\|H\circ \hat{G}_1\|^2_{{\cal L}_2({\cal H}_Z,\Omega_Y)}.
\end{eqnarray}

Then, the estimation problem reduces to

\begin{eqnarray}
&&\frac{1}{N_2}\sum_{i=1}^{N_2}(y_i^{(2)}-u_{0}(z_0)-{\boldsymbol \alpha}^T w(z))^2+\xi {\boldsymbol \alpha}^T{\bf K}_{XX}{\boldsymbol \alpha}+\lambda_3  {\boldsymbol \alpha}^T {\boldsymbol \alpha}\\
&=&\frac{1}{N_2}\|{\boldsymbol y}^{(2)}-{\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2\lambda_2 {\bf I})^{-1}{\bf K}_{Z^{(2)}z_0}\\
&&\hspace{0cm}-({\bf K}_{X^{(1)}X^{(1)}}({\bf K}_{Z^{(1)}Z^{(1)}}+N_1\lambda_1 {\bf I})^{-1}({\bf K}_{Z^{(1)}Z^{(2)}}-{\bf K}_{Z^{(1)}z_0}))^T{\boldsymbol \alpha}\|^2\\
&&+\xi {\boldsymbol \alpha}^T{\bf K}_{(X,{\boldsymbol W})^{(1)}(X,{\boldsymbol W})^{(1)}}{\boldsymbol \alpha}+\lambda_3  {\boldsymbol \alpha}^T {\boldsymbol \alpha},
\end{eqnarray}

and the solution to this optimization problem can be represented as

\begin{eqnarray}
&&\hat{\boldsymbol \alpha}=(\hat{\bf O}\hat{\bf O}^T+N_2\xi {\bf K}_{(X,{\boldsymbol W})^{(1)}(X,{\boldsymbol W})^{(1)}}+N_2\lambda_3{\bf I})^{-1}\hat{\bf O}\\
&&\hspace{3cm}\times ({\boldsymbol y}^{(2)}-{\boldsymbol y}^{(2)T}({\bf K}_{Z^{(2)}Z^{(2)}}+N_2\lambda_2 {\bf I})^{-1}{\bf K}_{Z^{(2)}z_0})\\
&&\hat{\bf O}={\bf K}_{(X,{\boldsymbol W})^{(1)}(X,{\boldsymbol W})^{(1)}}({\bf K}_{Z^{(1)}Z^{(1)}}+N_1\lambda_1 {\bf I})^{-1}({\bf K}_{Z^{(1)}Z^{(2)}}-{\bf K}_{Z^{(1)}z_0}).
\end{eqnarray}

Finally, RKHS CAPCE estimator of $(x,{\boldsymbol w})$ becomes
$\hat{\mathbb{E}}[\partial_x{Y}_{x}|{\boldsymbol w}]=\hat{\boldsymbol \alpha}^T{\bf K}_{(X,{\boldsymbol W})(x,{\boldsymbol w})}$.

§ CONSISTENCY OF SIEVE CAPCE ESTIMATOR

In this section, we  show that sieve CAPCE estimator is consistent under
assumptions similar to those guaranteeing the consistency of sieve NTSLS [44].

§.§ Notations

We introduce the notations for the assumptions.

Conditional Moment Restrictions.
The estimation problem reduces to the problem called conditional moment restrictions,
and properties of the estimator are well studied [44, 1], and it is widely used in machine learning fields [35].
Since $\mathbb{E}[Y|Z=z_0]-\mathbb{E}[Y|Z]=\mathbb{E}[\mathbb{E}[Y|Z=z_0]-Y|Z]$ and
$\mathbb{E}[\mathbbm{1}_{X\leq x,{\boldsymbol W}={\boldsymbol w}}|Z=z]-\mathbb{E}[\mathbbm{1}_{X\leq x,{\boldsymbol W}={\boldsymbol w}}|Z=z_0]=\mathbb{E}[\mathbbm{1}_{X\leq x,{\boldsymbol W}={\boldsymbol w}}-\mathbb{E}[\mathbbm{1}_{X\leq x,{\boldsymbol W}={\boldsymbol w}}|Z=z_0]|Z=z]$
, Theorem 3.1 reduces to

\begin{eqnarray}
\mathbb{E}\Big[(Y_{X_{z_0}}-Y)-\mathfrak{g}(X,X_{z_0},{\boldsymbol W},g)\Big|Z=z\Big]=0
\end{eqnarray}

where $\mathfrak{g}(X,X_{z_0},{\boldsymbol W},g)=\int_{\Omega_{\boldsymbol W}}\int_{\Omega_X}\{\mathbbm{1}_{X\leq x,{\boldsymbol W}={\boldsymbol w}}-\mathbbm{1}_{X_{z_0}\leq x,{\boldsymbol W}={\boldsymbol w}}\}g(X,{\boldsymbol W})dxd{\boldsymbol w}$.

We denote residual function $\rho(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)=(Y_{X_{z_0}}-Y)-\mathfrak{g}(X,{\boldsymbol W},g))$.

Then, the integral equation can be represented by $\mathbb{E}[{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)|Z]=0$.

Consistency of Sieve CAPCE Estimator.
First, we show consistency without compactness restriction. The Sieve CAPCE estimator reduces to the general form of the conditional moment restrictions method, which is well-studied in [44], as below:

\begin{eqnarray}
\hat{g}=\arg\min_{g \in {\cal G}}\sum_{i=1}^N\frac{1}{N}\hat{\rho}(z_i,g)^2,
\end{eqnarray}

where $\hat{\rho}(z_i,g)=\hat{c}_i-\hat{\boldsymbol d}_i{\boldsymbol \beta}$, and $\hat{\rho}(z_i,g)$ can be considered as the estimators of ${\mathbb{E}}[{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)|Z=z_i]$.

§.§ Assumptions

We make the following consistency assumptions introduced in [44].
We denote ${\cal G}_S=\{g \in{\cal G}: \|\mathfrak{g}_0(x,{\boldsymbol w})\|_{\tilde{W}^{l,2}}^2 \leq B_{S}\}$, and $\overline{{\cal G}_S}$ is a closure of ${\cal G}_S$.
[Uniqueness of $g$]

$g_0 \in {{\cal G}_S}$ is the only $g \in {{\cal G}_S}$ satisfying ${\mathbb{E}}[{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)|Z=z]={0}$.

[Completeness of Stage 1.]

Taking limits $P\rightarrow \infty$, $N \rightarrow \infty$ with $P/N\rightarrow 0$,
there exists ${\boldsymbol \pi}_P$ with $\mathbb{E}[\{b(z)-{\boldsymbol q}(z)^T{\boldsymbol \pi}_P\}^2]\rightarrow 0$
for any $b(z)$ with $\mathbb{E}[b(z)^2]<\infty$.

The above assumption

is for the completeness of parameter space used in Stage 1.
[Boundedness of ${\rho}$]

$\mathbb{E}[\|{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)\|^2|Z]$ is bounded and there exists $M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})$, $\nu>0$ such that for all $\tilde{g},g \in \overline{{\cal G}_S}$, $\|{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},\tilde{g})-{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)\|\leq M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})\|\tilde{g}- g\|_{W^{l,2}}^{\nu}$ and $\mathbb{E}[M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})^2|Z]$ is bounded.

The above assumption is for the boundness of the parameters used in stage 2.

Let ${\cal W}$ denote the domain of $\mathfrak{g}(x,{\boldsymbol w},g)$.
[Openness and Convexness of Restricted Parameter Space]

${\cal W}$ is open and convex.

The following lemma is shown in [44]:

If (i) $\Theta$ is a compact subset of a space with norm $\|\theta\|$: (ii) $\hat{Q}(\theta) \rightarrow_p Q(\theta)$ for all $\theta \in \Theta$: (iii) there is $v >0$ and $B_n O_p(1)$ such that for all $\tilde{\theta}, \theta \in \Theta$, $|\hat{Q}(\theta)-\hat{Q}(\tilde(\theta))| \leq B_n \Delta^v= B_n \epsilon/2M \leq \epsilon/2$ with a positive probability, then $Q(\theta)$ is continuous and $\sup_{\theta \in \Theta} |\hat{Q}(\theta)-Q(\theta)| \rightarrow_p 0$.

Theorem 4.1.
Under SCM ${\cal M}_{IV}$ and Assumptions 3.1, 3.2, 4.1, 4.2, 4.3,

B.1, B.2, B.3, and B.4,
letting $P \rightarrow \infty$ and $J\rightarrow \infty$, then $\|\hat{g}-g_0\|_{W^{l,\infty}}\xrightarrow{p} 0$.

From the Assumption <ref> and <ref>, the parameter space is compact subset.
From the Assumption <ref>, the following relation is satisfied:

\begin{eqnarray}
&&|{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},\tilde{g})-{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)|\\
&&\leq M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})\|\tilde{g}- g\|_{W^{l,2}}^{\nu}
\end{eqnarray}

From the Lemma <ref>,

\begin{eqnarray}
\|\tilde{g}- g\|_{W^{l,\infty}}  \rightarrow_p 0.
\end{eqnarray}

From Assumption <ref>, the limits of $\tilde{g}$ is $g_0$.

From the definition of ${W^{l,\infty}}$, this theorem means uniform convergence.

§ RATE OF CONVERGENCE OF SIEVE CAPCE ESTIMATOR

§.§ Notations

In this section, we explain the notations used in the assumptions for  Theorem 4.2 and Theorem 4.4.

Denote the estimation problem

\begin{eqnarray}
\inf_{g \in {\cal G}}\mathbb{E}\left[\mathfrak{g}(X,X_{z_0},{\boldsymbol W},g)^2\right]
\end{eqnarray}

and  introduce norm $\|\cdot\|_A$ as below:

\begin{eqnarray}
\|g_1-g_0\|_A=\sqrt{\mathbb{E}\left[\left(\frac{d\mathfrak{g}(X,X_{z_0},{\boldsymbol W},g_0)}{dg}\right)^2\right]}
\end{eqnarray}

\begin{eqnarray}
\frac{d{\rho}(Z,g_0)}{dg}[g-g_0]=\frac{d{\rho}(Z,(1-\tau)g_0+\tau g)}{d\tau} \text{ a.s. } Z
\end{eqnarray}

\begin{eqnarray}
\frac{d{\rho}(Z,g_0)}{dg}[g_1-g_2]=\frac{d{\rho}(Z,g_0)}{dg}[g_1-g_0]-\frac{d{\rho}(Z,g_0)}{dg}[g_2-g_0]
\end{eqnarray}

\begin{eqnarray}
\frac{d\mathfrak{g}(X,X_{z_0},{\boldsymbol W},g_0)}{dg}=\mathbb{E}\left[\frac{d{\rho}(Z,g_0)}{dg}[g_1-g_2] \Big|\{X,X_{z_0},{\boldsymbol W}\} \right].
\end{eqnarray}

These derivatives are called “pathwise derivatives." See [1] for details.

To evaluate the rate of convergence, we denote the number of the basis functions depending on sample size be $J_N$ and $P_N$.
Note that $N \rightarrow \infty$ implies $J_N \rightarrow \infty$ and $P_N\rightarrow \infty$.
We use more basis functions, ${\boldsymbol q}^{P_
N}=(q^1,\ldots,q^{P_N})$, as the sample size grows for the stage 1.

§.§ Assumptions

We make the following assumptions.
[Compactness of Domain]

$\Omega_{(X,X_{z_0},{\boldsymbol W})}$ is compact with non empty interior.

[Order of Convergence of Stage 1]

For  any $h \in {\cal G}_S$ with $\kappa>(1+d)/2$, there exists ${\boldsymbol q}^{P_
N}(X,X_{z_0},{\boldsymbol W})^T{\boldsymbol \pi}_{P_N} \in {\cal G}_S$, where ${\boldsymbol \pi}_{P_N}$ is $P_N$ vector, such that $\sup_{(X,X_{z_0},{\boldsymbol W}) \in \Omega_{(X,X_{z_0},{\boldsymbol W})}}|h(X,X_{z_0},{\boldsymbol W})-{\boldsymbol q}^{P_
N}(X,X_{z_0},{\boldsymbol W})^T{\boldsymbol \pi}_{P_N}|={\cal O}(P_N^{-\kappa/(1+d)})$ and $P_N^{-\kappa/(1+d)}={o}(N^{-1/4})$.

The above assumption guarantees the order of convergence of regression (basis functions) used in Stage 1.

[Order of Convergence of Stage 2]

There is a constant $\mu_1>0$ such  that for any $g \in {\cal G}$, there is $\Pi g \in {\cal G}$ satisfying $\|\Pi g-g\|={\cal O}(J_N^{-\kappa/(1+d)})$ and $J_N^{-\kappa/(1+d)}={o}(N^{-1/4})$. $\Pi$ is  the projections to ${\cal G}$.

The above assumption guarantees the order of convergence of regression (basis functions) used in Stage 2.
[Envelope condition]

Each element of ${\rho}(Z,g)$ satisfies the envelope condition in $g \in {\cal G}$;  and, each element of ${\rho}(Z,g) \in {\cal G}_S$ with $\kappa>(1+d)/2$.

The envelope condition is shown in [42].

Denote $\xi_N=\sup_{(X,X_{z_0},{\boldsymbol W})}\|{\boldsymbol q}^{P_
N}(X,X_{z_0},{\boldsymbol W})\|$.
[Condition of $J_N$]

$J_N\times ln(N)\times \xi_N\times N^{-1/2}=o(1)$

We denote $N(\epsilon^{1/k},{\cal G},\|\cdot\|_{W^{l,2}})$ as the minimal number of radius $\delta$ covering ball of ${\cal G}$.
[Condition of $J_N$]

$ln[N(\epsilon^{1/k},{\cal G},\|\cdot\|_{W^{l,2}})] \leq const. \times J_N \times ln(J_N/\epsilon)$

These assumptions show how to make the models complex depending on sample size.

[Convexness of Parameter Space]

${\cal G}$ is convex in $g$, and ${\rho}(Z,g)$ is pathwise differentiable at $g$; and, for some $c_1,c_2>0$,

\begin{eqnarray}
c_1\mathbb{E}[\hat{\rho}(Z,g)^2]\leq \|\hat{g}-g\|^2 \leq c_2\mathbb{E}[\hat{\rho}(Z,g)^2]
\end{eqnarray}

holds for all $\hat{g} \in {\cal G}$ with $\|\hat{g}-g\|_{W^{l,2}}^2=o(1)$

(A) Suppose Assumptions <ref> and <ref>, and the following are satisfied:

(i) there exists a constant $c_{1n}$ and a measurable function $c_1(Z): \Omega_Z \rightarrow [0,\infty)$ with $\mathbb{E}[c_1(Z)^p]<\infty$ for some $p \geq 4$ such that $|g(Z,\alpha)|\leq c_{1n}c_1(Z)$ for all $\alpha\in A_n$ and $Z \in \Omega_Z$.

(ii) there exists a constant $\kappa \in (0,1]$ and a measurable function $c_2(Z): \Omega_Z \rightarrow [0,\infty)$ with $\mathbb{E}[c_2(Z)^2] < \infty$ such that $|g(Z,\alpha_1)-g(Z,\alpha_2)|\leq c_2(Z) \|\alpha_1-\alpha_2\|^{\kappa}_s$ holds for all $Z \in \Omega_Z$ and $\alpha_1, \alpha_2 \in A_n$.

(iii) there exists a positive value $\delta_{1n}=o(1)$ such that

\begin{eqnarray}
\frac{n\delta_{1n}^2}{ln[(\frac{\xi_{1n}c_{1n}}{\delta_{1n}})^{d_x}N((\frac{\delta_{1n}}{\xi_{0n}})^{1/\kappa},A_n,\|\cdot\|_s)]\times \max\{\xi_{0n}^2,c_{1n}^2,\xi_{on}^{2+2/p},\delta_{1n}^{1-2/p},c_{1n}^{1+2/p}\}}\rightarrow + \infty
\end{eqnarray}

Then $p^{k_n}(X)^T(P^TP)^{-1}P^T\epsilon(\alpha)=o_p(\delta_{1n})$ uniformly over $(X,\alpha) \in \Omega_{X,{\boldsymbol W}} \times A_n$.

(B) Suppose Assumptions <ref> and <ref>, and the following are satisfied:

(i) there exists a positive value $\delta_{2n}=o(1)$ and coefficients $\pi(\alpha)$ such that $\mathbb{E}[g(Z,\alpha)|X]=p^{k_n}(X)^T\pi(\alpha)$ holds for all $X \in \Omega_{X,{\boldsymbol W}}$ and $\alpha \in A_n$.

Then $(1/n)\sum_{i=1}^n (p^{k_n}(X)^T(P^TP)^{-1}P^T\sum_{j=1}^np^{k_n}(X_j)\mathbb{E}[g(Z,\alpha)|X_j]-\mathbb{E}[g(Z,\alpha)|X_i])^2=o_p(\delta_{2n}^2)$ uniformly over $\alpha \in A_n$.

(C) Suppose Assumptions <ref> and <ref>, and the following are satisfied:

(v) For fixed $\overline{\alpha_n} \in \overline{A}$, and $\mathbb{E}[\epsilon(Z,\overline{\alpha_n})]\leq const$ for all $n \leq 1$ and $X \in \Omega_{X,{\boldsymbol W}}$.

Then $(1/n)\sum_{i=1}^n\{p^{k_n}(X)^T(P^TP)^{-1}P^T\sum_{j=1}^np^{k_n}(X_j)\epsilon(Z,\overline{\alpha_n})\}^2=O_p(k_n/n)$.

The following lemma holds [1]:

Under Assumptions <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, and <ref>,
(i) $\hat{L}_N(g)-L_N(g)=o_p(N^{-1/4})$ uniformly over $g \in {\cal G}$; and (ii) $\hat{L}_N(g)-\hat{L}_N(g_0)-\{L_N(g)-L_N(g_0)\}=o_p(\tau_N N^{-1/4})$ uniformly over $g \in {\cal G}$ with $\|g-g_0\|\leq o(\tau_N)$, where $\tau_N=N^{-\tau}$ with $\tau\leq 1/4$.

Theorem 4.2.
Under SCM ${\cal M}_{IV}$ and Assumptions 3.1, 3.2, 4.1, 4.2, 4.3,

<ref>, <ref>, <ref>, <ref>, <ref>, <ref>, and <ref>,

setting $N=N_1=N_2$, then $\|\hat{g}-{g}_0\|_{A}={o}_p(N^{-1/4})$.

\begin{eqnarray}
\hat{L}_{N}(g)=-\frac{1}{2N}\hat{\mathfrak{g}}(X,X_{z_0},{\boldsymbol W}.g)^2,\ \ \  {L}_{N}(g)=-\frac{1}{2N}{\mathfrak{g}}(X,X_{z_0},{\boldsymbol W},g)^2.
\end{eqnarray}

Then, Lemma <ref> implies

\begin{eqnarray}
\hat{L}_{N}(g)-\hat{L}_{N}(g_0)-\{{L}_{N}(g)-{L}_{N}(g_0)\}=o_p(N^{-1/4})
\end{eqnarray}

and this proves

\begin{eqnarray}
\|\hat{g}-g_0\|=o_p(N^{-1/4}).
\end{eqnarray}

§ CONSISTENCY OF PARAMETRIC CAPCE ESTIMATOR

In this section, we show the consistency property of the parametric CAPCE estimator.
We denote the functional space ${\cal G}$ be $\{g \in {\cal G}: g(x,{\boldsymbol w})=\sum_{k=1}^K \gamma_k \theta_k(x,{\boldsymbol w})\}$.

Consistency of Parametric CAPCE Estimator.
First, we show consistency without compactness restriction. The Parametric CAPCE estimator reduces to the general form of the conditional moment restrictions method, which is well-studied in [44], as below:

\begin{eqnarray}
\hat{{\boldsymbol \gamma}}=\arg\min_{{\boldsymbol \gamma}}\sum_{i=1}^N\frac{1}{N}\hat{\rho}(z_i,{\boldsymbol \gamma})^2,
\end{eqnarray}

where $\hat{\rho}(z_i,{\boldsymbol \gamma})=\hat{c}_i-\hat{\boldsymbol e}_i{\boldsymbol \gamma}$.
$\hat{\rho}(z_i,{\boldsymbol \gamma})$ can be considered as the estimators of ${\mathbb{E}}[{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})|Z=z_i]$.

§.§ Assumptions

We make the following  assumptions introduced in [44].
We denote ${\cal G}_P=\{{\boldsymbol \gamma}^T{\boldsymbol \gamma} \leq B_{P}\}$, and $\overline{{\cal G}_P}$ is the closure of ${\cal G}_P$.
[Uniqueness of $g$]

${\boldsymbol \gamma} \in {{\cal G}_P}$ is the only ${\boldsymbol \gamma} \in {{\cal G}_P}$ satisfying ${\mathbb{E}}[{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})|Z=z]={\boldsymbol 0}$.

[Completeness of ${\boldsymbol q}$]

Taking limits $P\rightarrow \infty$, $N \rightarrow \infty$ with $P/N\rightarrow 0$,
there exists ${\boldsymbol \pi}_P$ with $\mathbb{E}[\{b(z)-{\boldsymbol q}(z)^T{\boldsymbol \pi}_P\}^2]\rightarrow 0$
for any $b(z)$ with $\mathbb{E}[b(z)^2]<\infty$.

[Boundedness of ${\rho}$]

$\mathbb{E}[\|{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})\|^2|Z]$ is bounded and there exists $M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})$, $\nu>0$ such that for all $\tilde{{\boldsymbol \gamma}},{\boldsymbol \gamma} \in \overline{{\cal G}_P}$, $\|{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},\tilde{{\boldsymbol \gamma}})-{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},g)\|\leq M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})\|\tilde{{\boldsymbol \gamma}}- {\boldsymbol \gamma}\|^{\nu}$ and $\mathbb{E}[M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})^2|Z]$ is bounded.

Let ${\cal W}$ denote the domain of $\mathfrak{g}(x,{\boldsymbol w},{\boldsymbol \gamma})$.
[Openness and Convexness of Restricted Parameter Space]

${\cal W}$ is open and convex.

Theorem 4.3.
Under SCM ${\cal M}_{IV}$ and Assumptions 3.1, 3.2, 4.4,

D.1, D.2, D.3, and D.4,
letting $P \rightarrow \infty$, then $\|\hat{\boldsymbol \gamma}-{\boldsymbol \gamma}\|\xrightarrow{p} 0$.

From the Assumption <ref> and <ref>, the parameter space is compact subset.
From the Assumption <ref>, the following relation is satisfied:

\begin{eqnarray}
&&|{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},\tilde{{\boldsymbol \gamma}})-{\rho}(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})|\\
&&\leq M(Y,Y_{X_{z_0}},X,X_{z_0},{\boldsymbol W})\|\tilde{{\boldsymbol \gamma}}- {\boldsymbol \gamma}\|^{\nu}
\end{eqnarray}

From Lemma <ref>,

\begin{eqnarray}
\|\tilde{{\boldsymbol \gamma}}- {\boldsymbol \gamma}\|  \rightarrow_p 0.
\end{eqnarray}

From Assumption <ref>, the limits is ${\boldsymbol \gamma}_0$.

§ RATE OF CONVERGENCE OF PARAMETRIC CAPCE ESTIMATOR

§.§ Notations

Can this Notations section be deleted? Is it repeating the corresponding section in Appendix C? Only introduce new needed notations.

In this section, we explain the notations for the assumptions of Theorem 4.2 and 4.4.
We introduce more detailed notations for rate of convergence.
Denote the estimation problem

\begin{eqnarray}
\inf_{{\boldsymbol \gamma} \in {\cal G}}\mathbb{E}\left[\mathfrak{g}(X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})^2\right]
\end{eqnarray}

and  introduce norm $\|\cdot\|_A$ as below:

\begin{eqnarray}
\|{\boldsymbol \gamma}_1-{\boldsymbol \gamma}_0\|_A=\sqrt{\mathbb{E}\left[\left(\frac{d\mathfrak{g}(X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}\right)^2\right]}
\end{eqnarray}

\begin{eqnarray}
\frac{d{\rho}(Z,{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}[{\boldsymbol \gamma}-{\boldsymbol \gamma}_0]=\frac{d{\rho}(Z,(1-\tau){\boldsymbol \gamma}_0+\tau {\boldsymbol \gamma})}{d\tau} \text{ a.s. } Z
\end{eqnarray}

\begin{eqnarray}
\frac{d{\rho}(Z,{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}[{\boldsymbol \gamma}_1-{\boldsymbol \gamma}_2]=\frac{d{\rho}(Z,{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}[{\boldsymbol \gamma}_1-{\boldsymbol \gamma}_0]-\frac{d{\rho}(Z,{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}[{\boldsymbol \gamma}_2-{\boldsymbol \gamma}_0]
\end{eqnarray}

\begin{eqnarray}
\frac{d\mathfrak{g}(X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}=\mathbb{E}\left[\frac{d{\rho}(Z,{\boldsymbol \gamma}_0)}{d{\boldsymbol \gamma}}[{\boldsymbol \gamma}_1-{\boldsymbol \gamma}_2] \Big|\{X,X_{z_0},{\boldsymbol W}\} \right].
\end{eqnarray}

These derivatives are called “pathwise derivatives."

§.§ Assumptions

We make the following assumptions.
[Compactness of Domain]

$\Omega_{(X,X_{z_0},{\boldsymbol W})}$ is compact with non empty interior.

[Order of Convergence of Stage 1]

For  any $h \in {\cal G}_P$ with $\kappa>(1+d)/2$, there exists ${\boldsymbol q}^{P_
N}(X,X_{z_0},{\boldsymbol W})^T{\boldsymbol \pi}_{P_N} \in {\cal G}_P $, where ${\boldsymbol \pi_{P_N}}$ is $P_N$ vector, such that $\sup_{(X,X_{z_0},{\boldsymbol W}) \in \Omega_{(X,X_{z_0},{\boldsymbol W})}}|h(X,X_{z_0},{\boldsymbol W})-{\boldsymbol q}^{P_
N}(X,X_{z_0},{\boldsymbol W})^T{\boldsymbol \pi}_{P_N}|={\cal O}(P_N^{-\kappa/(1+d)})$ and $P_N^{-\kappa/(1+d)}={o}(N^{-1/4})$.

[Order of Convergence of Stage 2]

There is a constant $\mu_1>0$ such  that for any ${\boldsymbol \gamma} \in {\cal G}_P$, there is $\Pi {\boldsymbol \gamma} \in {\cal G}_P$

satisfying $\|\Pi {\boldsymbol \gamma}-{\boldsymbol \gamma}\|={\cal O}(1)$.

[Envelope condition]

Each element of ${\rho}(Z,{\boldsymbol \gamma})$ satisfies the envelope condition in ${\boldsymbol \gamma} \in {\cal G}_P$; and, each element of ${\rho}(Z,{\boldsymbol \gamma}) \in {\cal G}_P$ with $\kappa>(1+d)/2$, for all ${\boldsymbol \gamma} \in {\cal G}_P$.

The envelope condition is shown in [42].

[Convexness of Parameter Space]

${\cal G}_P$ is convex in ${\boldsymbol \gamma}$, and ${\rho}(Z,{\boldsymbol \gamma})$ is pathwise differentiable at ${\boldsymbol \gamma}$; and, for some $c_1,c_2>0$,

\begin{eqnarray}
c_1\mathbb{E}[\hat{\rho}(Z,{\boldsymbol \gamma})^2]\leq \|\hat{{\boldsymbol \gamma}}-{\boldsymbol \gamma}\|^2 \leq c_2\mathbb{E}[\hat{\rho}(Z,{\boldsymbol \gamma})^2]
\end{eqnarray}

holds for all $\hat{{\boldsymbol \gamma}} \in {\cal G}_P$ with $\|\hat{{\boldsymbol \gamma}}-{\boldsymbol \gamma}\|^2=o(1)$

The following lemma holds [1]:

Under Assumptions <ref>, <ref>, <ref>, <ref>, and <ref>,
(i) $\hat{L}_N({\boldsymbol \gamma})-L_N({\boldsymbol \gamma})=o_p(N^{-1/4})$ uniformly over ${\boldsymbol \gamma}$; and (ii) $\hat{L}_N({\boldsymbol \gamma})-\hat{L}_N({\boldsymbol \gamma}_0)-\{L_N({\boldsymbol \gamma})-L_N({\boldsymbol \gamma}_0)\}=o_p(\tau_N N^{-1/4})$ uniformly over ${\boldsymbol \gamma}$ with $\|{\boldsymbol \gamma}-{\boldsymbol \gamma}_0\|\leq o(\tau_N)$, where $\tau_N=N^{-\tau}$ with $\tau\leq 1/4$.

Theorem 4.4.
Under SCM ${\cal M}_{IV}$ and Assumptions 3.1, 3.2, 4.4,

E.1, E.2, E.3, E.4, and E.5,
setting $N=N_1=N_2$, then $\|\hat{\boldsymbol \gamma}-{\boldsymbol \gamma}\|={o}_p(N^{-1/4})$.

\begin{eqnarray}
\hat{L}_{N}({\boldsymbol \gamma})=-\frac{1}{2N}\hat{\mathfrak{g}}(X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})^2, \ \ \ {L}_{N}(g)=-\frac{1}{2N}{\mathfrak{g}}(X,X_{z_0},{\boldsymbol W},{\boldsymbol \gamma})^2.
\end{eqnarray}

Then, Lemma <ref> implies

\begin{eqnarray}
\hat{L}_{N}({\boldsymbol \gamma})-\hat{L}_{N}({\boldsymbol \gamma}_0)-\{{L}_{N}({\boldsymbol \gamma})-{L}_{N}({\boldsymbol \gamma}_0)\}=o_p(N^{-1/4})
\end{eqnarray}

and this proves

\begin{eqnarray}
\|\hat{{\boldsymbol \gamma}}-{\boldsymbol \gamma}_0\|=o_p(N^{-1/4}).
\end{eqnarray}

§ PROPERTIES OF RKHS CAPCE ESTIMATOR

We show the consistency and rate of convergence of RKHS CAPCE estimator following [50] when $\lambda_3$ is $0$.

§.§ Notations

We use the integral operator notations from the kernel methods literature. ${\cal L}_2(\Omega_Z,\mathfrak{p}_{Z})$ denotes a  ${\cal L}_2$ integrable function from $\Omega_{Z}$ to $\Omega_Y$ with respect to measure $\mathfrak{p}_{Z}$.

The stage 1 operators are

\begin{eqnarray}
&&S^*_1: {\cal H}_{Z} \rightarrow {\cal L}_2(\Omega_Z,\mathfrak{p}_{Z}), l \mapsto \left<l,\psi(\cdot)\right>_{{\cal H}_{\boldsymbol Z}}\\
&&S_1: {\cal L}_2(\Omega_Z,\mathfrak{p}_{Z}) \rightarrow {\cal H}_{Z}, \tilde{l} \mapsto \int \psi(z) \tilde{l}(z) \mathfrak{p}_{Z}(z) dz
\end{eqnarray}

and $T_1=S^*_1 \circ S_1$ is the uncentered covariance operator.
The details of the theory of vector-valued RKHS are shown in [50].

In addition, we denote

\begin{eqnarray}
{G_1}_{\rho}=\arg \min {\cal E}_1 (E), {\cal E}_1=\mathbb{E}[\pi(X,{\boldsymbol W})-{G_1}(\psi(Z))]_{{\cal H}_{X,{\boldsymbol W}}}^2,
\end{eqnarray}

\begin{eqnarray}
{G_1}_{\lambda}=\arg \min {\cal E}_1 ({G_1}), {\cal E}_1=\mathbb{E}[\pi(X,{\boldsymbol W})-{G_1}(\psi(Z))]_{{\cal H}_{X,{\boldsymbol W}}}^2+\lambda \|G_1\|_{{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})}^2,
\end{eqnarray}

\begin{eqnarray}
\hat{{G_1}}_{\lambda}=\arg \min {\cal E}_1 ({G_1}), {\cal E}_1=\hat{\mathbb{E}}[\pi(X,{\boldsymbol W})-{G_1}(\psi(Z))]_{{\cal H}_{X,{\boldsymbol W}}}^2+\lambda \|G_1\|_{{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})}^2,
\end{eqnarray}

\begin{eqnarray}
{G_2}_{\rho}=\arg \min {\cal E}_1 (E), {\cal E}_1=\mathbb{E}[Y-{G_2}(\psi(Z))]^2,
\end{eqnarray}

\begin{eqnarray}
{G_2}_{\lambda}=\arg \min {\cal E}_1 ({G_2}), {\cal E}_1=\mathbb{E}[Y-{G_2}(\psi(Z))]^2+\lambda \|G_2\|_{{\cal L}_2({\cal H}_Z,\Omega_Y)}^2,
\end{eqnarray}

\begin{eqnarray}
\hat{{G_2}}_{\lambda}=\arg \min {\cal E}_1 ({G_2}), {\cal E}_1=\hat{\mathbb{E}}[Y-{G_2}(\psi(Z))]^2+\lambda \|G_2\|_{{\cal L}_2({\cal H}_Z,\Omega_Y)}^2.
\end{eqnarray}

The stage 2 operators are

\begin{eqnarray}
&&S_2^*: {\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}}) \rightarrow {\cal L}_2({\cal H}_{X,{\boldsymbol W}},\mathfrak{p}_{{\cal H}_{X,{\boldsymbol W}}}),H \mapsto \Omega^*_{(\cdot)}H \\
&&S_2: {\cal L}_2({\cal H}_{X,{\boldsymbol W}},\mathfrak{p}_{{\cal H}_{X,{\boldsymbol W}}}) \rightarrow {\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}}),\\
&&\tilde{H} \mapsto \int \Omega_{\mu(z)-\mu(z_0)}\circ \tilde{H}\{\mu(z)-\mu(z_0)\} \mathfrak{p}_{{\cal H}_{X,{\boldsymbol W}}}(\mu(z))
\end{eqnarray}

and $T_2=S^*_2 \circ S_2$ is the uncentered covariance operator.

We denote

\begin{eqnarray}
H_{\rho}=\arg \min {\cal E} (H), {\cal E}(H)=\mathbb{E}[Y-\mu_2(z_0)-H(\mu(Z)-\mu(z_0))]_{{\cal H}_{X,{\boldsymbol W}}}^2,
\end{eqnarray}

\begin{eqnarray}
&&H_{\xi}=\arg \min {\cal E}_{\xi} (H),\\
&&{\cal E}(H)=\mathbb{E}[Y-\mu_2(z_0)-H(\mu(Z)-\mu(z_0))]_{{\cal H}_{X,{\boldsymbol W}}}^2+\xi \|H\|_{{\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)}^2,
\end{eqnarray}

\begin{eqnarray}
&&\hat{H}_{\xi}=\arg \min \hat{\cal E}_{\xi} (H),\\
&&\hat{\cal E}(H)=\hat{\mathbb{E}}[Y-\mu_2(z_0)-H(\mu(Z)-\mu(z_0))]_{{\cal H}_{X,{\boldsymbol W}}}^2+\xi \|H\|_{{\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)}^2.
\end{eqnarray}

§.§ Assumptions

Next, we show assumptions for Theorem 4.5.
[Restriction for the domains]

Suppose that $\Omega_{X,{\boldsymbol W}}$ and $\Omega_Z$ are Polish spaces, i.e., separable and completely metrizable topological spaces.

[Restriction for the feature functions]

Suppose that

* $k_{X,{\boldsymbol W}}$ and $k_{\boldsymbol Z}$ are continuous and bounded: $\sup_{x \in \Omega_{X,{\boldsymbol W}}} \|\pi(x,{\boldsymbol w})\|_{{\cal H}_{X,{\boldsymbol W}}} \leq Q$ and $\sup_{z \in \Omega_Z} \|\psi(z)\|_{{\cal H}_{\boldsymbol Z}} \leq \kappa$.

* $\pi$ and $\psi$ are measurable.

* $k_{X,{\boldsymbol W}}$ is characteristic.

Suppose that ${G_1}_{\rho} \in {\cal L}_2({\cal H}_{Z},{\cal H}_{\boldsymbol Z})$, then ${\cal E}_1({G_1}_{\rho})=\inf_{{G_1} \in {\cal H}_Z}{\cal E}_1({G_1})$.
Furthermore, suppose that ${G_2}_{\rho} \in {\cal L}_2({\cal H}_{Z},{\cal H}_{\boldsymbol Z})$, then ${\cal E}_1({G_2}_{\rho})=\inf_{{G_2} \in {\cal H}_{Z}}{\cal E}_1({G_2})$.

[Boundness of stage 1]

Fix $\zeta_1, \zeta_2 \leq \infty$. For given $c_1,c_2 \in (1,2]$, define the prior ${\cal P}(\zeta_1,c_1)$ and ${\cal P}(\zeta_2,c_2)$ as the set of the probability distributions on $\Omega_{X,{\boldsymbol W}} \times \Omega_Z$ such that a range space assumption is satisfied: $\exists C_1 \in {\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})$ such that $G_{1\rho}=T_1^{\frac{c_1-1}{2}} \circ C_1$ and $\|C_1\|_{{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})}^2 \leq \zeta_1$, and
$\exists C_2 \in {\cal L}_2({\cal H}_Z,\Omega_Y)$ such that $G_{2\rho}=T_1^{\frac{c_2-1}{2}} \circ C_2$ and $\|C_2\|_{{\cal L}_2({\cal H}_Z,\Omega_Y)}^2 \leq \zeta_2$.

Make Assumptions <ref>, <ref>, <ref> and <ref>. For all $\delta \in (0,1)$, the following holds w.p. $1-\delta$:

\begin{eqnarray}
&&    \|\hat{{G_1}}_{\lambda}-{G_1}_{\rho}\|_{{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})} \nonumber\\
&&\leq \frac{\sqrt{\zeta_1}(c_1+1)}{4^{\frac{1}{c_1+1}}}\left( \frac{4\kappa(Q+\kappa\|{G_1}_{\rho}\|_{{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}})})ln(2/\delta)}{\sqrt{n\zeta_1}(c_1-1)}\right)
\end{eqnarray}

Make Assumptions <ref>, <ref>, <ref> and <ref>. For all $\delta \in (0,1)$, the following holds w.p. $1-\delta$:

\begin{eqnarray}
&&    \|\hat{{G_2}}_{\lambda}-{G_2}_{\rho}\|_{{\cal L}_2({\cal H}_Z,\Omega_Z)} \nonumber\\
&&\leq \frac{\sqrt{\zeta_2}(c_2+1)}{4^{\frac{1}{c_2+1}}}\left( \frac{4\kappa(Q+\kappa\|{G_2}_{\rho}\|_{{\cal L}_2({\cal H}_Z,\Omega_Z)})ln(2/\delta)}{\sqrt{n\zeta_2}(c_2-1)}\right)
\end{eqnarray}

The proof is shown in [50].
The above lemma implies consistency of Stage 1 (A).

[Restriction of domain]

Suppose that $\Omega_Y$ is a Polish space, i.e., separable and completely metrizable topological spaces.

[Boundness of stage 2]

Suppose that

* The $\{\Psi_{\mu(z)-\mu(z_0)}\}$ operator family is uniformly bounded in Hilbert-Schmidt norm: $\exists B$ such that $\forall \mu(z)$, $\|\Psi_{\mu(z)-\mu(z_0)}\|^2_{{\cal L}_2(\Omega_Z,{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}}))}=Tr(\Psi^*_{\mu(z)-\mu(z_0)} \circ \Psi_{\mu(z)-\mu(z_0)}) \leq B$.

* The $\{\Psi_{\mu(z)-\mu(z_0)}\}$ operator family is Hölder continuous in operator norm: $\exists L > 0$, $\iota \in (0,1]$ such that $\forall \mu(z), \mu(z')$, $\|\Psi_{\mu(z)-\mu(z_0)}-\Psi_{\mu(z')-\mu(z_0)}\|_{L(\Omega_Z,{\cal L}_2({\cal H}_Z,{\cal H}_{X,{\boldsymbol W}}))} \leq L \|\mu(z)-\mu(z')\|^{\iota}_{{\cal H}_{X,{\boldsymbol W}}}$.

[Boundness of stage 2]

Suppose that

* $\left<H_{\rho},\cdot\ \right> \in {\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)$. Then, ${\cal E}(H_{\rho})=\inf_{H \in {\cal H}_{X,{\boldsymbol W}}}{\cal E}(H)$.

* $Y$ is bounded, i.e. $\exists C < \infty$ such that $\|Y\| \leq C$ almost surely.

[Boundness of stage 2]

Fix $\zeta < \infty$. For given $b \in (1,\infty]$ and $c \in (1,2]$, define the prior ${\cal P}(\zeta,b,c)$ as the set of probability distributions $\mathfrak{p}$ on ${\cal H}_{X,{\boldsymbol W}} \times \Omega_Y$ such that

* A range space assumption is satisfied: $\exists C \in {\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)$ such that $H_{\rho}=T_2^{\frac{c-1}{2}}\circ C$ and $\|C\|^2_{{\cal L}_2({\cal H}_{X,{\boldsymbol W}},\Omega_Y)} \leq \zeta$.

* In the spectral decomposition $T=\sum_{k=1}^{\infty}\lambda_ke_k\left< \cdot, e_k \right>_{{\cal H}_{X,{\boldsymbol W}}}$, where $\{e_k\}_{k=1}^\infty$ is a basis of $Ker(T)^{\perp}$, the eigenvalues satisfies $\alpha \leq k^b\lambda_k\leq \beta$ for some $\alpha, \beta>0$.

These assumptions are for the boundness of Stage 2.

Make Assumptions <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, <ref> and <ref>. Let $\lambda=N_1^{-\frac{1}{c_1+1}}$, $N_1=N_2^{\frac{a(c_1+1)}{\iota(c_1-1)}}$, $a>0$, and $\lambda_3=0$. We have

* if $a \leq \frac{b(c+1)}{bc+1}$ then ${\cal E}(\hat{H}_{\xi})-{\cal E}({H}_{\rho})={\cal O}_p(N_2^{-\frac{ac}{c+1}})$ with $\xi=N_2^{-\frac{a}{c+1}}$.

* if $a \geq \frac{b(c+1)}{bc+1}$ then ${\cal E}(\hat{H}_{\xi})-{\cal E}({H}_{\rho})={\cal O}_p(N_2^{-\frac{bc}{bc+1}})$ with $\xi=N_2^{-\frac{b}{bc+1}}$.

Lemma <ref> can be proved  from the proof of Theorem 4
in [50] by subsituting $\mu(z)$ with $\mu(z)-\mu(z_0)$.

Theorem 4.5.
Under SCM ${\cal M}_{IV}$ and Assumptions 3.1, 3.2,

<ref>, <ref>, <ref>, <ref>, <ref>, <ref>, <ref> and <ref>,

the  RKHS CAPCE estimator in (25)
converges pointwise to CAPCE
when $\lambda_3=0$.

Lemma <ref> implies consistency of RKHS CAPCE estimator by taking limit $N_2 \rightarrow \infty$.

§ ADDITIONAL INFORMATION ON EXPERIMENTS AND THE APPLICATION

In this section, we give additional information about the settings of  experiments and the application and detailed results.

§.§ Detailed settings of experiments

We present detailed settings of numerical experiments.

Setting of P-CAPCE  and PTSLS. We learn the conditional expectations of basis functions
$\mathbb{E}[Y|Z=z]$, $\mathbb{E}[X|Z=z]$, $\mathbb{E}[WX|Z=z]$ and $\mathbb{E}[X^2|Z=z]$
by the nonlinear model,

$b_0+ b_1Z+b_2Z^2$.

We used the basis terms $\{1,W,X\}$ for P-CAPCE and $\{1,W,X,WX,X^2\}$ for PTSLS, which match setting (A), and let $z_0=-1$.

Regularize value is determined by test error from $\{1,10^{-1},10^{-2},10^{-3}\}$.

Setting of NTSLS and sieve CAPCE. We learn the conditional expectations by the nonlinear model,

$b_0+ b_1Z+b_2Z^2+b_3Z^3$,

We consider the  basis terms $h_p(X)h_q(W)$ for $p=0,1,2$ and $q=0,1,2$, where $h_p$ is Hermite polynomial functions ($h_0(t)=1$, $h_1(t)=t$, $h_2(t)=t^2-1$ and $h_3(t)=t^3-3t$), and let $z_0=-1$.
Let $\kappa=2$, and we calculate $\hat{\Lambda}$ by Monte Carlo integration using uniform distribution $(x,w)=(U(-4,4),U(-2,2))$, where $\Omega_X \subseteq [-4,4]$ and $\Omega_X \subseteq [-2,2]$. Regularize value is determined by test error from $\{1,10^{-1},10^{-2},10^{-3}\}$.

We estimate CAPCE via differentiating estimated $\mathbb{E}[Y_{x}|{W}={w}]$.

Setting of kernel IV and RKHS CAPCE estimator. We use polynomial kernel function $k_Z(z,z')=(z^Tz'+C_1)^{C_2}$ and $k_{X,{W}}((x,{w})(x,{w})^T+C_3)^{C_4}$.
We select the kernel parameters $(C_1,C_2)$ and $(C_3,C_4)$ from $\{1,2,3,4,5\} \times \{1,2,3,4,5\}$, respectively.

We select the regularize values $\lambda_1$ and $\lambda_2$ from $\{1,10^{-1},10^{-2},10^{-3}\}$, respectively, and $(\lambda_3,\xi)$ is from Cartesian product set $\{100,10,1\} \times \{100,10,1\}$.

Show summary results of all estimators in a Table.
The means of coefficient by $100$ time simulations of PTSLS and P-CAPCE estimators are shown in Table <ref>.
The mean of test error of each estimator by $100$ time simulations are shown in Table <ref>.
We note that TSLS (PTSLS and NTSLS) and CAPCE estimators (parametric, sieve and RKHS) have different risk functions.
The estimator of coefficients by P-CAPCE and PTSLS have large SD when $N=1000$, shown in Appendix.
The means of coefficients of P-CAPCE estimators are closer to true coefficients than PTSLS; on the other hand, PTSLS is biased largely due to the violation of the separability when $N=10000$.

We plotted the true and estimated CAPCE curve given $w=1$ in Figure <ref>.
We plotted the true CAPCE surface and estimated CAPCE and estimated PTSLS surface of the setting (A) in Figure <ref>.
The true surface and estimated surfaces of the sieve NTSLS estimator, sieve CAPCE estimator, and RKHS estimator of the setting (B) are shown in Figure <ref>.
The sieve and RKHS CAPCE surface is closer to the true CAPCE surface than NTSLS.

By the experiments' results, we show that our proposed methods (P-CAPCE estimator, sieve CAPCE estimator, and RKHS estimator) are superior to the previous works (PTSLS and sieve NTSLS) in terms of unbiasedness.
In the setting (A), all CAPCE estimators are almost the same MSE; on the other hand, sieve and RKHS CAPCE estimators are superior to P-CAPCE estimator in the setting (B).
The RKHS CAPCE estimator requires much computational time and tuning a lot of regularization parameters.
Sieve CAPCE estimator sometimes requires so many basis functions if the tuple $(X,{\boldsymbol W})$ is high dimensional.

True CAPCE $\mathbb{E}[Y_{x}|{W}={w}]$ is $\text{exp}(x)\text{exp}(w)$, and the sample size are $N=10000$.

Setting of sieve CAPCE estimator. We learn $\mathbb{E}[Y|Z=z], \mathbb{E}[h_p(X)h_q(W)|Z=z]$ for any $p=0,1,2$ and $q=0,1,2$ by the nonlinear model,

$b_0+ b_1Z+b_2Z^2+b_3Z^3$,

where $h_p$ is Hermite polynomial functions ($h_0(t)=1$, $h_1(t)=t$, $h_2(t)=t^2-1$ and $h_3(t)=t^3-3t$), and let $z_0=-1$.
Let $\kappa=2$, and we calculate $\hat{\Lambda}$ by Monte Carlo integration [39] using uniform distridution $(x,w)=(U(-4,4),U(-2,2))$, where $\Omega_X \subseteq [-4,4]$ and $\Omega_X \subseteq [-2,2]$.
Regularize value is determined by test error from $\{1,10^{-1},10^{-2},10^{-3},\ldots\}$.
We regularize the matrix $\displaystyle \hat{\bf G}^T \hat{\bf G}$ by adding $10^{-2} \hat{\Lambda}$.

We estimate CAPCE via differentiating estimated $\mathbb{E}[Y_{x}|{W}={w}]$.

Setting of RKHS CAPCE estimator. We use polynomial kernel function $k_Z(z,z')=(z^Tz'+\zeta_1)^{\zeta_2}$, and we determined $(\zeta_1,\zeta_2)=(4,5)$.
We select the regularize values $\lambda_1,\lambda_2$ from $\{1,0.1,0.01,\ldots\}$, respectively, and $(\lambda_3,\xi)$ is from cartesian product $\{100,10,1\} \times \{100,10,1\}$ by Algorithm <ref>.
Then, we determine $(\lambda_1,\lambda_2,\lambda_3,\xi)=(0.01,0.01,1,100)$.

The true surface and estimated surfaces of the sieve NTSLS estimator, sieve CAPCE estimator, and RKHS estimator are shown in Figure <ref>.
We plotted the surfaces in the range of $-2 \leq x \leq 2$ and $-1 \leq w \leq 1$ since the edge of the domain of all nonparametric estimators is unstable.
The sieve and RKHS CAPCE surfaces are closer to the true CAPCE surface, and the sieve NTSLS surface is largely biased. Is there a numerical performance measure for function estimation that can be computed, e.g. root mean squared error (RMSE) or variants used in Machine learning for regression performance? Then a summary results table could be presented.

Basic statistics of the test error of the CAPCE estimator over 1000 runs for regularization value ($N=1000$); the bold number is the smallest.

$\lambda$      1     0.1     0.01    0.001    0.0001

Min.       14697         9625          3645          12107         624

1st Qu.        1142891       706462        613128        957617        889920

Median     2581357       1935108       1946924       1720578       2562135

3rd Qu.        5944603       6159807       4679010       5361448       6646507

Max.       30939470      26625863      22605166      29466366      19800472

Mean       4690293       4402619       3888198       3887842       4187281

SD         5725131.0     5355639.0     4880814.0     5250446.0     4508818.0

§.§ Additional information on experimental results in the body of paper

Results: Parametric setting (A). The basic statistics of estimated coefficients by $100$ time simulations of
PTSLS and P-CAPCE  are shown in Tables  <ref> and <ref>.
These tables supplement  Table 1 in the paper. The true and estimated CAPCE surfaces over $(X, W)$ are shown in Fig. <ref>.

Results: Nonparametric setting (B). The true and estimated CAPCE surfaces over $(X, W)$ are shown in Fig. <ref>.

Basic statistics of the P-CAPCE estimator over 1000 runs when $N=1000$ and $N=10000$ in setting (A).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -51.515     -18.715     -116.480

1st Qu.     -10.044     0.160       -10.809

Median      -1.849      3.035       17.691

3rd Qu.     7.926       12.239      50.007

Max.        40.458      109.791     213.306

Mean        -1.651      10.383      19.293

SD          14.707      22.309      50.957

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -13.466     -2.302     -23.863

1st Qu.     -2.324      -0.157     10.908

Median      1.138       0.543      22.075

3rd Qu.     4.643       1.568      29.509

Max.        14.988      11.017     59.559

Mean        1.226       0.963      19.971

SD          5.380       2.124      15.487

Basic statistics of the test error of the NTSLS over 1000 runs for regularization value ($N=1000$); the bold number is the smallest.

$\lambda$      1     0.1     0.01

Min.       15606279      13623674      14019980

1st Qu.        20943014      20609263      21674532

Median     23919469      23994604      24633018

3rd Qu.        27036646      27329930      28037611

Max.       38088863      40235204      41456719

Mean       24258777      24218756      25204835

SD         4837954.0     4749243.0     5104092.0

Basic statistics of the PTSLS over 1000 runs when $N=1000$ and $N=10000$ in setting (A).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -33.368     2.971       -113.509

1st Qu.     -4.258      28.662      0.518

Median      2.785       45.497      28.629

3rd Qu.     7.617       62.155      61.569

Max.        26.799      161.738     138.283

Mean        1.248       50.032      27.862

SD          11.374      29.523      46.388

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -12.117     30.794     -20.407

1st Qu.     -2.336      45.907     9.404

Median      1.704       50.716     19.347

3rd Qu.     4.494       57.017     30.498

Max.        8.952       78.573     54.494

Mean        1.101       51.181     19.763

SD          4.638       8.814      15.171

§ EXPERIMENT: ADDITIONAL INFORMATION OF NONPARAMETRIC SETTINGS

Basic statistics of the test error of the sieve CAPCE estimator over 1000 runs for regularization value ($N=1000$); the bold number is the smallest.

$\lambda$      1     0.1     0.01     0.001

Min.       30749         17490         11666         1162

1st Qu.        393222        267892        163843        213469

Median     868635        785864        459062        653862

3rd Qu.        2128704       1850865       1193130       1419073

Max.       11915998      10501820      6207421       9983759

Mean       1637283       1334405       943600        1255112

SD         2209637.0     1693960.0     1263695.0     1732494

Basic statistics of the test error of the NTSLS over 1000 runs for regularization value ($N=10000$); the bold number is the smallest.

$\lambda$      1     0.1     0.01     0.001     0.0001

Min.       21830        20971        13734        19512        24647

1st Qu.       742515       223184       178707       140371       153036

Median     1051622      431384       325818       277296       239295

3rd Qu.       1664026      655220       468745       413653       406530

Max.       3692342      1776898      1911188      1080841      1160372

Mean       1238307      496540       373658       312570       327906

SD         758360.0     356171.3     297606.8     227955.3     261267.5

Basic statistics of the test error of $\lambda_1$ of the RKHS CAPCE estimator over 10 runs for regularization value ($N=100$); the bold number is the smallest.

$\lambda$      1     0.1     0.01     0.001

Min.        353       363.2     334.9     345.7

1st Qu.     374.9     372.6     353.7     366.8

Median      386.1     384       366.5     387.3

3rd Qu.     407.4     397.4     371.2     406.6

Max.        432       432.6     409.5     414.9

Mean        392.2     387.8     365.8     385.6

SD          26.5      20.3      22.0      24.1

Basic statistics of the test error of $\lambda_2$ of the RKHS CAPCE estimator over 10 runs for regularization value ($N=100$); the bold number is the smallest.

$\lambda$      1     0.1     0.01     0.001

Min.        893       1053      1000      797.7

1st Qu.     1261      1141      1028      1220.9

Median      1476      1306      1144      1259.5

3rd Qu.     1581      1457      1246      1475.1

Max.        2189      1986      1545      1650.9

Mean        1466      1384      1169      1277.4

SD          361.5     322.8     170.2     261.3

Basic statistics of the test error of $(\xi,\lambda_3)$ of the RKHS CAPCE estimator over 10 runs for regularization value ($N=100$); the bold number is the smallest.

$(\xi,\lambda_3)$      (100,100))    (100,10)     (100,1)     (10,100))    (10,10)     (10,1)     (1,100))    (1,10)     (1,1)

Min.        763        728.2      448.2      544.2      711.9      527.9        801.7        863.2        779.5

1st Qu.     824.5      804.7      704.4      760.8      741.6      669.5        816          919.9        822.8

Median      900.3      949.3      887.7      881.6      925.4      820.5        886.3        1059.9       875.3

3rd Qu.     974.1      1080       927.3      963.8      1036.5     1099.5       942.3        1312.3       962.5

Max.        1575.5     1468.7     1022.8     1345.9     1826.6     1259.7       1132.3       1815.6       2006.3

Mean        958.8      975.2      810.4      887.5      998.7      871.4        903.7        1159.9       998.1

SD          234.1      222.7      176.4      244.5      346.4      253.3599     104.5766     321.6026     365.5905

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