(0.087) | (0.103) | (0.082) | (0.337) | (0.068) | (0.068) | (0.062) | (0.062) | (0.107) | (0.107)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.212 |  | 1.959 |  | -0.791 |  | 2.059 |  | 3.074
| (0.900) |  | (1.353) |  | (0.652) |  | (2.040) |  | (3.827)
GMM-Lasso |  | 0.983 |  | 1.904 |  | -0.785 |  | 1.083 |  | 0.818
| (1.099) |  | (1.348) |  | (0.956) |  | (1.844) |  | (1.164)
DRSMD-Lasso | 0.307** | 0.777 | 0.042 | -9.902 | -0.320*** | -1.448 | 0.470 | -1.725 | -0.048 | -0.281
(0.142) | (4.239) | (0.153) | (19.627) | (0.085) | (1.554) | (0.417) | (1.821) | (0.522) | (0.590)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B5: Heterogeneous Treatment Effects of Medicaid on Debt for Health

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ TANF.

Average Treatment Effects
---
Debt for Health | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.184** |  | -0.114 |  | -0.131*** |  | -0.177*** |  | -0.147**
| (0.074) |  | (0.083) |  | (0.050) |  | (0.050) |  | (0.068)
GMM-Lasso |  | -0.230 |  | -0.133 |  | -0.228** |  | -0.341*** |  | -0.293
| (0.156) |  | (0.151) |  | (0.111) |  | (0.115) |  | (0.182)
DRSMD-Lasso | -0.203** | -0.185 | -0.135 | -0.440 | -0.155** | -0.168** | -0.194*** | -0.207*** | -0.140 | -0.142
(0.087) | (0.146) | (0.083) | (0.542) | (0.068) | (0.072) | (0.061) | (0.062) | (0.107) | (0.107)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B6: Heterogeneous Treatment Effects of Medicaid on Debt for Health

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ TANF. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Debt for Health | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.174 |  | -0.152 |  | -0.464*** |  | -0.146 |  | -0.424**
| (0.169) |  | (0.143) |  | (0.105) |  | (0.102) |  | (0.179)
GMM-Lasso |  | 0.392 |  | 0.075 |  | -0.302** |  | 0.030 |  | -0.370
| (0.277) |  | (0.220) |  | (0.149) |  | (0.165) |  | (0.299)
DRSMD-Lasso | -0.326*** | -0.136 | -0.229** | -0.116 | -0.260*** | -0.416*** | -0.198*** | -0.131 | -0.194 | -0.403*
(0.110) | (0.195) | (0.100) | (0.166) | (0.095) | (0.119) | (0.075) | (0.115) | (0.137) | (0.210)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.007 |  | -0.010 |  | 0.168*** |  | -0.022 |  | 0.118
| (0.070) |  | (0.065) |  | (0.050) |  | (0.048) |  | (0.073)
GMM-Lasso |  | -0.280*** |  | -0.112 |  | 0.042 |  | -0.183** |  | 0.034
| (0.108) |  | (0.091) |  | (0.068) |  | (0.075) |  | (0.111)
DRSMD-Lasso | 0.049** | -0.035 | 0.040** | -0.009 | 0.052** | 0.135** | 0.001 | -0.032 | 0.023 | 0.117
(0.022) | (0.077) | (0.020) | (0.071) | (0.021) | (0.054) | (0.018) | (0.052) | (0.026) | (0.082)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B7: Heterogeneous Treatment Effects of Medicaid on Debt for Health

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ smoke.

Average Treatment Effects
---
Debt for Health | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.190*** |  | -0.175*** |  | -0.100** |  | -0.192*** |  | -0.141**
| (0.066) |  | (0.064) |  | (0.051) |  | (0.047) |  | (0.068)
GMM-Lasso |  | -0.267* |  | -0.185 |  | -0.211* |  | -0.363*** |  | -0.289
| (0.155) |  | (0.138) |  | (0.114) |  | (0.115) |  | (0.184)
DRSMD-Lasso | -0.210** | -0.220** | -0.135* | -0.136* | -0.146** | -0.123* | -0.197*** | -0.201*** | -0.138 | -0.123
(0.087) | (0.087) | (0.082) | (0.082) | (0.067) | (0.069) | (0.061) | (0.062) | (0.107) | (0.104)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B8: Heterogeneous Treatment Effects of Medicaid on Debt for Health

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ smoke. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

##  Appendix C Proofs of the Theoretical Results

###  C.1 Equivalence between the Objective Functions (2.5) and (2.6)

###### Proof.

The objective function (2.5) can be written as

$M_{\infty}(\theta,g)=\int_{\mathbb{R}^{q_{z}}}E[\epsilon_{j}(\theta,g)e^{it^{\prime}Z_{j}}]E[\epsilon_{l}(\theta,g)e^{-it^{\prime}Z_{l}}]d\mu(t)$

From Assumption 1, for all $j\neq l$,
$Cov(\epsilon_{j}(\theta,g)e^{it^{\prime}Z_{j}},\epsilon_{l}(\theta,g)e^{-it^{\prime}Z_{l}})=0$.
Thus, for all $j\neq l$, we have:

$\displaystyle M_{\infty}(\beta)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}E(\epsilon_{j}(\theta,g)e^{it^{\prime}Z_{j}}\epsilon_{l}(\theta,g)e^{-it^{\prime}Z_{l}})d\mu(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}E(\epsilon_{j}(\theta,g)\epsilon_{l}(\theta,g)e^{it^{\prime}(Z_{j}-Z_{l})})d\mu(t)$
$\displaystyle=$ $\displaystyle
E(\int_{\mathbb{R}^{q_{z}}}\epsilon_{j}(\theta,g)\epsilon_{l}(\theta,g)e^{it^{\prime}(Z_{j}-Z_{l})}d\mu(t))$

Thus, the objective function becomes

$M_{\infty}(\beta)=E(\epsilon_{j}(\theta,g)\epsilon_{l}(\theta,g)\kappa_{j,l})$

where
$\kappa_{j,l}=k(Z_{j}-Z_{l})=\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(Z_{j}-Z_{l})}d\mu(t)$.
And $k(u)$ is the inverse Fourier transform of $d\mu(t)$ with $u=Z_{j}-Z_{l}$.
∎

###  C.2 Proof of Proposition 1

The orthogonal FOC in Equation (2.9) implies that

$E\left[\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)(\tilde{y}_{l}-\tilde{P}_{l}^{\prime}\theta)\kappa_{j,l}\right]=0$
$E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{y}_{l}-\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\theta\right]=0$
$\theta_{0}=E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]^{-1}E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{y}_{l}\right]$

The proof for invertibility of
$E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]$
follows the same steps of proofs for identification in Antoine and Sun (2021).

###  C.3 Proof of Orthogonal Properties of Equation (2.7) and (2.9)

In this section, we check the orthogonal properties of the two FOCs.

$M_{\infty}(\theta,g)=-\frac{1}{2}E[\epsilon_{j}(\theta)\epsilon_{l}(\theta)\kappa_{j,l}]=-\frac{1}{2}E[(\tilde{y}_{j}-\tilde{P}_{j}^{\prime}\theta)(\tilde{y}_{l}-\tilde{P}_{l}^{\prime}\theta)\kappa_{j,l}]$

where $g$ stands for all nuisance parameters.

The FOC defined in Equation (2.7) is not orthogonal

The key parameter is $\theta$, the true value is $\theta_{0}$. The nuisance
parameters are $g_{P}(X_{i})$ and $g_{y}(X_{i})$. Their true values are
$E(P_{i}|X_{i})=g_{0,P}(X_{i})$ and $E(y_{i}|X_{i})=g_{0,y}(X_{i})$.

The first order condition is written as follows:

$E[(P_{j}-g_{P}(X_{j}))(y_{l}-g_{y}(X_{l})-[P_{l}-g_{P}(X_{l})]^{\prime}\theta)\kappa_{j,l}]=0$

If $P_{i}=[W_{i},f(W_{i},X_{i})]^{\prime}$ and there are only one variable in
$X_{i}$, $\partial_{\theta}\varphi(D;\theta,g)$ becomes:

$\partial_{\theta}\varphi(D;\theta,g)=\begin{bmatrix}(P_{1j}-g_{P_{1}}(X_{j}))[y_{l}-g_{y}(X_{l})-(P_{1l}-g_{P_{1}}(X_{l}))\theta_{1}-(P_{2l}-g_{P_{2}}(X_{l}))\theta_{2}]\kappa_{j,l}\\\
(P_{2j}-g_{P_{2}}(X_{j}))[y_{l}-g_{y}(X_{l})-(P_{1l}-g_{P_{1}}(X_{l}))\theta_{1}-(P_{2l}-g_{P_{2}}(X_{l}))\theta_{2}]\kappa_{j,l}\end{bmatrix}$

Prove that $\partial_{g}E[\partial_{\theta}\varphi(D;\theta_{0},g_{0})]\neq
0$.

###### Proof.

The first row of $\partial_{\theta}\varphi(D;\theta_{0},g_{0}+r(g-g_{0}))$ is
defined as $I$ where

$\displaystyle I=$
$\displaystyle[P_{1j}-g_{0,P_{1}}(X_{j})-r(g_{P_{1}}(X_{j})-g_{0,P_{1}}(X_{j}))]$
$\displaystyle[y_{l}-g_{0,y}(X_{l})-r(g_{y}(X_{l})-g_{0,y}(X_{l}))$
$\displaystyle-$
$\displaystyle(P_{1l}-g_{0,P_{1}}(X_{l})-r(g_{P_{1}}(X_{l})-g_{0,P_{1}}(X_{l})))\theta_{1}$
$\displaystyle-$
$\displaystyle(P_{2l}-g_{0,P_{2}}(X_{l})-r(g_{P_{2}}(X_{l})-g_{0,P_{2}}(X_{l})))\theta_{2}]\kappa_{j,l}$

According to the definition for
$\partial_{g}E[\partial_{\theta}\varphi(D;\theta_{0},g_{0})]\neq 0$ in
Chernozhukov et al. (2018), to show that
$\partial_{g}E[\partial_{\theta}\varphi(D;\theta_{0},g_{0})]\neq 0$ we need to
show $\partial_{r}E[I]|_{r=0}\neq 0$.

$\partial_{r}E[I]|_{r=0}=-I_{1}-I_{2}+I_{3}+I_{4}$ $\displaystyle I_{1}$
$\displaystyle=$ $\displaystyle
E\left[(g_{P_{1}}(X_{j})-g_{0,P_{1}}(X_{j}))(y_{l}-g_{0,y}(X_{l})-(P_{1l}-g_{0,P_{1}}(X_{l}))\theta_{0,1}-(P_{2l}-g_{0,P_{2}}(X_{l}))\theta_{0,2})\kappa_{j,l}\right]$
$\displaystyle=$ $\displaystyle
E\left[(g_{P_{1}}(X_{j})-g_{0,P_{1}}(X_{j}))\epsilon_{l}\kappa_{j,l}\right]=0$
$\displaystyle I_{2}$ $\displaystyle=$ $\displaystyle
E\left[(P_{1j}-g_{0,P_{1}}(X_{j}))(g_{y}(X_{l})-g_{0,y}(X_{l}))\kappa_{j,l}\right]\neq
0$ $\displaystyle I_{3}$ $\displaystyle=$ $\displaystyle
E\left[(P_{1j}-g_{0,P_{1}}(X_{j}))(g_{P_{1}}(X_{l})-g_{0,P_{1}}(X_{l}))\theta_{1}\kappa_{j,l}\right]\neq
0$ $\displaystyle I_{4}$ $\displaystyle=$ $\displaystyle
E\left[(P_{1j}-g_{0,P_{1}}(X_{j}))(g_{P_{2}}(X_{l})-g_{0,P_{2}}(X_{l}))\theta_{2}\kappa_{j,l}\right]\neq
0$

Hence, $\partial_{g}E[\partial_{\theta}\varphi(D;\theta_{0},g_{0})]\neq 0$

∎

The FOC defined in Equation (2.9) is orthogonal

With
$E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]=g_{0,\tilde{P}_{1,j}}(X_{l})$,
$E[\kappa_{m,l}|X_{l}]=g_{0,\kappa_{m,l}}(X_{l})$,

$\Psi(D;\theta,g)$

$=\begin{bmatrix}\left(P_{1j}-g_{P_{1}}(X_{j})-\frac{g_{\tilde{P}_{1,j}}(X_{l})}{g_{\kappa_{j,l}}(X_{l})}\right)[y_{l}-g_{y}(X_{l})-(P_{1l}-g_{P_{1}}(X_{l}))\theta_{1}-(P_{2l}-g_{P_{2}}(X_{l}))\theta_{2}]\kappa_{j,l}\\\
\left(P_{2j}-g_{P_{2}}(X_{j})-\frac{g_{\tilde{P}_{2,j}}(X_{l})}{g_{\kappa_{j,l}}(X_{l})}\right)[y_{l}-g_{y}(X_{l})-(P_{1l}-g_{P_{1}}(X_{l}))\theta_{1}-(P_{2l}-g_{P_{2}}(X_{l}))\theta_{2}]\kappa_{j,l}\end{bmatrix}$

Prove that $\partial_{g}E[\Psi(D;\theta_{0},g_{0})]=0$.

###### Proof.

The first row of $\Psi(D;\theta_{0},g_{0}+r(g-g_{0}))$ is $I^{\prime}$.

$\displaystyle I^{\prime}=$
$\displaystyle\left(P_{1j}-g_{0,P_{1}}(X_{j})-r(g_{P_{1}}(X_{j})-g_{0,P_{1}}(X_{j}))-\frac{g_{0,\tilde{P}_{1,j}}(X_{l})+r[g_{\tilde{P}_{1,j}}(X_{l})-g_{0,\tilde{P}_{1,j}}(X_{l})]}{g_{0,\kappa_{m,l}}(X_{l})+r[g_{\kappa_{j,l}}(X_{l})-g_{0,\kappa_{m,l}}(X_{l})]}\right)$
$\displaystyle[y_{l}-g_{0,y}(X_{l})-r(g_{y}(X_{l})-g_{0,y}(X_{l}))$
$\displaystyle-$
$\displaystyle(P_{1l}-g_{0,P_{1}}(X_{l})-r(g_{P_{1}}(X_{l})-g_{0,P_{1}}(X_{l})))\theta_{1}$
$\displaystyle-$
$\displaystyle(P_{2l}-g_{0,P_{2}}(X_{l})-r(g_{P_{2}}(X_{l})-g_{0,P_{2}}(X_{l})))\theta_{2}]\kappa_{j,l}$
$\partial_{r}E[I^{\prime}]|_{r=0}=-I_{1}-I_{2}^{\prime}+I_{3}^{\prime}+I_{4}^{\prime}-I_{5}$
$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle
E\left[(g_{P_{1}}(X_{j})-g_{0,P_{1}}(X_{j}))(y_{l}-g_{0,y}(X_{l})-(P_{1l}-g_{0,P_{1}}(X_{l}))\theta_{0,1}-(P_{2l}-g_{0,P_{2}}(X_{l}))\theta_{0,2})\kappa_{j,l}\right]$
$\displaystyle=$ $\displaystyle 0$ $\displaystyle I_{2}^{\prime}$
$\displaystyle=$ $\displaystyle
E\left[(P_{1j}-g_{0,P_{1}}(X_{j})-\frac{E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]}{E[\kappa_{m,l}|X_{l}]})(g_{y}(X_{l})-g_{0,y}(X_{l})\kappa_{j,l}\right]$
$\displaystyle=$ $\displaystyle
E\left(E\left[\left(P_{1j}-g_{0,P_{1}}(X_{j})-\frac{E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]}{E[\kappa_{m,l}|X_{l}]}\right)\kappa_{j,l}|X_{l}\right](g_{y}(X_{l})-g_{0,y}(X_{l})\right)$
$\displaystyle=$ $\displaystyle
E\left(E\left[\left(P_{1j}-g_{0,P_{1}}(X_{j})\right)\kappa_{j,l}-\frac{E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]}{E[\kappa_{m,l}|X_{l}]}\kappa_{j,l}|X_{l}\right](g_{y}(X_{l})-g_{0,y}(X_{l})\right)$
$\displaystyle=$ $\displaystyle 0$

because
$E\left[\left(P_{1j}-g_{0,P_{1}}(X_{j})\right)\kappa_{j,l}-\frac{E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]}{E[\kappa_{m,l}|X_{l}]}\kappa_{j,l}|X_{l}\right]=0$.

$\displaystyle I_{3}^{\prime}$ $\displaystyle=$ $\displaystyle
E\left[(P_{1j}-g_{0,P_{1}}(X_{j})-\frac{E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]}{E[\kappa_{m,l}|X_{l}]})(g_{P_{1}}(X_{l})-g_{0,P_{1}}(X_{l}))\theta_{1}\kappa_{j,l}\right]=0$
$\displaystyle I_{4}^{\prime}$ $\displaystyle=$ $\displaystyle
E\left[(P_{1j}-g_{0,P_{1}}(X_{j})-\frac{E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]}{E[\kappa_{m,l}|X_{l}]})(g_{P_{2}}(X_{l})-g_{0,P_{2}}(X_{l}))\theta_{2}\kappa_{j,l}\right]=0$
$\displaystyle I_{5}$ $\displaystyle=$ $\displaystyle
E\left[\frac{\left(g_{\tilde{P}_{1,j}}(X_{l})-g_{0,\tilde{P}_{1,j}}(X_{l})\right)g_{0,\kappa_{m,l}}(X_{l})-\left(g_{\kappa_{j,l}}(X_{l})-g_{0,\kappa_{m,l}}(X_{l})\right)g_{0,\tilde{P}_{1,j}}(X_{l})}{(g_{0,\kappa_{m,l}}(X_{l}))^{2}}\epsilon_{l}\kappa_{j,l}\right]=0$

Since $E(\epsilon_{l}|X_{l},Z_{l})=0$.

Hence, $\partial_{g}E[\Psi(D;\theta_{0},g_{0})]=0$.

∎

###  C.4 Proof of Proposition 2:

###### Proof.

$\displaystyle\tilde{\theta}_{n,o}=$
$\displaystyle\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]^{-1}$
$\displaystyle\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)(\tilde{P}_{l}\theta_{0}+\epsilon_{l})\right]$
$\displaystyle=$
$\displaystyle\theta_{0}+\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]^{-1}$
$\displaystyle\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\epsilon_{l}\right]$

Denote $A_{n}=\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}$
and

$B_{n}=\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\epsilon_{l}$.
We first show that $A_{n}$ is a U-statistic and find its probability limit.
Then we show that $B_{n}$ is also a U-statistic and find its probability
limit.
To show that $A_{n}$ is a U-statistic, notice that

$A_{n}=\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}$
$=\frac{1}{2}\frac{2}{n(n-1)}\sum_{j<l}^{n}\left(\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}+\kappa_{l,j}\left(\tilde{P}_{l}-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\right)\tilde{P}_{j}^{\prime}\right)$

According to WLLN for U-statistics under Assumption 1,

$A_{n}\xrightarrow{p}A\qquad\text{with}\qquad A\equiv
E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]$

and under Assumption 1, $A$ is nonsingular.
To show that $B_{n}$ is also a U-statistic, notice that:

$B_{n}=\frac{1}{n(n-1)}\sum_{j<l}^{n}\left(\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\epsilon_{l}+\kappa_{l,j}\left(\tilde{P}_{l}-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\right)\epsilon_{j}\right)$

Define
$h(\tilde{p}_{1},e_{1},z_{1},x_{1};\tilde{p}_{2},e_{2},z_{2},x_{2})=\kappa_{1,2}\tilde{p}_{1}e_{2}+\kappa_{2,1}\tilde{p}_{2}e_{1}-\kappa_{1,2}\frac{g_{0,\tilde{P}_{1}}(x_{2})}{g_{0,\kappa_{1,2}}(x_{2})}e_{2}-\kappa_{2,1}\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}e_{1}$.
Since $h$ is a symmetric function of observations 1 and 2, a U-statistic with
kernel $h$ is defined as

$B_{n}^{\prime}=\frac{2}{n(n-1)}\sum_{j<l}^{n}h(\tilde{P}_{j},\epsilon_{j},Z_{j},X_{j};\tilde{P}_{l},\epsilon_{l},Z_{l},X_{l})\qquad\text{and}\qquad
B_{n}=\frac{1}{2}B_{n}^{\prime}$

And we have:

$\displaystyle E(B_{n}^{\prime})$ $\displaystyle=$ $\displaystyle
E(\kappa_{j,l}\tilde{P}_{j}\epsilon_{l}+\kappa_{l,j}\tilde{P}_{l}\epsilon_{j})-\left(E\left(\kappa_{j,l}\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\epsilon_{l}\right)+E\left(\kappa_{l,j}\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\epsilon_{j}\right)\right)$
$\displaystyle=$ $\displaystyle
2E(\kappa_{j,l}\tilde{P}_{j}\epsilon_{l})-2E\left(\kappa_{j,l}\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\epsilon_{l}\right)$
$\displaystyle=$ $\displaystyle
2\int_{\mathbb{R}^{q_{z}}}E[\tilde{P}_{j}\epsilon_{l}e^{it^{\prime}(Z_{j}-Z_{l})}]d\mu(t)-2\int_{\mathbb{R}^{q_{z}}}E[\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\epsilon_{l}e^{it^{\prime}(Z_{j}-Z_{l})}]d\mu(t)$
$\displaystyle=$ $\displaystyle
2\int_{\mathbb{R}^{q_{z}}}E[\tilde{P}_{j}e^{it^{\prime}Z_{j}}\epsilon_{l}e^{-it^{\prime}Z_{l}}]d\mu(t)-2\int_{\mathbb{R}^{q_{z}}}E[\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}e^{it^{\prime}Z_{j}}\epsilon_{l}e^{-it^{\prime}Z_{l}}]d\mu(t)$
$\displaystyle=$ $\displaystyle
2\int_{\mathbb{R}^{q_{z}}}E[\tilde{P}_{j}e^{it^{\prime}Z_{j}}]E[\epsilon_{l}e^{-it^{\prime}Z_{l}}]d\mu(t)-2\int_{\mathbb{R}^{q_{z}}}E[e^{it^{\prime}Z_{j}}]E[\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\epsilon_{l}e^{-it^{\prime}Z_{l}}]d\mu(t)$
$\displaystyle=$ $\displaystyle 0\qquad\text{since
$E[\epsilon_{l}^{\prime}e^{-it^{\prime}Z_{l}}]=0$ and
$E[\epsilon_{l}|X_{l},Z_{l}]=0$. }$

Hence, $E(B_{n})=0$. According to WLLN for U statistics, we have
$B_{n}\xrightarrow{p}0$, and we conclude that $\tilde{\beta}_{n}$ is a
consistent estimator of $\beta_{0}$.

To derive the asymptotic normality, we first need to compute the asymptotic
variance for the U-statistic $B_{n}^{\prime}$, which means that we need to
find the variance for
$E(h(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1};\tilde{P}_{2},\epsilon_{2},Z_{2},X_{2})|\tilde{P}_{1}=\tilde{p}_{1},\epsilon_{1}=e_{1},Z_{1}=z_{1},X_{1}=x_{1})$.
Let $h_{1}(\tilde{p}_{1},e_{1},z_{1},x_{1})\equiv
E(h(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1};\tilde{P}_{2},\epsilon_{2},Z_{2},X_{2})|\tilde{P}_{1}=\tilde{p}_{1},\epsilon_{1}=e_{1},Z_{1}=z_{1},X_{1}=x_{1})$.
We have:

$h(\tilde{p}_{1},e_{1},z_{1},x_{1};\tilde{p}_{2},e_{2},z_{2},x_{2})=\kappa_{1,2}\tilde{p}_{1}e_{2}+\kappa_{2,1}\tilde{p}_{2}e_{1}-\kappa_{1,2}\frac{g_{0,\tilde{P}_{1}}(x_{2})}{g_{0,\kappa_{1,2}}(x_{2})}e_{2}-\kappa_{2,1}\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}e_{1}$

and

$\displaystyle h_{1}(\tilde{p}_{1},e_{1},z_{1},x_{1})$ $\displaystyle=$
$\displaystyle
E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\tilde{p}_{1}\epsilon_{2}+\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\tilde{P}_{2}e_{1}$
$\displaystyle-$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\frac{g_{0,\tilde{P}_{1}}(X_{2})}{g_{0,\kappa_{1,2}}(X_{2})}\epsilon_{2}-\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}e_{1}]$
$\displaystyle=$ $\displaystyle
E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\tilde{p}_{1}\epsilon_{2}]+E[\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\tilde{P}_{2}e_{1}]$
$\displaystyle-$ $\displaystyle
E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\frac{g_{0,\tilde{P}_{1}}(X_{2})}{g_{0,\kappa_{1,2}}(X_{2})}\epsilon_{2}]-E[\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}e_{1}]$

The first element of the right hand side is

$\displaystyle
E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\tilde{p}_{1}\epsilon_{2}]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}E[e^{it^{\prime}z_{1}}e^{-it^{\prime}Z_{2}}\tilde{p}_{1}\epsilon_{2}]d\mu(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}z_{1}}\tilde{p}_{1}E[e^{-it^{\prime}Z_{2}}\epsilon_{2}]d\mu(t)=0$

The third term is

$\displaystyle
E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\frac{g_{0,\tilde{P}_{1}}(X_{2})}{g_{0,\kappa_{1,2}}(X_{2})}\epsilon_{2}]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}E[e^{it^{\prime}z_{1}}e^{-it^{\prime}Z_{2}}\frac{g_{0,\tilde{P}_{1}}(X_{2})}{g_{0,\kappa_{1,2}}(X_{2})}\epsilon_{2}]d\mu(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}z_{1}}\tilde{p}_{1}E[e^{-it^{\prime}Z_{2}}\frac{g_{0,\tilde{P}_{1}}(X_{2})}{g_{0,\kappa_{1,2}}(X_{2})}\epsilon_{2}]d\mu(t)=0$

The second term of the right hand side is

$\displaystyle
E[\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\tilde{P}_{2}e_{1}]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}E[e^{it^{\prime}Z_{2}}e^{-it^{\prime}z_{1}}\tilde{P}_{2}e_{1}]d\mu(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}z_{1}}e_{1}E[e^{it^{\prime}Z_{2}}\tilde{P}_{2}]d\mu(t)$

The fourth term is

$\displaystyle
E[\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}e_{1}]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}E[e^{it^{\prime}Z_{2}}e^{-it^{\prime}z_{1}}\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}e_{1}]d\mu(t)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}z_{1}}e_{1}\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}E[e^{it^{\prime}Z_{2}}]d\mu(t)$

Hence,
$h_{1}(\tilde{p}_{1},e_{1},z_{1},x_{1})=\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}z_{1}}e_{1}E[e^{it^{\prime}Z_{2}}\tilde{P}_{2}]d\mu(t)-\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}z_{1}}e_{1}\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}E[e^{it^{\prime}Z_{2}}]d\mu(t)$.
$h_{1}(\tilde{p}_{1},e_{1},z_{1},x_{1})=\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}z_{1}}e_{1}\left(E[e^{it^{\prime}Z_{2}}\tilde{P}_{2}]-\frac{g_{0,\tilde{P}_{2}}(x_{1})}{g_{0,\kappa_{2,1}}(x_{1})}E[e^{it^{\prime}Z_{2}}]\right)d\mu(t)$
Since $E(h_{1}(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1}))=0$, we have:

$\displaystyle Var[h_{1}(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1})]$
$\displaystyle=$ $\displaystyle
Var\left[\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}Z_{1}}\epsilon_{1}\left(E[e^{it^{\prime}Z_{2}}\tilde{P}_{2}]-\frac{g_{0,\tilde{P}_{2}}(X_{1})}{g_{0,\kappa_{2,1}}(X_{1})}E[e^{it^{\prime}Z_{2}}]\right)d\mu(t)\right]$
$\displaystyle=$ $\displaystyle
Var\left[\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}Z_{j}}\epsilon_{j}\left(E[e^{it^{\prime}Z_{l}}\tilde{P}_{l}]-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}E[e^{it^{\prime}Z_{l}}]\right)d\mu(t)\right]$

Following Hoeffding (1948), the asymptotic distribution for U-statistics
yields:

$\sqrt{n}(B_{n}^{\prime}-0)\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,4Var[h_{1}(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1})])$

Thus,

$\sqrt{n}(\tilde{\theta}_{n,o}-\theta_{0})\xrightarrow{d}N\left(0,A^{-1}Var[h_{1}(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1})]\left(A^{-1}\right)^{\prime}\right)$

with
$A=E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]$.
∎

###  C.5 Proof of Theorem 3.2

###### Proof.

Recall that

$\widehat{\tilde{y}}_{i}=y_{i}-\hat{g}_{y}(X_{i})=y_{i}-E(y_{i}|X_{i})+E(y_{j}|X_{i})-\hat{g}_{y}(X_{i})$

For the first two terms of the right hand side, we have

$\displaystyle y_{i}-E(y_{i}|X_{i})$ $\displaystyle=$
$\displaystyle(P_{i}-E(P_{i}|X_{i}))^{\prime}\theta_{0}+\epsilon_{i}$
$\displaystyle=$
$\displaystyle(P_{i}-\hat{g}_{P}(X_{i}))^{\prime}\theta_{0}+(\hat{g}_{P}(X_{i})-E(P_{i}|X_{i}))^{\prime}\theta_{0}+\epsilon_{i}$

In matrix form, the feasible estimator writes:

$\hat{\theta}_{n,o}=\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}\right]^{-1}\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{y}}_{l}\right]$

Define $C_{n}=\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}$.
We get:

$\displaystyle\hat{\theta}_{n,o}$ $\displaystyle=$
$\displaystyle[C_{n}]^{-1}[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)[y_{l}-E(y_{l}|X_{l})+E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]]$
$\displaystyle=$
$\displaystyle[C_{n}]^{-1}[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)[(P_{l}-\hat{g}_{P}(X_{l}))^{\prime}\theta_{0}$
$\displaystyle+(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}+\epsilon_{l}+E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]]$
$\displaystyle=$
$\displaystyle\theta_{0}+[C_{n}]^{-1}[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)$
$\displaystyle[(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}+\epsilon_{l}+E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]]$

Consider now,

$\displaystyle A_{n}-C_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}$
$\displaystyle-\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}$
$\displaystyle=$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}$
$\displaystyle-\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}+\widehat{\tilde{P}}_{j}-\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}$
$\displaystyle=$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}[\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}$
$\displaystyle+\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)(-\widehat{\tilde{P}}_{l}+\tilde{P}_{l})]$
$\displaystyle\stackrel{{\scriptstyle P}}{{\rightarrow}}$ $\displaystyle 0$

Hence, we have $\verb'Plim'C_{n}=A$, since we showed in the proof of
Proposition 2 that $\verb'Plim'A_{n}=A$.

Define now the following quantities:

$\displaystyle D_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)[(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}+\epsilon_{l}+E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]$
$\displaystyle E_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle F_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\epsilon_{l}$
$\displaystyle G_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)[E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]$

We have, $D_{n}=E_{n}+F_{n}+G_{n}$. For consistency, we show that the
probability limits for $E_{n}$, $F_{n}$, and $G_{n}$ are all zero.

$\displaystyle E_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}+\widehat{\tilde{P}}_{j}-\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)$
$\displaystyle(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}+\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)$
$\displaystyle(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle=$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle+$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle\stackrel{{\scriptstyle P}}{{\rightarrow}}0$ $\displaystyle
F_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}+\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\epsilon_{l}$
$\displaystyle=$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\epsilon_{l}$
$\displaystyle+$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\epsilon_{l}$
$\displaystyle\stackrel{{\scriptstyle P}}{{\rightarrow}}0$ $\displaystyle
G_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)[E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]$
$\displaystyle+$ $\displaystyle\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)[E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]$
$\displaystyle\stackrel{{\scriptstyle P}}{{\rightarrow}}0$

All in all, we have $\verb'Plim'D_{n}=\verb'Plim'(E_{n}+F_{n}+G_{n})=0$, so
$\hat{\theta}_{n,o}\xrightarrow{P}\theta_{0}$.

In addition, we have:

$\sqrt{n}(\hat{\theta}_{n,o}-\theta_{0})=[C_{n}]^{-1}\sqrt{n}[E_{n}+F_{n}+G_{n}]$

And we study each term separately:

$\displaystyle\sqrt{n}F_{n}$ $\displaystyle=$
$\displaystyle\sqrt{n}B_{n}+\sqrt{n}\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\epsilon_{l}$
$\displaystyle\sqrt{n}E_{n}$ $\displaystyle=$
$\displaystyle\sqrt{n}\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle+$
$\displaystyle\sqrt{n}\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)(\hat{g}_{P}(X_{l})-E(P_{l}|X_{l}))^{\prime}\theta_{0}$
$\displaystyle\sqrt{n}G_{n}$ $\displaystyle=$
$\displaystyle\sqrt{n}\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)[E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]$
$\displaystyle+$
$\displaystyle\sqrt{n}\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\tilde{P}_{j}+\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)[E(y_{l}|X_{l})-\hat{g}_{y}(X_{l})]$

From Assumption 2, $\sqrt{n}[E_{n}+F_{n}+G_{n}]=\sqrt{n}B_{n}+o_{p}(1)$.

Hence,

$\sqrt{n}(\hat{\theta}_{n,o}-\theta_{0})=[C_{n}]^{-1}\sqrt{n}B_{n}+o_{p}(1)$

$\hat{\theta}_{n,o}$ and $\tilde{\theta}_{n,o}$ share the same asymptotic
distribution. ∎

###  C.6 Consistent Estimator of the Asymptotic Variance (3.2)

$\displaystyle
Var\left[\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}Z_{j}}\epsilon_{j}\left(E[e^{it^{\prime}Z_{l}}\tilde{P}_{l}]-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}E[e^{it^{\prime}Z_{l}}]\right)d\mu(t)\right]$
$\displaystyle=$ $\displaystyle
Var\left[E_{l}\left[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(Z_{l}-Z_{j})}\epsilon_{j}\left(\tilde{P}_{l}-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\right)d\mu(t)\right]\right]$
$\displaystyle=$ $\displaystyle
Var\left[E_{l}\left[k(Z_{l}-Z_{j})\left(\tilde{P}_{l}-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\right)\epsilon_{j}\right]\right]$
$\displaystyle=$ $\displaystyle
E\left[\left(E_{l}\left[k(Z_{l}-Z_{j})\left(\tilde{P}_{l}-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\right)\right]\right)\left(E_{l}\left[k(Z_{l}-Z_{j})\left(\tilde{P}_{l}-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}\right)^{\prime}\right]\right)[\epsilon_{j}]^{2}\right]$

Under heteroskedasticity, the estimator for variance of the feasible-RSMD is

$\displaystyle[n(n-1)C_{n}]^{-1}\sum^{n}_{j=1}((\sum^{n}_{l=1}\kappa_{j,l}\left(\widehat{\tilde{P}}_{l}-\frac{\widehat{g_{0,\tilde{P}_{l}}}(X_{j})}{\widehat{g_{0,\kappa_{m,j}}}(X_{j})}\right))(\sum^{n}_{l=1}\kappa_{j,l}\left(\widehat{\tilde{P}}_{l}-\frac{\widehat{g_{0,\tilde{P}_{l}}}(X_{j})}{\widehat{g_{0,\kappa_{m,j}}}(X_{j})}\right)^{\prime})\hat{\epsilon}_{j}^{2})[n(n-1)C_{n}^{\prime}]^{-1}$

##  Appendix D Identification

###  D.1 Summary

In this section, I provide a summary to illustrate the identification issue
for SMD (Smooth Minimum Distance), RSMD, and DRSMD using a single instrument
to estimate two parameters. If there are constants in the model, the
conditions for identification will be slightly different, but the logic will
remain the same.

I will start with the simplest model first. Then move on to the complicated
models. Models are provided with coherent simulation results. Those results
are included in the Table B9.

  1. 1.

Model (1):

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,2}+u_{i}$ $X_{i,1}=\pi_{1}W_{i}+v_{i,1}$
$X_{i,2}=\pi_{2}W_{i}+v_{i,2}$

SMD cannot identify $\beta_{1}$ and $\beta_{2}$ with continuous or discrete
$W_{i}$. To identify $\beta_{1}$ and $\beta_{2}$ with SMD, the model need to
have a nonlinear part of $W_{i}$ (continuous) or another control variable
$Z_{i}$ inside one of the equations for $X_{i}$.

  2. 2.

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,2}+u_{i}$
$X_{i,1}=\pi_{1,1}W_{i,1}+\pi_{1,2}W_{i,2}+v_{i,1}$
$X_{i,2}=\pi_{2,1}W_{i,1}+\pi_{2,2}W_{i,2}+v_{i,2}$

When (continuous or discrete) $W_{i,1}$ and $W_{i,2}$ are independent, we are
unable to identify both parameters using SMD with only one instrument
$W_{i,1}$ under some specific conditions for the mean of $E(W_{i,2})$,
$\pi_{1,1}$, $\pi_{1,2}$, $\pi_{2,1}$, and $\pi_{2,2}$. We are able to
identify both parameters $\beta_{1}$ and $\beta_{2}$ when these conditions are
not met.

  3. 3.

Model (3):

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,1}*Z_{i}+u_{i}$
$X_{i,1}=\pi_{1}W_{i}+v_{i,1}$
$X_{i,1}*Z_{i}=\pi_{1}W_{i}*Z_{i}+v_{i,1}*Z_{i}$

If $W_{i}$ and $Z_{i}$ are dependent, when $W_{i}$ is continuous, both
parameters are identified, and when $W_{i}$ is binary, using SMD we can
identify 0 or 1 parameter based on the value of $E(Z_{i}|W_{i})$. If $W_{i}$
and $Z_{i}$ are independent, using SMD we can identify 0 or 1 parameter based
on the value of $E(Z_{i})$. This is the same case for a continuous or discrete
$W_{i}$.

  4. 4.

Model (4):

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,1}*Z_{i}+\gamma_{1}Z_{i}+u_{i}$
$X_{i,1}=\pi_{1}W_{i}+v_{i,1}$
$X_{i,1}*Z_{i}=\pi_{1}W_{i}*Z_{i}+v_{i,1}*Z_{i}$

If $W_{i}$ and $Z_{i}$ are dependent, RSMD identifies both parameters using a
continuous or discrete $W_{i}$. If $W_{i}$ and $Z_{i}$ are independent, we can
use RSMD to identify 0 or 1 parameter based on the value of $E(Z_{i})$ with
continuous or discrete $W_{i}$.

  5. 5.

DRSMD has extra terms in its formula and it is not symmetric. It will identify
both parameters when RSMD cannot.

###  D.2 Proofs and Simulation Examples

Following that, we move on to sections with more detailed explanations,
proofs, and simulation studies to provide more information on the
identification issue for methods incorporating SMD (smooth minimum distance
approach), such as SMD, RSMD, and DRSMD.

  1. 1.

Prove the identification of the key parameters using the SMD method for the
simplest model in the following (Model 1).

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,2}+u_{i}$ $X_{i,1}=\pi_{1}W_{i}+v_{i,1}$
$X_{i,2}=\pi_{2}W_{i}+v_{i,2}$

###### Proof.

The SMD estimator is

$\begin{pmatrix}\beta_{1}\\\
\beta_{2}\end{pmatrix}=[E(\kappa_{j,l}X_{j}X_{l}^{\prime})]^{-1}E(\kappa_{j,l}X_{j}y_{l})=\left[E\left(\kappa_{j,l}\begin{pmatrix}X_{j,1}\\\
X_{j,2}\end{pmatrix}\begin{pmatrix}X_{l,1}\\\
X_{l,2}\end{pmatrix}^{\prime}\right)\right]^{-1}E(\kappa_{j,l}X_{j}y_{l})$

We need to prove that $E(\kappa_{j,l}X_{j}X_{l}^{\prime})$ is nonsingular. To
show that $E(\kappa_{j,l}X_{j}X_{l}^{\prime})$ is nonsingular, we consider the
associated quadratic form, and show that it is positive definite. For any $a$
real vector of size $p$, we have:

$\displaystyle E(a^{\prime}X_{j}X_{l}^{\prime}a\kappa_{j,l})$
$\displaystyle=E(\kappa_{j,l}a^{\prime}E(X_{j}|W_{j})E(X_{l}^{\prime}a|W_{l}))$
$\displaystyle=E(\int_{R^{q_{w}}}e^{it^{\prime}(W_{j}-W_{l})}d\mu(t)a^{\prime}E(X_{j}|W_{j})E(X_{l}^{\prime}|W_{l})a)$
$\displaystyle=\int_{R^{q_{w}}}E[e^{it^{\prime}(W_{j}-W_{l})}a^{\prime}E(X_{j}|W_{j})E(X_{l}^{\prime}|W_{l})a]d\mu(t)$
$\displaystyle=\int_{R^{q_{w}}}E[e^{it^{\prime}W_{j}}a^{\prime}E(X_{j}|W_{j})E(X_{l}^{\prime}|W_{l})ae^{-it^{\prime}W_{l}}]d\mu(t)$
$\displaystyle=\int_{R^{q_{w}}}E[a^{\prime}e^{it^{\prime}W_{j}}E(X_{j}|W_{j})]E[E(X_{l}^{\prime}|W_{l})ae^{-it^{\prime}W_{l}}]d\mu(t)$
$\displaystyle=\int_{R^{q_{w}}}E[a^{\prime}e^{it^{\prime}W_{j}}E(X_{j}|W_{j})]E[E(X_{j}^{\prime}|W_{j})ae^{-it^{\prime}W_{j}}]d\mu(t)$
$\displaystyle=\int_{R^{q_{w}}}|\left(\int_{R^{q_{w}}}a^{\prime}e^{it^{\prime}W_{j}}E(X_{j}|W_{j})f_{W}(W_{j})dW_{j}\right)|^{2}d\mu(t)$
$\displaystyle=(2\pi)^{2q_{w}}\int_{R^{q_{w}}}|\left(\mathcal{F}[a^{\prime}E(X_{j}|W_{j})f_{W}(W_{j})](t)\right)|^{2}d\mu(t)$
$\displaystyle\geq 0$

with $\mu$ strictly positive on $\mathbb{R}^{p}$ and $\mathcal{F}[g]$ the
Fourier transform of a well-defined function $g(.)$ on $\mathbb{R}^{q_{w}}$ is
formally defined as,

$\displaystyle\mathcal{F}[g](t)=\frac{1}{(2\pi)^{q_{w}}}\int\exp^{it^{\prime}u}g(u)du\,.$
(D.1)

We then have:

$\displaystyle a^{\prime}E(\kappa_{j,l}X_{j}X_{l}^{\prime})a=0$
$\displaystyle\Leftrightarrow$ $\displaystyle\exists\ a\neq 0\ s.t.\
a^{\prime}E(X_{j}|W_{j})f(W_{j})=0\ a.s.$ $\displaystyle\Leftrightarrow$
$\displaystyle\exists\ a\neq 0\ s.t.\ a^{\prime}E(X_{j}|W_{j})=0\ a.s.$

In the specific case,

$\Leftrightarrow\exists\ a\neq 0\ s.t.\
a^{\prime}E(X_{j}|W_{j})=a_{1}\pi_{1}W_{i}+a_{2}\pi_{2}W_{i}=(a_{1}\pi_{1}+a_{2}\pi_{2})W_{i}=0\
a.s.$

$\beta_{1}$ and $\beta_{2}$ are not identified. To identify $\beta_{1}$ and
$\beta_{2}$, the model must contain a nonlinear part of $W_{i}$ or another
control variable $Z_{i}$ inside one of the equations for $X_{i}$. ∎

  2. 2.

Prove the identification of the key parameters using the SMD method for the
model with an interaction term (Model (3)).

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,1}*Z_{i}+u_{i}$
$X_{i,1}=\pi_{1}W_{i}+v_{i,1}$
$X_{i,1}*Z_{i}=\pi_{1}W_{i}*Z_{i}+v_{i,1}*Z_{i}$

###### Proof.

We need to prove that $E(\kappa_{j,l}X_{j}X_{l}^{\prime})$ is nonsingular.
Follow the same logic as in the previous proof.

$a^{\prime}E(\kappa_{j,l}X_{j}X_{l}^{\prime})a=0\Leftrightarrow\exists\ a\neq
0\ s.t.\ a^{\prime}E(X_{j}|W_{j})=0\ a.s.$ $E(X_{i,1}|W_{i})=\pi_{1}W_{i}$
$\displaystyle E(X_{i,1}*Z_{i}|W_{i})$ $\displaystyle=$ $\displaystyle
E((\pi_{1}W_{i}+v_{i,1})*Z_{i}|W_{i})$ $\displaystyle=$ $\displaystyle
E(\pi_{1}W_{i}*Z_{i}+v_{i,1}*Z_{i}|W_{i})$ $\displaystyle=$
$\displaystyle\pi_{1}W_{i}E(Z_{i}|W_{i})+E(v_{i,1}*Z_{i}|W_{i})$
$a^{\prime}E(X_{i}|W_{i})=a_{1}\pi_{1}W_{i}+a_{2}\pi_{1}W_{i}E(Z_{i}|W_{i})+a_{2}E(v_{i,1}*Z_{i}|W_{i})$

If $v_{i,1}\perp(Z_{i},W_{i})$,

$E(v_{i,1}*Z_{i}|W_{i})=0$
$a^{\prime}E(X_{i}|W_{i})=a_{1}\pi_{1}W_{i}+a_{2}\pi_{1}W_{i}E(Z_{i}|W_{i})$

When $W_{i}$ and $Z_{i}$ are dependent, and $W_{i}$ is continuous, there are
no $a_{1}\neq 0$ and $a_{2}\neq 0$, $s.t.\ a^{\prime}E(X_{j}|W_{j})=0\ a.s.$.
Both parameters are identified.

When $W_{i}$ is binary, there will be a problem.

For instance,

$a^{\prime}E(X_{i}|W_{i}=0)=a_{1}\pi_{1}*0+a_{2}\pi_{1}*0*E(Z_{i}|W_{i}=0)=0$
$a^{\prime}E(X_{i}|W_{i}=1)=a_{1}\pi_{1}*1+a_{2}\pi_{1}*1*E(Z_{i}|W_{i}=1)=0$

When $E(Z_{i}|W_{i}=1)\neq 0$ we can find a $a_{1}\neq 0$ and $a_{2}\neq 0$,
$s.t.\ a^{\prime}E(X_{j}|W_{j})=0\ a.s.$

When $W_{i}$ is 3-value instrument, both parameters will be identified.

∎

Here are 2 examples for simulation illustration when using the SMD estimator.
I consider the case that $W_{i}$ and $Z_{i}$ are independent for the second
example.

    1. (a)

$y_{i}=2X_{i,1}+3X_{i,2}+u_{i}$ $X_{i,1}=4W_{i,1}+W_{i,2}+v_{i,1}$
$X_{i,2}=W_{i,1}+3W_{i,2}+v_{i,2}$

When $W_{i,1}$ and $W_{i,2}$ are independent, we are not able to identify both
parameters inside the model for $y_{i}$ using SMD with only one instrument
$W_{i,1}$ or $W_{i,2}$ when $E(W_{i,2})$ or $E(W_{i,1})$ are zero. Indeed,
identification of SMD with $W_{i,1}$ yields the following equation.

$a^{\prime}E(X_{i}|W_{i,1})=a_{1}E(4W_{i,1}+W_{i,2}+v_{i,1}|W_{i,1})+a_{2}E(W_{i,1}+3W_{i,2}+v_{i,2}|W_{i,1})$
$=a_{1}4W_{i,1}+a_{1}E(W_{i,2}|W_{i,1})+a_{2}W_{i,1}+a_{2}E(3W_{i,2}|W_{i,1})$

When $W_{i,1}$ and $W_{i,2}$ are independent,

$a^{\prime}E(X_{i}|W_{i,1})=a_{1}4W_{i,1}+a_{1}E(W_{i,2})+a_{2}W_{i,1}+a_{2}E(3W_{i,2})$
$a^{\prime}E(X_{i}|W_{i,1})=(4a_{1}+a_{2})W_{i,1}+(a_{1}+3a_{2})E(W_{i,2})$

      1. i.

When $W_{i,1}$ and $W_{i,2}$ are continuous, we need $E(W_{i,2})\neq 0$ to
identify both parameters.

      2. ii.

When $E(W_{i,2})=0$, there are infinite possible combinations for $a_{1}$ and
$a_{2}$ to have

$a^{\prime}E(X_{i}|W_{i,1})=0$

. In this case, neither parameter is identified.

      3. iii.

When $W_{i,1}$ and $W_{i,2}$ are binary ${0,1}$, $E(W_{i,2})\neq 0$. $a_{1}=0$
and $a_{2}=0$ to have

$a^{\prime}E(X_{i}|W_{i,1})=0$

. Thus, we will identify both parameters.

    2. (b)

$y_{i}=2X_{i,1}+3X_{i,1}*Z_{i}+u_{i}$ $X_{i,1}=2W_{i}+v_{i,1}$
$X_{i,1}*Z_{i}=2W_{i}*Z_{i}+v_{i,1}*Z_{i}$

When $W_{i}$ and $Z_{i}$ are independent and $E(Z_{i})=0$, we are able to
identify the first parameter. When $W_{i}$ and $Z_{i}$ are independent and
$E(Z_{i})\neq 0$, there is no identification for both parameters. Indeed,

$a^{\prime}E(X_{i}|W_{i})=a_{1}E(2W_{i}+v_{i,1}|W_{i})+a_{2}E(2W_{i}*Z_{i}+v_{i,1}*Z_{i}|W_{i})$
$=2a_{1}W_{i}+2a_{2}W_{i}*E(Z_{i}|W_{i})$

When $W_{i}$ and $Z_{i}$ are independent,

$a^{\prime}E(X_{i}|W_{i})=2a_{1}W_{i}+2a_{2}W_{i}*E(Z_{i})$

      1. i.

When $E(Z_{i})=0$, $a_{1}$ must be 0 for a continuous or discrete $W_{i}$.
$a_{2}$ can be any value. Thus, we can only identify the first parameter.

      2. ii.

When $E(Z_{i})=1$, there are infinite combinations for $(a_{1},a_{2})$ to make

$a^{\prime}E(X_{i}|W_{i})=0$

. Both parameters are not identified.

All of these examples are verified used simulation studies. The simulation
results are included in the Table B9.

  3. 3.

Prove the identification of the key parameters using the RSMD method for the
following model (Model (4)).

$y_{i}=\beta_{1}X_{i,1}+\beta_{2}X_{i,1}*Z_{i}+Z_{i}+u_{i}$
$X_{i,1}=\pi_{1}W_{i}+v_{i,1}$
$X_{i,1}*Z_{i}=\pi_{1}W_{i}*Z_{i}+v_{i,1}*Z_{i}$

###### Proof.

We need

$a^{\prime}E(\kappa_{j,l}\tilde{X}_{j}\tilde{X}_{l}^{\prime})a=0\Leftrightarrow\exists\
a\neq 0\ s.t.\ a^{\prime}E(\tilde{X}_{j}|W_{j})=0\ a.s.$

to show that the parameters are not identified.

$\displaystyle E(\tilde{X}_{i,1}|W_{i})$ $\displaystyle=$ $\displaystyle
E(X_{i,1}-E(X_{i,1}|Z_{i})|W_{i})$ $\displaystyle=$ $\displaystyle
E(\pi_{1}W_{i}+v_{i,1}-E(\pi_{1}W_{i}+v_{i,1}|Z_{i})|W_{i})$ $\displaystyle=$
$\displaystyle E(\pi_{1}W_{i}-E(\pi_{1}W_{i}|Z_{i})|W_{i})$ $\displaystyle=$
$\displaystyle\pi_{1}W_{i}-\pi_{1}E(E(W_{i}|Z_{i})|W_{i})$ $\displaystyle
E(X_{i,1}Z_{i}-E(X_{i,1}Z_{i}|Z_{i})|W_{i})$ $\displaystyle=$ $\displaystyle
E(X_{i,1}Z_{i}-E(X_{i,1}|Z_{i})Z_{i}|W_{i})$ $\displaystyle=$ $\displaystyle
E(\pi_{1}W_{i}Z_{i}+v_{i,1}Z_{i}-E(X_{i,1}|Z_{i})Z_{i}|W_{i})$
$\displaystyle=$ $\displaystyle
E(\pi_{1}W_{i}Z_{i}-E(\pi_{1}W_{i}+v_{i,1}|Z_{i})Z_{i}|W_{i})$
$\displaystyle=$ $\displaystyle
E(\pi_{1}W_{i}Z_{i}-E(\pi_{1}W_{i}|Z_{i})*Z_{i}|W_{i})$ $\displaystyle=$
$\displaystyle\pi_{1}W_{i}E(Z_{i}|W_{i})-\pi_{1}E(E(W_{i}|Z_{i})Z_{i}|W_{i})$
$\displaystyle a^{\prime}E(\tilde{X}_{i}|W_{i})$ $\displaystyle=$
$\displaystyle a_{1}(\pi_{1}W_{i}-\pi_{1}E(E(W_{i}|Z_{i})|W_{i}))$
$\displaystyle+$ $\displaystyle
a_{2}(\pi_{1}W_{i}E(Z_{i}|W_{i})-\pi_{1}E(E(W_{i}|Z_{i})Z_{i}|W_{i}))$

Thus, when $a^{\prime}E(\tilde{X}_{i}|W_{i})=0$,

$a_{1}W_{i}-a_{1}E(E(W_{i}|Z_{i})|W_{i})+a_{2}W_{i}E(Z_{i}|W_{i})-a_{2}E(E(W_{i}|Z_{i})Z_{i}|W_{i})=0$
(D.2)

    1. (a)

When the instrument $W_{i}$ is binary, if $W_{i}=0$, Equation D.2 becomes:

$-a_{1}E(E(W_{i}|Z_{i})|W_{i}=0)-a_{2}E(E(W_{i}|Z_{i})Z_{i}|W_{i}=0)=0$

If $W_{i}=1$, Equation D.2 becomes:

$a_{1}-a_{1}E(E(W_{i}|Z_{i})|W_{i}=1)+a_{2}E(Z_{i}|W_{i}=1)-a_{2}E(E(W_{i}|Z_{i})Z_{i}|W_{i}=1)=0$

We need to solve for $a_{1}$ and $a_{2}$. If both of them are 0, both
parameters will be identified. If one of them is 0, only one parameter will be
identified.

$\begin{pmatrix}E(E(W_{i}|Z_{i})|W_{i}=0)&E(E(W_{i}|Z_{i})Z_{i}|W_{i}=0)\\\
1-E(E(W_{i}|Z_{i})|W_{i}=1)&(E(Z_{i}|W_{i}=1)-E(E(W_{i}|Z_{i})Z_{i}|W_{i}=1))\end{pmatrix}\begin{pmatrix}a_{1}\\\
a_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}$

We will have a problem if $W_{i}\perp Z_{i}$. The determinant of the above
matrix is 0. In that scenario, there are many possible solutions for $a_{1}$
and $a_{2}$. We cannot identify both parameters.

Another straightforward way to illustrate that is in the following.

$a^{\prime}E(\tilde{X}_{i}|W_{i})=a_{1}E(\tilde{X}_{i,1}|W_{i})+a_{2}E(\tilde{X}_{i,1}*Z_{i}|W_{i})$
$a^{\prime}E(\tilde{X}_{i}|W_{i}=0)=a_{1}E(\tilde{X}_{i,1}|W_{i}=0)+a_{2}E(\tilde{X}_{i,1}*Z_{i}|W_{i}=0)=0$
$a^{\prime}E(\tilde{X}_{i}|W_{i}=1)=a_{1}E(\tilde{X}_{i,1}|W_{i}=1)+a_{2}E(\tilde{X}_{i,1}*Z_{i}|W_{i}=1)=0$
$\begin{pmatrix}E(\tilde{X}_{i,1}|W_{i}=0)&E(\tilde{X}_{i,1}*Z_{i}|W_{i}=0)\\\
E(\tilde{X}_{i,1}|W_{i}=1)&E(\tilde{X}_{i,1}*Z_{i}|W_{i}=1)\end{pmatrix}\begin{pmatrix}a_{1}\\\
a_{2}\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}$
$A\begin{pmatrix}a_{1}\\\ a_{2}\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix}$

If matrix A is invertible or $det(A)$ is not 0, then $a_{1}$ and $a_{2}$ are
zeros. Both parameters are identified. If $det(A)$ is 0, we cannot identify
both parameters.
$det(A)=E(\tilde{X}_{i,1}|W_{i}=0)E(\tilde{X}_{i,1}*Z_{i}|W_{i}=1)-E(\tilde{X}_{i,1}*Z_{i}|W_{i}=0)E(\tilde{X}_{i,1}|W_{i}=1)$.
∎

Additionally, with a binary instrument, the RSMD identifies 2 or less than 2
parameters. For instance, if we want to identify 3 parameters with 2
parameters for 2 interaction terms, RSMD can only identify 2 parameters at
most.

$a^{\prime}E(\tilde{X}_{i}|W_{i})=a_{1}E(\tilde{X}_{i}|W_{i})+a_{2}E(\tilde{X}_{i}Z_{i,1}|W_{i})+a_{3}E(\tilde{X}_{i}Z_{i,2}|W_{i})$
$a^{\prime}E(\tilde{X}_{i}|W_{i}=0)=a_{1}E(\tilde{X}_{i}|W_{i}=0)+a_{2}E(\tilde{X}_{i}Z_{i,1}|W_{i}=0)+a_{3}E(\tilde{X}_{i}Z_{i,2}|W_{i}=0)$
$a^{\prime}E(\tilde{X}_{i}|W_{i}=1)=a_{1}E(\tilde{X}_{i}|W_{i}=1)+a_{2}E(\tilde{X}_{i}Z_{i,1}|W_{i}=1)+a_{3}E(\tilde{X}_{i}Z_{i,2}|W_{i}=1)$

  4. 4.

Prove the identification of the key parameters using the DRSMD method for
Model (4).

The DRSMD estimator relies on the invertibility of the following matrix.
Because it is not symmetric, we cannot use the same method to prove the
identification problem. There is an alternative proof.

$E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]=E\left[\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime}\right]-E\left[\kappa_{j,l}\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\tilde{P}_{l}^{\prime}\right]$

The invertibility of the matrix is illustrated by a simple example. $X_{i}$
and $W_{i}$ are random variables. $P_{i}=[W_{i},W_{i}X_{i}]^{\prime}$. $W$ is
the treatment. $X$ is the control. $Z$ is the instrument.

$\tilde{P}_{j}\equiv\begin{pmatrix}\tilde{W}_{j}\\\
\tilde{W}_{j}X_{j}\end{pmatrix}=\begin{pmatrix}W_{j}-E(W_{j}|X_{j})\\\
W_{j}X_{j}-E(W_{j}|X_{j})X_{j}\end{pmatrix}$

$g_{0,\tilde{P}_{1,m}}(X_{l})=E[(P_{1m}-g_{0,P_{1}}(X_{m}))\kappa_{m,l}|X_{l}]=E[\tilde{W}_{j}\kappa_{j,l}|X_{l}]=a(X_{l})$,

$g_{0,\tilde{P}_{2,j}}(X_{l})=E[(P_{2j}-g_{0,P_{2}}(X_{j}))\kappa_{j,l}|X_{l}]=E[\tilde{W}_{j}X_{j}\kappa_{j,l}|X_{l}]=b(X_{l})$,

$g_{0,\kappa_{m,l}}(X_{l})=E[\kappa_{m,l}|X_{l}]=c(X_{l})$.

Based on the definition of $\tilde{P}_{j}$, $E(\tilde{P}_{j})=0$.

$E(\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime})=E\begin{bmatrix}\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}&\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{l}\\\
\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{j}&\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{l}X_{j}\end{bmatrix}$
$E\left[\kappa_{j,l}\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\tilde{P}_{l}^{\prime}\right]=E\begin{bmatrix}\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}&\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}\\\
\kappa_{j,l}\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}&\kappa_{j,l}\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}\end{bmatrix}$

$E(\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l})=E[\kappa_{j,l}E(\tilde{W}_{j}|Z_{j})E(\tilde{W}_{l}|Z_{l})]$

$=E[\kappa_{j,l}E(\tilde{W}_{j}|Z_{j})E(\tilde{W}_{l}|Z_{l})|Z_{j}=1,Z_{l}=1]P_{r}(Z_{j}=1,Z_{l}=1)$

$+E[\kappa_{j,l}E(\tilde{W}_{j}|Z_{j})E(\tilde{W}_{l}|Z_{l})|Z_{j}=1,Z_{l}=0]P_{r}(Z_{j}=1,Z_{l}=0)$

$+E[\kappa_{j,l}E(\tilde{W}_{j}|Z_{j})E(\tilde{W}_{l}|Z_{l})|Z_{j}=0,Z_{l}=1]P_{r}(Z_{j}=0,Z_{l}=1)$

$+E[\kappa_{j,l}E(\tilde{W}_{j}|Z_{j})E(\tilde{W}_{l}|Z_{l})|Z_{j}=0,Z_{l}=0]P_{r}(Z_{j}=0,Z_{l}=0)$

In the simulations, we use Gaussian CDF for $\mu(t)$ inside $\kappa_{j,l}$.
$\kappa_{j,l}=1$ when instrument variables $Z_{j}=Z_{l}$ and
$\kappa_{j,l}=a\sim 0.61$ when $Z_{j}\neq Z_{l}$.

$E(\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l})=E(\tilde{W}_{j}|Z_{j}=1)E(\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{j}=1)P_{r}(Z_{l}=1)$

$+aE(\tilde{W}_{j}|Z_{j}=1)E(\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{j}=1)P_{r}(Z_{l}=0)$

$+aE(\tilde{W}_{j}|Z_{j}=0)E(\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{j}=0)P_{r}(Z_{l}=1)$

$+E(\tilde{W}_{j}|Z_{j}=0)E(\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$

Because
$E(\tilde{W}_{j}|Z_{j}=1)E(\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{j}=1)P_{r}(Z_{l}=1)$

$+E(\tilde{W}_{j}|Z_{j}=1)E(\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{j}=1)P_{r}(Z_{l}=0)$

$=E(\tilde{W}_{j}|Z_{j}=1)P_{r}(Z_{j}=1)[E(\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{l}=1)+E(\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{l}=0)]$

$=0$,

$E(\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l})=(1-a)E(\tilde{W}_{j}|Z_{j}=1)E(\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{j}=1)P_{r}(Z_{l}=1)$

$+(1-a)E(\tilde{W}_{j}|Z_{j}=0)E(\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$

Denote $E(\tilde{W}_{l}|Z_{l}=1)$ as $\theta_{1}$, $E(\tilde{W}_{l}|Z_{l}=0)$
as $\theta_{2}$, and $P_{r}(Z_{j}=1)$ as $b$. We have

$E(\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l})=(1-a)(\theta_{1}^{2}b^{2}+\theta_{2}^{2}(1-b)^{2})$

Follow the same steps,

$E[\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{l}]=(1-a)E(\tilde{W}_{j}|Z_{j}=1)E(X_{l}\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{j}=1)P_{r}(Z_{l}=1)$

$+(1-a)E(\tilde{W}_{j}|Z_{j}=0)E(X_{l}\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$

Denote $E(X_{l}\tilde{W}_{l}|Z_{l}=1)$ as $\theta_{3}$,
$E(X_{l}\tilde{W}_{l}|Z_{l}=0)$ as $\theta_{4}$.

$E[\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{l}]=(1-a)(\theta_{1}\theta_{3}b^{2}+\theta_{2}\theta_{4}(1-b)^{2})$

$E[\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{l}X_{j}]=(1-a)E(X_{j}\tilde{W}_{j}|Z_{j}=1)E(X_{l}\tilde{W}_{l}|Z_{l}=1)P_{r}(Z_{j}=1)P_{r}(Z_{l}=1)$

$+(1-a)E(X_{j}\tilde{W}_{j}|Z_{j}=0)E(X_{l}\tilde{W}_{l}|Z_{l}=0)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$

$E[\kappa_{j,l}\tilde{W}_{j}\tilde{W}_{l}X_{l}X_{j}]=(1-a)(\theta_{3}^{2}b^{2}+\theta_{4}^{2}(1-b)^{2})$

$E(\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime})=\begin{bmatrix}(1-a)(\theta_{1}^{2}b^{2}+\theta_{2}^{2}(1-b)^{2})&(1-a)(\theta_{1}\theta_{3}b^{2}+\theta_{2}\theta_{4}(1-b)^{2})\\\
(1-a)(\theta_{1}\theta_{3}b^{2}+\theta_{2}\theta_{4}(1-b)^{2})&(1-a)(\theta_{3}^{2}b^{2}+\theta_{4}^{2}(1-b)^{2})\end{bmatrix}$

$\displaystyle
E\left[\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}\right]$
$\displaystyle=$ $\displaystyle
E\left[\kappa_{j,l}E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}\right)\right]$
$\displaystyle=$ $\displaystyle
E\left[\kappa_{j,l}E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}\right)|Z_{j}=1,Z_{l}=1\right]P_{r}(Z_{j}=1,Z_{l}=1)$
$\displaystyle+$ $\displaystyle
E\left[\kappa_{j,l}E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}\right)|Z_{j}=1,Z_{l}=0\right]P_{r}(Z_{j}=1,Z_{l}=0)$
$\displaystyle+$ $\displaystyle
E\left[\kappa_{j,l}E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}\right)|Z_{j}=0,Z_{l}=1\right]P_{r}(Z_{j}=0,Z_{l}=1)$
$\displaystyle+$ $\displaystyle
E\left[\kappa_{j,l}E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}\right)|Z_{j}=0,Z_{l}=0\right]P_{r}(Z_{j}=0,Z_{l}=0)$
$\displaystyle
E\left[\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}\right]$
$\displaystyle=$ $\displaystyle
E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)P_{r}(Z_{j}=1)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
aE\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)P_{r}(Z_{j}=1)P_{r}(Z_{l}=0)$
$\displaystyle+$ $\displaystyle
aE\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)P_{r}(Z_{j}=0)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$
$E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}\right)=E\left(\frac{a(X_{l})}{c(X_{l})}E(\tilde{W}_{l}|X_{l})\right)=0$

with
$\tilde{P}_{j}\equiv[\tilde{W}_{j},\tilde{W}_{j}X_{j}]^{\prime}=[W_{j}-E(W_{j}|X_{j}),W_{j}X_{j}-E(W_{j}|X_{j})X_{j}]$

$\displaystyle
E\left[\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}\right]$
$\displaystyle=$ $\displaystyle
P_{r}(Z_{j}=1)(1-a)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
aP_{r}(Z_{j}=1)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
aP_{r}(Z_{j}=1)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)P_{r}(Z_{l}=0)$
$\displaystyle+$ $\displaystyle
aE\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)P_{r}(Z_{j}=0)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$
$\displaystyle=$ $\displaystyle
P_{r}(Z_{j}=1)(1-a)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
P_{r}(Z_{j}=0)(1-a)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)P_{r}(Z_{l}=0)$

When we have $P_{r}(Z_{j}=1)=0.5$, $d_{1}=0$.

Denote $E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)$ as
$\delta_{1}$ and
$E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)$ as
$\delta_{2}$.

$E\left[\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}\right]=(1-a)(b^{2}\delta_{1}+(1-b)^{2}\delta_{2})$

Follow the same steps,

$\displaystyle
E\left[\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}\right]$
$\displaystyle=$ $\displaystyle
P_{r}(Z_{j}=1)(1-a)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=1\right)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
aP_{r}(Z_{j}=1)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=1\right)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
aP_{r}(Z_{j}=1)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=0\right)P_{r}(Z_{l}=0)$
$\displaystyle+$ $\displaystyle
aE\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=1\right)P_{r}(Z_{j}=0)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=0\right)P_{r}(Z_{j}=0)P_{r}(Z_{l}=0)$
$\displaystyle=$ $\displaystyle
P_{r}(Z_{j}=1)(1-a)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=1\right)P_{r}(Z_{l}=1)$
$\displaystyle+$ $\displaystyle
P_{r}(Z_{j}=0)(1-a)E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=0\right)P_{r}(Z_{l}=0)$

Denote $E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=1\right)$ as
$\delta_{3}$ and
$E\left(\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=0\right)$ as
$\delta_{4}$.

$E\left[\kappa_{j,l}\frac{a(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}\right]=(1-a)(b^{2}\delta_{3}+(1-b)^{2}\delta_{4})$

Denote $E\left(\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=1\right)$ as
$\delta_{5}$ and
$E\left(\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}|Z_{l}=0\right)$ as
$\delta_{6}$.

$\displaystyle
E\left[\kappa_{j,l}\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}\right]$
$\displaystyle=$ $\displaystyle(1-a)(b^{2}\delta_{5}+(1-b)^{2}\delta_{6})$

Denote $E\left(\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=1\right)$ as
$\delta_{7}$ and
$E\left(\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}|Z_{l}=0\right)$ as
$\delta_{8}$.

$\displaystyle
E\left[\kappa_{j,l}\frac{b(X_{l})}{c(X_{l})}\tilde{W}_{l}X_{l}\right]$
$\displaystyle=$ $\displaystyle(1-a)(b^{2}\delta_{7}+(1-b)^{2}\delta_{8})$

$E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]$
becomes:

$\begin{bmatrix}(1-a)(\theta_{1}^{2}b^{2}+\theta_{2}^{2}(1-b)^{2})-(1-a)(b^{2}\delta_{1}+(1-b)^{2}\delta_{2})&e_{1}\\\
(1-a)(\theta_{1}\theta_{3}b^{2}+\theta_{2}\theta_{4}(1-b)^{2})-(1-a)(b^{2}\delta_{5}+(1-b)^{2}\delta_{6})&e_{2}\end{bmatrix}$

$e_{1}=(1-a)(\theta_{1}\theta_{3}b^{2}+\theta_{2}\theta_{4}(1-b)^{2})-(1-a)(b^{2}\delta_{3}+(1-b)^{2}\delta_{4})$

$e_{2}=(1-a)(\theta_{3}^{2}b^{2}+\theta_{4}^{2}(1-b)^{2})-(1-a)(b^{2}\delta_{7}+(1-b)^{2}\delta_{8})$

Since a is not 1, the determinant of
$E\left[\kappa_{j,l}\left(\tilde{P}_{j}-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)\tilde{P}_{l}^{\prime}\right]$
depends on the exact values for these numbers.

  5. 5.

Simulation illustration for RSMD and DRSMD

Here are one simulation example for illustration (Model (4)).

$y_{i}=X_{i,1}+X_{i,1}*Z_{i}+Z_{i}+u_{i}$ $X_{i,1}=2W_{i}+v_{i,1}$
$X_{i,1}*Z_{i}=2W_{i}*Z_{i}+v_{i,1}*Z_{i}$

    1. (a)

For RSMD,

$\displaystyle a^{\prime}E(\tilde{X}_{i}|W_{i})$ $\displaystyle=$
$\displaystyle 2a_{1}W_{i}-2a_{1}E(E(W_{i}|Z_{i})|W_{i})$ $\displaystyle+$
$\displaystyle 2a_{2}W_{i}E(Z_{i}|W_{i})-2a_{2}E(E(W_{i}|Z_{i})Z_{i}|W_{i})$
$\displaystyle=$ $\displaystyle 0$

When $W_{i}$ and $Z_{i}$ are independent and $E(W_{i})$ is a number (for
instance, $E(W_{i})=1$),

$a_{1}W_{i}-a_{1}E(W_{i})+a_{2}W_{i}E(Z_{i})-a_{2}E(Z_{i})E(W_{i})=0$
$(a_{1}+a_{2}E(Z_{i}))W_{i}-(a_{1}+a_{2}E(Z_{i}))E(W_{i})=0$

      1. i.

When $E(Z_{i})=0$, $a_{1}$ must be 0 when $W_{i}$ is continuous or discrete.
$a_{2}$ can be any value. Hence, the first parameter is identified.

      2. ii.

When $E(Z_{i})=1$, there are infinite combinations for $(a_{1},a_{2})$ to make
$a^{\prime}E(X_{i}|W_{i})=0$. Both parameters are not identified.

    2. (b)

For DRSMD, because we have extra terms, it should identify both parameters
when RSMD cannot.

All of the simulation results are included in Table B9.

Estimator | Ins | Med.Bias | MAD | Med.SE | RR | Mean.Bias | Mean.SE | Mean.AHSE
---|---|---|---|---|---|---|---|---
Example 2(a)ii: Continuous instrument
SMD $\beta_{1}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.005 | 0.007 | 0.055 | 0.000 | 0.007 | 0.007
SMD $\beta_{2}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.006 | 0.009 | 0.052 | 0.000 | 0.009 | 0.009
SMD $\beta_{1}$ | $W_{i,1}$ | -0.011 | 0.034 | 0.065 | 0.026 | -0.021 | 1.598 | 37.825
SMD $\beta_{2}$ | $W_{i,1}$ | 0.042 | 0.132 | 0.256 | 0.041 | 0.087 | 6.394 | 149.586
SMD $\beta_{1}$ | $W_{i,2}$ | 0.018 | 0.106 | 0.209 | 0.022 | -0.050 | 18.259 | 783.448
SMD $\beta_{2}$ | $W_{i,2}$ | -0.007 | 0.036 | 0.070 | 0.014 | 0.008 | 5.521 | 238.702
TSLS $\beta_{1}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.004 | 0.006 | 0.063 | 0.000 | 0.006 | 0.006
TSLS $\beta_{2}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.006 | 0.008 | 0.055 | 0.000 | 0.008 | 0.008
Example 2(a)iii: Binary instrument
SMD $\beta_{1}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.008 | 0.012 | 0.052 | 0.000 | 0.012 | 0.012
SMD $\beta_{2}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.010 | 0.015 | 0.047 | 0.000 | 0.015 | 0.015
SMD $\beta_{1}$ | $W_{i,1}$ | 0.000 | 0.011 | 0.017 | 0.053 | 0.000 | 0.017 | 0.017
SMD $\beta_{2}$ | $W_{i,1}$ | -0.001 | 0.017 | 0.026 | 0.049 | 0.000 | 0.026 | 0.026
SMD $\beta_{1}$ | $W_{i,2}$ | 0.000 | 0.013 | 0.020 | 0.046 | 0.000 | 0.020 | 0.021
SMD $\beta_{2}$ | $W_{i,2}$ | 0.000 | 0.014 | 0.021 | 0.046 | 0.000 | 0.021 | 0.021
TSLS $\beta_{1}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.008 | 0.013 | 0.054 | 0.000 | 0.013 | 0.013
TSLS $\beta_{2}$ | $(W_{i,1},W_{i,2})$ | 0.000 | 0.011 | 0.017 | 0.048 | 0.000 | 0.017 | 0.017
Example 2(b)i: Binary instrument
SMD $\beta_{1}$ | $W_{i}$ | 0.000 | 0.013 | 0.019 | 0.036 | 0.000 | 0.024 | 0.027
SMD $\beta_{2}$ | $W_{i}$ | 0.000 | 0.043 | 0.077 | 0.039 | 0.019 | 0.738 | 0.596
Example 2(b)ii: Binary instrument
SMD $\beta_{1}$ | $W_{i}$ | 0.000 | 0.046 | 0.077 | 0.039 | -0.020 | 0.742 | 0.602
SMD $\beta_{2}$ | $W_{i}$ | 0.000 | 0.043 | 0.077 | 0.039 | 0.019 | 0.738 | 0.596
Example 5(a)i: Continuous instrument
RSMD $\beta_{1}$ | $W_{i}$ | -0.001 | 0.009 | 0.013 | 0.032 | 0.000 | 0.171 | 1.937
RSMD $\beta_{2}$ | $W_{i}$ | 0.000 | 0.056 | 0.138 | 0.006 | 0.020 | 4.528 | 77.709
DRSMD $\beta_{1}$ | $W_{i}$ | -0.001 | 0.009 | 0.013 | 0.041 | -0.001 | 0.021 | 0.114
DRSMD $\beta_{2}$ | $W_{i}$ | 0.000 | 0.041 | 0.066 | 0.029 | 0.026 | 1.093 | 7.282
Example 5(a)ii: Continuous instrument
RSMD $\beta_{1}$ | $W_{i}$ | -0.002 | 0.057 | 0.137 | 0.008 | -0.039 | 7.188 | 120.469
RSMD $\beta_{2}$ | $W_{i}$ | 0.001 | 0.055 | 0.137 | 0.005 | 0.038 | 7.331 | 122.732
DRSMD $\beta_{1}$ | $W_{i}$ | -0.001 | 0.043 | 0.067 | 0.031 | -0.002 | 0.970 | 2.922
DRSMD $\beta_{2}$ | $W_{i}$ | 0.002 | 0.041 | 0.066 | 0.029 | 0.001 | 0.970 | 2.919

Table B9: Identification Illustration

Notes: Mean.AHSE stands for Mean.Asympt.Heterosk.SE. There are 5000 rounds of
simulation and 2000 observations for each simulation study.

##  Appendix E R-SMD Estimator and its Properties from Antoine and Sun (2021)

Under Assumption 1 and FOC defined in Equation (2.8)

$\theta_{0}=[E(\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime})]^{-1}E(\kappa_{j,l}\tilde{P}_{j}\tilde{y}_{l})$

$\tilde{\theta}_{n,u}=\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime}\right]^{-1}\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\tilde{P}_{j}\tilde{y}_{l}\right]$

Additionally, we show the similar consistency and Asymptotic normality for
$\tilde{\theta}_{n,u}$. Under Assumption 1 and iid assumption for the sample,
$\tilde{\theta}_{n,u}$ is consistent for $\theta_{0}$, and the asymptotic
distribution is:

$\sqrt{n}(\tilde{\theta}_{n,u}-\theta_{0})\xrightarrow{d}N\left(0,E\left[\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime}\right]^{-1}Var[h^{\prime}_{1}(\epsilon_{1},Z_{1},X_{1})]\left(E\left[\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime}\right]^{-1}\right)^{\prime}\right)$

where
$h^{\prime}_{1}(e_{1},z_{1},x_{1})=E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\tilde{p}_{1}\epsilon_{2}+\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\tilde{P}_{2}e_{1}$.

Replace every nuisance parameter with its estimate, we have the feasible
estimator $\hat{\theta}_{n,u}$, the R-SMD estimator is in the following.

$\displaystyle\hat{\theta}_{n,u}=\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\widehat{\tilde{P}}_{j}\widehat{\tilde{P_{l}}}^{\prime}\right]^{-1}\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\widehat{\tilde{P}}_{j}\widehat{\tilde{y}}_{l}\right]$

since $E[\kappa_{j,l}\widehat{\tilde{P}}_{j}\widehat{\tilde{P_{l}}}^{\prime}]$
is invertible.

Under Assumptions 1 \- 2, $\hat{\theta}_{n,u}$ is consistent for $\theta_{0}$,
and the asymptotic distribution is:

$\sqrt{n}(\hat{\theta}_{n,u}-\theta_{0})\xrightarrow{d}N\left(0,E\left[\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime}\right]^{-1}Var[h^{\prime}_{1}(\epsilon_{1},Z_{1},X_{1})]\left(E\left[\kappa_{j,l}\tilde{P}_{j}\tilde{P}_{l}^{\prime}\right]^{-1}\right)^{\prime}\right)$

where
$h^{\prime}_{1}(e_{1},z_{1},x_{1})=E[\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(z_{1}-Z_{2})}d\mu(t)\tilde{p}_{1}\epsilon_{2}+\int_{R^{q_{z}}}e^{it^{\prime}(Z_{2}-z_{1})}d\mu(t)\tilde{P}_{2}e_{1}$.

## Supplementary Appendix

### S1. Robustness Check: When the Model is not Sparse

All of the other simulation results are based on the sparsity assumption. In
this section, Table S1 reports the results when the sparsity assumption is
unsatisfied. We choose the DGP where all of the $\beta_{1q}$ and $\beta_{2q}$
are not 0. When $q\leq 5$, $\beta_{1q}=1$ and $\beta_{2q}=3$. When $q>5$,
$\beta_{1q}=\beta_{2q}=0.05$. The D-RSMD with the Lasso method as the nuisance
parameter estimator is affected by the violation of the assumption. It is no
surprise since we need the sparsity assumption to conduct the Lasso method.
The magnitude of the Med.Bias, MAD and Med.SE are not changing. RRs are
higher.

Instrument with 3 values ($P=30$)
---
|  | Estimator | Ins | Med.Bias | MAD | Med.SE | RR
|  | D-RSMD | $Z_{2}$ | -0.045 | 0.123 | 0.158 | 0.063
$n=3000$ | $\theta_{w0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.019 | 0.102 | 0.119 | 0.057
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | 0.000 | 0.089 | 0.130 | 0.056
|  | D-RSMD | $Z_{2}$ | -0.032 | 0.162 | 0.238 | 0.076
$n=3000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.053 | 0.069 | 0.084 | 0.094
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | 0.000 | 0.048 | 0.071 | 0.052

Table S1: Instruments with 3 values

Note: Simulation Results for $\theta_{w0}$ and $\theta_{wx0}$ in the benchmark
model using D-RSMD estimator 5000 replications. We report the Monte-Carlo
Median Bias (Med.Bias), Median Absolute Deviation (MAD), median of asymptotic
standard error under heteroskedasticity (Med.SE), and the Rejection Rate (RR)
using a 5% t-test.

### S2. Robustness Check: Other Results of Oregon Health Insurance Experiment

Panel A: Heterogeneous treatment effects
---
Employ | Age: 21 - 35 | Age: 36 - 50 | Age: 51 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.048 |  | -0.001 |  | -0.029
| (0.053) |  | (0.037) |  | (0.039)
GMM-Lasso |  | 0.042 |  | -0.030 |  | -0.062
| (0.086) |  | (0.067) |  | (0.075)
DRSMD-Lasso | 0.095 | 0.047 | 0.016 | -0.029 | 0.091 | 0.030
(0.067) | (0.061) | (0.057) | (0.045) | (0.067) | (0.049)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.080 |  | 0.046 |  | 0.073
| (0.092) |  | (0.083) |  | (0.090)
GMM-Lasso |  | -0.039 |  | 0.086 |  | 0.191
| (0.123) |  | (0.122) |  | (0.148)
DRSMD-Lasso | -0.147*** | -0.042 | -0.027 | 0.092 | -0.074** | 0.074
(0.034) | (0.099) | (0.029) | (0.092) | (0.037) | (0.100)
Panel B: Homogeneous treatment effects
GMM | 0.012 | 0.024 | 0.017 | 0.009 | 0.001 | -0.012
(0.047) | (0.045) | (0.038) | (0.034) | (0.042) | (0.036)
GMM-Lasso | 0.022 | 0.037 | 0.006 | -0.033 | 0.019 | -0.065
(0.098) | (0.086) | (0.089) | (0.067) | (0.103) | (-0.065)
DRSMD-Lasso | 0.027 | 0.025 | 0.006 | 0.007 | 0.059 | 0.064
(0.06) | (0.063) | (0.052) | (0.053) | (0.059) | (0.061)
N | 5962 | 6693 | 5917

Table S2: Heterogeneous Treatment Effects of Medicaid on Employ

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line.

Average Treatment Effects
---
Employ | Age: 21 - 35 | Age: 36 - 50 | Age: 51 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.000 |  | 0.026 |  | 0.017
| (0.052) |  | (0.047) |  | (0.054)
GMM-Lasso |  | 0.018 |  | 0.020 |  | 0.058
| (0.105) |  | (0.103) |  | (0.127)
DRSMD-Lasso | 0.008 | 0.022 | 0.000 | 0.024 | 0.044 | 0.076
(0.059) | (0.066) | (0.050) | (0.065) | (0.057) | (0.073)
N | 5962 | 6693 | 5917

Table S3: Heterogeneous Treatment Effects of Medicaid on Employ

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line. The
expression for LATE is $\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.134* |  | -0.042 |  | 0.007 |  | -0.063 |  | 0.022
| (0.071) |  | (0.067) |  | (0.048) |  | (0.040) |  | (0.064)
GMM-Lasso |  | 0.143 |  | -0.125 |  | -0.069 |  | -0.105 |  | -0.063
| (0.112) |  | (0.118) |  | (0.089) |  | (0.077) |  | (0.123)
DRSMD-Lasso | 0.255*** | 0.159** | 0.014 | -0.067 | 0.010 | -0.001 | 0.033 | -0.058 | 0.086 | 0.072
(0.093) | (0.081) | (0.087) | (0.085) | (0.071) | (0.060) | (0.066) | (0.049) | (0.103) | (0.086)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.070 |  | 0.066 |  | -0.025 |  | 0.159 |  | -0.046
| (0.128) |  | (0.120) |  | (0.100) |  | (0.104) |  | (0.127)
GMM-Lasso |  | -0.014 |  | 0.198 |  | 0.120 |  | 0.304* |  | 0.157
| (0.179) |  | (0.145) |  | (0.150) |  | (0.162) |  | (0.193)
DRSMD-Lasso | -0.233*** | -0.003 | -0.092** | 0.073 | 0.006 | 0.040 | -0.083** | 0.169 | -0.018 | 0.012
(0.043) | (0.145) | (0.039) | (0.135) | (0.037) | (0.113) | (0.036) | (0.113) | (0.041) | (0.153)
Panel B: Homogeneous treatment effects
GMM | 0.105 | 0.115 | -0.012 | -0.023 | -0.003 | 0.001 | -0.007 | -0.035 | -0.001 | 0.007
(0.063) | (0.060) | (0.061) | (0.058) | (0.047) | (0.043) | (0.044) | (0.038) | (0.063) | (0.056)
GMM-Lasso | 0.135 | 0.142 | -0.013 | -0.110 | -0.018 | -0.068 | 0.011 | -0.115 | 0.021 | -0.059
(0.132) | (0.112) | (0.137) | (0.118) | (0.111) | (0.089) | (0.104) | (0.076) | (0.166) | (0.124)
DRSMD-Lasso | 0.162** | 0.158* | -0.032 | -0.026 | 0.013 | 0.015 | 0.003 | 0.005 | 0.078 | 0.079
(0.083) | (0.086) | (0.085) | (0.088) | (0.066) | (0.067) | (0.059) | (0.060) | (0.099) | (0.105)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S4: Heterogeneous Treatment Effects of Medicaid on Employ

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line.

Average Treatment Effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.094 |  | -0.001 |  | -0.007 |  | 0.029 |  | -0.009
| (0.071) |  | (0.071) |  | (0.056) |  | (0.060) |  | (0.077)
GMM-Lasso |  | 0.135 |  | 0.000 |  | 0.000 |  | 0.071*** |  | 0.043
| (0.146) |  | (0.143) |  | (0.127) |  | (0.013) |  | (0.185)
DRSMD-Lasso | 0.122 | 0.157* | -0.044 | -0.021 | 0.014 | 0.021 | -0.015 | 0.040 | 0.074 | 0.080
(0.079) | (0.095) | (0.085) | (0.093) | (0.064) | (0.078) | (0.056) | (0.076) | (0.099) | (0.116)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S5: Heterogeneous Treatment Effects of Medicaid on Employ

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line. The
expression for LATE is $\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.211*** |  | -0.072 |  | 0.056 |  | -0.059 |  | -0.171
| (0.073) |  | (0.075) |  | (0.057) |  | (0.050) |  | (0.106)
GMM-Lasso |  | 0.196 |  | -0.150 |  | 0.011 |  | -0.154 |  | -0.331**
| (0.133) |  | (0.116) |  | (0.076) |  | (0.087) |  | (0.156)
DRSMD-Lasso | 0.183** | 0.081 | -0.028 | -0.321** | -0.017 | -0.150** | -0.011 | 0.566 | 0.013 | 0.001
(0.072) | (0.176) | (0.079) | (0.138) | (0.065) | (0.061) | (0.059) | (0.529) | (0.107) | (0.106)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.002* |  | 0.001 |  | -0.001 |  | 0.001 |  | 0.003
| (0.001) |  | (0.001) |  | (0.001) |  | (0.001) |  | (0.002)
GMM-Lasso |  | -0.001 |  | 0.002 |  | 0.000 |  | 0.003 |  | 0.006
| (0.001) |  | (0.001) |  | (0.001) |  | (0.002) |  | (0.004)
DRSMD-Lasso | 0.000 | -0.003 | 0.000 | 0.003 | 0.000 | 0.007*** | 0.000 | -0.029 | 0.000 | 0.001
(0.000) | (0.004) | (0.000) | (0.002) | (0.000) | (0.002) | (0.000) | (0.026) | (0.000) | (0.002)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S6: Heterogeneous Treatment Effects of Medicaid on employment

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ hhincome.

Average Treatment Effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.093 |  | -0.007 |  | -0.039 |  | 0.024 |  | 0.127
| (0.070) |  | (0.077) |  | (0.067) |  | (0.068) |  | (0.141 )
GMM-Lasso |  | 0.147 |  | -0.025 |  | -0.013 |  | 0.095 |  | 0.202
| (0.141) |  | (0.139) |  | (0.115) |  | (0.130) |  | (0.288)
DRSMD-Lasso | 0.169* | -0.116 | -0.046 | -0.057 | 0.000 | 0.345** | -0.023 | -1.679 | 0.046 | 0.070
(0.087) | (0.278) | (0.083) | (0.137) | (0.064) | (0.152) | (0.059) | (1.433) | (0.096) | (0.159)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S7: Heterogeneous Treatment Effects of Medicaid on employment

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ hhincome. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.113* |  | -0.014 |  | -0.007 |  | -0.008 |  | 0.000
| (0.064) |  | (0.060) |  | (0.047) |  | (0.044) |  | (0.063)
GMM-Lasso |  | 0.139 |  | -0.017 |  | -0.010 |  | 0.029 |  | 0.009
| (0.140) |  | (0.130) |  | (0.101) |  | (0.103) |  | (0.166)
DRSMD-Lasso | 0.172** | 0.199 | -0.042 | -0.097 | -0.011 | -0.007 | -0.019 | -0.019 | 0.033 | 0.033
(0.084) | (0.129) | (0.081) | (0.181) | (0.063) | (0.063) | (0.058) | (0.058) | (0.099) | (0.099)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.323 |  | -1.006 |  | 0.199 |  | -0.513 |  | -1.173***
| (0.876) |  | (1.033) |  | (0.558) |  | (1.510) |  | (0.378)
GMM-Lasso |  | 0.361 |  | -1.490 |  | 0.160 |  | -0.432 |  | -0.574***
| (0.979) |  | (1.113) |  | (0.882) |  | (1.703) |  | (0.049)
DRSMD-Lasso | -0.029 | -3.115 | -0.018 | 4.228 | 0.333** | -0.331 | 0.070 | 0.294 | -0.475 | -0.574
(0.129) | (7.497) | (0.163) | (9.724) | (0.159) | (1.291) | (0.480) | (1.229) | (0.456) | (0.492)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S8: Heterogeneous Treatment Effects of Medicaid on employment

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ TANF.

Average Treatment Effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.126* |  | -0.057 |  | -0.003 |  | -0.012 |  | -0.003
| (0.073) |  | (0.074) |  | (0.048) |  | (0.045) |  | (0.063)
GMM-Lasso |  | 0.154 |  | -0.081 |  | -0.006 |  | 0.025 |  | 0.008
| (0.148) |  | (0.140) |  | (0.102) |  | (0.106) |  | (0.166)
DRSMD-Lasso | 0.171** | 0.070 | -0.042 | 0.087 | -0.003 | -0.014 | -0.018 | -0.016 | 0.031 | 0.031
(0.084) | (0.231) | (0.082) | (0.275) | (0.063) | (0.066) | (0.058) | (0.057) | (0.099) | (0.099)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S9: Heterogeneous Treatment Effects of Medicaid on employment

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ TANF. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.121 |  | -0.047 |  | -0.046 |  | -0.008 |  | -0.030
| (0.170) |  | (0.146) |  | (0.103) |  | (0.098) |  | (0.163)
GMM-Lasso |  | 0.185 |  | -0.089 |  | -0.017 |  | -0.040 |  | -0.040
| (0.267) |  | (0.216) |  | (0.138) |  | (0.153) |  | (0.263)
DRSMD-Lasso | 0.282*** | 0.193 | 0.000 | -0.054 | 0.005 | -0.026 | -0.002 | -0.024 | 0.017 | -0.096
(0.107) | (0.192) | (0.099) | (0.168) | (0.088) | (0.114) | (0.069) | (0.109) | (0.125) | (0.188)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -0.002 |  | 0.009 |  | 0.020 |  | -0.001 |  | 0.012
| (0.070) |  | (0.064) |  | (0.048) |  | (0.046) |  | (0.068)
GMM-Lasso |  | -0.017 |  | 0.021 |  | 0.004 |  | 0.033 |  | 0.021
| (0.102) |  | (0.087) |  | (0.063) |  | (0.068) |  | (0.101)
DRSMD-Lasso | -0.047** | -0.022 | -0.018 | 0.005 | -0.005 | 0.007 | -0.008 | 0.000 | 0.006 | 0.043
(0.022) | (0.074) | (0.020) | (0.070) | (0.020) | (0.051) | (0.017) | (0.050) | (0.023) | (0.075)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S10: Heterogeneous Treatment Effects of Medicaid on employment

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ smoke.

Average Treatment Effects
---
Employ | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 0.118* |  | -0.025 |  | -0.003 |  | -0.009 |  | -0.001
| (0.065) |  | (0.063) |  | (0.048) |  | (0.045) |  | (0.064)
GMM-Lasso |  | 0.144 |  | -0.040 |  | -0.007 |  | 0.031 |  | 0.011
| (0.142) |  | (0.131) |  | (0.105) |  | (0.104) |  | (0.168)
DRSMD-Lasso | 0.171** | 0.142* | -0.043 | -0.044 | -0.007 | -0.010 | -0.020 | -0.024 | 0.032 | 0.006
(0.084) | (0.083) | (0.081) | (0.080) | (0.062) | (0.064) | (0.058) | (0.058) | (0.099) | (0.096)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S11: Heterogeneous Treatment Effects of Medicaid on employment

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ smoke. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Out of Pocket Cost | Age: 21 - 35 | Age: 36 - 50 | Age: 51 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -249.086*** |  | -94.061* |  | -171.578***
| (73.769) |  | (50.838) |  | (53.230)
GMM-Lasso |  | -465.791*** |  | -236.344*** |  | -320.039***
| (110.733) |  | (95.697) |  | (96.416)
DRSMD-Lasso | -250.542** | -309.857*** | -75.370 | -67.352 | 1.626 | -162.346**
(102.629) | (90.142) | (87.700) | (65.262) | (114.941) | (67.412)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 127.498 |  | -109.671 |  | 193.479
| (141.823) |  | (128.054) |  | (144.174)
GMM-Lasso |  | 295.686 |  | -63.730 |  | 254.041
| (187.704) |  | (188.740) |  | (261.915)
DRSMD-Lasso | 29.129 | 163.879 | -46.060 | -60.297 | -112.079** | 263.280
(52.022) | (149.725) | (35.699) | (142.875) | (55.790) | (170.840)
Panel B: Homogeneous treatment effects
GMM | -191.55*** | -210.785*** | -136.74** | -118.182** | -91.278 | -127.007**
(68.965) | (63.796) | (57.490) | (48.792) | (64.973) | (52.175)
GMM-Lasso | -323.057** | -431.615*** | -263.160** | -234.630** | -211.509 | -323.868***
(141.975) | (112.434) | (133.556) | (94.993) | (170.007) | (95.266)
DRSMD-Lasso | -237.074*** | -222.744*** | -92.681 | -90.985 | -46.131 | -41.349
(91.39) | (95.524) | (80.402) | (81.631) | (100.07) | (103.067)
N | 5962 | 6693 | 5917

Table S12: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line.

Average Treatment Effects
---
Out of Pocket Cost | Age: 21 - 35 | Age: 36 - 50 | Age: 51 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -173.233** |  | -157.835** |  | -50.003
| (79.133) |  | (74.153) |  | (88.032)
GMM-Lasso |  | -289.877* |  | -273.403* |  | -160.41
| (156.242) |  | (156.520) |  | (217.11)
DRSMD-Lasso | -233.213*** | -212.36** | -102.154 | -102.415 | -68.800 | 3.088
(89.090) | (100.468) | (77.14) | (100.419) | (94.184) | (126.818)
N | 5962 | 6693 | 5917

Table S13: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line. The
expression for LATE is $\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -238.166** |  | -294.692*** |  | -32.148 |  | -149.770*** |  | -153.110*
| (94.947) |  | (92.333) |  | (69.362) |  | (53.074) |  | (87.341)
GMM-Lasso |  | -545.537*** |  | -569.684*** |  | -276.342** |  | -321.387*** |  | -465.763**
| (148.534) |  | (141.359) |  | (131.711) |  | (91.886) |  | (189.849)
DRSMD-Lasso | -373.155*** | -250.372** | -153.656 | -376.780*** | -55.406 | 18.780 | -162.227 | -185.345*** | 262.158 | -49.106
(136.072) | (119.078) | (142.101) | (126.053) | (99.185) | (84.691) | (101.883) | (68.418) | (197.082) | (114.506)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -101.939 |  | 384.290** |  | -269.238* |  | 155.502 |  | 173.800
| (193.546) |  | (181.372) |  | (156.752) |  | (165.046) |  | (211.235)
GMM-Lasso |  | 226.006 |  | 558.392** |  | -77.050 |  | 307.058 |  | 477.385
| (245.546) |  | (243.326) |  | (209.610) |  | (276.655) |  | (370.165)
DRSMD-Lasso | 104.666* | -179.667 | 46.142 | 496.881** | -72.394* | -242.417 | 6.897 | 72.042 | -111.454* | 502.058*
(62.096) | (214.131) | (53.675) | (214.081) | (43.318) | (167.287) | (45.935) | (181.569) | (57.602) | (295.269)
Panel B: Homogeneous treatment effects
GMM | -281.393*** | -265.709*** | -118.835 | -183.904** | -140.436* | -100.078 | -94.476 | -122.298** | -68.536 | -98.839
(89.881) | (82.886) | (90.944) | (82.150) | (72.634) | (64.576) | (67.326) | (53.027) | (100.771) | (82.629)
GMM-Lasso | -447.852** | -532.224*** | -255.052 | -527.148*** | -308.891* | -276.433** | -204.110 | -330.655*** | -210.799 | -453.139**
(176.244) | (142.177) | (206.913) | (151.105) | (159.359) | (131.643) | (163.024) | (89.334) | (302.765) | (192.478)
DRSMD-Lasso | -331.473*** | -330.392*** | -130.819 | -98.119 | -83.096 | -81.152 | -159.723* | -158.614* | 208.194 | 232.779
(119.712) | (123.117) | (138.478) | (145.535) | (91.477) | (93.811) | (92.494) | (93.832) | (186.861) | (198.671)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S14: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line.

Average Treatment Effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -296.380*** |  | -52.748 |  | -187.091** |  | -59.762 |  | -36.809
| (104.514) |  | (108.950) |  | (89.257) |  | (95.636) |  | (128.929)
GMM-Lasso |  | -416.474** |  | -218.127 |  | -320.684* |  | -143.655 |  | -146.312
| (197.893) |  | (220.931) |  | (179.511) |  | (212.015) |  | (347.923)
DRSMD-Lasso | -313.384*** | -352.973*** | -124.606 | -63.950 | -97.068 | -120.728 | -158.234* | -143.646 | 187.576 | 286.854
(113.461) | (137.349) | (138.286) | (154.905) | (88.592) | (110.318) | (87.948) | (122.867) | (183.656) | (227.628)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S15: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid $\times$ Above 50% Federal Poverty Line. The
expression for LATE is $\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -157.173* |  | -338.762*** |  | 74.768 |  | -197.276** |  | -120.243
| (95.566) |  | (115.896) |  | (99.214) |  | (76.924) |  | (198.875)
GMM-Lasso |  | -438.811*** |  | -605.793*** |  | -260.345** |  | -526.900*** |  | -483.119
| (163.018) |  | (177.487) |  | (123.764) |  | (122.174) |  | (305.685)
DRSMD-Lasso | -309.745*** | -145.185 | -187.617 | -237.79 | -81.096 | 58.587 | -182.139** | -949.979 | 213.045 | -201.021
(102.007) | (260.982) | (125.011) | (204.272) | (88.358) | (87.128) | (90.764) | (663.448) | (195.506) | (123.915)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -2.089 |  | 3.889* |  | -4.594* |  | 2.234 |  | 0.994
| (1.294) |  | (2.183) |  | (2.486) |  | (1.992) |  | (4.800)
GMM-Lasso |  | -0.208 |  | 5.190* |  | 0.064 |  | 6.601** |  | 8.209
| (1.349) |  | (2.858) |  | (2.175) |  | (3.138) |  | (8.435)
DRSMD-Lasso | -0.268 | -6.179 | 0.737* | -1.490 | -0.374 | -2.399 | 0.140 | 37.031 | -1.034** | 5.204**
(0.543) | (5.497) | (0.428) | (3.529) | (0.582) | (2.276) | (0.377) | (31.714) | (0.496) | (2.432)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S16: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ hhincome.

Average Treatment Effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -311.890** |  | -28.504 |  | -260.520** |  | -26.410 |  | -31.122
| (100.413) |  | (123.187) |  | (121.165) |  | (113.466) |  | (260.191)
GMM-Lasso |  | -454.203** |  | -191.738 |  | -255.659 |  | -22.092 |  | 252.781
| (186.231) |  | (219.071) |  | (166.964) |  | (218.677) |  | (558.109)
DRSMD-Lasso | -329.612*** | -602.840** | -128.813 | -356.685* | -108.424 | -116.457 | -171.432* | 1881.963 | 120.352 | 265.458
(126.950) | (279.909) | (132.003) | (194.470) | (92.768) | (148.272) | (90.212) | (1774.876) | (172.107) | (215.602)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S17: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ hhincome. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -274.917*** |  | -127.619 |  | -144.510 |  | -103.605 |  | -69.921
| (91.821) |  | (88.876) |  | (73.094) |  | (66.604) |  | (100.480)
GMM-Lasso |  | -408.080** |  | -217.524 |  | -248.498* |  | -158.129 |  | -16.772
| (187.386) |  | (194.537) |  | (146.186) |  | (160.274) |  | (302.841)
DRSMD-Lasso | -328.977*** | -361.671** | -144.073 | -142.501 | -92.484 | -115.336 | -174.760* | -171.099* | 158.324 | 158.070
(122.904) | (145.690) | (129.297) | (169.190) | (88.281) | (89.513) | (89.460) | (90.094) | (180.997) | (181.520)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -360.531 |  | 438.043 |  | 341.411 |  | 3124.245 |  | 160.364
| (595.971) |  | (1474.901) |  | (692.758) |  | (3527.129) |  | (681.382)
GMM-Lasso |  | -2695.988* |  | -1430.592 |  | -751.087 |  | 3216.942 |  | 620.170
| (1567.443) |  | (2019.704) |  | (1111.768) |  | (4538.626) |  | (383.799)
DRSMD-Lasso | 142.752 | 2647.014 | 67.449 | 3.327 | -429.745 | 1630.753 | -178.469 | -1515.266 | 741.691 | 1129.482
(97.306) | (5599.135) | (196.456) | (6734.098) | (364.850) | (1734.534) | (358.992) | (2158.009) | (699.283) | (1056.448)
Panel B: Homogeneous treatment effects
GMM | -280.010*** | -276.059*** | -121.970 | -126.700 | -140.874* | -143.04** | -94.666 | -101.783 | -69.635 | -69.778
(91.172) | (91.62) | (90.067) | (88.798) | (72.829) | (72.905) | (67.076) | (66.588) | (100.400) | (100.438)
GMM-Lasso | -454.731** | -389.029** | -239.756 | -211.541 | -256.614* | -253.578* | -156.003 | -176.968 | -15.345 | -22.411
(187.884) | (186.352) | (198.492) | (194.269) | (145.706) | (145.766) | (160.649) | (158.521) | (303.056) | (302.061)
DRSMD-Lasso | -326.593*** | -332.779*** | -143.005 | -142.455 | -98.117 | -101.027 | -175.448** | -175.753** | 160.566 | 162.147
(122.625) | (122.35) | (129.513) | (128.689) | (88.763) | (88.614) | (89.326) | (89.209) | (181.463) | (181.422)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S18: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ TANF.

Average Treatment Effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -289.837*** |  | -108.616 |  | -136.800* |  | -75.073 |  | -69.429
| (92.479) |  | (107.224) |  | (73.492) |  | (73.867) |  | (100.347)
GMM-Lasso |  | -519.644*** |  | -279.585 |  | -265.460* |  | -128.750 |  | -14.868
| (202.286) |  | (217.774) |  | (146.215) |  | (171.223) |  | (303.081)
DRSMD-Lasso | -323.070*** | -252.134 | -141.147 | -142.357 | -102.189 | -78.509 | -176.390** | -184.937** | 160.600 | 161.537
(122.184) | (197.461) | (130.029) | (221.220) | (89.195) | (93.360) | (89.177) | (89.035) | (181.463) | (181.433)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S19: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ TANF. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for $\theta_{w0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -534.801* |  | -285.780 |  | -263.638* |  | -241.214 |  | -254.837
| (278.120) |  | (225.302) |  | (144.724) |  | (146.954) |  | (241.115)
GMM-Lasso |  | -555.430 |  | -374.526 |  | -440.806*** |  | -566.475** |  | -578.634
| (361.855) |  | (341.910) |  | (167.480) |  | (230.489) |  | (427.954)
DRSMD-Lasso | -425.270*** | -685.989** | -165.238 | -521.033** | -62.497 | -236.941 | -126.876 | -338.461** | 326.003 | -214.314
(159.063) | (299.677) | (162.906) | (264.138) | (125.410) | (171.558) | (105.255) | (164.009) | (218.319) | (299.180)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | 109.750 |  | 74.501 |  | 60.194 |  | 71.647 |  | 80.074
| (108.556) |  | (99.719) |  | (73.097) |  | (70.949) |  | (105.337)
GMM-Lasso |  | 43.733 |  | 57.688 |  | 93.144 |  | 197.556* |  | 247.086
| (138.335) |  | (137.141) |  | (93.057) |  | (109.936) |  | (176.298)
DRSMD-Lasso | 42.055 | 151.033 | 9.669 | 151.900 | -17.395 | 74.186 | -23.615 | 70.550 | -71.546** | 160.300
(29.563) | (111.229) | (28.358) | (110.100) | (26.269) | (77.875) | (20.306) | (76.311) | (31.867) | (119.734)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S20: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ smoke.

Average Treatment Effects
---
Out of Pocket Cost | Age: 21 - 29 | Age: 30 - 38 | Age: 39 - 47 | Age: 48 - 56 | Age: 57 - 64
Estimator for LATE | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
GMM |  | -276.588*** |  | -112.372 |  | -133.077* |  | -87.451 |  | -62.898
| (90.822) |  | (92.812) |  | (76.291) |  | (68.967) |  | (102.980)
GMM-Lasso |  | -452.537** |  | -240.252 |  | -238.778 |  | -142.495 |  | 13.630
| (188.316) |  | (198.422) |  | (155.644) |  | (165.046) |  | (313.414)
DRSMD-Lasso | -326.325*** | -330.648*** | -142.732 | -167.472 | -100.227 | -76.031 | -177.557** | -187.052** | 154.507 | 169.926
(122.502) | (120.503) | (129.249) | (127.050) | (87.164) | (89.525) | (89.010) | (91.782) | (180.620) | (168.551)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table S21: Heterogeneous Treatment Effects of Medicaid on Out of Pocket Cost

Note: *** Significant at 1%, ** at 5%, * at 10%. Each row shows the estimates
and robust standard errors for the same type of estimator. In the columns, we
present the instruments these estimators used and the age groups. The
interaction term is Medicaid$\times$ smoke. The expression for LATE is
$\theta_{w0}+\theta_{wx0}E(X)$.