# Estimation of Heterogeneous Treatment Effects Using a Conditional Moment
Based Approach

Xiaolin Sun111I am very grateful for the guidance and helpful comments from
Bertille Antoine at Simon Fraser University. I would also like to thank Chris
Bidner, Irene Botosaru, Sam Hwang, Dongwoo Kim, Brian Krauth, Chris Muris,
Krishna Pendakur, Kevin Schnepel, Hitoshi Shigeoka, Kevin Song, Purevdorj
Tuvaandorj, and Simon Woodcock for constructive discussions and suggestions.
Simon Fraser University
<EMAIL_ADDRESS>
###### Abstract

We propose a new estimator for heterogeneous treatment effects in a partially
linear model (PLM) with many exogenous covariates and a possibly endogenous
treatment variable. The PLM has a parametric part that includes the treatment
and the interactions between the treatment and exogenous characteristics, and
a nonparametric part that contains those characteristics and many other
covariates. The new estimator is a combination of a Robinson transformation to
partial out the nonparametric part of the model, the Smooth Minimum Distance
(SMD) approach to exploit all the information of the conditional mean
independence restriction, and a Neyman-Orthogonalized first-order condition
(FOC). With the SMD method, our estimator using only one valid binary
instrument identifies both parameters. With the sparsity assumption, using
regularized machine learning methods (i.e., the Lasso method) allows us to
choose a relatively small number of polynomials of covariates. The Neyman-
Orthogonalized FOC reduces the effect of the bias associated with the
regularization method on estimates of the parameters of interest. Our new
estimator allows for many covariates and is less biased, consistent, and
$\sqrt{n}$-asymptotically normal under standard regularity conditions. Our
simulations show that our estimator behaves well with different sets of
instruments, but the GMM type estimators do not. We estimate the heterogeneous
treatment effects of Medicaid on individual outcome variables from the Oregon
Health Insurance Experiment. We find using our new method with only one valid
instrument produces more significant and more reliable results for
heterogeneous treatment effects of health insurance programs on economic
outcomes than using GMM type estimators.

##  1 Introduction

Heterogeneous treatment effects have been a focus in literature since Imbens
and Angrist (1994). Heterogeneous treatment effects models are those where the
treatment effects of policies, programs, or other variables on some outcome of
interest may vary across individuals with different characteristics. To
measure heterogeneous treatment effects, many papers add interaction terms
between the treatment variable and some covariates to the standard linear
model (see Wooldridge (2010) and Imbens and Rubin (2015)). The basic linear
model becomes linear in the treatment, the interaction term, and the
covariates. The parameters of interest are the parameter for the treatment and
that for the interaction term.

Unfortunately, the true model is unknown. If the model is nonlinear in the
covariates, OLS, 2SLS, or GMM with a linear form will suffer from model
misspecification, and estimates will not be consistent and normally
distributed. With the correct model form, the OLS, 2SLS, or GMM will work.
However, if the nonlinear part of the model is unknown, we will need to use
other methods to estimate the parameters. There are at least three challenges
associated with the estimation of both parameters in the model with a
parametric part and unknown nonlinear part, i.e., the partially linear model
(PLM).

One challenge happens when the treatment variable is endogenous, we need to
use valid instruments to estimate parameters for the treatment and for the
interaction term. The usual approach would be to partial out the unknown and
nonlinear part of the model via a Robinson (Robinson (1988)) transformation
and then use the GMM estimator with one instrument and the interaction term
between the instrument and the covariate to estimate both parameters. When we
only have one valid instrument and the covariate inside the interaction term
has little variation or is not very informative, the GMM estimator will not be
consistent and its variance will have a large magnitude. This is similar to a
weak identification problem. Additionally, the selection of instruments will
affect the interpretation of the treatment (see Dieterle and Snell (2016) for
related discussions).

A second challenge is that traditional estimation methods do not work when the
number of covariates is possibly larger than the number of observations. This
challenge exists if there are a large number of covariates in the structural
model or if many covariates must be included to ensure that the instrument is
valid.

The third challenge is that if the covariates enter the model in an unknown
(and possibly nonlinear) way, estimation methods assuming linearity will
suffer from specification bias.

Our new estimator solves the first challenge by using only the valid
instrument (e.g., the random assignment) to estimate both parameters. Our
method does not use the interaction term between the instrument and the
covariate to identify model parameters. We do this by first applying the
Robinson transformation to remove the nonlinear and unknown part, and then
transforming the conditional moment restriction: $E(\epsilon_{i}|Z_{i})=0$
where $\epsilon_{i}$ is the error term inside the model and $Z_{i}$ is the
instrument. Usually, GMM uses the unconditional moments generated by
$E(\epsilon_{i}|Z_{i})=0$ to construct objective functions. But, with the
i.i.d assumption we can also generate objective function as
$E(\epsilon_{i}|Z_{i})E(\epsilon_{j}|Z_{j})$. As written, this type of
objective function is hard to use. To make it tractable, we apply both the
i.i.d. assumption and the Fourier transform to obtain a new objective function
related with $E(\epsilon_{i}|Z_{i})E(\epsilon_{j}|Z_{j})$. This is tractable.
In technical terms, the Fourier transform and the new objective function are
based on the Smooth Minimum Distance (SMD) approach (Lavergne and Patilea
(2013)). The SMD-based approach exploits all the information of the
conditional mean independence restriction without having to specify the first
stage, or to select a finite number of unconditional moments (e.g., using
transformations of $Z_{i}$).

To tackle the last two challenges, our new estimator uses regularized Machine
Learning (i.e., the Lasso method) to estimate the nuisance parameters
generated by Robinson transformation (Robinson (1988)) and a Neyman-
Orthogonalized (Chernozhukov et al. (2018)) First-Order Condition (FOC). With
the Robinson transformation, we do not need to specify how these covariates
enter the model; it partials out the unknown (and nonlinear) part of the
partially linear model, but it also introduces nuisance parameters which need
to be estimated. The Neyman-Orthogonalized FOC reduces the effect of the bias
associated with the estimation of these nuisance parameters. When there are
many covariates, assuming sparsity and using regularized machine learning
methods, such as the Lasso method, allows us to choose a relatively small
number of covariates for the estimation of nuisance parameters. Thus, our new
estimator is the D-RSMD estimator (Debiased Robinson-SMD).

To apply the new estimator we only need one tuning parameter. If we are using
the Lasso method, the tuning parameter is inside the penalty term for the
estimation of the nuisance parameter. In practice, choosing Lasso tuning
parameters is well-understood (see van de Geer (2016)) and we use cross-
validation (Chernozhukov et al. (2018)). When there are few covariates (and
the model is sparse by design), the Nadaraya–Watson estimator is an
alternative method to estimate the nuisance parameter. This estimation
procedure also needs one tuning parameter, that is, the bandwidth. The
bandwidth can be chosen by the rule of thumb or cross-validation in practice.
We show that our D-RSMD estimator is consistent and $\sqrt{n}$ asymptotically
normal under mild regularity conditions. Monte-Carlo simulations show that the
D-RSMD estimator outperforms the R-SMD and GMM-type estimators in terms of
much smaller bias and lower standard errors.

To illustrate the value of our method, we estimate both parameters to check
the effects of Medicaid from the Oregon Health Insurance Experiment222I would
like to thank A. Colin Cameron for bringing the Oregon Health Insurance
Experiment to my attention. and to extend the estimation results in Card
(1993). In the Oregon Health Insurance Experiment, using our estimator with
only one valid instrument produces statistically significant results for
heterogeneous treatment effects when the GMM estimator does not. We use the
Card application to show the differences of estimators using various sets of
instruments. It shows that our new method generates more precise estimates in
comparison to GMM.

Our work is an extension of Antoine and Sun (2021) who propose the R-SMD
estimator which combines a Robinson transformation and a smooth minimum
distance estimator in a partially linear model333See Robinson (1988) and Li
and Racine (2006) for combining Robinson transformation and 2SLS estimators..
However, when we use the Nadaraya-Watson estimator to estimate the conditional
means, the dimension of covariates has to be less than four (see chapter 7 of
Li and Racine (2006)). To extend the R-SMD estimator to the big data setting
and relax the restriction on the dimensions of covariates, our work applies
machine learning methods, such as the Lasso. To reduce the effect of bias
introduced by Lasso, our work employs the Neyman-orthogonal first-order
condition. The method also extends the Neyman-orthogonal estimator
(Chernozhukov et al. (2018)) to the U-statistic setting. Our new estimator
contributes to the literature on the partially linear model and conditional
moment restriction models in the same way as the R-SMD estimator does. Both
the partially linear model and conditional moment restriction model involve
the problem of choosing the model form. For a partially linear model, some
researchers use sieves for the nonlinear part. For conditional moment
restriction models, some generate a finite number of unconditional moments. In
comparison, our method does not need to choose the number of polynomials for
the sieve method or the number of unconditional moments. Our method only needs
to choose one tuning parameter.

The paper is organized as follows. Section 2 introduces our framework,
motivation, and our estimator. Section 3 states the large sample properties
for our estimator. Section 4 presents the simulation results for finite
samples with and without a large number of covariates. Section 5 uses the
D-RSMD estimator to estimate the effects of education on earnings and effects
of Enrollment in Medicaid on outcomes of interest, based on the Oregon
Medicaid health experiment. Additional results and proofs are in the Appendix
and Supplementary Appendix.

##  2 Framework and Motivation

Our framework is based on a partially linear model with a treatment variable
$W_{i}$ which is binary. The heterogeneous treatment effects enter the model
through the linear term on $W_{i}$ and some interaction terms between $W_{i}$
and $X_{i}$, a vector of $P$ covariates with
$X_{i}=[X_{i1}^{\prime},X_{i2}^{\prime}]^{\prime}$. The other part of the
model is an unknown function of $X_{i}$.

$y_{i}=\theta_{w0}W_{i}+W_{i}\cdot
X_{i1}^{\prime}\theta_{wx0}+f_{0,1}(X_{i})+\epsilon_{i}$ (2.1)

The treatment is not always exogenous. For example, in the Oregon Medicaid
health experiment, the treatment is the enrollment in Medicaid. Enrollment in
Medicaid is endogenous because it is based on the choices of the lottery
winners. Only a fraction of lottery winners decided to enroll in Medicaid.

$W_{i}\cdot X_{i1}$ is the interaction term. The formal definition for
$W_{i}\cdot X_{i1}^{\prime}\theta_{wx0}$ is

$\sum_{k=1}^{p_{1}}W_{i}X_{i1,k}\times\theta_{wx0,k}$

where $X_{i1,k}$ is a $k$-th element inside the vector $X_{i1}$. The treatment
part of interest is $W_{i}+W_{i}\cdot X_{i1}$. The parameters that measure
heterogeneous treatment effects, i.e. $\theta_{w0}$ and $\theta_{wx0}$, are
our key parameters. The control vector $X_{i}$ in our setting is exogenous and
appears in an unknown function $f_{0,1}(X_{i})$. $f_{0,1}(.)$ is a nuisance
parameter that we are not interested in.

$X_{i1}$ is a subvector of $X_{i}$ of dimension $p_{1}$ with $p_{1}\leq P$. We
can have $X_{i1}=X_{i}$ if the dimension of $X_{i2}$, $p_{2}$, is zero. That
is, $W_{i}\cdot X_{i1}$ can be the interaction term using only a small set of
covariates, while $f_{0,1}(X_{i})$ is the unknown function of the whole set of
exogenous controls. The dimension of $X_{i}$ is not restricted and can be
either high or low-dimensional. However, we restrict the dimension of
$X_{i1}$. And we also maintain the exogeneity of all the controls $X_{i}$. The
outcome variable $y_{i}$ is not necessarily binary. For a binary outcome
variable, a large number of covariates, and similar restricted dimension
setting on estimating heterogeneous treatment effects, see Nekipelov et al.
(2018).

The unknown form of $f_{0,1}(X_{i})$ allows $X_{i}$ to enter the model in a
flexible way. To avoid estimating the nuisance parameter $f_{0,1}(X_{i})$, a
Robinson transformation is introduced to eliminate the unknown
$f_{0,1}(X_{i})$. The Robinson transformation consists of subtracting the
conditional expectation of $y_{i}$ with respect to the controls $X_{i}$. After
such a Robinson transformation, $f_{0,1}(X_{i})$ will disappear. In this case,
we do not need to assume the form of $f_{0,1}(X_{i})$. It can be sparse and
may be nonlinear. With Robinson transformation and the exogenous assumption on
$X_{i}$, Equation 2.1 becomes

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

where $P_{i}=[W_{i},W_{i}\cdot X_{i1}^{\prime}]^{\prime}$ and
$\theta_{0}=[\theta_{w0},\theta_{wx0}^{\prime}]^{\prime}$.

Denote

$\epsilon_{j}(\theta,g_{0})=\tilde{y}_{j}-\tilde{P}_{j}\theta$ (2.2)

where $\tilde{y}_{j}\equiv y_{j}-E(y_{j}|X_{j})$ and $\tilde{P}_{j}\equiv
P_{j}-E(P_{j}|X_{j})$. $g_{0}$ stands for all of the unknown real values of
nuisance parameters. $g_{0}$ is a vector of $g_{0,y}(X_{j})$ (or
$E(y_{j}|X_{j})$) and $g_{0,P}(X_{j})$ (or $E(P_{j}|X_{j})$) at this stage.

As in the previous discussion, $W_{i}$ and the interaction term between
$W_{i}$ and $X_{i1}$ are all endogenous. Following the classic endogenous
variable estimation, a vector of instruments, including the random assignment
$Z_{i}$, is introduced. For instance, in the Oregon Medicaid health
experiment, the instrument is the lottery outcome444In the experiment, the
lottery winners are allowed to enroll in the Medicaid program. Not every
lottery winner enrolled. The treatment variable is the actual enrollment
status.. We maintain the conditional mean independence assumption for the
random assignment. With valid instrument and exogenous control variables, the
conditional moment restriction is

$E(\epsilon_{i}|X_{i},Z_{i})=0\ a.s.$ (2.3)

In our setting, we look at the single treatment case where the controls may be
correlated with the instruments. With traditional estimators such as GMM or
2SLS, $1+p_{1}$ instruments are needed to identify (and estimate) the slope
parameters associated with $W_{i}$ and the interaction terms $W_{i}\cdot
X_{i1}$. GMM uses $Z_{i}$ (the random assignment) and the interaction terms
$Z_{i}\cdot X_{i1}$ to estimate the key parameters.

However, if there is little variation in $X_{i1}$, the $Z_{i}\cdot X_{i1}$ is
less informative, or if we use a second instrument that is (highly) correlated
with the valid $Z_{i}$, the GMM method will fail to provide reliable
estimates: intuitively, it is similar to a weak instrument problem. With only
one valid instrument, GMM encounters the problem of how to generate new
moments when we need to estimate more than one parameter. With only one valid
instrument and two parameters to estimate, there is under-identification, and
GMM cannot be implemented. We will show that our estimation strategy, which
only relies on using one valid instrument (e.g. the random assignment
$Z_{i}$), delivers reliable inference on both parameters.

Further, when it comes to the interpretation of the heterogeneous treatment
effects, when using two or more instruments, additional assumptions are needed
because the traditional monotonicity assumption (as in Imbens and Angrist
(1994)) is only for one instrument. For our setting, in the simplest case
where $X_{i1}$ is one dimensional (with values -1 or 1), if we use instruments
$Z_{i}$ (with values 0 or 1) and $Z_{i}\cdot X_{i1}$, then there are two types
of compliers. The first kind of compliers are the individuals who will accept
the treatment if and only if $Z_{i}$ is one regardless of the values of
$X_{i1}$. The second kind of compliers are the individuals who will accept the
treatment if and only if $Z_{i}$ is one and $X_{i1}$ is positive.

Hence, if we want to use LATE interpretation as in Imbens and Angrist (1994)
then we cannot include the interaction term to measure the heterogeneous
treatment effects. If we need to estimate the interaction term, then we need
to use the interaction term between the instrument and covariates as extra
instruments. As a consequence, the types of compliers will affect the
interpretation of LATE. This interpretation problem is discussed further after
our formal Assumption 1 is introduced.

This dilemma can be solved by employing the conditional moment restriction
directly. The conditional moment restriction contains all of the information
no matter the number of instruments inside. Hence, the conditional moment
based approach enables us to use one instrument $Z_{i}$ (e.g. the random
assignment) only to identify and estimate slope parameters associated with
both $W_{i}$ and $W_{i}\cdot X_{i1}$. The Bierens type estimator is a
conditional moment based approach which transforms the conditional moment into
an infinite number of unconditional moments using complex exponential
functions. See Bierens (1982), Antoine and Lavergne (2014), and Antoine and
Sun (2021).

$\displaystyle
E[\epsilon_{j}(\theta_{0},g_{0})e^{it^{\prime}Z_{j}}]=0\;\forall
t\in\mathbb{R}^{q_{z}}\qquad\Longleftrightarrow\qquad
E(\epsilon_{j}(\theta_{0},g_{0})|Z_{j})=0\ a.s.$ (2.4)

The population objective function defined in Equation (2.5) below is based on
the previous equation. Inside the objective function, $\mu(t)$ is a strictly
positive measure on the vector $t$. $Z_{i}$ stands for the vector of
instruments. 555$Z_{i}$ denotes the general vector of instruments. If we only
use one instrument, $Z_{i}$ will be the random assignment. If we use a vector
of instruments, $Z_{i}$ will be the random assignment and the interaction term
between random assignment and covariates.

$\displaystyle
M_{\infty}(\theta,g)=\int_{\mathbb{R}^{q_{z}}}|E[\epsilon_{j}(\theta,g)e^{it^{\prime}Z_{j}}]|^{2}d\mu(t)$
(2.5)

The objective function involves the norm of a complex function. To estimate
$\theta_{0}$, we need to find its derivative, which is difficult to compute.
Under the independent assumption for the population, the objective function
has an alternative expression.

$\forall j\neq l$,

$\displaystyle
M_{\infty}(\theta,g)=E[\epsilon_{j}(\theta,g)\epsilon_{l}(\theta,g)\kappa_{j,l}]\;\;\text{with}\;\;\kappa_{j,l}=\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(Z_{j}-Z_{l})}d\mu(t)$
(2.6)

The objective function defined in Equation (2.6) is a function of the
parameters $\theta$ we are interested in and the nuisance parameters $g$ as in
the Equation (2.2) after the Robinson transformation. With Regularity
Assumptions provided in the following, $\theta_{0}$ is the unique minimizer of
the objective function $M_{\infty}(\theta,g_{0})$ where $g_{0}$ is a vector of
$g_{0,y}(X_{j})$ (or $E(y_{j}|X_{j})$) and $g_{0,P}(X_{j})$ (or
$E(P_{j}|X_{j})$) at this stage.

######  Assumption 1.

(Regularity assumptions)
(i) $E(\epsilon_{i}|X_{i},Z_{i})=0$.
(ii) $E(\tilde{P}_{i}|Z_{i})\neq 0$ a.s. (with probability 1) with
$\tilde{P}_{i}=P_{i}-E(P_{i}|X_{i})$.
(iii) $E(\tilde{P}_{i}\tilde{P}_{i}^{\prime})$ is nonsingular.
(iv) Let $f_{Z}(.)$ denote the density function of $Z_{j}$. We assume that
$E(\tilde{P}_{j}|Z_{j}=.)f_{Z}(.)$ is $L_{q}$ for some $1\leq q\leq 2$.
(v) for all possible $z_{1},z_{2}$, either $E(W_{i}|Z_{i}=z_{1})\geq
E(W_{i}|Z_{i}=z_{2})$ for all $i$, or $E(W_{i}|Z_{i}=z_{1})\leq
E(W_{i}|Z_{i}=z_{2})$ for all $i$.
(vi) $(y_{l},W_{l},X_{l},Z_{l})$ is an independent and identical copy of
$(y_{j},W_{j},X_{j},Z_{j})$.
(vii) Let $\mu$ be a given strictly positive measure on $\mathbb{R}^{q_{z}}$.
Let $k(.)$ be the Fourier transform induced by $\mu$,
$k(Z_{j}-Z_{l})=\int_{\mathbb{R}^{q_{z}}}e^{it^{\prime}(Z_{j}-Z_{l})}d\mu(t)$.
We assume that $k(.)$ is a symmetric bounded density function on
$\mathbb{R}^{q_{z}}$ and that its Fourier transform is strictly positive.

Assumption 1 is the exogeneity assumption for the controls and instruments. In
the potential outcomes setting, Assumption 1 is also the Random Assignment
Assumption. The model form implies the Exclusion Restriction Assumption as in
Imbens and Rubin (2015), that is, the value of the instrument does not affect
the potential outcomes directly. Assumption 1 is the relevant instrument
assumption. Assumption 1 and 1 guarantee the identification of the parameters
that we are interested in. Robinson (1988) also imposes the same assumption as
Assumption 1. Assumption 1 is for the measure $\mu(.)$. These conditions in
Assumption 1 are not very restrictive. There are many different available
measures. We use the CDF of Gaussian distribution in simulations and
applications.

The Monotonicity Assumption (or Assumption 1) needs more investigation.
Assumption 1 is a multivariate extension of the Monotonicity Assumption from
Imbens and Angrist (1994). It is a condition that assumes all individuals make
the same choice if they are given the same options. That is, even if there are
two types of compliers in reality, Assumption 1 assumes that only one type
exists. Recall that when individuals accept the treatment based on the value
of the covariate inside the interaction term, they are different compliers. If
we only use one instrument, Assumption 1 will simply be the Monotonicity
Assumption. The interpretation of the parameters $\theta_{w0}$ and
$\theta_{wx0}$ are associated with the average treatment effect of the
compliers who make their choices based on the result of the random assignment.
If $Z_{i}$ is a vector of instruments, Assumption 1 still ensures that there
is only one type of compliers. The interpretation of the parameters
$\theta_{w0}$ and $\theta_{wx0}$ is connected with the average treatment
effect of that kind of compliers. We will show that our estimator can be
reliably implemented with only one instrument. In our simulations and
applications, we will provide results using one and multiple instruments for
comparison.

Under Assumption 1, $\theta_{0}$ is the unique minimizer of

$M_{\infty}(\theta,g)=0$

when $g=g_{0}$, because $E(\epsilon_{i}|Z_{i})=0$ with probability 1. Without
replacing the nuisance parameter $g$ with its true value $g_{0}$ the First
Order Condition of $M_{\infty}(\theta,g)$ is

$\displaystyle
E(\left(P_{j}-g_{P}(X_{j})\right)[y_{l}-g_{y}(X_{l})-(P_{l}-g_{P}(X_{l}))^{\prime}\theta]\kappa_{j,l})=0$
(2.7)

When $g=g_{0}$, $g_{P}(X_{j})$ becomes $g_{0,P}(X_{j})$ or $E(P_{j}|X_{j})$
and $g_{y}(X_{l})$ is $g_{0,y}(X_{l})$ or $E(y_{l}|X_{l})$, the FOC becomes:

$\displaystyle
E[\tilde{P}_{j}(\tilde{y}_{l}-\tilde{P}_{l}^{\prime}\theta)\kappa_{j,l}]=0$
(2.8)

$\theta_{0}$ is identified under a strong identification assumption and
Assumption 1. We leave the formal discussion to Proposition 1. The FOC defined
in Equation (2.8) provides an explicit form for $\theta_{0}$. The sample
analog of the explicit form for $\theta_{0}$ under Equation (2.8) is a direct
extension from Antoine and Sun (2021) which allows for an interaction term
inside the parametric part of the model. The expression for $\theta_{0}$ is
provided in Section E of the Appendix. Antoine and Sun (2021) provide the
estimator for the parameter in front of the treatment. It is a special case of
Lavergne and Patilea (2013) when the bandwidth inside $\kappa_{j,l}$ is fixed.
Lavergne and Patilea (2013) show that such an estimator is consistent and
asymptotically normal. However, since the nuisance parameter $g_{0}$ is
unknown, the estimator based on the sample analogue of Equation (2.8) is
infeasible and denoted as $\tilde{\theta}_{n,u}$. To obtain the feasible
estimator, we need to estimate the nuisance parameters in the first step.
Antoine and Sun (2021) replace their nuisance parameters with Nadaraya-Watson
estimators, and show that with proper assumption on the bandwidth, the
infeasible and feasible estimators share the same asymptotic properties. To
satisfy the assumption on bandwidth, the Nadaraya-Watson estimator imposes one
constraint on the number of covariates. We will discuss this constraint in the
later sections about feasible estimators. Using Nadaraya-Watson estimators or
other non-parametric estimators, with regularized conditions, such as the
Lasso method, the bias introduced in estimating the nuisance parameters may
cause bias in the second stage where we estimate the key parameters.

We propose a new FOC to estimate the $\theta_{0}$, which extends the Neyman-
orthogonal estimator (Chernozhukov et al. (2018)) to the U-statistic setting.
Our original FOC defined in Equation (2.7) is not orthogonal to the nuisance
parameter, so the bias in the first step will affect the estimate of the key
parameter. This is shown in the Appendix. The Neyman-orthogonal method
constructs a new FOC which is orthogonal to the nuisance parameters introduced
in the first step. The new FOC has all partial derivatives with respect to the
nuisance parameters equal to zero, in this way it is orthogonal to the bias of
the nuisance parameter (or function). The definition of the partial derivative
with respect to a function is provided in Chernozhukov et al. (2018). A
similar method is used in Nekipelov et al. (2018).

After Neyman-orthogonalization, the new FOC for $\theta_{0}$ becomes
insensitive to the bias of the estimator for nuisance parameters. It is
sensitive to the square of the bias. As long as the bias is $o_{p}(n^{-1/4})$,
we still have a $\sqrt{n}$-asymptotically normally distributed estimator for
the key parameter. This order for the bias is not an issue, because there are
still many estimation methods to choose from, for instance, the Lasso, Sieves,
Random forest, and so on.

The new FOC for $\theta_{0}$ corresponding to Equation (2.7) is

$\displaystyle E[\Psi(D;\theta,g_{0})]$ $\displaystyle\equiv$ $\displaystyle
E\left[\left(P_{j}-g_{0,P}(X_{j})-\frac{g_{0,\tilde{P}_{m}}(X_{l})}{g_{0,\kappa_{m,l}}(X_{l})}\right)[y_{l}-g_{0,y}(X_{l})-(P_{l}-g_{0,P}(X_{l}))^{\prime}\theta]\kappa_{j,l}\right]$
(2.9) $\displaystyle=$ $\displaystyle 0$

with $g_{0,\tilde{P}_{m}}(X_{l})\equiv
E[(P_{m}-g_{0,P}(X_{m}))\kappa_{m,l}|X_{l}]$ and
$g_{0,\kappa_{m,l}}(X_{l})\equiv E[\kappa_{m,l}|X_{l}]$.

$g_{0,\tilde{P}_{m}}(X_{l})$ and $g_{0,\kappa_{m,l}}(X_{l})$ are two
additional parameters inside the nuisance parameter vector. At this stage,
there are four nuisance parameters inside the vector. With these two extra
nuisance parameters, the FOC defined in Equation (2.9) has a partial
derivative with respect to all nuisance parameters equal to zero. This is
shown in the Appendix.

Equation (2.9) gives us the identification of the true key parameter and the
forms of the infeasible and feasible estimators.

######  Proposition 1.

(Identification of $\theta_{0}$ using the orthogonalized FOC)
Under Assumption 1 and FOC defined in Equation (2.9)

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

In Section D of the Appendix, we illustrate the identification issue for the
group of SMD estimators, such as SMD, RSMD, and D-RSMD. We consider cases
involving various types of instruments and models.

If we plug Equation (2.2) into the formula for $\theta^{*}_{0}$ with error
term $\epsilon_{l}$, we will have $\theta^{*}_{0}=\theta_{0}$. The sample
analog of $\theta^{*}_{0}$ under Equation (2.9) (in Proposition 1) delivers an
estimator for $\theta_{0}$. Because we have two distinct individuals inside
the expectation (e.g., j and l), we need to replace the expectation by the
average of a double summation to obtain the infeasible estimator under
Equation (2.9). The closed-form expression for the infeasible estimator,
$\tilde{\theta}_{n,o}$, is:

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

To distinguish the infeasible estimators under two different FOCs, the
infeasible estimator under orthogonal FOC (Equation (2.9)) is denoted as
$\tilde{\theta}_{n,o}$. Recall that the infeasible estimator under Equation
(2.8) (from Antoine and Sun (2021)) is $\tilde{\theta}_{n,u}$ and the
expression is shown in Section E of the Appendix.

##  3 Large Sample Theory

In this section, we show the asymptotic properties of the infeasible
estimators under the Neyman-Orthogonalized FOC. Then, we introduce the D-RSMD
estimator (Debiased Robinson-SMD) as the feasible estimator under the Neyman-
Orthogonalized FOC, which shares the same asymptotic properties as the
infeasible estimator. The asymptotic properties of the infeasible and feasible
estimators under the non-orthogonal FOC from Antoine and Sun (2021) are in the
Appendix.

###  3.1 Asymptotic Properties of the Infeasible Estimators

The infeasible estimators under both FOCs have explicit forms and are linear
in $\tilde{y}_{l}$. The SMD estimator introduced in Lavergne and Patilea
(2013) has a general form for the asymptotic properties even if there are no
explicit forms for these infeasible estimators. Our work here extends Lavergne
and Patilea (2013) to allow for nuisance parameters and introduces the
debiased method to limit the impact of their estimation.

######  Proposition 2.

(Consistency and Asymptotic normality of $\tilde{\theta}_{n,o}$)
Under Assumption 1 and iid assumption for the sample, $\tilde{\theta}_{n,o}$
is consistent for $\theta_{0}$, that is
$\tilde{\theta}_{n,o}\stackrel{{\scriptstyle p}}{{\rightarrow}}\theta_{0}$,
and asymptotically normally distributed,

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

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

$\displaystyle\text{and}\quad\verb'Var'\left[h_{1}(\tilde{P}_{j},\epsilon_{j},Z_{j},X_{j})\right]\equiv\verb'Var'\left[\int_{\mathbb{R}^{q_{z}}}e^{-it^{\prime}Z_{j}}\epsilon_{j}\left(E[e^{it^{\prime}Z_{l}}\tilde{P}_{l}]-\frac{g_{0,\tilde{P}_{m}}(X_{j})}{g_{0,\kappa_{m,j}}(X_{j})}E[e^{it^{\prime}Z_{l}}]\right)d\mu(t)\right]$

$\verb'Var'$ stands for the variance-covariance matrix for the vector
$h_{1}(\tilde{P}_{1},\epsilon_{1},Z_{1},X_{1})$, which is a conditional mean
function defined in Hoeffding (1948). The expression of
$h_{1}(\tilde{P}_{1},\epsilon_{1},$ $Z_{1},X_{1})$ is shown in the proof
section of the Appendix. The asymptotic properties of $\tilde{\theta}_{n,o}$
are based on the corresponding properties of U-statistics introduced by
Hoeffding (1948). See Theorem 7.1. The expression for the middle term of the
asymptotic variance looks complicated because the $\kappa_{j,l}$ is expressed
explicitly. If we show the asymptotic variance with $\kappa_{j,l}$, the form
will be simple. The form for the asymptotic variance is shown in the Appendix.

Additionally, we show the similar consistency and asymptotic normality
properties of $\tilde{\theta}_{n,u}$ in the Appendix. Because
$\tilde{\theta}_{n,u}$ is a direct extension from Antoine and Sun (2021), we
only list the expressions and theorems. The detailed proofs for the properties
of $\tilde{\theta}_{n,u}$ are shown in the Appendix of Antoine and Sun (2021).
The comparison between Neyman orthogonal estimators and non-orthogonal
estimators is discussed in the next section.

###  3.2 Feasible Estimator

The infeasible estimator $\tilde{\theta}_{n,o}$ depends on unknown nuisance
parameters: $g_{0,\tilde{P}_{m}}(X_{l})$, $g_{0,\kappa_{m,l}}(X_{l})$,
$E(y_{l}|X_{l})$, and $E(P_{l}|X_{l})$. All of the nuisance parameters are
conditional expectation functions on covariates. Hence, in practice, we need
to find estimators for these conditional expectation functions to obtain the
feasible estimator. Replacing every nuisance parameter $g_{0}$ with estimators
$\hat{g}$ such that $\hat{g}$ converges to $g_{0}$ at a rate of
$o_{p}(n^{-1/4})$ will deliver the feasible estimator on $\theta_{0}$ with
$\sqrt{n}$ asymptotic normality.

In this section, we are concerned with models with $P$ variables inside
$X_{l}$ where $P$ can be large. If $P$ is large, we need to assume sparsity
for the nuisance parameters, that is, the conditional means can be described
with only a few non-zero parameters in front of $X_{l}$. The number of non-
zero parameters, $s_{0}$, is allowed to grow at the rate of
$o_{p}(n^{1/2}/log(P))$. Under the assumed rate for $s_{0}$, the Lasso
estimation has the following property (see van de Geer (2016)) on the order of
mean square error

$||X(\hat{\beta}-\beta_{0})||_{2}^{2}/n=\mathcal{O}_{p}\left(\frac{s_{0}log(P)}{n}\right)$

where $\hat{\beta}$ is the Lasso estimator (Chernozhukov et al. (2018) and van
de Geer (2016)) and $||.||_{2}$ is the $\mathcal{L}_{2}$ norm. When $3<P<n$,
we can still use the Lasso method with a higher assumed rate for $s_{0}$ to
select variables. If $P\leq 3$, we can use Nadaraya–Watson estimator to
estimate the nuisance parameter with a second degree kernel. $P\leq 3$ is a
constraint which is unlikely to hold in practice. Our estimation procedure
allows us to handle larger values of $P$ unlike previous literature: for
instance, Li and Racine (2006), Robinson (1988), and Antoine and Sun (2021).

After replacing every nuisance parameter with its estimate, we have the
feasible estimator $\hat{\theta}_{n,o}$, that is, D-RSMD estimator:

$\hat{\theta}_{n,o}=\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}\right]^{-1}\left[\frac{1}{n(n-1)}\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{y}}_{l}\right]$
(3.1)

since
$E[\kappa_{j,l}\left(\widehat{\tilde{P}}_{j}-\frac{\widehat{g_{0,\tilde{P}_{m}}}(X_{l})}{\widehat{g_{0,\kappa_{m,l}}}(X_{l})}\right)\widehat{\tilde{P_{l}}}^{\prime}]$
is invertible.

This leads to the algorithm of our D-RSMD estimation procedure.

######  Algorithm 3.1.

(Implementation of the D-RSMD estimation procedure)

  * (i)

conduct Robinson transformation. This step is to estimate the conditional
means inside $\widehat{\tilde{P}}_{j}$ and $\widehat{\tilde{y}}_{l}$. Any
estimators $\hat{g}$ with $\hat{g}$ converges to $g_{0}$ at a rate of
$o_{p}(n^{-1/4})$ can be applied, for instance, the Lasso method.

  * (ii)

calculate the $\kappa_{j,l}$. Inside $\kappa_{j,l}$, $\mu(.)$ is the CDF of
the Gaussian distribution. Because the Fourier transform of the Gaussian is
still Gaussian, $\kappa_{j,l}$ is easy to compute.

  * (iii)

estimate the nuisance parameters $\widehat{g_{0,\tilde{P}_{m}}}(X_{l})$ and
$\widehat{g_{0,\kappa_{m,l}}}(X_{l})$ inside the orthogonal FOC. The
orthogonal FOC is also orthogonal to the nuisance parameters, so we use the
Lasso method to estimate these parameters.

  * (iv)

calculate the estimate based on Equation (3.1) for $\hat{\theta}_{n,o}$.

In the approach from the algorithm, all of the nuisance parameters for D-RSMD
estimators in the later sections are estimated by the Lasso method with cross
validation. The maximum degree of the polynomial for the nuisance parameters
is 5, which guarantees that the nuisance parameters can be approximated by 5
degree polynomials in all controls and Lasso with cross validation helps us
select the controls and their polynomials.

######  Assumption 2.

$\hat{g}$ converges to $g_{0}$ at a rate of $o_{p}(n^{-1/4})$. For the Lasso
method, the number of non-zero parameters $s_{0}$ grows at the rate of
$o_{p}(n^{1/2}/log(P))$. For the Nadaraya-Watson estimator,
$\sqrt{n}\left(\sum_{s=1}^{q_{z}}h_{s}^{4}+\left[\frac{1}{nh_{1}...h_{q_{z}}}\right]\right)=o(1)$
where $h$ is the bandwidth.

######  Theorem 3.2.

(Consistency and Asymptotic normality of the D-RSMD estimator:
$\hat{\theta}_{n,o}$) Under Assumptions 1 \- 2 and iid assumption for the
sample, $\hat{\theta}_{n,o}$ is consistent, and has an asymptotically normal
distribution, that is,

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

From Theorem 3.2, the feasible and infeasible estimators share the same
asymptotic distribution and are consistent. It is because under Assumption 2,
the bias introduced in the first step will not affect the second step
substantially. This is the same convergence rate assumption used in
Chernozhukov et al. (2018). We expand these properties to Bierens type
estimators.

The asymptotic properties for the feasible version of $\tilde{\theta}_{n,u}$
or the R-SMD estimator from Antoine and Sun (2021) are in Section E of the
Appendix.

The asymptotic distributions of the D-RSMD and R-SMD estimators are not the
same. When there are many covariates, the D-RSMD estimator using the Lasso for
the nuisance parameters works. R-SMD is generated under the condition that the
number of covariates ($P$) is small. When $P$ is small, the D-RSMD estimator
is less biased than the R-SMD estimator if they both estimate the nuisance
parameters using the same Nadaraya-Watson estimator. There are no analytical
results for comparing the asymptotic distributions. In Chernozhukov et al.
(2018), the comparison between Neyman orthogonal estimators and non-orthogonal
estimators is shown with simulations. We also present the results on bias in
Section 4, i.e., the simulation section.

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

$\displaystyle[n(n-1)C_{n}]^{-1}\sum^{n}_{j=1}((\sum^{n}_{l=1}\kappa_{j,l}\left(\widehat{\tilde{P}}_{l}-\frac{\widehat{g_{0,\tilde{P}_{l}}}(X_{j})}{\widehat{g_{0,\kappa_{m,j}}}(X_{j})}\right))(\sum^{n}_{l=1}\kappa_{j,l}\left(\widehat{\tilde{P}}_{l}-\frac{\widehat{g_{0,\tilde{P}_{l}}}(X_{j})}{\widehat{g_{0,\kappa_{m,j}}}(X_{j})}\right)^{\prime})\hat{\epsilon}_{j}^{2})[n(n-1)C_{n}^{\prime}]^{-1}$
(3.2)

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

##  4 Simulation Study

In this section, we conduct 5000 Monte-Carlo replications for two types of
Data Generating Process (DGP) to investigate the properties of our D-RSMD
estimators for the parameters in the heterogeneous treatment effects of a
partially linear model when the number of controls in $X_{i}$ is greater than
or equal to 1. Recall that the partially linear model with heterogeneous
treatment effects is in the following.

$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}W_{i}+W_{i}\cdot
X_{i1}^{\prime}\theta_{wx0}+f_{0,1}(X_{i})+\epsilon_{i}$ (4.1) $\displaystyle
W_{i}$ $\displaystyle=$ $\displaystyle I(f_{0,2}(X_{i},Z_{i})>v_{i})$

$W_{i}\cdot X_{i1}$ is the interaction term between the treatment and a
subvector of covariates. The benchmark model has a nonlinear $f_{0,1}(X_{i})$
and a nonlinear function $f_{0,2}(X_{i},Z_{i})$. $I(.)$ is the indicator
function. It will take the value one if the statement inside is true and zero
otherwise. In all of the simulations, the key parameters $\theta_{w0}$ and
$\theta_{wx0}$ are 2 and 3 respectively, and we use the same benchmark model.

With the chosen parameter values, the benchmark model is

$\displaystyle X_{i}$ $\displaystyle=$ $\displaystyle X_{i}^{*}+0.4Z_{i}$
$\displaystyle W_{i}$ $\displaystyle=$ $\displaystyle
I(3Z_{i}+4Z_{i}^{3}+\sum_{q=1}^{P}\alpha_{1q}X_{qi}+\sum_{q=1}^{P}\alpha_{3q}X_{qi}^{3}>-v_{i})$
$\displaystyle y_{i}$ $\displaystyle=$ $\displaystyle
2W_{i}+3W_{i}X_{1i}+\sum_{q=1}^{P}\beta_{1q}X_{qi}+\sum_{q=1}^{P}\beta_{2q}X_{qi}^{2}+\epsilon_{i}$

where $\alpha_{3q}=2$, $\beta_{2q}=-3$, $\alpha_{1q}=\beta_{1q}=1$ for $q\leq
S$ with $S$ the number of non-zero parameters and $P$ the number of
covariates. Because there are nonlinear terms in the model, $S$ is chosen to
be a half of $s_{0}$, the sparsity level.
$\alpha_{3q}=\beta_{2q}=\alpha_{1q}=\beta_{1q}=0$ when $q>S$.

If $\beta_{2q}$ is 0 for all $q$, $y_{i}$ is linear in $X_{i}$. Otherwise, the
model is partially linear, because $X_{i}$ enters nonlinearly. The covariate
$X_{i}$ is the sum of random variables $X_{i}^{*}$ generated by a standard
normal distribution (or multivariate standard normal distribution) and a
fraction of the instrument because our model allows for the correlation
between covariate $X_{i}$ and $Z_{i}$. In the benchmark model, we choose the
fraction level to be 0.4. The errors $(\epsilon,v)$ are bivariate normal
distributed with mean 0 and covariance matrix $\Sigma_{1}$ such that
$\Sigma_{1}=\left(\begin{array}[]{cc}1&4/9\\\ 4/9&1\end{array}\right)$. The
correlation between $\epsilon$ and $v$ makes the treatment $W_{i}$ an
endogenous variable. In the DGP, $y_{i}$ is not dependent on $Z_{i}$ directly.
The exclusion restrictions assumption is satisfied. Also, we generate
$X_{i}^{*}$ and $Z_{i}$ separately and independently because $X_{i}^{*}$ is
continuous and $Z_{i}$ is categorical.

When there is only one control variable $X_{i}$, the two nuisance parameters
are

$\displaystyle f_{0,1}(X_{i})$ $\displaystyle=$
$\displaystyle\beta_{11}X_{i}+\beta_{21}X_{i}^{2}$ $\displaystyle
f_{0,2}(X_{i},Z_{i})$ $\displaystyle=$ $\displaystyle
3Z_{i}+4Z_{i}^{3}+\alpha_{11}X_{i}+\alpha_{21}X_{i}^{3}$

We consider two types of DGPs which mainly differ in how the treatment
variable is generated, e.g. using either a categorical instrument with three
values, or a binary instrument. Both are motivated by our applications. The
first kind of DGP uses a categorical instrument to generate a treatment
variable. Specifically, the instrument variable has three values, 0, 1, and 2,
and can be interpreted as the sum of two binary variables. This is motivated
by our application based on Card (1993) where we construct a similar
instrument by adding an indicator for proximity to a four-year college to an
indicator for two-year college.

The second type of DGP employs a binary instrument to generate a treatment
variable. Correspondingly in Card (1993), this is the indicator for proximity
to a four-year college 666In the Oregon Health Insurance experiment the
lottery variable is also a binary instrument.. We generate the binary
instrument based on the distribution of the indicator for proximity to a four-
year college.

For each DGP, the benchmark case considers 3000 observations and 30 control
variables. This is once again in line with both applications. For instance, in
Card (1993), there are 3010 valid observations and 27 covariates777Analyzing
the Oregon Health insurance experiment, we split the data set into three
groups based on age. Each group has about 6000 observations and 21
covariates.. We will also consider different sample sizes and a small number
of controls.

We report and compare the simulation results for the following estimators.

  * (i)

the D-RSMD estimator proposed in this paper.

  * (ii)

the R-SMD estimator (from Antoine and Sun (2021)) when the number of controls
is less than 3.

  * (iii)

the R-GMM estimator which combines Robinson Transformation with GMM. In the
Application section, it is called GMM-Lasso, where we use the Lasso method to
estimate the nuisance parameters after the Robinson transformation.

  * (iv)

the GMM estimator that treats $f_{0,1}(X_{i})$ as a linear function in
$X_{i}$.

  * (v)

the GMM (Oracle) estimator that uses the true $f_{0,1}(X_{i})$.

The nuisance parameters in the first step of the Algorithm for the D-RSMD
estimator are estimated by the Lasso method because we will use the same
estimator when the number of controls is 30. The tuning parameter of the Lasso
method (e.g. the penalty term) is selected by cross-validation (see e.g. van
de Geer (2016) and Chernozhukov et al. (2018)).

The R-SMD estimator can only be implemented in the first simulation design
when the number of control variables is small (e.g. equal to 1). We apply the
Nadaraya-Watson estimator, with the rule of thumb bandwidth
$h=\sigma_{x}n^{-0.2}$ to estimate the nuisance parameters in the first step
of the R-SMD estimator. D-RSMD and R-SMD estimators both use the CDF of a
standard Gaussian distribution as $\mu(.)$ inside
$\kappa(Z_{j}-Z_{l})=\int_{R^{q_{z}}}e^{it^{\prime}(Z_{j}-Z_{l})}d\mu(t)$. The
choice of $\mu(.)$ satisfies Assumption 1.

For each DGP, we report the results for the D-RSMD estimator using only one
instrument and two instruments in the following subsection. For the R-SMD
estimator, we also report the results for both cases. For GMM type estimators,
to estimate two parameters, we need to use at least two instruments. The
following is the list of available instrument sets we use for estimators.

  * (i)

$Z_{1}$: the binary instrument.

  * (ii)

$Z_{2}$: the sum of $Z_{1}$ and another binary variable. There are three
values in $Z_{2}$.

  * (iii)

$(Z_{1},Z_{1}X_{1})$: $Z_{1}$ and the interaction term between $Z_{1}$ and the
covariate.

  * (iv)

$(Z_{2},Z_{2}X_{1})$: $Z_{2}$ and the interaction term between $Z_{2}$ and the
covariate.

  * (v)

$(Z_{1},Z_{2})$: $Z_{1}$ and $Z_{2}$. The correlation between $Z_{1}$ and
$Z_{2}$ is around 0.7.

In all simulation designs, we report the Monte-Carlo Median Bias (Med.Bias),
Median Absolute Deviation (MAD), the median of asymptotic standard error under
heteroskedasticity (Med.SE), and the Rejection Rate (RR).

###  4.1 Results for the Model with a Categorical Instrument

In this section, instrument $Z$ is a categorical instrument with three values.
That is, for the benchmark model, we generate the treatment by using $Z_{2}$.
In the simulation, we set the probability distribution for the instrument
$Z_{2}$ equal to the observed distribution of the instrument in the Card
(1993) data to provide insights for the empirical results. Additionally, we
use consistent notations are throughout the simulation and application parts.

In Table 1, we report the simulation results of four D-RSMD, four R-SMD, and
six GMM type estimators for $\theta_{w0}$ and $\theta_{wx0}$. The sample size
is 3,000 and there are 5,000 replications with only one control, that is,
$P=1$. The table contains the results for D-RSMD and R-SMD estimators using
one instrument, that is, $Z_{1}$ or $Z_{2}$, or two instruments, i.e.,
$(Z_{1},Z_{1}X_{1})$ or $(Z_{2},Z_{2}X_{1})$. $Z_{2}$ is the instrument used
in the DGP, so R-SMD using $Z_{2}$ or $(Z_{2},Z_{2}X_{1})$ will produce better
results than it with $Z_{1}$ or $(Z_{1},Z_{1}X_{1})$. This is also shown in
the table. The Nadaraya-Watson estimators for nuisance parameters inside R-SMD
and R-GMM estimators when $n=3000$ and $P=1$ use a bandwidth of
$0.202\hat{\sigma}_{x}$. GMM type estimators are generated using two
instruments $(Z_{1},Z_{1}X_{1})$ or $(Z_{2},Z_{2}X_{1})$.

| $\theta_{w0}$ | $\theta_{wx0}$
---|---|---
Estimator | Instrument | Med.Bias | MAD | Med.SE | RR | Med.Bias | MAD | Med.SE | RR
D-RSMD | $Z_{1}$ | -0.007 | 0.069 | 0.103 | 0.047 | 0.019 | 0.036 | 0.052 | 0.067
D-RSMD | $Z_{2}$ | -0.017 | 0.129 | 0.211 | 0.035 | 0.034 | 0.313 | 0.523 | 0.035
D-RSMD | $(Z_{1},Z_{1}X_{1})$ | 0.002 | 0.077 | 0.112 | 0.050 | -0.008 | 0.089 | 0.130 | 0.044
D-RSMD | $(Z_{2},Z_{2}X_{1})$ | 0.001 | 0.045 | 0.065 | 0.048 | -0.006 | 0.055 | 0.077 | 0.054
R-SMD | $Z_{1}$ | -0.207 | 0.072 | 0.236 | 0.007 | 0.424 | 0.065 | 0.449 | 0.000
R-SMD | $Z_{2}$ | 0.027 | 0.172 | 0.280 | 0.022 | -0.060 | 0.416 | 0.693 | 0.018
R-SMD | $(Z_{1},Z_{1}X_{1})$ | 0.064 | 0.077 | 0.127 | 0.052 | -0.112 | 0.083 | 0.220 | 0.003
R-SMD | $(Z_{2},Z_{2}X_{1})$ | 0.028 | 0.039 | 0.072 | 0.031 | -0.053 | 0.048 | 0.076 | 0.079
R-GMM | $(Z_{1},Z_{1}X_{1})$ | -0.044 | 0.076 | 0.182 | 0.003 | -0.239 | 0.089 | 0.245 | 0.026
R-GMM | $(Z_{2},Z_{2}X_{1})$ | -0.015 | 0.042 | 0.105 | 0.001 | -0.172 | 0.053 | 0.156 | 0.043
GMM | $(Z_{1},Z_{1}X_{1})$ | 2.571 | 0.318 | 0.472 | 1.000 | -6.003 | 0.415 | 0.618 | 1.000
GMM | $(Z_{2},Z_{2}X_{1})$ | 2.176 | 0.231 | 0.322 | 1.000 | -5.233 | 0.319 | 0.454 | 1.000
GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | 0.000 | 0.076 | 0.113 | 0.048 | -0.003 | 0.093 | 0.137 | 0.045
GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | 0.000 | 0.042 | 0.061 | 0.049 | 0.000 | 0.053 | 0.078 | 0.057

Table 1: Categorical instrument when $P=1$ (n=3000)

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), the median of
asymptotic standard error under heteroskedasticity (Med.SE), and Rejection
Rate (RR) using a 5% t-test.

Recall that $Z_{2}$ is the instrument generating the treatment variable.
Intuitively, estimators applying $Z_{2}$ or $(Z_{2},Z_{2}X_{1})$ will produce
better results than those using $Z_{1}$ or $(Z_{1},Z_{1}X_{1})$ do. Indeed,
this is the case for D-RSMD with $(Z_{2},Z_{2}X_{1})$, R-SMD, R-GMM, GMM, and
GMM (Oracle). From Table 1, comparing D-RSMD using $(Z_{2},Z_{2}X_{1})$ and
GMM (Oracle), we find that all results are close. It suggests that D-RSMD with
$(Z_{2},Z_{2}X_{1})$ is as good as GMM (Oracle). D-RSMD with
$(Z_{2},Z_{2}X_{1})$ has a lower median bias than R-SMD with
$(Z_{2},Z_{2}X_{1})$, which means that the Debiased part in D-RSMD works.
Applying the debiased procedure reduces the effect of bias on the estimate
introduced by the nuisance parameters. In the table, D-RSMD and R-SMD with
$Z_{2}$ also work. MAD and Med.SE for D-RSMD are smaller than those for R-SMD.
RR for D-RSMD is closer to 5% than R-SMD. The Med.Bias is higher for D-RSMD
with $Z_{2}$, which suggests that the bias introduced by the new nuisance
parameters in the debiased procedure has a larger effect on the estimate than
D-RSMD with $(Z_{2},Z_{2}X_{1})$. This is reasonable because the new nuisance
parameters are conditional expectations of the $\kappa(Z_{j}-Z_{l})$, which is
a function of the instruments.

When we use the sets of instruments not in the DGP, e.g., D-RSMD with valid
$Z_{1}$ or $(Z_{1},Z_{1}X_{1})$, the D-RSMD type estimators still work. Some
of the results are better than the D-RSMD with $Z_{2}$ or
$(Z_{2},Z_{2}X_{1})$. For instance, The MAD and Med.SE for D-RSMD with $Z_{1}$
are smaller than those for D-RSMD with $Z_{2}$ for both parameters. This
property disappears in the later sections, so it can be case-specific. This
also implies that D-RSMD works with various instruments. With valid $Z_{1}$ or
$(Z_{1},Z_{1}X_{1})$, R-SMD or GMM type estimators for $\theta_{w0}$ and
$\theta_{wx0}$ have much higher median biases than the same estimators with
valid $Z_{2}$ or $(Z_{2},Z_{2}X_{1})$. This suggests that choosing the
instruments outside of the DGP causes problems for R-SMD or GMM type
estimators.

In the empirical application, there are more than 20 control variables. Hence,
the benchmark model in the simulation considers 30 controls to mimic the
situation. When there are 30 controls, the controls are generated by a
multivariate normal distribution with mean 0 and the covariance matrix set to
the identity matrix. Following the sparsity assumption in Assumption 2, when
$P=30$, we choose sparsity level to be 10, that is, $\alpha_{3q}=2$,
$\beta_{2q}=-3$, $\alpha_{1q}=\beta_{1q}=1$ when $q$ is less than or equal to
5. We still use the benchmark model in the DGP.

We report the table of robustness check when the model is not sparse, that is,
all $\beta_{2q}$ are not zero, in Supplementary Appendix.

Instrument with 3 values ($P=30$)
---
|  | Estimator | Ins | Med.Bias | MAD | Med.SE | RR
|  | D-RSMD | $Z_{2}$ | -0.041 | 0.115 | 0.146 | 0.064
$n=2000$ | $\theta_{w0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.059 | 0.119 | 0.144 | 0.072
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | -0.006 | 0.110 | 0.160 | 0.050
|  | D-RSMD | $Z_{2}$ | -0.019 | 0.106 | 0.158 | 0.064
$n=2000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.033 | 0.079 | 0.101 | 0.064
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | 0.001 | 0.059 | 0.087 | 0.053
|  | D-RSMD | $Z_{1}$ | 0.139 | 0.765 | 1.140 | 0.012
|  | D-RSMD | $Z_{2}$ | -0.018 | 0.119 | 0.158 | 0.044
|  | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.115 | 0.557 | 0.843 | 0.025
$n=3000$ | $\theta_{w0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.028 | 0.100 | 0.118 | 0.055
|  | D-RSMD | $(Z_{1},Z_{2})$ | -0.019 | 0.189 | 0.267 | 0.036
|  | RSMD | $Z_{1}$ | -0.476 | 0.773 | 1.173 | 0.024
|  | RSMD | $Z_{2}$ | -0.134 | 0.178 | 0.313 | 0.083
|  | RSMD | $(Z_{1},Z_{1}X_{1})$ | 0.082 | 0.790 | 1.158 | 0.015
|  | RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.202 | 0.111 | 0.190 | 0.157
|  | RSMD | $(Z_{1},Z_{2})$ | -0.128 | 0.267 | 0.411 | 0.073
|  | GMM | $(Z_{2},Z_{2}X_{1})$ | 3.379 | 0.897 | 1.354 | 0.715
|  | GMM | $(Z_{1},Z_{2})$ | 12.082 | 4.798 | 7.355 | 0.230
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | 0.000 | 0.089 | 0.130 | 0.056
|  | GMM (Oracle) | $(Z_{1},Z_{2})$ | 0.003 | 0.506 | 0.802 | 0.004
|  | D-RSMD | $Z_{1}$ | -0.013 | 0.091 | 0.158 | 0.020
|  | D-RSMD | $Z_{2}$ | -0.019 | 0.145 | 0.236 | 0.048
|  | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.015 | 0.105 | 0.154 | 0.037
$n=3000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.023 | 0.068 | 0.083 | 0.059
|  | D-RSMD | $(Z_{1},Z_{2})$ | -0.018 | 0.274 | 0.435 | 0.034
|  | RSMD | $Z_{1}$ | -0.049 | 0.101 | 0.161 | 0.052
|  | RSMD | $Z_{2}$ | -0.052 | 0.231 | 0.421 | 0.058
|  | RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.279 | 0.135 | 0.260 | 0.071
|  | RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.074 | 0.054 | 0.085 | 0.129
|  | RSMD | $(Z_{1},Z_{2})$ | -0.058 | 0.374 | 0.602 | 0.045
|  | GMM | $(Z_{2},Z_{2}X_{1})$ | -5.957 | 0.509 | 0.726 | 1.000
|  | GMM | $(Z_{1},Z_{2})$ | -26.653 | 9.961 | 15.090 | 0.312
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | 0.000 | 0.048 | 0.071 | 0.052
|  | GMM (Oracle) | $(Z_{1},Z_{2})$ | 0.015 | 1.065 | 1.712 | 0.002
|  | D-RSMD | $Z_{2}$ | -0.005 | 0.397 | 0.694 | 0.021
$n=5000$ | $\theta_{w0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.016 | 0.082 | 0.096 | 0.053
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | -0.001 | 0.068 | 0.101 | 0.050
|  | D-RSMD | $Z_{2}$ | -0.028 | 0.714 | 1.267 | 0.022
$n=5000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.013 | 0.054 | 0.066 | 0.056
|  | GMM (Oracle) | $(Z_{2},Z_{2}X_{1})$ | -0.001 | 0.037 | 0.055 | 0.049

Table 2: Instrument 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 Rejection Rate (RR)
using a 5% t-test.

Table 2 reports the simulation results of D-RSMD, RSMD, and GMM type
estimators for the key parameters $\theta_{w0}$ and $\theta_{wx0}$ when the
sample sizes are 2,000, 3000, or 5000. All of the simulation studies have 5000
replications. The table contains the results for D-RSMD estimators using one
instrument $Z_{1}$ or $Z_{2}$ or two instruments, i.e., $(Z_{1},Z_{1}X_{1})$,
$(Z_{2},Z_{2}X_{1})$, and $(Z_{1},Z_{2})$. For the GMM type estimators, we
report the results using two instruments, that is, $(Z_{2},Z_{2}X_{1})$ or
$(Z_{1},Z_{2})$.

When relying on one instrument (e.g., the categorical instrument) in the
estimation procedure, only D-RSMD type estimators work. We expect that
estimators using $Z_{2}$ will have generally better performance than the same
estimators using $Z_{1}$ in terms of smaller Med.Bias, MAD and Med.SE. RR
should be closer to 5% for estimators using $Z_{2}$. Indeed, for
$\theta_{w0}$, D-RSMD with $Z_{2}$ follows this expectation. Although for
$\theta_{w0}$, D-RSMD using $Z_{1}$ has lower Med.Bias, MAD and Med.SE, the RR
is downward bias and not that close to 5%.

When we work with two instruments, i.e., $(Z_{1},Z_{1}X_{1})$,
$(Z_{2},Z_{2}X_{1})$, or $(Z_{1},Z_{2})$, results for D-RSMD and GMM type
estimators are in the 3000 observations panels of Table 2. The GMM (Oracle)
estimator is a GMM estimator using the correct second degree polynomial for
the nonlinear $f_{0,1}(X_{i})$. Thus, it has relatively better performance
with instrument $(Z_{2},Z_{2}X_{1})$, in terms of much smaller bias, lower
standard errors and higher t-statistic. It is also shown in the table.
However, RR is slightly higher. This higher RR disappears when we have a
larger number of observations. See the panel with $n=5000$.

It is no surprise to see that using another set of instruments
$(Z_{1},Z_{2})$, the GMM (Oracle) estimator does not work so well. It is
because the instruments in $(Z_{1},Z_{2})$ have a higher correlation. If we
check the raw estimation results generated by GMM (Oracle) with
$(Z_{1},Z_{2})$, we will see many extreme cases. The same issue happens to GMM
as well. However, it is not a big problem for the D-RSMD estimator. D-RSMD
estimators using any instrument has relatively stable performance. The only
problem with the instrument selection problem happens to the
$(Z_{2},Z_{2}X_{1})$ where the RR is higher than 5%. This rejection rate
decreases when we have a bigger dataset. See the last six rows with $n=5000$
in Table 2.

With a categorical instrument in the DGP, the D-RSMD estimators have the
proper size, much smaller bias than GMM estimators. As the sample size grows,
the size distortions decrease for $(Z_{2},Z_{2}X_{1})$. D-RSMD allows us to
use any possible instrument set to estimate heterogeneous treatment effects.
When the correlation between the instruments inside the instrument vector is
high, D-RSMD still works, while GMM generates extreme results.

###  4.2 Results for the Model with a Binary Instrument

The DGP in this section also follows the benchmark model for the categorical
instrument variable. The only difference is that when we generate the
treatment variable, $Z_{i}$ is a binary instrument instead of a categorical
instrument. Hence, in this section, $Z_{1}$ is the true instrument variable.
Instrument sets $(Z_{1},Z_{1}X_{1})$ and $(Z_{1},Z_{2})$ contain the true
instrument. We report results in Table 3.

The first two panels of Table 3 contain the preliminary results when $P=3$.
D-RSMD using $Z_{1}$ has the lowest MAD and Med.SE among all of the
estimators. Its RR are close to 5% for $\theta_{w0}$ and slightly oversized
for $\theta_{wx0}$. Using the instrument $Z_{2}$ (not in the DGP) for D-RSMD
will generate higher MAD and Med.SE than utilizing the correct instrument.
D-RSMD estimator also works when employing $(Z_{1},Z_{1}X_{1})$ and
$(Z_{2},Z_{2}X_{1})$. When comparing D-RSMD estimators, we see that using the
instrument from the DGP or the instrument set including the instrument will
generate results with lower MADs and Med.SEs. It is reasonable because working
with the correct variable will increase the precision of the estimate. The
Med.Bias column shows that for $\theta_{w0}$ the D-RSMD estimators using
$(Z_{1},Z_{1}X_{1})$ and $(Z_{2},Z_{2}X_{1})$ have a higher bias than the
other D-RSMD estimators. It suggests that using the instrument from the DGP
directly will generate a lower bias. This property of D-RSMD also shows in the
case where we have 30 controls in the model.

In the simulation, when $P=30$, we see similar results as the case with a
categorical instrument inside DGP. For D-RSMD estimators, using $Z_{1}$ or
$Z_{2}$ already provides similar or better results than the GMM (Oracle) in
terms of lower MAD and Med.SE. Because the DGP in this section uses the binary
instrument, the D-RSMD results with $Z_{1}$ should be better than the other
D-RSMD estimators in terms of the magnitude of the Med.Bias, MAD, Med.SE.
Indeed this is the case. For the RR column, if we compare the $Z_{1}$ results
with $Z_{2}$ ones, we see that the RR is higher in $Z_{1}$. It is reasonable
because the instrument $Z_{1}$ has only two values and $Z_{2}$ has three
values and $Z_{2}$ is the sum of $Z_{1}$ and another binary instrument.
$Z_{2}$ contains more information. The size distortion problem decreases when
we generate 5000 observations for each replication.

Comparing the results for instrument sets $(Z_{1},Z_{1}X_{1})$ and
$(Z_{1},Z_{2})$, we see similar stories to before. For GMM type estimators,
$(Z_{1},Z_{1}X_{1})$ is the better choice. For instance, the GMM (Oracle)
using $(Z_{1},Z_{1}X_{1})$ is the best results for both $\theta_{w0}$ and
$\theta_{wx0}$ with lowest Med.Bias. GMM type estimators using $(Z_{1},Z_{2})$
still generate scattered results and many extreme cases. The RR for GMM and
GMM (Oracle) with $(Z_{1},Z_{2})$ is close to 0 for $\theta_{wx0}$. Even with
the GMM (Oracle) estimator, the correlation between the instruments inside
$(Z_{1},Z_{2})$ still causes GMM estimators a big problem.

If we compare D-RSMD results with GMM (Oracle) estimator, we find that D-RSMD
results with $Z_{1}$ are close to the Oracle ones for $\theta_{w0}$ in terms
of MAD, Med.SE, and RR, that suggests that with $Z_{1}$ D-RSMD is good enough.
For $\theta_{wx0}$ estimation, D-RSMD results with $Z_{1}$ is better than the
GMM (Oracle) using $(Z_{1},Z_{1}X_{1})$ in terms of lower MAD and Med.SE. It
is reasonable, because we use all of the information from the conditional
moment, while GMM (Oracle) only uses two unconditional moments.

Using one binary instrument for D-RSMD generates better results than using two
instruments for D-RSMD estimators. It suggests that for D-RSMD, the number of
instruments is not a big problem.

Binary instrument
---
|  | Estimator | Ins | Med.Bias | MAD | Med.SE | RR
|  | D-RSMD | $Z_{1}$ | -0.006 | 0.146 | 0.215 | 0.048
|  | D-RSMD | $Z_{2}$ | -0.004 | 0.212 | 0.314 | 0.043
$n=3000$ | $\theta_{w0}$ | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.026 | 0.164 | 0.247 | 0.048
$P=3$ |  | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.035 | 0.230 | 0.340 | 0.049
|  | GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | -0.004 | 0.164 | 0.250 | 0.048
|  | D-RSMD | $Z_{1}$ | -0.005 | 0.039 | 0.057 | 0.052
|  | D-RSMD | $Z_{2}$ | -0.004 | 0.110 | 0.182 | 0.050
$n=3000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.004 | 0.124 | 0.186 | 0.047
$P=3$ |  | D-RSMD | $(Z_{2},Z_{2}X_{1})$ | -0.001 | 0.200 | 0.301 | 0.040
|  | GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | -0.001 | 0.126 | 0.191 | 0.047
|  | D-RSMD | $Z_{1}$ | 0.028 | 0.288 | 0.427 | 0.045
|  | D-RSMD | $Z_{2}$ | 0.033 | 0.524 | 0.773 | 0.030
$n=3000$ | $\theta_{w0}$ | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.259 | 0.291 | 0.431 | 0.082
$P=30$ |  | D-RSMD | $(Z_{1},Z_{2})$ | 0.041 | 0.333 | 0.493 | 0.041
|  | RSMD | $Z_{1}$ | 0.128 | 0.408 | 0.563 | 0.072
|  | RSMD | $Z_{2}$ | 0.154 | 0.555 | 0.790 | 0.039
|  | RSMD | $(Z_{1},Z_{1}X_{1})$ | 0.230 | 0.463 | 0.572 | 0.110
|  | RSMD | $(Z_{1},Z_{2})$ | 0.140 | 0.438 | 0.605 | 0.060
|  | GMM | $(Z_{1},Z_{1}X_{1})$ | 7.471 | 2.968 | 4.368 | 0.356
|  | GMM | $(Z_{1},Z_{2})$ | 6.857 | 5.691 | 11.948 | 0.043
|  | GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | 0.007 | 0.266 | 0.392 | 0.048
|  | GMM (Oracle) | $(Z_{1},Z_{2})$ | -0.011 | 0.635 | 1.424 | 0.003
|  | D-RSMD | $Z_{1}$ | -0.012 | 0.045 | 0.059 | 0.064
|  | D-RSMD | $Z_{2}$ | -0.012 | 0.126 | 0.206 | 0.037
$n=3000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | 0.005 | 0.111 | 0.163 | 0.049
$P=30$ |  | D-RSMD | $(Z_{1},Z_{2})$ | -0.013 | 0.181 | 0.287 | 0.039
|  | RSMD | $Z_{1}$ | -0.020 | 0.053 | 0.065 | 0.105
|  | RSMD | $Z_{2}$ | -0.024 | 0.147 | 0.302 | 0.018
|  | RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.337 | 0.137 | 0.273 | 0.144
|  | RSMD | $(Z_{1},Z_{2})$ | -0.020 | 0.195 | 0.328 | 0.033
|  | GMM | $(Z_{1},Z_{1}X_{1})$ | -6.735 | 0.953 | 1.430 | 0.999
|  | GMM | $(Z_{1},Z_{2})$ | -4.444 | 21.525 | 47.007 | 0.000
|  | GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | -0.004 | 0.107 | 0.157 | 0.049
|  | GMM (Oracle) | $(Z_{1},Z_{2})$ | 0.114 | 2.502 | 5.548 | 0.000
|  | D-RSMD | $Z_{1}$ | 0.003 | 0.209 | 0.314 | 0.052
$n=5000$ | $\theta_{w0}$ | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | -0.158 | 0.210 | 0.308 | 0.083
$P=30$ |  | GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | 0.001 | 0.203 | 0.304 | 0.051
|  | D-RSMD | $Z_{1}$ | -0.005 | 0.029 | 0.040 | 0.062
$n=5000$ | $\theta_{wx0}$ | D-RSMD | $(Z_{1},Z_{1}X_{1})$ | 0.006 | 0.078 | 0.120 | 0.045
$P=30$ |  | GMM (Oracle) | $(Z_{1},Z_{1}X_{1})$ | 0.002 | 0.080 | 0.121 | 0.045

Table 3: Binary Instrument

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 Rejection Rate (RR)
using a 5% t-test.

##  5 Empirical Application

###  5.1 Estimating the Returns of Education on Wages

To illustrate the proposed process, we use the data set from Card (1993) to
estimate the heterogeneous returns to education. Many works have used the same
data set, for instance, Yanagi (2019), Kitagawa (2015), and Ashenfelter and
Rouse (1998). The data is from the National Longitudinal Survey for young men.
The detailed information for the variables is in Card (1993). In this paper,
we use the same dependent variable, log hourly wages and the same covariates
in the baseline model. The education variable in the original work is the
years of education. To investigate the treatment effect of college education,
we construct the college indicator based on whether the years of education are
higher than 14 years. The treatment can be considered as a two-year college
degree or higher. It is used in Yanagi (2019) as well. We also conduct the
same analysis for years of education. In Kitagawa (2015) the author also
treats the education variable as an indicator.

The instrument variable used in Card (1993) is a dummy for growing up near a
local four-year college ($Z_{1}$). It is used as an instrument because it is
not correlated with the individual’s ability and increases the probability of
attending college. From Kitagawa (2015), the validity of the instrument is not
rejected once the covariates are in the estimation. In our models and
estimators, all of the covariates are included.

We use the covariates in the original study by Card, for instance, experience,
experience squared, age, and so on. In Card (1993), the variables on the
family background are an important group of control variables, including the
parents’ years of education, classes of education, and two indicators for
family structure. We incorporate those variables in the analysis. To
illustrate the heterogeneous treatment effects of interest, we use the
parents’ education to generate the interaction term. The parents’ education is
the average of father’s and mother’s years of education. A similar control
variable is in Ashenfelter and Rouse (1998). The minimum of parents’ education
is 0. Three individuals’ parents have no education. Twenty-four fathers and
fifteen mothers have zero years of education. The average value of parents’
education is 10.16.

To investigate the properties of different instrument sets, we construct five
instrument sets. In three of them, we utilize the information from the dummy
of proximity to a two-year college. For instance, $Z_{2}$ is the sum of two
dummies: growing up near a four-year college and a two-year college. $Z_{2}$
generally implies that the more local colleges are, the higher the value is.
There are three values, 0, 1, and 2, in $Z_{2}$. There are two reasons why
$Z_{2}$ is generated. The first is that we will use a categorical instrument
$Z_{2}$ to demonstrate the properties of the new estimator. We also use
$Z_{1}$ and $Z_{2}$ as an instrument set to show that the new method works
well with highly correlated instruments. The sets of instruments we considered
in this subsection are as follows:

  * (i)

$Z_{1}$: the indicator for proximity to a four-year college.

  * (ii)

$Z_{2}$: the sum of the indicator for proximity to a two-year college and
$Z_{1}$. There are 3 values in $Z_{2}$.

  * (iii)

$(Z_{1},Z_{1}X_{1})$: $Z_{1}$ and the interaction term between parents’
education and $Z_{1}$.

  * (iv)

$(Z_{1},Z_{2})$: $Z_{1}$ and $Z_{2}$. The correlation between $Z_{1}$ and
$Z_{2}$ is around 0.729.

We examine four models in the following. Because we have two types of D-RSMD
estimators in this section, we denote the one used in simulations as DRSMD-
Lasso. The other one is DRSMD-2SOLS.

  * (i)

DRSMD-Lasso estimators:

$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}W_{i}+W_{i}\cdot
X_{i1}^{\prime}\theta_{wx0}+f_{0,1}(X_{i})+\epsilon_{i}$ $\displaystyle W_{i}$
$\displaystyle=$ $\displaystyle I(f_{0,2}(X_{i},Z_{i})>v_{i})$

  * (ii)

DRSMD-2SOLS estimators:

$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}W_{i}+W_{i}\cdot
X_{i1}^{\prime}\theta_{wx0}+\sum_{p=1}^{27}\beta_{p}X_{i}+\epsilon_{i}$
$\displaystyle W_{i}$ $\displaystyle=$ $\displaystyle
I(f_{0,2}(X_{i},Z_{i})>v_{i})$

  * (iii)

GMM estimators:

$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}W_{i}+W_{i}\cdot
X_{i1}^{\prime}\theta_{wx0}+\sum_{p=1}^{27}\beta_{p}X_{i}+\epsilon_{i}$
$\displaystyle W_{i}$ $\displaystyle=$ $\displaystyle
I(\alpha_{z}Z_{i}+\sum_{p=1}^{27}\alpha_{p}X_{i}>v_{i})$

  * (iv)

GMM-Lasso estimators:

$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}W_{i}+W_{i}\cdot
X_{i1}^{\prime}\theta_{wx0}+f_{0,1}(X_{i})+\epsilon_{i}$ $\displaystyle W_{i}$
$\displaystyle=$ $\displaystyle
I(\alpha_{z}Z_{i}+\sum_{p=1}^{27}\alpha_{p}X_{i}>v_{i})$

Our new method, the DRSMD-Lasso method allows the covariates $X_{i}$ to enter
the model through an unknown function $f_{0,1}(.)$ and $f_{0,2}(.)$. Because
the true function form is unknown, our method will generate more reliable
results. The DRSMD-2SOLS estimator allows a nonlinear and unknown
$f_{0,2}(.)$, but $f_{0,1}(.)$ needs to be linear. The GMM estimator assumes
both $f_{0,1}(.)$ and $f_{0,2}(.)$ to be linear in $X_{i}$. The GMM-Lasso
estimator imposes that $f_{0,2}(.)$ is linear. All of the results in the
subsection are generated with the college indicator as treatment. 888The
treatment effects of years of education are reported in Table A1 in Appendix
A. The estimation results for 2SLS from Card (1993) are around 0.132 to 0.140.
The DRSMD-Lasso estimator produces similar results. Table A3 in Appendix A
reports heterogeneous treatment effects of years of education.

Table 4 reports the results for heterogeneous treatment effects of college
($W_{i}$). $X_{i1}$ is the parents’ education. If we use GMM estimates with
$(Z_{1},Z_{1}X_{1})$ and $(Z_{1},Z_{2})$, no results are statistically
significant. The GMM results for $(Z_{1},Z_{2})$ have a large magnitude. There
are two moments in $(Z_{1},Z_{2})$ including the valid instrument. It suggests
that using the extra moment is not a good idea. In Table 3 of Section 4, the
GMM estimators with and without the oracle features using $(Z_{1},Z_{2})$ also
generate unreliable results. In this application, the results for GMM and GMM-
Lasso using $(Z_{1},Z_{1}X_{1})$ show that the heterogeneous treatment effects
are not statistically significant at 5% level. The results from GMM are
affected by instruments.

For DRSMD-Lasso estimators, we have obtained different results. DRSMD-Lasso
and DRSMD-2SOLS methods with $(Z_{1},Z_{2})$ as instruments generate more
reliable results. Using $(Z_{1},Z_{1}X_{1})$ the DRSMD-Lasso and DRSMD-2SOLS
produce similar statistically significant results for $\theta_{w0}$ and
$\theta_{wx0}$ at 5% level. The estimate changes across different sets of
instruments, but the magnitude does not change too much. It is reasonable
because different instruments imply distinct local average treatment effects
on the corresponding groups of compliers. When we compare all of the estimates
for the overall treatment effect, in Table 5, we see that for DRSMD type
estimators, the treatment effects are higher than those estimated by GMM type
estimators. This difference suggests that the nonlinear part in the
$f_{0,2}(.)$ is responsible for the gap between DRSMD and GMM type estimators.

The interpretation of estimates of the parameter $\theta_{w0}$ and
$\theta_{wx0}$ is straightforward. Recall that three individuals’ parents have
no education. For DRSMD-Lasso with $(Z_{1},Z_{1}X_{1})$, when the parents’
education is 0, having a college degree increases the average hourly earnings
by 299 log points holding other variables constant. When parents’ education is
10, having a college degree increases the average hourly earnings by 108 log
points. This is not surprising. It means that before 1981, having a college
degree or higher almost doubled the average hourly wage ($e^{1.08}-1$). It is
also rare that parents’ education is 0 and the child has a college degree. If
that is the case, having a college degree helps find a job with higher
payments significantly. When we compare the overall effects of heterogeneous
treatment effect with parents’ education being the mean with the homogeneous
treatment effect in Appendix B, we find that they are very close, which means
that the new method generates reliable results.

Heterogeneous treatment effects
---
Estimator for $\theta_{w0}$ | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM |  |  | 1.380 | -20.348
|  | (0.858) | (127.277)
GMM-Lasso |  |  | -0.821 | -13.471
|  | (1.825) | (50.295)
DRSMD-Lasso | 0.731 | 0.457 | 2.988** | 0.641**
(0.513) | (0.339) | (1.244) | (0.313)
DRSMD-2SOLS | 0.058 | -0.505 | 2.691** | -0.039
(0.262) | (0.950) | (1.180) | (0.257)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM |  |  | -0.053 | 1.882
|  | (0.058) | (11.288)
GMM-Lasso |  |  | 0.149 | 1.320
|  | (0.148) | (4.700)
DRSMD-Lasso | 0.086 | 0.159 | -0.191* | 0.065
(0.078) | (0.173) | (0.099) | (0.052)
DRSMD-2SOLS | 0.161 | 0.302 | -0.152** | 0.140
(0.163) | (0.386) | (0.076) | (0.101)

Table 4: Heterogeneous Treatment Effects of college education on the log wage

Note: *** Significant at 1%, ** at 5%, * at 10%. $\theta_{w0}$ is the
parameter in front of the treatment and $\theta_{wx0}$ is the parameter for
$College\times Parents^{\prime}education$. 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. Every regression contains 3010
observations.

Average Treatment Effects
---
Estimator for $\theta_{w0}+\theta_{wx0}E(X)$ | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM |  |  | 0.844** | -1.221
|  | (0.385) | (12.645)
GMM-Lasso |  |  | 0.697 | -0.054
|  | (0.711) | (3.192)
DRSMD-Lasso | 1.607 | 2.074 | 1.042* | 1.299*
(1.261) | (1.924) | (0.628) | (0.727)
DRSMD-2SOLS | 1.698 | 2.560 | 1.150 | 1.384*
(1.45) | (2.998) | (0.749) | (0.828)

Table 5: Average Treatment Effects of college education on the log wage

Note: 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. Every regression contains 3010 observations.

###  5.2 Oregon Health Insurance Experiment

In early 2008, Oregon used a lottery to select low-income uninsured adults to
expand Medicaid enrollment (health coverage). In this process, low-income
uninsured adults first register for the waiting list. Then, from the waiting
list, the names will be drawn by a lottery. The draw (or the lottery) was
random. The randomness provided by the lottery allows researchers to analyze
the effects of health insurance on medical, financial, and labour market
outcomes. This experiment is a well-known Randomized Control Trial and is
studied in many articles, for instance, Baicker et al. (2014) and Finkelstein
et al. (2012).

Many studies focus on the treatment effects of health insurance (Medicaid) on
health care, employment, debt for health, and many other outcome variables. In
those works, the treatment effects are assumed to be homogeneous. In
Finkelstein et al. (2012), authors conduct their analysis assuming homogeneity
and also use regression analysis to check whether there are heterogeneous
treatment effects. They are unable to make precise inferences using
traditional estimators like GMM or 2SLS. Studying the heterogeneous treatment
effects using only one valid instrument variable needs a more advanced method.

The new estimator, the D-RSMD estimator, is generated to estimate the
heterogeneous treatment effects using a conditional moment restriction
directly. With an instrument variable that satisfies the conditional moment
restriction, the estimator uses all of the information from the restriction to
allow us to estimate more than one key parameter inside the model. The new
estimator is designed for a partially linear model—that is, a model that
contains a linear part and a nonparametric part.

In this empirical study, we include the interaction terms inside the model to
account for heterogeneity. The new estimator is used to estimate both
parameters within the heterogeneous treatment effects using the only valid
instrument variable without generating new variables. The traditional
estimation procedure, such as GMM or 2SLS, needs two moments to estimate both
parameters within the heterogeneous treatment effects. Without the generated
instrument variable, the traditional estimator will face an under-
identification problem; that is, one of the key parameters will not be
identified. The new estimator with the only valid instrument can identify
both. With the new estimation method, we compare the results between the new
and traditional estimation methods using or not using the generated instrument
variable with reliable inference.

To illustrate the existence of heterogeneous treatment effects in some
studies, we revisit the Oregon Health Insurance Experiment and estimate both
parameters within the heterogeneous treatment effects to check the impacts of
Medicaid from the Oregon Health Insurance Experiment. The public website for
this Oregon Medicaid Health Experiment contains all of the related work, data
sets, and data descriptions. 999See https://www.nber.org/research/data/oregon-
health-insurance-experiment-data The original data sets contain several files.
In this paper, we use the data set derived from the original data sets.
101010I would like to thank A. Colin Cameron for the data set and for bringing
the Oregon Health Insurance Experiment to my attention.

In the Oregon Health Insurance Experiment, using our estimator with only one
valid instrument produces statistically significant results for heterogeneous
treatment effects when the GMM estimator does not. Also, if the generated
instrument variable is not reliable, using our estimator with only one valid
instrument generates more reliable results, and the GMM estimator cannot.

The section contributes another empirical study for the new D-RSMD estimator
and provides a clear illustration of the advantages of the new estimator in
estimates and standard errors. It also contributes to the literature on the
Oregon Health Insurance Experiment. With a new estimation procedure, we
identify both parameters inside the heterogeneous treatment effects and verify
that there are heterogeneous treatment effects, while the traditional
estimation method does not.

The original data sets are available on the public website, and the detailed
description is on the website and in Finkelstein et al. (2012). There are
18572 observations in the dataset we use. Using the dataset, we want to
estimate the effects of Medicaid (health coverage) on various outcomes.
Because the health conditions, employment, and other outcomes are closely
related to the age of the individuals, we believe that age is one of the
sources of heterogeneity. We generate the age group variable from the year of
the birth variable. First, we calculate the age of each individual. The range
of ages is from 21 to 64. Second, we split the original dataset into three or
five subsets based on age to create an age group variable. After the split,
there are around 5900 to 6900 observations in each subsample. Individuals in
the same age group have more similarities. In Section 5.2.1, we report the
results under three age groups (e.g., 21 to 35, 35 to 50, and 50 to 64).
111111See Section B of the Appendix for tables with five age groups.

Additionally, because age is a source of heterogeneity, we can include the age
group variable inside the linear part of the partially linear model. The
results for this regression framework are provided in Section 5.2.2.

There are also many control variables inside the dataset. We use similar
controls to Baicker et al. (2014) and Finkelstein et al. (2012). These
controls include household controls, lottery and survey wave indicators, and
individual characteristic. We are interested in the heterogeneous treatment
effects from the interaction terms generated by an indicator for household
income above 50% of the federal poverty line in 2008, household income
(hhincome), TANF (cash welfare assistance to low-income families), and
cigarette smoking level (smoke). The dependent variables are the current
employment indicator, constructed by three indicators on hours of employment
(employment), the total out-of-pocket spending on medical care (Out of Pocket
Cost), and whether the individual is currently owing money to a health care
provider (Debt for Health). All of the dependent variables are from a mail
survey starting in July 2009 and ending in March 2010. They are outcomes
obtained approximately one year after the treatment. The out-of-pocket cost is
in dollars, and hhincome is the household income as a percent of the federal
poverty line.

The D-RSMD estimator is an estimation method that combines regularized machine
learning methods, Smooth Minimum Distance (SMD) estimation, and Robinson
transformation. With the Robinson transformation, the D-RSMD estimator
provides the estimation results without assuming the function form for the
nonparametric part of the partially linear model. For the linear part of the
model, we introduce an interaction term between the treatment and a covariate
to account for the heterogeneity of the treatment. With the SMD estimation
method, the D-RSMD estimator delivers the results of both parameters inside
the heterogeneous treatment effects. The parameters we are interested in are
the one in front of the treatment Medicaid and the one in front of the
interaction term between the treatment and the control variable $X_{1}$.

In this model, $X_{1}$ represents the vector of control variables that are the
sources of the heterogeneity. In Section 5.2.1, it is the indicator of the
income level, and the estimation results are provided. Section 5.2.2 presents
the D-RSMD estimation results when $X_{1}$ is a vector of control variables.
The detailed information will be discussed in the later sections.

To create a better illustration, we compare the D-RSMD estimator and the
traditional estimator, for instance, the GMM estimator, in terms of framework.
The D-RSMD estimator is reliable in a nonparametric first stage and a
partially linear second stage. The framework is in the following:

$\displaystyle Debt_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}Medicaid_{i}+\theta_{wx0}Medicaid_{i}\times
X_{i1}+f_{0,1}(X_{i})+\epsilon_{i}$ $\displaystyle Medicaid_{i}$
$\displaystyle=$ $\displaystyle I(f_{0,2}(X_{i},Lottery_{i})>v_{i})$

For one of the traditional estimators, for example, GMM estimators, the
framework for the GMM estimator in this section is linear first stage (for the
treatment: Medicaid) and linear second stage (for the dependent variable: debt
for health). The framework is shown as follows.

$\displaystyle Debt_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}Medicaid_{i}+ \theta_{wx0}Medicaid_{i}\times X_{i1} +
X_{i}^{\prime}\beta_{x}+\epsilon_{i}$ $\displaystyle Medicaid_{i}$
$\displaystyle=$ $\displaystyle
I(\alpha_{z}Lottery_{i}+X_{i}^{\prime}\alpha_{x}>v_{i})$

The GMM estimator is not the only estimator that we compare the D-RSMD
estimator to. There are other estimators. For instance, the GMM-Lasso
estimator combines the Robinson transformation, a regularized machine learning
method, and the traditional GMM method. The GMM-Lasso estimator is reliable
when the first stage is linear and the second stage is partially linear. The
framework for this estimator is as follows.

$\displaystyle Debt_{i}$ $\displaystyle=$
$\displaystyle\theta_{w0}Medicaid_{i}+ \theta_{wx0}Medicaid_{i}\times X_{i1} +
f_{0,1}(X_{i})+\epsilon_{i}$ $\displaystyle Medicaid_{i}$ $\displaystyle=$
$\displaystyle I(\alpha_{z}Lottery_{i}+X_{i}^{\prime}\alpha_{x}>v_{i})$

The expression of the D-RSMD is shown in Section 3. In this estimator, the
regularized machine learning method we choose to use is still Lasso. Since the
D-RSMD estimation method can utilize many kinds of machine learning methods
when the convergence rate of these methods satisfy the assumption we provided
in the Section 3, we add ”Lasso” to the name of the estimator to indicate we
use the Lasso method here.

####  5.2.1 Results

In this section, we discuss the possible heterogeneous treatment effects of
Medicaid on debt for health when household income is the source of the
heterogeneity, after splitting the dataset in 3 subsets by age (see Page 9 for
specific details). This procedure allows for straightforward intuition and
interpretation. 121212We also consider the full sample (not split by age) and
consider possible heterogeneous treatment effects based on income and age: see
Page • ‣ 5.2.2 or Section • ‣ 5.2.2. The heterogeneous treatment effects
inside the model is presented by the non-zero coefficient for the interaction
term between the indicator of household income above 50% federal poverty line
and Medicaid. The whole table that contains all of the results for the effects
of the treatment (with covariate household income above 50% federal poverty
line inside the interaction term) is contained at the end of the section. The
additional results for the other dependent variables (or with other covariates
inside the interaction term) are in the Appendix. Some necessary robustness
checks are contained in Section 5.2.2.

We look at two sets of instruments: the lottery (only for D-RSMD) and the
lottery and its interaction with $X_{1}$ (income greater than 50% of the
poverty line). Because D-RSMD is the only estimator that uses the lottery only
to estimate both parameters, the D-RSMD results will contain two parts. The
first part of the results uses the lottery only, and the second part uses the
lottery and its interaction term with $X_{1}$. In this way, we can compare the
results of the two different instrument sets for the D-RSMD estimation
procedure. The GMM and GMM-Lasso estimators can only use the lottery and its
interaction term as instruments, which suggests that we can compare the
results from D-RSMD and GMM estimators with the lottery and its interaction
term as instruments.

Age, in this example, is also considered a source of the heterogeneity. In
this section, we split our sample by age into three age groups. In Section
5.2.2 we run the regressions for the whole dataset for D-RSMD and GMM
estimators, with the age variable agegroup serving as a categorical variable
with three values to mimic the three age groups. In this way, we can compare
the results in Section 5.2.1 and Section 5.2.2 to check the properties of
estimators with more than one interaction term inside.

The following table (Table 6) contains the results for individuals between 36
and 50 years old. In this age group, the total number of observations is 6693.
Please keep in mind that $X_{1}$ in this table of results stands for ”Income
above the 50% Federal Poverty Line.” Table 6 is a subset of Table 9. Notice
that DRSMD-Lasso is the D-RSMD estimator.

Debt for Health | Estimator for $\theta_{w0}$ | Estimator for $\theta_{wx0}$
---|---|---
| Lottery | (Lottery, Lottery$\times X_{i1}$) | Lottery | (Lottery, Lottery$\times X_{i1}$)
GMM |  | -0.213*** |  | 0.108
| (0.041) |  | (0.085)
DRSMD-Lasso | -0.232*** | -0.232*** | 0.091*** | 0.088
(0.062) | (0.050) | (0.031) | (0.096)

Table 6: results for Individuals Aged 36 to 50

Note: *** Significant at 1%, ** at 5%, * at 10%.

We find heterogeneous treatment effects for individuals between 36 and 50.
Estimates obtained by D-RSMD (with the lottery as an instrument or the lottery
and its interaction with $X_{1}$ as instruments) are very similar. They are
also very close to the GMM estimator (with the lottery and its interaction
with $X_{1}$ as instruments). It suggests that the interaction between the
lottery and $X_{1}$ is strong and reliable. One important difference between
D-RSMD and GMM is that heterogeneous effects are found to be statistically
significant only with D-RSMD estimation procedure. In Table 6, the age group
reports statistically significant results for both $\theta_{w0}$ and
$\theta_{wx0}$ when we use our new estimator with the valid lottery
instrument. It suggests the effect of Medicaid on debt for health depends on
the income level. Hence, there is heterogeneity.

The interpretation of the estimates is that for people with an income 50%
below federal poverty line, Medicaid enrollment decreases their probability of
owning money to health providers by 23.2 log points on average, ceteris
paribus. For people with an income above 50% federal poverty line, Medicaid
enrollment reduces their probability of owning money, but not by that much.
This is reasonable because enrollment in Medicaid is not that critical in
reducing the debt for individuals with higher incomes compared with
individuals with low incomes.

Comparing the results between the two instrument sets (the lottery as an
instrument set and the lottery and its interaction with $X_{1}$ as the other
instrument set), we find that the estimates are close for both parameters, but
the standard errors for the $\theta_{wx0}$ using the new method are
substantially lower, suggesting that using one valid instrument works better.
Table 3 in the Simulation Section also shows that the standard errors of the
DRSMD-Lasso estimator for $\theta_{wx0}$ are the lowest among all of the
estimators.

Table 7 has the results for individuals aged 21 to 35. In this age group, the
number of observations is 5962. The covariate remains $X_{1}$, indicating
whether income is greater than 50% of the Federal Poverty Line.

Debt for Health | Estimator for $\theta_{w0}$ | Estimator for $\theta_{wx0}$
---|---|---
| Lottery | (Lottery, Lottery$\times X_{i1}$) | Lottery | (Lottery, Lottery$\times X_{i1}$)
GMM |  | -0.069 |  | -0.204**
| (0.055) |  | (0.096)
DRSMD-Lasso | -0.199*** | -0.053 | 0.058 | -0.252**
(0.071) | (0.065) | (0.036) | (0.105)

Table 7: Results for Individuals Aged 21 to 35

Note: *** Significant at 1%, ** at 5%, * at 10%.

For the age group 21–35, all estimators generate similar results using the
second instrument set, that is, the lottery and its interaction term with the
indicator for household income. The coefficients for $\theta_{w0}$ are not
statistically significant, and the ones for $\theta_{wx0}$ are statistically
significant. It suggests that when households’ incomes are above 50% federal
poverty line, individuals with Medicaid will be less likely to own money to
the health providers. It also suggests that Medicaid helps people with higher
income levels more than it helps people with lower incomes. Using only the
lottery as the instrument, our new procedure generates the opposite results.
The interpretation is that for people with lower incomes, Medicaid enrollment
decreases their probability of owing money to health providers by 19.9 log
points on average, holding other variables constant. For people with higher
incomes, the effect of Medicaid decreases.

Based on the D-RSMD results using only the lottery, we do not find support in
the data to say that there are heterogeneous treatment effects for individuals
between 21 and 35 years old. Furthermore, we discover that the results for
individuals aged 21–35 differ significantly between the two instrument sets.
It suggests that the generated interaction $Lottery\times X_{1}$ is invalid.

Next, we re-estimate a homogeneous model. In a homogeneous model, because we
only want to estimate the homogeneous treatment effects, the interaction term
is not included in the regression model. The traditional estimation method
only needs one instrument to estimate one parameter. When we look at the
homogeneous treatment effects using both instrument sets in Table 8, all
estimators generate similar results as the DRSMD-Lasso using only the lottery
as the instrument in the heterogeneous treatment effects panel. It also
suggests that using a valid instrument to estimate both parameters provides a
more reliable outcome. DRSMD-Lasso is reliable under both heterogeneous and
homogeneous conditions. Since it is unclear in practice whether the model is
homogeneous or not, DRSMD-Lasso appears to be extremely valuable.

Homogeneous Treatment Effects of Medicaid on Debt for Health (Age: 21 - 35)
---
Estimator for $\theta_{w0}$ | GMM | DRSMD-Lasso
Lottery | -0.161*** | -0.172***
(0.049) | (0.063)

Table 8: Homogeneous Treatment Effects of Medicaid (Age: 21 - 35)

Note: *** Significant at 1%, ** at 5%, * at 10%.

Table 6-8 are subsets of Table 9. Table 9 is provided later. The estimates and
standard errors of average treatment effects in Table 10 are calculated based
on Table 9.

Panel A: Heterogeneous treatment effects
---
Debt for Health | 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 |  | -0.069 |  | -0.213*** |  | -0.193***
| (0.055) |  | (0.041) |  | (0.046)
GMM-Lasso |  | -0.091 |  | -0.259*** |  | -0.245***
| (0.093) |  | (0.078) |  | (0.090)
DRSMD-Lasso | -0.199*** | -0.053 | -0.232*** | -0.232*** | -0.182** | -0.190***
(0.071) | (0.065) | (0.062) | (0.050) | (0.072) | (0.056)
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 |  | -0.204** |  | 0.108 |  | 0.069
| (0.096) |  | (0.085) |  | (0.094)
GMM-Lasso |  | -0.184 |  | 0.008 |  | -0.081
| (0.131) |  | (0.128) |  | (0.153)
DRSMD-Lasso | 0.058 | -0.252** | 0.091*** | 0.088 | 0.076* | 0.096
(0.036) | (0.105) | (0.031) | (0.096) | (0.040) | (0.103)
Panel B: Homogeneous treatment effects
GMM | -0.161*** | -0.130*** | -0.170*** | -0.189*** | -0.164*** | -0.177***
(0.049) | (0.046) | (0.040) | (0.037) | (0.044) | (0.041)
GMM-Lasso | -0.180* | -0.113 | -0.256*** | -0.260*** | -0.279** | -0.244***
(0.104) | (0.092) | (0.097) | (0.078) | (0.112) | (0.089)
DRSMD-Lasso | -0.172*** | -0.187*** | -0.198*** | -0.198*** | -0.150** | -0.145**
(0.063) | (0.066) | (0.056) | (0.057) | (0.063) | (0.064)
N | 5962 | 6693 | 5917

Table 9: 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 $\times$ Above 50% Federal Poverty Line.

Average Treatment Effects
---
Debt for Health | 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 |  | -0.190*** |  | -0.150*** |  | -0.149***
| (0.055) |  | (0.048) |  | (0.055)
GMM-Lasso |  | -0.201* |  | -0.255** |  | -0.296**
| (0.112) |  | (0.110) |  | (0.133)
DRSMD-Lasso | -0.164*** | -0.202*** | -0.179*** | -0.181*** | -0.135** | -0.129*
(0.061) | (0.069) | (0.054) | (0.067) | (0.060) | (0.075)
N | 5962 | 6693 | 5917

Table 10: 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 $\times$ Above 50% Federal Poverty Line. The
expression for LATE is $\theta_{w0}+\theta_{wx0}E(X)$.

####  5.2.2 Robustness Check

In this section, we look into the two different kinds of variations of the
previous estimation. In Section 5.2.1, the covariate inside the interaction
term is the indicator of whether the household income is higher than 50%
Federal Poverty Line. To provide more information about these regression
results in Section 5.2.1, we need to conduct several different robustness
checks. In the first subsection, we check the estimation results when the
covariate is a dummy. The dummy has a value of 1 when the household income is
higher than 100% or 150% Federal Poverty Line. In the second subsection, we
will not split the dataset into three groups, and we will include the age
group variable and its interaction term with the treatment in the model
directly. In the last subsection, we will briefly discuss the other robustness
checks.

  * •

Estimation Results with Other Covariates

The results are included in Table 11 when $X_{1}$ is an indicator for ”Income
Above 100% Federal Poverty Line”. Comparing Table 11 with Table 9, we find
that the results are quite similar. For individuals between 36 and 50 years
old, there are heterogeneous treatment effects. The estimates show that they
benefit more from Medicaid health coverage when they have lower incomes. We do
not find support in the data to say that there are heterogeneous treatment
effects for individuals between 21 and 35 years old, and the DRSMD-Lasso
esimator using only one valid instrument variable generates more reliable
results.

Heterogeneous Treatment Effects of Medicaid on Debt for Health
---
Debt for Health | Age: 21 - 35 | Age: 36 - 50
Estimator for $\theta_{w0}$ | Lottery | (Lottery, Lottery$\times X_{i1}$) | Lottery | (Lottery, Lottery$\times X_{i1}$)
GMM |  | -0.123** |  | -0.179***
| (0.048) |  | (0.038)
DRSMD-Lasso | -0.178*** | -0.123** | -0.202*** | -0.200***
(0.063) | (0.061) | (0.057) | (0.049)
Estimator for $\theta_{wx0}$ | Lottery | (Lottery, Lottery$\times X_{i1}$) | Lottery | (Lottery, Lottery$\times X_{i1}$)
GMM |  | -0.200 |  | 0.072
| (0.148) |  | (0.159)
DRSMD-Lasso | 0.023 | -0.212 | 0.102*** | 0.032
(0.033) | (0.154) | (0.038) | (0.192)
N | 5962 | 6693

Table 11: Income above 100% Federal Poverty Line

Note: *** Significant at 1%, ** at 5%, * at 10%.

Table 12 reports the outcomes when $X_{1}$ is a indicator for “Income Above
150% Federal Poverty Line”. Results in Table 12 and 11 are similar in
estimates but different in standard errors. In the table, estimation results
from using a dummy for income above 150% Federal Poverty Line show that the
treatment effects of Medicaid on debt for people between 36 and 50 are not
supported by the data to be heterogeneous, because the estimate for the
parameter in front of the interaction term is not statistically significant at
the 5% significance level.

The effects of Medicaid on debt for people between 21 and 35 are heterogeneous
because the estimate is statistically significant. The difference in the
results using two instrument sets for the DRSMD-Lasso estimators suggests that
using only the lottery variable as an instrument generates more reliable
results. The difference between Table 12 and 11 shows that the covariate
inside the interaction is important for the estimation results. It can be
explained by the fact that there are only 808 individuals with incomes above
150% Federal Poverty Line for people between 36 and 50, and 794 individuals
for people between 21 and 35 in this data set.

Heterogeneous Treatment Effects of Medicaid on Debt for Health
---
Debt for Health | Age: 21 - 35 | Age: 36 - 50
Estimator for $\theta_{w0}$ | Lottery | (Lottery, Lottery$\times X_{i1}$) | Lottery | (Lottery, Lottery$\times X_{i1}$)
GMM |  | -0.145*** |  | -0.179***
| (0.048) |  | (0.038)
DRSMD-Lasso | -0.181*** | -0.149** | -0.207*** | -0.205***
(0.062) | (0.062) | (0.056) | (0.053)
Estimator for $\theta_{wx0}$ | Lottery | (Lottery, Lottery$\times X_{i1}$) | Lottery | (Lottery, Lottery$\times X_{i1}$)
GMM |  | -0.204 |  | 0.195
| (0.245) |  | (0.340)
DRSMD-Lasso | 0.095* | -0.254 | 0.067 | 0.030
(0.055) | (0.292) | (0.061) | (0.453)
N | 5962 | 6693

Table 12: Income above 150% Federal Poverty Line

Note: *** Significant at 1%, ** at 5%, * at 10%.

  * •

Estimation Results Using the Whole Data Set

This subsection provides the estimation results with an age group variable
included in the model, instead of splitting the data set by the age group
variable. The difference between the framework in Section 5.2.1 and the
framework in this subsection lies in the covariate $X_{1}$. In this section,
$X_{1}$ has more than one variable inside. In Section 5.2.1, age is also a
source of heterogeneity, so the data set is split into 3 groups based on the
age of the individuals in the data. In this subsection, through an interaction
term, the heterogeneity from age is included in the framework. For the D-RSMD
estimation method, including more than two interactions will still work.

The framework in this section also means that we move the $X_{1}$ from the
nonparametric part of the model to the linear part. Hence, there are three
kinds of coefficients in the linear part of the model. The first type of
coefficient is the coefficient in front of the treatment. The second type of
coefficients is the parameters in front of $X_{1}$, such as the parameters
measuring the effects of age and income. And the third type of parameters is
the parameters in front of the interaction terms. The first and third types of
parameters are our key parameters since we want to measure the heterogeneous
treatment effects of the treatment.

The lottery is still the instrument in this framework. For the traditional
method, the GMM estimators, it uses the instrument set, including lottery,
income, age, the interaction term between lottery and income, and the
interaction term between lottery and age. That is, to estimate five
parameters, the GMM estimators use the instrument set, which includes five
instruments, for instance, exogenous variables, lottery, and generated
instruments.

The framework for DRSMD-Lasso estimators is summarised in the following
equation.

$\displaystyle Debt_{i}$
$\displaystyle=\theta_{w0}Medicaid_{i}+\theta_{wx0}Medicaid_{i}\times
income_{i}+\theta_{x10}income_{i}$
$\displaystyle+\theta_{x20}agegroup_{i}+\theta_{x30}Medicaid_{i}\times
agegroup_{i}+f_{0,1}(X_{i})+\epsilon_{i}$ $\displaystyle Medicaid_{i}$
$\displaystyle=I(f_{0,2}(X_{i},Lottery_{i})>v_{i})$

The framework for GMM estimators includes a linear first stage within the
indicator function, as well as a linear second stage for the dependent
variable.

$\displaystyle Debt_{i}$
$\displaystyle=\theta_{w0}Medicaid_{i}+\theta_{wx0}Medicaid_{i}\times
income_{i}+\theta_{x10}income_{i}$
$\displaystyle+\theta_{x20}agegroup_{i}+\theta_{x30}Medicaid_{i}\times
agegroup_{i}+X_{i}^{\prime}\beta_{x}+\epsilon_{i}$ $\displaystyle
Medicaid_{i}$
$\displaystyle=I(\alpha_{z}Lottery_{i}+X_{i}^{\prime}\alpha_{x}>v_{i})$

The model for GMM-Lasso estimators includes a linear first stage for the
indicator function and a partially-linear second stage for the dependent
variable.

$\displaystyle Debt_{i}$
$\displaystyle=\theta_{w0}Medicaid_{i}+\theta_{wx0}Medicaid_{i}\times
income_{i}+\theta_{x10}income_{i}$
$\displaystyle+\theta_{x20}agegroup_{i}+\theta_{x30}Medicaid_{i}\times
agegroup_{i}+f_{0,1}(X_{i})+\epsilon_{i}$ $\displaystyle Medicaid_{i}$
$\displaystyle=I(\alpha_{z}Lottery_{i}+X_{i}^{\prime}\alpha_{x}>v_{i})$

In Table 13, we show the results for three estimators from the regressions
when we extract income outside the nonparametric part of the model. For DRSMD-
Lasso estimator, the estimates for Medicaid, the interaction between Medicaid
and income are statistically significant at 5% significance level when using
only one instrument, and the estimates for income and the interaction term
between Medicaid and age are statistically significant at 15% significance
level when using only one instrument. These results allow us to draw similar
conclusions as in Section 5.2.1. That is, using only the valid instrument, the
model shows that the treatment of Medicaid is heterogeneous, and the
heterogeneity comes from age and income. After analyzing the outcomes from the
GMM estimator or the GMM-Lasso using the instrument set $(Z_{1},Z_{1}X_{1})$,
we would incorrectly conclude that there is no heterogeneity in the treatment
effects. This shows that the generated instruments are problematic.

Heterogeneous treatment effects
---
Debt for Health | GMM | GMM-Lasso | DRSMD-Lasso
Variables | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$ | $Z_{1}$ | $(Z_{1},Z_{1}X_{1})$
Medicaid |  | -0.148*** |  | -0.206* | -0.187*** | -0.178***
| (0.044) |  | (0.115) | (0.037) | (0.046)
Medicaid*income |  | 0.003 |  | -0.045 | 0.074*** | -0.018
| (0.053) |  | (0.068) | (0.020) | (0.056)
income |  | 0.003 |  | 0.014 | -0.025. | 0.005
| (0.020) |  | (0.062) | (0.016) | (0.025)
agegroup |  | -0.007 |  | -0.010 | -0.006 | -0.007
| (0.010) |  | (0.021) | (0.005) | (0.009)
Medicaid*agegroup |  | -0.019 |  | -0.016 | -0.018. | -0.016
| (0.032) |  | (0.054) | (0.011) | (0.028)

Table 13: Heterogeneous Treatment Effects for Robustness Check (5 parameters)

Note: *** Significant at 1%, ** at 5%, * at 10%, . at 15%.

  * •

Other Robustness Checks

For other robustness check, we report the results when the covariate $X_{1}$
is TANF in the appendix. TANF variable is known as a variable with little
variation. For instance, there are only 2% of individuals on TANF. We want to
see whether this will affect our results. Table B5 reports the case where
$y_{i}$ is still Debt for Health and there are five age groups. The DRSMD-
Lasso with $Z_{1}$ estimate for $\theta_{w0}$ is -0.215 and for $\theta_{wx0}$
is 0.307. Both of them are statistically significant at 5% level. It suggests
that for an individual between 21 and 29 years old, TANF will decrease the
negative effect of Medicaid on the probability of owning money. We also see
that the new method using the valid instrument $Z_{1}$ still generates
reliable results with the lowest standard error.

Table B1 contains the results after splitting the sample into 5 age groups.
Table S4 and Table S14 report the results for the effects of treatment on
Employment and Out of Pocket Cost correspondingly. Both of the two tables show
that there are heterogeneous treatment effects for young individuals (between
21 and 29 years old) when $X_{1}$ is a dummy for “Income Above 50% Federal
Poverty Line”.

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##  Appendix A Tables for the Returns of Education

Homogeneous treatment effects
---
Estimator | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM | 0.138** | 0.223** | 0.138** | 0.144***
(0.055) | (0.092) | (0.055) | (0.055)
GMM-Lasso | 0.142 | 0.244 | 0.156 | 0.064
(0.108) | (0.207) | (0.111) | (0.082)
DRSMD-Lasso | 0.187* | 0.574 | 0.128* | 0.203**
(0.104) | (0.728) | (0.078) | (0.098)
DRSMD-2SOLS | 0.214 | 0.738 | 0.137* | 0.234**
(0.131) | (1.226) | (0.075) | (0.121)

Table A1: Homogeneous Treatment Effects of years of education on the log wage

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. Every regression contains 3010
observations.

Homogeneous treatment effects
---
Estimator | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM | 0.788** | 1.025** | 0.656** | 0.867**
(0.349) | (0.447) | (0.309) | (0.349)
GMM-Lasso | 0.793 | 1.085 | 1.017 | 0.648
(0.676) | (0.967) | (0.715) | (0.609)
DRSMD-Lasso | 1.757 | 2.474 | 0.914 | 1.394*
(1.457) | (2.646) | (0.620) | (0.820)
DRSMD-2SOLS | 2.194 | 3.677 | 0.985 | 1.642
(2.269) | (5.765) | (0.728) | (1.084)

Table A2: Homogeneous Treatment Effects of education on the log wage

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. Every regression contains 3010
observations.

Heterogeneous treatment effects
---
Estimator for $\theta_{w0}$ | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM |  |  | 0.137 | -0.524
|  | (0.085) | (0.659)
GMM-Lasso |  |  | -0.095 | -0.547
|  | (0.202) | (0.732)
DRSMD-Lasso | 0.187 | 0.095 | 0.382** | 0.363**
(0.124) | (0.079) | (0.167) | (0.181)
DRSMD-2SOLS | 0.149* | -0.022 | 0.192** | 0.190**
(0.083) | (0.092) | (0.090) | (0.093)
Estimator for $\theta_{wx0}$ | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM |  |  | 0.000 | 0.064
|  | (0.006) | (0.063)
GMM-Lasso |  |  | 0.024 | 0.070
|  | (0.018) | (0.083)
DRSMD-Lasso | 0.000 | 0.030 | -0.026 | -0.015
(0.002) | (0.035) | (0.017) | (0.011)
DRSMD-2SOLS | 0.006 | 0.043 | -0.007 | 0.004
(0.004) | (0.047) | (0.011) | (0.008)

Table A3: Heterogeneous Treatment Effects of years of education on the log
wage

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. Every regression contains 3010
observations.

Heterogeneous treatment effects
---
Estimator for $\theta_{w0}+\theta_{wx0}E(X)$ | $Z_{1}$ | $Z_{2}$ | $(Z_{1},Z_{1}X_{1})$ | $(Z_{1},Z_{2})$
GMM |  |  | 0.138** | 0.129
|  | (0.055) | (0.083)
GMM-Lasso |  |  | 0.149 | 0.164
|  | (0.116) | (0.172)
DRSMD-Lasso | 0.187* | 0.403 | 0.119 | 0.216**
(0.104) | (0.349) | ( 0.080) | (0.107)
DRSMD-2SOLS | 0.209* | 0.414 | 0.124 | 0.228**
(0.124) | (0.388) | (0.084) | (0.112)

Table A4: Heterogeneous Treatment Effects of years of education on the log
wage

Note: 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. Every regression contains 3010 observations.

##  Appendix B Results of Oregon Health Insurance Experiment

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 |  | -0.088 |  | -0.122 |  | -0.200*** |  | -0.226*** |  | -0.020
| (0.076) |  | (0.077) |  | (0.062) |  | (0.053) |  | (0.116)
GMM-Lasso |  | -0.101 |  | -0.121 |  | -0.205** |  | -0.318*** |  | -0.251
| (0.141) |  | (0.122) |  | (0.088) |  | (0.095) |  | (0.159)
DRSMD-Lasso | -0.203*** | -0.108 | -0.177** | -0.071 | -0.146** | -0.167** | -0.241*** | -1.131 | -0.162 | -0.278**
(0.075) | (0.252) | (0.080) | (0.148) | (0.071) | (0.078) | (0.063) | (0.691) | (0.116) | (0.126)
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 |  | 0.002 |  | 0.001 |  | -0.003
| (0.001) |  | (0.001) |  | (0.001) |  | (0.001) |  | (0.003)
GMM-Lasso |  | -0.002 |  | -0.001 |  | 0.000 |  | -0.001 |  | -0.001
| (0.001) |  | (0.001) |  | (0.001) |  | (0.002) |  | (0.004)
DRSMD-Lasso | 0.000 | -0.007 | 0.001** | -0.004* | 0.000 | 0.003 | 0.001*** | 0.045 | 0.000 | 0.005**
(0.000) | (0.006) | (0.000) | (0.002) | (0.000) | (0.002) | (0.000) | (0.034) | (0.000) | (0.002)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B1: 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 $\times$ hhincome.

Average Treatment Effects (Local)
---
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 |  | -0.216*** |  | -0.196** |  | -0.078 |  | -0.166** |  | -0.251*
| (0.072) |  | (0.079) |  | (0.069) |  | (0.066) |  | (0.144)
GMM-Lasso |  | -0.248** |  | -0.195 |  | -0.223* |  | -0.362*** |  | -0.318
| (0.149) |  | (0.145) |  | (0.124) |  | (0.133) |  | (0.281)
DRSMD-Lasso | -0.211** | -0.613** | -0.124 | -0.427** | -0.156** | 0.064 | -0.170*** | 2.293 | -0.122 | 0.189
(0.090) | (0.276) | (0.084) | (0.174) | (0.069) | (0.185) | (0.062) | (1.886) | (0.103) | (0.163)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B2: 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 $\times$ hhincome. 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 |  | -0.048 |  | -0.125* |  | -0.210*** |  | -0.190*** |  | -0.244***
| (0.072) |  | (0.069) |  | (0.053) |  | (0.046) |  | (0.077)
GMM-Lasso |  | -0.130 |  | -0.087 |  | -0.271*** |  | -0.274*** |  | -0.489***
| (0.121) |  | (0.122) |  | (0.101) |  | (0.091) |  | (0.156)
DRSMD-Lasso | -0.251** | -0.064 | -0.156** | -0.085 | -0.155** | -0.218*** | -0.269*** | -0.215*** | -0.141 | -0.237**
(0.098) | (0.086) | (0.087) | (0.086) | (0.076) | (0.066) | (0.071) | (0.055) | (0.112) | (0.102)
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 |  | -0.330** |  | -0.109 |  | 0.223** |  | 0.004 |  | 0.192
| (0.135) |  | (0.124) |  | (0.104) |  | (0.106) |  | (0.137)
GMM-Lasso |  | -0.312 |  | -0.112 |  | 0.070 |  | -0.158 |  | 0.327
| (0.196) |  | (0.150) |  | (0.155) |  | (0.169) |  | (0.204)
DRSMD-Lasso | 0.059 | -0.409*** | 0.025 | -0.119 | 0.039 | 0.196* | 0.137*** | -0.013 | 0.027 | 0.237
(0.045) | (0.158) | (0.039) | (0.139) | (0.039) | (0.118) | (0.039) | (0.116) | (0.047) | (0.164)
Panel B: Homogeneous treatment effects
GMM | -0.188*** | -0.138** | -0.175*** | -0.156*** | -0.120** | -0.154*** | -0.191*** | -0.190*** | -0.151** | -0.184***
(0.065) | (0.062) | (0.063) | (0.059) | (0.049) | (0.046) | (0.046) | (0.042) | (0.068) | (0.064)
GMM-Lasso | -0.265* | -0.148 | -0.150 | -0.096 | -0.242** | -0.271*** | -0.334*** | -0.269*** | -0.315* | -0.481***
(0.140) | (0.120) | (0.142) | (0.121) | (0.120) | (0.101) | (0.114) | (0.090) | (0.183) | (0.156)
DRSMD-Lasso | -0.228*** | -0.246*** | -0.144* | -0.152* | -0.140** | -0.138** | -0.219*** | -0.219*** | -0.128 | -0.104
(0.087) | (0.090) | (0.085) | (0.089) | (0.070) | (0.071) | (0.063) | (0.063) | (0.107) | (0.112)
N | 3504 | 3596 | 3941 | 4599 | 2932

Table B3: 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 $\times$ Above 50% Federal Poverty Line.

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 |  | -0.237*** |  | -0.193** |  | -0.082 |  | -0.192*** |  | -0.116
| (0.076) |  | (0.073) |  | (0.058) |  | (0.060) |  | (0.080)
GMM-Lasso |  | -0.308* |  | -0.158 |  | -0.231* |  | -0.365*** |  | -0.271
| (0.160) |  | (0.148) |  | (0.134) |  | (0.138) |  | (0.200)
DRSMD-Lasso | -0.217*** | -0.298*** | -0.140* | -0.160* | -0.132* | -0.106 | -0.189*** | -0.222*** | -0.123 | -0.078
(0.083) | (0.102) | (0.085) | (0.094) | (0.068) | (0.082) | (0.059) | (0.079) | (0.107) | (0.123)

Table B4: 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 $\times$ Above 50% Federal Poverty Line. The
expression for LATE is $\theta_{w0}+\theta_{wx0}E(X)$.

Panel A: Heterogeneous treatment effects
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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 |  | -0.193*** |  | -0.199*** |  | -0.113** |  | -0.196*** |  | -0.157**
| (0.065) |  | (0.061) |  | (0.049) |  | (0.046) |  | (0.068)
GMM-Lasso |  | -0.270* |  | -0.216 |  | -0.211* |  | -0.351*** |  | -0.295
| (0.147) |  | (0.136) |  | (0.110) |  | (0.112) |  | (0.182)
DRSMD-Lasso | -0.215** | -0.217** | -0.137* | -0.010 | -0.148** | -0.135** | -0.199*** | -0.192*** | -0.139 | -0.141