# Inference of Sample Complier Average Causal Effects in Completely Randomized
Experiments

Zhen Zhong
Yau Mathematical Science Center, Tsinghua University

Per Johansson
Department of Statistics, Uppsala University
and Yau Mathematical Science Center, Tsinghua University

Junni L. Zhang*
National School of Development and Center for Statistical Science,
Peking University

###### Abstract

In randomized experiments with non-compliance scholars have argued that the
complier average causal effect (CACE) ought to be the main causal estimand.
The literature on inference of the complier average treatment effect (CACE)
has focused on inference about the population CACE. However, in general
individuals in the experiments are volunteers. This means that there is a risk
that individuals partaking in a given experiment differ in important ways from
a population of interest. It is thus of interest to focus on the sample at
hand and have easy to use and correct procedures for inference about the
sample CACE. We consider a more general setting than in the previous
literature and construct a confidence interval based on the Wald estimator in
the form of a finite closed interval that is familiar to practitioners.
Furthermore, with the access of pre-treatment covariates, we propose a new
regression adjustment estimator and associated methods for constructing
confidence intervals. Finite sample performance of the methods is examined
through a Monte Carlo simulation and the methods are used in an application to
a job training experiment.

Keywords: non-compliance, Wald estimator, regression adjustment estimator,
finite population

##  1 Introduction

One requirement for the inference to an average treatment effect from a
randomized experiment to be valid is that all experimental units comply to
their treatment assignments. In practice, incomplete compliance to the
assigned treatment is common.

One standard approach is to ignore the information on compliance behavior and
to focus on the intention-to-treat (ITT) analysis. It is argued that the ITT-
effect may be most policy-relevant, since one cannot in general force people
to take a treatment. However, the ITT-effect may be misleading, for example,
when a treatment with negative side effects may appear less effective simply
because subjects adhere to it to a greater extent. The international
guidelines for good clinical practice (Dixon,, 1999), for example, suggest
that the analysis of negative side effects should be according to treatment
received. Yet the “as-treated” analysis that compares those who receive
treatment with those who receive control generally yields a biased estimate of
the treatment effect, because the treatment and control groups are no longer
similar.

When information on treatment received is available, which often is the case,
the effect for those who comply to the assigned treatment, i.e., the
compliers, can be identified under certain conditions, and scholars have
argued that the complier average causal effect (CACE) ought to be the main
causal estimand (see e.g. McNamee, (2009); Shrier et al., (2014, 2017); Steele
et al., (2015)). In the literature, often the experimental units are regarded
as being independently and randomly sampled from a super-population, and
inference is made about the population CACE. However, in general individuals
in the experiments are volunteers which means that there is a risk that they
differ in important ways from a population of interest.

This paper focuses on the finite population consisting of the experimental
units, and thus on inference about the sample CACE. As Athey and Imbens,
(2017) points out, the line of research that only focuses on the sample at
hand originates in the work by Neyman, (1923) (translated to English in 1990)
and Neyman et al., (1935). Neyman was interested in estimating the average
effect of treatment assigned for the sample at hand. In this paper we extend
this line of research to estimate the average effect of treatment received for
the compliers in the sample at hand.

When pre-treatment covariates are not considered, we show that the Wald
estimator which is typically used to estimate population CACE can also be used
to estimate sample CACE. For construction of a confidence interval based on
the Wald estimator, Li and Ding, (2017) assumes that the individual treatment
effect is homogeneous among compliers. This allows them to derive four
possible forms of the confidence sets for sample CACE, including an empty set,
a finite closed interval, the whole real line, and a union of two infinite
half-open intervals. The homogeneous assumption, or linear instrumental
variable model, is overly restrictive. Our framework derives finite confidence
intervals familiar to practitioner that puts no restriction on the individual
treatment effects among compliers.

To improve efficiency, we also consider adjusting for pre-treatment
covariates, and propose a regression adjustment estimator and associated
methods for constructing confidence intervals.

The next section defines the sample CACE. Section 3 discusses the Wald
estimator and methods for constructing confidence intervals based on the Wald
estimator. Section 4 discusses the regression adjustment estimator and methods
for constructing confidence intervals based on the regression adjustment
estimator. Asymptotic properties are given in both of these sections. Section
5 presents a Monte Carlo study that compares the small sample performance of
different methods. Section 6 presents an application to a job training dataset
with non-compliance. The paper concludes with a brief discussion in Section 7.

##  2 Sample Complier Average Causal Effect

Consider a completely randomized experiment with $n$ units, where $n_{1}$
units are randomly assigned to treatment and the remaining $n_{0}$ units are
assigned to control. Let $Z_{i}\in\\{0,1\\}$, $i=1,\cdots,n$, denote the
indicator of treatment assigned for unit $i$. Let $\bm{Z}$ denote the vector
of $Z_{i}$ for the $n$ units. Not everyone assigned treatment/control receives
treatment/control, however. Due to this, one can only estimate the average
treatment effect under the restrictive assumption of homogeneous treatment
effects. However, with heterogeneous effects it is possible to estimate the
average treatment effect for those who comply with the treatment assignment.

We assume Stable Unit Treatment Value Assumption (SUTVA) (Rubin,, 1978), that
is, there is only one version of treatment or control and there is no
interference between units. Let $W_{i}(1)$ and $W_{i}(0)$, $i=1,\cdots,n$, be
the potential treatment received if assigned to treatment and control for unit
$i$. Let $\bm{W}(z)$, $z=0,1$, be the vector of $W_{i}(z)$ for the $n$ units.
For $i=1,\cdots,n$, also define the four potential outcomes $Y_{i}(z,w)$ for
which the treatment assigned and treatment received are fixed at $z=0,1$ and
$w=0,1.$ For each individual only two of these potential outcomes,
$Y_{i}(0)=Y_{i}(0,W_{i}(0))$ or $Y_{i}(1)=Y_{i}(1,W_{i}(1))$, can possibly be
observed. Let $\bm{Y}(z)$, $z=0,1$, be the vector of $Y_{i}(z)$ for the $n$
units. We further assume $Y_{i}(z,w)=Y_{i}(z^{\prime},w)$ for all
$z,z^{\prime}$ and all $w$, that is, potential outcomes only depend on
treatment received. This is known as the exclusion restriction assumption
(Angrist et al.,, 1996).

There are four possible latent types of units: always-takers (at) who take
treatment regardless of whether being assigned to treatment or control, with
$W_{i}(1)=W_{i}(0)=1$; compliers (co) take treatment when assigned to
treatment and take control when assigned to control, with $W_{i}(1)=1$ and
$W_{i}(0)=0$, or equivalently $W_{i}(1)-W_{i}(0)=1$; defiers (de) who would
take control when assigned to treatment and take treatment when assigned to
control, with $W_{i}(1)=0$ and $W_{i}(0)=1$, or equivalently
$W_{i}(1)-W_{i}(0)=-1$; and never-takers (nt) who take control regardless of
whether being assigned to treatment or control, with $W_{i}(1)=W_{i}(0)=0$.

If we regard units in the experiment as being independently and randomly
sampled from a super-population, we can define the population complier average
causal effect (CACE) as

$\tau_{CACE}^{pop}=\text{E}(Y_{i}(1)-Y_{i}(0)|W_{i}(1)-W_{i}(0)=1).$

The fraction of compliers in the super-population is
$p_{co}^{pop}=\text{Pr}(W_{i}(1)-W_{i}(0)=1)$. Angrist et al., (1996) show
that under further assumptions: (i) $P(W_{i}(1)=1)\neq P(W_{i}(0)=1)$ and (ii)
$W_{i}(1)\geq W_{i}(0)$, it is possible to non-parametrically identify
$\tau_{CACE}^{pop}$. Assumption (i) says that treatment assigned affects
treatment received, and hence there exist compliers or defiers in the super-
population. Assumption (ii) rules out defiers. Under assumption (ii),
$p_{co}^{pop}=\text{E}(W_{i}(1)-W_{i}(0))$.

In this paper we do not assume that units in the experiment are independently
and randomly sampled from a super-population, and instead focus only on the
finite population consisting of these units. Thus, we treat $\bm{W}(0)$,
$\bm{W}(1)$, $\bm{Y}(0)$ and $\bm{Y}(1)$ all as fixed, and treat only $\bm{Z}$
as random.

We rephrase assumptions (i) and (ii) in the finite population setting: (i’)
$W_{i}(1)\neq W_{i}(0)$ for some unit $i$ and (ii’) $W_{i}(1)\geq W_{i}(0)$.
Under these two assumptions, there is at least one complier and no defiers
among the units in the study. Let $G_{i},\ i=1,\cdots,n$ denote the latent
group for unit $i$. Since $W_{i}(1)$ and $W_{i}(0)$ are fixed, $G_{i}$ is also
fixed. The sample complier average causal effect (sample CACE) is defined as

$\tau_{CACE}=\frac{1}{n_{co}}\sum_{G_{i}=co}(Y_{i}(1)-Y_{i}(0)),$

where $n_{co}$ is the number of compliers in the sample. Under assumption
(ii’) $n_{co}=\sum_{i=1}^{n}(W_{i}(1)-W_{i}(0))$. The fraction of compliers in
the sample is $p_{co}=n_{co}/n$. Further under the assumption that potential
outcomes only depend on treatment received, for always-takers and never-takers
$Y_{i}(1)-Y_{i}(0)=0$. Therefore,
$\sum_{G_{i}=co}(Y_{i}(1)-Y_{i}(0))=\sum_{i=1}^{n}(Y_{i}(1)-Y_{i}(0))$.

Define the sample average effect of treatment assigned on $W$ as

$\tau_{W}=\frac{1}{n}\sum_{i=1}^{n}(W_{i}(1)-W_{i}(0))=p_{co},$

and the sample average effect of treatment assigned on $Y$ as

$\tau_{Y}=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}(1)-Y_{i}(0)).$

We can then get another form of the sample CACE:

$\tau_{CACE}=\frac{\tau_{Y}}{\tau_{W}}.$ (1)

$\tau_{CACE}$ is a parameter for the finite population of units in the
experiment, for which point and interval estimates can be constructed.

##  3 The Wald Estimator

###  3.1 Definition and Consistency of The Wald Estimator

Let the observed indicator of treatment received be $W_{i}=W_{i}(Z_{i})$,
$i=1,...,n$. Let $\bm{W}$ be the vector of $W_{i}$ for all $n$ units. Let the
observed outcome be $Y_{i}=Y_{i}(Z_{i})$. Let $\bm{Y}$ be the vector of
$Y_{i}$ for all $n$ units.

For any set of quantities $Q_{i}$ for $i=1,\cdots,n$, let
$\overline{Q}_{1}=\frac{1}{n_{1}}\sum_{i:Z_{i}=1}Q_{i}$ denote the mean for
those assigned to treatment, let
$\overline{Q}_{0}=\frac{1}{n_{0}}\sum_{i:Z_{i}=0}Q_{i}$ denote the mean for
those assigned to control, and define the difference-in-means between these
two groups as

$\widehat{\tau}_{Q}=\overline{Q}_{1}-\overline{Q}_{0}.$

Let $\widehat{\tau}_{Y}$ and $\widehat{\tau}_{W}$ denote the difference-in-
means for $Y_{i}$ and $W_{i}$. When $\widehat{\tau}_{W}\neq 0$, the Wald
estimator is defined as

$\widehat{\tau}^{Wald}=\frac{\widehat{\tau}_{Y}}{\widehat{\tau}_{W}}=\frac{\overline{Y}_{1}-\overline{Y}_{0}}{\overline{W}_{1}-\overline{W}_{0}}.$
(2)

In the case when the units are regarded as being independently and randomly
sampled from a superpopulation and treatment assignment is completely random,
$\widehat{\tau}^{Wald}$ is a asymptotically unbiased and normally distributed
estimator for $\tau_{CACE}^{pop}$, and the delta method can be used to
estimate its asymptotic variance (see e.g. Imbens and Rubin, (2015), Chapter
23). We will study the properties of using $\widehat{\tau}^{Wald}$ to estimate
$\tau_{CACE}$.

The following theorem says that $\widehat{\tau}^{Wald}$ is a consistent
estimator of $\tau_{CACE}$ under Condition S1 (details in the Supplemetary
Material).

###### Theorem 1

Under Condition S1, $\widehat{\tau}^{Wald}-\tau_{CACE}=o_{p}(1)$ as
$n\rightarrow\infty$.

###  3.2 Construction of Confidence Interval Using the Wald Estimator

For later use, we introduce some new notation. For any set of quantities
$\\{Q_{i}:\ i=1,\cdots,n\\}$, let
$\overline{Q}=\frac{1}{n}\sum_{i=1}^{n}Q_{i}$ denote the mean for all $n$
units, and define the finite population variance as

$\mathbb{S}_{Q}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(Q_{i}-\overline{Q}\right)^{2}.$

For any two sets of quantities $\\{Q_{i}:\ i=1,\cdots,n\\}$ and
$\\{Q^{\prime}_{i}:\ i=1,\cdots,n\\}$, define the finite population covariance
as

$\mathbb{S}_{Q,Q^{\prime}}=\frac{1}{n-1}\sum_{i=1}^{n}\left(Q_{i}-\overline{Q}\right)\left(Q_{i}^{\prime}-\overline{Q}^{\prime}\right).$

To set our contribution into context we first discuss the approach given in Li
and Ding, (2017).

####  3.2.1 The Approach in Li and Ding, (2017)

Li and Ding, (2017) assumes that $Y_{i}(1)-Y_{i}(0)=\beta(W_{i}(1)-W_{i}(0))$
for all $i$. Under this linear instrumental variable model, for each complier,
$W_{i}(1)-W_{i}(0)=1$ and hence $Y_{i}(1)-Y_{i}(0)=\beta$, which says that the
individual treatment effect is homogeneous among compliers. It follows that
$\tau_{CACE}=\beta$.

Define the adjusted outcome $A_{i}=Y_{i}-\beta W_{i}$ with potential outcomes
$A_{i}(z)=Y_{i}(z)-\beta W_{i}(z)$ for $z=0,1$. Let $\widehat{\tau}_{A}$ be
the difference-in-means for $A_{i}$. Let $\mathbb{S}_{A}^{2}$,
$\mathbb{S}_{A(1)}^{2}$ and $\mathbb{S}_{A(0)}^{2}$ be the finite population
variances for $A_{i}$, $A_{i}(1)$ and $A_{i}(0)$. Let
$\mathbb{S}_{A(1)-A(0)}^{2}$ be the finite population variance for
$A_{i}(1)-A_{i}(0)$. If $\\{A_{i}:\ i=1,\cdots,n\\}$ satisfy Condition (4) in
Theorem 1 of Li and Ding, (2017), then $\widehat{\tau}_{A}$ converges to a
normal distribution with mean
$\tau_{A}=\frac{1}{n}\sum_{i=1}^{n}(A_{i}(1)-A_{i}(0))$ and variance

$Var(\widehat{\tau}_{A})=\frac{\mathbb{S}_{A(1)}^{2}}{n_{1}}+\frac{\mathbb{S}_{A(0)}^{2}}{n_{0}}-\frac{\mathbb{S}_{A(1)-A(0)}^{2}}{n}.$

Because $Y_{i}(1)-\beta W_{i}(1)=Y_{i}(0)-\beta W_{i}(0)$ under the linear
instrumental variable model, the adjusted outcome satisfies
$A_{i}=A_{i}(1)=A_{i}(0)$. Therefore $\tau_{A}=0$,
$\mathbb{S}_{A(1)}^{2}=\mathbb{S}_{A(0)}^{2}=\mathbb{S}_{A}^{2}$,
$\mathbb{S}_{A(1)-A(0)}=0$, and
$Var(\widehat{\tau}_{A})=(\frac{1}{n_{1}}+\frac{1}{n_{0}})\mathbb{S}_{A}^{2}$.
Let $\mathbb{S}_{Y}^{2}$ and $\mathbb{S}_{W}^{2}$ be the finite population
variances for $Y_{i}$ and $W_{i}$, and let $\mathbb{S}_{Y,W}$ be the finite
population covariance between $Y_{i}$ and $W_{i}$. Because $Y_{i}$ and $W_{i}$
are observed, $\mathbb{S}_{Y}^{2}$, $\mathbb{S}_{W}^{2}$ and
$\mathbb{S}_{Y,W}$ are all observed. Due to the linear relationship between
$A_{i}$ and $(Y_{i},W_{i})$, we have

$Var(\widehat{\tau}_{A})=\left(\frac{1}{n_{1}}+\frac{1}{n_{0}}\right)\left(\mathbb{S}_{Y}^{2}+\beta^{2}\mathbb{S}_{W}^{2}-2\beta\mathbb{S}_{Y,W}\right).$

Let $\nu_{1-\alpha/2}$ be the $(1-\alpha/2)$th quantile of the standard normal
distribution. Based on the normal approximation, the $1-\alpha$ confidence
interval for $\beta$ consists of the values satisfying
$|\widehat{\tau}_{A}/\sqrt{Var(\widehat{\tau}_{A})}|\leq\nu_{1-\alpha/2}$, or
$\widehat{\tau}_{A}^{2}\leq\nu_{1-\alpha/2}^{2}Var(\widehat{\tau}_{A})$. Due
to the linear relationship between $A_{i}$ and $(Y_{i},W_{i})$,
$\widehat{\tau}_{A}=\widehat{\tau}_{Y}-\beta\widehat{\tau}_{W}$. Therefore the
$1-\alpha$ confidence interval for $\beta$ is the solution to the following
inequality:

$\left(\widehat{\tau}_{Y}-\beta\widehat{\tau}_{W}\right)^{2}\leq\nu_{1-\alpha/2}^{2}\times\left(\frac{1}{n_{1}}+\frac{1}{n_{0}}\right)\left(\mathbb{S}_{Y}^{2}+\beta^{2}\mathbb{S}_{W}^{2}-2\beta\mathbb{S}_{Y,W}\right).$

For different observed data, this solution can be an empty set, a finite
interval $[c_{1},c_{2}]$, the whole real line, or the union of two disjoint
infinite intervals $(-\infty,c_{1}]\cup[c_{2},\infty]$. Infinite confidence
sets suggest little or no identification due to weak instruments, and an empty
set suggests possible violation of the linear instrumental variable model.

The linear instrumental variable model is overly restrictive. Also, the four
possible forms of confidence sets are not handy for practical use. In the next
sub-section we derive under heterogeneous effects a finite confidence interval
for $\tau_{CACE}$ that are familiar to practitioners.

####  3.2.2 A Finite Confidence Interval under Heterogeneous Effects

Define a new adjusted outcome $B_{i}=Y_{i}-\tau_{CACE}W_{i}$ with potential
outcomes $B_{i}(z)=Y_{i}(z)-\tau_{CACE}W_{i}(z)$ for $z=0,1$. Let
$\widehat{\tau}_{B}$ be the difference-in-means for $B_{i}$. We have

$\widehat{\tau}_{B}=\widehat{\tau}_{Y}-\tau_{CACE}\widehat{\tau_{W}}.$ (3)

Let $\mathbb{S}_{B(1)}^{2}$ and $\mathbb{S}_{B(0)}^{2}$ be the finite
population variances for $B_{i}(1)$ and $B_{i}(0)$. Let
$\mathbb{S}_{B(1)-B(0)}^{2}$ be the finite population variance for
$B_{i}(1)-B_{i}(0)$. Under Condition S1, we can apply Theorem 4 of Li and
Ding, (2017) to obtain that
$(\widehat{\tau}_{B}-\tau_{B})/\sqrt{Var(\widehat{\tau}_{B})}$ converges to
the standard normal distribution, with
$\tau_{B}=\frac{1}{n}\sum_{i=1}^{n}(B_{i}(1)-B_{i}(0))$ and

$Var(\widehat{\tau}_{B})=\frac{\mathbb{S}_{B(1)}^{2}}{n_{1}}+\frac{\mathbb{S}_{B(0)}^{2}}{n_{0}}-\frac{\mathbb{S}_{B(1)-B(0)}^{2}}{n}.$
(4)

It is easy to see that $\tau_{B}=\tau_{Y}-\tau_{CACE}\tau_{W}$, which equals 0
according to (1). As $n\rightarrow\infty$, we have

$Pr\left(\widehat{\tau}_{B}-\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{B})}\leq
0\leq\widehat{\tau}_{B}+\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{B})}\right)\rightarrow
1-\alpha.$

Taking into account (3) and (2), when $\widehat{\tau}_{W}\neq 0$, we have

$\displaystyle
Pr\left(\widehat{\tau}_{B}-\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{B})}\leq
0\leq\widehat{\tau}_{B}+\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{B})}\right)$
(5) $\displaystyle=$ $\displaystyle
Pr\left(\frac{\widehat{\tau}_{B}}{|\widehat{\tau}_{W}|}-\nu_{1-\alpha/2}\frac{\sqrt{Var(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}\leq
0\leq\frac{\widehat{\tau}_{B}}{|\widehat{\tau}_{W}|}+\nu_{1-\alpha/2}\frac{\sqrt{Var(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}\right)$
$\displaystyle=$ $\displaystyle
Pr\left(\frac{\widehat{\tau}_{Y}}{|\widehat{\tau}_{W}|}-\nu_{1-\alpha/2}\frac{\sqrt{Var(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}\leq\frac{\widehat{\tau}_{W}}{|\widehat{\tau}_{W}|}\tau_{CACE}\leq\frac{\widehat{\tau}_{Y}}{|\widehat{\tau}_{W}|}+\nu_{1-\alpha/2}\frac{\sqrt{Var(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}\right)$
$\displaystyle=$ $\displaystyle
Pr\left(\widehat{\tau}^{Wald}-\nu_{1-\alpha/2}\frac{\sqrt{Var(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}\leq\tau_{CACE}\leq\widehat{\tau}^{Wald}+\nu_{1-\alpha/2}\frac{\sqrt{Var(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}\right),$

where the last equality can be obtained by considering each of the two cases
with $\widehat{\tau}_{W}>0$ or $\widehat{\tau}_{W}<0$. Therefore, as
$n\rightarrow\infty$,

$\widehat{\tau}^{Wald}\pm\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{B})}/|\widehat{\tau}_{W}|$
(6)

has $(1-\alpha)$ coverage rate for $\tau_{CACE}$.

If we can find an asymptotically conservative estimator for
$Var\left(\widehat{\tau}_{B}\right)$, we can construct an asymptotically
conservative $(1-\alpha)$ confidence interval for $\tau_{CACE}$. Replacing
$\tau_{CACE}$ in the expression of $B_{i}$ with its estimator
$\widehat{\tau}^{Wald}$, we define
$\widehat{B}_{i}=Y_{i}-\widehat{\tau}^{Wald}W_{i}$, which is observed. We can
use the sample variance of $\widehat{B}_{i}$ for those assigned to treatment
arm $z$,

$S_{\widehat{B}(z)}^{2}=\frac{1}{n_{z}-1}\sum_{i:Z_{i}=z}\left(\widehat{B}_{i}-\sum_{i:Z_{i}=z}\widehat{B}_{i}/n_{z}\right)^{2},$

to estimate $\mathbb{S}_{B(z)}^{2}$ for $z=0,1$. We propose the following
estimator for $Var\left(\widehat{\tau}_{B}\right)$:

$\widehat{Var}(\widehat{\tau}_{B})=\frac{S_{\widehat{B}(1)}^{2}}{n_{1}}+\frac{S_{\widehat{B}(0)}^{2}}{n_{0}}.$
(7)

The following theorem says that the confidence interval constructed using
$\widehat{Var}(\widehat{\tau}_{B})$ is asymptotically conservative under
Condition S1.

###### Theorem 2

Under Condition S1, the confidence interval

$\widehat{\tau}^{Wald}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}(\widehat{\tau}_{B})}/\left|\widehat{\tau}_{W}\right|$
(8)

has at least $(1-\alpha)$ coverage rate for $\tau_{CACE}$ as
$n\rightarrow\infty$.

Proposition S1 in the Supplemetary Material shows that this confidence
interval is the same as the super-population confidence interval for
$\tau_{CACE}^{pop}$ obtained by the delta method.

##  4 The Regression Adjustment Estimator

###  4.1 Definition and Consistency of The Regression Adjustment Estimator

The Wald estimator does not account for any covariates. In practical
experiments, covariates are often available, and we can adjust for them to
improve efficiency.

Let $\bm{x}_{i}$, $i=1,\cdots,n$, be the $K\times 1$ vector of covariates for
unit $i$. These covariates are all observed before treatment assignment and
are treated as fixed. Without loss of generality, we assume that the
covariates have been centered around zero, that is, $\overline{\bm{x}}\equiv
1/n\sum_{i=1}^{n}\bm{x}_{i}=\bm{0}$. Let $\bm{X}$ be the $n\times K$ matrix of
covariates for the $n$ units.

For any set of quantities $\\{Q_{i}:\ i=1,\cdots,n\\}$, let
$\widehat{\alpha}_{Q,z}$ and $\widehat{\bm{\beta}}_{Q,z}$ be the intercept and
the vector of coefficients on $\bm{x}_{i}$ in the linear projection of $Q_{i}$
on $\bm{x}_{i}$ using units with $Z_{i}=z$ for $z=0,1$. That is,

$\left(\widehat{\alpha}_{Q,z},\widehat{\bm{\beta}}_{Q,z}\right)=\arg\min_{\alpha,\bm{\beta}}\sum_{i:\
Z_{i}=z}\left(Q_{i}-\alpha-\bm{\beta}^{\top}\bm{x}_{i}\right)^{2}.$

For quantities $\\{W_{i}:\ i=1,\cdots,n\\}$, $\\{Y_{i}:\ i=1,\cdots,n\\}$ and
$\\{B_{i}:\ i=1,\cdots,n\\}$, we can thus define $\widehat{\bm{\beta}}_{W,z}$,
$\widehat{\bm{\beta}}_{Y,z}$ and $\widehat{\bm{\beta}}_{B,z}$ ($z=0,1)$. Note
that, $\widehat{\bm{\beta}}_{W,z}$ and $\widehat{\bm{\beta}}_{Y,z}$ are
obtained by the Ordinary Least squares (OLS) regression of $Y_{i}$ and $W_{i}$
on $\bm{x}_{i}$ using units with $Z_{i}=z$.

According to the results of Lin, (2013), under complete randomization,
$\tau_{W}$ or $\tau_{Y}$ can be consistently estimated by the estimated
coefficient on $Z_{i}$ (denoted by $\widehat{\tau}_{W}^{Reg}$ or
$\widehat{\tau}_{Y}^{Reg}$) in the OLS regression of $W_{i}$ or $Y_{i}$ on
$Z_{i}$, $\bm{x}_{i}$ and $Z_{i}\bm{x}_{i}$ using all $n$ units.

Define $\overline{\bm{x}}_{1}=\sum_{i:\ Z_{i}=1}\bm{x}_{i}/n_{1}$ and
$\overline{\bm{x}}_{0}=\sum_{i:\ Z_{i}=0}\bm{x}_{i}/n_{0}$. Applying
Proposition S2 in the Supplemetary Material, we can write
$\widehat{\tau}_{W}^{Reg}$ and $\widehat{\tau}_{Y}^{Reg}$ as

$\displaystyle\widehat{\tau}_{W}^{Reg}$
$\displaystyle=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(W_{i}-\widehat{\bm{\beta}}_{W,1}\bm{x}_{i}\right)-\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(W_{i}-\widehat{\bm{\beta}}_{W,0}\bm{x}_{i}\right)$ (9)
$\displaystyle=\widehat{\tau}_{W}-\widehat{\bm{\beta}}_{W,1}\overline{\bm{x}}_{1}+\widehat{\bm{\beta}}_{W,0}\overline{\bm{x}}_{0}$

and

$\displaystyle\widehat{\tau}_{Y}^{Reg}$
$\displaystyle=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(Y_{i}-\widehat{\bm{\beta}}_{Y,1}\bm{x}_{i}\right)-\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(Y_{i}-\widehat{\bm{\beta}}_{Y,0}\bm{x}_{i}\right)$ (10)
$\displaystyle=\widehat{\tau}_{Y}-\widehat{\bm{\beta}}_{Y,1}\overline{\bm{x}}_{1}+\widehat{\bm{\beta}}_{Y,0}\overline{\bm{x}}_{0}.$

The following theorem says that the estimator,

$\widehat{\tau}^{Reg}=\frac{\widehat{\tau}_{Y}^{Reg}}{\widehat{\tau}_{W}^{Reg}},$
(11)

is a consistent estimator of $\tau_{CACE}$ under condition Condition S2
(details in the Supplemetary Material).

###### Theorem 3

Under Condition S2, $\widehat{\tau}^{Reg}-\tau_{CACE}=o_{p}(1)$ as
$n\rightarrow\infty$.

###  4.2 Construction of Confidence Interval Using the Regression Adjustment
Estimator

Because $B_{i}$ is a linear combination of $Y_{i}$ and $W_{i}$, we have

$\widehat{\bm{\beta}}_{B,z}=\widehat{\bm{\beta}}_{Y,z}-\tau_{CACE}\widehat{\bm{\beta}}_{W,z}.$
(12)

Define a new adjusted outcome

$D_{i}=B_{i}-\widehat{\bm{\beta}}_{B,Z_{i}}^{\top}\bm{x}_{i},$

with potential outcomes
$D_{i}(z)=B_{i}(z)-\widehat{\bm{\beta}}_{B,z}^{\top}\bm{x}_{i}$. Let
$\widehat{\tau}_{D}$ be the difference-in-means for $D_{i}$. Let
$Var(\widehat{\tau}_{D})$ denote the variance of $\widehat{\tau}_{D}$. The
following theorem says that $\widehat{\tau}_{D}$ asymptotically has a normal
distribution under Condition S2.

###### Theorem 4

Under Condition S2, as $n\rightarrow\infty$,

$\widehat{\tau}_{D}/\sqrt{Var(\widehat{\tau}_{D})}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1).$ (13)

With Theorem 4, as $n\rightarrow\infty$, we have

$Pr\left(\widehat{\tau}_{D}-\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{D})}\leq
0\leq\widehat{\tau}_{D}+\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{D})}\right)\rightarrow
1-\alpha.$

Combining the definitions of $D_{i}$ and $B_{i}$ with (9), (10) and (12), we
have

$\displaystyle\widehat{\tau}_{D}=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}D_{i}-\frac{1}{n_{0}}\sum_{i:\ Z_{i}=0}D_{i}$ (14) $\displaystyle=$
$\displaystyle\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(B_{i}-\widehat{\bm{\beta}}_{B,1}^{\top}\bm{x}_{i}\right)-\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(B_{i}-\widehat{\bm{\beta}}_{B,0}^{\top}\bm{x}_{i}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(Y_{i}-\tau_{CACE}W_{i}-\left(\widehat{\bm{\beta}}_{Y,1}-\tau_{CACE}\widehat{\bm{\beta}}_{W,1}\right)^{\top}\bm{x}_{i}\right)$
$\displaystyle-\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(Y_{i}-\tau_{CACE}W_{i}-\left(\widehat{\bm{\beta}}_{Y,0}-\tau_{CACE}\widehat{\bm{\beta}}_{W,0}\right)^{\top}\bm{x}_{i}\right)$
$\displaystyle=$
$\displaystyle\widehat{\tau}_{Y}^{Reg}-\tau_{CACE}\widehat{\tau}_{W}^{Reg}.$

Note that $\widehat{\tau}_{D}$ has a similar form as $\widehat{\tau}_{B}$ in
(3). Following similar arguments to those in Section 3.2.2 that lead to (6)
having $(1-\alpha)$ coverage rate for $\tau_{CACE}$, we obtain that as
$n\rightarrow\infty$,

$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{Var(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|$

has $(1-\alpha)$ coverage rate for $\tau_{CACE}$.

If we can find an asymptotically conservative estimator for
$Var\left(\widehat{\tau}_{D}\right)$, we can construct an asymptotically
conservative $(1-\alpha)$ confidence interval for $\tau_{CACE}$. Applying
Proposition S2 in the Supplementary Material, we know that
$\widehat{\tau}_{D}$ equals the coefficient on $Z_{i}$ in the linear
projection of $B_{i}$ on $Z_{i}$, $\bm{x}_{i}$ and $Z_{i}\bm{x}_{i}$ using all
$n$ units. If $B_{i}$ were known, we could use the EHW robust variance
estimator to estimate $Var(\widehat{\tau}_{D})$. However, as $B_{i}$ contains
the unknown quantity $\tau_{CACE}$, in practice we replace $\tau_{CACE}$ with
$\widehat{\tau}^{Reg}$, and consider instead the linear projection of

$\widehat{B}_{i}=Y_{i}-\widehat{\tau}^{Reg}W_{i},$

that is, the OLS regression of $\widehat{B}_{i}$ since $\widehat{B}_{i}$ is
observed.

Let $\widehat{Var}^{EHW}(\widehat{\tau}_{D})$ denote the EHW variance
estimator of the estimated coefficient on $Z_{i}$ in the OLS regression of
$\widehat{B}_{i}$ on $Z_{i}$, $\bm{x}_{i}$ and $Z_{i}\bm{x}_{i}$ using all $n$
units. We can also use the series of HC variance estimators that can be used
as finite sample corrections of the EHW variance estimator in usual OLS
regressions (MacKinnon,, 2012). Specifically, let
$\widehat{Var}^{HC2}(\widehat{\tau}_{D})$ and
$\widehat{Var}^{HC3}(\widehat{\tau}_{D})$ denote the HC2 and HC3 variance
estimators of the estimated coefficient on $Z_{i}$ in the OLS regression of
$\widehat{B}_{i}$ on $Z_{i}$, $\bm{x}_{i}$ and $Z_{i}\bm{x}_{i}$ using all $n$
units. The following theorem says that the confidence interval constructed
using $\widehat{Var}^{EHW}(\widehat{\tau}_{D})$,
$\widehat{Var}^{HC2}(\widehat{\tau}_{D})$ or
$\widehat{Var}^{HC3}(\widehat{\tau}_{D})$ is asymptotically conservative under
Condition S2.

###### Theorem 5

Under Condition S2, each of the following confidence intervals,

$\displaystyle\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{EHW}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|,$
(15)
$\displaystyle\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{HC2}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|,$
(16) and
$\displaystyle\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{HC3}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|,$
(17)

is asymptotically equivalent and has at least $(1-\alpha)$ coverage rate for
$\tau_{CACE}$ as $n\rightarrow\infty$.

###  4.3 Efficiency Gain Using Regression Adjustment

To characterize the efficiency gain using regression adjustment, we
investigate the percentage by which regression adjustment reduces the length
of the asymptotic $1-\alpha$ confidence interval, which we refer to as the
percent reduction in length (PRIL).

For $z=0,1$, let $\mathbb{S}_{B(z)|\bm{x}}^{2}$ denote the finite population
variance of the linear projection of $B_{i}(z)$ on $\bm{x}_{i}$,
$\alpha_{B,z}+\bm{\beta}_{B,z}^{\top}\bm{x}_{i}$, where $\alpha_{B,z}$ and
$\bm{\beta}_{B,z}$ are the intercept and the vector of coefficients on
$\bm{x}_{i}$ in the linear projection of $B_{i}(z)$ on $\bm{x}_{i}$ using all
$n$ units. Define

$\displaystyle Var(\widehat{\tau}_{B})^{+}$
$\displaystyle=\frac{\mathbb{S}_{B(1)}^{2}}{n_{1}}+\frac{\mathbb{S}_{B(0)}^{2}}{n_{0}},$
(18) $\displaystyle Var(\widehat{\tau}_{B\mid\bm{x}})^{+}$
$\displaystyle=\frac{\mathbb{S}_{B(1)\mid\bm{x}}^{2}}{n_{1}}+\frac{\mathbb{S}_{B(0)\mid\bm{x}}^{2}}{n_{0}},$
(19)

###### Theorem 6

Under Condition S2, compared with (8), the PRIL of (15), (16) or (17)
converges in probability to

$1-\sqrt{1-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}/Var\left(\widehat{\tau}_{B}\right)^{+}}.$

For $z=0,1$, since $\mathbb{S}_{B(z)|\bm{x}}^{2}$ is the finite population
variance of the linear projection of $B_{i}(z)$ on $\bm{x}_{i}$, we have
$\mathbb{S}_{B(z)\mid\bm{x}}^{2}\leq{}\mathbb{S}_{B(z)}^{2}$. Hence
$Var(\widehat{\tau}_{B\mid\bm{x}})^{+}\leq Var(\widehat{\tau}_{B})^{+}$, and
the PRIL is asymptotically nonnegative. Furthermore, recall that
$B_{i}(z)=Y_{i}(z)-\tau_{CACE}W_{i}(z)$. If the covariates are more predictive
of $Y_{i}(z)$ or $W_{i}(z)$, $\mathbb{S}_{B(z)\mid\bm{x}}^{2}$ is closer to
$\mathbb{S}_{B(z)}^{2}$, then $Var(\widehat{\tau}_{B\mid\bm{x}})^{+}$ is
closer to $Var(\widehat{\tau}_{B})^{+}$, and the limit of the PRIL is larger.

##  5 Monte Carlo Simulation

The focus of the Monte Carlo simulation is to compare the performance of the
methods. We use “Wald” to denote the Wald estimator. For confidence intervals
constructed using the Wald estimator, we use “Wald-LD” to denote the
confidence set constructed using the approach in Li and Ding, (2017), and use
“Wald-Delta” to denote the confidence interval constructed using our approach.
We use “Reg” to denote the regression adjustment estimator. For confidence
intervals constructed using the regression adjustment estimator, we use “Reg-
EHW”, “Reg-HC2” and “Reg-HC3” to denote the confidence intervals constructed
using the EHW, HC2 or HC3 variance estimators.

###  5.1 Setup of the Monte Carlo Simulation

We set $n_{1}=n_{0}=n/2$. Each covariate independently follows a $N(0,1)$
distribution. Noting that potential outcomes only depend on treatment
received, we use $Y_{i}^{W}(w)$ ($w=0,1$) to denote the potential outcome if
receiving treatment arm $w$. The potential outcomes and the potential latent
variables are generated using the following data generating process:

$\displaystyle Y_{i}^{W}(0)$
$\displaystyle=\bm{\xi}^{\top}\bm{x}_{i}+\epsilon_{0i},$ $\displaystyle
Y_{i}^{W}(1)$
$\displaystyle=\bm{\xi}^{\top}\bm{x}_{i}+\bm{\eta}^{\top}\bm{x}_{i}+\epsilon_{1i},$
$\displaystyle L_{i}(0)$
$\displaystyle=\delta_{0}+\bm{\psi}^{\top}\bm{x}_{i}+u_{i},$ $\displaystyle
L_{i}(1)$ $\displaystyle=\delta_{1}+L_{i}(0).$

We then set $W_{i}(z)=I(L_{i}(z)>0)$ for $z=0,1$. Here $\bm{\xi}$, $\bm{\eta}$
and $\bm{\psi}$ are each a $K\times 1$ vector with all elements being 1.

For a given value of $p_{co}$, we set
$\delta_{0}=((p_{co}-0.5)/0.35+1)\sqrt{K}$, and choose the value of
$\delta_{1}$ such that the fraction of compliers equals $p_{co}$. We fix
$K=5$, and consider $n\in\\{200,400\\}$ and $p_{co}\in\\{0.85,0.5,0.15\\}$.

The error terms are generated according to

$\left(\begin{array}[]{c}\kappa_{0}\epsilon_{i0}\\\ \kappa_{1}\epsilon_{i1}\\\
\kappa_{2}u_{i}\end{array}\right)\sim
N\left(\bm{0},\left(\begin{array}[]{ccc}1&0&\rho\\\ 0&1&\rho\\\
\rho&\rho&1\end{array}\right)\right)$

with $\rho\in\\{0,0.5\\}$. The values of $\kappa_{0}$, $\kappa_{1}$ and
$\kappa_{2}$ are chosen such that the squared multiple correlation coefficient
is $0.5$ in each of the equations for $Y_{i}^{W}(0)$, $Y_{i}^{W}(1)$ and
$L_{i}(0)$.

The values of
$(\bm{x}_{i},L_{i}(0),L_{i}(1),W_{i}(0),W_{i}(1),Y_{i}(0),Y_{i}(1))$ are
fixed, and only the values of $Z_{i}$ are randomly generated $1,000$ times
under complete randomization. The observed values are
$(\bm{x}_{i},Z_{i},W_{i},Y_{i})$. We thus have $1,000$ datasets for each of 12
scenarios with different combinations of $n$, $p_{co}$ and error distribution.

###  5.2 Performance Measures

If $\widehat{\tau}_{W}=0$ or $\widehat{\tau}_{W}^{Reg}=0$, the point estimate
is infinite, and the confidence interval is also infinite, regardless of which
method is used to constructed the confidence interval. These results are
regarded as being abnormal. In addition, for the Wald-LD method, we view the
three forms of confidence sets other than finite confidence intervals as being
abnormal. We remove these results in the following comparison.

For the $r$th dataset and the $m$th method, let $\widehat{\tau}_{rm}$ denote
the point estimate, let $\mathcal{C}_{rm}$ denote the $95\%$ interval
estimate, and let $L_{rm}$ denote the length of $\mathcal{C}_{rm}$. We
consider three performance measures for method $m$: (1) median absolute error

$\text{MAE}_{m}=\text{median of }|\widehat{\tau}_{rm}-\tau_{CACE}|,$

(2) coverage rate of 95% intervals

$\text{CRate}_{m}=\text{mean of
}I\left(\tau_{CACE}\in\mathcal{C}_{rm}\right),$

and (3) median length of 95% intervals

$\text{Len}_{m}=\text{median of }L_{rm}.$

###  5.3 Results

In our simulation, abnormal results only appear when $p_{co}=0.15$. Table 1
presents the proportions of abnormal confidence intervals for different
methods when $p_{co}=0.15$. The Wald-LD method has rather high proportions of
abnormal confidence intervals, all in the form of the whole real line or the
union of two disjoint infinite intervals, indicating poor identification due
to weak instruments. These proportions are especially high when the sample
size is small. The Wald-Delta method has zero or very low proportion of
abnormal results with $\widehat{\tau}_{W}=0$. The regression adjustment
methods have no abnormal results with $\widehat{\tau}_{W}^{Reg}=0$.

When $\widehat{\tau}_{W}$ or $\widehat{\tau}_{W}^{Reg}$ is nonzero but close
to zero, the Wald method or the regression adjustment method can be unstable.
The point estimate can be quite large in absolute value, and the corresponding
confidence interval can be quite wide. To illustrate this point, for the 1,000
data sets generated with $n=400$, $p_{co}=0.15$ and $\rho=0$, Figure 1
presents histograms of point estimates and lengths of 95% confidence
intervals, with abnormal results excluded. It appears that the regression
adjustment method is much stabler than the Wald method. The histogram of
lengths of the 95% confidence intervals has a longer tail for the Wald-Delta
method than for the Wald-LD method. This is because for the Wald-LD method
13.1% of the datasets with infinite intervals have been excluded, whereas for
the Wald-Delta method no datasets with infinite intervals have been excluded.

(a) point estimates

(b) lengths of 95% intervals

Figure 1: Histograms of point estimates and lengths of 95% intervals with
$n=400$, $p_{co}=0.15$ and $\rho=0$.

With abnormal results excluded, Table 2 presents median absolute error, Table
3 presents coverage rate of 95% intervals measured in percentage differences
from the nominal level 0.95, $100\times(\text{CRate}_{m}-0.95)$, and Table 4
presents median length of 95% intervals.

Table 1: Proportions of abnormal confidence intervals for different methods when $p_{co}=0.15$. method | $\rho=0$ | $\rho=0.5$
---|---|---
| $n=200$
Wald-LD | 0.395 | 0.425
Wald-Delta | 0.004 | 0.004
Reg-EHW, Reg-HC2, Reg-HC3 | 0.000 | 0.000
| $n=400$
Wald-LD | 0.131 | 0.119
Wald-Delta | 0.000 | 0.000
Reg-EHW, Reg-HC2, Reg-HC3 | 0.000 | 0.000
Table 2: Median absolute error for different methods. | $p_{co}=0.85$ | $p_{co}=0.5$ | $p_{co}=0.15$
---|---|---|---
method | $\rho=0$ | $\rho=0.5$ | $\rho=0$ | $\rho=0.5$ | $\rho=0$ | $\rho=0.5$
| $n=200$
Wald | 0.450 | 0.517 | 0.855 | 0.897 | 3.031 | 3.028
Reg | 0.301 | 0.362 | 0.591 | 0.643 | 2.205 | 2.239
| $n=400$
Wald | 0.377 | 0.350 | 0.778 | 0.682 | 2.597 | 2.288
Reg | 0.245 | 0.222 | 0.484 | 0.419 | 1.749 | 1.557
Table 3: Coverage rate of 95% intervals for different methods, measured in percentage differences from the nominal level 0.95, $100\times(\text{CRate}_{m}-0.95)$. | $p_{co}=0.85$ | $p_{co}=0.5$ | $p_{co}=0.15$
---|---|---|---
method | $\rho=0$ | $\rho=0.5$ | $\rho=0$ | $\rho=0.5$ | $\rho=0$ | $\rho=0.5$
| $n=200$
Wald-LD | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 5.0
Wald-Delta | 2.4 | 1.4 | 1.3 | 1.4 | -0.2 | 0.3
Reg-EHW | 4.1 | 3.0 | 1.8 | 2.5 | 3.3 | 3.6
Reg-HC2 | 4.4 | 3.5 | 2.1 | 2.7 | 3.5 | 4.0
Reg-HC3 | 4.5 | 3.7 | 2.2 | 3.1 | 3.7 | 4.1
| $n=400$
Wald-LD | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 | 5.0
Wald-Delta | 2.5 | 3.3 | 0.6 | 2.5 | 2.5 | 1.5
Reg-EHW | 3.6 | 3.4 | 2.0 | 2.8 | 2.9 | 2.7
Reg-HC2 | 3.9 | 3.7 | 2.3 | 3.0 | 3.4 | 2.9
Reg-HC3 | 4.1 | 3.8 | 2.3 | 3.2 | 3.5 | 3.0
Table 4: Median length of 95% intervals for different methods. | $p_{co}=0.85$ | $p_{co}=0.5$ | $p_{co}=0.15$
---|---|---|---
method | $\rho=0$ | $\rho=0.5$ | $\rho=0$ | $\rho=0.5$ | $\rho=0$ | $\rho=0.5$
| $n=200$
Wald-LD | 3.486 | 4.027 | 6.477 | 7.124 | 30.035 | 30.327
Wald-Delta | 3.079 | 3.504 | 5.341 | 5.906 | 18.367 | 18.722
Reg-EHW | 2.231 | 2.403 | 3.846 | 4.021 | 13.091 | 13.722
Reg-HC2 | 2.298 | 2.490 | 3.966 | 4.163 | 13.494 | 14.233
Reg-HC3 | 2.369 | 2.580 | 4.090 | 4.311 | 13.906 | 14.790
| $n=400$
Wald-LD | 2.828 | 2.721 | 5.339 | 4.965 | 23.131 | 21.812
Wald-Delta | 2.434 | 2.404 | 4.421 | 4.195 | 14.165 | 13.573
Reg-EHW | 1.711 | 1.676 | 3.103 | 2.925 | 10.697 | 9.532
Reg-HC2 | 1.738 | 1.702 | 3.151 | 2.971 | 10.864 | 9.675
Reg-HC3 | 1.765 | 1.730 | 3.201 | 3.018 | 11.033 | 9.820

In term of the median absolute error, compared to the Wald method, regression
adjustment has a reduction of 26.1% to 38.6% across different scenarios. In
terms of coverage rate, none of the methods has under coverage. However, the
Wald-LD method is too conservative as its coverage rates are 100% in all
scenarios. In terms of median length of 95% intervals, (i) compared to the
Wald-LD method, the Wald-Delta method has a reduction of 11.7% to 38.9%; (ii)
compared to the Wald-Delta method, the Reg-EHW method has a reduction of 24.5%
to 31.9%; (iii) compared to the Reg-EHW method, the Reg-HC2 method has an
increase of 1.5% to 3.7%, and the Reg-HC3 method has an increase of 3.0% to
7.8%.

To summarize, we recommend the Wald-Delta method when covariates are
unavailable, and the Reg-EHW method when covariates are available.

##  6 Application to a Job Training Experiment with Non-compliance

The Job Search Intervention Study (JOBS II) dataset comes from a field
experiment designed and conducted by Vinokur et al., (1995) that investigates
the efficacy of a job training intervention on unemployed workers.
Participants were randomly selected to attend the JOBS II training program
that taught job-search skills and coping strategies for dealing with setbacks
in the job-search process. In the dataset, 372 participants were selected to
attend the training program and actually attended, 228 participants were
selected to attend the training program but did not attend, and 299
participants were not selected to attend the training program and did not
attend. One outcome of interest is a continuous variable measuring the level
of job-search self-efficacy with values from 1 to 5. The covariates include
age, gender, ethnicity, marital status, monthly income and educational
attainment. This data set has been analyzed in Chapter 21 of Ding, (2023).

Since no one could attend the training program without being selected, there
were no always-takers or defiers. Those who were selected to attend the
training program and actually attended were compliers, those who were selected
to attend the training program but did not attend were never-takers, and those
who were not selected to attend the training program and did not attend were a
mixture of compliers and never-takers. Obviously assumptions (i’) and (i”) in
Section 2 are satisfied.

For different methods presented in this paper, Table 5 presents point
estimates of $p_{co}$, and point estimates and 95% intervals of $\tau_{CACE}$.
The estimated fraction of compliers is around 0.62 for all methods. The points
estimates of $\tau_{CACE}$ from different methods are similar. The Reg-EHW
method has a slightly shorter 95% interval than the Wald-Delta method. The
Reg-HC2 and Reg-HC3 methods have slightly longer 95% intervals than the Wald-
Delta method. The Wald-LD method has a much longer 95% interval than other
methods, which is overly conservative.

Table 5: Point estimates of $p_{co}$, and point estimates and 95% intervals of $\tau_{CACE}$ from different methods for the JOBS II dataset | point estimate | point estimate | 95% interval
---|---|---|---
method | of $p_{co}$ | of $\tau_{CACE}$ | of $\tau_{CACE}$
Wald-LD | 0.620 | 0.109 | [-0.141, 0.358]
Wald-Delta | 0.620 | 0.109 | [-0.050, 0.268]
Reg-EHW | 0.616 | 0.118 | [-0.039, 0.274]
Reg-HC2 | 0.616 | 0.118 | [-0.042, 0.278]
Reg-HC3 | 0.616 | 0.118 | [-0.046, 0.281]

##  7 Discussion

In pragmatic randomized experiments, incomplete adherence and/or incomplete
compliance to the assigned treatment is common. This paper focuses on
inference about the sample complier average causal effect. We propose to use
the Wald estimator and the regression adjustment estimator, and also propose
methods for construction of confidence intervals using these two estimators.
We establish their asymptotic properties, and compare their small sample
performance using a Monte Carlo simulation.

The results from the Monte Carlo simulation show that when covariates are
available, compared to the Wald method, the regression adjustment method is
more stable, does not yield abnormal results, and reduces median absolute
error of point estimates and median length of 95% confidence intervals,
without sacrificing coverage rate of the confidence intervals. When covariates
are not available, in terms of construction of confidence intervals based on
the Wald estimator, our proposed method yields a lot less abnormal results and
greatly reduces median length of 95% confidence intervals, compared to a
previous method in Li and Ding, (2017). We also present an application to the
Job Search Intervention Study (JOBS II) dataset.

SUPPLEMENTARY MATERIAL

Conditions and Proofs for Theorems and Propositions:

Including Conditions S1 and S2, Propositions S1 and S2 and their proof, and
proof for Theorem 1-6. (.pdf file)

R-package SCACE:

The R package SCACE contains code that implements the Wald estimator
(“WaldLD.R” and “WaldDelta.R”) and the regression adjustment estimator
(“Reg.R”). It also contains code for reproducing the Monte Carlo simulation
(“genData.R” and “mc.R”) and code for analyzing the JOBS II dataset
(“example.R”, which uses the “jobs” dataset in R package “mediation”). A
readme document is also included. (.zip file)

## Supplementary Material: Conditions for Theorems and Propositions

Recall that for $z=0,1$, $B_{i}(z)=Y_{i}(z)-\tau_{CACE}W_{i}(z)$. For $z=0,1$,
let $\overline{B}(z)$ be the mean of $B_{i}(z)$ for all $n$ units, and let
$\mathbb{S}_{B(z)}^{2}$ denote the finite population variance of $B_{i}(z)$.
Extending the conditions in Li and Ding, (2017), we introduce Condition S1 to
study the asymptotic properties of the Wald estimator in Theorems 1 and 2.

###### Condition S1

As $n\rightarrow\infty$, (1) the limit inferior of $p_{co}$ is positive; (2)
the proportion of units under treatment, $n_{1}/n$, has a limit in $(0,1)$;
(3) the limit superiors of $\mathbb{S}_{B(0)}^{2}$ and $\mathbb{S}_{B(1)}^{2}$
are finite; (4) for $z=0,1$, the following Lindeberg-Feller type condition
holds:

$\frac{\max_{1\leq i\leq
n}\left(B_{i}(z)-\overline{B}(z)\right)^{2}}{\min(n_{1},n_{0})^{2}\cdot
Var\left(\widehat{\tau}_{B}\right)}\rightarrow 0.$ (20)

Let $\mathbb{S}_{B(1)-B(0)}^{2}$ denote the finite population variance of
$B_{i}(1)-B_{i}(0)$. Let $\mathbb{S}_{\bm{x}\bm{x}}$ denote the finite
population covariance matrix of the covariates,

$\mathbb{S}_{\bm{x}\bm{x}}=\frac{1}{n-1}\sum_{i=1}^{n}(\bm{x}_{i}-\overline{\bm{x}})(\bm{x}_{i}-\overline{\bm{x}})^{\top}=\frac{1}{n-1}\sum_{i=1}^{n}\bm{x}_{i}\bm{x}_{i}^{\top}.$

For $z=0,1$, let $\mathbb{S}_{B(z),\bm{x}}$ be the finite population
covariance vector between $B_{i}(z)$ and $\bm{x}_{i}$,

$\mathbb{S}_{B(z),\bm{x}}=\frac{1}{n-1}\sum_{i=1}^{n}(B_{i}(z)-\overline{B}(z))(\bm{x}_{i}-\overline{\bm{x}}).$

Let $\left\|\bm{x}_{i}\right\|$ denote the Euclidean distance between
$\bm{x}_{i}$ and $\overline{\bm{x}}=\bm{0}$. Extending the conditions in Li
and Ding, (2020), we introduce Condition S2 to study the asymptotic properties
of the regression adjustment estimator in Theorems 3-5.

###### Condition S2

As $n\rightarrow\infty$, (1) the limit inferior of $p_{co}$ is positive; (2)
the proportion of units under treatment, $n_{1}/n$, has a limit in $(0,1)$;
(3) $\mathbb{S}_{B(z)}^{2}$ ($z=0,1$), $\mathbb{S}_{B(1)-B(0)}^{2}$,
$\mathbb{S}_{\bm{x}\bm{x}}$ and $\mathbb{S}_{B(z),\bm{x}}$ ($z=0,1$) have
finite limiting values, and the limit of $\mathbb{S}_{\bm{x}\bm{x}}$ is non-
singular; (4) $\max_{1\leq i\leq
n}\left|B_{i}(z)-\overline{B}(z)\right|^{2}/n\rightarrow 0$ ($z=0,1$) and
$\max_{1\leq i\leq n}\left\|\bm{x}_{i}\right\|^{2}/n$ $\rightarrow 0$.

Remark. With (4), we have

$\displaystyle\frac{\max_{1\leq i\leq
n}\left(B_{i}(z)-\overline{B}(z)\right)^{2}}{\min(n_{1},n_{0})^{2}\cdot
Var\left(\widehat{\tau}_{B}\right)}$ $\displaystyle=$
$\displaystyle\frac{n^{2}}{\min(n_{1},n_{0})^{2}}\cdot\frac{\max_{1\leq i\leq
n}\left|B_{i}(z)-\overline{B}(z)\right|^{2}/n}{n\left(\mathbb{S}_{B(1)}/n_{1}+\mathbb{S}_{B(0)}/n_{0}-\mathbb{S}_{B(1)-B(0)}/n\right)}$
$\displaystyle=$
$\displaystyle\left(\frac{n}{\min(n_{1},n_{0})}\right)^{2}\cdot\frac{\max_{1\leq
i\leq
n}\left|B_{i}(z)-\overline{B}(z)\right|^{2}/n}{n/n_{1}\cdot\mathbb{S}_{B(1)}+n/n_{0}\cdot\mathbb{S}_{B(0)}-\mathbb{S}_{B(1)-B(0)}}.$

Under assumption (2) of Condition S2, as $n\rightarrow\infty$,
$n/\min(n_{1},n_{0})$, $n/n_{1}$ and $n/n_{0}$ all have a finite limit.
Together with assumptions (3) and (4) of Condition S2, we have (20). Also,
assumptions (1)-(3) of Condition S2 imply assumptions (1)-(3) of Condition S1.
Hence, Condition S2 implies Condition S1, and conclusions obtained under
Condition S1 automatically hold under Condition S2.

## S.1 Propositions and Their Proof

For any set of potential outcomes $\\{(Q_{i}(0),Q_{i}(1)):\ i=1,\cdots,n\\}$,
let $Q_{i}=Q_{i}(Z_{i})$ be the realized outcome under the assigned treatment
arm. For units with $Z_{i}=z$ ($z=0,1$), let $\overline{Q}_{z}$ denote the
sample mean of $Q_{i}=Q_{i}(z)$: $\overline{Q}_{z}=\sum_{i:\
Z_{i}=z}Q_{i}/n_{z}$, and let $S_{Q(z)}^{2}$ denote the sample variance of
$Q_{i}=Q_{i}(z)$:

$S_{Q(z)}^{2}=\frac{1}{n_{z}-1}\sum_{Z_{i}=z}\left(Q_{i}-\overline{Q}_{z}\right)^{2}.$

Also, for any two sets of potential outcomes $\\{(Q_{i}(0),Q_{i}(1)):\
i=1,\cdots,n\\}$ and $\\{(Q_{i}^{\prime}(0),Q_{i}^{\prime}(1)):\
i=1,\cdots,n\\}$, for units with $Z_{i}=z$ ($z=0,1$), let
$S_{Q(z),Q^{\prime}(z)}$ denote the sample covariance between $Q_{i}=Q_{i}(z)$
and $Q^{\prime}_{i}=Q^{\prime}_{i}(z)$:

$S_{Q(z),Q^{\prime}(z)}=\frac{1}{n_{z}-1}\sum_{Z_{i}=z}\left(Q_{i}-\overline{Q}_{z}\right)\left(Q_{i}^{\prime}-\overline{Q}^{\prime}_{z}\right).$

Finally, for units with $Z_{i}=z$ ($z=0,1$), let $\bm{S}_{Q(z),\bm{x}}$
($z=0,1$) denote the vector of sample covariances between $Q_{i}=Q_{i}(z)$ and
$\bm{x}_{i}$:

$\bm{S}_{Q(z),\bm{x}}=\frac{1}{n_{z}-1}\sum_{Z_{i}=z}\left(Q_{i}-\overline{Q}_{z}\right)\left(\bm{x}_{i}-\overline{\bm{x}}_{z}\right)^{\top},$

where $\overline{\bm{x}}_{z}=\sum_{i:\ Z_{i}=z}\bm{x}_{i}/n_{z}$.

###### Proposition S1

The confidence interval in (8),

$\widehat{\tau}^{Wald}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}(\widehat{\tau}_{B})}/\widehat{\tau}_{W},$

is the same as the super-population confidence interval for
$\tau_{CACE}^{pop}$ obtained by the delta method.

### S.1.1 Proof of Proposition S1:

For units with $Z_{i}=z$ ($z=0,1$), let $S_{Y(z)}^{2}$ be the sample variance
of $Y_{i}$, $S_{W(z)}^{2}$ the sample variance of $W_{i}$, and $S_{Y(z),W(z)}$
the sample covariance between $Y_{i}$ and $W_{i}$. Because
$\widehat{B}_{i}=Y_{i}-\widehat{\tau}^{Wald}W_{i}=Y_{i}-(\widehat{\tau}_{Y}/\widehat{\tau}_{W})W_{i}$,
we have

$S_{\widehat{B}(z)}^{2}=S_{Y(z)}^{2}+\frac{\widehat{\tau}_{Y}^{2}}{\widehat{\tau}_{W}^{2}}S_{W(z)}^{2}-2\frac{\widehat{\tau}_{Y}}{\widehat{\tau}_{W}}S(Y(z),W(z)).$

Combining this with (7), we have

$\displaystyle\widehat{Var}(\widehat{\tau}_{B})=$
$\displaystyle\left(\frac{S_{Y(1)}^{2}}{n_{1}}+\frac{S_{Y(0)}^{2}}{n_{0}}\right)+\frac{\widehat{\tau}_{Y}^{2}}{\widehat{\tau}_{W}^{2}}\left(\frac{S_{W(1)}^{2}}{n_{1}}+\frac{S_{W(0)}^{2}}{n_{0}}\right)$
$\displaystyle-2\frac{\widehat{\tau}_{Y}}{\widehat{\tau}_{W}}\left(\frac{S_{Y(1),W(1)}}{n_{1}}+\frac{S_{Y(0),W(0)}}{n_{0}}\right).$

To construct confidence interval for $\tau_{CACE}^{pop}$, according to the
delta method (see e.g. Imbens and Rubin, (2015), Ch. 23), in large samples,
$\widehat{\tau}^{Wald}$ is approximately normally distributed with mean
$\tau_{CACE}^{pop}$ and variance

$Var_{sp}\left(\widehat{\tau}^{Wald}\right)=\frac{1}{\tau_{W}^{2}}\left(Var_{sp}(\widehat{\tau}_{Y})+\frac{\tau_{Y}^{2}}{\tau_{W}^{2}}Var_{sp}(\widehat{\tau}_{W})-2\frac{\tau_{Y}}{\tau_{W}}Cov_{sp}(\widehat{\tau}_{Y},\widehat{\tau}_{W})\right),$

where the subscript “sp” indicates that the variances and covariance are for
super-population. Since units are assumed to be independently and randomly
sampled from the super-population, it is easy to see that the estimate for the
super-population variance of $\widehat{\tau}_{Y}$ is

$\widehat{Var}_{sp}(\widehat{\tau}_{Y})=\widehat{Var}_{sp}(\overline{Y}_{1}-\overline{Y}_{0})=\frac{S_{Y(1)}^{2}}{n_{1}}+\frac{S_{Y(0)}^{2}}{n_{0}}.$

Similarly, the estimates for the super-population variance of
$\widehat{\tau}_{W}$ and the super-population covariance between
$\widehat{\tau}_{Y}$ and $\widehat{\tau}_{W}$ are

$\displaystyle\widehat{Var}_{sp}(\widehat{\tau}_{W})$
$\displaystyle=\frac{S_{W(1)}^{2}}{n_{1}}+\frac{S_{W(0)}^{2}}{n_{0}},$
$\displaystyle Cov_{sp}(\widehat{\tau}_{Y},\widehat{\tau}_{W})$
$\displaystyle=\frac{S_{Y(1),W(1)}}{n_{1}}+\frac{S_{Y(0),W(0)}}{n_{0}}.$

Further with $\widehat{\tau}_{Y}$ and $\widehat{\tau}_{W}$ as the estimates
for $\tau_{Y}$ and $\tau_{W}$, it is easy to see that the estimate of
$Var_{sp}\left(\widehat{\tau}^{Wald}\right)$ can be written as
$\widehat{Var}_{sp}\left(\widehat{\tau}^{Wald}\right)=\widehat{Var}(\widehat{\tau}_{B})/\widehat{\tau}_{W}^{2}$.
The super-population confidence interval for $\tau_{CACE}^{pop}$,
$\widehat{\tau}^{Wald}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}_{sp}\left(\widehat{\tau}^{Wald}\right)}$,
is thus the same as that in (8).

###### Proposition S2

For any set of quantities $\\{Q_{i}:\ i=1\ldots,n\\}$, the coefficient on
$Z_{i}$ (denoted by $\widehat{\tau}_{Q}^{Reg}$) in the linear projection of
$Q_{i}$ on $Z_{i}$, $\bm{x}_{i}$ and $Z_{i}\bm{x}_{i}$ using all $n$ units
equals to

$\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(Q_{i}-\widehat{\bm{\beta}}_{Q,1}^{\top}\bm{x}_{i}\right)-\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(Q_{i}-\widehat{\bm{\beta}}_{Q,0}^{\top}\bm{x}_{i}\right).$

### S.1.2 Proof of Proposition S2:

In the linear projection of $Q_{i}$ on $Z_{i}$, $\bm{x}_{i}$ and
$Z_{i}\bm{x}_{i}$, the intercept and the slope coefficients are

$\displaystyle\left(\widehat{\beta}_{0},\widehat{\tau}_{Q}^{Reg},\widehat{\bm{\beta}}_{\bm{x}},\widehat{\bm{\beta}}_{Z\bm{x}}\right)$
(21) $\displaystyle=$
$\displaystyle\text{argmin}_{\beta_{0},\tau_{Q}^{Reg},\bm{\beta}_{\bm{x}},\bm{\beta}_{Z\bm{x}}}\sum_{i=1}^{n}\left(Q_{i}-\beta_{0}-\tau_{Q}^{Reg}Z_{i}-\bm{\beta}_{\bm{x}}^{\top}\bm{x}_{i}-\bm{\beta}_{Z\bm{x}}^{\top}(Z_{i}\bm{x}_{i})\right)^{2}$
$\displaystyle=$
$\displaystyle\text{argmin}_{\beta_{0},\tau_{Q}^{Reg},\bm{\beta}_{\bm{x}},\bm{\beta}_{Z\bm{x}}}\left(\sum_{i:\
Z_{i}=0}\left(Q_{i}-\beta_{0}-\bm{\beta}_{\bm{x}}^{\top}\bm{x}_{i}\right)^{2}+\right.$
$\displaystyle\left.\sum_{i:\
Z_{i}=1}\left(Q_{i}-\beta_{0}-\tau_{Q}^{Reg}-\bm{\beta}_{\bm{x}}^{\top}\bm{x}_{i}-\bm{\beta}_{Z\bm{x}}^{\top}\bm{x}_{i}\right)^{2}\right)$

In order to minimize the part in the big parenthesis in (21), we can take the
following two steps: (1) find $\widehat{\beta}_{0}$ and
$\widehat{\bm{\beta}}_{\bm{x}}$ that minimize the first term in the big
parenthesis, $\sum_{i:\
Z_{i}=0}\left(Q_{i}-\beta_{0}-\bm{\beta}_{\bm{x}}^{\top}\bm{x}_{i}\right)^{2}$;
(2) fix $\widehat{\beta}_{0}$ and $\widehat{\bm{\beta}}_{\bm{x}}$, and find
$\widehat{\tau}_{Q}^{Reg}$ and $\widehat{\bm{\beta}}_{Z\bm{x}}$ that minimize
the second term in the big parenthesis,
$\sum_{i:Z_{i}=1}\left(Q_{i}-\widehat{\beta}_{0}-\tau_{Q}^{Reg}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}-\bm{\beta}_{Z\bm{x}}^{\top}\bm{x}_{i}\right)^{2}$.
From step 1, it is easy to see that
$\widehat{\bm{\beta}}_{Q,0}=\widehat{\bm{\beta}}_{\bm{x}}$. Step 2 is
equivalent to finding $\widehat{\alpha}_{Q,1}$ and
$\widehat{\bm{\beta}}_{Q,1}$ that minimize
$\sum_{i:Z_{i}=1}\left(Q_{i}-\alpha_{Q,1}-\bm{\beta}_{Q,1}^{\top}\bm{x}_{i}\right)^{2}$,
with $\widehat{\alpha}_{Q,1}=\widehat{\beta}_{0}+\widehat{\tau}_{Q}^{Reg}$ and
$\widehat{\bm{\beta}}_{Q,1}=\widehat{\bm{\beta}}_{\bm{x}}+\widehat{\bm{\beta}}_{Z\bm{x}}$.

With the linear projection, we can write

$Q_{i}=\widehat{\beta}_{0}+\widehat{\tau}_{Q}^{Reg}Z_{i}+\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}+\widehat{\bm{\beta}}_{Z\bm{x}}^{\top}(Z_{i}\bm{x}_{i})+\widehat{\varepsilon}_{Q,i}.$

For units with $Z_{i}=0$,
$\widehat{\varepsilon}_{Q,i}=Q_{i}-\widehat{\beta}_{0}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}$;
for units with $Z_{i}=1$,
$\widehat{\varepsilon}_{Q,i}=Q_{i}-\widehat{\beta}_{0}-\widehat{\tau}_{Q}^{Reg}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}-\widehat{\bm{\beta}}_{Z\bm{x}}\bm{x}_{i}$.
Due to properties of the linear projection, $\widehat{\varepsilon}_{Q,i}$ is
orthogonal to the constant 1, and also orthogonal to $Z_{i}$. Therefore
$\sum_{i=1}^{n}\widehat{\varepsilon}_{Q,i}=0$ and
$\sum_{i=1}^{n}Z_{i}\widehat{\varepsilon}_{Q,i}=0$. Because
$\sum_{i=1}^{n}\widehat{\varepsilon}_{Q,i}=\sum_{i:\
Z_{i}=0}\widehat{\varepsilon}_{Q,i}+\sum_{i:\
Z_{i}=1}\widehat{\varepsilon}_{Q,i}$ and
$\sum_{i=1}^{n}Z_{i}\widehat{\varepsilon}_{Q,i}=\sum_{i:\
Z_{i}=1}\widehat{\varepsilon}_{Q,i}$, we have $\sum_{i:\
Z_{i}=0}\widehat{\varepsilon}_{Q,i}=\sum_{i:\
Z_{i}=1}\widehat{\varepsilon}_{Q,i}=0$.

Plugging in the expressions of $\widehat{\varepsilon}_{Q,i}$ for units with
$Z_{i}=0$ and $Z_{i}=1$, we have

$\displaystyle\sum_{i:\
Z_{i}=0}\left(Q_{i}-\widehat{\beta}_{0}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}\right)=0$
$\displaystyle\implies$
$\displaystyle\widehat{\beta}_{0}=\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(Q_{i}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}\right)=\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(Q_{i}-\widehat{\bm{\beta}}_{Q,0}^{\top}\bm{x}_{i}\right)$

and

$\displaystyle\sum_{i:\
Z_{i}=1}\left(Q_{i}-\widehat{\beta}_{0}-\widehat{\tau}_{Q}^{Reg}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}-\widehat{\bm{\beta}}_{Z\bm{x}}^{\top}\bm{x}_{i}\right)=0$
$\displaystyle\implies$
$\displaystyle\widehat{\beta}_{0}+\widehat{\tau}_{Q}^{Reg}=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(Q_{i}-\widehat{\bm{\beta}}_{\bm{x}}^{\top}\bm{x}_{i}-\widehat{\bm{\beta}}_{Z\bm{x}}^{\top}\bm{x}_{i}\right)=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(Q_{i}-\widehat{\bm{\beta}}_{Q,1}^{\top}\bm{x}_{i}\right)$

Hence $\widehat{\tau}_{Q}^{Reg}=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(Q_{i}-\widehat{\bm{\beta}}_{Q,1}^{\top}\bm{x}_{i}\right)-\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(Q_{i}-\widehat{\bm{\beta}}_{Q,0}^{\top}\bm{x}_{i}\right)$.

## S.2 Proof of Theorems

### S.2.1 Proof of Theorem 1

Let $\mathbb{S}^{2}_{W(z)}$ ($z=0,1$) denote the finite population variance of
$W_{i}(z)$, which is bounded since $W_{i}(z)$ is binary. Let
$\mathbb{S}^{2}_{W(1)-W(0)}$ denote the finite population variance of
$W_{i}(1)-W_{i}(0)$. Applying Theorem 3 in Li and Ding, (2017), over all
randomizations, $\widehat{\tau}_{W}$ has mean $\tau_{W}=p_{co}$ and variance
$Var(\widehat{\tau}_{W})=\mathbb{S}^{2}_{W(1)}/n_{1}+\mathbb{S}^{2}_{W(0)}/n_{0}-\mathbb{S}^{2}_{W(1)-W(0)}/n\leq\mathbb{S}^{2}_{W}(1)/n_{1}+\mathbb{S}^{2}_{W}(0)/n_{0}$.
Since $n_{1}/n$ has a limit in $(0,1)$, we have
$\lim_{n\rightarrow\infty}1/n_{1}=\lim_{n\rightarrow\infty}1/n_{0}=0$. By
Chebyshev’s inequality, for any $\epsilon>0$,

$\displaystyle\lim_{n\rightarrow\infty}Pr\left(|\widehat{\tau}_{W}-p_{co}|\geq\epsilon\right)$
$\displaystyle\leq$
$\displaystyle\lim_{n\rightarrow\infty}\frac{Var(\widehat{\tau}_{W})}{\epsilon^{2}}\leq\lim_{n\rightarrow\infty}\frac{1}{\epsilon^{2}}\left(\frac{\mathbb{S}^{2}_{W(1)}}{n_{1}}+\frac{\mathbb{S}^{2}_{W(0)}}{n_{0}}\right)=0,$

that is, $\widehat{\tau}_{W}-p_{co}=o_{p}(1)$. Similarly, given that the limit
superiors of $\mathbb{S}^{2}_{B(0)}$ and $\mathbb{S}^{2}_{B(1)}$ are finite,
we have $\widehat{\tau}_{B}-\tau_{B}=\widehat{\tau}_{B}=o_{p}(1)$ (noting that
$\tau_{B}=0$ from Section 3.2.2).

Combining (2) and (3), we have

$\frac{\widehat{\tau}_{B}}{\widehat{\tau}_{W}}=\widehat{\tau}^{Wald}-\tau_{CACE}.$
(22)

Since $p_{co}$ has a positive limit inferior, $\widehat{\tau}_{W}$ has a
positive limit inferior in probability. Hence
$\widehat{\tau}_{B}/\widehat{\tau}_{W}=o_{p}(1)$, which implies that
$\widehat{\tau}^{Wald}-\tau_{CACE}=o_{p}(1)$.

### S.2.2 Proof of Theorem 2

For $z=0,1$, define

$Var(\widehat{\tau}_{B})^{+}=\frac{\mathbb{S}_{B(1)}^{2}}{n_{1}}+\frac{\mathbb{S}_{B(0)}^{2}}{n_{0}},$

and also define

$\widetilde{Var}(\widehat{\tau}_{B})=\frac{S_{B(1)}^{2}}{n_{1}}+\frac{S_{B(0)}^{2}}{n_{0}},$

where $S_{B(z)}^{2}$ is the sample variance of $B_{i}=B_{i}(z)$ for units with
$Z_{i}=z$. We first prove that
$|\widehat{Var}(\widehat{\tau}_{B})-\widetilde{Var}(\widehat{\tau}_{B})|/Var\left(\widehat{\tau}_{B}\right)^{+}=o_{p}(1)$
as $n\rightarrow\infty$. (Recall that
$\widehat{Var}(\widehat{\tau}_{B})=S_{\widehat{B}(1)}^{2}/n_{1}+S_{\widehat{B}(0)}^{2}/n_{0}$.)

According to the expressions of $B_{i}$ and $\widehat{B}_{i}$,
$\widehat{B}_{i}=B_{i}-\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)W_{i}$.
Let $S_{W(z)}^{2}$ be sample variance of $W_{i}$ for units with $Z_{i}=z$, and
let $S_{B(z),W(z)}$ be the sample covariance between $B_{i}$ and $W_{i}$ for
units with $Z_{i}=z$. We have

$\displaystyle
S_{\widehat{B}(z)}^{2}=S_{B(z)}^{2}-2\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)S_{B(z),W(z)}+\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}S_{W(z)}^{2}$

for $z=0,1$. The absolute difference between
$\widehat{Var}(\widehat{\tau}_{B})$ and $\widetilde{Var}(\widehat{\tau}_{B})$
can be bounded as follows:

$\displaystyle\left|\widehat{Var}(\widehat{\tau}_{B})-\widetilde{Var}(\widehat{\tau}_{B})\right|$
(23) $\displaystyle=$
$\displaystyle\left|\frac{-2\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)S_{B(1),W(1)}+\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}S^{2}_{W(1)}}{n_{1}}\right.$
$\displaystyle\left.+\frac{-2\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)S_{B(0),W(0)}+\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}S^{2}_{W(0)}}{n_{0}}\right|$
$\displaystyle\leq$
$\displaystyle\frac{2\left|\widehat{\tau}^{Wald}-\tau_{CACE}\right|\left|S_{B(1),W(1)}\right|+\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}S^{2}_{W(1)}}{n_{1}}$
$\displaystyle+\frac{2\left|\widehat{\tau}^{Wald}-\tau_{CACE}\right|\left|S_{B(0),W(0)}\right|+\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}S^{2}_{W(0)}}{n_{0}}.$

Because $W_{i}$ is binary, we have

$S_{W(z)}^{2}=\frac{1}{n_{z}-1}\sum_{i:\
Z_{i}=z}(W_{i}-\overline{W}_{z})^{2}\leq\frac{n_{z}}{n_{z}-1}$ (24)

and

$\displaystyle\left|S_{B(z),W(z)}\right|$
$\displaystyle=\left|\frac{1}{n_{z}-1}\sum_{i:\
Z_{i}=z}(B_{i}-\overline{B}_{z})(W_{i}-\overline{W}_{z})\right|$ (25)
$\displaystyle\leq\frac{n_{z}}{n_{z}-1}\max_{i:Z_{i}=z}\left|B_{i}(z)-\overline{B}_{z}\right|$
$\displaystyle\leq\frac{n_{z}}{n_{z}-1}\left(\max_{i:Z_{i}=z}\left|B_{i}(z)-\overline{B}(z)\right|+\left|\overline{B}(z)-\overline{B}_{z}\right|\right)$
$\displaystyle=\frac{n_{z}}{n_{z}-1}\left(\max_{i:Z_{i}=z}\left|B_{i}(z)-\overline{B}(z)\right|+\left|\overline{B}(z)-\frac{1}{n_{z}}\sum_{i:\
Z_{i}=z}B_{i}(z)\right|\right)$
$\displaystyle\leq\frac{n_{z}}{n_{z}-1}\left(\max_{i:Z_{i}=z}\left|B_{i}(z)-\overline{B}(z)\right|+\frac{1}{n_{z}}\sum_{i:Z_{i}=z}\left|\overline{B}(z)-{B}_{i}(z)\right|\right)$
$\displaystyle\leq\frac{2n_{z}}{n_{z}-1}\max_{i:Z_{i}=z}\left|B_{i}(z)-\overline{B}(z)\right|$
$\displaystyle\leq\frac{2n_{z}}{n_{z}-1}\max_{1\leq{}i\leq{}n}\left|B_{i}(z)-\overline{B}(z)\right|$

for $z=0,1$. Plugging (24) and (25) into (23), we have

$\displaystyle\left|\widehat{Var}(\widehat{\tau}_{B})-\widetilde{Var}(\widehat{\tau}_{B})\right|$
(26) $\displaystyle\leq$
$\displaystyle\frac{4\left|\widehat{\tau}^{Wald}-\tau_{CACE}\right|\max_{1\leq
i\leq
n}\left|B_{i}(1)-\overline{B}(1)\right|}{n_{1}-1}+\frac{\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}}{n_{1}-1}$
$\displaystyle+\frac{4\left|\widehat{\tau}^{Wald}-\tau_{CACE}\right|\max_{1\leq
i\leq
n}\left|B_{i}(0)-\overline{B}(0)\right|}{n_{0}-1}+\frac{\left(\widehat{\tau}^{Wald}-\tau_{CACE}\right)^{2}}{n_{0}-1}.$

Substituting (22) into (26) and dividing by
$Var\left(\widehat{\tau}_{B}\right)^{+}$, we have

$\displaystyle\left|\widehat{Var}(\widehat{\tau}_{B})-\widetilde{Var}(\widehat{\tau}_{B})\right|/Var\left(\widehat{\tau}_{B}\right)^{+}$
(27) $\displaystyle\leq$ $\displaystyle\frac{4\max_{1\leq i\leq
n}\left|B_{i}(1)-\overline{B}(1)\right|}{(n_{1}-1)\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}}}\times|\widehat{\tau}_{W}^{-1}|\times\left|\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}}}\right|$
$\displaystyle+\frac{\widehat{\tau}_{W}^{-2}}{n_{1}-1}\left(\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}}}\right)^{2}$
$\displaystyle+\frac{4\max_{1\leq i\leq
n}\left|B_{i}(0)-\overline{B}(0)\right|}{(n_{0}-1)\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}}}\times|\widehat{\tau}_{W}^{-1}|\times\left|\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}}}\right|$
$\displaystyle+\frac{\widehat{\tau}_{W}^{-2}}{n_{0}-1}\left(\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}}}\right)^{2}$
$\displaystyle\leq$ $\displaystyle\frac{8\max_{1\leq i\leq
n}|B_{i}(1)-\overline{B}(1)|}{\min(n_{1},n_{0})\sqrt{Var\left(\widehat{\tau}_{B}\right)}}\times|\widehat{\tau}_{W}^{-1}|\times\left|\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)}}\right|$
$\displaystyle+\frac{8\max_{1\leq i\leq
n}|B_{i}(0)-\overline{B}(0)|}{\min(n_{1},n_{0})\sqrt{Var\left(\widehat{\tau}_{B}\right)}}\times|\widehat{\tau}_{W}^{-1}|\times\left|\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)}}\right|$
$\displaystyle+\frac{4\widehat{\tau}_{W}^{-2}}{\min(n_{1},n_{0})}\left(\frac{\widehat{\tau}_{B}}{\sqrt{Var\left(\widehat{\tau}_{B}\right)}}\right)^{2},$

where the last inequality is derived using the following inequalities:

$\displaystyle\frac{1}{n_{z}-1}\leq\frac{2}{n_{z}}\leq\frac{2}{\min(n_{0},n_{1})},\quad
z=0,1,$ $\displaystyle Var\left(\widehat{\tau}_{B}\right)^{+}\geq
Var\left(\widehat{\tau}_{B}\right).$

(Assumption (2) in Condition S1 implies that $n_{z}\rightarrow\infty$ as
$n\rightarrow\infty$, and hence we can assume $n_{z}\geq 2$, which implies
that $\frac{1}{n_{z}-1}\leq\frac{2}{n_{z}}$.)

Equation (20) in Condition S1 implies that

$\lim_{n\rightarrow\infty}\frac{\max_{1\leq i\leq
n}\left|B_{i}(z)-\overline{B}(z)\right|}{\min(n_{1},n_{0})\cdot\sqrt{Var\left(\widehat{\tau}_{B}\right)}}=0,\quad\text{for
}z=0,1.$

According to assumption (1) in Condition S1, the limit inferior of $p_{co}$ is
positive. Since $\widehat{\tau}_{W}$ is a consistent estimator of $p_{co}$,
$\widehat{\tau}_{W}$ has a positive limit inferior in probability, and
therefore $\widehat{\tau}_{W}^{-1}$ has a positive limit superior in
probability. Assumption (2) in Condition S1 implies that
$\min(n_{1},n_{0})\rightarrow\infty$ as $n\rightarrow\infty$. Under Condition
S1, we can apply Theorem 4 of Li and Ding, (2017) to have
$\widehat{\tau}_{B}/\sqrt{Var\left(\widehat{\tau}_{B}\right)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,1)$ as $n\rightarrow\infty$. We then apply Slutsky’s
theorem and the continuous mapping theorem to conclude that the right hand
side of (27) converges to 0 in probability. Hence
$|\widehat{Var}(\widehat{\tau}_{B})-\widetilde{Var}(\widehat{\tau}_{B})|/Var\left(\widehat{\tau}_{B}\right)^{+}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$ as $n\rightarrow\infty$.

Under Condition S1, we can apply Proposition 1 of Li and Ding, (2017) to have
$\widetilde{Var}(\widehat{\tau}_{B})/Var\left(\widehat{\tau}_{B}\right)^{+}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}1$ as $n\rightarrow\infty$. It follows that
$\widehat{Var}(\widehat{\tau}_{B})/Var(\widehat{\tau}_{B})^{+}$
$\stackrel{{\scriptstyle p}}{{\longrightarrow}}1$ as $n\rightarrow\infty$.
Because $Var\left(\widehat{\tau}_{B}\right)^{+}\geq
Var\left(\widehat{\tau}_{B}\right)$, $\widehat{Var}(\widehat{\tau}_{B})$ is an
asymptotically conservative estimator of $Var\left(\widehat{\tau}_{B}\right)$.
Combining this with the statement that (6) has $(1-\alpha)$ coverage rate for
$\tau_{CACE}$ as $n\rightarrow\infty$, Theorem 2 is proved.

### S.2.3 Proof of Theorem 3

For $z=0,1$, for units with $Z_{i}=z$, let

$\bm{S}_{\bm{x}\bm{x},z}=\frac{1}{n_{z}-1}\sum_{i:Z_{i}=z}(\bm{x}_{i}-\overline{\bm{x}}_{z})(\bm{x}_{i}-\overline{\bm{x}}_{z})^{\top}$

denote the sample covariance matrix for the covariates, with
$\overline{\bm{x}}_{z}=\frac{1}{n_{z}}\sum_{i:Z_{i}=z}\bm{x}_{i}$ being the
mean of covariates for units with $Z_{i}=z$. Let
$\bm{S}_{\bm{x}\bm{x},z}=\bm{C}^{\top}\bm{C}$ be its Cholesky decomposition.
Note that $\widehat{\bm{\beta}}_{W,z}$ is the vector of coefficients on
$\bm{x}$ in the linear projection of $W_{i}=W_{i}(z)$ on $\bm{x}_{i}$ using
units with $Z_{i}=z$. For units with $Z_{i}=z$, the sample variance of the
linear projection $\widehat{\bm{\beta}}_{W,z}^{\top}\bm{x}$, which equals
$\widehat{\bm{\beta}}_{W,z}^{\top}\bm{S}_{\bm{x}\bm{x},z}\widehat{\bm{\beta}}_{W,z}=\widehat{\bm{\beta}}_{W,z}^{\top}\bm{C}^{\top}\bm{C}\widehat{\bm{\beta}}_{W,z}$,
is no larger than the sample variance of $W_{i}=W_{i}(z)$, $S^{2}_{W(z)}$.
Hence

$\displaystyle\left(\widehat{\bm{\beta}}_{W,z}^{\top}\overline{\bm{x}}_{z}\right)^{2}=\left(\left(\widehat{\bm{\beta}}_{W,z}^{\top}\bm{C}^{\top}\right)\left(\bm{C}\bm{S}_{\bm{x}\bm{x},z}^{-1}\overline{\bm{x}}_{z}\right)\right)^{2}$
(28) $\displaystyle\leq$
$\displaystyle\left(\widehat{\bm{\beta}}_{W,z}^{\top}\bm{C}^{\top}\bm{C}\widehat{\bm{\beta}}_{W,z}\right)\left(\overline{\bm{x}}_{z}^{\top}\bm{S}_{\bm{x}\bm{x},z}^{-1}\bm{C}^{\top}\bm{C}\bm{S}_{\bm{x}\bm{x},z}^{-1}\overline{\bm{x}}_{z}\right)$
$\displaystyle\leq$ $\displaystyle\
S^{2}_{W(z)}\left(\overline{\bm{x}}_{z}^{\top}\bm{S}_{\bm{x}\bm{x},z}^{-1}\overline{\bm{x}}_{z}\right)\leq\frac{n_{z}}{n_{z}-1}\overline{\bm{x}}_{z}^{\top}\bm{S}_{\bm{x}\bm{x},z}^{-1}\overline{\bm{x}}_{z},$

where the first inequality is obtained using Cauchy-Schwarz inequality and the
last is obtained using (24).

Let the $k$th covariate of unit $i$ be $x_{i}^{(k)}$, let
$\overline{x}_{z}^{(k)}$ ($z=0,1$) denote the mean of $x_{i}^{(k)}$ for units
with $Z_{i}=z$, and let $\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}$ denote the $k$th
diagonal element of $\mathbb{S}_{\bm{x}\bm{x}}$, that is, the finite
population variance of $x_{i}^{(k)}$. We have

$\displaystyle\frac{\max_{1\leq i\leq
n}\left(x_{i}^{(k)}\right)^{2}}{\min(n_{1},n_{0})\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}}$
$\displaystyle=\frac{\max_{1\leq i\leq
n}\left(x_{i}^{(k)}\right)^{2}}{n}\cdot\frac{n}{\min(n_{1},n_{0})}\cdot\frac{1}{\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}}$
$\displaystyle\leq\frac{\max_{1\leq i\leq
n}\left\|\bm{x}_{i}\right\|^{2}}{n}\cdot\frac{n}{\min(n_{1},n_{0})}\cdot\frac{1}{\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}}.$

Under assumption (4) of Condition S2, $\max_{1\leq i\leq
n}\left\|\bm{x}_{i}\right\|^{2}/n\rightarrow 0$; under assumption (2) of
Condition S2, $n/\min(n_{1},n_{0})$ has a finite limiting value; under
assumption (3) of Condition S2, $1/\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}$ has a
finite limiting value. Hence

$\displaystyle\frac{\max_{1\leq i\leq
n}\left(x_{i}^{(k)}\right)^{2}}{\min(n_{1},n_{0})\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}}\rightarrow
0.$ (29)

Applying Theorem 1 of Li and Ding, (2017), we have that, with (29),
$\overline{x}_{z}^{(k)}/\sqrt{(1/n_{z}-1/n)\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}}$
converges to a standard normal distribution. Assumption (2) of Condition S2
implies that $1/n_{z}-1/n=o_{p}(1)$, together with the assumption that
$\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}$ has a finite limiting value (implied by
assumption (3) of Condition S2), we obtain that
$\overline{x}_{z}^{(k)}=o_{p}(1)$.

With the fact that $\bm{x}_{i}$ can be regarded as potential values of the
covariates under both $Z_{i}=0$ and $Z_{i}=1$, applying Proposition 3 of Li
and Ding, (2017), we have
$\bm{S}_{\bm{x}\bm{x},z}-\mathbb{S}_{\bm{x}\bm{x}}=o_{p}(1)$, where
$\mathbb{S}_{\bm{x}\bm{x}}$ has a finite non-singular limiting value according
to assumption (3) of Condition S2. Therefore, applying Slusky’s theorem
together with the continuous mapping theorem to the right hand side of (28),
we conclude that
$\widehat{\bm{\beta}}_{W,z}^{\top}\overline{\bm{x}}_{z}=o_{p}(1)$. In the
proof of Theorem 1 we have obtained $\widehat{\tau}_{W}-p_{co}=o_{p}(1)$. From
(9) and the conclusions above, we know that
$\widehat{\tau}_{W}^{Reg}-p_{co}=o_{p}(1)+o_{p}(1)+o_{p}(1)=o_{p}(1)$.

Recall that $D_{i}=B_{i}-\widehat{\beta}_{B,Z_{i}}^{\top}\bm{x}_{i}$. Hence

$\displaystyle\widehat{\tau}_{D}$
$\displaystyle=\overline{D}_{1}-\overline{D}_{0}$
$\displaystyle=\overline{B}_{1}-\overline{B}_{0}-\widehat{\bm{\beta}}_{B,1}^{\top}\overline{\bm{x}}_{1}+\widehat{\bm{\beta}}_{B,0}^{\top}\overline{\bm{x}}_{0}$
$\displaystyle=\widehat{\tau}_{B}-\widehat{\bm{\beta}}_{B,1}^{\top}\overline{\bm{x}}_{1}+\widehat{\bm{\beta}}_{B,0}^{\top}\overline{\bm{x}}_{0}.$

Following the proof in Theorem 1,
$\widehat{\tau}_{B}-\tau_{B}=\widehat{\tau}_{B}=o_{p}(1)$. Following arguments
similar as those for proving that
$\widehat{\bm{\beta}}_{W,z}^{\top}\overline{\bm{x}}_{z}=o_{p}(1)$ for $z=0,1$,
we have $\widehat{\bm{\beta}}_{B,z}^{\top}\overline{\bm{x}}_{z}=o_{p}(1)$ for
$z=0,1$. Hence $\widehat{\tau}_{D}=o_{p}(1)+o_{p}(1)+o_{p}(1)=o_{p}(1)$.

Equations (14) and (11) imply that

$\widehat{\tau}^{Reg}-\tau_{CACE}=\frac{\widehat{\tau}_{D}}{\widehat{\tau}_{W}^{Reg}}.$

Since $p_{co}$ has a positive limit inferior and
$\widehat{\tau}_{W}^{Reg}-p_{co}=o_{p}(1)$, $\widehat{\tau}_{W}^{Reg}$ has a
positive limit inferior in probability. With $\widehat{\tau}_{D}=o_{p}(1)$, we
have $\widehat{\tau}_{D}/\widehat{\tau}_{W}^{Reg}=o_{p}(1)$. Therefore
$\widehat{\tau}^{Reg}-\tau_{CACE}=o_{p}(1)$ as $n\rightarrow\infty$.

### S.2.4 Proof of Theorem 4

Let $\widetilde{\alpha}_{B,z}$ and $\widetilde{\bm{\beta}}_{B,z}$ ($z=0,1$) be
the intercept and the vector of coefficients on $\bm{x}_{i}$ in the linear
projection of $B_{i}(z)$ on $\bm{x}_{i}$ using all units. That is,

$\left(\widetilde{\alpha}_{B,z},\widetilde{\bm{\beta}}_{B,z}\right)=\arg\min_{\alpha,\bm{\beta}}\sum_{i=1}^{n}\left(B_{i}(z)-\alpha-\bm{\beta}^{\top}\bm{x}_{i}\right)^{2}.$

Define $\widetilde{D}_{i}=B_{i}-\widetilde{\bm{\beta}}_{B,Z_{i}}\bm{x}_{i}$.
Let $\widehat{\tau}_{\widetilde{D}}$ be the difference-in-means for
$\widetilde{D}_{i}$. According to Example 9 in Li and Ding, (2017), under
Condition S2, $\widehat{\tau}_{\widetilde{D}}$ has mean $\tau_{B}$(=0) and is
asymptotically normal, and $\widehat{\tau}_{D}$ has the same asymptotic
distribution with $\widehat{\tau}_{\widetilde{D}}$. Hence,
$Var(\widehat{\tau}_{\widetilde{D}})-Var(\widehat{\tau}_{D})=o_{p}(n^{-1})$
and

$\widehat{\tau}_{D}/\sqrt{Var(\widehat{\tau}_{D})}\stackrel{{\scriptstyle
d}}{{\rightarrow}}N(0,1).$

### S.2.5 Proof of Theorem 5

We divide the proof into three parts.

Part 1. Define
$\widehat{D}_{i}=\widehat{B}_{i}-\widehat{\bm{\beta}}_{\widehat{B},Z_{i}}^{\top}\bm{x}_{i}$.
For units with $Z_{i}=z$ ($z=0,1$), let $S_{\widehat{D}(z)}^{2}$ be the sample
variance of $\widehat{D}_{i}$. Define

$\widehat{Var}(\widehat{\tau}_{D})=\frac{S_{\widehat{D}(1)}^{2}}{n_{1}}+\frac{S_{\widehat{D}(0)}^{2}}{n_{0}}.$

We show that

$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|$
(30)

is an asymptotically conservative $(1-\alpha)$ confidence interval for
$\tau_{CACE}$.

Part 2. We show that

$\widehat{Var}^{EHW}(\widehat{\tau}_{D})=\widehat{Var}(\widehat{\tau}_{D})+o_{p}(n^{-1}).$
(31)

Hence

$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{EHW}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|$

is an asymptotically conservative $(1-\alpha)$ confidence interval for
$\tau_{CACE}$.

Part 3. We show that $\widehat{Var}^{HC2}(\widehat{\tau}_{D})$ and
$\widehat{Var}^{HC3}(\widehat{\tau}_{D})$ are both asymptotically equivalent
to $\widehat{Var}^{EHW}(\widehat{\tau}_{D})$. Therefore
$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{HC2}(\widehat{\tau}_{D})}/\widehat{\tau}_{W}^{Reg}$
and
$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{HC3}(\widehat{\tau}_{D})}/\widehat{\tau}_{W}^{Reg}$
are also asymptotically conservative $(1-\alpha)$ confidence intervals for
$\tau_{CACE}$.

Details of the proofs are as follows.

Proof of Part 1: Let $\widehat{\bm{\beta}}_{\widehat{B},z}$ denote the vector
of coefficients on $\bm{x}_{i}$ in the linear projection of $\widehat{B}_{i}$
on $\bm{x}_{i}$ using units with $Z_{i}=z$. Recall that
$\widehat{B}_{i}=Y_{i}-\widehat{\tau}^{Reg}W_{i}$. Due to the linear
relationship between $\widehat{B}_{i}$ and $(Y_{i},W_{i})$, we have
$\widehat{\bm{\beta}}_{\widehat{B},z}=\widehat{\bm{\beta}}_{Y,z}-\widehat{\tau}^{Reg}\widehat{\bm{\beta}}_{W,z}$.
Define $V_{i}=W_{i}-\widehat{\beta}_{W,Z_{i}}^{\top}\bm{x}_{i}$. For units
with $Z_{i}=z$ ($z=0,1$), let $\overline{V}_{z}$ be the sample mean of
$V_{i}$, $S_{V(z)}^{2}$ be the sample variance of $V_{i}$, $S_{D(z)}^{2}$ the
sample variance of $D_{i}$, and $S_{D(z),V(z)}$ be the sample covariance
between $D_{i}$ and $V_{i}$.

Recall that $B_{i}=Y_{i}-\tau_{CACE}W_{i}$,
$D_{i}=B_{i}-\widehat{\bm{\beta}}_{B,Z_{i}}^{\top}\bm{x}_{i}$ and
$\widehat{\bm{\beta}}_{B,z}=\widehat{\bm{\beta}}_{Y,z}-\tau_{CACE}\widehat{\bm{\beta}}_{W,z}$.
For $z=0,1$,

$\displaystyle\widehat{D}_{i}-D_{i}$
$\displaystyle=\left(\widehat{B}_{i}-B_{i}\right)-\left(\widehat{\bm{\beta}}_{\widehat{B},Z_{i}}-\widehat{\bm{\beta}}_{B,Z_{i}}\right)^{\top}\bm{x}_{i}$
$\displaystyle=-\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)W_{i}+\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)\widehat{\bm{\beta}}_{W,z}^{\top}\bm{x}_{i}$
$\displaystyle=-\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)V_{i}.$

Hence we have

$\displaystyle
S_{\widehat{D}(z)}^{2}=S_{D(z)}^{2}-2\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)S_{D(z),V(z)}+\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)^{2}S_{V(z)}^{2}.$

The absolute difference between $S_{\widehat{D}(z)}^{2}$ and $S_{D(z)}^{2}$
can be bounded by

$\displaystyle\left|S_{\widehat{D}(z)}^{2}-S_{D(z)}^{2}\right|$ (32)
$\displaystyle=$
$\displaystyle\left|-2\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)S_{D(z),V(z)}+\left(\tau_{CACE}-\widehat{\tau}^{Reg}\right)^{2}S_{V(z)}^{2}\right|$
$\displaystyle\leq$ $\displaystyle
2\left|\widehat{\tau}^{Reg}-\tau_{CACE}\right|\left|S_{D(z),V(z)}\right|+\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)^{2}S_{V(z)}^{2}.$

Because $V_{i}$ is the residual of the linear projection of $W_{i}$ on
$\bm{x}_{i}$, we have

$S_{V(z)}^{2}\leq{}S_{W(z)}^{2}\leq\frac{n_{z}}{n_{z}-1},$ (33)

where the second inequality is due to (24). Because $D_{i}$ is the residual of
the linear projection of $B_{i}$ on $\bm{x}_{i}$, we have

$S_{D(z)}^{2}\leq{}S_{B(z)}^{2}.$ (34)

Combining (33) and (34) with Cauchy-Schwarz inequality, we have

$|S_{D(z),V(z)}|\leq\sqrt{S^{2}_{D(z)}S^{2}_{V(z)}}\leq\sqrt{\frac{n_{z}}{n_{z}-1}S_{B(z)}^{2}}.$
(35)

Substituting (33) and (35) into (32), we obtain the following inequality

$\displaystyle\left|S_{\widehat{D}(z)}^{2}-S_{D(z)}^{2}\right|$ (36)
$\displaystyle\leq$ $\displaystyle
2\sqrt{\frac{n_{z}}{n_{z}-1}S_{B(z)}^{2}}\left|\widehat{\tau}^{Reg}-\tau_{CACE}\right|+\frac{n_{z}}{n_{z}-1}\left(\widehat{\tau}^{Reg}-\tau_{CACE}\right)^{2}.$

Applying Proposition 1 of Li and Ding, (2017), we have that under Condition
S2, $S_{B(z)}^{2}-\mathbb{S}_{B(z)}^{2}=o_{p}(1)$, where
$\mathbb{S}_{B(z)}^{2}$ has a finite limiting value. According to Theorem 3,
$\widehat{\tau}^{Reg}-\tau_{CACE}=o_{p}(1)$ as $n\rightarrow\infty$. Applying
the continuous mapping theorem, the right hand side of (36) converges to 0 in
probability, which means $S_{\widehat{D}(z)}^{2}-S_{D(z)}^{2}=o_{p}(1)$ as
$n\rightarrow\infty$.

Define

$\widetilde{Var}(\widehat{\tau}_{D})=\frac{S_{D(1)}^{2}}{n_{1}}+\frac{S_{D(0)}^{2}}{n_{0}}.$

Under assumption (2) of Condition S2, $n_{1}/n$ and $n_{0}/n$ both have a
limit in (0,1). Hence
$\widehat{Var}(\widehat{\tau}_{D})-\widetilde{Var}(\widehat{\tau}_{D})=o_{p}(n^{-1})$.
According to Example 9 in Li and Ding, (2017), the interval

$\widehat{\tau}_{D}\pm\nu_{1-\alpha/2}\sqrt{\widetilde{Var}(\widehat{\tau}_{D})}$

has at least $1-\alpha$ coverage rate for $\tau_{B}=0$ as
$n\rightarrow\infty$. Hence, the interval

$\widehat{\tau}_{D}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}(\widehat{\tau}_{D})}$

also has at least $1-\alpha$ coverage rate for $0$ as $n\rightarrow\infty$.
Following similar arguments to those in Section 4.2, we obtain that an
equivalent statement is that

$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|$
(37)

is an asymptotically conservative $(1-\alpha)$ confidence interval for
$\tau_{CACE}$.

Proof of Part 2: Let $\bm{\Omega}$ denote the $n\times(2(K+1))$ design matrix
in which the $i$th row is
$\bm{\Omega}_{i}^{\top}=(1,Z_{i},\bm{x}_{i},Z_{i}\bm{x}_{i})^{\top}$. Let
$\widehat{u}_{i}$ denote the residual in the OLS regression of
$\widehat{B}_{i}$ on $Z_{i}$, $\bm{x}_{i}$ and $Z_{i}\bm{x}_{i}$ using all
units. The EHW estimator for the covariance matrix of the coefficients is

$\widehat{\bm{V}}^{EHW}=\left(\sum_{i=1}^{n}\bm{\Omega}_{i}\bm{\Omega}_{i}^{\top}\right)^{-1}\left(\sum_{i=1}^{n}\widehat{u}_{i}^{2}\bm{\Omega}_{i}\bm{\Omega}_{i}^{\top}\right)\left(\sum_{i=1}^{n}\bm{\Omega}_{i}\bm{\Omega}_{i}^{\top}\right)^{-1}.$
(38)

$\widehat{Var}^{EHW}(\widehat{\tau}_{D})$ is the diagonal element of
$\widehat{\bm{V}}^{EHW}$ that corresponds to the coefficient for $Z_{i}$, that
is, the second diagonal element of $\widehat{\bm{V}}^{EHW}$.

Denote

$\displaystyle\bm{G}$ $\displaystyle\equiv
n^{-1}\sum_{i=1}^{n}\bm{\Omega}_{i}\bm{\Omega}_{i}^{\top}$ (39)
$\displaystyle=\left(\begin{array}[]{cccc}1&\sum_{i=1}^{n}Z_{i}/n&\sum_{i=1}^{n}\bm{x}_{i}^{\top}/n&\sum_{i=1}^{n}Z_{i}\bm{x}_{i}^{\top}/n\\\
\sum_{i=1}^{n}Z_{i}/n&\sum_{i=1}^{n}Z_{i}^{2}/n&\sum_{i=1}^{n}Z_{i}\bm{x}_{i}^{\top}/n&\sum_{i=1}^{n}Z_{i}^{2}\bm{x}_{i}^{\top}/n\\\
\sum_{i=1}^{n}\bm{x}_{i}/n&\sum_{i=1}^{n}Z_{i}\bm{x}_{i}/n&\sum_{i=1}^{n}\bm{x}_{i}\bm{x}_{i}^{\top}/n&\sum_{i=1}^{n}Z_{i}\bm{x}_{i}\bm{x}_{i}^{\top}/n\\\
\sum_{i=1}^{n}Z_{i}\bm{x}_{i}/n&\sum_{i=1}^{n}Z_{i}^{2}\bm{x}_{i}/n&\sum_{i=1}^{n}Z_{i}\bm{x}_{i}\bm{x}_{i}^{\top}/n&\sum_{i=1}^{n}Z_{i}^{2}\bm{x}_{i}\bm{x}_{i}^{\top}/n\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\bm{G}_{11}&\bm{G}_{12}\\\
\bm{G}_{21}&\bm{G}_{22}\end{array}\right),$

where

$\displaystyle\bm{G}_{11}=\left(\begin{array}[]{cc}1&n_{1}/n\\\
n_{1}/n&n_{1}/n\end{array}\right),\quad\bm{G}_{12}=\left(\begin{array}[]{cc}\bm{0}&n_{1}/n\cdot\overline{\bm{x}}_{1}^{\top}\\\
n_{1}/n\cdot\overline{\bm{x}}_{1}^{\top}&n_{1}/n\cdot\overline{\bm{x}}_{1}^{\top}\end{array}\right)$
$\displaystyle\bm{G}_{21}=\bm{G}_{12}^{\top},\quad\bm{G}_{22}=n^{-1}\sum_{i=1}^{n}\left(\begin{array}[]{cc}\bm{x}_{i}\bm{x}_{i}^{\top}&Z_{i}\bm{x}_{i}\bm{x}_{i}^{\top}\\\
Z_{i}\bm{x}_{i}\bm{x}_{i}^{\top}&Z_{i}\bm{x}_{i}\bm{x}_{i}^{\top}\end{array}\right),$

taking into account that $\sum_{i=1}^{n}Z_{i}=n_{1}$,
$\sum_{i=1}^{n}\bm{x}_{i}/n=\overline{\bm{x}}=\bm{0}$,
$\sum_{i=1}^{n}Z_{i}\bm{x}_{i}/n=n_{1}/n\cdot\sum_{i:\
Z_{i}=1}\bm{x}_{i}/n_{1}=n_{1}/n\cdot\overline{\bm{x}}_{1}$ and
$Z_{i}^{2}=Z_{i}$.

Denote

$\bm{H}\equiv
n^{-2}\sum_{i=1}^{n}\widehat{u}_{i}^{2}\bm{\Omega}_{i}\bm{\Omega}_{i}^{\top}=\left(\begin{array}[]{cc}\bm{H}_{11}&\bm{H}_{12}\\\
\bm{H}_{21}&\bm{H}_{22}\end{array}\right),$

where

$\displaystyle\bm{H}_{11}=n^{-2}\sum_{i=1}^{n}\left(\begin{array}[]{cc}\widehat{u}_{i}^{2}&Z_{i}\widehat{u}_{i}^{2}\\\
Z_{i}\widehat{u}_{i}^{2}&Z_{i}\widehat{u}_{i}^{2}\end{array}\right),\quad\bm{H}_{12}=n^{-2}\sum_{i=1}^{n}\left(\begin{array}[]{cc}\widehat{u}_{i}^{2}\bm{x}_{i}^{\top}&Z_{i}\widehat{u}_{i}^{2}\bm{x}_{i}^{\top}\\\
Z_{i}\widehat{u}_{i}^{2}\bm{x}_{i}^{\top}&Z_{i}\widehat{u}_{i}^{2}\bm{x}_{i}^{\top}\end{array}\right)$
$\displaystyle\bm{H}_{21}=\bm{H}_{12}^{\top},\quad\bm{H}_{22}=n^{-2}\sum_{i=1}^{n}\left(\begin{array}[]{cc}\widehat{u}_{i}^{2}\bm{x}_{i}\bm{x}_{i}^{\top}&Z_{i}\widehat{u}_{i}^{2}\bm{x}_{i}\bm{x}_{i}^{\top}\\\
Z_{i}\widehat{u}_{i}^{2}\bm{x}_{i}\bm{x}_{i}^{\top}&Z_{i}\widehat{u}_{i}^{2}\bm{x}_{i}\bm{x}_{i}^{\top}\end{array}\right)$

According to (38), we have

$\widehat{\bm{V}}^{EHW}=\bm{G}^{-1}\bm{H}\bm{G}^{-1}.$ (40)

In the proof of Theorem 3, we have shown that
$\overline{x}_{z}^{(k)}/\sqrt{(1/n_{z}-1/n)\mathbb{S}_{\bm{x}\bm{x}}^{(k,k)}}$
converges to a standard normal distribution for $k=1,\ldots,K$. Since
$\mathbb{S}_{\bm{x}\bm{x}}$ has a finite non-singular limit and $n_{1}/n$ and
$n_{0}/n$ both have a limit in $(0,1)$, we have
$\overline{\bm{x}}_{z}=O_{p}(n^{-1/2})$ and
$\overline{\bm{x}}_{z}\overline{\bm{x}}_{z}^{\top}=O_{p}(n^{-1})$. According
to the expression of $\bm{G}_{12}$, we have $\bm{G}_{12}=O_{p}(n^{-1/2})$.
Noting that

$\displaystyle\bm{S}_{\bm{x}\bm{x},z}$
$\displaystyle=\frac{1}{n_{z}-1}\sum_{i:\
Z_{i}=z}(\bm{x}_{i}-\overline{\bm{x}}_{z})(\bm{x}_{i}-\overline{\bm{x}}_{z})^{\top}$
$\displaystyle=\frac{1}{n_{z}-1}\left(\sum_{i:\
Z_{i}=z}\bm{x}_{i}\bm{x}_{i}^{\top}-n_{z}\overline{\bm{x}}_{z}\overline{\bm{x}}_{z}^{\top}\right),$

and recalling that
$\bm{S}_{\bm{x}\bm{x},z}=\mathbb{S}_{\bm{x}\bm{x}}+o_{p}(1)$ in the proof of
Theorem 3, we also have

$\displaystyle\sum_{i:\ Z_{i}=z}\bm{x}_{i}\bm{x}_{i}^{\top}/n$
$\displaystyle=(n_{z}-1)/n\cdot\bm{S}_{\bm{x}\bm{x},z}+n_{z}/n\cdot\overline{\bm{x}}_{z}\overline{\bm{x}}_{z}^{\top}$
$\displaystyle=(n_{z}-1)/n\cdot\mathbb{S}_{\bm{x}\bm{x}}+o_{p}(1)+O_{p}(n^{-1})$
$\displaystyle=(n_{z}-1)/n\cdot\mathbb{S}_{\bm{x}\bm{x}}+o_{p}(1).$

Hence, $\sum_{i=1}^{n}\bm{x}_{i}\bm{x}_{i}^{\top}/n=\sum_{i:\
Z_{i}=1}\bm{x}_{i}\bm{x}_{i}^{\top}/n+\sum_{i:\
Z_{i}=0}\bm{x}_{i}\bm{x}_{i}^{\top}/n$ converges in probability to a non-
singular matrix, and
$\sum_{i=1}^{n}Z_{i}\bm{x}_{i}\bm{x}_{i}^{\top}/n=\sum_{i:\
Z_{i}=1}\bm{x}_{i}\bm{x}_{i}^{\top}/n$ also converges in probability to a non-
singular matrix. Using the expression of $\bm{G}_{22}$, we obtain that
$\bm{G}_{22}$ converges in probability to a non-singular matrix. Since
$n_{1}/n$ has a limit in $(0,1)$, $\bm{G}_{11}$ also converges to a non-
singular matrix. Overall, $\bm{G}$ converges in probability to a non-singular
matrix.

Let

$\bm{G}_{diag}=\left(\begin{array}[]{cc}\bm{G}_{11}&\bm{0}\\\
\bm{0}&\bm{G}_{22}\end{array}\right),\quad\bm{G}_{diff}=\left(\begin{array}[]{cc}\bm{0}&\bm{G}_{12}\\\
\bm{G}_{21}&\bm{0}\end{array}\right).$

We have $\bm{G}=\bm{G}_{diag}+\bm{G}_{diff}$. According to previous results,
$\bm{G}_{diff}=O_{p}(n^{-1/2})$ and $\bm{G}_{diag}$ converges in probability
to a non-singular matrix. According to Hua’s identity, if two square matrices
$\bm{M}$ and $\bm{N}$ are non-singular and $\bm{M}-\bm{N}^{-1}$ is also non-
singular, then $(\bm{M}-\bm{N}^{-1})^{-1}-\bm{M}^{-1}$ is non-singular, with
$(\bm{M}-\bm{N}^{-1})^{-1}-\bm{M}^{-1}=(\bm{M}\bm{N}\bm{M}-\bm{M})^{-1}$.
Plugging in $\bm{M}=\bm{G}_{diag}$ and $\bm{N}=-\bm{G}_{diff}^{-1}$, we have

$\bm{G}_{diag}^{-1}-\bm{G}^{-1}=(\bm{G}_{diag}+\bm{G}_{diag}\bm{G}_{diff}^{-1}\bm{G}_{diag})^{-1}=O_{p}(n^{-1/2}).$
(41)

Consider the linear projection of $\widehat{B}_{i}$ on $Z_{i}$, $\bm{x}_{i}$
and $Z_{i}\bm{x}_{i}$. Let $\widehat{\beta}_{0}^{*}$,
$\widehat{\beta}_{Z}^{*}$, $\widehat{\bm{\beta}}_{\bm{x}}^{*}$ and
$\widehat{\bm{\beta}}_{Z\bm{x}}^{*}$ denote respectively the intercept, the
coefficient on $Z_{i}$, the vector of coefficients on $\bm{x}_{i}$ and the
vector of coefficients on $Z_{i}\bm{x}_{i}$. Recall that
$\widehat{\bm{\beta}}_{\widehat{B},z}$ is the vector of coefficients on
$\bm{x}_{i}$ in the linear projection of $\widehat{B}_{i}$ on $\bm{x}_{i}$
using units with $Z_{i}=z$ for $z=0,1$. According to the proof of Proposition
A2, we have
$\widehat{\bm{\beta}}_{\widehat{B},0}=\widehat{\bm{\beta}}_{\bm{x}}^{*}$,
$\widehat{\bm{\beta}}_{\widehat{B},1}=\widehat{\bm{\beta}}_{\bm{x}}^{*}+\widehat{\bm{\beta}}_{Z\bm{x}}^{*}$,
$\sum_{i:\ Z_{i}=0}\widehat{u}_{i}=\sum_{i:\ Z_{i}=1}\widehat{u}_{i}=0$.
Recall that
$\widehat{D}_{i}=\widehat{B}_{i}-\widehat{\bm{\beta}}_{\widehat{B},Z_{i}}^{\top}\bm{x}_{i}$.
According to the definition of $\widehat{u}_{i}$, we have that for units with
$Z_{i}=0$,
$\widehat{u}_{i}=\widehat{B}_{i}-\widehat{\beta}_{0}^{*}-(\widehat{\bm{\beta}}_{\bm{x}}^{*})^{\top}\bm{x}_{i}=\widehat{D}_{i}-\widehat{\beta}_{0}^{*}$,
and that for units with $Z_{i}=1$,
$\widehat{u}_{i}=\widehat{B}_{i}-\widehat{\beta}_{0}^{*}-\widehat{\beta}_{Z}^{*}-(\widehat{\bm{\beta}}_{\bm{x}}^{*})^{\top}\bm{x}_{i}-(\widehat{\bm{\beta}}_{Z\bm{x}}^{*})^{\top}\bm{x}_{i}=\widehat{D}_{i}-\widehat{\beta}_{0}^{*}-\widehat{\beta}_{Z}^{*}$.
Hence,

$\displaystyle\overline{\widehat{D}}_{0}$
$\displaystyle\equiv\frac{1}{n_{0}}\sum_{i:\ Z_{i}=0}\widehat{D}_{i}$
$\displaystyle=\frac{1}{n_{0}}\sum_{i:\
Z_{i}=0}\left(\widehat{u}_{i}+\widehat{\beta}_{0}^{*}\right)=\widehat{\beta}_{0}^{*},$
$\displaystyle\overline{\widehat{D}}_{1}$
$\displaystyle\equiv\frac{1}{n_{1}}\sum_{i:\ Z_{i}=1}\widehat{D}_{i}$
$\displaystyle=\frac{1}{n_{1}}\sum_{i:\
Z_{i}=1}\left(\widehat{u}_{i}+\widehat{\beta}_{0}^{*}+\widehat{\beta}_{Z}^{*}\right)=\widehat{\beta}_{0}^{*}+\widehat{\beta}_{Z}^{*},$
$\displaystyle S_{\widehat{D}(0)}^{2}$
$\displaystyle\equiv\frac{1}{n_{0}-1}\sum_{i:\
Z_{i}=0}\left(\widehat{D}_{i}-\overline{\widehat{D}}_{0}\right)^{2}$
$\displaystyle=\frac{1}{n_{0}-1}\sum_{i:\
Z_{i}=0}\left(\widehat{u}_{i}+\widehat{\beta}_{0}^{*}-\widehat{\beta}_{0}^{*}\right)^{2}=\frac{1}{n_{0}-1}\sum_{i:\
Z_{i}=0}\widehat{u}_{i}^{2},$ $\displaystyle S_{\widehat{D}(1)}^{2}$
$\displaystyle\equiv\frac{1}{n_{1}-1}\sum_{i:\
Z_{i}=1}\left(\widehat{D}_{i}-\overline{\widehat{D}}_{0}\right)^{2}$
$\displaystyle=\frac{1}{n_{1}-1}\sum_{i:\
Z_{i}=1}\left(\widehat{u}_{i}+\widehat{\beta}_{0}^{*}+\widehat{\beta}_{Z}^{*}-\widehat{\beta}_{0}^{*}-\widehat{\beta}_{Z}^{*}\right)^{2}=\frac{1}{n_{1}-1}\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}.$

Therefore we have for $z=0,1$,

$n_{z}^{-1}\sum_{i:\
Z_{i}=z}\widehat{u}_{i}^{2}=n_{z}^{-1}(n_{z}-1)S^{2}_{\widehat{D}(z)}.$ (42)

For $z=0,1$, recall that $S^{2}_{\widehat{D}(z)}-S^{2}_{D(z)}=o_{p}(1)$.
According to Example 9 of Li and Ding, (2017),
$S^{2}_{D(z)}-\mathbb{S}^{2}_{\widetilde{D}(z)}=o_{p}(1)$, where
$\mathbb{S}^{2}_{\widetilde{D}(z)}$ is the population variance of
$\widetilde{D}_{i}(z)=B_{i}(z)-\widetilde{\bm{\beta}}_{B,z}\bm{x}_{i}$ for
$z=0,1$ ($\widetilde{\bm{\beta}}_{B,z}$ ($z=0,1$) is defined at the beginning
of the proof of Theorem 4 to be the vector of coefficients on $\bm{x}_{i}$ in
the linear projection of $B_{i}(z)$ on $\bm{x}_{i}$ using all units).
Combining these results with (42), we have

$n_{z}^{-1}\sum_{i:\
Z_{i}=z}\widehat{u}_{i}^{2}=S^{2}_{D(z)}+o_{p}(1)=\mathbb{S}^{2}_{\widetilde{D}(z)}+o_{p}(1),\quad{}z=0,1.$

Since $\widetilde{D}_{i}(z)$ is the projection residual of $B_{i}(z)$, we have
$\mathbb{S}^{2}_{\widetilde{D}(z)}\leq\mathbb{S}^{2}_{B(z)}$, with
$\mathbb{S}^{2}_{B(z)}$ having a finite limit for $z=0,1$. Under assumption
(2) of Condition S2, $n_{1}/n$ and $n_{0}/n$ both have a limit in $(0,1)$.
Thus, $\sum_{i:\ Z_{i}=z}\widehat{u}_{i}^{2}=O_{p}(n)$ for $z=0,1$. It follows
that $\sum_{i=1}^{n}\widehat{u}_{i}^{2}=\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}+\sum_{i:\ Z_{i}=0}\widehat{u}_{i}^{2}=O_{p}(n)$,
and $\sum_{i=1}^{n}Z_{i}\widehat{u}_{i}^{2}=\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}=O_{p}(n)$. According to the definition of
$\bm{H}_{11}$, we have $\bm{H}_{11}=O_{p}(n^{-1})$.

Assumption (4) of Condition S2 implies that
$\max_{1\leq{}i\leq{}n}|x_{i}^{(k)}|=o_{p}(n^{1/2})$. According to the
definition of $\bm{H}_{12}$, each element of $\bm{H}_{12}$ is either in the
form of $\sum_{i=1}^{n}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}=\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}+\sum_{i:\
Z_{i}=0}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}$, or in the form of
$\sum_{i=1}^{n}Z_{i}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}=\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}$. For $z=0,1$, we have

$\displaystyle\left|\frac{1}{n^{2}}\sum_{i:\
Z_{i}=z}\widehat{u}_{i}^{2}x_{i}^{(k)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\left(\sum_{i:\
Z_{i}=z}\widehat{u}_{i}^{2}\right)\left(\max_{1\leq{}i\leq{}n}|x_{i}^{(k)}|\right)$
$\displaystyle=$ $\displaystyle\frac{1}{n^{2}}\cdot O_{p}(n)\cdot
o_{p}(n^{1/2})=o_{p}(n^{-1/2}).$

Hence $\sum_{i=1}^{n}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}=o_{p}(n^{-1/2})$,
and $\sum_{i=1}^{n}Z_{i}\widehat{u}_{i}^{2}x_{i}^{(k)}/n^{2}=o_{p}(n^{-1/2})$.
Therefore, $\bm{H}_{12}=o_{p}(n^{-1/2})$.

According to the definition of $\bm{H}_{22}$, each element of $\bm{H}_{22}$ is
either in the form of
$\sum_{i=1}^{n}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}=\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}+\sum_{i:\
Z_{i}=0}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}$, or in the form of
$\sum_{i=1}^{n}Z_{i}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}=\sum_{i:\
Z_{i}=1}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}$. For $z=0,1$, we have

$\displaystyle\left|\frac{1}{n^{2}}\sum_{i:\
Z_{i}=z}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\left(\sum_{i:\
Z_{i}=z}\widehat{u}_{i}^{2}\right)\left(\max_{1\leq{}i\leq{}n}|x_{i}^{(k)}|\right)\left(\max_{1\leq{}i\leq{}n}|x_{i}^{(l)}|\right)$
$\displaystyle=$ $\displaystyle\frac{1}{n^{2}}\cdot O_{p}(n)\cdot
o_{p}(n^{1/2})\cdot o_{p}(n^{1/2})=o_{p}(1).$

Hence
$\sum_{i=1}^{n}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}=o_{p}(1)$, and
$\sum_{i=1}^{n}Z_{i}\widehat{u}_{i}^{2}x_{i}^{(k)}x_{i}^{(l)}/n^{2}=o_{p}(1)$.
Therefore, $\bm{H}_{22}=o_{p}(1)$. In summary, we have $\bm{H}=o_{p}(1)$.

With (40), we have

$\displaystyle\widehat{\bm{V}}^{EHW}-\bm{G}_{diag}^{-1}\bm{H}\bm{G}_{diag}^{-1}$
$\displaystyle=$
$\displaystyle\bm{G}^{-1}\bm{H}\bm{G}^{-1}-\bm{G}_{diag}^{-1}\bm{H}\bm{G}_{diag}^{-1}$
$\displaystyle=$
$\displaystyle(\bm{G}^{-1}-\bm{G}_{diag}^{-1})\bm{H}\bm{G}^{-1}+\bm{G}_{diag}^{-1}\bm{H}(\bm{G}^{-1}-\bm{G}_{diag}^{-1})$
$\displaystyle=$
$\displaystyle(\bm{G}^{-1}-\bm{G}_{diag}^{-1})\bm{H}\bm{G}_{diag}^{-1}+\bm{G}_{diag}^{-1}\bm{H}(\bm{G}^{-1}-\bm{G}_{diag}^{-1})$
$\displaystyle+(\bm{G}^{-1}-\bm{G}_{diag}^{-1})\bm{H}(\bm{G}^{-1}-\bm{G}_{diag}^{-1})$

According to (41), we have $\bm{G}^{-1}-\bm{G}_{diag}^{-1}=O_{p}(n^{-1/2})$.
We then have
$(\bm{G}^{-1}-\bm{G}_{diag}^{-1})\bm{H}(\bm{G}^{-1}-\bm{G}_{diag}^{-1})=O_{p}(n^{-1/2})\cdot
o_{p}(1)\cdot O_{p}(n^{-1/2})=o_{p}(n^{-1})$. Since $\bm{G}_{diag}$ is a
block-diagonal matrix, we have

$\bm{G}_{diag}^{-1}=\left(\begin{array}[]{cc}\bm{G}_{11}^{-1}&\bm{0}\\\
\bm{0}&\bm{G}_{22}^{-1}\end{array}\right).$

Hence, we obtain that

$\displaystyle\widehat{\bm{V}}^{EHW}-\bm{G}_{diag}^{-1}\bm{H}\bm{G}_{diag}^{-1}=$
$\displaystyle
O_{p}(n^{-1/2})\left(\begin{array}[]{cc}\bm{H}_{11}\bm{G}_{11}^{-1}&\bm{H}_{12}\bm{G}_{22}^{-1}\\\
\bm{H}_{21}\bm{G}_{11}^{-1}&\bm{H}_{22}\bm{G}_{22}^{-1}\end{array}\right)$
(43)
$\displaystyle+\left(\begin{array}[]{cc}\bm{G}_{11}^{-1}\bm{H}_{11}&\bm{G}_{11}^{-1}\bm{H}_{12}\\\
\bm{G}_{22}^{-1}\bm{H}_{21}&\bm{G}_{22}^{-1}\bm{H}_{22}\end{array}\right)O_{p}(n^{-1/2})+o_{p}(n^{-1})$

Let $\widehat{\bm{V}}^{EHW}_{[2,2]}$ denote the submatrix of
$\widehat{\bm{V}}^{EHW}$ that consists of the first two rows and the first two
columns. It is easy to see that the corresponding submatrix of
$\bm{G}_{diag}^{-1}\bm{H}\bm{G}_{diag}^{-1}$ equals
$\bm{G}_{11}^{-1}\bm{H}_{11}\bm{G}_{11}^{-1}$. Taking the submatrix of (43),
we have

$\displaystyle\widehat{\bm{V}}^{EHW}_{[2,2]}-\bm{G}_{11}^{-1}\bm{H}_{11}\bm{G}_{11}^{-1}=O_{p}(n^{-1/2})\cdot\bm{H}_{11}\bm{G}_{11}^{-1}+\bm{G}_{11}^{-1}\bm{H}_{11}\cdot
O_{p}(n^{-1/2})+o_{p}(n^{-1})$

Since $\bm{G}_{11}$ converges in probability to a non-singular matrix and
$\bm{H}_{11}=O_{p}(n^{-1})$, we obtain that
$\widehat{\bm{V}}^{EHW}_{[2,2]}-\bm{G}_{11}^{-1}\bm{H}_{11}\bm{G}_{11}^{-1}=o_{p}(n^{-1})$.

Plugging in the expressions for $\bm{G}_{11}$ and $\bm{H}_{11}$, we have

$\displaystyle\bm{G}_{11}^{-1}\bm{H}_{11}\bm{G}_{11}^{-1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}1/n_{0}&-1/n_{0}\\\
-1/n_{0}&n/(n_{1}n_{0})\end{array}\right)\sum_{i=1}^{n}\left(\begin{array}[]{cc}\widehat{u}_{i}^{2}&Z_{i}\widehat{u}_{i}^{2}\\\
Z_{i}\widehat{u}_{i}^{2}&Z_{i}\widehat{u}_{i}^{2}\end{array}\right)\left(\begin{array}[]{cc}1/n_{0}&-1/n_{0}\\\
-1/n_{0}&n/(n_{1}n_{0})\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}n_{0}^{-2}\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}&-n_{0}^{-2}\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}\\\
-n_{0}^{-2}\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}&n_{1}^{-2}\sum_{i:Z_{i}=1}\widehat{u}_{i}^{2}+n_{0}^{-2}\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}\end{array}\right).$

By definition, $\widehat{Var}^{EHW}(\widehat{\tau}_{D})$ is the element of
$\widehat{\bm{V}}^{EHW}_{[2,2]}$ that is in the second row and second column.
Hence we have

$\widehat{Var}^{EHW}(\widehat{\tau}_{D})=\frac{1}{n_{1}^{2}}\sum_{i:Z_{i}=1}\widehat{u}_{i}^{2}+\frac{1}{n_{0}^{2}}\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}+o_{p}(n^{-1}).$
(44)

By definition,
$\widehat{Var}(\widehat{\tau}_{D})=S_{\widehat{D}(1)}^{2}/n_{1}+S_{\widehat{D}(0)}^{2}/n_{0}$.
Combining this with (42), we have

$\widehat{Var}(\widehat{\tau}_{D})=\frac{\sum_{i:Z_{i}=1}\widehat{u}_{i}^{2}}{n_{1}(n_{1}-1)}+\frac{\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}}{n_{0}(n_{0}-1)}.$
(45)

Combining this with (44), we have

$\widehat{Var}^{EHW}(\widehat{\tau}_{D})=\widehat{Var}(\widehat{\tau}_{D})-\frac{\sum_{i:Z_{i}=1}\widehat{u}_{i}^{2}}{n_{1}^{2}(n_{1}-1)}-\frac{\sum_{i:Z_{i}=0}\widehat{u}_{i}^{2}}{n_{0}^{2}(n_{0}-1)}+o_{p}(n^{-1}).$
(46)

Recall that $\sum_{Z_{i}=z}\widehat{u}_{i}^{2}=O_{p}(n)$ for $z=0,1$. Also
$n_{1}/n$ and $n_{0}/n$ both have a limit in $(0,1)$. Hence we have

$\widehat{Var}^{EHW}(\widehat{\tau}_{D})=\widehat{Var}(\widehat{\tau}_{D})+o_{p}(n^{-1}).$

Combining this with the statement that (37) is an asymptotically conservative
$(1-\alpha)$ confidence interval for $\tau_{CACE}$, we obtain that

$\widehat{\tau}^{Reg}\pm\nu_{1-\alpha/2}\sqrt{\widehat{Var}^{EHW}(\widehat{\tau}_{D})}/\left|\widehat{\tau}_{W}^{Reg}\right|$

is an asymptotically conservative $(1-\alpha)$ confidence interval for
$\tau_{CACE}$.

Proof of Part 3: Let $h_{i}$ denote the $i$th diagonal element of
$\bm{\Omega}(\bm{\Omega}^{\top}\bm{\Omega})^{-1}\bm{\Omega}^{\top}$. The HC2
estimator for the covariance matrix of the coefficients replaces
$\widehat{u}_{i}$ with $\widehat{u}_{i}/\sqrt{1-h_{i}}$, and the HC3 estimator
for the covariance matrix of the coefficients replaces $\widehat{u}_{i}$ with
$\widehat{u}_{i}/(1-h_{i})$. According to the definition of $\bm{G}$ in (39),
we have $\bm{\Omega}^{\top}\bm{\Omega}=n\bm{G}$. Previously we have proved
that $\bm{G}$ converges in probability to a non-singular matrix. Hence
$(\bm{\Omega}^{\top}\bm{\Omega})^{-1}=O_{p}(n^{-1})$. Also, assumption (4) of
Condition S2 implies that $\bm{x}_{i}=o(n^{1/2})$, and hence
$\bm{\Omega}=o(n^{1/2})$. Therefore,
$h_{i}=o(n^{1/2})\cdot{}O_{p}(n^{-1})\cdot{}o(n^{1/2})=o_{p}(1)$. It follows
that the scale multiplier $1/\sqrt{1-h_{i}}$ in the HC2 estimator and the
scale multiplier $1/(1-h_{i})$ in the HC3 estimator both converge to 1 as
$n\rightarrow\infty$. Hence, $\widehat{Var}^{HC2}(\widehat{\tau}_{D})$ and
$\widehat{Var}^{HC3}(\widehat{\tau}_{D})$ are both asymptotically equivalent
to $\widehat{Var}^{EHW}(\widehat{\tau}_{D})$.

### S.2.6 Proof of Theorem 6

In the proof of Theorem 2 (the last paragraph in that proof), we have shown
$\widehat{Var}(\widehat{\tau}_{B})/Var(\widehat{\tau}_{B})^{+}$
$\stackrel{{\scriptstyle p}}{{\longrightarrow}}1$ as $n\rightarrow\infty$.
Under assumptions (2) and (3) of Condition S2,
$n\cdot{}Var(\widehat{\tau}_{B})^{+}$ has a finite limiting value as
$n\rightarrow\infty$, implying that
$Var(\widehat{\tau}_{B})^{+}=O_{p}(n^{-1})$ and
$\widehat{Var}(\widehat{\tau}_{B})-Var(\widehat{\tau}_{B})^{+}=o_{p}(n^{-1})$.
Furthermore,

$\displaystyle\left|\frac{\sqrt{\widehat{Var}(\widehat{\tau}_{B})}}{|\widehat{\tau}_{W}|}-\frac{\sqrt{Var(\widehat{\tau}_{B})^{+}}}{p_{co}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\sqrt{\widehat{Var}(\widehat{\tau}_{B})}-\sqrt{Var(\widehat{\tau}_{B})^{+}}}{|\widehat{\tau}_{W}|}-\frac{\sqrt{Var(\widehat{\tau}_{B})^{+}}(|\widehat{\tau}_{W}|-p_{co})}{|\widehat{\tau}_{W}|{}p_{co}}\right|$
$\displaystyle\leq$
$\displaystyle\left|\frac{\sqrt{\widehat{Var}(\widehat{\tau}_{B})}-\sqrt{Var(\widehat{\tau}_{B})^{+}}}{|\widehat{\tau}_{W}|}\right|+\left|\frac{\sqrt{Var(\widehat{\tau}_{B})^{+}}(|\widehat{\tau}_{W}|-p_{co})}{|\widehat{\tau}_{W}|{}p_{co}}\right|.$

We also know from the proof of Theorem 1 that
$\widehat{\tau}_{W}-p_{co}=o_{p}(1)$. Under assumption (1) of Condition S2,
$\widehat{\tau}_{W}$ and $\widehat{\tau}_{W}p_{co}$ have positive limit
inferiors in probability as $n\rightarrow\infty$. Hence
$|\widehat{\tau}_{W}|-p_{co}=o_{p}(1)$, and $|\widehat{\tau}_{W}|$ and
$|\widehat{\tau}_{W}|p_{co}$ have positive limit inferiors in probability as
$n\rightarrow\infty$. It follows from Slutsky’s theorem and the continous
mapping theorem that

$\sqrt{\widehat{Var}(\widehat{\tau}_{B})}/|\widehat{\tau}_{W}|-\sqrt{Var(\widehat{\tau}_{B})^{+}}/p_{co}=o_{p}(n^{-1/2}).$
(47)

For $z=0,1$, let $\mathbb{S}^{2}_{\widetilde{D}(z)}$ be the finite population
variance of $\widetilde{D}_{i}(z)$ in treatment arm $z$. As
$\widetilde{D}_{i}(z)$ is the projection residual of $B_{i}(z)$, we have

$\displaystyle\mathbb{S}^{2}_{\widetilde{D}(z)}$
$\displaystyle=\mathbb{S}^{2}_{B(z)}-\mathbb{S}^{2}_{B(z)\mid\bm{x}}$ (48)
$\displaystyle=\mathbb{S}^{2}_{B(z)}-\mathbb{S}_{B(z),\bm{x}}\mathbb{S}_{\bm{x}\bm{x}}^{-1}\mathbb{S}_{B(z),\bm{x}}^{\top}.$

(Recall that $\mathbb{S}^{2}_{B(z)}$ is the finite population variance of
$B_{i}(z)$, $\mathbb{S}^{2}_{B(z)\mid\bm{x}}$ is the finite population
variance of the linear projection of $B_{i}(z)$ on $\bm{x}_{i}$,
$\mathbb{S}_{B(z),\bm{x}}$ is the finite population covariance vector between
$B_{i}(z)$ and $\bm{x}_{i}$, and $\mathbb{S}_{\bm{x}\bm{x}}$ is the finite
population covariance matrix for $\bm{x}_{i}$.) It follows from assumption (3)
of Condition S2 that $\mathbb{S}^{2}_{\widetilde{D}(z)}$ has a finite limiting
value.

According to Example 9 of Li and Ding, (2017),
$S^{2}_{D(z)}-\mathbb{S}^{2}_{\widetilde{D}(z)}=o_{p}(1)$. It is also shown in
the proof of Theorem 5 that $S^{2}_{\widehat{D}(z)}-S^{2}_{D(z)}=o_{p}(1)$.
Hence $S^{2}_{\widehat{D}(z)}-\mathbb{S}^{2}_{\widetilde{D}(z)}=o_{p}(1)$. By
definition,
$\widehat{Var}(\widehat{\tau}_{D})=S_{\widehat{D}(1)}^{2}/n_{1}+S_{\widehat{D}(0)}^{2}/n_{0}$.
As $n_{1}/n$ and $n_{0}/n$ have finite limits in $(0,1)$ according to
assumption (2) of Condition S2, we have

$\widehat{Var}(\widehat{\tau}_{D})-\left(\frac{\mathbb{S}^{2}_{\widetilde{D}(1)}}{n_{1}}+\frac{\mathbb{S}^{2}_{\widetilde{D}(0)}}{n_{0}}\right)=o_{p}(n^{-1}).$

Combining the first line of (48) with (18) and (19), we have

$\frac{\mathbb{S}^{2}_{\widetilde{D}(1)}}{n_{1}}+\frac{\mathbb{S}^{2}_{\widetilde{D}(0)}}{n_{0}}=Var\left(\widehat{\tau}_{B}\right)^{+}-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}.$

Hence

$\widehat{Var}(\widehat{\tau}_{D})-\left(Var\left(\widehat{\tau}_{B}\right)^{+}-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}\right)=o_{p}(n^{-1}).$

Following similar arguments that lead to (47), we have

$\sqrt{\widehat{Var}(\widehat{\tau}_{D})}/|\widehat{\tau}_{W}^{Reg}|-\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}}/p_{co}=o_{p}(n^{-1/2}).$
(49)

With (47) and (49), we have

$\displaystyle
1-\frac{2\sqrt{\widehat{Var}(\widehat{\tau}_{D})}/|\widehat{\tau}_{W}^{Reg}|}{2\sqrt{\widehat{Var}(\widehat{\tau}_{B})}/|\widehat{\tau}_{W}|}$
(50) $\displaystyle=$ $\displaystyle
1-\frac{\sqrt{Var\left(\widehat{\tau}_{B}\right)^{+}-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}}/p_{co}+o_{p}(n^{-1/2})}{\sqrt{Var(\widehat{\tau}_{B})^{+}}/p_{co}+o_{p}(n^{-1/2})}$
$\displaystyle=$ $\displaystyle
1-\sqrt{1-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}/Var(\widehat{\tau}_{B})^{+}}+o_{p}(1)\quad\text{as
}n\rightarrow\infty.$

By definition, the PRIL of (15) compared with (8) is

$1-\frac{2\sqrt{\widehat{Var}^{EHW}(\widehat{\tau}_{D})}/|\widehat{\tau}_{W}^{Reg}|}{2\sqrt{\widehat{Var}(\widehat{\tau}_{B})}/|\widehat{\tau}_{W}|}$

In the proof of Theorem 5, it has been shown that
$\widehat{Var}^{EHW}(\widehat{\tau}_{D})$ is asymptotically equivalent to
$\widehat{Var}(\widehat{\tau}_{D})$. Combining this with (50), we obtain that
the PRIL of (15) has a limit
$1-\sqrt{1-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}/Var(\widehat{\tau}_{B})^{+}}$.
Similarly, the PRIL of (16) or (17) also has a limit
$1-\sqrt{1-Var(\widehat{\tau}_{B\mid\bm{x}})^{+}/Var(\widehat{\tau}_{B})^{+}}$.

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