# A GMM Approach for Non-monotone Missingness on Both Treatment and Outcome
Variables

Shenshen Yang Ma Yinchu School of Economics, Tianjin University. Email:
<EMAIL_ADDRESS>I am very grateful to Jason Abrevaya and Sukjin Han
for their guidance and support, and I would like thank Brendan Kline, Haiqing
Xu, Stephen Donald, Vasiliki Skreta, Daniel Ackerberg, Dean Spears, Stephen
Trejo, Isaiah Andrews, James Brand, Shaofei Jiang, Xue Li, Jiangang Zeng, and
the participants at the UT Austin writing seminar, 2021 Asian Meeting of the
Econometrics Society, 2021 European Winter Meeting of the Econometric Society
for their helpful comments and suggestions.

(This Draft: )

###### Abstract

I examine the common problem of multiple missingness on both the endogenous
treatment and outcome variables. Two types of dependence assumptions for
missing mechanisms are proposed for identification, based on which a two-step
AIPW GMM estimator is proposed. This estimator is unbiased and more efficient
than the previously used estimation methods. Statistical properties are
discussed case by case. This method is applied to the Oregon Health Insurance
Experiment and shows the significant effects of enrolling in the Oregon Health
Plan on improving health-related outcomes and reducing out-of-pocket costs for
medical care. There is evidence that simply dropping the incomplete data
creates downward biases for some of the chosen outcome variables. Moreover,
the estimator proposed in this paper reduced standard errors by 6-24% of the
estimated effects of the Oregon Health Plan.

JEL Numbers: C13, C14, C26, C83

Keywords: Multiple missingness, missing outcome, missing covariates, GMM,
endogeneity.

##  1 Introduction

Missingness in multiple variables is common in practice. This phenomenon is
caused by various reasons, including sub-sampling and a combination of
different data sets; a typical example is missingness in both the endogenous
treatment status and the outcome. This problem usually appears in empirical
works but has not drawn enough attention. Researchers often observe
missingness on both the endogenous treatment and outcome variables in a data
set collected by surveys in both observational studies and field experiments.
One approach that is regularly used by practitioners is to drop all
observations in incomplete data; this is referred to as the complete case (CC)
analysis. It is well known that the CC approach creates an inefficient
estimator, and when the missing mechanism depends on endogenous variables, it
is also biased (Little and Rubin (2002); Qi and Sun (2014)). This paper
proposes consistent and more efficient estimators with multiple missingness,
which allows for the missing mechanism to be endogenous.

When there are two missing variables, the missing patterns can be divided into
monotone missingness and strict non-monotone missingness; both are examples of
a general non-monotone missing pattern.111The general non-monotone missing
pattern also includes univariate missingness. Monotone missingness has been
thoroughly studied in the literature (Tsiatis (2007); Barnwell and Chaudhuri
(2018); Chaudhuri (2020)). It refers to the situation where the missingness of
one indicates the missingness of the other variable. In our framework, this
means that missingness in treatment implies missingness in the outcome. The
monotone missing pattern can be caused by data attrition when once the survey
participants drop out, they never return. However, in survey-collected data,
missing variables can be caused by a broader range of reasons; as a result,
strict non-monotone missingness often appears in a data set, which means that
researchers can observe the outcome status for some observations, despite the
missing treatment status. Survey data frequently suffers from strict non-
monotone missingness; therefore, this paper focuses on strictly non-monotone
missingness and discusses how to generate the proposed approach for more
general missing patterns in the appendix.

There are two sources of endogeneity in the model. First, the full model with
no missing values is endogenous because of the endogenous treatment variable.
Second, the missingness can be endogenous and correlated with latent variables
in the model. The endogeneity in the full model is often addressed using an
exogenous instrumental variable, which is correlated with the endogenous
regressor, but does not directly affect the outcome. This strategy has been
widely used in the two-stage least squares (2SLS) estimation procedure, the
generalized method of moments (GMM) model, and a more general nonparametric IV
model (Theil (1971); Angrist and Imbens (1995); Baum et al. (2003); Newey and
Powell (2003); Newey (2013)). The other source of endogeneity is from
selective missingness. Prior studies in univariate missingness literature have
shown that if the Missing at Random (MAR) assumption is satisfied—meaning that
the missing mechanism is independent with missing values conditional on fully
observed variables—the endogeneity of missingness can be solved (Cheng (1994);
Seaman et al. (2013)). However, the non-monotone missing pattern complicates
this assumption, because simultaneous dependence between multiple missingness
creates challenges in identifying the joint missing mechanism.

However, if some MAR-type of assumptions are satisfied, the missing mechanisms
can still be identified. One simple way to modify the original MAR assumption
is to assume that both missing mechanisms of the treatment and outcome
variables are independent with partially missing variables conditional on some
fully observed variables under every missing type. To keep it consistent, we
still refer to it as MAR assumption. This assumption brings many desired
properties, but can be violated in some cases with multiple stages of missing
variables because it excludes any possible channels through which variables
realized at previous stage can affect the missing mechanism at next stage.
When data are collected sequentially (i.e., the treatment variable is
collected before the outcome variable), the simultaneous dependence
relationship can also be avoided. This happens when the realization of outcome
status takes time and is collected via a later follow-up survey. Motivated by
the sequential data collection process, we propose the alternative identifying
assumption allows for the later-stage missingness (i.e., missingness on the
outcome) to rely on partially observed variables in the previous stage (i.e.,
the partially observed treatment variable), which we describe as the
sequentially updating feature of the missing mechanism; due to this feature,
we name the assumption the Sequential MAR (SMAR) assumption.

Under either identifying assumption, the missing mechanism (i.e., the
propensity of each missing pattern) is identified and utilized in an augmented
inverse propensity weighted (AIPW) GMM estimator for the primary model
coefficients. The moment function is composed of an inverse propensity
weighted (IPW) moment function and an augmentation term chosen to make full
use of the observed data and achieve higher efficiency; this function is
equivalent to a two-step backward AIPW imputation approach, in which the
missing outcome is initially imputed, followed by the imputation of the
missing treatment.

The estimation strategy depends on a first-stage estimation of nuisance
parameters, which are the propensities of missing patterns and models for
incomplete data; we use sieve estimation for the first stage nuisance
parameters. Even though AIPW has been shown to maintain double robustness and
semiparametric efficiency in many cases (e.g., Robins et al. (1994);
Scharfstein et al. (1999); Glynn and Quinn (2010); Chen et al. (2008);
Cattaneo (2010)), the desired properties usually fail under non-monotone
missingness (e.g., Tsiatis (2007); Chaudhuri and Guilkey (2016); Seaman and
Vansteelandt (2018); Sun and Tchetgen Tchetgen (2018)). We show that under the
MAR assumption, the double robustness and semiparametric efficiency properties
are maintained, and this result is consistent with the findings of Chaudhuri
and Guilkey (2016). The estimator is robust under the SMAR assumption, as long
as the missing mechanism is correctly specified; as a result, the asymptotic
variance is affected by the first stage missing mechanism estimation. We
provide asymptotic variance for the estimator under the SMAR assumption and
show that the estimator become more efficient than previously used estimators
by incorporating the incomplete data.

The AIPW-GMM approach is used in the Oregon Health Insurance Experiment to
estimate the effect of enrolling in the Oregon Health Plan (OHP) on health-
related outcomes. The endogenous treatment variable and outcomes are collected
via different-stage follow-up surveys. There are two reasons behind the
missingness: non-response to surveys and non-response to treatment-and-
outcome-related questions among survey responders. There exist participants
who did not respond to the first follow-up survey but responded to the final
survey, and some participants did not answer questions on treatment status;
therefore, we observe non-monotone missing patterns. The data shows evidence
that missing mechanisms of outcome variables are correlated with the
endogenous treatment variable. As a result, the CC analysis yield biased
estimation results. The regression results show significant effects of the OHP
on reducing out-of-pocket costs for medical care, reducing the number of days
when physical health is not good, and improving self-evaluated health
conditions; these results suggest that CC estimators tend to overestimate the
effect, with downward biases for out-of-pocket costs and self-evaluated health
conditions outcomes, while the IPW estimator results are closer to those of
the AIPW estimators. Furthermore, of the three different estimation
strategies, the AIPW-GMM estimators achieve the smallest standard errors for
all estimated coefficients. For the estimated effects of OHP, the AIPW
approach reduces the standard errors by up to 24%. These results are
consistent with the findings in the Monte Carlo simulation.

This paper contributes to the literature in the following aspects. First, it
considers the problem of missing endogenous treatment variables and outcome
variables, which lacks adequate attention. Related literature includes the
partial identification approach developed by Horowitz and Manski (2000), when
there exist multiple missing covariates and outcomes in randomized
experiments. By making assumptions that can be plausible in many cases of
program evaluation, this paper proposes a way to point identify the missing
mechanism and to use it in the construction of a consistent and more efficient
estimator. Second, this study contributes to the literature on non-monotone
missingness (Robins and Gill (1997); Gad (2011); Sun and Tchetgen Tchetgen
(2018); Tchetgen et al. (2018); Sun et al. (2018); Chaudhuri and Guilkey
(2016)) by adapting the sequentially updating feature of the missing
mechanism222Similar sequential feature has only been applied when the missing
pattern is monotone (Chaudhuri (2020)). and allowing the missing mechanism to
depend on a partially observed variable in a non-monotone missing pattern.
Finally, this paper derives variance under the dynamic updating feature and
provides sufficient conditions under which the closed-form efficient influence
function is available under non-monotone missingness. The difficulties in
achieving desired statistical properties hinder the usage of assumptions other
than the MAR assumption, therefore, deriving closed-form variance can be
useful in many scenarios, especially when data are collected via multiple
surveys.

The rest of this paper is organized as follows. Section 2 introduces the
model; Section 3 missing mechanism and the assumptions and gives the
identification result; Section 4 proposes the AIPW moment condition and GMM
estimator based on the assumptions introduced in the previous sections;
Section 5 discusses the asymptotic properties of the AIPW estimator; Section 6
illustrates the performance of the AIPW-GMM estimator through the Monte Carlo
simulation results; Section 7 offers an empirical example using the Oregon
Health Insurance Experiment Data; and Section 8 will provide our conclusions.
Generalization of current method and proofs are listed in the appendix.

##  2 Model and Assumptions

We consider the following model:

$Y_{i}=g(D_{i},X_{i};\beta)+\epsilon_{i}$ (2.1)

where $Y_{i}$ denotes the outcome of individual $i$ and is missing for some
observations; and $D_{i}$ is a partially observed endogenous treatment
variable. For purposes of this study, we only consider one partially missing
endogenous treatment variable. $X_{i}$ is a vector of $K$ fully observed
regressors, and we do not exclude the possibility that $X_{i}$ contains
endogenous variables.

We are interested in consistently estimating the parameter vector $\beta$, and
we assume that there exists a vector of valid instrument variables $Z_{i}$ for
$[D_{i},X_{i}]$ with $d_{Z}\geq d_{D}+d_{X}$, such that

$\displaystyle E[Z_{i}\epsilon_{i}]$ $\displaystyle=0$ (2.2) $\displaystyle
cov(Z_{i},[D_{i},X_{i}])$ $\displaystyle\neq 0$ (2.3)

Equations 2.2 and 2.3 are the exogeneity and relevance conditions, under which
$Z_{i}$ is a valid instrument variable. We assume that $Z_{i}$ plays a role in
determining $[D_{i},X_{i}]$. We do not place structural restrictions on the
first-stage model; therefore, the regressors can be either discrete or
continuous.333If more structures are added to the first stage, we could obtain
additional moment conditions; as an example, a linear specification on $D_{i}$
such that:  $D_{i}=Z_{i}^{\prime}\gamma+u_{i}$ provides an extra moment
condition besides Equation 2.4:  $E[Z_{i}u_{i}]=0$ This extra information
potentially helps to improve the precision and efficiency of the estimator
(Rai 2020). In the full model, the moment condition follows directly from the
exogeneity of $Z_{i}$:

$E[Z_{i}(Y_{i}-g(D_{i},X_{i};\beta))]=0$ (2.4)

Under the standard regularity conditions444The regularity conditions can be
found in Newey and McFadden (1994). for $g$, the parameter value of interest
$\beta^{0}$ is identified by the moment conditions:

$\beta=\beta^{0}\text{ if and only if
}E[Z_{i}(Y_{i}-g(D_{i},X_{i};\beta))]=0\text{ for }\beta\in\mathcal{B}$

We consider an environment where the instrument variables $Z_{i}$ and fully
observed covariates $X_{i}$ are fully observed, but the endogenous treatment
$D_{i}$ and the outcome variable $Y_{i}$ are partially missing. We use
$R_{i}^{D}$ and $R_{i}^{Y}$to indicate observing $D_{i}$ and $Y_{i}$, such
that

$R_{i}^{D}=\begin{cases}1&D_{i}\text{ is observed}\\\ 0&D_{i}\text{ is
unobserved}\end{cases}$ (2.5)

$R_{i}^{Y}=\begin{cases}1&Y_{i}\text{ is observed}\\\ 0&Y_{i}\text{ is
unobserved}\end{cases}$ (2.6)

In the following section, we will discuss the missing patterns and dependence
between $R^{D}$, $R^{Y}$, and the other variables.

##  3 Patterns of Missingness

We first capture the missing mechanism and give two sets of assumptions to
identify the propensity of missingness. The commonly used assumptions for
univariate missing values are the Missing Completely at Random (MCAR) and the
Missing at Random (MAR) assumptions. The MCAR assumes that missingness is
independent from any variable in the data set, and MAR assumes that
missingness is independent from the missing values conditioning on a set of
observables.555These two assumptions can be seen as strong ignorability and
conditional ignorability assumptions. Missing mechanisms that fail the MCAR
and MAR assumptions are called Missing Not at Random (MNAR); these allow the
missingness to be correlated with the unobserved missing values, and therefore
creates difficulties in identifying the missing mechanism. Extra parametric
assumptions are usually imposed when the missing pattern is MNAR (Scharfstein
et al. (1999); Andrea et al. (2001)).

The existence of multiple partially missing variables complicates the missing
mechanism. Based on the features of the missing patterns, prior studies have
described multiple missing patterns as monotone missingness versus non-
monotone missingness. Monotone missingness is defined as the situation where
missingness happens gradually, and missingness in one variable indicates
missingness in other variables. In our framework, it is implied that $Y_{i}$
is missing when $D_{i}$ is missing or vice versa. We focus on the former
possibility, since it is a more reasonable scheme compared to the latter. The
missing mechanism is formally defined as monotone if
$(1-R_{i}^{D})R_{i}^{Y}=0$ almost surely.

For monotone missingness, each stage can be seen as a subsample of the last
step, and the missingness depends on the fully observed variables from the
previous stage. One reason behind this pattern is data attrition; when the
program participants choose to drop out of the survey, they do not return to
the following surveys of their outcomes. Another example is data composed of
multi-phase sampling, from which the researchers choose a subsample of the
previous phase to collect information on, due to budget constraints and survey
design.

Non-monotone missingness is a more general pattern that allows for the
possibility of observing $Y_{i}$ while $D_{i}$ is missing; it is also referred
to as general missingness (Van Buuren (2018)), and includes well-discussed
univariate missingness,666Discussions on univariate missingness on the
regressor can be found in Little (1992), Abrevaya and Donald (2017), Breunig
and Haan (2018); Wang et al. (2010), Groenwold et al. (2012), Abrevaya (2019)
studied on methods dealing with missing outcome variables.  monotone
missingness, and strict non-monotone missingness. Non-monotone missingness
arises in various scenarios other than data attrition, including general
survey non-responses, panel studies in which participants drop out but return
in later surveys, and general sub-sampling in each stage. In Figure 1, we
illustrate the missing patterns included in non-monotone missingness; among
the patterns, strict non-monotone missingness is the most frequently seen and
interesting case, so we will focus on this case.

Figure 1: Examples of monotone missing patterns

With a non-monotone missing pattern, the previous MAR assumption is difficult
to justify (Robins (1997); Robins and Gill (1997); Little and Rubin (2002);
Sun and Tchetgen Tchetgen (2018))777The statistical literature has shown that
for commonly used estimation process, it is difficult to include all features
allowed by the previous MAR assumption. In this paper, we focus more on the
challenge in identification, and show some features allowed by the previous
MAR assumption must be excluded to identify the missing mechanism., and it
creates challenges in identification. The challenge was first introduced by
Robins and Gill (1997) when they considered a case where two partially missing
variables exist and showed that MAR assumption implicitly implies the MCAR
assumption in a logistic model; when there are other fully observed variables,
the traditional MAR becomes a stronger assumption of conditional ignorability
for the same observables. In our framework, because we have two potentially
missing variables in the environment, we can observe four missing patterns in
the data. We use $M$ to denote the four patterns and introduce difficulties in
point identifying the distribution of the missing patterns. We suppress the
index $i$ for the following arguments:

$\displaystyle M$ $\displaystyle=M_{1}:\text{Observe both D and Y}$
$\displaystyle M$ $\displaystyle=M_{2}:\text{Observe D but not Y}$
$\displaystyle M$ $\displaystyle=M_{3}:\text{Observe Y but not D}$
$\displaystyle M$ $\displaystyle=M_{4}:\text{Observe neither D nor Y}$

The MAR assumption assumes mean independence between missingness and missing
values conditioning on the observables under each missing pattern and can be
written as:

$\begin{split}\Pr[M=M_{1}|Z,X,D,Y]=&\Pr[M=M_{1}|Z,X,D,Y]\\\
\Pr[M=M_{2}|Z,X,D,Y]=&\Pr[M=M_{2}|Z,X,D]\\\
\Pr[M=M_{3}|Z,X,D,Y]=&\Pr[M=M_{3}|Z,X,Y]\\\
\Pr[M=M_{4}|Z,X,D,Y]=&\Pr[M=M_{4}|Z,X]\end{split}$ (3.1)

The first equation is intentionally redundant to show that no information is
lost under the pattern $M_{1}$. Equation 3.1 implies Equation 3.2888Derivation
is omiited, and the intuition behind is that each missing pattern depends on
the observed variables under that pattern. Therefore, when $R^{Y}=1$,
$\Pr[R^{D}=1|Z,X,D,Y,R^{Y}=1]$, as a part of the missing mechanism, depends on
$Y$; and when $R^{Y}=0$, $\Pr[R^{D}=1|Z,X,D,Y,R^{Y}=1]$ does not depend on $Y$
anymore. Therefore, $E[R^{D}|Z,X,D,Y,R^{Y}]=E[R^{D}|Z,X,R^{Y}Y,R^{Y}]$; same
argument can be applied on $E[R^{Y}|Z,X,D,Y,R^{D}]$.:

$\begin{split}E[R^{D}|Z,X,D,Y,R^{Y}]=E[R^{D}|Z,X,R^{Y}Y,R^{Y}]\\\
E[R^{Y}|Z,X,D,Y,R^{D}]=E[R^{Y}|Z,X,R^{D}D,R^{D}]\end{split}$ (3.2)

One way to model Equation 3.1 is using the threshold crossing model. To
illustrate how the simultaneous relationship affects identification, we can
rewrite the assumption in Equation 3.2 into the following simultaneous binary
model999The equivalence between Equations 3.2 and 3.3 follows from the
property of binary variables.:

$\begin{split}R^{D}&=1\left[f(Z,X,R^{Y}Y,R^{Y})\geq\mu\right]\\\
R^{Y}&=1\left[g(Z,X,R^{D}D,R^{D})\geq\nu\right]\end{split}$ (3.3)

for some unknown functions $f$ and $g$, with $\mu$ and $\nu$ as uniformly
distributed latent variables. The simultaneous feature in the binary model
creates a problem in the identification of
$\Pr\left[R^{D}=1,R^{Y}=1|Z,X,D,Y\right]$. A similar problem has been widely
discussed in the entry game model, and there is a well-known challenge to
identify joint distribution of the simultaneous bivariate model. Prior studies
provided partial identification strategies for the simultaneous binary model
(e.g., Tamer (2003); Ciliberto and Tamer (2009); Balat and Han (2018)) and
point identification under additional restrictions. Even though identifying
the missing mechanism is an important step to identifying the main model, that
is not our primary interest. As an intermediate step, we want to avoid either
partial identification or too-complex structural assumptions. Therefore, we
provide two sets of ignorability-type assumptions that could be satisfied
under many settings and rule out the simultaneity, and we provide
justifications for these assumptions. These assumptions are used to point
identify the missing mechanism.

###  3.1 Missing at Random Assumption

The first assumption is a stronger version of the MAR used in the univariate
missingness literature by assuming the ignorability of multiple missingness
conditioning on a common set of fully observed variables.101010Prior studies
have implicitly shown that the MAR assumption in a univariate missingness
setting is justified under the strong version of MAR; this can be displayed
using the example from Sun and Tchetgen Tchetgen (2018), except for minor
differences.

######  Assumption MAR.

$\displaystyle R^{D}$ $\displaystyle\perp\\!\\!\\!\perp(D,Y)|Z,X\text{\hskip
28.45274pt}(1)$ $\displaystyle R^{Y}$
$\displaystyle\perp\\!\\!\\!\perp(D,Y)|Z,X,R^{D}\;\;(2)$

We refer to this assumption as the MAR assumption for convenience, following
the name used in prior studies. Chaudhuri and Guilkey (2016) applied similar
assumptions, and one important example is non-monotonically missing
instrumental variables, assuming the missingness was independent with
unobserved values conditioning on fully observed variables.

If there are no endogenous variables in $X$, the MAR assumption can be
interpreted as exogenous missingness, assuming that the joint missingness is
independent with unobserved values conditioning on fully observed exogenous
covariates. Missingness is determined by instrument variables (e.g., random
assignment) and personal characteristics (e.g., age, education, distance to
research center).

######  Example 1.

In some field experiments, experimenters collect data via face-to-face visits.
Missingness is believed to be caused by missed experimenter visits, either
accidentally or by design, instead of self-selection of the participants.
Therefore, missingness on $D$ and $Y$ depends on survey participant
characteristics (e.g., location, gender, age, etc.) and instrument variables
(e.g., random assignment into treatment group), instead of the treatment and
outcome status.

###  3.2 Sequential Missing at Random Assumption

The MAR assumption introduced above is easy to use and interpret, and under
it, the estimator satisfies many desired statistical properties. Therefore,
this type of assumptions are welcomed in statistical literature. However, it
is sometimes violated in applied research. One test is to check the
correlation between the correlation between $Y$ and $R^{D}$ when $Y$ is
observed. When the data are collected from different stages of surveys, it is
highly possible that the treatment variable $D$ realized at the previous stage
plays some roles in affecting the missing mechanism at later stage $R^{Y}$.
The MAR assumption introduced above does not allow this kind of dependence
relation, and it is unclear how does the missing mechanism of outcome variable
depends on $R^{D}$. We therefore propose a different assumption that allows
for dependence between partially observed $D$ and the missingness mechanism on
$Y$, which occurs after the realization of $R^{D}$.

When missingness happens sequentially, the missing process has the analogous
“dynamic updating” feature introduced in Chaudhuri (2020), and this feature is
applied to the monotone missing pattern. Despite the non-monotone pattern,
this dynamic feature is still allowed in our framework. We can therefore make
a Sequential MAR (SMAR) assumption that missingness on $D$ is independent with
unobserved values conditional on the fully observed variables, and missingness
on $Y$ is independent with the unobserved values conditional on fully observed
variables and the partially observed $D$.

The SMAR assumption is formalized below:

######  Assumption SMAR.

111111Both the assumption MAR and the assumption SMAR can be relaxed to
conditional mean independence condition.

$\displaystyle R^{D}$ $\displaystyle\perp\\!\\!\\!\perp(D,Y)|Z,X\hskip
56.9055pt\;(1)$ $\displaystyle R^{Y}$
$\displaystyle\perp\\!\\!\\!\perp(D,Y)|Z,X,R^{D}D,R^{D}\;\;(2)$

and $(2)$ is equivalent to

$\displaystyle R^{Y}$
$\displaystyle\perp\\!\\!\\!\perp((1-R^{D})D,Y)|Z,X,R^{D}D,R^{D}\;\;(2^{\prime})$

The SMAR assumption allows $R^{Y}$ to be correlated with $D$; on the other
hand, $R^{Y}$ can be correlated with $Y$ through the correlation with $D$.
Therefore, it assumes that $R^{Y}$ is independent with $Y$ conditioning on the
fully observed variables assumed in the MAR assumption, as well as $R^{D}D$.
Under the SMAR assumption, we need to take $R^{D}D$ into consideration when
capturing the mechanism behind $R^{Y}$, unlike the MAR assumption; and the
other crucial feature assumed in SMAR is that $D$ does not affect $R^{Y}$ if
$D$ is not observed. In the following, we will provide an example of when
these features hold.

######  Example 2.

In survey-collected data sets, missingness can be due to both attrition and
self-selection. For the self-selected missingness, we make an extra assumption
that survey participants are honest reporters to exclude distortion from
misreporting. Besides the participants whose missing mechanisms are determined
by the fully observed variables, we assume there exists the other group of
participants who choose to report if they have the information. Therefore,
missing indicators $R^{D}$ and $R^{Y}$ can be interpreted as indicators of
awareness, or an effort to acquire the status of $D$ and $Y$.

Knowledge of treatment status $D_{i}$ can be caused by participants’
characteristics, as well as the instrument variables. Age, living area, and
education level can all affect participants’ motivation to acquire their
treatment status. The instrument variables can also affect $R^{D}$. One
classic instrument variable used in the field-experiment literature is the
random assignment of treatment, with non-perfect compliance as the
instrumented endogenous variable. Researchers are likely to visit participants
in the treatment group more than the control group, which might cause lack of
awareness of treatment status for the control group. Moving to the next stage,
if participants have no knowledge of their treatment status (i.e., $R^{D}=0$),
$D$ does not enter their information set and thus does not affect their
motivation to attain knowledge about their outcome status (i.e., $D$ does not
affect $R^{Y}$). This explains the assumed non-correlation between $R^{Y}$ and
$D$ when $R^{D}=0$.

On the other hand, for participants who already have information about $D$,
this enters their information set and plays a role in determining whether or
not they learn their outcome status. This is consistent with the feature that
$R^{Y}$ depends on $D$ when $R^{D}=1$, which is allowed in the SMAR assumption
but not allowed in MAR assumption; as a result, the missing mechanism diverges
for those with and without information about $D$. One example of this is that
people with health insurance tend to care more about their health status
because it affects their premium in the next billing cycle.

Figure 2 illustrates the sequential missing procedure. We use $I_{1}$ to
denote the initial information set, which includes the fully observed
variables. Variables in $I_{1}$ play roles in determining $R^{D}$; when
$R^{D}=1$, $D$ is contained in the information set $I_{2}$, and when
$R^{D}=0$, $D$ is not included in $I_{2}$. $I_{2}$ contains the variables
determining $R^{Y}$. The missing mechanisms of $Y$ diverge at $I_{2}$, and $D$
only affects $R^{Y}$ when $R^{D}=1$. Therefore, the missing mechanism at each
stage diverge across different observed variables in the previous stage.

$I_{1}=\\{Z,X\\}$ $I_{2}=\\{Z,X,D,R^{D}\\}$
$M_{1}:R^{D}=1,R^{Y}=1$observed$R^{Y}=1$$M_{2}:R^{D}=1,R^{Y}=0$missing$R^{Y}=0$observed$R^{D}=1$
$I_{2}=\\{Z,X,R^{D}\\}$
$M_{3}:R^{D}=0,R^{Y}=1$observed$R^{Y}=1$$M_{4}:R^{D}=0,R^{Y}=0$missing$R^{Y}=0$missing$R^{D}=0$
Figure 2: Sequential Missing Mechanism

######  Remark 3.1.

The SMAR and MAR assumption are two parallel assumptions, assuming different
conditioning variables for conditional independence, and we can not tell which
is weaker directly. Depending on characteristics of the data sets, we can
choose the most suitable assumption and achieve identification.

Assumptions MAR and SMAR differ when it comes to the hypothetical missing
mechanism on $R^{Y}$. We define $p_{d}$ and $p_{y}$ as:

$\displaystyle p_{d}$ $\displaystyle\equiv\Pr[R^{D}|Z,X,D,Y]=\Pr[R^{D}|Z,X]$
(3.4) $\displaystyle p_{y}$
$\displaystyle\equiv\Pr[R^{Y}|Z,X,D,Y,R^{D}]=\Pr[R^{Y}|\mathcal{C},R^{D}]$
(3.5)

The second equality in Equation 3.4 holds under either assumption MAR or
assumption SMAR. $\mathcal{C}$ represents the conditioning variables for
ignorability of $R^{Y}$; under the assumption MAR, $\mathcal{C}=(Z,X)$; and
under the assumption SMAR, $\mathcal{C}=(R^{D}D,Z,X)$. The distribution of
missing patterns derives from the multiplication of $p_{d}$ and $p_{y}$. For
each missing pattern $M_{m}$, $m=1,2,3,4$, the probability of observing it is
equal to

$\Pr[M_{m}|Z,X,D,Y]=\Pr[R^{Y}=M_{m}^{Y}|\mathcal{C},R^{D}]\times\Pr[R^{D}=M_{m}^{D}|Z,X]$
(3.6)

for $M_{m}^{D},$ $M_{m}^{Y}$ being the corresponding value of $R^{D}$ and
$R^{Y}$ under missing pattern $M_{m}$.

######  Proposition 3.1.

If either the Assumption MAR or the Assumption SMAR hold, the probability of
missing patterns in 3.6 is identified.

The identification result follows directly from selection on observables.

##  4 AIPW-GMM Estimator

The IPW estimator has been widely applied in the missing data literature
(Rosenbaum and Rubin (1983); Wooldridge (2007); Seaman and White (2013)). It
reweights the sample and inflates the underrepresented subsample due to
missingness to provide a consistent estimator. Compared to the imputation
method (e.g., expectation-maximization, multiple imputations), the IPW
estimator avoids making parametric assumptions on the model for an incomplete
subsample and is easier to compute. The efficiency of the IPW estimator can be
improved by adding an augmentation term, and this approach is referred to as
the Augmented IPW (AIPW) approach. The AIPW estimator takes full advantage of
the data set by incorporating dropped information in the IPW estimation into
the augmentation term and achieves semiparametric efficiency bounds under
certain conditions (Robins et al. (1994); Robins (1997); Carpenter et al.
(2006); Tsiatis (2007); Chen et al. (2008); Glynn and Quinn (2010)).

###  4.1 AIPW Moment Condition

First, we make an overlap assumption:

######  Assumption Overlap.

(a) $p_{d}\in[\kappa_{d},1)$ almost surely in $(Z,X)$

(b) $p_{y}\in[\kappa_{y},1)$ almost surely in $(\mathcal{C},R^{D})$ for
$\kappa_{d}>0,$$\kappa_{y}>0$.

This assumption requires the possibility that each of the four missing
patterns is positive (i.e., the strict non-monotone missing pattern).121212A
weaker version of the overlap assumption is introduced in Section A, when a
more general missing pattern is allowed.

We further introduce the notation of probability from observing both the
treatment and outcome, which is defined as:

$p_{11}(X,Z,D)=\Pr[R^{Y}=1|\mathcal{C},R^{D}=1]\times\Pr[R^{D}=1|Z,X]$

We construct a moment function $m_{aipw}(\beta)$:

$m_{aipw}(\beta)=\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))+\phi(Z,X,R^{D},R^{Y},R^{D}D,R^{Y}Y,\beta)$
(4.1)

The moment function above is composed of an IPW moment function and an
augmenting term $\phi$, which is determined by fully observed variables and
coefficient parameter $\beta$. We set the augmentation term to be:

$\displaystyle\phi$
$\displaystyle=\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$
(4.2)
$\displaystyle=\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[Y-E\left[Y|Z,X\right]\right]$
(4.3)
$\displaystyle\;\;+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
(4.4)
$\displaystyle\;\;+\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$
(4.5)

The augmentation term is composed of observed variables, as well as two sets
of nuisance parameters, the missing mechanism $(p_{d},p_{y},p_{11})$, and the
imputed value for the unobserved model:
$(E\left[g(D,X;\beta)|Z,X\right],E\left[Y|D,Z,X\right],E\left[Y|Z,X\right])$131313The
last two terms in the augmentation term can be re-arranged into the
augmentation term used with monotone missingness, introduced in Chaudhuri
(2020); and the first term is added to take into consideration observations
with $R^{D}=0,$while $R^{Y}=1$.. Identification of the missing mechanisms was
shown in the previous section. The assumptions of MAR or SMAR identifies the
second set of nuisance parameters; under either assumption, these can be
identified by141414Under Assumption MAR, $E\left[Y|Z,X\right]$ can also be
identified by $E\left[Y|Z,X,R^{D}=1,R^{Y}=1\right]$ and
$E\left[Y|Z,X,R^{Y}=1\right]$. :

$\displaystyle
E\left[g(D,X;\beta)|Z,X\right]=E\left[g(D,X;\beta)|Z,X,R^{D}=1\right]$
$\displaystyle E\left[Y|D,Z,X\right]=E\left[Y|D,Z,X,R^{D}=1,R^{Y}=1\right]$
$\displaystyle E\left[Y|Z,X\right]=E\left[Y|Z,X,R^{D}=0,R^{Y}=1\right]$

The moment function above can be interpreted as a two-step AIPW imputation.
With full data, the moment function is:

$m_{full}(\beta)=Z\epsilon=Z\left(Y-g(D,X;\beta)\right)$

The goal is to construct an unbiased estimator of $\epsilon$ under missingness
on $D$ and $Y$. Without partially missing $D$, the AIPW estimator for
$\epsilon$ is written as:

$\epsilon_{aipw}=\frac{R^{Y}}{p_{y}}\left(Y-g(D,X;\beta)\right)+\left(1-\frac{R^{Y}}{p_{y}}\right)\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)$

When $D$ is partially missing, $g(D,X;\theta)$ and $E\left[Y|D,Z,X\right]$ are
not fully observed as functions of $D$. We again use the AIPW strategy on the
functions of $D$ and obtain:

$\displaystyle\epsilon_{aipw}$
$\displaystyle=\frac{R^{Y}}{p_{y}}\left[Y-\left(\frac{R^{D}}{p_{d}}g(D,X;\beta)+(1-\frac{R^{D}}{p_{d}})E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(1-\frac{R^{Y}}{p_{y}}\right)\left[\frac{R^{D}}{p_{d}}\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)+\left(1-\frac{R^{D}}{p_{d}}\right)E\left\\{E\left[Y|D,Z,X\right]-g(D,X;\beta)|Z,X\right\\}\right]$
(4.6)

######  Proposition 4.1.

If $\hat{\epsilon}$ is defined as in 4.6, $m_{aipw}(\beta)=Z\epsilon_{aipw}$.

The moment function is used to construct a moment condition, following the
theorem below:

######  Theorem 4.1.

Suppose Assumption MAR or SMAR, and Assumption Overlap hold,
$E\left[m_{aipw}(\beta)\right]=0$ when $\beta=\beta^{0}$.

The equality in this theorem will be used as the AIPW moment condition.

###  4.2 Estimation Strategy

The estimation procedure is composed of two steps. In the first step, we
construct appropriate estimators for $(p_{d},p_{y},p_{11})$ and
$(E[Y|D,Z,X],E[Y|Z,X],E[g(D,X;\beta)|Z,X])$, and denote the estimators as
$(\hat{p}_{d},\hat{p}_{y},\hat{p}_{11})$ and
$(\hat{E}[Y|D,Z,X],\hat{E}[Y|Z,X],\hat{E}[g(D,X;\beta)|Z,X])$. Estimation
strategies on the nuisance parameters depend on the researcher’s prior belief
about model structures. If the correct specification of the nuisance parameter
is not clear, nonparametric estimation could be applied to avoid models that
are too restrictive. We will apply the series estimation strategy on the
models, and we will show the corresponding asymptotic properties thereof in
the following section.

After constructing the nuisance parameters, we plug them back into the GMM
estimation equation and solve:

$\hat{\beta}_{AIPW-
GMM}=argmin_{\beta}\hat{m}_{aipw,i}(\beta)^{{}^{\prime}}\hat{W}\hat{m}_{aipw,i}$

where

$\hat{m}_{aipw}=\frac{1}{N}\sum_{i=1}^{N}\frac{R_{i}^{D}R_{i}^{Y}}{\hat{p}_{11,i}}Z_{i}\left(Y_{i}-g\left(D_{i},X_{i};\beta\right)\right)+\hat{\phi_{i}}$

$\displaystyle\hat{\phi_{i}}$
$\displaystyle=\left(\frac{R_{i}^{Y}}{\hat{p}_{y,i}}-\frac{R_{i}^{D}R_{i}^{Y}}{\hat{p}_{11,i}}\right)Z_{i}\left[\left(Y_{i}-\hat{E}\left[g(D_{i},X_{i};\beta)|Z_{i},X_{i}\right]\right)-\left(\hat{E}\left[Y_{i}|Z_{i},X_{i}\right]-\hat{E}\left[g(D_{i},X_{i};\beta)|Z_{i},X_{i}\right]\right)\right]$
$\displaystyle\;\;+\left(\frac{R_{i}^{D}}{\hat{p}_{d,i}}-\frac{R_{i}^{D}R_{i}^{Y}}{\hat{p}_{11,i}}\right)Z_{i}\left[\left(\hat{E}\left[Y_{i}|D_{i},Z_{i},X_{i}\right]-g(D_{i},X_{i};\beta)\right)-\left(\hat{E}\left[Y_{i}|Z_{i},X_{i}\right]-\hat{E}\left[g(D_{i},X_{i};\beta)|Z_{i},X_{i}\right]\right)\right]$
$\displaystyle\;\;+\left(1-\frac{R_{i}^{D}R_{i}^{Y}}{\hat{p}_{11,i}}\right)Z_{i}\left(\hat{E}\left[Y_{i}|Z_{i},X_{i}\right]-\hat{E}\left[g(D_{i},X_{i};\beta)|Z_{i},X_{i}\right]\right)$

is the estimator of the augmentation term $\phi$ and $\hat{W}$ is the
estimator of chosen weight matrix $W$.

##  5 Asymptotic Properties

Though the AIPW estimator is known to have the efficient property in many
cases, the efficiency usually fails with non-monotone missingness. One
critical condition to produce closed-form semiparametric efficient bound is
the independence of different missing indicators condition on the same set of
variables (Chaudhuri and Guilkey (2016)); this condition implies that the
asymptotic variances should be different under Assumption MAR and Assumption
SMAR. Under the MAR assumption, desired statistical properties hold such that
the AIPW-GMM estimator maintains the double robustness property and achieves
the efficient semiparametric bound. Under the SMAR assumption, double
robustness fail, and we are not able to prove efficiency; but it shows
efficiency improvement from incorporating observations with missing values.

###  5.1 Asmptotic Properties

We make the standard assumptions for asymptotic normality in the GMM model:

######  Assumption M.

(1)
$(Z_{i},X_{i},D_{i},Y_{i},R_{i}^{D}D_{i},R_{i}^{Y}Y_{i},R_{i}^{D},R_{i}^{Y})$
are i.i.d;

(2)$E\left[m_{aipw}(\beta)\right]$ is differentiable with respect to $\beta\in
int(\mathcal{B})$;

(3) Define $G(\beta)=\frac{\partial}{\partial\beta}E\left[m_{aipw}\right]$,
$G(\beta)$ has full rank at $\beta=\beta^{0}$;

(4) $V_{MAR}$ and $V_{SMAR}$ are bounded and positive semidefinite;

We then derive the variance of the moment functions under different
assumptions, and use these derivations to construct the asymptotic
distribution of the AIPW-GMM estimator. For a semi-parametric GMM estimation,
the variance closely relates to robustness of the estimator, thereby
indicating how the first-stage nuisance parameters estimators affect the
variance of the primary model.

Under the Assumption MAR, the ignorabilities of $R^{D}$ and $R^{Y}$ depend on
the same set of conditioning variables. The assumption is powerful enough that
the double robustness property of AIPW estimator is maintained.

######  Theorem 5.1.

If Assumption MAR and Overlap hold, $E\left[m_{aipw}(\beta)\right]=0$ when
either $(\hat{p}_{d},\hat{p}_{y},\hat{p}_{11})$, or
$(\hat{E}\left[g(D,X;\beta)|X,Z\right],\hat{E}[g(D,X;\beta)|X,Z,Y],\hat{E}\left[Y|X,Z,D\right],\hat{E}\left[Y|X,Z\right])$
are correctly specified.

As a direct result of double robustness, the first-stage estimators of the
nuisance parameters do not affect the variance of the primary model.

######  Theorem 5.2.

Let $V_{MAR}$ denote $Var\left(m_{aipw}\right)$ under assumption MAR and
Overlap, then

$V_{MAR}=E\left[\frac{1}{p_{11}}Var\left(m_{full}|Z,X\right)+E\left[m_{full}|Z,X\right]E\left[m_{full}|Z,X\right]^{{}^{\prime}}\right]-\Delta$

where

$\displaystyle\Delta$
$\displaystyle=Var\left(\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;+Var\left(\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|Y,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;+2E\left[\left(1-\frac{1}{p_{y}}-\frac{1}{p_{d}}+\frac{1}{p_{11}}\right)Cov\left(ZY,E\left[m_{full}|D,Z,X\right]|Z,X\right)\right.$
$\displaystyle\;\;\;\;\;\;\;\left.+\left(\frac{1}{p_{y}}-\frac{1}{p_{11}}\right)Cov\left(Zg(D,X;\beta),ZY|Z,X\right)\right]$

The first term in $V_{MAR}$ is similar to the $\Omega_{\beta}$ that was
introduced in Chen et al. (2008), and $\Delta$ captures improvements in
efficiency when we include the partially observed variables in the moment
function.

Under the Assumption SMAR, the double robustness property no longer holds with
non-monotone missingness. However, the estimator remains robust as long as the
missing mechanism is correctly specified.

######  Theorem 5.3.

If Assumption SMAR and Overlap hold, $E\left[m_{aipw}(\beta)\right]=0$ when
$(\hat{p}_{d},\hat{p}_{y},\hat{p}_{11})$ is correctly specified.

Proof of Theorem 5.3 follows directly from the proof of Theorem 4.1.

The double robustness fails under the SMAR assumption because parts of the
missing mechanisms depend on partially missing variables. As a result, the
consistency of some parameters in the first-step estimation of nuisance
parameters affects the consistency and efficiency of the primary model and has
a direct effect on the variance. Luckily, only the estimator $\hat{p}_{11}$
affects the consistency through the first component in the augmenting term,
and we can therefore construct a correction term following Newey (1994).

######  Theorem 5.4.

Let $V_{SMAR}$ denote $Var\left(m_{aipw}\right)$ under assumption SMAR and
Overlap, then

$V_{SMAR}=E\left[\frac{1}{p_{11}}Var\left(\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|D,Z,X\right)\right]+Var\left(E\left[m_{full}|D,Z,X\right]\right)-\Delta$

where

$\displaystyle\Delta$
$\displaystyle=Var\left(\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;+Var\left((1-p_{d})(\frac{R^{D}R^{Y}}{p_{11}}-1)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\right)$
$\displaystyle\;\;+E\left[\left(\frac{1-p_{d}^{2}}{p_{11}}-\frac{(1-p_{d})^{2}}{p_{01}}\right)Var\left(Y|D,Z,X\right)\right]$
$\displaystyle\;\;+2E\left[(1-p_{d})Cov\left(E\left[m_{full}|D,Z,X\right],E\left[Y|D,Z,X\right]|Z,X\right)\right]$

Let $n$ denote the sample size. Under certain regularity conditions,
$\sqrt{n}$ convergence can be maintained with a non-parametric estimation in
the first stage. We assume the well-established regularity conditions for
Sieve basis functions that are presented in the literature (Newey (1994); Chen
et al. (2008); Cattaneo (2010); Chaudhuri and Guilkey (2016)), and we
construct $\sqrt{n}$ normality.

######  Theorem 5.5.

Let $\hat{E}(w)$ denote a vector of sieve estimation of first-stage nuisance
estimators. For each component $e\in E$, suppose $e$ is a function of $d_{e}$
elements and is $s_{e}$ times differentiable. Let $\eta=1$ for power series
basis; and $\eta=\frac{1}{2}$ for spline basis. Let $K$ denote the terms in
the series estimator, $n$ denote the sample size, and $K=n^{\nu}$ such that:

$4\eta+2<\frac{1}{\nu}<4\frac{s_{e}}{d_{e}}-6\eta$

then,

(1) Under Assumption MAR, Overlap, and M,

$\sqrt{n}(\hat{\beta}-\beta)\rightarrow_{d}N(0,(G^{\prime}WG)^{-1}G^{\prime}WV_{MAR}WG(G^{\prime}WG)^{-1})$

Furthermore, if $W=V_{MAR}^{-1}$,

$\sqrt{n}(\hat{\beta}-\beta)\rightarrow_{d}N(0,(G^{\prime}V_{MAR}^{-1}G)^{-1})$

(2) Under Assumption SMAR, Overlap, and M,

$\sqrt{n}(\hat{\beta}-\beta)\rightarrow_{d}N(0,(G^{\prime}WG)^{-1}G^{\prime}WV_{SMAR}WG(G^{\prime}WG)^{-1})$

Furthermore, if $W=V_{SMAR}^{-1}$,

$\sqrt{n}(\hat{\beta}-\beta)\rightarrow_{d}N(0,(G^{\prime}V_{SMAR}^{-1}G)^{-1})$

The regularity conditions used here are no different from the conditions used
in the work of Cattaneo (2010), Chaudhuri and Guilkey (2016). These conditions
restrict the estimation of nuisance parameters to converge fast enough that
they do not affect the convergence rate of the second-step estimation.

###  5.2 Efficiency

The well-known efficiency property of the AIPW estimator usually fails with
the non-monotone missing pattern (Tsiatis (2007); Chaudhuri and Guilkey
(2016)). We show two sufficient conditions in which the estimator maintains
the efficiency property.

######  Lemma 5.1.

Suppose Assumption SMAR and Overlap hold, and $Y\perp\\!\\!\\!\perp D|Z,X$,
the correction term for $V_{SMAR}$ is equivalent to zero.

######  Theorem 5.6.

Suppose Assumption Overlap, Assumption M, and one of the following assumptions
hold:

(i) Assumption MAR

(ii) Assumption SMAR and $Y\perp\\!\\!\\!\perp D|Z,X$

then for $\beta^{0}$, the asymptotic variance lower bound for
$\sqrt{N}(\hat{\beta}-\beta^{0})$ of any regular estimator $\hat{\beta}$ is
given by $\Omega=(G^{\prime}V_{MAR}^{-1}G)^{-1}.$ An estimator with an
asymptotic variance that is equal to $\Omega$ has the following asymptotic
linear representation:

$\sqrt{N}(\hat{\beta}-\beta^{0})=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\psi(Z_{i},X_{i},R_{i}^{D},R_{i}^{Y},R_{i}^{D}D_{i},R_{i}^{Y}Y_{i})$

where

$\psi(Z,X,R^{D},R^{Y},R^{D}D,R^{Y}Y)=-\Omega^{-1}G^{\prime}V_{MAR}^{-1}m_{aipw}(Z,X,R^{D},R^{Y},R^{D}D,R^{Y}Y;\beta^{0})$

Although there are two sufficient conditions under which semiparametric
efficiency bound holds, condition (ii) only holds when $D$ does not affect the
$Y$ conditional of $(Z,X)$ and is an unreasonable assumption. In the settings
with different missing variables, however, (ii) can be more useful. One
example is when there exists multiple missing treatments at different stages,
and the treatments themselves are independent conditional on fully observed
variables, then the SMAR assumption can be applied and achieves the closed-
form semiparametric efficiency bound; the other example is when the fully
observed variables are $D$ and $X$, and the partially missing variables are
$Z$ and $Y$, and it is reasonable to assume that $Z\perp\\!\\!\\!\perp Y|D,X$.

##  6 Monte Carlo Simulation

The previous results suggest that the AIPW-GMM approach yields consistent and
more efficient results. This section provides numerical evidence of these
properties.

We consider the full model:

$\displaystyle Y_{i}$ $\displaystyle=\alpha D_{i}+\beta
X_{i}+\epsilon_{i}\equiv 0.3D_{i}+0.5X_{i}+\epsilon_{i}$ $\displaystyle D_{i}$
$\displaystyle=1(0.1+0.3Z_{i}+0.1X_{i}\geq u_{i})$

where $D_{i}$ is a single-value endogenous variable; $X_{i}$ is an exogenous
variable; $Z_{i}$ is an instrument variable; and $\epsilon_{i}$ and $u_{i}$
are jointly normally distributed with $\gamma=corr(\epsilon_{i},u_{i})$.
$R^{D}$ is determined by fully observed $Z,X$; and $R^{Y}$ is determined by
$Z,X$ and partially observed $D$. These variables are determined via the
binary model stated below:

$\displaystyle R^{D}$ $\displaystyle=1(p_{d}\geq u_{rd})$ $\displaystyle
R^{Y}$ $\displaystyle=1(p_{y}\geq u_{ry})$

where

$\displaystyle p_{d}$ $\displaystyle=0.2+0.2X+0.3Z$ $\displaystyle p_{y}$
$\displaystyle=0.3-0.05X+0.2Z+0.3R_{D}D$

and $u_{rd}$, $u_{ry}$ are correlated. Since $R^{Y}$ is determined by the
endogenous variable $D$, and is conditional on $Z,X$; $u_{ry}$ is correlated
with $\epsilon$, which is allowed in the SMAR assumption.

Table 1: Monte Carlo Simulation with Different Values for $corr(\epsilon,u)$ | $\alpha=0.3$ | $\beta=0.5$
---|---|---
| $\hat{\alpha}$ | Mean Bias | RMSE | $\hat{\beta}$ | Mean Bias | RMSE
N = 1000, R = 500, $corr(\epsilon,u)=0.8$
Complete Case | 0.1842 | 0.1158 | 0.1537 | 0.4973 | 0.0027 | 0.1180
IPW | 0.3223 | 0.0223 | 0.1255 | 0.5017 | 0.0017 | 0.1901
AIPW | 0.3246 | 0.0246 | 0.0878 | 0.4798 | 0.0202 | 0.0863
N = 1000, R = 500, $corr(\epsilon,u)=0.5$
Complete Case | 0.2178 | 0.0822 | 0.1405 | 0.4949 | 0.0051 | 0.1252
IPW | 0.2974 | 0.0026 | 0.1285 | 0.5050 | 0.0050 | 0.1803
AIPW | 0.2993 | 0.0007 | 0.0932 | 0.4903 | 0.0097 | 0.1144
N = 1000, R = 500, $corr(\epsilon,u)=0.3$
Complete Case | 0.2552 | 0.0448 | 0.1162 | 0.4992 | 0.0008 | 0.1293
IPW | 0.3136 | 0.0136 | 0.1498 | 0.5405 | 0.0405 | 0.3948
AIPW | 0.3028 | 0.0028 | 0.0915 | 0.4792 | 0.0208 | 0.0919

Table 1 shows the simulation results with different values for the correlation
between $\epsilon$ and $u$. Because $D$ affects $R^{Y}$, and $D$ is correlated
with $\epsilon$ when it is an endogenous treatment variable, this implies that
$R^{Y}$ is also endogenous in the sense that $R^{Y}$ is correlated with
$\epsilon$. The higher the correlation between $\epsilon$ and $u$ is, the more
endogenous $R^{Y}$ is; the endogeneity of $R^{Y}$ will affect the consistency
of the estimator derived from the complete case analysis because the moment
condition no longer holds. As is shown in the Table 1, the complete case
estimator for $\alpha$ is more biased when $corr(\epsilon,u)$ is higher, while
the other two estimators remain consistent. The other finding is that in all
exercises, the AIPW estimators have smaller RMSE compared to the other
estimation strategies. The IPW estimators have higher RMSE than the CC
estimator in some cases, because they not only drop all the observations with
incomplete data, but also run a nonparametric estimation with the limited data
on the nuisance parameters.

Table 2: Monte Carlo Simulation with Misspecified Imputed Values | $\alpha=0.3$ | $\beta=0.5$
---|---|---
| $\hat{\alpha}$ | Mean Bias | RMSE | $\hat{\beta}$ | Mean Bias | RMSE
N = 1000, R = 500, misspecified $E[Y|Z,X],E[Y|D,Z,X]$
Complete Case | 0.1840 | 0.1160 | 0.1523 | 0.4904 | 0.0096 | 0.1253
IPW | 0.3277 | 0.0277 | 0.1491 | 0.5344 | 0.0344 | 0.1039
AIPW | 0.3067 | 0.0067 | 0.0857 | 0.4812 | 0.0188 | 0.0867
N = 1000, R = 500, misspecified $E[D|Z,X]$
Complete Case | 0.2178 | 0.0822 | 0.1405 | 0.4949 | 0.0051 | 0.1252
IPW | 0.2997 | 0.0003 | 0.1295 | 0.5076 | 0.0076 | 0.1714
AIPW | 0.2990 | 0.0010 | 0.0923 | 0.4925 | 0.0075 | 0.1145
N = 1000, R = 500, misspecified $p_{y}$
Complete Case | 0.1716 | 0.1284 | 0.1665 | 0.4914 | 0.0086 | 0.1206
IPW | 0.1705 | 0.1295 | 0.1674 | 0.5035 | 0.0035 | 0.1416
AIPW | 0.3649 | 0.0649 | 0.1166 | 0.4716 | 0.0284 | 0.1324

Table 2 shows the simulation results when the imputed values
$\hat{E}[Y|Z,X],\hat{E}[Y|D,Z,X]$, or $\hat{E}[D|Z,X]$ are misspecified. The
AIPW-GMM estimator was shown to be robust when the missing mechanisms were
correctly specified in Section 5, and misspecified imputed values should not
affect the performance of the estimator. We can observe some evidence of the
theoretical result on robustness in Table 2. The final exercise illustrates
how the AIPW estimator can be biased when $\hat{p}_{y}$ is misspecified and is
captured by a function of $Z,X$ without the partially observed $D$; this is
the result when the missing mechanism is wrongly specified, and also when the
correlation between the treatment status and the missing mechanism of the
outcome is ignored.

##  7 Application

The Oregon Health Insurance Experiment is a large scale social experiment, for
which most of the data were collected via surveys. In 2008, a group of low-
income individuals was randomly selected for the opportunity to apply for the
Oregon Health Plan (OHP) Standard, which is a Medicaid-extension program to
cover low-income adults who are not eligible for the OHP Plus, which covers
children, pregnant women, and families enrolled in the Temporary Assistance to
Needy Families Program. The OHP standard program was not open for applicants
until 2008. Participants registered for the lottery and were randomly assigned
to win, conditional on the number of household members on the waiting list.
The lottery winners were asked to return their application form. Only 60.82%
of the lottery winners chose to return their application forms, and only some
of those applications were approved. As such, this produced an endogenous non-
compliance problem.

The data sets used here were composed of four parts. The descriptive data set
recorded lottery participants’ basic information and administrative data on
the lotteries. The researchers conducted three follow-up surveys to collect
information on health insurance, healthcare needs, experiments, and costs. The
initial follow-up survey was conducted right after the experiment during the
period of June to November 2008 and included 58,405 survey participants. The
intermediate survey was conducted six months after the experiment for a
subsample of initial survey participants and included 11,756 participants. The
final survey was conducted a year after the experiment for the same group of
people who participated in the initial survey. These three surveys were
referred to as the 0m, 6m, and 12m surveys, respectively.

We select the variables from the descriptive data, the 0m survey data, and the
12m survey data. The summary statistics of the variables used in this example
are showed in Table 3. In the experiment, 50.66% of the survey participants
were randomly selected to win the lottery, and selection into the lottery was
used as the instrument variable, which is random conditional on the number of
people in a household. For the number-of-household- members variable, 1
represented a household with a single member, while 2 and 3 represent
households with two and more than two members. The age varies from 20 to 63;
therefore, the influence of Medicare is excluded. The genders are balanced in
the experiment; approximately 55% of the lottery participants are female. Most
of the respondents are from the metropolitan statistical area, and less than
10% of the participants required a non-English questionnaire.

The second block in the table records partially observed variables. There are
7,611 participants with observed treatment statuses. We choose enrollment into
the OHP program, including both the OHP standard and the OHP plus, as the
treatment variable.151515We did not choose the recorded enrollment for the OHP
Standard in the administrative data as the treatment variable for the
following reasons. First, the application’s approval took a long time, it took
277 days for some people to get their application forms approved, and this was
just before the final round of the survey; therefore, the administrative data
do not show precise enrollment statuses during the first round of the survey.
Moreover, even though some people were granted late approval, the effective
date for the insurance card did not change; the late receivers needed to renew
their insurance card to get coverage, which involved another endogenous self-
selection problem. This directly results in the fact that even though they
were approved and notified before the final round survey, approximately 1,400
participants chose the “Not Covered” option in the final round of the survey,
and approximately 160 participants did not know their exact status OHP
insurance, even though they were already notified about the application
decision. A previous study (Finkelstein et al. (2012)) used external
administrative data to recover the treatment status and avoid the missing
problem. Since our goal is to show how the survey data can be used to get a
consistent estimate, we choose to use the variables recorded in the survey,
instead of in other data sets.

There are three outcomes from the final stage survey, including out-of-pocket
costs for medical care, the number of days when physical health was not good,
and how physical health had changed in the past six months. We refer to the
health change as “Worse Health” in the later regression table, because for
this variable, the higher the value, the worse the health status has become.

Table 3: Summary Statistics in the OHIE Data | Count | Mean | Sd | Min | Max
---|---|---|---|---|---
$Z$: Selected in the lottery | 58405 | 0.5066 | 0.5000 | 0 | 1
$X$: Number of people in household | 58405 | 1.2926 | .4608 | 1 | 3
$X$: Age | 58405 | 39.8863 | 12.1226 | 20 | 63
$X$: Female | 58405 | 0.5464 | 0.4978 | 0 | 1
$X$: Zip code in a metropolitan statistical area | 58405 | 0.7658 | 0.4235 | 0 | 1
$X$: Individual requested English-language materials | 58405 | .9072 | .2902 | 0 | 1
$D$: Currently have OHP insurance | 7611 | 0.1566 | 0.3634 | 0 | 1
$Y$: Total out-of-pocket costs for medical care, last 6 months | 22539 | 230.5913 | 539.7317 | 0 | 4740
$Y$: Number of days (out of the past 30) when physical health not good | 21415 | 9.5070 | 10.8559 | 0 | 30
$Y$: How has your health changed: past 6 months | 23443 | 2.1503 | .6058 | 1 | 3
Observations | 58405 |  |  |  |

###  7.1 Missing Pattern

We first show the missing pattern of the chosen treatment and outcome
variables to confirm the non-monotone missing pattern; the main reason for
missingness in this data set is non-response to the surveys. The non-response
rates for the initial (0m) and final (12m) surveys are shown in Figure 3. For
both the 0m and 12m surveys, the response rates were less than 50%.
Furthermore, 16.88% of participants responded to the 0m but not to the 12m
survey, so it is highly possible to observe their treatment statuses, but not
their outcome statuses; for the group that responded to the 12m but not the 0m
survey, it is possible to observe the opposite.

Figure 3: Non-response to the surveys

Another important source of missingness is non-response to survey questions
among responders. We choose the survey participants who returned both the 0m
and the 12m surveys, and the missing rate of answers for the questions on
treatment and outcome status are shown in Figure 4. 16,566 participants
returned both stage surveys. However, we could not confirm the treatment
status for 8.73% of the respondents after correction of the variable.161616If
the participants mentioned that they had been successfully enrolled in the OHP
Standard, and the administrative data showed consistent enrollment status, the
OHP enrollment variable is corrected to value 1, whether it was missing or
not.For the outcome variables, the non-response rates vary from 1.35% to 9.53%
among survey responders. The question related to subjective healthiness (worse
health) suffers the least from missingness, while the question that asks for
clear memories of the exact number of days when physical health is not good
has the highest percentage of missing.

Figure 4: Non-response to survey questions

The non-response to survey and survey questions result in non-monotone missing
patterns, and these are shown in Figure 5. We observe four missing types for
all three outcomes, and this is consistent with the strict non-monotone
missingness.

Figure 5: Non-monotone Missing Pattern

Next, we show evidence of a violation of the MAR assumption in this example.
The MAR assumption assumes away the correlation between missingness in the
outcomes and the partially observed treatment variable. We run a regression as
a simple test on the correlation between them. As is shown in 4, when
$R^{D}=1$ (i.e., enrollment status in the OHP program is observed), there
exists a significant correlation between $R^{Y}$ and $D$. To use the SMAR
assumption, it is also necessary to confirm that $R^{D}$ does not depend on
partially observe $Y$, so we run the same regression for the outcomes, and we
find a small and insignificant correlation between $R^{D}$ and $Y$ when
$R^{Y}=1$; the results are recorded in Table 5.

Table 4: Regression of $R^{Y}$ on $D$ when $R^{D}=1$ | Out-of-Pocket Costs | Days of Bad Health | Worse Health
---|---|---|---
Currently have OHP insurance |  -0.0382∗∗∗ |  -0.0393∗∗∗ |  -0.0379∗∗∗
| (0.00865) | (0.00880) | (0.00854)
Selected in the lottery |  -0.0197∗∗ | -0.0124 |  -0.0207∗∗
| (0.00657) | (0.00668) | (0.00648)
Number of people in household | 0.0107 |  0.0150∗ | 0.00953
| (0.00706) | (0.00718) | (0.00697)
Female |  0.0419∗∗∗ |  0.0391∗∗∗ |  0.0437∗∗∗
| (0.00644) | (0.00655) | (0.00636)
Age |  0.00451∗∗∗ |  0.00422∗∗∗ |  0.00501∗∗∗
| (0.000261) | (0.000265) | (0.000257)
Zip code in a metropolitan statistical area | 0.00360 | 0.000263 | -0.00327
| (0.00729) | (0.00741) | (0.00719)
Individual requested English-language materials | -0.0123 |  0.0370∗∗ | -0.0242
| (0.0127) | (0.0129) | (0.0125)
Constant |  0.424∗∗∗ |  0.350∗∗∗ |  0.438∗∗∗
| (0.0219) | (0.0222) | (0.0216)
Observations | 23140 | 23140 | 23140
Standard errors in parentheses
∗ $p<0.05$, ∗∗ $p<0.01$, ∗∗∗ $p<0.001$

Table 5: Regression of $R^{D}$ on $Y$ when $R^{Y}=1$ | Out-of-Pocket Costs | Days of Bad Health | Worse Health
---|---|---|---
Selected in the lottery | -0.00729 | -0.00602 | -0.00857
| (-1.15) | (-0.92) | (-1.37)
Number of people in household | 0.00168 | -0.00226 | 0.00365
| (0.24) | (-0.31) | (0.53)
Female | 0.0464∗∗∗ | 0.0495∗∗∗ | 0.0454∗∗∗
| (7.20) | (7.46) | (7.14)
Age | 0.00420∗∗∗ | 0.00421∗∗∗ | 0.00414∗∗∗
| (16.08) | (15.41) | (15.98)
Zip code in a metropolitan statistical area | -0.00584 | -0.00500 | -0.00734
| (-0.80) | (-0.66) | (-1.02)
Individual requested English-language materials | 0.0951∗∗∗ | 0.0898∗∗∗ | 0.0998∗∗∗
| (8.05) | (7.12) | (8.61)
Total out of pocket costs for medical care, last 6 months | 1.64e-08 |  |
| (0.81) |  |
Number of days (out of the past 30) when physical health not good |  | -0.000401 |
|  | (-1.31) |
How has your health changed: past 6 months |  |  | -0.00820
|  |  | (-1.57)
Constant | 0.351∗∗∗ | 0.363∗∗∗ | 0.366∗∗∗
| (16.60) | (16.45) | (16.00)
Observations | 22766 | 21415 | 23443
t statistics in parentheses
∗ $p<0.05$, ∗∗ $p<0.01$, ∗∗∗ $p<0.001$

###  7.2 Regression Results

Table 6 shows the regression results. The nuisance estimators are estimated by
the sieve with B-spline basis functions, and the number of knots was selected
through cross-validation. The estimated effects of OHP are shown in the first
row. For the out-of-pocket costs, the CC estimator gives a lower number than
the other two estimation strategies, while the IPW and AIPW estimators are
closer to each other. The same pattern happens for the outcome of worse
health, and the confidence interval of the CC estimator does not include the
estimated values from IPW and AIPW approaches. This is the evidence that the
CC estimator can be biased in some circumstances; for the two outcomes
introduced above, the CC estimator tends to overestimate the effect.

We also find that AIPW estimators have smaller standard errors for all
coefficient estimates than the other two estimation strategies; this is
because the AIPW estimator tends to be significant at a higher significance
level when the estimates are the same. Using the estimated coefficients on the
OHP enrollment as an example, the standard errors were improved by 6–24% for
different outcomes, compared to the CC and IPW estimators. For the days-of-
bad-health outcome, the AIPW estimator is significant with a 99.9%
significance level, while the other two estimators are significant at a lower
significance level. However, this result was not apparent in the primary
table, because the sample size was still as large as 14,500 when all
observations with incomplete information are dropped. Therefore, we present
another example in which the data are restricted to people who were 35–40
years of age, the days-of-bad-health outcome is shown in the table 7. When the
sample size is small, the differences between the three estimates are more
significant. The AIPW estimator is the only significant estimator with a
significance level being 99%. The AIPW estimator gives an estimate lower than
the other two estimation methods, while the IPW is closer to AIPW, compared to
the CC estimator.

The results above show that enrolling in the OHP program reduced out-of-pocket
costs by $199.7, reduced the number of days when physical health was not good
by 3.3 days, and improved the health index by an average of 0.31. For survey
participants who were 35–40 years of age, the effect of OHP enrollment on
reducing days of bad physical health was greater than the overall population,
and on average, OHP enrollees tended to have five fewer days when their
physical health was not good.

Table 6: Regression Results | Out-of-Pocket Costs | Days of Bad Health | Worse Health
---|---|---|---
| (1) | (2) | (3) | (1) | (2) | (3) | (1) | (2) | (3)
| CC GMM | IPW GMM | AIPW GMM | CC GMM | IPW GMM | AIPW GMM | CC GMM | IPW GMM | AIPW GMM
OHP |  -225.3∗∗∗ |  -191.0∗∗∗ |  -199.7∗∗∗ |  -3.059∗∗ |  -2.626∗∗ |  -3.295∗∗∗ |  -0.380∗∗∗ |  -0.312∗∗∗ |  -0.310∗∗∗
| (38.59) | (35.11) | (32.99) | (1.053) | (0.890) | (0.805) | (0.0579) | (0.0508) | (0.0456)
Female |  59.73∗∗∗ |  46.96∗∗∗ |  49.45∗∗∗ | -0.0935 | 0.199 | 0.140 | -0.00485 | 0.00492 | 0.00734
| (6.963) | (10.75) | (6.044) | (0.191) | (0.193) | (0.149) | (0.0103) | (0.0107) | (0.00841)
Number of Household Members | -2.252 | 6.856 | -5.612 |  -1.434∗∗∗ |  -0.834∗∗∗ |  -1.390∗∗∗ |  -0.0418∗∗∗ | 0.00311 |  -0.0351∗∗∗
| (7.596) | (8.566) | (6.665) | (0.201) | (0.235) | (0.157) | (0.0108) | (0.0126) | (0.00900)
Age |  1.079∗∗∗ | 0.196 |  1.284∗∗∗ |  0.164∗∗∗ |  0.195∗∗∗ |  0.168∗∗∗ |  0.00457∗∗∗ |  0.00629∗∗∗ |  0.00558∗∗∗
| (0.290) | (0.730) | (0.254) | (0.00750) | (0.00813) | (0.00609) | (0.000421) | (0.000453) | (0.000354)
MSA |  -16.64∗ | -1.405 |  -26.33∗∗∗ |  -0.972∗∗∗ |  -0.657∗∗ |  -0.957∗∗∗ |  -0.0376∗∗ | -0.0102 |  -0.0225∗
| (7.803) | (7.431) | (7.089) | (0.218) | (0.240) | (0.177) | (0.0115) | (0.0126) | (0.00981)
English-Speaking |  32.60∗ |  39.81∗ |  21.17∗ |  1.646∗∗∗ |  2.427∗∗∗ |  1.400∗∗∗ |  0.159∗∗∗ |  0.216∗∗∗ |  0.186∗∗∗
| (13.65) | (16.87) | (20.54) | (0.319) | (0.403) | (0.259) | (0.0171) | (0.0209) | (0.0149)
Constant |  124.2∗∗∗ |  68.29∗∗ |  152.4∗∗∗ |  4.112∗∗∗ | 0.856 |  4.112∗∗∗ |  1.950∗∗∗ |  1.733∗∗∗ |  1.850∗∗∗
| (24.75) | (25.69) | (21.01) | (0.659) | (0.816) | (0.521) | (0.0357) | (0.0422) | (0.0306)
Observations | 14500 | 14500 | 58396 | 13696 | 13696 | 58393 | 14933 | 14933 | 58358
Standard errors in parentheses
∗ $p<0.05$, ∗∗ $p<0.01$, ∗∗∗ $p<0.001$

Table 7: Table: Effect of OHP on Days of Bad Health, for Age 35-40 | Days of Bad Health: Age 35-40
---|---
| (1) | (2) | (3)
| CC GMM | IPW GMM | AIPW GMM
OHP | -3.305 | -3.822 | -5.855∗∗
| (3.397) | (3.547) | (2.271)
Female | 0.834 | 0.971 | 0.825
| (0.587) | (0.650) | (0.460)
Number of Household Members | -1.472∗ | -1.070 | -1.624∗∗∗
| (0.582) | (0.812) | (0.472)
Age | 0.412∗ | 1.333 | 0.472∗∗
| (0.200) | (1.111) | (0.163)
MSA | -0.286 | 0.165 | -1.270∗
| (0.708) | (0.759) | (0.589)
English Speaking | 0.106 | 0.589 | -0.348
| (0.798) | (0.991) | (0.697)
Constant | -4.605 | -40.06 | -4.792
| (7.449) | (42.26) | (6.096)
Observations | 1305 | 1305 | 6630
Standard errors in parentheses
∗ $p<0.05$, ∗∗ $p<0.01$, ∗∗∗ $p<0.001$

##  8 Conclusion

This paper studies the problem of two missing variables, and we focus on the
missing treatment and outcome as an example, and we include a discussion of
how to extend the current framework to more than two missing variables in
Section A. The first thing to do is to find the appropriate assumptions on the
missing mechanism so that it can be identified. We propose the MAR assumption
and the SMAR assumption, the difference between which lies in whether the
correlation between the missing mechanism and partially observed variables is
allowed; the identified missing mechanism is used in constructing an AIPW-GMM
estimator.

Even though there are many desirable asymptotic properties recorded in the
literature for the AIPW approach, many of these fail with non-monotone
missingness. We find that these properties are maintained under the MAR
assumption; they only hold under the SMAR assumption when the treatment
variable has no direct effect on the outcome. The Monte Carlo simulation shows
that the AIPW-GMM estimator performs better than the previously used CC and
IPW estimators, in the sense that it is consistent and has the smallest
standard error compared to the other two approaches. The performance is also
verified in the empirical example of estimating the treatment effect of OHP on
health-related outcomes. The AIPW-GMM estimator reduced the standard error by
6-24% on the estimated coefficients for the treatment variables, compared to
the CC and IPW estimators, and showed a significant effect of enrolling in the
OHP on reducing out-of-pocket costs for medical care, reducing the number of
days when physical health is not good, and improving health status.

##  Appendix A Discussion

The paper majorly focused on missing treatment and outcome variables with a
strict non-monotone missing pattern; this method can be generalized to a more
general arrangement. This section will discuss ways to extend the current
approach.

###  A.1 Generalization to General Missing Patterns

The previous sections only considered the case where the missing pattern is
strictly non-monotone and can be extended to a more general case that allows
for monotone missingness, as well as univariate missingness. First, we propose
a weaker overlap assumption.

######  Assumption Weak Overlap.

$p_{11}\in[\kappa_{11},1)$ almost surely in $(D,Z,X)$.

We provide a new AIPW moment function denoted by $\tilde{m}$ such that:

$\tilde{m}_{aipw}=\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))+\tilde{\phi}(Z,X,R^{D},R^{Y},R^{D}D,R^{Y}Y,\beta)$
(A.1)

where

$\displaystyle\tilde{\phi}$
$\displaystyle=p_{01}\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$
(A.2)

######  Theorem A.1.

Suppose Assumption MAR or SMAR, and Assumption Weak Overlap hold,
$E\left[\tilde{m}_{aipw}\right]=0$ when $\beta=\beta^{0}$.

Proof of Theorem A.1 follows the same argument as in the proof of Theorem 4.1.

For the previous moment condition to hold, we require $p_{d}$ to be strictly
positive in $Z,X$; and $p_{y}$ to be strictly positive in $Z,X$ when $R^{D}=0$
and in $Z,X,D$ when $R^{D}=1$. Therefore, we need:

$\displaystyle p_{01}$ $\displaystyle=\Pr\left[R^{D}=0,R^{Y}=1|Z,X\right]>0$
$\displaystyle p_{11}$ $\displaystyle=\Pr\left[R^{D}=1,R^{Y}=1|Z,X,D\right]>0$

To allow for the case where $p_{01}=0$, we multiply the augmentation term
involving $p_{y}$ by $p_{01}$. Adding in $p_{01}$ does not affect the
consistency of the estimator; thus, the first-stage estimation therefore does
not affect the efficiency of the primary model.

###  A.2 Generalization to More than Two Missing Variables

This method can be extended to a situation in which there exists additional
missing variables. We take missing IV, missing treatment status, and missing
outcome as an example to illustrate the idea of extending the current method
to more than two missing values. We use $R^{Z}$ as the indicator of observing
$Z$:

$R^{Z}=\begin{cases}1&Z\text{ is observed}\\\ 0&Z\text{ is
unobserved}\end{cases}$

For simplicity, we only develop the moment condition under the SMAR-like
assumption. Suppose there exists a vector of fully observed variables $X$, and
we make two analogous ignorability assumptions on the missing mechanism, and
the overlap assumption is:

######  Assumption A.1.

$\displaystyle R^{Z}$ $\displaystyle\perp\\!\\!\\!\perp(D,Y)|X$ $\displaystyle
R^{D}$ $\displaystyle\perp\\!\\!\\!\perp(D,Y)|X,R^{Z}Z,R^{Z}$ $\displaystyle
R^{Y}$ $\displaystyle\perp\\!\\!\\!\perp(D,Y)|X,R^{Z}Z,R^{D}D,R^{Z},R^{D}$

######  Assumption A.2.

(a) $p_{z}\in[\kappa_{z},1)$ almost surely in $(X)$

(b) $p_{d}\in[\kappa_{d},1)$ almost surely in $(X,R^{Z}Z,R^{Z})$

(c) $p_{y}\in[\kappa_{y},1)$ almostly surely in
$(X,R^{Z}Z,R^{D}D,R^{Z},R^{D})$ for
$\kappa_{z}>0,\kappa_{d}>0,$$\kappa_{y}>0$.

The moment function is correspondingly:

$m_{aipw}=\frac{R^{Z}R^{D}R^{Y}}{p_{111}}Z\left(Y-g(D,X;\beta)\right)+\phi(Z,X,R^{Z},R^{D},R^{Y},R^{Z}Z,R^{D}D,R^{Y}Y,\beta)$

where

$\displaystyle\phi$
$\displaystyle=\left(1-\frac{R^{Z}}{p_{z}}\right)\frac{R^{D}}{p_{d}}\frac{R^{Y}}{p_{y}}\left[E\left[Z|X\right]\left(Y-g(D,X;\beta)\right)-E\left[m_{full}|X\right]\right]$
$\displaystyle\;\;+\frac{R^{Z}}{p_{z}}\left(1-\frac{R^{D}}{p_{d}}\right)\frac{R^{Y}}{p_{y}}\left[Z\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-E\left[m_{full}|X\right]\right]$
$\displaystyle\;\;+\frac{R^{Z}}{p_{z}}\frac{R^{D}}{p_{d}}\left(1-\frac{R^{Y}}{p_{y}}\right)\left[Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-E\left[m_{full}|X\right]\right]$
$\displaystyle\;\;+\left(1-\frac{R^{Z}}{p_{z}}\right)\left(1-\frac{R^{D}}{p_{d}}\right)\frac{R^{Y}}{p_{y}}\left[E\left[Z|X\right]\left(Y-E\left[g(D,X;\beta)|X\right]\right)-E\left[m_{full}|X\right]\right]$
$\displaystyle\;\;+\left(1-\frac{R^{Z}}{p_{z}}\right)\frac{R^{D}}{p_{d}}\left(1-\frac{R^{Y}}{p_{y}}\right)\left[E\left[Z|X\right]\left(E\left[Y|D,X\right]-g(D,X;\beta)\right)-E\left[m_{full}|X\right]\right]$
$\displaystyle\;\;+\frac{R^{Z}}{p_{z}}\left(1-\frac{R^{D}}{p_{d}}\right)\left(1-\frac{R^{Y}}{p_{y}}\right)\left[Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)-E\left[m_{full}|X\right]\right]$
$\displaystyle\;\;+\left(1-\text{$\frac{R^{Z}R^{D}R^{Y}}{p_{111}}$}\right)E\left[m_{full}|X\right]$

and

$\displaystyle p_{111}$
$\displaystyle=\Pr\left[R^{Z}=1,R^{D}=1,R^{Y}=1|X,Z,D\right]$

In the case where there are two partially missing variables, the moment
function can be derived through a three-step AIPW imputation by imputing the
functions of $Z$ using the AIPW strategy after the second step.

######  Theorem A.2.

Suppose Assumptions A.1 and A.2 hold, $E\left[m_{aipw}\right]=0$ when
$\beta=\beta^{0}$.

The proof is analogous to the proof of Theorem 4.1.

##  Appendix B Proofs

###  B.1 Proof of Proposition 4.1;

$\hat{\epsilon}$ in 4.6 is defined as:

$\displaystyle\hat{\epsilon}$
$\displaystyle=\frac{R^{Y}}{p_{y}}\left[Y-\left(\frac{R^{D}}{p_{d}}g(D,X;\beta)+(1-\frac{R^{D}}{p_{d}})E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(1-\frac{R^{Y}}{p_{y}}\right)\left[\frac{R^{D}}{p_{d}}\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)+\left(1-\frac{R^{D}}{p_{d}}\right)E\left\\{E\left[Y|D,Z,X\right]-g(D,X;\beta)|Z,X\right\\}\right]$

We expand the right hand side and get

$\displaystyle\hat{\epsilon}$
$\displaystyle=\frac{R^{D}R^{Y}}{p_{d}p_{y}}\left(Y-g(D,X;\beta)\right)+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{d}p_{y}}\right)\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)$
$\displaystyle\;\;+\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{d}p_{y}}\right)\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)$
$\displaystyle\;\;-+\left(1-\frac{R^{D}}{p_{d}}\right)\left(1-\frac{R^{Y}}{p_{y}}\right)\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)$
$\displaystyle=\frac{R^{D}R^{Y}}{p_{d}p_{y}}\left(Y-g(D,X;\beta)\right)+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{d}p_{y}}\right)\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)$
$\displaystyle\;\;+\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{d}p_{y}}\right)\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)$
$\displaystyle\;\;+\left[(1-\frac{R^{D}R^{Y}}{p_{d}p_{y}})-\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{d}p_{y}}\right)-\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{d}p_{y}}\right)\right]\left(E\left[Y|D,Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$
$\displaystyle=\frac{R^{D}R^{Y}}{p_{11}}\left(Y-g(D,X;\beta)\right)$
$\displaystyle\;\;+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|D,Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$

$\displaystyle\;\;+\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|D,Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\text{\;\;}+\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[Y|D,Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$

In the third equality, $p_{11}$ is replaced by $p_{d}p_{y}$, because
$\frac{R^{D}R^{Y}}{p_{d}p_{y}}=0,\forall Z,X$ if $R^{D}=0$. Therefore,

$\frac{R^{D}R^{Y}}{p_{d}p_{y}}\equiv\frac{R^{D}R^{Y}}{p_{d}(Z,X)p_{y}(Z,X,R^{D}D,R^{D})}=\frac{R^{D}R^{Y}}{p_{d}(Z,X)p_{y}(Z,X,D,R^{D}=1)}=\frac{R^{D}R^{Y}}{p_{11}}$
(B.1)

###  B.2 Proof of Theorem 4.1

We proceed the proof term by term, we first show the expected value of the
first term equals to zero:

$E\left[\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))\right]=E\left[Z(Y-g(D,X;\beta))\right]=0$

where the first equality sign follows
$E\left[\frac{R^{D}R^{Y}}{p_{11}}|Z,X,D,Y\right]=1$, and the second equality
sign follows the population moment condition.

Analogously, the zero expectation of the latter three terms can be proved by

$\displaystyle
E\left[\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)|Z,X,D,Y\right]=0$
$\displaystyle
E\left[\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)|Z,X,D,Y\right]=0$
$\displaystyle
E\left[\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)|Z,X,D,Y\right]=0$

Thus, $E\left[m_{aipw}(\beta)\right]=0$.

###  B.3 Proof of Theorem 5.1

The AIPW estimator maintains the property of double robustness under the
assumption MAR. In this section, we give a detailed discussion to show the
moment condition holds when either the missing mechanism or the imputed values
of missing variables is correctly specified.

First, we show the moment condition specified in Theorem 4.1 holds when the
missing mechanism is correctly specified. The expectation of moment function
is written as:

$\displaystyle E\left[m_{aipw}(\beta)\right]$
$\displaystyle=E\left[\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))\right]+E\left[\phi(Z,X,R^{D},R^{Y},R^{D}D,R^{Y}Y,\beta)\right]$
(B.2)

The first term in equation B.2 equals to the expectation of full data moment
function, and equals to zero:

$\displaystyle E\left[\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))\right]$
$\displaystyle=E\left\\{E\left[\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))|Z,X,D,Y\right]\right\\}$
$\displaystyle=E\left\\{E\left[\frac{R^{D}R^{Y}}{p_{11}}|Z,X,D,Y\right]Z(Y-g(D,X;\beta))\right\\}$
$\displaystyle=E\left\\{\frac{\Pr\left[R^{D}=1,R^{Y}=1|Z,X,D,Y\right]}{p_{11}}Z(Y-g(D,X;\beta))\right\\}$
$\displaystyle=E\left[Z(Y-g(D,X;\beta))\right]=0$

The last equality follows the full data moment condition defined in equation
2.4.

The second term in equation B.2 equals to zero following the analogous
argument. We show the expectation of each element in $\phi$ equals to zero:

$\displaystyle
E\left\\{\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right\\}$
$\displaystyle=E\left\\{E\left[\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)a(Z,X,Y)|Z,X,D,Y\right]\right\\}$
$\displaystyle=E\left\\{E\left[\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)|Z,X,D,Y\right]a(Z,X,Y)\right\\}$
$\displaystyle=E\left\\{\left(\frac{\Pr\left[R^{Y}=1|Z,X,D,Y\right]}{p_{y}}-\frac{\Pr\left[R^{D}=1,R^{Y}=1|Z,X,D,Y\right]}{p_{11}}\right)a(Z,X,Y)\right\\}$
$\displaystyle=0$

where $a$ is a function of the observables such that

$a(Z,X,Y)=Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X,Y\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$

The last equality holds if $p_{11},p_{y}$ are correctly specified.

The second term follows the same argument.

$E\left[\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)|Z,X,D,Y\right]=\frac{\Pr\left[R^{D}=1|Z,X,D,Y\right]}{p_{d}}-\frac{\Pr\left[R^{D}=1,R^{Y}=1|Z,X,D,Y\right]}{p_{11}}=0$

if $p_{d}$ is also correctly specified.

And the third term equals to 0, following

$E\left[\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)|Z,X,D,Y\right]=1-\frac{\Pr\left[R^{D}=1,R^{Y}=1|Z,X,D,Y\right]}{p_{11}}=0$

Therefore, the moment condition holds if $(p_{d},p_{y,}p_{11})$ are correctly
specified.

Next, we show the moment condition holds if $E\left[g(D,X;\beta)|Z,X\right]$,
$E\left[Y|D,Z,X\right]$,$E\left[Y|Z,X\right]$ are correctly specified. We
first consider the first component of the augmentation term $\phi$:

$\displaystyle
E\left\\{\left(\frac{R^{Y}}{p_{11}+p_{01}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right\\}$
$\displaystyle=E\left\\{\left(\frac{R^{Y}}{p_{11}+p_{01}}-\frac{R^{D}R^{Y}}{p_{11}}\right)E\left[a(Z,X,Y)|Z,X\right]\right\\}$
$\displaystyle=0$

following
$E\left[Y-E\left[g(D,X;\beta)|Z,X,Y\right]|Z,X\right]=E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]$.

The same proof can be applied on the second component of $\phi$, following

$E\left[E\left[Y|D,Z,X\right]-g(D,X;\beta)|Z,X\right]=E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]$

Next, we combine the last term in $\phi$ and the IPW moment condition and get:

$\displaystyle
E\left\\{\frac{R^{D}R^{Y}}{p_{11}}\left[(Y-g(D,X;\beta))-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right.$
$\displaystyle\left.+\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right\\}$
$\displaystyle=0+E\left\\{E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right\\}$
$\displaystyle=E[m_{aipw}(\beta)]=0$

The second equality follows:

$E\left[Y-g(D,X;\beta)|Z,X\right]=E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]$

and the last equality follows the iterative law of expectations.

###  B.4 Proof of Theorem 5.2

We use $m_{full}$ to denote the original population level moment function,
such that:

$m_{full}(\beta)=Z\left(Y-g(D,X;\beta)\right)$

We reorganize the moment function into:

$\displaystyle m_{aipw}(\beta)$
$\displaystyle=\underbrace{\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)}_{(1)}+\underbrace{E\left[m_{full}|Z,X\right]}_{(2)}$
$\displaystyle\;\;+\underbrace{\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)}_{(3)}$
$\displaystyle\;\;+\underbrace{\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left(Y-E\left[Y|Z,X\right]\right)}_{(4)}$

We discuss the variance of each term.

The variance of the first term equals to:

$\displaystyle Var\left((1)\right)$
$\displaystyle=E\left[Var\left(\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|Z,X\right)\right]$
$\displaystyle\;\;+Var\left(E\left[\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|Z,X\right]\right)$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|Z,X\right)\right]=E\left[\frac{1}{p_{11}}Var\left(m_{full}|Z,X\right)\right]$

The variance of the second term equals to:

$Var\left((2)\right)=E\left[E\left[m_{full}|Z,X\right]E\left[m_{full}|Z,X\right]^{{}^{\prime}}\right]$

The covariance between $(1)$ and $(2)$ equals to $0$ following:

$\displaystyle Cov\left((1),(2)\right)$
$\displaystyle=E\left[\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)E\left[m_{full}|Z,X\right]\right]$
$\displaystyle=E\left[\left(E\left[m_{full}|Z,X\right]-E\left[m_{full}|Z,X\right]\right)E\left[m_{full}|Z,X\right]\right]=0$

The covariance between $(1)$ and $(3)$ is equivalent to the negative of
$Var\left((3)\right)$, following

$\displaystyle Cov\left((1),(3)\right)$
$\displaystyle=Cov\left((1),\frac{R^{D}}{p_{d}}\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;-Cov\left((1),\frac{R^{D}R^{Y}}{p_{11}}\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle=E\left[\frac{1}{p_{d}}\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)^{2}\right]$
$\displaystyle\;\;-E\left[\frac{1}{p_{11}}\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)^{2}\right]$
$\displaystyle=-Var\left((3)\right)$

Also, $Cov\left((2),(3)\right)=Cov\left((2),(4)\right)=0$ following similar
argument as in the proof for $Cov\left((1),(2)\right)$.

Next, we derive the correlation between $(3)$ and the other terms.

$\displaystyle Cov\left((1),(4)\right)$
$\displaystyle=E\left[\frac{R^{D}R^{Y}}{p_{11}p_{y}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)Z\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle\;\;-E\left[\frac{R^{D}R^{Y}}{p_{11}^{2}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)Z\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[\left(\frac{1}{p_{y}}-\frac{1}{p_{11}}\right)E\left[\left(m_{full}-E\left[m_{full}|Z,X\right]\right)Z\left(Y-E\left[Y|Z,X\right]\right)|Z,X\right]\right]$
$\displaystyle=-Var\left((4)\right)-E\left[\left(\frac{1}{p_{y}}-\frac{1}{p_{11}}\right)Z^{2}Cov\left(g(D,X;\beta),Y|Z,X\right)\right]$

$\displaystyle Cov\left((3),(4)\right)$
$\displaystyle=E\left[\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right.$
$\displaystyle\;\;\;\;\;\left.Z\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[\left(1-\frac{1}{p_{y}}-\frac{1}{p_{d}}+\frac{1}{p_{11}}\right)Cov\left(ZE\left[Y|D,Z,X\right],E\left[m_{full}|D,Z,X\right]|Z,X\right)\right]$

Therefore,

$\displaystyle V_{MAR}$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(m_{full}|Z,X\right)+E\left[m_{full}|Z,X\right]E\left[m_{full}|Z,X\right]^{{}^{\prime}}\right]$
$\displaystyle\;\;-Var\left(\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;-Var\left(\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|Y,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;-2E\left[\left(1-\frac{1}{p_{y}}-\frac{1}{p_{d}}+\frac{1}{p_{11}}\right)Cov\left(ZE\left[Y|D,Z,X\right],E\left[m_{full}|D,Z,X\right]|Z,X\right)\right]$
$\displaystyle\;\;-2E\left[\left(\frac{1}{p_{y}}-\frac{1}{p_{11}}\right)Z^{2}Cov\left(g(D,X;\beta),Y|Z,X\right)\right]$

###  B.5 Proof of Theorem 5.4

The first part of this proof heavily depend on the results from Newey (1994)
and Newey and McFadden (1994). Two takeaways from Newey (1994) are: (a) the
methods of estimating the nuisance parameters do not affect the asymptotic
variance of the estimator; (b) the nuisance parameters do not affect the
variance of the primary model if it does not affect consistency of the model.

First, we rewrite the moment function as:

$\displaystyle m_{aipw}(\beta)$
$\displaystyle=\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))$
$\displaystyle\;\;+(1-p_{d})\left(\frac{(1-R^{D})R^{Y}}{p_{01}}-\frac{R^{D}R^{Y}}{p_{11}}\right)$
$\displaystyle\;\;\;\;\times
Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\;\;+\left(1-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$

by replacing

$\displaystyle\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}$
$\displaystyle=\frac{R^{D}R^{Y}}{p_{y}}+\frac{(1-R^{D})R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}$
$\displaystyle=\frac{p_{d}R^{D}R^{Y}}{p_{11}}+\frac{(1-p_{d})(1-R^{D})R^{Y}}{p_{01}}-\frac{R^{D}R^{Y}}{p_{11}}$
$\displaystyle=(1-p_{d})\left(\frac{(1-R^{D})R^{Y}}{p_{01}}-\frac{R^{D}R^{Y}}{p_{11}}\right)$

in the second component in $m_{aipw}$.

The nuisance parameters include the probability of observing $D$, $Y$, and
both, conditional on the observables. Recall that the imputed values for
incomplete model, i.e, $E\left[g(D,X;\beta)|Z,X\right]$,
$E\left[Y|Z,X\right]$, $E\left[Y|D,Z,X\right]$ do not affect the moment
condition, they do not have effect on the variance of $V_{SMAR}$. On the other
hand, among the propensities, only $p_{11}$ affect consistency of $m_{aipw}$
via the first component, therefore, despite existence of multiple nuisance
parameters in four components, we need to only create the correction term for
one of them. For simplicity of notation, denote

$\displaystyle\phi_{c}$
$\displaystyle=-(1-p_{d})\frac{R^{D}R^{Y}}{p_{11}}Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle=-(1-p_{d})\frac{R^{D}R^{Y}}{p_{11}}Z\left[Y-E\left[Y|Z,X\right]\right]$

Derivation of the correction term closely followsNewey (1994). Note that
$p_{11}$ can be seen as a conditional expectation of $R^{D}R^{Y}$. Results
from Section 4 in Newey (1994) can be applied. We first need to find a
linearization of $\phi_{c}.$ Because $p_{11}$ only affects $\phi_{c}$ through
its values instead of functional form, we apply equation (3.17) from Newey
(1994) directly, and derive the linearization of $\phi_{c}$ as:

$D(\mathcal{O},\phi_{c})=E\left[(1-p_{d})\frac{1}{p_{11}}\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\hat{p}_{11}(D,Z,X)\right]$

and from result in (4.5),

$\displaystyle\frac{\partial
E\left[D(\mathcal{O},\phi_{c})\right]}{\partial\theta}$
$\displaystyle=E\left\\{(1-p_{d})\frac{1}{p_{11}}\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\left(R^{D}R^{Y}-p_{11}\right)\mathcal{S}(\mathcal{O})\right\\}$
$\displaystyle=E\left[(1-p_{d})(\frac{R^{D}R^{Y}}{p_{11}}-1)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\mathcal{S}(\mathcal{O})\right]$

Therefore, the correction term to be incorporated is
$(1-p_{d})(1-\frac{R^{D}R^{Y}}{p_{11}})\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)$.

Now, we proceed calculation of the $V_{SMAR}$ term by term. We reorganize the
influence function as in Section B.4:

$\displaystyle\phi_{SMAR}(\beta)$
$\displaystyle=\underbrace{\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)}_{(1)}+\underbrace{E\left[m_{full}|Z,X\right]}_{(2)}$
$\displaystyle\;\;+\underbrace{\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)}_{(3)}$
$\displaystyle\;\;+\underbrace{\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(Y-E\left[Y|Z,X\right]\right)}_{(4)}$
$\displaystyle\;\;+\underbrace{(1-p_{d})(\frac{R^{D}R^{Y}}{p_{11}}-1)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)}_{(5)}$

where $(5)$ is the correction term added for nuisance parameter used in $(3).$

The variance of the first term equals to:

$\displaystyle Var\left((1)\right)$
$\displaystyle=E\left[Var\left(\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|D,Z,X\right)\right]$
$\displaystyle\;\;+Var\left(E\left[\frac{R^{D}R^{Y}}{p_{11}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|D,Z,X\right]\right)$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|D,Z,X\right)\right]+Var\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|D,Z,X\right)\right]+Var\left(E\left[m_{full}|D,Z,X\right]\right)$
$\displaystyle\;\;+Var\left(E\left[m_{full}|Z,X\right]\right)-2Cov\left(E\left[m_{full}|D,Z,X\right],E\left[m_{full}|Z,X\right]\right)$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(m_{full}|D,Z,X\right)\right]+Var\left(E\left[m_{full}|D,Z,X\right]\right)-Var\left(E\left[m_{full}|Z,X\right]\right)$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(m_{full}|D,Z,X\right)\right]+Var\left(E\left[m_{full}|D,Z,X\right]\right)-Var\left((2)\right)$

The $-Var((2))$ cancels out with the variance of the second term.

Now we consider the covariance between different terms:

$Cov\left((1),(2)\right)=0,Cov\left((2),(3)\right)=0,Cov\left((2),(4)\right)=0$

$Cov\left((2),(5)\right)$ also equals to zero following analogous argument and
the fact that $p_{11}=E\left[R^{D}R^{Y}|Z,X,D,Y\right]$.

$Cov\left((1),(3)\right)=-Var((3))$

following analogous argument in the Section B.4.

Now we consider the covariance between $(1)$ and $(4)$:

$\displaystyle Cov\left((1),(4)\right)$
$\displaystyle=E\left[-(1-p_{d})\frac{R^{D}R^{Y}}{p_{11}^{2}}\left(m_{full}-E\left[m_{full}|Z,X\right]\right)\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[-\frac{1-p_{d}}{p_{11}}E\left[\left(m_{full}-E\left[m_{full}|Z,X\right]\right)\left(Y-E\left[Y|Z,X\right]\right)|D,Z,X\right]\right]$

We notice that the covariance between $(1)$ and $(5)$ equals to:

$\displaystyle Cov\left((1),(5)\right)$
$\displaystyle=E\left[(1-p_{d})\left(\frac{R^{D}R^{Y}}{p_{11}^{2}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(m_{full}-E\left[m_{full}|Z,X\right]\right)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[\left(1-p_{d}\right)\left(\frac{1}{p_{11}}-1\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\right]$

Therefore, some parts in $Cov\left((1),(4)\right)$ and
$Cov\left((1),(5)\right)$ cancels with each other, and get:

$\displaystyle Cov\left((1),(4)\right)+Cov\left((1),(5)\right)$
$\displaystyle=-E\left[\frac{1-p_{d}}{p_{11}}Var\left(Y|D,Z,X\right)\right]$
$\displaystyle\;\;-E\left[(1-p_{d})Cov\left(E\left[m_{full}|D,Z,X\right],E\left[Y|D,Z,X\right]|Z,X\right)\right]$

The variance of $(4)$ equals to:

$Var\left((4)\right)=E\left[(1-p_{d})^{2}\left(\frac{1}{p_{01}}+\frac{1}{p_{11}}\right)Var\left(Y|D,Z,X\right)\right]$

Next, consider covariance between $(3),(4),(5)$

$\displaystyle Cov\left((3),(4)\right)$
$\displaystyle=E\left[-(1-p_{d})\left(\frac{R^{D}R^{Y}}{p_{d}p_{11}}-\frac{R^{D}R^{Y}}{p_{11}^{2}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[-(1-p_{d})\left(\frac{1}{p_{d}}-1\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[-\frac{(1-p_{d})^{2}}{p_{d}}Cov\left(E\left[m_{full}|D,Z,X\right],E\left[Y|D,Z,X\right]|Z,X\right)\right]$

$\displaystyle Cov\left((3),(5)\right)$
$\displaystyle=E\left[(1-p_{d})\left(\frac{R^{D}R^{Y}}{p_{d}p_{11}}-\frac{R^{D}R^{Y}}{p_{11}^{2}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle\;\;-E\left[(1-p_{d})\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[\text{$\frac{(1-p_{d})^{2}}{p_{d}}$}Cov\left(E\left[m_{full}|D,Z,X\right],E\left[Y|D,Z,X\right]|Z,X\right)\right]$

The two covariances cancel with each other.

$\displaystyle Cov\left((4),(5)\right)$
$\displaystyle=E\left[-(1-p_{d})^{2}(\frac{R^{D}R^{Y}}{p_{11}^{2}}-\frac{R^{D}R^{Y}}{p_{11}})\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\left(Y-E\left[Y|Z,X\right]\right)\right]$
$\displaystyle=E\left[(1-p_{d})^{2}(1-\frac{1}{p_{11}})Var\left(E\left[Y|D,Z,X\right]|Z,X\right)\right]$
$\displaystyle=-Var\left((5)\right)$

Thus, $V_{SMAR}$ is derived as follows:

$\displaystyle V_{SMAR}$
$\displaystyle=E\left[\frac{1}{p_{11}}Var\left(\left(m_{full}-E\left[m_{full}|Z,X\right]\right)|D,Z,X\right)\right]+Var\left(E\left[m_{full}|D,Z,X\right]\right)$
$\displaystyle\;\;-Var\left(\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left(E\left[m_{full}|D,Z,X\right]-E\left[m_{full}|Z,X\right]\right)\right)$
$\displaystyle\;\;-Var\left((1-p_{d})(\frac{R^{D}R^{Y}}{p_{11}}-1)\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)\right)$
$\displaystyle\;\;-E\left[\left(\frac{1-p_{d}^{2}}{p_{11}}-\frac{(1-p_{d})^{2}}{p_{01}}\right)Var\left(Y|D,Z,X\right)\right]$
$\displaystyle\;\;-2E\left[(1-p_{d})Cov\left(E\left[m_{full}|D,Z,X\right],E\left[Y|D,Z,X\right]|Z,X\right)\right]$

where the two terms come from
$Var\left((4)\right)+2Cov\left((1),(4)\right)+2Cov\left((1),(5)\right)$.

###  B.6 Proof of Theorem 5.5

The sketch of the proof follows Cattaneo (2010), Chaudhuri and Guilkey (2016),
Newey (1994), Newey (1997) closely, and most of the results have been proved
in the proof of Proposition 2.3 in Chaudhuri and Guilkey (2016).

First, by the results showed in Theorem 5 and Theorem 8 in Cattaneo (2010),
under the conditions listed in Theorem 5.5,

$\displaystyle||\hat{p}-p||_{\infty}$ $\displaystyle=o_{p}(N^{-\frac{1}{4}})$
(B.3) $\displaystyle||\hat{q}-q||_{\infty}$
$\displaystyle=o_{p}(N^{-\frac{1}{4}})$ (B.4)

where $p$ stands for the missing mechanism parameters $(p_{d},p_{y},p_{11})$,
and $q$ stands for the imputed missing values
$(E\left[D|Z,X\right],E\left[Y|Z,X\right],E\left[Y|Z,X,D\right])$.

Then, what left is to show the multiplication of the nuisance parameters
converges to zero in probability. We show the sketch for

$\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$

and we use analogous conditions to the conditions (32) and (33) from the proof
of proposition 2.3 in Chaudhuri and Guilkey (2016), what left to show is that

$\displaystyle o_{p}(1)$
$\displaystyle=\sqrt{N}\left(\bar{\xi}_{N}(\beta^{0},\hat{p},\hat{q}(\beta^{0}))-\bar{\xi}_{N}(\beta^{0},p,q(\beta^{0}))\right)$
(B.5) $\displaystyle o_{p}(1)$
$\displaystyle=\sup_{|\beta-\beta_{0}|\leq\delta_{N}}\frac{\sqrt{N}|\bar{\xi}_{N}\left(\text{$\beta,\hat{p},\hat{q}(\beta)$}\right)-E\left[\bar{\xi}_{N}(\beta,p,q(\beta))\right]-\bar{\xi}_{N}\left(\beta^{0},\hat{p},\hat{q}(\beta^{0})\right)|}{1+C\sqrt{N}|\beta-\beta^{0}|}$
(B.6)

for all positive sequences $\delta_{N}=o(1)$ and a generic constant $C>0$, and

$\displaystyle\bar{\xi}_{N}(\beta,\hat{p},\hat{q})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\hat{\nu}_{i}\hat{\tau}_{i}$
$\displaystyle\bar{\xi}_{N}(\beta,p\hat{q})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\nu_{i}\tau_{i}$

where

$\nu_{i}=\left(\frac{R_{i}^{D}}{p_{d,i}}-\frac{R_{i}^{D}R_{i}^{Y}}{p_{11,i}}\right)$

$\tau_{i}=\left[\left(E\left[Y_{i}|D_{i},Z_{i},X_{i}\right]-g(D_{i},X_{i};\beta)\right)-\left(E\left[Y_{i}|Z_{i},X_{i}\right]-E\left[g(D_{i},X_{i};\beta)|Z_{i},X_{i}\right]\right)\right]$

Condition B.5 holds follows the fact that $\frac{1}{N}\sum_{i=1}^{N}|\nu_{i}|$
and $\frac{1}{N}\sum_{i=1}^{N}|\tau_{i}|$ are $O_{p}(1)$, which follows the
conditions B.3 and B.4.

The proof of the condition B.6 also follows the proof of the analogous
condition in Chaudhuri and Guilkey (2016). First,
$E\left[\bar{\xi}_{N}(\beta,p,q(\beta))\right]=0$ under either the Assumption
MAR or the Assumption SMAR, therefore, the condition B.6 reduces to:

$\displaystyle o_{p}(1)$
$\displaystyle=\sup_{|\beta-\beta_{0}|\leq\delta_{N}}\frac{\sqrt{N}|\bar{\xi}_{N}\left(\text{$\beta,\hat{p},\hat{q}(\beta)$}\right)-\bar{\xi}_{N}\left(\beta^{0},\hat{p},\hat{q}(\beta^{0})\right)|}{1+C\sqrt{N}|\beta-\beta^{0}|}$
$\displaystyle=\sup_{|\beta-\beta_{0}|\leq\delta_{N}}\frac{|\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\hat{\nu}_{i}\left(\hat{\tau}_{i}(\beta)-\hat{\tau}_{i}(\beta^{0})\right)|}{1+C\sqrt{N}|\beta-\beta^{0}|}$
(B.7)

Proof of equation B.7 can be found in proof of Proposition 2.3 and Proposition
2.4 in Chaudhuri and Guilkey (2016), by setting $\omega_{i}=c$ for some
constant $c$ in their corresponding condition. The key condition used in the
proof is that $E\left[\tau_{i}(\beta)\right]=0$ for any $\beta$, and it holds
from the definition of $\tau_{i}(\beta)$.

The analogous proof sketch can be used on the component for $R^{Y}=1$, i.e.,

$\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$

Next, we combine the IPW moment condition and the last component in the
augmentation term into:

$\frac{R^{D}R^{Y}}{p_{11}}\left[\left(Y-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]+\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$
(B.8)

and proof of convergence of equation B.8 can be found in Theorem 8 in Cattaneo
(2010).

###  B.7 Proof of Lemma 5.1

Suppose Assumption SMAR and Overlap hold, and $Y\perp\\!\\!\\!\perp D|Z,X$,
then the AIPW estimator maintains double-robustness, following the fact

$E\left[Y|Z,X\right]=E\left[Y|D,Z,X\right]$

and therefore estimation on $p_{11}$ does not affect consistency of the
primary estimator. Then, there is no correction term needed for the AIPW
estimator, under the SMAR assumption. Or, the correction term derived in the
section B.5 equals to 0 following:

$\displaystyle(1-p_{d})(1-\frac{R^{D}R^{Y}}{p_{11}})\left(E\left[Y|D,Z,X\right]-E\left[Y|Z,X\right]\right)$
$\displaystyle=(1-p_{d})(1-\frac{R^{D}R^{Y}}{p_{11}})\left(E\left[Y|D,Z,X\right]-E\left[Y|D,Z,X\right]\right)=0$

Therefore, $V_{SMAR}=V_{MAR}$.

###  B.8 Proof of Theorem 5.6

We proceed the proof the efficiency following the classical three steps.

#### Step 1

We denote the fully observed variables as $O=(Z,X,R^{D},R^{Y},R^{D}D,R^{Y}Y)$,
and consider a class of parametric submodels indexed by $\theta$ such that the
distribution of $O$ can be expressed as:

$\displaystyle f_{\theta}(O)$
$\displaystyle=\left[p_{\theta,d}(Z,X)p_{\theta,y}^{1}(Z,X,D)f_{\theta}(Y|Z,X,D)f_{\theta}(D|Z,X)\right]^{R^{D}R^{Y}}$
$\displaystyle\;\;\times\left[p_{\theta,d}(Z,X)\left(1-p_{\theta,y}^{1}(Z,X,D)\right)f_{\theta}(D|Z,X)\right]^{R^{D}(1-R^{Y})}$
$\displaystyle\;\;\times\left[\left(1-p_{\theta,d}(Z,X)\right)p_{\theta,y}^{0}(Z,X)f_{\theta}(Y|Z,X)\right]^{(1-R^{D})R^{Y}}$
$\displaystyle\;\;\times\left[\left(1-p_{\theta,d}^{0}(Z,X)\right)\left(1-p_{\theta,y}(Z,X)\right)\right]^{(1-R^{D})(1-R^{Y})}f_{\theta}(Z,X)$

where $p_{\theta,y}^{1}(D,Z,X)$ is defined as the probability of observing $Y$
given $R^{D}=1$; and $p_{\theta,d}^{0}(Z,X)$ defined as the probability of
observing $Y$ given $R^{D}=0$. They are defined formally as:

$\displaystyle p_{\theta,y}^{1}(D,Z,X)$
$\displaystyle=\Pr\left[R^{Y}=1|Z,X,D,R^{D}=1\right]$ $\displaystyle
p_{\theta,y}^{0}(Z,X)$ $\displaystyle=\Pr\left[R^{Y}=1|Z,X,R^{D}=0\right]$

The score function is defined as:

$\displaystyle\mathcal{S}_{\theta}(O)$
$\displaystyle=s_{\theta}(Z,X)+R^{D}R^{Y}s_{\theta}(Y,D|Z,X)+R^{D}(1-R^{Y})s_{\theta}(D|Z,X)+(1-R^{D})R^{Y}s_{\theta}(Y|Z,X)$
$\displaystyle\;\;+\left\\{R^{D}R^{Y}\left(\frac{\dot{p}_{\theta,d}(Z,X)}{p_{\theta,d}(Z,X)}+\frac{\dot{p}_{\theta,y}^{1}(Z,X,D)}{p_{\theta,y}^{1}(Z,X,D)}\right)+R^{D}(1-R^{Y})\left(\frac{\dot{p}_{\theta,d}(Z,X)}{p_{\theta,d}(Z,X)}-\frac{\dot{p}_{\theta,y}^{1}(Z,X,D)}{1-p_{\theta,y}^{1}(Z,X,D)}\right)\right.$
$\displaystyle\;\;+(1-R^{D})R^{Y}\left(\frac{\dot{p}_{\theta,y}^{0}(Z,X)}{p_{\theta,y}^{0}(Z,X)}-\frac{\dot{p}_{\theta,d}(Z,X)}{1-p_{\theta,d}(Z,X)}\right)$
$\displaystyle\;\left.+(1-R^{D})(1-R^{Y})\left(-\frac{\dot{p}_{\theta,y}^{0}(Z,X)}{1-p_{\theta,y}^{0}(Z,X)}-\frac{\dot{p}_{\theta,d}(Z,X)}{1-p_{\theta,d}(Z,X)}\right)\right\\}$
$\displaystyle=s_{\theta}(Z,X)+R^{D}R^{Y}s_{\theta}(Y,D|Z,X)+R^{D}(1-R^{Y})s_{\theta}(D|Z,X)+(1-R^{D})R^{Y}s_{\theta}(Y|Z,X)$
$\displaystyle\;\;+\left\\{\frac{R^{D}-p_{\theta,d}(Z,X)}{p_{\theta,d}(Z,X)(1-p_{\theta,d}(Z,X))}\dot{p}_{\theta,d}(Z,X)+R^{D}\frac{R^{Y}-p_{\theta,y}^{1}(D,Z,X)}{p_{\theta,y}^{1}(D,Z,X)(1-p_{\theta,y}^{1}(D,Z,X))}\dot{p}_{\theta,y}^{1}(D,Z,X)\right.$
$\displaystyle\;\left.(1-R^{D})\frac{R^{Y}-p_{\theta,y}^{0}(Z,X)}{p_{\theta,y}^{0}(D,Z,X)(1-p_{\theta,y}^{0}(Z,X))}\dot{p}_{\theta,y}^{0}(Z,X)\right\\}$

where $s_{\theta}(Z,X)\equiv\frac{\partial}{\partial\theta}f_{\theta}(Z,X)$,
$s_{\theta}(Y,D|Z,X)\equiv\frac{\partial}{\partial\theta}f_{\theta}(Y,D|Z,X)$
, $s_{\theta}(D|Z,X)\equiv\frac{\partial}{\partial\theta}f_{\theta}(D|Z,X)$ ,
$s_{\theta}(Y|Z,X)\equiv\frac{\partial}{\partial\theta}f_{\theta}(Y|Z,X)$, and
$\dot{p}_{\theta,d}(Z,X)=\frac{\partial}{\partial\theta}p_{\theta,d}(Z,X)$,
$\dot{p}_{\theta,y}^{1}(Z,X,D)=\frac{\partial}{\partial\theta}p_{\theta,y}^{1}(Z,X,D)$,
$\dot{p}_{\theta,y}^{0}(Z,X)=\frac{\partial}{\partial\theta}p_{\theta,y}^{0}(Z,X)$.

The tangent set is characterized by:

$\displaystyle\mathcal{T}$ $\displaystyle\equiv
R^{D}R^{Y}f_{11}(Y,D,Z,X)+R^{D}(1-R^{Y})f_{10}(D,Z,X)+(1-R^{D})R^{Y}f_{01}(Y,Z,X)+f_{0}(Z,X)$
$\displaystyle\;\;+R^{D}R^{Y}\frac{b_{11}(D,Z,X)}{c_{11}(D,Z,X)}+R^{D}(1-R^{Y})\frac{b_{10}(D,Z,X)}{c_{10}(D,Z,X)}+(1-R^{D})R^{Y}\frac{b_{01}(D,Z,X)}{c_{01}(D,Z,X)}$
$\displaystyle\;\;+(1-R^{D})(1-R^{Y})\frac{b_{00}(Z,X)}{c_{00}(Z,X)}$

where

$\displaystyle f_{11}(Y,D,Z,X)\text{$\in
L_{0}^{2}\left(F\left(Y,D|Z,X\right)\right),$}f_{10}(D,Z,X)$ $\displaystyle\in
L_{0}^{2}\left(F(D|Z,X)\right)$ $\displaystyle f_{01}(Y,Z,X)\text{$\in
L_{0}^{2}\left(F\left(Y|Z,X\right)\right),$}f_{0}(Z,X)$ $\displaystyle\in
L_{0}^{2}\left(F(Z,X)\right)$

#### Step 2

The full data moment condition is written as:

$AE\left[m_{full}(Z,X,D,Y;\text{$\beta^{0}$})\right]=0$

for any matrix $A$ of size $d_{\theta}\times d_{m}$, where $d_{\theta}$ is the
dimension of unknown parameters while $d_{m}$ is the number of moment
conditions. The matrix $A$ is added to convert an over-identified system of
moment conditions into the just-identified moment conditions. In our
framework, the moment condition $m_{full}$ is defined to be
$m_{full}(Z,X,D,Y;\beta^{0})\equiv Z\left(Y-g(D,X;\beta^{0})\right)$.

$\displaystyle\frac{\partial}{\partial\theta}\beta^{0}(\theta_{0})$
$\displaystyle=-(AG)^{-1}AE\left[m_{full}(Z,X,D,Y;\beta^{0})\frac{\partial}{\partial\theta}logf_{\theta_{0}}(Z,X,D,Y)\right]$
$\displaystyle=-(AG)^{-1}AE\left[m_{full}(Z,X,D,Y;\beta^{0})\left(s(Z,X)^{\prime}+s(D|Z,X)^{\prime}+s(Y|D,Z,X)^{\prime}\right)\right]$

Then, we conjecture such that

$E\left[\text{$\varphi\mathcal{S}_{O}^{\prime}$}\right]=E\left[m_{full}(Z,X,D,Y;\beta^{0})\left(s(Z,X)^{\prime}+s(D|Z,X)^{\prime}+s(Y|D,Z,X)^{\prime}\right)\right]$

Then, we confirm term by term, and the first term equals

$\displaystyle
E\left[\frac{R^{D}R^{Y}}{p_{11}}Z(Y-g(D,X;\beta))\mathcal{S}_{O}\right]$
$\displaystyle=E\left\\{\frac{R^{D}R^{Y}}{p_{11}}Z\left(Y-g(D,X;\beta)\right)\times\left(s_{\theta}(Z,X)+s_{\theta}(Y,D|Z,X)+\frac{R^{D}-p_{\theta,d}(Z,X)}{p_{\theta,d}(Z,X)(1-p_{\theta,d}(Z,X))}\dot{p}_{\theta,d}(Z,X)\right.\right.$
$\displaystyle\left.\left.\;\;\;\;+R^{D}\frac{R^{Y}-p_{\theta,y}^{1}(D,Z,X)}{p_{\theta,y}^{1}(D,Z,X)(1-p_{\theta,y}^{1}(D,Z,X))}\dot{p}_{\theta,y}^{1}(Z,X)\right)\right\\}$
$\displaystyle=E\left\\{Z\left(Y-g(D,X;\beta)\right)\times\left(s_{\theta}(Z,X)+s_{\theta}(Y,D|Z,X)+\frac{R^{D}-p_{\theta,d}(Z,X)}{p_{\theta,d}(Z,X)(1-p_{\theta,d}(Z,X))}\dot{p}_{\theta,d}(Z,X)\right.\right.$
$\displaystyle\left.\left.\;\;\;\;+R^{D}\frac{R^{Y}-p_{\theta,y}^{1}(D,Z,X)}{p_{\theta,y}^{1}(D,Z,X)(1-p_{\theta,y}^{1}(D,Z,X))}\dot{p}_{\theta,y}^{1}(D,Z,X)\right)\right\\}$
$\displaystyle=E\left[m_{full}(Z,X,D,Y;\beta)\left(s(Z,X)^{\prime}+s(D|Z,X)^{\prime}+s(Y|D,Z,X)^{\prime}\right)\right]$

The second equality follows law of iterated expectation and that
$\frac{E\left[R^{D}R^{Y}|Z,X,D,Y\right]}{p_{11}}=1$; the last equality follows
the definition of $m_{full}$ and the fact the SMAR assumption such that

$\displaystyle E\left[R^{D}|Z,X,D,Y\right]-p_{\theta,d}(Z,X)$
$\displaystyle=0$ $\displaystyle
E\left[R^{Y}|Z,X,D,Y,R^{D}=1\right]-p_{\theta,y}^{1}(D,Z,X)$ $\displaystyle=0$

Then we need to show, the expectation of interaction between $\mathcal{S}_{O}$
and the augmenting terms in $\varphi$ equals to zero. For the second term, we
have:

$\displaystyle
E\left\\{\left(\frac{R^{Y}}{p_{y}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\theta)|Z,X\right]\right)\right]\mathcal{S}_{O}\right\\}$
$\displaystyle=E\left\\{\left(1-p_{d}\right)\left(\frac{(1-R^{D})R^{Y}}{p_{01}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\right.$
$\displaystyle\;\;\;\;\;\times
Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\left.\;\;\;\;\times\left(R^{D}R^{Y}s_{\theta}(Y,D|Z,X)+(1-R^{D})R^{Y}s_{\theta}(Y|Z,X)\right)\right\\}$
$\displaystyle=E\left[(1-p_{d})Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right.$
$\displaystyle\;\;\left.\;\;\times\left(s_{\theta}(Y|Z,X)-s_{\theta}(Y,D|Z,X)\right)\right]$
$\displaystyle=E\left[(1-p_{d})Z\left[\left(Y-E\left[g(D,X;\beta)|Z,X\right]\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]s_{\theta}(D|Z,X,Y)\right]=0$

The third term equals to

$\displaystyle
E\left\\{\left(\frac{R^{D}}{p_{d}}-\frac{R^{D}R^{Y}}{p_{11}}\right)Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\mathcal{S}_{O}\right\\}$
$\displaystyle=E\left\\{\left(\frac{R^{D}}{p_{11}+p_{10}}-\frac{R^{D}R^{Y}}{p_{11}}\right)\right.$
$\displaystyle\;\;\;\;\;\times
Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]$
$\displaystyle\left.\;\;\;\;\times\left(R^{D}R^{Y}s_{\theta}(Y,D|Z,X)+R^{D}(1-R^{Y})s_{\theta}(D|Z,X)\right)\right\\}$
$\displaystyle=E\left[\frac{p_{10}}{p_{d}}\tilde{Z}\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right.$
$\displaystyle\;\;\left.\;\;\times\left(s_{\theta}(D|Z,X)-s_{\theta}(Y,D|Z,X)\right)\right]$
$\displaystyle=E\left[-\frac{p_{10}}{p_{d}}Z\left[\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]s_{\theta}(Y|D,Z,X)\right]=0$

The last term equals to

Then, we confirm that $\varphi$ is in the tangent set, we can rewrite
$\varphi$ into the form:

$\displaystyle\varphi$
$\displaystyle=R^{D}R^{Y}\left\\{\frac{1}{p_{11}}\left[Z(Y-g(D,X;\beta))-Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)\right]+\frac{1-p_{d}}{p_{11}}\left[Z\left(Y-E\left[Y|Z,X\right]\right)\right]\right.$
$\displaystyle\;\;\left.+\frac{1}{p_{d}}\left[Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right\\}$
$\displaystyle\;\;+R^{D}(1-R^{Y})\left\\{\frac{1}{p_{d}}\left[Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\right\\}$
$\displaystyle\;\;+(1-R^{D})R^{Y}\left\\{\frac{1-p_{d}}{p_{01}}Z\left(Y-E\left[Y|Z,X\right]\right)\right\\}+Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)$

Set $b_{11},b_{01},b_{10},b_{00}$ to be zero, and we can easily confirm that

$\displaystyle
Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\theta)|Z,X\right]\right)\in
L_{0}^{2}\left(F(Z,X)\right)$
$\displaystyle\frac{1-p_{d}(Z,X)}{p_{01}(Z,X)}Z\left(Y-E\left[Y|Z,X\right]\right)\in
L_{0}^{2}\left(F(Y|Z,X)\right)$
$\displaystyle\frac{1}{p_{d}(Z,X)}\left[Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]\in
L_{0}^{2}\left(F(D|Z,X)\right)$

The last to confirm is that the function following $R^{D}R^{Y}$ is in
$L_{0}^{2}\left(F(Y,D|Z,X)\right)$, and this is the case when either condition
$(1)$ or $(2)$ in Theorem 5.6 holds. Under condition $(1)$:

$\displaystyle
E\left\\{\frac{1}{p_{11}}\left[Z(Y-g(D,X;\beta))-Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)\right]+\frac{1-p_{d}}{p_{11}}\left[Z\left(Y-E\left[Y|Z,X\right]\right)\right]\right.$
$\displaystyle\;\;\;\left.+\frac{1}{p_{d}}\left[Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]|Z,X\right\\}=0$

because all the missing probabilities depend on the fully observed $Z,X$.

Under condition $(2)$:

$\displaystyle
E\left\\{\frac{1}{p_{11}}\left[Z(Y-g(D,X;\beta))-Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)\right]+\frac{1-p_{d}}{p_{11}}\left[Z\left(Y-E\left[Y|Z,X\right]\right)\right]|Z,X,D\right\\}$
$\displaystyle=E\left\\{\frac{1}{p_{11}}\left[Z(Y-g(D,X;\beta))-Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)\right]\right.$
$\displaystyle\;\;\left.\;\;+\frac{1-p_{d}}{p_{11}}\left[Z\left(Y-E\left[Y|D,Z,X\right]\right)\right]|Z,X,D\right\\}$
$\displaystyle=0$

and can be interpreted as a function in
$L_{0}^{2}\left(F\left(Y|D,Z,X\right)\right)$.

$E\left[\frac{1}{p_{d}}\left[Z\left(E\left[Y|D,Z,X\right]-g(D,X;\beta)\right)-Z\left(E\left[Y|Z,X\right]-E\left[g(D,X;\beta)|Z,X\right]\right)\right]|Z,X\right]=0$

and can be interpreted as a function in
$L_{0}^{2}\left(F\left(D|Z,X\right)\right)$.

Note that $s_{\theta}(Y,D|Z,X)=s_{\theta}(Y|D,Z,X)+s_{\theta}(D|Z,X)$.
Therefore, $\varphi\in\mathcal{T}$.

Therefore, given $A$, the efficient influence function is
$-(AG)^{-1}A\varphi$, and the variance of it is
$(AG)^{-1}AV_{MAR}A^{\prime}(AG)^{-1^{\prime}}$. The efficient influence
function involves $A$ affecting the variance, and we choose the variance
minimizer to be $A=G^{\prime}V_{MAR}^{-1}$, then the efficiency bound is
$\Omega=(G^{\prime}V_{MAR}^{-1}G)^{-1}$.

## References

  * Abrevaya [2019] Jason Abrevaya.  Missing dependent variables in fixed-effects models.  _Journal of econometrics_ , 211(1):151–165, 2019\.
  * Abrevaya and Donald [2017] Jason Abrevaya and Stephen G Donald.  A gmm approach for dealing with missing data on regressors.  _Review of Economics and Statistics_ , 99(4):657–662, 2017.
  * Andrea et al. [2001] Rotnitzky Andrea, Daniel Scharfstein, Ting-Li Su, and James Robins.  Methods for conducting sensitivity analysis of trials with potentially nonignorable competing causes of censoring.  _Biometrics_ , 57(1):103–113, 2001.
  * Angrist and Imbens [1995] Joshua D Angrist and Guido W Imbens.  Two-stage least squares estimation of average causal effects in models with variable treatment intensity.  _Journal of the American statistical Association_ , 90(430):431–442, 1995.
  * Balat and Han [2018] Jorge F Balat and Sukjin Han.  Multiple treatments with strategic interaction.  _Available at SSRN 3182766_ , 2018.
  * Barnwell and Chaudhuri [2018] Jean-Louis Barnwell and Saraswata Chaudhuri.  Efficient estimation in sub and full populations with monotonically missing at random data.  Technical report, Technical report, McGill University, 2018.
  * Baum et al. [2003] Christopher F Baum, Mark E Schaffer, and Steven Stillman.  Instrumental variables and gmm: Estimation and testing.  _The Stata Journal_ , 3(1):1–31, 2003.
  * Breunig and Haan [2018] Christoph Breunig and Peter Haan.  Nonparametric regression with selectively missing covariates.  _arXiv preprint arXiv:1810.00411_ , 2018.
  * Carpenter et al. [2006] James R Carpenter, Michael G Kenward, and Stijn Vansteelandt.  A comparison of multiple imputation and doubly robust estimation for analyses with missing data.  _Journal of the Royal Statistical Society: Series A (Statistics in Society)_ , 169(3):571–584, 2006.
  * Cattaneo [2010] Matias D Cattaneo.  Efficient semiparametric estimation of multi-valued treatment effects under ignorability.  _Journal of Econometrics_ , 155(2):138–154, 2010\.
  * Chaudhuri [2020] Saraswata Chaudhuri.  On efficiency gains from multiple incomplete subsamples.  _Econometric Theory_ , 36(3):488–525, 2020.
  * Chaudhuri and Guilkey [2016] Saraswata Chaudhuri and David K Guilkey.  Gmm with multiple missing variables.  _Journal of Applied Econometrics_ , 31(4):678–706, 2016.
  * Chen et al. [2008] Xiaohong Chen, Han Hong, Alessandro Tarozzi, et al.  Semiparametric efficiency in gmm models with auxiliary data.  _The Annals of Statistics_ , 36(2):808–843, 2008\.
  * Cheng [1994] Philip E Cheng.  Nonparametric estimation of mean functionals with data missing at random.  _Journal of the American statistical association_ , 89(425):81–87, 1994.
  * Ciliberto and Tamer [2009] Federico Ciliberto and Elie Tamer.  Market structure and multiple equilibria in airline markets.  _Econometrica_ , 77(6):1791–1828, 2009.
  * Finkelstein et al. [2012] Amy Finkelstein, Sarah Taubman, Bill Wright, Mira Bernstein, Jonathan Gruber, Joseph P Newhouse, Heidi Allen, Katherine Baicker, and Oregon Health Study Group.  The oregon health insurance experiment: evidence from the first year.  _The Quarterly journal of economics_ , 127(3):1057–1106, 2012.
  * Gad [2011] Ahmed M Gad.  A selection model for longitudinal data with non-ignorable non-monotone missing values.  _Journal of Data Science_ , 9(171-180), 2011.
  * Glynn and Quinn [2010] Adam N Glynn and Kevin M Quinn.  An introduction to the augmented inverse propensity weighted estimator.  _Political analysis_ , pages 36–56, 2010.
  * Groenwold et al. [2012] Rolf HH Groenwold, A Rogier T Donders, Kit CB Roes, Frank E Harrell Jr, and Karel GM Moons.  Dealing with missing outcome data in randomized trials and observational studies.  _American journal of epidemiology_ , 175(3):210–217, 2012.
  * Horowitz and Manski [2000] Joel L Horowitz and Charles F Manski.  Nonparametric analysis of randomized experiments with missing covariate and outcome data.  _Journal of the American statistical Association_ , 95(449):77–84, 2000.
  * Little [1992] Roderick JA Little.  Regression with missing x’s: a review.  _Journal of the American statistical association_ , 87(420):1227–1237, 1992.
  * Little and Rubin [2002] Roderick JA Little and Donald B Rubin.  Bayes and multiple imputation.  _Statistical analysis with missing data_ , pages 200–220, 2002.
  * Newey and McFadden [1994] KW Newey and Daniel McFadden.  Large sample estimation and hypothesis.  _Handbook of Econometrics, IV, Edited by RF Engle and DL McFadden_ , pages 2112–2245, 1994.
  * Newey [1994] Whitney K Newey.  The asymptotic variance of semiparametric estimators.  _Econometrica: Journal of the Econometric Society_ , pages 1349–1382, 1994.
  * Newey [1997] Whitney K Newey.  Convergence rates and asymptotic normality for series estimators.  _Journal of econometrics_ , 79(1):147–168, 1997\.
  * Newey [2013] Whitney K Newey.  Nonparametric instrumental variables estimation.  _American Economic Review_ , 103(3):550–56, 2013\.
  * Newey and Powell [2003] Whitney K Newey and James L Powell.  Instrumental variable estimation of nonparametric models.  _Econometrica_ , 71(5):1565–1578, 2003.
  * Qi and Sun [2014] Li Qi and Yanqing Sun.  Missing data approaches for probability regression models with missing outcomes with applications.  _Journal of statistical distributions and applications_ , 1(1):23, 2014.
  * Robins [1997] James M Robins.  Non-response models for the analysis of non-monotone non-ignorable missing data.  _Statistics in medicine_ , 16(1):21–37, 1997\.
  * Robins and Gill [1997] James M Robins and Richard D Gill.  Non-response models for the analysis of non-monotone ignorable missing data.  _Statistics in medicine_ , 16(1):39–56, 1997\.
  * Robins et al. [1994] James M Robins, Andrea Rotnitzky, and Lue Ping Zhao.  Estimation of regression coefficients when some regressors are not always observed.  _Journal of the American statistical Association_ , 89(427):846–866, 1994.
  * Rosenbaum and Rubin [1983] Paul R Rosenbaum and Donald B Rubin.  The central role of the propensity score in observational studies for causal effects.  _Biometrika_ , 70(1):41–55, 1983.
  * Scharfstein et al. [1999] Daniel O Scharfstein, Andrea Rotnitzky, and James M Robins.  Adjusting for nonignorable drop-out using semiparametric nonresponse models.  _Journal of the American Statistical Association_ , 94(448):1096–1120, 1999.
  * Seaman et al. [2013] Shaun Seaman, John Galati, Dan Jackson, and John Carlin.  What is meant by” missing at random”?  _Statistical Science_ , pages 257–268, 2013.
  * Seaman and Vansteelandt [2018] Shaun R Seaman and Stijn Vansteelandt.  Introduction to double robust methods for incomplete data.  _Statistical science: a review journal of the Institute of Mathematical Statistics_ , 33(2):184, 2018.
  * Seaman and White [2013] Shaun R Seaman and Ian R White.  Review of inverse probability weighting for dealing with missing data.  _Statistical methods in medical research_ , 22(3):278–295, 2013.
  * Sun and Tchetgen Tchetgen [2018] BaoLuo Sun and Eric J Tchetgen Tchetgen.  On inverse probability weighting for nonmonotone missing at random data.  _Journal of the American Statistical Association_ , 113(521):369–379, 2018.
  * Sun et al. [2018] BaoLuo Sun, Neil J Perkins, Stephen R Cole, Ofer Harel, Emily M Mitchell, Enrique F Schisterman, and Eric J Tchetgen Tchetgen.  Inverse-probability-weighted estimation for monotone and nonmonotone missing data.  _American journal of epidemiology_ , 187(3):585–591, 2018.
  * Tamer [2003] Elie Tamer.  Incomplete simultaneous discrete response model with multiple equilibria.  _The Review of Economic Studies_ , 70(1):147–165, 2003.
  * Tchetgen et al. [2018] Eric J Tchetgen Tchetgen, Linbo Wang, and BaoLuo Sun.  Discrete choice models for nonmonotone nonignorable missing data: Identification and inference.  _Statistica Sinica_ , 28(4):2069–2088, 2018.
  * Theil [1971] Henri Theil.  Principles of econometrics.  Technical report, 1971.
  * Tsiatis [2007] Anastasios Tsiatis.  _Semiparametric theory and missing data_.  Springer Science & Business Media, 2007.
  * Van Buuren [2018] Stef Van Buuren.  _Flexible imputation of missing data_.  CRC press, 2018.
  * Wang et al. [2010] Lu Wang, Andrea Rotnitzky, and Xihong Lin.  Nonparametric regression with missing outcomes using weighted kernel estimating equations.  _Journal of the American Statistical Association_ , 105(491):1135–1146, 2010.
  * Wooldridge [2007] Jeffrey M Wooldridge.  Inverse probability weighted estimation for general missing data problems.  _Journal of econometrics_ , 141(2):1281–1301, 2007.