# Instrument-based estimation of full treatment effects with movers

Didier Nibbering Department of Econometrics and Business Statistics, Monash
University, Melbourne, Australia<EMAIL_ADDRESS>Matthijs
Oosterveen †† We thank Massimo Anelli, Akanksha Negi, Denni Tommasi, and
Dinand Webbink for valuable comments. The authors have no relevant or material
financial interests that relate to the research described in this paper. All
omissions and errors are our own.  Department of Economics, Lisbon School of
Economics and Management, and Advance/CSG, University of Lisbon, Lisbon
1200-781, Portugal<EMAIL_ADDRESS>
###### Abstract

The effect of the full treatment is a primary parameter of interest in policy
evaluation, while often only the effect of a subset of treatment is estimated.
We partially identify the local average treatment effect of receiving full
treatment (LAFTE) using an instrumental variable that may induce individuals
into only a subset of treatment (movers). We show that movers violate the
standard exclusion restriction, necessary conditions on the presence of movers
are testable, and partial identification holds under a double exclusion
restriction. We identify movers in four empirical applications and estimate
informative bounds on the LAFTE in three of them.

JEL: C36, D04
Keywords: Instrumental variables, Local average treatment effects, Movers,
Exclusion restriction

##  1 Introduction

This paper develops an instrumental variable (IV) framework that partially
identifies the local average treatment effect (LATE) of receiving full
treatment compared to no treatment. The effect of the full treatment program
is a primary parameter of interest in policy evaluation. However, in many
instances, policy evaluations yield estimates of only a subset of the
treatment program. For instance, when the effect of college completion may be
of interest, the effect of college enrollment is estimated. This is a problem
for two main reasons. First, most treatment programs are designed to be
received in full and the effects of separate treatment parts may be less
informative. Second, there is often non-compliance with randomized treatment
assignment and hence identification of treatment effects in an IV framework
requires an exclusion restriction. This restriction may be violated by
individuals moving in or out of treatment after the start of the program.

Individuals that move in or out of treatment are present in many policy
evaluation settings. For instance, Leu (2017) reports that approximately 30%
of college students changes major, Sugar et al. (2021) describe the high
prevalence of Medicaid coverage disruptions, often referred to as churning,
due to income fluctuations, and Heckman et al. (2000) discuss the incidence of
control group substitution and treatment group dropout across various job
market training programs. This implies that instead of staying in or out of
treatment during the whole program, there are late-adopters missing the first
part of treatment or dropouts missing the final part of treatment.

The IV exclusion restriction rules out certain types of movers depending on
the definition of the treatment variable. For instance, Silliman and Virtanen
(2022), Grosz (2020), Burde and Linden (2013), and Hoekstra (2009) take
enrollment at the start of an educational program as treatment indicator. In
this case, the exclusion restriction does not allow for movers that are
induced by the instrument to only take the second part of the treatment. Ketel
et al. (2016) and Zimmerman (2014) use enrollment at the end of an educational
program as treatment indicator. In this case, the exclusion restriction does
not allow for movers that are induced by the instrument to only take the first
part of the treatment.

Researchers are familiar with the challenging restrictions imposed by the
exclusion restriction. For instance, Silliman and Virtanen (2022) estimate the
returns to vocational secondary education enrollment, and discuss how student
dropout may violate the exclusion restriction. Grosz (2020) discusses how
movers in and out of a nursing program affect the exclusion restriction with
different definitions of treatment, such as enrolling at the start or ever
enrolling, in the nursing program. Finkelstein et al. (2012) suggest that
different definitions for the treatment variable of medicaid insurance can be
used to provide a lower and upper bound for a LATE using the largest and
smallest first stage, respectively.

This paper partially identifies the LATE of receiving full treatment compared
to no treatment under a double exclusion restriction. This parameter is
referred to as the local average full treatment effect (LAFTE). Our IV
framework includes the binary potential treatment status for the first and
second part of treatment with only one binary instrument. The instrument may
induce individuals to obtain full treatment, or only the first or second part
of the treatment. We refer to the latter group of individuals as movers.
Within this framework, movers do not violate the IV exclusion restriction and
LATEs are only identified under additional assumptions. The proposed double
exclusion restriction extends this exclusion restriction –the instrument can
only affect the outcome variable through treatment enrollment– with the
condition that the instrument can only affect treatment enrollment in the
second part through enrollment in the first part. We provide a procedure for
testing necessary conditions of the presence of movers and of the double
exclusion restriction.

The partial identification of the LAFTE is achieved with nonparametric sharp
bounds. The bounds formalize the intuition that different definitions for the
treatment variable can be utilized to identify the effect of receiving a full
treatment program, while allowing for movers. We show that the bounds can be
tightened using the monotone treatment response and monotone treatment
selection assumptions of Manski (1997) and Manski and Pepper (2000) in
addition to the double exclusion restriction. Less informative bounds rely
solely on double exclusion and a bounded response.

The double exclusion restriction holds if a delayed treatment response to
treatment assignment is absent, as it rules out the movers that are induced by
the instrument to only take the second part of treatment. It now follows that
the common practice of using IV with enrollment in the first part of treatment
as treatment indicator identifies the LATE of taking the first part of the
treatment. Our double exclusion restriction is similar to, albeit weaker than,
the dynamic exclusion restriction of Angrist et al. (2022). The dynamic
exclusion restriction imposes the additional condition that treatment
enrollment in the first part can only affect the outcome variable though
enrollment in the second part. This either rules out movers, restricts
treatment effect heterogeneity, or a combination of these two. In this case,
LAFTE can be point-identified using the IV method of Imbens and Angrist
(1994).

The policy relevance of the LAFTE is illustrated with four empirical
applications from different fields of applied economics: a health program by
Finkelstein et al. (2012); Finkelstein et al. (2016), a labour program by
Wheeler et al. (2022), an educational program by Anelli (2020), and a
development program by Burde and Linden (2013). For instance, Finkelstein et
al. (2016) use a randomized opportunity to apply for Medicaid as an instrument
for Medicaid enrollment. They find that half a year of Medicaid enrollment
increases the probability of an emergency department visit by 0.088 percentage
points for the compliers. However, the short-term impact of Medicaid on health
care utilization may be small, and hence policymakers make efforts to reduce
Medicaid coverage disruptions (Sugar et al., 2021). Indeed, our estimated
LAFTE shows that continuous Medicaid enrollment for two years increases the
probability of an emergency department visit between 0.081 and 0.224
percentage points for the compliers. This suggests that policies encouraging
continuous enrollment result in larger effects on health care utilization. We
discuss similar policy implications in the other applications.

The applications also underline the empirical applicability of our methods.
The four studies all fit our IV framework, with data available on both
enrollment in the first and second part of the treatment. By testing the
necessary conditions, we find evidence for the presence of movers in all four
studies. In the health program, the movers even establish the majority of the
compliers, consistent with the high prevalence of Medicaid coverage
disruptions. Except for the development program, the necessary conditions for
the double exclusion restriction cannot be rejected. In these three
applications, the lower bounds on the LAFTEs are statistically significantly
different from zero, and two of the three upper bounds are informative on the
size of the LAFTE.

There is an extensive literature that bounds LATEs under violations of the
exclusion restriction. For instance, Flores and Flores-Lagunes (2013)
establishes partial identification using one treatment variable. Instead of
our approach of using treatment indicators for the first and second part of
the treatment, Mealli and Pacini (2013) use two outcome variables to construct
bounds. Conley et al. (2012) propose Bayesian approaches for identification
under relaxed exclusion restrictions. Swanson et al. (2018) provide an
overview on treatment bounds for settings with a binary instrument, binary
treatment, and a binary outcome. Heckman et al. (2000) bound average treatment
effects in the presence of control group substitution and treatment group
dropping out, without employing instrumental variables.

The identification of causal parameters with multiple treatment parts has been
studied in similar settings. First, we study the identification of the LAFTE
by indexing the potential outcomes with the full treatment history. Similar
potential outcome models are considered in the dynamic treatment literature.
For instance, Lechner (2009) identifies treatment effects using conditional
independence assumptions and Ding and Lehrer (2010) employ a structural
economic model. Blackwell (2017) identifies the treatment effects for
different types of movers between two sequential treatments with one
instrument for each treatment. Heckman et al. (2016) also identify dynamic
treatment effects using multiple instruments. In contrast, our identification
approach relies on only one instrument and does not require a conditionally
exogenous treatment or an underlying structural model.

Second, instead of estimating LATEs, recent difference-in-differences
techniques are used to estimate the intention to treat in the presence of
movers. For instance, Hull (2018) identifies mover average treatment effects
for the individuals that move in or out of treatment. Verdier (2020)
extrapolates treatment effects for movers to stayers: individuals whose
treatment status does not change. Specific types of movers are studied in, for
example, Athey and Imbens (2022) who only allow for late-adopters in a
staggered adoption framework.

Third, instead of estimating the LAFTE, causal mediation analysis identifies
the direct and indirect effects of enrollment in the first part of treatment.
Our proposed double exclusion restriction imposes that all effects of the
instrument on the outcome variable go only through enrollment in the first
part of treatment. However, the effect of enrollment in the first part on the
outcome can either be direct or via treatment enrollment in the second part of
the treatment. Hence, enrollment in the second part can be considered a
mediator, and Huber (2019) provides an overview of estimation methods for
direct and indirect effects of enrollment in the first part of treatment.

The outline of this paper is as follows. Section 2 defines the causal
parameter of interest. Section 3 discusses our IV framework with movers.
Section 4 introduces the double exclusion restriction, partial identification
of the LAFTE, and testable necessary conditions for the presence of movers and
the double exclusion restriction. Section 5 applies the proposed methods to
four empirical treatment evaluation settings. Section 6 concludes.

##  2 Full treatment effect and instrumental variables

Assume a setting in which individuals are randomly assigned to treatment,
after which full treatment or only a subset of treatment can be received. The
binary instrumental variable $Z\in\\{0,1\\}$ equals one if an individual is
randomly assigned to treatment. The treatment indicator $D_{t}\in\\{0,1\\}$
equals one if treatment part $t$ is received, where $t=1$ corresponds to, for
instance, college enrollment and $t=2$ to college graduation. When compliance
with the treatment assignment is not mandatory, the treatment indicators may
take different values than the instrument. After assignment, individuals may
obtain full treatment $\\{D_{1}=1,D_{2}=1\\}$ or no treatment
$\\{D_{1}=0,D_{2}=0\\}$. Moreover, $D_{1}$ may be different from $D_{2}$ in
settings that allow for late enrollment into treatment $\\{D_{1}=0,D_{2}=1\\}$
and/or dropping out of treatment $\\{D_{1}=1,D_{2}=0\\}$. Finally, let the
variable $Y$ be the observed outcome of interest.

###  2.1 Causal parameters of interest

A causal parameter of interest to policy makers is the average treatment
effect (ATE) of receiving full treatment $\mathbb{E}[Y(1,1)-Y(0,0)]$, where
$Y(D_{1},D_{2})$ is an individual’s potential outcome for values of $D_{1}$
and $D_{2}$. This ATE is the average difference between receiving both parts
of the treatment ($Y(1,1)$) compared to no treatment at all ($Y(0,0)$). The
ATE of the first part of treatment $\mathbb{E}[Y(1,d_{2})-Y(0,d_{2})]$ or the
second part $\mathbb{E}[Y(d_{1},1)-Y(d_{1},0)]$ may be of interest for the
evaluation of which treatment period is most effective. However, most
treatments are designed to be received in full. For instance, the first part
of a training program builds knowledge that prepares for the second part, and
the second part builds upon the knowledge obtained in the first. Hence, the
average treatment effect of receiving full treatment is often of primary
interest to policy makers.

Since we only observe one $Y$ for each individual and individuals may not
comply with their assigned treatment, ATEs are in general not identified. In
this case it is common to report local ATEs (LATEs) that can be identified by
using the random treatment assignment as an instrumental variable for
treatment enrollment. The LATE equals the average treatment effect for the
individuals who are induced into treatment by the instrument, and is therefore
the ATE for a subpopulation.

###  2.2 The standard IV framework

The standard IV framework introduced by Imbens and Angrist (1994) defines an
IV estimand and shows that it identifies a LATE of a treatment variable $D$ on
the outcome of interest $Y$ using the potential outcome framework and four
assumptions on the instrumental variable $Z$. The IV estimand is defined as

$\displaystyle\beta_{IV}(D)=\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D|Z]},$
(1)

where $\Delta\mathbb{E}[A|Z]=\mathbb{E}[A|Z=1]-\mathbb{E}[A|Z=0]$. The
potential outcome framework links the observed treatment indicator $D$ to the
potential treatment status $D(Z)$ as $D=D(1)Z+D(0)(1-Z)$, and the observed
outcome $Y$ to the potential outcomes $Y(Z,D)$ as
$Y=Y(1,1)ZD+Y(1,0)Z(1-D)+Y(0,1)(1-Z)D+Y(0,0)(1-Z)(1-D)$.

The four instrumental variable assumptions are:

######  Assumption 1 (Instrumental variable assumptions).

  1. 1.

(Independence) $Y(z,d),D(z)\perp Z$ for all $z,d$.

  2. 2.

(Relevance) $\Delta\mathbb{E}[D|Z]>0$.

  3. 3.

(Monotonicity) $D(1)\geq D(0)$.

  4. 4.

(Exclusion) $Y(z,d)=Y(d)$ for all $z,d$.

Under 1, the IV estimand in (1) identifies a LATE if the treatment can be
summarized by a single binary variable $D$:
$\beta_{IV}=\mathbb{E}[Y(1)-Y(0)|C]$, where $C=\\{D(1)-D(0)=1\\}$ defines the
individuals induced into treatment by the instrument. These individuals are
referred to as compliers.

Table 1 shows four ways the treatment indicator $D$ can be constructed when
$\\{Z,D_{1},D_{2},Y\\}$ is observed. The treatment indicator can be defined as
enrollment at the start of the treatment ($D_{1}$), enrollment at the end of
the treatment ($D_{2}$), received full treatment ($D_{\land}$), or received at
least one part of the treatment ($D_{\lor}$). The third column in Table 1
shows that different definitions of the treatment indicator are used in the
applied economics literature.

Table 1: Treatment indicator specifications

Definition $D$ | Treatment indicator |  Examples in empirical work
---|---|---
| for enrollment in … |
$D_{1}$ | first part |  Silliman and Virtanen (2022), Grosz (2020), Zimmerman (2014), Burde and Linden (2013), Hoekstra (2009)
$D_{2}$ | second part |  Ketel et al. (2016), Zimmerman (2014)
$D_{\land}=D_{1}D_{2}$ | both parts |
$D_{\lor}=D_{1}+D_{2}-D_{1}D_{2}$ | at least one part |  Wheeler et al. (2022), Anelli (2020), Finkelstein et al. (2012)

  * Notes: this table provides an overview of four treatment indicators that can be constructed when $\\{D_{1},D_{2}\\}$ is observed and gives examples of research articles using each treatment indicator.

##  3 Instrumental variables with movers

When the instrument induces individuals to obtain only a subset of the
treatment, all four definitions for $D$ may result in violations of the
exclusion restriction made by 1.4. These violations arise when $Z$ affects $Y$
while $D$ remains constant. For instance, if the researcher uses $D=D_{1}$ and
the instrument induces some individuals into only the second part of
treatment, $Z$ may affect $Y$ through $D_{2}$ whereas $D_{1}$ is not affected.
Similarly, if the researcher uses $D=D_{2}$, the exclusion restriction may be
violated if the instrument induces some individuals into only the first part
of treatment. An instrument inducing individuals from no treatment at all to
only part of the treatment, or an instrument inducing individuals from only
part of the treatment to full treatment, may run into problems with
$D=D_{\land}$ or $D=D_{\lor}$ respectively.

This section extends the standard IV framework to include the potential
treatment status at the first and second part of the treatment. Within this
framework, five different groups of compliers can be distinguished. The four
additional groups of compliers cause the violations of the exclusion
restriction described above. We show that in the presence of these four
groups, LATEs can only be point identified under additional strict
assumptions.

###  3.1 Model and notation

We extend the potential outcome model to include two potential treatments
$D_{1}(Z)$ and $D_{2}(Z,D_{1})$ and a potential outcome $Y(Z,D_{1},D_{2})$.
Potential and observed treatments and outcomes are linked as follows,

$\displaystyle D_{1}=$ $\displaystyle D_{1}(1)Z+D_{1}(0)(1-Z),$ (2)
$\displaystyle D_{2}=$
$\displaystyle[D_{2}(1,1)D_{1}+D_{2}(1,0)(1-D_{1})]Z+[D_{2}(0,1)D_{1}+D_{2}(0,0)(1-D_{1})](1-Z),$
(3) $\displaystyle Y=$
$\displaystyle[Y(1,1,1)D_{1}D_{2}+Y(1,0,1)(1-D_{1})D_{2}+Y(1,1,0)D_{1}(1-D_{2})+$
(4)
$\displaystyle\,\,Y(1,0,0)(1-D_{1})(1-D_{2})]Z+[Y(0,1,1)D_{1}D_{2}+Y(0,0,1)(1-D_{1})D_{2}+$
$\displaystyle\,\,Y(0,1,0)D_{1}(1-D_{2})+Y(0,0,0)(1-D_{1})(1-D_{2})](1-Z).$

2 replaces 1 for the potential outcome framework defined above:

######  Assumption 2 (Instrumental variable assumptions).

  1. 1.

(Independence) $Y(z,d_{1},d_{2}),D_{1}(z),D_{2}(z,d_{1})\perp Z$ for all
$z,d_{1},d_{2}$.

  2. 2.

(Relevance) $\Delta\mathbb{E}[D|Z]>0$ for $D=D_{1},D_{2},D_{\land},D_{\lor}$.

  3. 3.

(Monotonicity) $D_{1}(1)\geq D_{1}(0)$ and $D_{2}(1,D_{1}(1))\geq
D_{2}(0,D_{1}(0))$.

  4. 4.

(Exclusion) $Y(z,d_{1},d_{2})=Y(d_{1},d_{2})$ for all $z,d_{1},d_{2}$.

These assumptions hold if $Z$ is randomly assigned (2.1), if there are
individuals that are induced into full treatment by the instrument (2.2), and
there are no individuals induced to move out of any part of treatment by the
instrument (2.3). 2.4 implies that

$\displaystyle
Y=Y(1,1)D_{1}D_{2}+Y(0,1)(1-D_{1})D_{2}+Y(1,0)D_{1}(1-D_{2})+Y(0,0)(1-D_{1})(1-D_{2}),$
(5)

which shows that $Y$ is a function of the two treatment parts, as defined in
Section 2.1.

In contrast, the exclusion restriction in 1.4 imposes that
$Y(z,d_{1},d_{2})=Y(d)$ for all $z,d_{1},d_{2}$, which only depends on one
summary of the treatment $D=d$. 1.4 is more restrictive as it does not allow
$Z$ to affect $Y$ through both treatment parts, whereas 2.4 does. For
instance, with $D=D_{1}$, 1.4 imposes that $Y(z,d_{1},d_{2})=Y(d_{1})$ and
therefore $Y$ does not depend on the second treatment part.

The potential outcome model in (5) implies that there are different types of
compliers. Since individuals can comply with the instrument in the first part,
the second part, or the full treatment, the group of compliers $C$ consists of
five different complier groups:
$C=\\{\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\},\\{N_{1},C_{2}\\},\\{A_{1},C_{2}\\}\\}$.
These groups are defined as follows:

  1. 1.

$\\{C_{1},C_{2}\\}=\\{D_{1}(1)-D_{1}(0)=1,D_{2}(1,1)-D_{2}(0,0)=1\\},$

  2. 2.

$\\{C_{1},N_{2}\\}=\\{D_{1}(1)-D_{1}(0)=1,D_{2}(1,1)=D_{2}(0,0)=0\\},$

  3. 3.

$\\{C_{1},A_{2}\\}=\\{D_{1}(1)-D_{1}(0)=1,D_{2}(1,1)=D_{2}(0,0)=1\\},$

  4. 4.

$\\{N_{1},C_{2}\\}=\\{D_{1}(1)=D_{1}(0)=0,D_{2}(1,0)-D_{2}(0,0)=1\\},$

  5. 5.

$\\{A_{1},C_{2}\\}=\\{D_{1}(1)=D_{1}(0)=1,D_{2}(1,1)-D_{2}(0,1)=1\\},$

where $C_{t}$, $N_{t}$, and $A_{t}$ refer to respectively compliers, never-
takers, and always-takers in treatment part $t$. The first group of compliers
$\\{C_{1},C_{2}\\}$ is induced into full treatment by the instrument, and we
refer to this group as full compliers. We define the individuals that are
induced to obtain only part of the treatment by the instrument as movers.
Hence, there are four types of movers: $\\{C_{1},N_{2}\\}$ and
$\\{C_{1},A_{2}\\}$ are induced into only the first part of treatment, whereas
$\\{N_{1},C_{2}\\}$ and $\\{A_{1},C_{2}\\}$ are induced into only the second
part of treatment.

Our definition of movers concerns individuals who change treatment status
after the start of the program depending on the value of the instrument. The
two mover types $\\{C_{1},N_{2}\\}$ and $\\{A_{1},C_{2}\\}$ dropout from
treatment and miss the second part of the program when $Z=1$ and $Z=0$,
respectively. The two mover types $\\{C_{1},A_{2}\\}$ and $\\{N_{1},C_{2}\\}$
are late-adopters and miss the first part of the program when $Z=0$ and $Z=1$,
respectively. Individuals with $\\{N_{1},A_{2}\\}$ or $\\{A_{1},N_{2}\\}$ are
not included, as they do not affect the LATE identification. Hence, our mover
group is different from the more general definition of movers as the set of
all individuals with $D_{1}\neq D_{2}$, as used by, for instance, Hull (2018)
in difference-in-differences estimation.

Using the potential outcome framework in (5) and the definition of the
complier group $C$, the local average full treatment effect (LAFTE) can now be
expressed as $\mathbb{E}[Y(1,1)-Y(0,0)|C]$.

###  3.2 What IV can identify with movers

The following lemma shows what the first stages $\Delta\mathbb{E}[D|Z]$ for
different definitions of $D$ and the reduced form $\Delta\mathbb{E}[Y|Z]$
identify when the potential outcome framework takes both treatment parts into
account.

######  Lemma 1 (First stages and reduced form).

Under 2 it holds that

  1. 1.

$\Delta\mathbb{E}[D_{1}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]$.

  2. 2.

$\Delta\mathbb{E}[D_{2}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]$.

  3. 3.

$\Delta\mathbb{E}[D_{\land}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[A_{1},C_{2}]$.

  4. 4.

$\Delta\mathbb{E}[D_{\lor}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[N_{1},C_{2}]$.

  5. 5.

$\begin{aligned}
\Delta\mathbb{E}[Y|Z]=&\,\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+\\\
&\,\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+\\\
&\,\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}].\end{aligned}$

The proof is deferred to Appendix A.

Lemma 1 shows that the first stages represent the proportion of different
complier types depending on the definition of $D$. Each first stage captures
the proportion of the full compliers plus the proportions of two mover types
that comply with the corresponding $D$. For instance, with $D=D_{1}$, the
movers $\\{C_{1},N_{2}\\}$ and $\\{C_{1},A_{2}\\}$ comply in the first part of
treatment.

The reduced form coefficient $\Delta\mathbb{E}[Y|Z]$ equals a weighted sum of
LATEs, in which each weight corresponds to the proportion of one of the five
complier types. Each LATE corresponds to a different treatment effect for a
complier type. The first term of the reduced form in Lemma 1 corresponds to
the LAFTE, but only for the full compliers $\\{C_{1},C_{2}\\}$.

For each of the definitions of $D$ in Table 1, the IV estimand in (1)
identifies a weighted average of LATEs plus a bias term. For instance, for
$D=D_{1}$ we have

$\displaystyle\left.\begin{aligned}
\beta_{IV}(D_{1})=&\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]w(C_{1},C_{2})+\\\
&\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]w(C_{1},N_{2})+\\\
&\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]w(C_{1},A_{2})+\end{aligned}\right\\}\text{
weighted LATEs}$ (6) $\displaystyle\left.\begin{aligned}
&\mathbb{E}[Y(1,1)-Y(1,0)|N_{1},C_{2}]w(N_{1},C_{2})+\\\
&\mathbb{E}[Y(0,1)-Y(0,0)|A_{1},C_{2}]w(A_{1},C_{2}),\end{aligned}\right\\}\text{bias
\hskip 59.26192pt}$

with
$w(G_{1},G_{2})=\frac{\mathbb{P}[G_{1},G_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}$,
and where $\\{G_{1},G_{2}\\}$ can represent any of the five complier groups.
The bias terms reflect the effect of $D_{2}$ on $Y$ while keeping $D_{1}$
constant. It follows that the IV estimand does not have a clear causal
interpretation:

######  Proposition 1 (No causal interpretation of $\beta_{IV}(D)$).

Under 2 it holds that the IV estimand in (1) has no causal interpretation with
either $D=D_{1},D_{2},D_{\land},D_{\lor}$ as defined in Table 1.

The proof follows directly from Lemma 1.

Figure 1 visualizes the identification problem, where each arrow represents a
possible effect among $\\{Z,D_{1},D_{2},Y\\}$ under 2. The instrument induces
the full compliers $\\{C_{1},C_{2}\\}$ into full treatment, but also induces
the mover groups into only a subset of the treatment. The figure suggests that
the LAFTE can be identified under two different additional assumptions. The
first assumption rules out the presence of all mover groups. The second
assumes that the treatment effects for movers are identical to their full
treatment effect. Corollary 1 and 2 below formalize this intuition.

Figure 1: An overview of the possible effects in IV with
movers.$Z$$D_{1}(z)$$D_{2}(z,d_{1})$$Y(d_{1},d_{2})$$\\{C_{1},C_{2}\\}$ Double
exclusion$\\{N_{1},C_{2}\\},\\{A_{1},C_{2}\\}$$\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}$
Notes: under 2, $Z$ cannot be affected by any variable, $D_{1}(z)$ can only be
affected by $Z$, $D_{2}(z,d_{1})$ can only be affected by both $Z$ and
$D_{1}$, and $Y(d_{1},d_{2})$ can only be affected by both $D_{1}$ and
$D_{2}$. The dashed arrow represents the effect prevented by 3.

######  Corollary 1 (No movers).

Under 2 and
$\mathbb{P}[C_{1},N_{2}]=\mathbb{P}[C_{1},A_{2}]=\mathbb{P}[N_{1},C_{2}]=\mathbb{P}[A_{1},C_{2}]=0$,
it holds that

$\displaystyle\beta_{IV}(D)=\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D|Z]}=\mathbb{E}[Y(1,1)-Y(0,0)|C],$
(7)

for $D=D_{1},D_{2},D_{\land},D_{\lor}$.

The proof follows directly from Lemma 1. Corollary 1 may suit a setting in
which, for instance, treatment completion is mandatory after treatment
enrollment and treatment completion is impossible without treatment
enrollment. It follows that all compliers are full compliers
$C=\\{C_{1},C_{2}\\}$.

######  Corollary 2 (Homogeneous treatment effects movers).

Under 2 and the treatment effect homogeneity assumptions in Appendix B, it
holds that the IV estimand in (1) with $D=D_{1},D_{2},D_{\land},D_{\lor}$
identifies the LAFTE for three complier groups.

The proof is deferred to Appendix B. Corollary 2 may suit a setting in which,
for instance, $\mathbb{E}[Y(1,d_{2})-Y(0,d_{2})|G_{1},G_{2}]$ does not depend
on $d_{2}$ for the movers $\\{G_{1},G_{2}\\}$. This setting implies
homogeneous treatment effects within mover types.

The LAFTE can also be identified by a combination of assumptions on the
presence of certain mover types and the homogeneity of certain treatment
effects. This is formalized by the dynamic exclusion model discussed by
Angrist et al. (2022):

$\displaystyle D_{1}=$ $\displaystyle\delta+\pi Z+\eta,$ (8) $\displaystyle
D_{2}=$ $\displaystyle\alpha+\psi D_{1}+\xi,$ (9) $\displaystyle Y=$
$\displaystyle\beta+\mu D_{2}+\varepsilon,$ (10)

where $\delta$, $\pi$, $\alpha$, $\psi$, $\beta$, and $\mu$ are coefficients,
$\eta$, $\xi$, and $\varepsilon$ are error terms, and we exclude additional
covariates. From (9) follows that $D_{2}$ does not depend on $Z$, and
consequently the movers $\\{N_{1},C_{2}\\}$ and $\\{A_{1},C_{2}\\}$ in Figure
1 do not exist. Similarly, (10) shows that $Y$ does not depend on $D_{1}$.
Figure 1 shows that this is the case if either $\\{C_{1},N_{2}\\}$ and
$\\{C_{1},A_{2}\\}$ do not exist, or if $Y(0,d_{2})=Y(1,d_{2})$ for all
$d_{2}$.

##  4 Movers with a double exclusion restriction

This section proposes a partial identification strategy for the LAFTE. We
replace the single exclusion restriction in 2.4, by a double exclusion
restriction:

######  Assumption 3 (Double exclusion).

  1. $Y(z,d_{1},d_{2})=Y(d_{1},d_{2})$ for all $z,d_{1},d_{2}$ and $D_{2}(z,d_{1})=D_{2}(d_{1})$ for all $z,d_{1}$.

The first part of 3 is identical to 2.4, and the second part states that $Z$
must not have a direct effect on $D_{2}$ other than through $D_{1}$. These two
parts together impose a double exclusion restriction on $Z$.

Figure 1 shows that the double exclusion allows for $Z$ to have a direct
effect on $D_{1}$, and for $Z$ to have an effect on $D_{2}$ through $D_{1}$.
Hence, compliers in the second part of treatment have to be compliers in the
first part, and the group of compliers now consists of only three types:
$C=\\{\\{C_{1},A_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},C_{2}\\}\\}$. 3 holds under
(9), but does not impose (10), and hence does not exclude all mover types or
imposes homogeneous treatment effects.

The double exclusion restriction imposes that $Z$ randomly assigns individuals
to the first part of treatment but not the second part of treatment if the
first part stays constant. This holds in settings in which a delayed treatment
response to treatment assignment can be ruled out. For instance, in the
Medicaid experiment analysed by Finkelstein et al. (2012), individuals with a
lottery draw of $Z=1$ could only obtain Medicaid at the start of the
treatment, and hence mover type $\\{N_{1},C_{2}\\}$ is likely to be absent.
Since individuals with $Z=0$ had little to no opportunity to obtain medicaid
coverage at the start of the treatment, mover type $\\{A_{1},C_{2}\\}$ is also
likely to be absent. In case the first stage estimate for enrollment at the
start of treatment is close to one, for instance in a carefully conducted
randomized controlled trial (Duflo et al., 2007; De Ree et al., 2023), the
double exclusion may also hold.

###  4.1 Partial identification of the LAFTE

Since the double exclusion restriction rules out two complier types,
$\\{A_{1},C_{2}\\}$ and $\\{N_{1},C_{2}\\}$, the first stages and reduced form
in Lemma 1 can be simplified.

######  Lemma 2 (First stages and reduced form with double exclusion).

Under 2.1-2.3 and 3 it holds that

  1. 1.

$\Delta\mathbb{E}[D_{1}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]$.

  2. 2.

$\Delta\mathbb{E}[D_{2}|Z]=\mathbb{P}[C_{1},C_{2}]$.

  3. 3.

$\Delta\mathbb{E}[D_{\land}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]$.

  4. 4.

$\Delta\mathbb{E}[D_{\lor}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]$.

  5. 5.

$\begin{aligned}
\Delta\mathbb{E}[Y|Z]=&\,\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+\\\
&\,\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}].\end{aligned}$

The proof is deferred to Appendix C. Lemma 2 shows that under the double
exclusion restriction the proportions of all three complier groups are
identified.

The identification of the proportions of the complier groups allows for the
partial identification of LAFTE under additional assumptions:

######  Theorem 1 (Partial identification LAFTE).

Under 2.1-2.3 and 3, positive response $\mathbb{E}[Y(0,0)|C_{1},A_{2}]\geq 0$,
monotone treatment responses $\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]\geq
0\text{ and }\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]\geq 0,$ and monotone
treatment selections
$\mathbb{E}[Y(1,1)|C_{1},A_{2}]\geq\mathbb{E}[Y(1,1)|C_{1},N_{2}]\text{ and
}\mathbb{E}[Y(1,1)|C_{1},C_{2}]\geq\mathbb{E}[Y(1,1)|C_{1},N_{2}]$, it holds
that

$\displaystyle\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D_{1}|Z]}\leq\mathbb{E}[Y(1,1)-Y(0,0)|C]\leq\frac{\Delta\mathbb{E}[D_{\land}Y|Z]}{\Delta\mathbb{E}[D_{\land}|Z]}+\frac{\Delta\mathbb{E}[(1-D_{1})(1-D_{2})Y|Z]}{\Delta\mathbb{E}[D_{1}|Z]}.$
(11)

The proof is deferred to Appendix D. Since the bounds in Theorem 1 collapse to
the LAFTE for the full compliers in absence of movers, the bounds are sharp.

Assuming that the LATEs $\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]$ and
$\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]$ are nonnegative, the difference
between the reduced form of the LAFTE and the reduced form in Lemma 2 is
positive. Hence, the latter can be used as a lower bound on the LAFTE. The
assumption of positive expected treatment effects have been used by, for
instance, Flores and Flores-Lagunes (2013) for the partial identification of
LATEs with a single treatment indicator. Manski (1997) introduces the monotone
treatment response (MTR) assumption on the individual level instead of in
expectation.

The reduced form in Lemma 2 only differs from the reduced form of the LAFTE by
the two potential outcomes $\mathbb{E}[Y(0,0)|C_{1},A_{2}]$ and
$\mathbb{E}[Y(1,1)|C_{1},N_{2}]$. The upper bound on the LAFTE can be obtained
by bounding these potential outcomes using the following assumptions: The
potential outcome $\mathbb{E}[Y(0,0)|C_{1},A_{2}]$ is non-negative, and the
potential outcome of obtaining full treatment $Y(1,1)$ is smaller for
$\\{C_{1},N_{2}\\}$ than for $\\{C_{1},C_{2}\\}$ and $\\{C_{1},A_{2}\\}$.
Outcomes can generally be rescaled so that the assumption of a non-negative
potential outcome is harmless, for instance when the response has a lower
bound. The second assumption is invoked in expectation, and therefore weaker
than the monotone treatment selection (MTS) assumption of Manski and Pepper
(2000).

A combination of assumptions similar to the ones in Theorem 1 have been used
for the partial identification of treatment effects by, for instance, Molinari
(2010), De Haan (2011), and Kreider et al. (2012). In general, partial
identification is common in the analysis of treatment effect identification
problems. For instance, Kreider and Pepper (2007), Battistin and Sianesi
(2011), Tommasi and Zhang (2020), and Calvi et al. (2022) derive bounds on
treatment effects when the treatment is misreported.

When the MTR and MTS assumptions are considered too strong, the following
bounds can be derived assuming only a bounded response:

######  Corollary 3 (Partial identification LAFTE with bounded response).

Under 2.1-2.3 and 3, with bounded response: $Y\in[Y_{\min},Y_{\max}]$, it
holds that

$\displaystyle\frac{\Delta\mathbb{E}[(1-D_{1}-D_{2}+2D_{1}D_{2})Y|Z]+Y_{\min}\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]-Y_{\max}\Delta\mathbb{E}[D_{\land}-D_{2}|Z]}{\Delta\mathbb{E}[D_{1}|Z]}$
(12) $\displaystyle\leq\mathbb{E}[Y(1,1)-Y(0,0)|C]\leq$
$\displaystyle\frac{\Delta\mathbb{E}[(1-D_{1}-D_{2}+2D_{1}D_{2})Y|Z]+Y_{\max}\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]-Y_{\min}\Delta\mathbb{E}[D_{\land}-D_{2}|Z]}{\Delta\mathbb{E}[D_{1}|Z]}.$

The proof is deferred to Appendix E.

The bounded response allows us to replace $Y(1,0)$ for $\\{C_{1},N_{2}\\}$ and
$Y(0,1)$ for $\\{C_{1},A_{2}\\}$ with respectively $Y_{\min}$ and $Y_{\max}$
($Y_{\max}$ and $Y_{\min}$) to construct a lower (upper) bound on the LAFTE.
The difference between the bounds equals
$(y_{\max}-y_{\min})(\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}])/(\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}])$.
The bounds equal $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]$ when no movers are
present, but can be wide when the response bounds are wide and the proportion
of movers is large.

A complier group that may be of particular interest are the full compliers
$\\{C_{1},C_{2}\\}$. This group is induced to obtain both parts of the
treatment by the instrument. However, with only one instrument, it is not
possible to affect the full compliers without also affecting the other
complier types. Hence, the full compliers are mostly relevant for settings in
which the policymaker has an instrument for both the first and second part of
treatment. Blackwell (2017) shows that with two binary instruments, and an
additional treatment exclusion restriction that each instrument only affects
its own treatment, the LAFTE for the full compliers can be identified.

###  4.2 Testing assumptions

Our identification strategy for the LAFTE relies on restrictions on the
presence of mover types. This section derives testable necessary conditions
for these restrictions.

First, Corollary 1 shows that the IV method introduced by Imbens and Angrist
(1994) point-identifies the LAFTE when no movers are present. This assumption
results in the following necessary conditions.

######  Proposition 2 (Necessary conditions no movers).

Under 2 it holds that

$\displaystyle\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]$
$\displaystyle=\mathbb{P}[C_{1},N_{2}]-\mathbb{P}[A_{1},C_{2}],$ (13)
$\displaystyle\Delta\mathbb{E}[D_{\land}-D_{2}|Z]$
$\displaystyle=\mathbb{P}[C_{1},A_{2}]-\mathbb{P}[N_{1},C_{2}],$ (14)

which both equal zero if $\mathbb{P}[C_{1},N_{2}]=\mathbb{P}[A_{1},C_{2}]$ and
$\mathbb{P}[C_{1},A_{2}]=\mathbb{P}[N_{1},C_{2}]$.

Under 2 and $p_{1}=\mathbb{P}[C_{1},N_{2}]=\mathbb{P}[A_{1},C_{2}]$ and
$p_{2}=\mathbb{P}[C_{1},A_{2}]=\mathbb{P}[N_{1},C_{2}]$, it holds that

$\displaystyle\Delta\mathbb{E}[(D_{\lor}-D_{2})Y|Z]$
$\displaystyle=(\mathbb{E}[Y(1,0)|C_{1},N_{2}]-\mathbb{E}[Y(1,0)|A_{1},C_{2}])p_{1},$
(15) $\displaystyle\Delta\mathbb{E}[(D_{\land}-D_{2})Y|Z]$
$\displaystyle=(\mathbb{E}[Y(0,1)|C_{1},A_{2}]-\mathbb{E}[Y(0,1)|N_{1},C_{2}])p_{2},$
(16)

which both equal zero if
$\mathbb{P}[C_{1},N_{2}]=\mathbb{P}[A_{1},C_{2}]=\mathbb{P}[C_{1},A_{2}]=\mathbb{P}[N_{1},C_{2}]=0$
and/or
$\mathbb{E}[Y(1,0)|C_{1},N_{2}]=\mathbb{E}[Y(1,0)|A_{1},C_{2}]$ and
$\mathbb{E}[Y(0,1)|C_{1},A_{2}]-\mathbb{E}[Y(0,1)|N_{1},C_{2}]$.

The proof is deferred to Appendix F.

The first two necessary conditions (13) and (14) in Proposition 2 can be used
to test the null-hypothesis that
$\mathbb{P}[C_{1},N_{2}]=\mathbb{P}[A_{1},C_{2}]$ and
$\mathbb{P}[N_{1},C_{2}]=\mathbb{P}[C_{1},A_{2}]$. A failure to reject this
null-hypothesis does not necessarily imply the absence of movers, as it may
also indicate mover types with identical proportions. Hence, in case of a
failure to reject (13) and (14), the necessary conditions (15) and (16) can be
considered. These conditions are equal to zero if there are no movers or if
potential outcomes are homogeneous.

Provided that potential outcomes are heterogeneous, Proposition 2 provides a
two-step testing procedure for the presence of movers. If the null-hypothesis
that (13) and (14) both equal zero is rejected, movers may be present. If this
null-hypothesis cannot be rejected, the null-hypothesis that (15) and (16)
both equal zero has to be tested. In case of a rejection, we still conclude
that there may be movers. However, a failure to reject indicates that there
are no movers. So if both sets of necessary conditions cannot be rejected, we
recommend using standard IV approaches to identify the LAFTE for the full
compliers $\\{C_{1},C_{2}\\}$.

Second, Theorem 1 shows that in the absence of certain mover types, the LAFTE
is partially identified. In particular, the double exclusion restriction in 3
only allows for $\\{C_{1},A_{2}\\}$ and $\\{C_{1},N_{2}\\}$. This has sign
implications for (13) and (14):

######  Proposition 3 (Necessary condition double exclusion restriction).

Under 2 it holds that

$\displaystyle\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]$
$\displaystyle=\mathbb{P}[C_{1},N_{2}]-\mathbb{P}[A_{1},C_{2}],$ (17)
$\displaystyle\Delta\mathbb{E}[D_{\land}-D_{2}|Z]$
$\displaystyle=\mathbb{P}[C_{1},A_{2}]-\mathbb{P}[N_{1},C_{2}],$ (18)

which are both nonnegative if 3 holds.

The proof follows directly from Proposition 2.

The sign conditions in Proposition 3 are necessary for the absence of the
movers ruled out by the double exclusion restriction. Since it could be the
case that $\mathbb{P}[C_{1},N_{2}]>\mathbb{P}[A_{1},C_{2}]>0$ and
$\mathbb{P}[C_{1},A_{2}]>\mathbb{P}[N_{1},C_{2}]>0$, the conditions are not
sufficient for the double exclusion restriction to hold. Hence, a failure to
reject the null-hypothesis that the sign restrictions hold need not imply the
double exclusion restriction, but a rejection implies that the double
exclusion restriction cannot be invoked.

###  4.3 Causal estimands under weaker assumptions

Theorem 1 provides sharp nonparametric bounds for the LAFTE. However, point-
identification may be required or in some empirical settings the assumptions
may be deemed too strong. For these cases, we discuss three alternative causal
objects that can be identified under weaker assumptions, and from which two
can be point-identified.

First, Lemma 2 shows that under 2.1-2.3 and 3, the IV estimand in (1) with
$D=D_{1}$ identifies the LATE of taking the first part of the treatment:

$\displaystyle\beta_{IV}(D_{1})=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},C_{2}]w[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},C_{2}]w[C_{1},C_{2}]+$
(19)
$\displaystyle\,\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]w[C_{1},N_{2}]+\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]w[C_{1},A_{2}],$

with
$w[G_{1},G_{2}]=\frac{\mathbb{P}[G_{1},G_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}$.

The weighted average of LATEs in (19) can be interpreted as a decomposition of
the total effect of $D_{1}$ on $Y$ using $D_{2}$ as a mediator. The mediation
literature, see Huber (2019) for a review, defines the direct effect as the
effect of $D_{1}$ on $Y$ while the mediator $D_{2}$ is constant, and the
indirect effect as the effect of $D_{2}$ on on $Y$ while $D_{1}$ is constant.
Hence, the first term in (19) is the indirect effect, and the remaining terms
are direct effects.

Table 1 shows that the IV estimand in (19) is often used in the literature.
This paper shows that this is a valid estimator of the effect of $D_{1}$ under
the double exclusion restriction. In addition, we argue that the effect of
receiving full treatment is also a causal parameter of interest. Adding
$\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]w[C_{1},N_{2}]$ and
$\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]w[C_{1},A_{2}]$ to (19) results in the
effect of receiving full treatment. The additional MTR assumptions in Theorem
1 guarantee that these two average treatment effects of receiving the second
part of treatment are positive. It follows that the lower bound on the LAFTE
equals (19) and the LAFTE is equal to or larger than the LATE of taking the
first part of the treatment.

Second, the double exclusion restriction may be considered too strong, or its
necessary conditions may be rejected for the setting at hand. Lemma 2 shows
that under only 2, the IV estimand in (1) with the multivalued treatment
$D=D_{1}+D_{2}\in\\{0,1,2\\}$ equals

$\displaystyle\beta_{IV}(D_{1}+D_{2})=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},C_{2}]w[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},C_{2}]w[C_{1},C_{2}]$
(20) $\displaystyle+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]w[C_{1},N_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]w[N_{1},C_{2}]$
$\displaystyle+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]w[C_{1},A_{2}]+\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]w[A_{1},C_{2}],$

with
$w[G_{1},G_{2}]=\frac{\mathbb{P}[G_{1},G_{2}]}{2\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[A_{1},C_{2}]}$.
This results in a causal interpretation of a weighted average of causal
effects of receiving one part of the treatment, instead of the causal effect
of receiving full treatment.

The IV estimand $\beta_{IV}(D_{1}+D_{2})$ is related to the average causal
response (ACR) as introduced by Angrist and Imbens (1995). The ACR is also a
weighted average of the effects of unit changes in treatment, but considers a
multivalued treatment that does not distinguish between
$\\{D_{1}=1,D_{2}=0\\}$ and $\\{D_{1}=0,D_{2}=1\\}$. Without late-adopters who
miss the first part of the program, the mover types $\\{C_{1},A_{2}\\}$ and
$\\{N_{1},C_{2}\\}$ are absent and (20) is equal to the ACR.

Summarising the multivalued treatment by one binary treatment indicator may
also lead to violations of the exclusion restriction similar to the ones
discussed in Section 3. Andresen and Huber (2021) show that these violations
can also be ruled out by restricting treatment effect heterogeneity or mover
types, where movers with a multivalued treatment are defined by the
individuals induced by the instrument to change treatment status from and to
treatment values that are both below or above the binarisation threshold.

The ACR type estimand in (20) considers shifting from no treatment to the
first part and from the first part to full treatment, as separate treatment
effects for the full compliers. Therefore, it double counts the full compliers
in the denominator of the weighting function $w[G_{1},G_{2}]$. An alternative
is to combine the two effects for the full compliers and to interpret it as
the full treatment effect. 2 allows for the partial identification of this
alternative weighted average of LATEs:

######  Corollary 4 (Partial identification weighted average of LATEs).

Under 2 a weighted average of LATEs denoted by $\tau$ is bounded by

$\displaystyle\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D_{1}+D_{2}|Z]}\leq\tau\leq\frac{\Delta\mathbb{E}[Y|Z]}{\max(\Delta\mathbb{E}[D_{1}|Z],\Delta\mathbb{E}[D_{2}|Z],\Delta\mathbb{E}[D_{\land}|Z],\Delta\mathbb{E}[D_{\lor}|Z])},$
(21)

where $\tau$ is defined as

$\displaystyle\tau=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]w[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]w[C_{1},N_{2}]+$
(22)
$\displaystyle\,\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]w[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]w[N_{1},C_{2}]+$
$\displaystyle\,\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]w[A_{1},C_{2}],$

with
$w(G_{1},G_{2})=\frac{\mathbb{P}[G_{1},G_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}$.

The proof is based on Lemma 1 and deferred to Appendix G. The weights in (22)
are nonnegative and add up to one. Hence $\tau$ is a convex combination of
LATEs and has a causal interpretation. However, this interpretation may not
directly be policy relevant as it measures an average across different
treatment effects and different groups.

Intuitively, the treatment indicator with the largest (smallest) first stage
estimate may provide a lower (upper) bound on a treatment effect of interest.
For instance, Finkelstein et al. (2012) apply this intuition to bound the
effects of Medicaid insurance. Corollary 4 shows that this intuition does not
apply to $\tau$: Instead of bounding $\tau$ from below, the treatment
indicator with the largest first stage is an upper bound on $\tau$.

##  5 Empirical applications

This section illustrates the empirical relevance of our methods for treatment
evaluation. We consider the LAFTE of a health program, a labour program, an
educational program, and a development program. Additional details on the
empirical specifications are deferred to Appendix H. This section focuses on
the main results following from Theorem 1 and Proposition 2 and 3. Appendix I
shows additional empirical results, such as the bounds in Corollary 3, which
are in general wide and cannot reject that the LAFTEs are equal to zero.

###  5.1 The Oregon health insurance experiment

This application studies the LAFTE of medicaid coverage on health care
utilization. Policy makers actively aim to reduce coverage disruptions by
continuous enrollment policies. As a recent example, the Families First
Coronavirus Recovery Act requires Medicaid programs to keep individuals
enrolled for the duration of the public health emergency (Sugar et al., 2021).
To better understand the impact of such policies, the effect of continuous
Medicaid coverage across the full study period is of particular interest.

####  5.1.1 Context and empirical specification

In 2008, a group of uninsured low-income adults in Oregon was randomly given
the opportunity to apply for Medicaid. The state opened a waiting list for
10,000 Medicaid spots and subsequently drew names by lottery from the 89,924
individuals that placed themselves on this list. The Medicaid program provided
comprehensive benefits with no consumer cost sharing. The monthly enrollment
premiums ranged from $0 to $20 depending on income.

Finkelstein et al. (2012) use the randomized opportunity to apply for Medicaid
as an instrument for Medicaid enrollment and analyze the effects on health
care utilization, financial strain, and health. The Oregon Health Insurance
Experiment (OHIE) was subsequently used in a series of papers to study the
effects of Medicaid on other outcomes. For example, Finkelstein et al. (2016)
study the impact on emergency department use over time.

The data in Finkelstein et al. (2016) contains Medicaid coverage and emergency
department use for 24,646 individuals with Portland-area zip codes across four
time periods: day 0 to 180, 181 to 360, 361 to 540, and 541 to 720 after
lottery notification. Our $D_{1}$ equals one if an individual was enrolled in
Medicaid in the first year after lottery notification, and $D_{2}$ equals one
if an individual was enrolled in the second year after lottery notification.
The instrument equals one if an individual was randomly given the opportunity
to apply for Medicaid. The binary outcome variable $Y$ equals one if an
individual had any emergency department visit during the two years after the
lottery notification.

####  5.1.2 Results

Table 2 reports the estimated necessary conditions for the presence of movers
and the validity of 3. Column (1) and (2) in Panel A show, respectively, the
results of a regression from $D_{\lor}-D_{2}$ and $D_{\land}-D_{2}$ upon $Z$.
First, since both estimates are significantly different from zero at the 1%
level, we reject that movers are absent. Second, since both estimates are
positive, we do not reject the necessary conditions of 3.

Table 2: Necessary conditions for the double exclusion restriction and no
movers

widercol | (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
|  $D_{\lor}-D_{2}$ |  $D_{\land}-D_{2}$ |  $(D_{\lor}-D_{2})Y$ |  $(D_{\land}-D_{2})Y$ |  $D_{2}$
Panel A: The Oregon health insurance experiment, $N=24646$
$Z$ | 0.078 | 0.060 | 0.032 | 0.029 | 0.115
| (0.003) | (0.003) | (0.002) | (0.002) | (0.006)
Panel B: The LinkedIn job opportunities experiment, $N=988$
$Z$ | 0.036 | 0.042 | 0.027 | 0.023 | 0.315
| (0.039) | (0.011) | (0.032) | (0.006) | (0.077)
Panel C: The elite university education natural experiment, $N=645$
$Z$ | 0.047 | 0.537 | 0.446 | 4.772 | 0.095
| (0.012) | (0.043) | (0.116) | (0.387) | (0.061)
Panel D: The village-based schools experiment, $N=1181$
$Z$ | 0.000 | -0.041 | 0.000 | 0.016 | 0.483
| (.) | (0.008) | (.) | (0.006) | (0.080)

  * Notes: columns (1) to (4) show the estimates for the necessary conditions in Proposition 2 and 3, corresponding to (13)-(16) respectively. Column (5) shows the estimate of a regression from $D_{2}$ upon $Z$. The standard errors are in parentheses. The panels correspond to the empirical applications in Section 5, where $N$ indicates the number of observations.

Under the double exclusion restriction, column (1) implies that
$\widehat{\mathbb{P}[C_{1},N_{2}]}=0.078$. This group includes movers who drop
out of treatment. Since individuals had to recertify eligibility every six
months, they could have lost coverage due to, for instance, income
fluctuations. Column (2) implies that
$\widehat{\mathbb{P}[C_{1},A_{2}]}=0.060$. These movers likely used the
opportunity to gain eligibility in the second time period: 14 months after the
lottery, Oregon took lottery draws from a new Medicaid waiting list. Column
(5) reports the results of a regression from $D_{2}$ upon $Z$. Under the
double exclusion restriction this estimate equals
$\widehat{\mathbb{P}[C_{1},C_{2}]}=0.115$, and hence the movers compose a
larger proportion of the data than the full compliers.

The double exclusion restriction imposes that
$\mathbb{P}[N_{1},C_{2}]=\mathbb{P}[A_{1},C_{2}]=0$. The first mover type is
likely absent since individuals with $Z=1$ had to apply for Medicaid within 45
days after the state made them aware of the lottery draw. Individuals that did
not apply during this window could not apply after. In general, Finkelstein et
al. (2012) describe that the mechanisms to obtain Medicaid coverage are
limited to the initial lottery and the lottery 14 months later. They also show
that only 2% of the individuals with $Z=0$ obtained Medicaid coverage up unto
one year after the lottery. This may explain the absence of the second mover
type.

Figure 2 shows the estimated bounds for the LAFTE from Theorem 1. Panel A
identifies a statistically significant LAFTE: The 95% confidence interval of
the lower bound, $[0.035,0.128]$, does not include zero. The bounds rely on
the assumption that the effect of Medicaid in the second year on emergency
department use is non-negative (MTR), and that always takers and compliers of
Medicaid in the second year are not less likely to use the emergency
department than never takers (MTS). Based on these bounds, we conclude that
Medicaid enrollment during both years increases the probability of an
emergency department visit between 0.081 and 0.224 percentage points for the
individuals induced to enroll by the lottery.

The estimated bounds imply that the LAFTE may be up to three times as large as
the LATE of Medicaid enrollment in the first period. Hence, policy makers may
expect larger treatment effects in combination with continuous enrollment
requirements, which may inform the design of future Medicaid enrollment
policies.

Figure 2: Estimated bounds on the LAFTE Notes: this figure shows the lower and
upper bound from Theorem 1 for the LAFTE. The lines are 95% confidence
intervals. The precise estimates and standard errors are reported in Table 4
of Appendix I.

###  5.2 The LinkedIn job opportunities experiment

This application studies the LAFTE of a LinkedIn training on employment. Job
market training programs, such as a training that stimulates LinkedIn usage,
are often characterized by high incidence of control group substitution and
treatment group dropout (Heckman et al., 2000). Since the programs are
carefully designed to receive and follow in full, the treatment effect of
LinkedIn usage for a longer period should be of particular interest to policy
makers.

####  5.2.1 Context and empirical specification

Wheeler et al. (2022) run and evaluate a randomly assigned program that trains
work seekers to join and use LinkedIn. Their study sample includes 30 cohorts
from existing job readiness training programs in four large South African
cities. They randomly assign 15 cohorts to four hours of LinkedIn training
during their job readiness training program. The intervention trains
participants, among others, to open accounts, build their profiles, and search
and apply for jobs. The study examines the effect of the LinkedIn training on
a range of outcomes. Although the experiment is not specifically designed to
identify the causal effect of LinkedIn usage, one of the analysis uses the
random assignment to LinkedIn training as an instrument for LinkedIn usage to
study its effect on employment.

The data includes LinkedIn usage and employment outcomes across three time
periods: directly, six months, and twelve months after the end of the job
readiness training program. Our $D_{1}$ equals one if a participant has a
Linkedin account at the end of the job training and $D_{2}$ equals one if a
participant has a LinkedIn account six months later. The instrument $Z$ equals
one if the participant is randomly assigned to the LinkedIn training. The
binary outcome variable $Y$ equals one if the participant is employed twelve
months after the program ended. The estimation sample includes 988 of the
1,638 participants across the 30 cohorts.

####  5.2.2 Results

The estimated necessary conditions in column (1) and (2) of Panel B in Table 2
are positive, and the latter estimate is also significantly different from
zero at the 1% level. Hence, we reject that movers are absent and do not
reject the necessary conditions of 3.

Under the double exclusion restriction, column (1) implies that
$\widehat{\mathbb{P}[C_{1},N_{2}]}=0.036$, which is not significantly
different from zero. This estimate refers to individuals that were induced to
open a LinkedIn account by the LinkedIn training but deleted it after the job
training program. Column (2) implies that
$\widehat{\mathbb{P}[C_{1},A_{2}]}=0.042$. This suggests that several treated
individuals would have also opened a LinkedIn account after the job training
program if assigned to the control. Hence, the LinkedIn training only
accelerated them to create an account. Column (5) implies that the two groups
of movers are small compared to the full compliers.

The double exclusion restriction imposes that
$\mathbb{P}[N_{1},C_{2}]=\mathbb{P}[A_{1},C_{2}]=0$. Since the treatment group
was incentivized to immediately open a LinkedIn account in the first week of
the job training program, the first mover type can be absent. However, if the
LinkedIn training succeeds in explaining the benefits of LinkedIn in the long
term, both mover types could be present. Hence, even though they are not
detected by the necessary conditions, this empirical setting may include
movers that violate the double exclusion restriction.

Panel B of Figure 2 shows a statistically significant LAFTE: The 95%
confidence interval of the lower bound, $[0.047,0.305]$, does not include
zero. The bounds rely on the assumption that the effect of a LinkedIn account
six months after the job training program on employment is non-negative (MTR),
and that always takers and compliers of a LinkedIn account six monthts after
the job training are not less likely to find a job than never takers (MTS).
Based on these bounds, we conclude that having a Linkedin account in both time
periods increases the probability to find a job between 0.176 and 0.235
percentage points for individuals induced to open an account due to the
LinkedIn training. The estimated bounds imply that the LAFTE is similar to the
LATE of the first part of the LinkedIn training, which suggests that having a
Linkedin account is most effective shortly after the job training program.

###  5.3 The elite university education natural experiment

This application studies the LAFTE of a university education on income. School
curricula, including the design and sequencing of learning content, are
developed to receive in full. Moreover, students enroll in a program with the
aim to graduate, which makes the LAFTE a valuable piece of information for
their study choice. Hence, for both university educators and students the
effect of following a full university program is of interest.

####  5.3.1 Context and empirical specification

Anelli (2020) studies the returns to a selective, expensive, and private
university offering business, economics, and law degrees in a large Italian
city. Admission to the elite university is based on a uni-dimensional
application score. Every year, the number of admitted students is fixed, and
the university strictly offers admission to the students with the highest
application score. This procedure generates a cutoff in the application score,
where students above the cutoff are offered admission. A score above the
cutoff is used as an instrument for ever enrolled at the elite university to
study the effect of ever enrolled on income.

The main sample is restricted to the elite university applicants between 1995
and 2000. For 645 of these applicants the dataset contains information on the
application score, ever being enrolled at the elite university between 1995
and 2005, graduation from any university up unto the year 2005, and yearly
income in the year 2005. If an applicant retook the admission test, the score
refers to the first observed application.

Our $D_{1}$ equals one if an applicant was ever enrolled at the elite
university between 1995 and 2005. This is the treatment variable in Anelli
(2020). Our $D_{2}$ equals one if an applicant graduated from any university
up unto the year 2005. The dataset does not contain information on whether the
applicant graduated from the elite university. The instrument $Z$ equals one
if the applicant scores above to application score cutoff. The outcome
variable $Y$ is the logarithm of taxable income in 2005, measured before taxes
and after deductions.

####  5.3.2 Results

The estimated necessary conditions in column (1) and (2) of Panel C in Table 2
are positive and significantly different from zero at the 1% level. Hence, we
can reject that movers are absent and do not reject the necessary conditions
of 3.

Under the double exclusion restriction, column (1) implies that
$\widehat{\mathbb{P}[C_{1},N_{2}]}=0.047$. This mover type may refer to
students who were induced to enroll at the elite university but never
graduate. Column (2) implies that $\widehat{\mathbb{P}[C_{1},A_{2}]}=0.537$.
Recall that our $D_{2}$ equals one if a student graduated from any university,
not just the elite institution, and so this group is relatively large because
students with $Z=0$ may graduate from other universities.

The double exclusion restriction imposes that
$\mathbb{P}[N_{1},C_{2}]=\mathbb{P}[A_{1},C_{2}]=0$. The first mover type is
absent if university graduation for the never takers of elite university
enrollment is not affected by scoring above or below the cutoff. This group of
movers could be present if scoring above the cutoff has some positive
psychological effect that persists even if a student does not enroll at the
elite institution. The large estimate in column (2) of Table 2 suggests that
such psychological effects are close to zero. The second mover type is likely
to be absent since Anelli (2020) reports that across all application rounds
only 23 students retook the admission test after scoring below the cutoff.
Hence the possible number of always takers with enrollment in the elite
university is small.

Panel C of Figure 2 shows a statistically significant LAFTE: the estimated
bounds equal 0.532 and 7.545, and the 95% confidence interval of the lower
bound, $[0.076,0.988]$, does not include zero. The bounds rely on the
assumption that the effect of university graduation on income is non-negative,
and that always takers and compliers of university graduation have higher
wages than never takers. The upper bound is large due to the large group of
movers $\\{C_{1},A_{2}\\}$. Based on the bounds, we conclude that the full
treatment of enrolling at the elite and graduating at any university increases
annual income between 53 and 750 log points for the individuals induced to
enroll by scoring above the cutoff. The estimated bounds imply that the LAFTE
may be much larger than the LATE of university enrollment. This suggests that
consecutive parts of a university degree are complements and students may
particularly benefit from fully completing a degree.

###  5.4 The village-based schools experiment

This application aims to study the LAFTE of a development program. The
implementation of development programs often requires careful design,
planning, and coordination with local partners, and can be expensive (Duflo et
al., 2007). Hence, the full treatment effect is of interest to the
stakeholders.

####  5.4.1 Context and empirical specification

Burde and Linden (2013) conduct and evaluate the randomized opening of
village-based schools in Afghanistan, where primary-school participation rates
are low. Their study sample includes 31 villages in rural Afghanistan. In the
summer of 2007, they randomly opened schools in 13 of these villages. Village-
based schools are public schools that are designed to deliver the official
national curriculum to children living in close proximity to the school. They
use the random assignment of village-based schools as an instrument for school
enrollment to estimate the effect of enrollment on academic performance.

Their household survey data contains information on school enrollment and math
and language test scores in two time periods: four months after and eight
months after opening the village-based schools. Our $D_{1}$ equals one if a
child was enrolled in school four months after the opening and $D_{2}$ equals
one if a child was enrolled in school eight months after the opening. The
binary instrument $Z$ equals one if a child lives in a village that randomly
received a school. The outcome variable $Y$ is the standardized test score
eight months after the opening of the schools. The final estimation sample
includes 1,181 children, of in total 1,490 school-age children across the 31
villages.

####  5.4.2 Results

Panel D in Table 2 shows the estimated necessary conditions. Since the
observed $D_{2}=1$ if $D_{1}=1$, the variable $D_{\lor}-D_{2}$ is zero for
each individual, and hence the estimates in column (1) and (3) are zero. This
indicates the absence of the mover types $\\{C_{1},N_{2}\\}$ and
$\\{A_{1},C_{2}\\}$. The estimate in column (2) is negative and statistically
significant at the 1% level. This estimate implies that
$\mathbb{P}[N_{1},C_{2}]>0$, which violates the double exclusion restriction.
This mover type may arise if households need time to change their beliefs
towards allowing their children to enroll in school. Burde and Linden (2013)
discuss that the conservative beliefs in Afghanistan may be an impediment for
school enrollment, in particular for girls.

Appendix I shows the estimates for the parameters that can be identified when
the double exclusion restriction is violated: the ACR type estimand in (20)
and the weighted average of LATEs in Corollary 4. To conclude, this
application shows that it depends on the empirical context whether the double
exclusion restriction holds, and that the testable necessary conditions are
able to help researchers detect violations of this assumption.

##  6 Conclusion

Policy evaluation has to deal with individuals that only take a subset of a
treatment program. This poses the question: Which treatment effect is of
interest to the policymaker? For instance, are the outcomes of the individuals
receiving the full treatment or any individual that came in contact with the
treatment of interest? Does the control group consists of individuals who
obtained zero treatment or did not complete the full treatment? This paper
defines the effect of full treatment versus no treatment at all as the
parameter of interest.

We develop an IV framework in the presence of individuals that are induced by
the instrument to obtain only a subset of the treatment, and refer to these
individuals as movers. We show that these movers violate the exclusion
restriction in the standard IV framework, that necessary conditions on the
presence of movers are testable, and that under a double exclusion restriction
the average treatment effect of receiving full treatment for the individuals
induced into treatment by the instrument is partially identifiable. We refer
to this treatment effect as the local average full treatment effect (LAFTE).

We study the LAFTE in empirical applications from four different fields of
applied economics. Our methods find evidence for the presence of movers in all
four studies. These types of movers align with the intuition provided by the
setting at hand. In three of these applications, the necessary conditions for
the double exclusion restriction cannot be rejected, and the bounds identify a
statistically significant LAFTE. We discuss the potential policy implications
within the empirical context of the applications.

We allow the instrument to induce individuals into dropping out or late-
adoption of treatment by taking into account two treatment parts. In settings
in which the instrument additionally affects dropping out or late-adoption
across more treatment parts, the exclusion restriction in our IV framework may
also be violated. In case these settings include the observed treatment status
across these treatment parts, our framework can be extended to more specific
mover types that also take these violations into account.

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##  Appendix A Proof Lemma 1

The first stage with $D=D_{1}$ equals

$\displaystyle\Delta\mathbb{E}[D_{1}|Z]=$
$\displaystyle\mathbb{E}[D_{1}(1)|Z=1]-\mathbb{E}[D_{1}(0)|Z=0]=\mathbb{E}[D_{1}(1)-D_{1}(0)]=\mathbb{P}[D_{1}(1)-D_{1}(0)=1]$
(23) $\displaystyle=$
$\displaystyle\mathbb{P}[C_{1}]=\mathbb{P}[C_{1},D_{2}(1,1)-D_{2}(0,0)=1]+\mathbb{P}[C_{1},D_{2}(1,1)=D_{2}(0,0)=0]+$
$\displaystyle\mathbb{P}[C_{1},D_{2}(1,1)=D_{2}(0,0)=1]$ $\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}],$

using (2), 2.1 and 2.3, respectively.

The first stage with $D=D_{2}$ equals

$\displaystyle\Delta\mathbb{E}[D_{2}|Z]=$
$\displaystyle\mathbb{E}[D_{2}(1,1)D_{1}+D_{2}(1,0)(1-D_{1})|Z=1]-$ (24)
$\displaystyle\mathbb{E}[D_{2}(0,1)D_{1}+D_{2}(0,0)(1-D_{1})|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[D_{2}(1,1)D_{1}(1)+D_{2}(1,0)(1-D_{1}(1))|Z=1]-$
$\displaystyle\mathbb{E}[D_{2}(0,1)D_{1}(0)+D_{2}(0,0)(1-D_{1}(0))|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[D_{2}(1,1)D_{1}(1)+D_{2}(1,0)(1-D_{1}(1))-D_{2}(0,1)D_{1}(0)-D_{2}(0,0)(1-D_{1}(0))]$
$\displaystyle=$
$\displaystyle\mathbb{E}[D_{2}(1,1)-D_{2}(0,0)|D_{1}(1)-D_{1}(0)=1]\mathbb{P}[D_{1}(1)-D_{1}(0)=1]+$
$\displaystyle\mathbb{E}[D_{2}(1,0)-D_{2}(0,0)|D_{1}(1)=D_{1}(0)=0]\mathbb{P}[D_{1}(1)=D_{1}(0)=0]+$
$\displaystyle\mathbb{E}[D_{2}(1,1)-D_{2}(0,1)|D_{1}(1)=D_{1}(0)=1]\mathbb{P}[D_{1}(1)=D_{1}(0)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[D_{2}(1,1)-D_{2}(0,0)=1|D_{1}(1)-D_{1}(0)=1]\mathbb{P}[D_{1}(1)-D_{1}(0)=1]+$
$\displaystyle\mathbb{P}[D_{2}(1,0)-D_{2}(0,0)=1|D_{1}(1)=D_{1}(0)=0]\mathbb{P}[D_{1}(1)=D_{1}(0)=0]+$
$\displaystyle\mathbb{P}[D_{2}(1,1)-D_{2}(0,1)=1|D_{1}(1)=D_{1}(0)=1]\mathbb{P}[D_{1}(1)=D_{1}(0)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[D_{2}(1,1)-D_{2}(0,0)=1,D_{1}(1)-D_{1}(0)=1]+$
$\displaystyle\mathbb{P}[D_{2}(1,0)-D_{2}(0,0)=1,D_{1}(1)=D_{1}(0)=0]+$
$\displaystyle\mathbb{P}[D_{2}(1,1)-D_{2}(0,1)=1,D_{1}(1)=D_{1}(0)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}],$

using (3), (2), 2.1 and 2.3, respectively.

The first stage with $D=D_{\land}=D_{1}D_{2}$ can be rewritten using (2) and
(3):

$\displaystyle D_{\land}=$
$\displaystyle[D_{1}(1)Z+D_{1}(0)(1-Z)]\times[D_{2}(1,1)(D_{1}(1)Z+D_{1}(0)(1-Z))Z+$
(25) $\displaystyle D_{2}(1,0)(1-D_{1}(1)Z-D_{1}(0)(1-Z))Z+$ $\displaystyle
D_{2}(0,1)(D_{1}(1)Z+D_{1}(0)(1-Z))(1-Z)+$ $\displaystyle
D_{2}(0,0)(1-D_{1}(1)Z-D_{1}(0)(1-Z))(1-Z)]$ $\displaystyle=$
$\displaystyle[D_{1}(1)Z+D_{1}(0)(1-Z)]\times[D_{2}(1,1)D_{1}(1)Z+D_{2}(1,0)(1-D_{1}(1))Z+$
$\displaystyle D_{2}(0,1)D_{1}(0)(1-Z)+D_{2}(0,0)(1-D_{1}(0))(1-Z)]$
$\displaystyle=$ $\displaystyle D_{1}(1)D_{2}(1,1)Z+D_{1}(0)D_{2}(0,1)(1-Z),$

so that

$\displaystyle\Delta\mathbb{E}[D_{\land}|Z]=$
$\displaystyle\mathbb{E}[D_{1}(1)D_{2}(1,1)|Z=1]-\mathbb{E}[D_{1}(0)D_{2}(0,1)|Z=0]$
(26) $\displaystyle=$
$\displaystyle\mathbb{E}[D_{1}(1)D_{2}(1,1)-D_{1}(0)D_{2}(0,1)]$
$\displaystyle=$
$\displaystyle\mathbb{P}[D_{1}(1)D_{2}(1,1)-D_{1}(0)D_{2}(0,1)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[D_{1}(1)-D_{1}(0)=1,D_{2}(1,1)-D_{2}(0,1)=1]+$
$\displaystyle\mathbb{P}[D_{1}(1)-D_{1}(0)=1,D_{2}(1,1)=D_{2}(0,1)=1]+$
$\displaystyle\mathbb{P}[D_{1}(1)=D_{1}(0)=1,D_{2}(1,1)-D_{2}(0,1)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[A_{1},C_{2}],$

using 2.1 and 2.3, respectively.

The first stage with $D=D_{\lor}=D_{1}+D_{2}-D_{1}D_{2}$ equals

$\displaystyle\Delta\mathbb{E}[D_{\lor}|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{1}|Z]+\Delta\mathbb{E}[D_{2}|Z]-\Delta\mathbb{E}[D_{\land}|Z]$
(27) $\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[N_{1},C_{2}].$

For the reduced form, we use 2.4 and rewrite (5) to

$\displaystyle Y=$ $\displaystyle[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)]D_{1}D_{2}+$
(28) $\displaystyle[Y(1,0)-Y(0,0)]D_{1}+[Y(0,1)-Y(0,0)]D_{2}+Y(0,0),$

to write

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\Delta\mathbb{E}[[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)]D_{1}D_{2}|Z]+$
(29)
$\displaystyle\Delta\mathbb{E}[[Y(1,0)-Y(0,0)]D_{1}|Z]+\Delta\mathbb{E}[[Y(0,1)-Y(0,0)]D_{2}|Z]$
$\displaystyle=$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0))D_{1}D_{2}|Z=1]-$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0))D_{1}D_{2}|Z=0]+$
$\displaystyle\mathbb{E}[(Y(1,0)-Y(0,0))D_{1}|Z=1]-\mathbb{E}[(Y(1,0)-Y(0,0))D_{1}|Z=0]+$
$\displaystyle\mathbb{E}[(Y(0,1)-Y(0,0))D_{2}|Z=1]-\mathbb{E}[(Y(0,1)-Y(0,0))D_{2}|Z=0].$

It follows from the results in (23), (24), and (26) that

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\},\\{A_{1},C_{2}\\}]\times$
(30)
$\displaystyle\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\},\\{A_{1},C_{2}\\}]+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]\times$
$\displaystyle\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]+$
$\displaystyle\mathbb{E}[Y(0,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{N_{1},C_{2}\\},\\{A_{1},C_{2}\\}]\times$
$\displaystyle\mathbb{P}[\\{C_{1},C_{2}\\},\\{N_{1},C_{2}\\},\\{A_{1},C_{2}\\}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}].$

##  Appendix B Proof Corollary 2

For an instrumental variable analysis with treatment indicator $D=D_{1}$, it
follows from Lemma 1 that the first stage and reduced form is equal to

$\displaystyle\Delta\mathbb{E}[D_{1}|Z]=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]$
(31) $\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}].$

We write the reduced form as,

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
(32)
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]-\mathbb{E}[Y(1,1)-Y(0,1)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}]-\mathbb{E}[Y(1,0)-Y(0,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}],$

and substitute in the homogeneity assumptions for $D=D_{1}$ in Table 3:

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
(33)
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]-\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}]-\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}].$

Then we can write

$\displaystyle\beta_{IV}(D_{1})=\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D_{1}|Z]}=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}].$
(34)

The proofs for $D=D_{2},D_{\land},D_{\lor}$ use the same strategy.

Table 3: Treatment effect homogeneity assumptions for movers

$D$ | Homogeneity assumption | IV estimand
---|---|---
$D_{1}$ | $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]=\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]$ | $\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]$
| $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]=\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]=\mathbb{E}[Y(1,1)-Y(0,1)|N_{1},C_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]=\mathbb{E}[Y(1,0)-Y(0,0)|A_{1},C_{2}]$ |
$D_{2}$ | $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]=\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]$ | $\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{N_{1},C_{2}\\},\\{A_{1},C_{2}\\}]$
| $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]=\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]=\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]=\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]$ |
$D_{\land}$ | $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]=\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]$ | $\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\},\\{A_{1},C_{2}\\}]$
| $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]=\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]=\mathbb{E}[Y(1,1)-Y(0,1)|N_{1},C_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]=\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]$ |
$D_{\lor}$ | $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]=\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]$ | $\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{N_{1},C_{2}\\}]$
| $\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]=\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|N_{1},C_{2}]=\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]$ |
| $\mathbb{E}[Y(1,1)-Y(0,0)|A_{1},C_{2}]=\mathbb{E}[Y(1,0)-Y(0,0)|A_{1},C_{2}]$ |

  * Notes: this table provides an overview of the homogeneity assumptions in column (2) that are required for the IV estimand in Equation (1) to identify the causal parameter in column (3) when the treatment indicator in column (1) is used as endogenous treatment variable.

##  Appendix C Proof Lemma 2

Since 3 only applies to $D_{2}$, $\Delta\mathbb{E}[D_{1}|Z]$ is equal to (23):

$\displaystyle\Delta\mathbb{E}[D_{1}|Z]=$
$\displaystyle\mathbb{P}[C_{1}]=\mathbb{P}[C_{1},D_{2}(1)-D_{2}(0)=1]+\mathbb{P}[C_{1},D_{2}(1)=D_{2}(0)=0]+$
(35) $\displaystyle\mathbb{P}[C_{1},D_{2}(1)=D_{2}(0)=1]$ $\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}],$

using 2.1 and 2.3.

3 allows us to link observed to potential outcomes for $D_{2}$ as follows,

$\displaystyle D_{2}=$ $\displaystyle D_{2}(1)D_{1}+D_{2}(0)(1-D_{1})$ (36)
$\displaystyle=$ $\displaystyle
D_{2}(1)[D_{1}(1)Z+D_{1}(0)(1-Z)]+D_{2}(0)[1-D_{1}(1)Z-D_{1}(0)(1-Z)]$
$\displaystyle=$ $\displaystyle
D_{2}(0)+(D_{2}(1)-D_{2}(0))(D_{1}(1)Z+D_{1}(0)(1-Z)),$

so that the first stage with $D=D_{2}$ equals

$\displaystyle\Delta\mathbb{E}[D_{2}|Z]=$
$\displaystyle\mathbb{E}[D_{2}(0)+(D_{2}(1)-D_{2}(0))(D_{1}(1)Z+D_{1}(0)(1-Z))|Z=1]-$
(37)
$\displaystyle\mathbb{E}[D_{2}(0)+(D_{2}(1)-D_{2}(0))(D_{1}(1)Z+D_{1}(0)(1-Z))|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[(D_{1}(1)-D_{1}(0))(D_{2}(1)-D_{2}(0))]=\mathbb{P}[(D_{1}(1)-D_{1}(0))(D_{2}(1)-D_{2}(0))=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[D_{1}(1)-D_{1}(0)=1,D_{2}(1)-D_{2}(0)=1]$
$\displaystyle=$ $\displaystyle\mathbb{P}[C_{1},C_{2}],$

using 2.1 and 2.3.

The first stage with $D=D_{1}D_{2}=D_{\land}$ uses that under 3

$\displaystyle D_{\land}=$
$\displaystyle[D_{1}(1)Z+D_{1}(0)(1-Z)]\times[D_{2}(0)+(D_{2}(1)-D_{2}(0))(D_{1}(1)Z+D_{1}(0)(1-Z))]$
(38) $\displaystyle=$ $\displaystyle
D_{1}(1)D_{2}(0)Z+(D_{2}(1)-D_{2}(0))D_{1}(1)Z+$ $\displaystyle
D_{1}(0)D_{2}(0)(1-Z)+(D_{2}(1)-D_{2}(0))D_{1}(0)(1-Z)$ $\displaystyle=$
$\displaystyle D_{1}(1)D_{2}(1)Z+D_{1}(0)D_{2}(1)(1-Z),$

so that the first stage equals

$\displaystyle\Delta\mathbb{E}[D_{\land}|Z]=$
$\displaystyle\mathbb{E}[D_{1}(1)D_{2}(1)Z+D_{1}(0)D_{2}(1)(1-Z)|Z=1]-$ (39)
$\displaystyle\mathbb{E}[D_{1}(1)D_{2}(1)Z+D_{1}(0)D_{2}(1)(1-Z)|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[(D_{1}(1)-D_{1}(0))D_{2}(1)]=\mathbb{P}[(D_{1}(1)-D_{1}(0))D_{2}(1)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[D_{1}(1)-D_{1}(0)=1,D_{2}(1)-D_{2}(0)=1]+\mathbb{P}[D_{1}(1)-D_{1}(0)=1,D_{2}(1)=D_{2}(0)=1]$
$\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}],$

using 2.1 and 2.3.

The first stage with $D=D_{\lor}=D_{1}+D_{2}-D_{1}D_{2}$ equals

$\displaystyle\Delta\mathbb{E}[D_{\lor}|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{1}|Z]+\Delta\mathbb{E}[D_{2}|Z]-\Delta\mathbb{E}[D_{\land}|Z]$
(40) $\displaystyle=$
$\displaystyle\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}].$

For the reduced form, the proof of Lemma 1 shows that

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0))D_{1}D_{2}|Z=1]-$ (41)
$\displaystyle\mathbb{E}[(Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0))D_{1}D_{2}|Z=0]+$
$\displaystyle\mathbb{E}[(Y(1,0)-Y(0,0))D_{1}|Z=1]-\mathbb{E}[(Y(1,0)-Y(0,0))D_{1}|Z=0]+$
$\displaystyle\mathbb{E}[(Y(0,1)-Y(0,0))D_{2}|Z=1]-\mathbb{E}[(Y(0,1)-Y(0,0))D_{2}|Z=0],$

using 2.4. It follows from the results above in (23), (37), and (39) that

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\}]\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\}]+$
(42)
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]+$
$\displaystyle\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)-Y(1,0)+Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}].$

##  Appendix D Proof Theorem 1

First, we collect some relations that will be useful for this proof and the
proof of Corollary 3. Note that from Lemma 2 we have the following relations:

$\displaystyle\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]$
$\displaystyle=\Delta\mathbb{E}[D_{1}|Z]-\Delta\mathbb{E}[D_{\land}|Z]=\mathbb{P}[C_{1},N_{2}].$
(43) $\displaystyle\Delta\mathbb{E}[D_{\land}-D_{2}|Z]$
$\displaystyle=\Delta\mathbb{E}[D_{\land}|Z]-\Delta\mathbb{E}[D_{2}|Z]=\mathbb{P}[C_{1},A_{2}].$
(44)

Using (5), we can write

$\displaystyle D_{1}Y$ $\displaystyle=[Y(1,1)-Y(1,0)]D_{1}D_{2}+Y(1,0)D_{1},$
(45) $\displaystyle D_{2}Y$
$\displaystyle=[Y(1,1)-Y(0,1)]D_{1}D_{2}+Y(0,1)D_{2},$ (46) $\displaystyle
D_{1}D_{2}Y$ $\displaystyle=Y(1,1)D_{1}D_{2},$ (47)

and it follows from the results in Lemma 2 that

$\displaystyle\Delta\mathbb{E}[D_{1}Y|Z]=$
$\displaystyle\Delta\mathbb{E}[[Y(1,1)-Y(1,0)]D_{1}D_{2}|Z]+\Delta\mathbb{E}[Y(1,0)D_{1}|Z]$
(48) $\displaystyle=$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(1,0))D_{1}D_{2}|Z=1]-\mathbb{E}[(Y(1,1)-Y(1,0))D_{1}D_{2}|Z=0]+$
$\displaystyle\mathbb{E}[(Y(1,0)D_{1}|Z=1]-\mathbb{E}[(Y(1,0)D_{1}|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
$\displaystyle\mathbb{E}[Y(1,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]$
$\displaystyle\mathbb{E}[Y(1,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}],$

and

$\displaystyle\Delta\mathbb{E}[D_{2}Y|Z]=$
$\displaystyle\Delta\mathbb{E}[[Y(1,1)-Y(0,1)]D_{1}D_{2}|Z]+\Delta\mathbb{E}[Y(0,1)D_{2}|Z]$
(49) $\displaystyle=$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(0,1))D_{1}D_{2}|Z=1]-\mathbb{E}[(Y(1,1)-Y(0,1))D_{1}D_{2}|Z=0]+$
$\displaystyle\mathbb{E}[Y(0,1)D_{2}|Z=1]-\mathbb{E}[Y(0,1)D_{2}|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
$\displaystyle\mathbb{E}[Y(0,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}],$

and

$\displaystyle\Delta\mathbb{E}[D_{\land}Y|Z]=$
$\displaystyle\Delta\mathbb{E}[Y(1,1)D_{1}D_{2}|Z]$ (50) $\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)D_{1}D_{2}|Z=1]-\mathbb{E}[Y(1,1)D_{1}D_{2}|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}].$

From this we can construct,

$\displaystyle\Delta\mathbb{E}[(D_{\lor}-D_{2})Y|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{1}Y|Z]-\Delta\mathbb{E}[D_{\land}Y|Z]$ (51)
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}],$
$\displaystyle\Delta\mathbb{E}[(D_{\land}-D_{2})Y|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{\land}Y|Z]-\Delta\mathbb{E}[D_{2}Y|Z]$ (52)
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}],$
$\displaystyle\Delta\mathbb{E}[(1-D_{1})(1-D_{2})Y|Z]=$
$\displaystyle\Delta\mathbb{E}[Y|Z]+\Delta\mathbb{E}[D_{\land}Y|Z]-\Delta\mathbb{E}[D_{1}Y|Z]-\Delta\mathbb{E}[D_{2}Y|Z]$
(53) $\displaystyle=$
$\displaystyle-\mathbb{E}[Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]-\mathbb{E}[Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}].$

Second, we construct the lower and upper bounds. Assume the following monotone
treatment response: $\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]\geq 0$ and
$\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]\geq 0$. Using the result in Lemma 2, we
can construct the lower bound as

$\displaystyle\Delta\mathbb{E}[Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+$
(54)
$\displaystyle\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]-$
$\displaystyle\mathbb{E}[Y(1,1)-Y(1,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]-\mathbb{E}[Y(0,1)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}](\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}].$

And from Lemma 2 follows that
$\Delta\mathbb{E}[D_{1}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]$.

Assume the following monotone treatment selection:
$\mathbb{E}[Y(1,1)|C_{1},A_{2}]\geq\mathbb{E}[Y(1,1)|C_{1},N_{2}]$ and
$\mathbb{E}[Y(1,1)|C_{1},C_{2}]\geq\mathbb{E}[Y(1,1)|C_{1},N_{2}]$. Using the
results above, it follows that

$\displaystyle\frac{\Delta\mathbb{E}[D_{\land}Y|Z]}{\Delta\mathbb{E}[D_{\land}|Z]}=$
$\displaystyle\frac{\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]}$
(55) $\displaystyle\geq$
$\displaystyle\frac{\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[C_{1},N_{2}]}+$
$\displaystyle\frac{\mathbb{E}[Y(1,1)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[C_{1},N_{2}]}.$

Next, assume the following positive response:
$\mathbb{E}[Y(0,0)|C_{1},A_{2}]\geq 0$. Using the results above, it follows
that

$\displaystyle\frac{\Delta\mathbb{E}[(1-D_{1})(1-D_{2})Y|Z]}{\Delta\mathbb{E}[D_{1}|Z]}=$
$\displaystyle\frac{-\mathbb{E}[Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]-\mathbb{E}[Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}$
(56) $\displaystyle\geq$
$\displaystyle\frac{-\mathbb{E}[Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]-\mathbb{E}[Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}-$
$\displaystyle\frac{\mathbb{E}[Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}.$

We can construct the upper bound as

$\displaystyle\frac{\Delta\mathbb{E}[D_{\land}Y|Z]}{\Delta\mathbb{E}[D_{\land}|Z]}+\frac{\Delta\mathbb{E}[(1-D_{1})(1-D_{2})Y|Z]}{\Delta\mathbb{E}[D_{1}|Z]}\geq$
(57)
$\displaystyle\frac{\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}+$
$\displaystyle\frac{\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}.$

##  Appendix E Proof Corollary 3

Using the results in Appendix D, construct the lower bound as

$\displaystyle\Delta\mathbb{E}[(1-D_{1}-D_{2}+2D_{1}D_{2})Y|Z]+Y_{\min}\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]-Y_{\max}\Delta\mathbb{E}[D_{\land}-D_{2}|Z]=$
(58)
$\displaystyle\Delta\mathbb{E}[Y|Z]+\Delta\mathbb{E}[(D_{\land}-D_{2})Y|Z]-\Delta\mathbb{E}[(D_{\lor}-D_{2})Y|Z]+$
$\displaystyle
Y_{\min}\mathbb{P}[C_{1},N_{2}]-Y_{\max}\mathbb{P}[C_{1},A_{2}]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y_{\min}-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y_{\max}|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]\leq$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}].$

Similarly, the upper bound equals

$\displaystyle\Delta\mathbb{E}[(1-D_{1}-D_{2}+2D_{1}D_{2})Y|Z]+Y_{\max}\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]-Y_{\min}\Delta\mathbb{E}[D_{\land}-D_{2}|Z]=$
(59)
$\displaystyle\Delta\mathbb{E}[Y|Z]+\Delta\mathbb{E}[(D_{\land}-D_{2})Y|Z]-\Delta\mathbb{E}[(D_{\lor}-D_{2})Y|Z]+$
$\displaystyle
Y_{\max}\mathbb{P}[C_{1},N_{2}]-Y_{\min}\mathbb{P}[C_{1},A_{2}]=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y_{\max}-Y(0,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)-Y_{\min}|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]\geq$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}].$

And from Lemma 2 follows that
$\Delta\mathbb{E}[D_{1}|Z]=\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]$.

##  Appendix F Proof Proposition 2

Note that we do not invoke 3 in this proof. Hence, we use the expressions in
Lemma 1 to show that

$\displaystyle\Delta\mathbb{E}[D_{\lor}-D_{2}|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{1}|Z]-\Delta\mathbb{E}[D_{\land}|Z]=\mathbb{P}[C_{1},N_{2}]-\mathbb{P}[A_{1},C_{2}],$
(60) $\displaystyle\Delta\mathbb{E}[D_{\land}-D_{2}|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{\land}|Z]-\Delta\mathbb{E}[D_{2}|Z]=\mathbb{P}[C_{1},A_{2}]-\mathbb{P}[N_{1},C_{2}].$
(61)

Using (45), (46), and (47), and using the results in (23) and (26), we have

$\displaystyle\Delta\mathbb{E}[D_{1}Y|Z]=$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(1,0))D_{1}D_{2}+Y(1,0)D_{1}|Z=1]-$ (62)
$\displaystyle\mathbb{E}[(Y(1,1)-Y(1,0))D_{1}D_{2}+Y(1,0)D_{1}|Z=0]$
$\displaystyle=$
$\displaystyle\mathbb{E}[(Y(1,1)-Y(1,0))|\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\},\\{A_{1},C_{2}\\}]\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},A_{2}\\},\\{A_{1},C_{2}\\}]+$
$\displaystyle\mathbb{E}[Y(1,0)|\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]\mathbb{P}[\\{C_{1},C_{2}\\},\\{C_{1},N_{2}\\},\\{C_{1},A_{2}\\}]$
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]+$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}].$

If we additionally use (24), we similarly have that

$\displaystyle\Delta\mathbb{E}[D_{2}Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(0,1)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}]+$
(63)
$\displaystyle\mathbb{E}[Y(1,1)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}]+\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}],$
$\displaystyle\Delta\mathbb{E}[D_{\land}Y|Z]=$
$\displaystyle\mathbb{E}[Y(1,1)|C_{1},C_{2}]\mathbb{P}[C_{1},C_{2}]+\mathbb{E}[Y(1,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]+$
(64) $\displaystyle\mathbb{E}[Y(1,1)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}],$

from which follows that

$\displaystyle\Delta\mathbb{E}[(D_{\lor}-D_{2})Y|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{1}Y|Z]-\Delta\mathbb{E}[D_{\land}Y|Z]$ (65)
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(1,0)|C_{1},N_{2}]\mathbb{P}[C_{1},N_{2}]-\mathbb{E}[Y(1,0)|A_{1},C_{2}]\mathbb{P}[A_{1},C_{2}],$
$\displaystyle\Delta\mathbb{E}[(D_{\land}-D_{2})Y|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{\land}Y|Z]-\Delta\mathbb{E}[D_{2}Y|Z]$ (66)
$\displaystyle=$
$\displaystyle\mathbb{E}[Y(0,1)|C_{1},A_{2}]\mathbb{P}[C_{1},A_{2}]-\mathbb{E}[Y(0,1)|N_{1},C_{2}]\mathbb{P}[N_{1},C_{2}].$

##  Appendix G Proof Corollary 4

Corollary 4 defines the weighted average of LATEs

$\displaystyle\tau=$
$\displaystyle\mathbb{E}[Y(1,1)-Y(0,0)|C_{1},C_{2}]w[C_{1},C_{2}]+\mathbb{E}[Y(1,0)-Y(0,0)|C_{1},N_{2}]w[C_{1},N_{2}]+$
(67)
$\displaystyle\,\mathbb{E}[Y(1,1)-Y(0,1)|C_{1},A_{2}]w[C_{1},A_{2}]+\mathbb{E}[Y(0,1)-Y(0,0)|N_{1},C_{2}]w[N_{1},C_{2}]+$
$\displaystyle\,\mathbb{E}[Y(1,1)-Y(1,0)|A_{1},C_{2}]w[A_{1},C_{2}],$

with
$w(G_{1},G_{2})=\frac{\mathbb{P}[G_{1},G_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}$.
From Lemma 1 follows that

$\displaystyle\Delta\mathbb{E}[Y|Z]=\tau(\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]).$
(68)

From this we can construct an upper bound on $\tau$:

$\displaystyle\beta_{IV}(D_{1})=\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D_{1}|Z]}=$
$\displaystyle\tau\frac{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}$
(69) $\displaystyle=$
$\displaystyle\tau\Big{(}1+\frac{\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]}\Big{)}\geq\tau.$

However, if other specifications for $D$ give a smaller second stage estimate,
this is a tighter upper bound.

From Lemma 1 also follows that

$\displaystyle\Delta\mathbb{E}[D_{1}+D_{2}|Z]=$
$\displaystyle\Delta\mathbb{E}[D_{1}|Z]+\Delta\mathbb{E}[D_{2}|Z]$ (70)
$\displaystyle=$ $\displaystyle
2\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}].$

From this we can construct a lower bound on $\tau$:

$\displaystyle\beta_{IV}(D_{1}+D_{2})=$
$\displaystyle\frac{\Delta\mathbb{E}[Y|Z]}{\Delta\mathbb{E}[D_{1}+D_{2}|Z]}$
(71) $\displaystyle=$
$\displaystyle\tau\frac{\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}{2\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}$
$\displaystyle=$
$\displaystyle\tau\Big{(}1-\frac{\mathbb{P}[C_{1},C_{2}]}{2\mathbb{P}[C_{1},C_{2}]+\mathbb{P}[C_{1},N_{2}]+\mathbb{P}[C_{1},A_{2}]+\mathbb{P}[N_{1},C_{2}]+\mathbb{P}[A_{1},C_{2}]}\Big{)}\leq\tau.$

##  Appendix H Additional details on the empirical specifications

###  H.1 The Oregon health insurance experiment

Similar to Finkelstein et al. (2012); Finkelstein et al. (2016), the outcome
variable $Y$ is measured during the same period as the treatment variables
$D_{1}$ and $D_{2}$. For IV estimation to be valid in this specification,
emergency department use should not affect Medicaid coverage. We find
qualitatively similar results if we measure $D_{1}$ in the first year, $D_{2}$
in the first half of the second year, and $Y$ in the second year after
treatment.

We follow the specification of the controls and standard errors in Finkelstein
et al. (2016). We include three control variables in our analysis. First,
since the entire household of any selected individual had the opportunity to
apply for Medicaid, we include household size dummies. Second, since the
lottery draws were done in eight rounds, we include lottery round dummies.
Third, to improve precision, we control for the outcome variable $Y$ measured
during the 14 months prior to the first lottery draw. Standard errors are
clustered at the household level.

###  H.2 The LinkedIn job opportunities experiment

LinkedIn usage is measured via extractions from the Linkedin administrative
data at the end of the job training program and again six (and twelve) months
later. Employment six and twelve months after the job training program is
measured via phone surveys. The response to the phone survey is, respectively,
68% and 60% six and twelve months after the program ended. This response rate
is balanced across the treatment and control group. The final estimation
sample is restricted to the 988 participants for whom we observe employment
status twelve months after the program.

The random assignment took place on the cohort level. In particular, the 30
cohorts were randomly split into 15 treatment and 15 control groups using
within-city, sequentially paired randomization. We follow the specification of
the controls and standard errors in Wheeler et al. (2022). We include cohort-
pair dummies as control variables. Standard errors are clustered at the cohort
level.

###  H.3 The elite university education natural experiment

The application score equally weighs a combination of high school grades and
an admission test score administered by the elite university. Due to the short
time span between the admission test and enrollment deadline, the elite
university does not have a waiting list but sets the number of admitted
students slightly above capacity based upon previous years’ enrollment rates.

Anelli (2020) compares applicants just above and below the application score
cutoffs via a fuzzy Regression Discontinuity Design (RDD). The number of
students, pre-treatment characteristics, and the probability of observing zero
or missing income are smooth across the cutoff. The sample minimum and maximum
of log income ($Y$) is equal to $2.398$ and $12.441$ respectively.

The polynomial and bandwidth are important considerations for an RD analysis.
Anelli (2020) uses a polynomial of degree 0, which is often referred to as the
local randomization approach (Cattaneo et al., 2023), and uses four different
bandwidths: no bandwidth using all 645 applicants (Table 4, column (2) of
Anelli (2020)), the MSE-optimal bandwidth using 162 to 175 applicants
depending on the specification (Table 4, column (3)), and twice and three
times the MSE-optimal bandwidth using 280 to 303 and 382 to 409 applicants
respectively (Table 4, column (4) and (5)). Results are qualitatively similar
across the four bandwidths.

Our results are based on the specification without bandwidth. The RD
specification, absent polynomial and bandwidth, follows a standard IV setup
with a binary instrument $Z$ that equals one if the applicant had a score
above the cutoff. We follow the specification of the controls and standard
errors in Anelli (2020). We include five control variables: parental house
value, a female dummy, a dummy for students commuting from outside the city,
fixed effects for the year of application, and a (potentially endogenous)
dummy for whether the individual is enrolled in university in 2005, which is
the year when income is observed. Standard errors are clustered at the cohort
high-school level.

Ideally our $D_{2}$ would measure whether the student graduated from the elite
university. In this case, the bounds would partially identify the LAFTE of the
elite university. Since students that were induced to enroll at the elite
university may never graduate from this institution, it is still likely that
$\mathbb{P}[C_{1},N_{2}]>0$. In turn, with this $D_{2}$ it may be that
$\mathbb{P}[C_{1},A_{2}]=\mathbb{P}[N_{1},C_{2}]=\mathbb{P}[A_{1},C_{2}]=0$.
The first two mover types would likely be absent since a student cannot
graduate from the elite university without ever being enrolled. The third
mover type would be absent since there are very few always takers with elite
enrollment.

###  H.4 The village-based schools experiment

The village-based schools were opened in the summer of 2007. The researchers
conducted surveys of all available households in the control and treatment
villages in the fall of 2007, four months after the opening, and in the spring
of 2008, eight months after the opening. Both surveys determined school
enrollment status of each school-age child living in the household and
administered a short test covering math and language. Survey coverage rates
were similar across treatment and control villages.

We follow Burde and Linden (2013) and exclude a small number of extremely
large and wealthy households and further restrict the estimation sample to the
children for whom we observe enrollment in both surveys and test scores in the
spring 2008 survey. The final sample includes 1,181 children, of in total
1,490 school-age children across the 31 villages. The sample minimum and
maximum of the standardized test scores in the spring 2008 survey ($Y$) is
equal to $-1.263$ and $2.662$ respectively.

We follow the specification of the controls and standard errors in Burde and
Linden (2013). We control for a single dummy variable that equals one if the
village is located in the Chagcharan district. Since the 31 villages were
grouped into 11 equally sized village groups, randomization took place at the
group level, and standard errors are clustered at the village-group level.

###  H.5 Sensitivity analysis controls

The empirical results in this paper are based on the control specification
used in the original papers. Here we analyse the robustness of our results
against excluding the controls. We find in all four empirical applications
identical mover types and identical results for the necessary conditions of
the double exclusion restriction. The bounds of Theorem 1 are qualitatively
similar except for inflated standard errors in all applications, and a lower
bound close to zero in the Oregon health insurance experiment.

##  Appendix I Additional empirical results

Column (1) and (2) of Table 4 report the precise estimates and standard errors
of the bounds from Theorem 1, which are visually reported in Figure 2. Column
(3) and (4) show the bounds from Corollary 3 based on the bounded response.
The latter two columns show that, for all four empirical applications, bounds
based on the bounded response assumption are wide and cannot reject that the
LAFTEs are equal to zero.

Table 5 reports the first stage and second stage estimates corresponding to
different treatment variables. Column (1) shows the results for $D_{1}+D_{2}$,
corresponding to the IV estimand in (20). Columns (2) to (5) show the results
for the treatment indicators $D_{1}$, $D_{2}$, $D_{\land}$, and $D_{\lor}$,
respectively. Note that the estimate in column (1) does not require the double
exclusion restriction and equals the lower bound for $\tau$ from Corollary 4.
The upper bound for $\tau$ equals the smallest second stage IV estimate in
column (2) to (5).

The double exclusion restriction is rejected for the village-based schools
experiment in panel D. Column (1) allows us to conclude, however, that four
months of education increases test scores by 0.679 standard deviations for the
compliers. Subsequently, using the smallest second stage estimate in column
(2) to (5), we can conclude that the estimated bounds for $\tau$ are
$[0.679,1.301]$.

Table 4: Estimated bounds on the LAFTE

|  | Theorem 1 bounds | Corollary 3 bounds
---|---|---|---
|  | (1) | (2) | (3) | (4)
|  | lower | upper | lower | upper
Panel A: Health insurance | Estimate | 0.081 | 0.224 | -0.170 | 0.375
| Std. Error. | 0.024 | 0.026 | 0.025 | 0.023
Panel B: LinkedIn training | Estimate | 0.176 | 0.235 | 0.061 | 0.260
| Std. Error. | 0.066 | 0.078 | 0.086 | 0.082
Panel C: Elite university | Estimate | 0.532 | 7.545 | -2.772 | 5.865
| Std. Error. | 0.233 | 0.760 | 0.337 | 0.580

  * Notes: this table shows the estimated lower and upper bounds from Theorem 1 and Corollary 3 with their corresponding standard errors, for the empirical applications in Section 5 for which the double exclusion restriction cannot be rejected. The upper bound from Theorem 1 is a linear function of two coefficients from two different second stage IV regressions. To obtain the standard error for this upper bound, we use a stacking approach that duplicates the data and estimates the two second stage IV coefficients via a single second stage IV regression. We cluster standard errors on the cluster variable that is similarly duplicated.

Table 5: The first and second stage estimates corresponding to different
treatment variables

| (1) | (2) | (3) | (4) | (5)
---|---|---|---|---|---
|  $D_{1}+D_{2}$ |  $D_{1}$ |  $D_{2}$ |  $D_{\land}$ |  $D_{\lor}$
Panel A: The Oregon health insurance experiment, $N=24646$
| First stage
$Z$ | 0.369 | 0.253 | 0.115 | 0.176 | 0.193
| (0.011) | (0.006) | (0.006) | (0.006) | (0.006)
| Second stage
$D$ from top row | 0.056 | 0.081 | 0.179 | 0.117 | 0.107
| (0.016) | (0.024) | (0.052) | (0.034) | (0.031)
Panel B: The LinkedIn job opportunities experiment, $N=988$
| First stage
$Z$ | 0.707 | 0.393 | 0.315 | 0.357 | 0.351
| (0.148) | (0.076) | (0.077) | (0.079) | (0.074)
| Second stage
$D$ from top row | 0.098 | 0.176 | 0.220 | 0.194 | 0.197
| (0.039) | (0.066) | (0.096) | (0.079) | (0.077)
Panel C: The elite university education natural experiment, $N=645$
| First stage
$Z$ | 0.775 | 0.680 | 0.095 | 0.633 | 0.143
| (0.106) | (0.053) | (0.061) | (0.051) | (0.063)
| Second stage
$D$ from top row | 0.466 | 0.532 | 3.796 | 0.571 | 2.534
| (0.215) | (0.233) | (3.079) | (0.254) | (1.605)
Panel D: The village-based schools experiment , $N=1181$
| First stage
$Z$ | 0.926 | 0.443 | 0.483 | 0.443 | 0.483
| (0.158) | (0.078) | (0.080) | (0.078) | (0.080)
| Second stage
$D$ from top row | 0.679 | 1.420 | 1.301 | 1.420 | 1.301
| (0.090) | (0.191) | (0.172) | (0.191) | (0.172)

  * Notes: this table provides the first and second stage estimates corresponding to different treatment variables. The numbers represent the estimated coefficients and the numbers in parentheses the corresponding standard errors. The panels correspond to the empirical applications in Section 5, where $N$ indicates the number of observations.