# Analysis of Randomized Experiments
with Network Interference and Noncompliance

Bora Kim

###### Abstract

Randomized experiments have become a standard tool in economics. In analyzing
randomized experiments, the traditional approach has been based on the Stable
Unit Treatment Value (SUTVA: Rubin (1990)) assumption which dictates that
there is no interference between individuals. However, the SUTVA assumption
fails to hold in many applications due to social interaction, general
equilibrium, and/or externality effects. While much progress has been made in
relaxing the SUTVA assumption, most of this literature has only considered a
setting with perfect compliance to treatment assignment. In practice, however,
noncompliance occurs frequently where the actual treatment receipt is
different from the assignment to the treatment. In this paper, we study causal
effects in randomized experiments with network interference and noncompliance.
Spillovers are allowed to occur at both treatment choice stage and outcome
realization stage. In particular, we explicitly model treatment choices of
agents as a binary game of incomplete information where resulting equilibrium
treatment choice probabilities affect outcomes of interest. Outcomes are
further characterized by a random coefficient model to allow for general
unobserved heterogeneity in the causal effects. After defining our causal
parameters of interest, we propose a simple control function estimator and
derive its asymptotic properties under large-network asymptotics. We apply our
methods to the randomized subsidy program of Dupas (2014) where we find
evidence of spillover effects on both short-run and long-run adoption of
insecticide-treated bed nets. Finally, we illustrate the usefulness of our
methods by analyzing the impact of counterfactual subsidy policies.

Keywords: causal inference, interference, spillover, networks, games of
incomplete information, control function

##  Introduction

Randomized experiments have become a standard tool for causal inference in
economics. In analyzing randomized experiments, the traditional approach is
based on the Stable Unit Treatment Value (SUTVA: Rubin (1990)) assumption
which dictates that there is no interference between individuals. However,
there are many settings where the SUTVA assumption fails to hold. For
instance, deworming treatment given to some student may affect academic
achievements of other students through externality effects (See for instance,
Miguel and Kremer (2004)). In labor market, Crépon et al. (2013) show that a
large-scale job placement program affects non-participant’s employment
probability through general equilibrium effects. Ferracci et al. (2014) also
report similar results. In such cases, there is interference or spillover
effect where an individual’s behavior either directly or indirectly affects
others’ outcomes through social interactions, externalities, or general
equilibrium effects.

In recent years, there has been substantial progress in relaxing the SUTVA
assumption in causal inference framework. Examples include Manski (2013),
Hudgens and Halloran (2008), Leung (2020a), Vazquez-Bare (2020), and Baird et
al. (2018). Much of the literature, however, has been built on the restrictive
assumption of perfect compliance to intervention in which experimental units
perfectly comply with their assignment of treatment. In practice,
noncompliance occurs commonly — some units assigned to treatment group may opt
out of the treatment, while some units assigned to control group may decide to
take the treatment. In studies of labor market, for example, Crépon et al.
(2013) report that only 35$\%$ of those who were offered intensive job
counseling actually took up the offer. While instrumental variables (IV)
methods are widely used to address the noncompliance problem, these methods
are developed based on the assumption that rules out interference between
units (Imbens and Angrist (1994)).

The goal of this paper is to develop a formal framework to conduct causal
inference in randomized experiments with _both_ spillovers and noncompliance.
In the presence of noncompliance, spillovers can occur at two stages: at the
treatment decision stage, and at the outcome realization stage. In the first
stage in which each agent _chooses_ their treatment status, spillovers may
occur if the utility from choosing treatment depends on the treatment choices
of others. In the second stage where outcomes (or responses) are realized,
agent’s outcome can be affected not only by their own treatment choice, but
also by treatment choices of others either directly or indirectly. While most
of existing literature has only addressed the spillover effects at the outcome
level (i.e., at the second stage), we allow for spillover effects both at the
treatment choice (first stage) and at the outcome (second stage).

To model spillovers, we take a game-theoretic approach. We consider a first
stage model in which agents play a binary game of incomplete information. Such
binary games of incomplete information have been used in various economic
applications, e.g., in empirical industrial organization literature (Bajari et
al. (2010)), to model binary choices under peer effects (Brock and Durlauf
(2001), Brock and Durlauf (2007) and Xu (2018)), and recently, to model
network formation process (Leung (2015), and Ridder and Sheng (2020)). We
apply the method to the problem of endogenous treatment choices in the
presence of spillovers. Specifically, we assume that agents simultaneously
choose their treatment status as to maximize their expected utilities, given
beliefs about anticipated treatment choices of their neighbors. In
equilibrium, agents’ subjective beliefs coincide with objective choice
probabilities. Assuming that the unique equilibrium exists, the reduced-form
model of agent’s treatment choice can be written as a single threshold-
crossing model where the threshold is a function of agent’s own treatment
assignment and the average equilibrium treatment choice probability of their
neighbors. In the second stage, outcomes are modeled as being a function of
agent’s own treatment choice and the equilibrium average treatment choice
probability of their neighbors, as it is determined in the first stage game.
As in the first stage choice model, spillovers are captured by the equilibrium
treatment choice probabilities.

In our model, therefore, equilibrium treatment choice probabilities work as a
mediator of spillover effects. This is different from the existing literature
which often models the spillover at the outcome level by the proportion of
treated neighbors. See for instance Hudgens and Halloran (2008), Leung
(2020a), and Vazquez-Bare (2020). As we show later, when the outcome of
interest represents a choice or behavior of individuals, their formulation
implicitly assumes that the proportion of treated neighbors is fully
observable to agents, i.e., agents possess a complete information over
behaviors of their peers. However, the assumption of complete information is
unrealistic especially in a single large network setting as ours where each
individual has a considerable number of peers.111In our application, for
instance, agents have 17 neighbors on average. In such cases, it is more
reasonable to assume that agents face uncertainty over others’ behavior,
making an incomplete information framework more adequate approximation of
reality.

We then characterize outcomes as a random coefficient model to allow for
general unobserved heterogeneity. Our parameters of interest are average
causal effects which include an average direct effect of own treatment take-up
and an average spillover effect from direct neighbors. After rigorously
defining our parameters of interest, we show our identification result. We
first note that under general unobserved heterogeneity, the conventional
instrumental variables (IV) methods do not identify the causal parameters when
we allow for general heterogeneity in the outcome. We therefore propose our
alternative identification based on a control function approach.

We then propose a simple two-step estimator where the first step estimates the
payoff parameters of treatment choice games using nested fixed-point maximum-
likelihood estimation and the second step estimates the average potential
outcome functions using control function regression. Our estimator extends
canonical Heckman (1979) sample selection estimator (“Heckit”) to incorporate
possible spillover effects. We show that the estimators are
$\sqrt{n}$-consistent and asymptotically normal under the “large-network”
asymptotics in which a number of individuals connected in a single network
increases to infinity. We study finite-sample properties of our estimators
through Monte Carlo simulation.

Our methods are applied to the randomized subsidy program of Dupas (2014).
While the use of insecticide-treated nets (ITNs) has been shown to be
effective in controlling malaria, the rate of adoption remains low. Given that
the mosquito nets need to be re-purchased and replaced regularly,
understanding the factors affecting household’s short-run and long-run
decision to purchase the bednet is an important task to achieve sufficiently
high equilibrium adoption rate. In our application, we study the effect of
short-run purchase of the bednet on the long-run purchase decision while
incorporating possible spillovers from neighbors defined by geographical
proximity. The treatment is a binary is a binary indicator for purchasing a
mosquito net in the short-run (in Phase 1) and the outcome is a binary
indicator for purchasing a mosquito net in the long-run (in Phase 2).

We find evidence of positive spillover effects in the short-run bednet
purchase decision. More specifically, in Phase 1, households were more likely
to purchase the bednet when the average expected purchase rate of their
neighbors is higher. On the contrary, we find the evidence of negative
spillover effects in the long run although the statistical power is limited.
Specifically, households were less likely to purchase the bednet in Phase 2
when the average expected purchase rate in Phase 1 was higher. Our results
also suggest that the average direct effect of the bednet purchase in Phase 1
on the purchase in Phase 2 declines monotonically with respect to the expected
neighborhood purchase rate in Phase 1. When the Phase-1 neighborhood purchase
rate was $0\%$ (no spillover), households who purchased the bednet in Phase 1
were 36.9 percentage points more likely to purchase the bednet in Phase 2
compared to those who did not purchase the bednet in Phase 1. Such effect
becomes almost to zero at another extreme where the neighborhood purchase rate
was $100\%$ (full spillover). Ignoring spillover effects leads to the
misleading conclusion that the average direct effect of the short-run purchase
on the long-run purchase is almost zero when in fact, the effect varies from
$0\%$ to $36\%$ depending on the degree of spillovers.

Our structural modeling allows researchers to analyze the impact of
counterfactual policies on the outcome of interest. We illustrate this by
analyzing the impact of counterfactual subsidy program on the long-run
adoption in which a policy-maker implements a means-tested subsidy rule where
the subsidy is given only when the household’s income level is below some pre-
specified threshold. We predict the average long-term adoption rate under
different subsidy regimes defined by different values of the eligibility
threshold. We find that even under the very generous subsidy regime where
almost everyone in the sample receives the subsidy, the average long-run
adoption rate does not exceed $20\%$, due to the large negative spillover in
the long-run.

##### Related Literature

Recent works on causal inference under spillovers mainly concentrate on the
case with random treatment, i.e., they do not address treatment choice
endogeneity. Examples include Hudgens and Halloran (2008), Leung (2020a), and
Vazquez-Bare (2020).

In causal inference literature, game-theoretic models have been used in
several papers. Lazzati (2015) proposes a structural model of treatment
responses using games of complete information. However, the paper does not
address the endogeneity of treatment choices. Balat and Han (2019) allow
spillovers at both choice and outcome stages using game theoretic approach.
Their model is different from ours in that they model treatment choice by a
binary game of complete (perfect) information. Also, Balat and Han (2019)
consider an interaction within groups while we consider an interaction under
general network. While the assumption of complete information may be
appropriate under interactions in a relatively small group, incomplete
information assumption is more reasonable under network interactions,
especially when the network size is large. Jackson et al. (2020) model
treatment choices as a binary game of incomplete information. However, they do
not consider spillovers at the outcome level while we are interested in
separately identifying the individual treatment effect and spillover effect.

Meanwhile a literature from statistics has started to incorporate spillovers
and noncompliance in network setting. See Imai et al. (2020) for the most
recent progress. Unlike our game-theoretic model, their model is reduced-form
in nature and consequently, important aspects of economic mechanism behind
treatment choices such as utility maximization are largely ignored.

##### Outline

We describe our model in Section 2. We first outline our model of treatment
choices and then the model of potential outcomes. Parameters of interest are
also discussed. Section 3 discusses identification of parameters of interest.
We first show that the conventional IV methods are not valid in the presence
of treatment effect heterogeneity. We then show how to use control function
approach to achieve point identification. In Section 4, we propose a simple
two-stage estimation procedure. Asymptotic properties are derived and
simulation results are also presented. Section 5 applies our methods to
empirical setting.

##  Model of Treatment Choices and Outcomes

In this section, we first describe our treatment choice models as a binary
game under incomplete information. We then describe our model of treatment
responses under spillovers.

Let $\mathcal{N}_{n}=\\{1,\cdots,n\\}$ denote a set of agents. $n$-many agents
are connected through a single, large network. Let $G$ be a symmetric $n\times
n$ adjacency matrix where $ij$th entry ($G_{ij}$) represents a connection or
link between agents. Specifically, $G_{ij}=1$ if agent $i$ and $j$ are
connected and $G_{ij}=0$ otherwise. We assume $G_{ii}=0$ for all
$i\in\mathcal{N}_{n}$ (no self-link). When $G_{ij}=1$, we say that $i$ and $j$
are (direct) _peers_ or _neighbors_. Let $\mathcal{N}_{i}$ be a set of $i$’s
peers, i.e., $\mathcal{N}_{i}=\\{j\in\mathcal{N}_{n}:G_{ij}=1\\}$. The number
of $i$’s neighbors or _degree_ of $i$ is denoted as $|\mathcal{N}_{i}|$.

###  Treatment Choice Model with Spillovers

We consider a game theoretic model of treatment choice. Specifically, we
characterize a realized treatment choice as a solution to a binary game under
incomplete information played by agents in a given network. In this framework,
agents simultaneously choose their treatment status in order to maximize their
expected utility, given beliefs about the anticipated behaviors of their
peers.

##### Utility

Each agent $i$ has a vector of observed characteristics $X_{i}\in\mathcal{X}$
and an unobserved utility shock $v_{i}\in\mathbb{R}$. Throughout the paper, we
assume that $\mathcal{X}$ is a bounded subset of $\mathbb{R}^{k}$. In
addition, each $i$ is randomly assigned to treatment. Let $Z_{i}\in\\{0,1\\}$
represent $i$’s randomized treatment assignment where $Z_{i}=1$ if $i$ is
assigned to treatment and $Z_{i}=0$ if $i$ is assigned to control. Let
$Z=(Z_{i})_{i\in\mathcal{N}_{n}}$ and $X=(X_{i})_{i\in\mathcal{N}_{n}}$. There
is noncompliance if $Z\not=D$, i.e., for some $i$, the treatment assignment is
different from the actual treatment received. There are two possible cases for
this: $(Z_{i},D_{i})=(1,0)$ and $(Z_{i},D_{i})=(0,1)$. The former indicates
that $i$ who was assigned to treatment group has refused to take the
treatment. The latter indicates that $i$ has received the treatment even when
$i$ was assigned to control group. In this paper, we allow for both cases,
i.e., we consider a setting with two-sided noncompliance.

Unlike $Z_{i}$, $D_{i}$ is self-selection. We assume that each $i$ chooses
$D_{i}\in\\{0,1\\}$ by utility maximization where the utility that $i$
receives depends on the choices of $i$’s peers. Let the utility function of
agent $i$ be $\pi(D_{i},D_{-i},X_{i},Z_{i},v_{i})$ where
$D_{-i}\in\\{0,1\\}^{n-1}$ is a vector of treatment choices of agents except
for $i$. We specify the utility function as the following linear model:

$\displaystyle\pi(D_{i},D_{-i},X_{i},Z_{i},v_{i})=\begin{cases}X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}D_{j}-v_{i}&\textrm{if
}D_{i}=1\\\ 0&\textrm{if }D_{i}=0.\end{cases}$ (1)

First note that the utility from choosing $D_{i}=0$ is normalized as zero.
This is without loss of generality as only difference in utilities is
identified. Utility of choosing $D_{i}=1$ depends on other agents’ treatment
choices through the term $\sum_{j\in\mathcal{N}_{i}}D_{j}/|\mathcal{N}_{i}|$,
the fraction of peers taking up the treatment. This term represents social
interactions or spillover effects in treatment choice. When $\theta_{3}=0$,
there are no spillovers and the model becomes a usual single-agent binary
choice model as in McFadden (1984). When $\theta_{3}>0$, we have positive
spillovers where the utility of choosing $D_{i}=1$ is higher when members of
$i$’s reference group (directed neighbors in our specification) behave
similarly. $\theta_{3}>0$ thus implies that agents have preference for
conformity. On the other hand, when $\theta_{3}<0$, we conclude that there are
negative spillovers in treatment choice.

We assume that $v_{i}$ is a private information, i.e., $v_{i}$ is known only
to $i$, and other agents cannot observe $v_{i}$. Therefore agents have
incomplete information over others’ choices. In other words, $i$ cannot
observe other players’ treatment choices at the time their choice is made.
Instead, each agent $i$ chooses an action that maximizes their expected
utility given their beliefs on
$\sum_{j\in\mathcal{N}_{i}}D_{j}/|\mathcal{N}_{i}|$. Beliefs are formed under
the information set available to $i$. Let $\tau_{i}$ denote $i$’s information
set. We specify $\tau_{i}$ as follows:

######  Assumption 1 (informational structure).

Let $G=(G_{ij})_{i,j\in\mathcal{N}_{n}}$, $X=(X_{i})_{i\in\mathcal{N}_{n}}$
and $Z=(Z_{i})_{i\in\mathcal{N}_{n}}$. We assume that $(G,X,Z)$ is a _public
information_ , i.e., every agent knows the entire network structure ($G$), the
vector of observed characteristics ($X$) and the vector of treatment
assignment ($Z$). On the other hand, $v_{i}$ is a _private information_ of $i$
where its value is only known to $i$. Therefore $\tau_{i}=(G,X,Z,v_{i})$
summarizes the information available to $i$.

The assumption 1 is standard in the literature on games of incomplete
information. Let $S=(G,X,Z)$ be the set of public information. This is often
called a _public state variable_ as well. For private information $v_{i}$, we
make the following assumption:

######  Assumption 2 (unobserved heterogeneity).

For all $i\in\mathcal{N}_{n}$, a private information $v_{i}$ is

  1. (i)

i.i.d. with a standard normal cdf $\Phi$ and

  2. (ii)

independent of $S$.

As in the standard single-agent binary choice models, distribution of $v_{i}$
must be known up to a finite-dimensional parameter. We use the normal
distribution only for convenience. Other distributional assumptions such as
logit can be used as well. The assumption that $v_{i}$’s are independent to
each other is critical for our identification analysis. This assumption
implies that the knowledge of $v_{i}$ does not help predicting $v_{j}$ for any
$j\not=i$. To our knowledge, identification of incomplete information games
with correlated private information in a general network setting is an open
question. Assumption 2 (ii) is trivially satisfied if we treat $S$ as fixed.
Consequently, we do not address the issue of network endogeneity as it is not
a focus of this paper.

##### Strategy

Let $D_{i}(\tau_{i},\theta)$ denote $i$’s pure strategy which maps $i$’s
information set $\tau_{i}=(S,v_{i})$ to a treatment choice $D_{i}\in\\{0,1\\}$
given a parameter value $\theta=(\theta_{1},\theta_{2},\theta_{3})$. Agent $i$
chooses her optimal action by maximizing her expected utility
$\mathbf{E}[\pi(D_{i},D_{-i},X_{i},Z_{i},v_{i})|\tau_{i}]$ where the
expectation is taken with respect to $D_{-i}$ given her belief about $D_{-i}$.
Let $\sigma_{j,i}$ be $i$’s belief over the event $\\{D_{j}=1\\}$ given the
information $\tau_{i}$. Then

$\displaystyle\sigma_{j,i}$ $\displaystyle=^{def}$
$\displaystyle\Pr(D_{j}=1|\tau_{i})$ (2) $\displaystyle=$
$\displaystyle\Pr(D_{j}(\tau_{j},\theta)=1|\tau_{i})$ (3) $\displaystyle=$
$\displaystyle\Pr(D_{j}(S,v_{j},\theta)=1|S,v_{i})$ (4) $\displaystyle=$
$\displaystyle\Pr(D_{j}(S,v_{j},\theta)=1)$ (5) $\displaystyle=$
$\displaystyle\sigma_{j}(S,\theta)$ (6)

where the fourth equality follows from the Assumption 2. From the last
equality, we see that $\sigma_{j,i}=\sigma_{j}$ for all $i\not=j$, i.e., every
agent shares a common belief on $j$’s choice. This common belief should be
consistent with actual probability of $j$ choosing $D_{j}=1$ under rational
expectations as we show below.

##### Equilibrium

Given the belief profile of $\\{\sigma_{j}(S,\theta)\\}_{j\not=i}$, agent $i$
calculates the expected utility he gets when choosing $D_{i}=1$ as follows:

$\displaystyle\mathbf{E}\big{[}\pi(1,D_{-i},X_{i},Z_{i},v_{i})|\tau_{i}\big{]}$
$\displaystyle=$
$\displaystyle\mathbf{E}\big{[}X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}D_{j}-v_{i}\big{|}S,v_{i}\big{]}$
(7) $\displaystyle=$ $\displaystyle
X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\underbrace{\Pr(D_{j}=1|S)}_{=\sigma_{j}(S,\theta)}-v_{i}$
(8) $\displaystyle=$ $\displaystyle
X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}(S,\theta)-v_{i}.$
(9)

Agent $i$ would choose $D_{i}=1$ if
$\mathbf{E}\big{[}\pi(1,D_{-i},X_{i},Z_{i},v_{i})|\tau_{i}\big{]}\geq 0$.
Therefore,

$D_{i}=\mathbf{1}\Big{\\{}v_{i}\leq
X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}(S,\theta)\Big{\\}}.$

Bayes-Nash equilibrium (BNE) is defined by a vector of choice probabilities
$\sigma^{*}(S,\theta)=\big{(}\sigma_{i}^{*}(S,\theta)\big{)}_{i\in\mathcal{N}_{n}}$
that is consistent with the observed decision rule in the sense that it
satisfies the following system of equations:

$\displaystyle\sigma_{i}^{*}(S,\theta)$ $\displaystyle=$
$\displaystyle\Pr\big{(}v_{i}\leq
X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)\big{)},\quad\forall
i\in\mathcal{N}_{n}$ (10) $\displaystyle=$
$\displaystyle\Phi\Big{(}X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)\Big{)},\quad\forall
i\in\mathcal{N}_{n}.$ (11)

Here we use the superscript $[*]$ to emphasize that $\sigma^{*}(S,\theta)$ is
an equilibrium quantity. In other words, Bayes-Nash equilibrium given
$(S,\theta)$ is a vector $\sigma^{*}(S,\theta)$ which is defined as a fixed
point to the system of equations above. By the implicit function theorem, it
can be shown easily that $\sigma^{*}(S,\theta)$ is smooth in both $S$ and
$\theta$. Therefore the existence of a fixed point is guaranteed due to
Brouwer’s fixed point theorem for any realized data $S$ and parameter value
$\theta$. However, there can be many fixed points $\sigma^{*}(S,\theta)$
solving the system. We show that a unique equilibrium exists if we restrict
the value of $\theta_{3}$ to be sufficiently mild. Formally,

######  Theorem 1 (unique equilibrium).

Let the pdf of $v_{i}$ be $\phi(v)$. Define
$\lambda=|\theta_{3}|\sup_{u}\phi(u)$. For any $S$ and $\theta$, there exists
a unique equilibrium $\\{\sigma^{*}_{j}(S,\theta)\\}_{j\in\mathcal{N}_{n}}$ if
$\lambda<1$.

See appendix A for proof. When $v_{i}$ is normally distributed, we have
$\sup_{u}\phi(u)=1/\sqrt{2\pi}$. Therefore $\lambda<1$ is equivalent to
$|\theta_{3}|<\sqrt{2\pi}\approx 2.5$. Throughout the paper we assume that
$\lambda<1$ so that the degree of interaction is not too strong to breed
multiple equilibria.

######  Assumption 3 (unique equilibrium).

$|\theta_{3}|<\sqrt{2\pi}$.

Under the unique equilibrium, agent’s treatment choice can be written as the
following reduced-form equation:

$\displaystyle D_{i}=\mathbf{1}\Big{\\{}v_{i}\leq
X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)\Big{\\}}$
(12) $\displaystyle\Longleftrightarrow$ $\displaystyle
D_{i}=\mathbf{1}\Big{\\{}\Phi(v_{i})\leq\Phi\Big{(}X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)\Big{)}\Big{\\}}$
(13) $\displaystyle\Longleftrightarrow$ $\displaystyle
D_{i}=\mathbf{1}\Big{\\{}\Phi(v_{i})\leq\sigma_{i}^{*}(S,\theta)\Big{\\}}$
(14)

where the last step follows from 11.

The story goes like this: For given $S$ and $\theta$, the equilibrium choice
probabilities $\sigma_{i}^{*}(S,\theta),\forall i\in\mathcal{N}_{n}$ are
realized. Observing this equilibrium, each agent chooses their treatment
status according to either 12,13 or 14.

###  Potential Outcomes Model with Spillovers

In this section, we propose our model of treatment response in settings with
spillovers. Previous research on treatment response has been based on the
SUTVA assumption which requires that an individual’s outcome depends only on
their own treatment status. Under the SUTVA assumption, $i$’s outcome or
response $Y_{i}$ can be written as $Y_{i}=Y_{i}(D_{i})$. Let $d\in\\{0,1\\}$
be the possible treatment value that agents can get. Potential outcome under
the SUTVA assumption is denoted by $Y_{i}(d)$, which delivers the response of
$i$ when assigned to $D_{i}=d$. Unlike the SUTVA case, however, there is no
obvious way to model spillovers in the treatment response. As Manski (2013)
and Kline and Tamer (2020) show, there are many ways to relax the SUTVA
assumption, each of which is based on different restrictions on the nature of
interference between agents.

In our paper, we assume that $i$’s outcome is a function of a _direct effect_
from own treatment status and an _indirect effect_ or _spillover effect_ from
$i$’s neighbors. Spillover effects are assumed to be mediated by
$\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)/|\mathcal{N}_{i}|$. For
notational simplicity, let us define
$\pi_{i}^{*}(S,\theta)=\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)/|\mathcal{N}_{i}|$.
Also, $\pi_{i}^{*}$ and $\pi_{i}^{*}(S,\theta)$ will be used interchangeably.
Thus, we write the realized outcome of $i$ as follows:

$Y_{i}=Y_{i}(D_{i},\pi_{i}^{*})$

where $\pi_{i}^{*}=\pi_{i}^{*}(S,\theta)$ is the average of equilibrium
treatment choice probabilities of $i$’s neighbor. From now on, we simply refer
to $\pi_{i}^{*}$ as $i$’s “ _neighborhood (propensity) score_ ”. This is the
average value of propensity scores of $i$’s direct neighbors where each score
measures the probability of taking up the treatment given the public
information $S$. Jackson et al. (2020) have termed the same object as “peer-
influenced propensity score”.

Let $\pi\in[0,1]$ be the possible value that $\pi_{i}^{*}$ can take. The
potential outcome $Y_{i}(d,\pi)$ represents $i$’s response when we exogenously
assign $D_{i}=d$ and $\pi_{i}^{*}=\pi$. Concretely, $Y_{i}(1,\pi)$ represents
$i$’s outcome when $i$ is required to be treated and $i$’s neighborhood score
has been exogenously set to $\pi$. Similarly $Y_{i}(0,\pi)$ is $i$’s outcome
when $i$ is forbidden to be treated and $i$’s neighborhood score has been
exogenously set to $\pi$. Underlying assumption is that it is possible to
manipulate the value of $D_{i}$ and $\pi_{i}^{*}$. Since $\pi_{i}^{*}$ is a
function of the public state variable $S=(G,X,Z)$, we can conceivably
manipulate the value of $\pi_{i}^{*}$ by changing $Z$ for a given $(G,X)$,
which is assumed to be predetermined and non-manipulable. Thus $Y_{i}(d,\pi)$
can be realized through changing $Z$ profile in the population in a way that
it induces $\pi_{i}^{*}=\pi$ as an equilibrium in the first-stage and then
requiring $i$ to choose $D_{i}=d$. 222Note that some combination $(d,\pi)$ may
represent off-the-equilibrium quantity. Thus, the resulting $Y_{i}(d,\pi)$ may
not be a policy-relevant counterfactual. Nevertheless, to define causal
effects rigorously, we need to consider every possible combinations of
$(d,\pi)\in\\{0,1\\}\times[0,1]$.

##### Comparison to other approaches

The existing literature with interference often models potential outcomes as a
function of own treatment status and the proportion of treated neighbors or
the number of treated neighbors (e.g. Hudgens and Halloran (2008), Leung
(2020a), Vazquez-Bare (2020)). Define
$\bar{D}_{i}\equiv\sum_{j\in\mathcal{N}_{i}}D_{j}/|\mathcal{N}_{i}|$ with a
generic value $\bar{d}\in[0,1]$. Such models then write the realized outcome
as $Y_{i}=Y_{i}(D_{i},\bar{D}_{i})$ and the potential outcomes as
$Y_{i}(d,\bar{d})$. Our model differs from theirs in that we model spillovers
via _ex ante_ (anticipated) expectation of $\bar{D}_{i}$ rather than _ex post_
realization of $\bar{D}_{i}$ itself. Recall that
$\pi_{i}^{*}(S,\theta)=\mathbf{E}[\bar{D}_{i}|S]$. Since the difference
between $\bar{D}_{i}$ and $\pi_{i}^{*}(S,\theta)$ has a mean zero (i.e.,
$\mathbf{E}[\bar{D}_{i}-\pi_{i}^{*}(S,\theta)|S]=0$), in practice the values
of these two quantities may not be too different, especially when
$|\mathcal{N}_{i}|$ is large.

Nevertheless, they are based on two different behavioral assumptions. Suppose
that the outcome of interest represents decision or behavior of agents. Then
the formulation $Y_{i}=Y_{i}(D_{i},\bar{D}_{i})$ is derived under the
assumption that agents base their decisions on $\bar{D}_{i}$ rather than
expected $\bar{D}_{i}$. This is realistic only when $\bar{D}_{i}$ is fully
observed at the time decision on $Y_{i}$ is made. Thus, the model could be
interpreted as a model with complete or perfect information. On the other
hand, our specification $Y_{i}=Y_{i}(D_{i},\pi_{i}^{*})$ assumes that agents
do not fully observe $\bar{D}_{i}$ when they decide their $Y_{i}$. Thus agents
face an intrinsic uncertainty over others’ treatment choices even at the
second-stage. This is plausible when the reference group is relatively large
so that it is not easy for agents to fully observe the value of $\bar{D}_{i}$.
Also, there are settings where agents are reluctant to reveal their treatment
status — For instance when treatment represents learning about their HIV
status as in Godlonton and Thornton (2012). In such cases, it may be more
realistic to assume that agents have private information even in the second
stage. Unlike $\bar{D}_{i}$, the equilibrium neighborhood score $\pi_{i}^{*}$
is always observable to agents as it is a function of public information $S$.
Thus it is plausible that agents base their decisions on the equilibrium
quantity $\pi_{i}^{*}$ which signals a priori prevalence of treatment adoption
in the neighborhood.

##### Random Coefficients Model of Potential Responses

We put more structure on $Y_{i}(d,\pi)$ by using random coefficients model
where we allow for a correlation between individual treatment status and
random coefficients. Therefore our model can be seen as a correlated random
coefficient model as in Masten and Torgovitsky (2016) and Wooldridge (2003).

######  Assumption 4 (random coefficient model).

(i) For any $i\in\mathcal{N}_{n}$, $d\in\\{0,1\\}$ and $\pi\in[0,1]$, we have

$\displaystyle Y_{i}(1,\pi)=\alpha_{1i}+\beta_{1i}\pi,\quad
Y_{i}(0,\pi)=\alpha_{0i}+\beta_{0i}\pi$

where $(\alpha_{1i},\beta_{1i})$ and $(\alpha_{0i},\beta_{0i})$ are unit-
specific coefficients.
(ii) For $S=(G,X,Z)$, unit-specific coefficients satisfy the following
restrictions:

$\displaystyle\mathbf{E}[\alpha_{1i}|S]=\mathbf{E}[\alpha_{1i}|X_{i}]=X_{i}^{\prime}\alpha_{1},\quad\&\quad\mathbf{E}[\beta_{1i}|S]=\mathbf{E}[\beta_{1i}|X_{i}]=X_{i}^{\prime}\beta_{1}$

and similarly,

$\displaystyle\mathbf{E}[\alpha_{0i}|S]=\mathbf{E}[\alpha_{0i}|X_{i}]=X_{i}^{\prime}\alpha_{0},\quad\&\quad\mathbf{E}[\beta_{0i}|S]=\mathbf{E}[\beta_{0i}|X_{i}]=X_{i}^{\prime}\beta_{0}.$

Recall that $Y_{i}(1,\pi)$ represent $i$’s response when $i$ is given the
treatment and $i$’s neighborhood score had been exogenously set to $\pi$.
Under the Assumption 4 (i), such response is assumed to be linear in $\pi$
with the intercept $\alpha_{1i}$ and the slope $\beta_{1i}$ that are allowed
to be different across agents. Similarly, $Y_{i}(0,\pi)$ is assumed to be
linear in $\pi$ with the intercept $\alpha_{0i}$ and the slope $\beta_{0i}$.
Note that unit-specific coefficients under the treatment,
$(\alpha_{1i},\beta_{1i})$, are allowed to be different from those without the
treatment, $(\alpha_{0i},\beta_{0i})$ for generality.

The assumption that $\pi$ affects the potential outcomes $Y_{i}(1,\pi)$ and
$Y_{i}(0,\pi)$ in a linear way is only for convenience. It is straightforward
to extend our model to include higher-order terms such as $\pi^{2}$, e.g.,
$Y_{i}(d,\pi)=\alpha_{d,i}+\beta_{d,i}\pi+\gamma_{d,i}\pi^{2}$ for
$d\in\\{0,1\\}$.

Unit-specific coefficients are unobservable random variables that are
potentially dependent on unit’s observed covariates. By Assumption 4 (ii), we
assume that the observed parts of the coefficients depend on the public state
variable $S=(G,X,Z)$ only through $X_{i}$. Importantly, this assumption
implies that $Z$ is irrelevant for the random coefficients. This rules out the
case that the treatment assignment vector $Z=(Z_{i},Z_{-i})$ directly affects
$Y_{i}$. This is the standard exclusion restriction of instruments. Therefore
under this assumption, $Z$ is given a status of an instrumental variable.

The assumption that $G$ is redundant is only for convenience as we can always
include network statistics such as the number of direct peers in $X_{i}$.
Finally, that the conditional expectation is linear in $X_{i}$ is also for
convenience as we can always allow $X_{i}$ to include nonlinear functions of
underlying covariates.

Under Assumption 4 (ii), we can decompose the unit-specific coefficients into
its mean part given $X_{i}$, and its deviation from mean as follows:

$\displaystyle\alpha_{1i}=X_{i}^{\prime}\alpha_{1}+u_{1i},$
$\displaystyle\mathbf{E}[u_{1i}|S]=0,$
$\displaystyle\beta_{1i}=X_{i}^{\prime}\beta_{1}+e_{1i},$
$\displaystyle\mathbf{E}[e_{1i}|S]=0.$

Analogously for $D_{i}=0$ as well:

$\displaystyle\alpha_{0i}=X_{i}^{\prime}\alpha_{0}+u_{0i},$
$\displaystyle\mathbf{E}[u_{0i}|S]=0,$
$\displaystyle\beta_{0i}=X_{i}^{\prime}\beta_{0}+e_{0i},$
$\displaystyle\mathbf{E}[e_{0i}|S]=0.$

Therefore the potential outcomes can be written as

$\displaystyle
Y_{i}(1,\pi)=X_{i}^{\prime}\alpha_{1}+u_{1i}+\pi\big{(}X_{i}^{\prime}\beta_{1}+e_{1i}\big{)},\quad\mathbf{E}[u_{1i}|S]=\mathbf{E}[e_{1i}|S]=0,$
$\displaystyle
Y_{i}(0,\pi)=X_{i}^{\prime}\alpha_{0}+u_{0i}+\pi\big{(}X_{i}^{\prime}\beta_{0}+e_{0i}\big{)},\quad\mathbf{E}[u_{0i}|S]=\mathbf{E}[e_{0i}|S]=0,$

while the observed outcome is given as follows:

$\displaystyle
Y_{i}=Y_{i}(D_{i},\pi_{i}^{*})=\begin{cases}X_{i}^{\prime}\alpha_{1}+u_{1i}+\pi_{i}^{*}\big{(}X_{i}^{\prime}\beta_{1}+e_{1i}\big{)}&\textrm{if
$D_{i}=1$}\\\
X_{i}^{\prime}\alpha_{0}+u_{0i}+\pi_{i}^{*}\big{(}X_{i}^{\prime}\beta_{0}+e_{0i}\big{)}&\textrm{if
$D_{i}=0$}\end{cases}$

Our model contains the four-dimensional error term:
$\eta_{i}=(u_{1i},e_{1i},u_{0i},e_{0i})$. By construction, $\eta_{i}$ are
uncorrelated with $S$, i.e., $\mathbf{E}[\eta_{i}|S]=0$. By having $\eta_{i}$,
random coefficients are allowed to be heterogeneous even after controlling for
relevant observed characteristics $X_{i}$. The importance of allowing for such
unobserved heterogeneity has been emphasized in the modern program evaluation
literature (See, e.g., Heckman (2001), Heckman et al. (2006) and Imbens
(2007)).

###  Parameters of Interest

In this section, we formally define our parameters of interest, the class of
average casual effects. For this purpose, let us first study average potential
outcomes functions.

##### Average potential outcomes

Under our specifications, _average potential outcomes_ for agents with
$X_{i}=x$ are computed as follows: for $\pi\in[0,1]$,

$\displaystyle\mathbf{E}[Y_{i}(1,\pi)|X_{i}=x]=x^{\prime}\alpha_{1}+(x^{\prime}\beta_{1})\pi,\quad\mathbf{E}[Y_{i}(0,\pi)|X_{i}=x]=x^{\prime}\alpha_{0}+(x^{\prime}\beta_{0})\pi.$

Integrating them over identically distributed $X_{i}$ gives the unconditional
average potential outcomes. Letting $\mu_{X}=\mathbf{E}[X_{i}]$,

$\displaystyle\mathbf{E}[Y_{i}(1,\pi)]$ $\displaystyle=$
$\displaystyle\mu_{X}^{\prime}\alpha_{1}+(\mu_{X}^{\prime}\beta_{1})\pi$ (15)
$\displaystyle=$ $\displaystyle\alpha_{1m}+\beta_{1m}\pi,$ (16)
$\displaystyle\mathbf{E}[Y_{i}(0,\pi)]$ $\displaystyle=$
$\displaystyle\mu_{X}^{\prime}\alpha_{0}+(\mu_{X}^{\prime}\beta_{0})\pi$ (17)
$\displaystyle=$ $\displaystyle\alpha_{0m}+\beta_{0m}\pi$ (18)

where
$(\alpha_{1m},\beta_{1m},\alpha_{0m},\beta_{0m})=(\mu_{X}^{\prime}\alpha_{1},\mu_{X}^{\prime}\beta_{1},\mu_{X}^{\prime}\alpha_{0},\mu_{X}^{\prime}\beta_{0})$.
Since $\mu_{X}$ is identifiable from the data, identification of
$(\alpha_{1m},\beta_{1m},\alpha_{0m},\beta_{0m})$ requires one to identify
$(\alpha_{1},\beta_{1},\alpha_{0},\beta_{0})$.

$(\alpha_{1m},\alpha_{0m})$ represent the baseline mean potential outcomes
when we set $\pi=0$, i.e.,
$(\alpha_{1m},\alpha_{0m})=(\mathbf{E}[Y_{i}(1,0)],\mathbf{E}[Y_{i}(0,0)])$.
Effect of $\pi$ is captured by $(\beta_{1m},\beta_{0m})$.

On the other hand, $(\alpha_{1},\beta_{1},\alpha_{0},\beta_{0})$ measures the
heterogeneous effect of $X_{i}$ on the mean potential outcomes. To see this,
notice that the following equations hold:

$\displaystyle\mathbf{E}[Y_{i}(1,\pi)|X_{i}=x]$ $\displaystyle=$
$\displaystyle x^{\prime}\alpha_{1}+\pi x^{\prime}\beta_{1}$ $\displaystyle=$
$\displaystyle\mathbf{E}[Y_{i}(1,\pi)]+(x-\mu_{X})^{\prime}\alpha_{1}+\pi(x-\mu_{X})^{\prime}\beta_{1},$
$\displaystyle\mathbf{E}[Y_{i}(1,\pi)|X_{i}=x]$ $\displaystyle=$
$\displaystyle x^{\prime}\alpha_{0}+\pi x^{\prime}\beta_{0}$ $\displaystyle=$
$\displaystyle\mathbf{E}[Y_{i}(0,\pi)]+(x-\mu_{X})^{\prime}\alpha_{0}+\pi(x-\mu_{X})^{\prime}\beta_{0}.$

Therefore for $d\in\\{0,1\\}$, $(\alpha_{d},\beta_{d})$, without constant
coefficients parts, explains the difference between
$\mathbf{E}[Y_{i}(d,\pi)|X_{i}=x]$ and $\mathbf{E}[Y_{i}(d,\pi)]$.

##### Average causal effects

Given the average response functions, we now define average causal effects,
which are our parameters of interest. Let us define the _average direct effect
(ADE)_ of own treatment under $\pi$ as follows:

$ADE(\pi)=\mathbf{E}[Y_{i}(1,\pi)-Y_{i}(0,\pi)].$

$ADE(\pi)$ measures the average change in outcomes under the regime in which
$i$ is required to choose $D_{i}=1$, compared to the regime in which $i$ is
forbidden to choose $D_{i}=1$ while $i$’s neighborhood score is fixed to
$\pi$. Under our random coefficients specification, $ADE(\pi)$ can be written
as

$\displaystyle ADE(\pi)=\alpha_{1m}-\alpha_{0m}+(\beta_{1m}-\beta_{0m})\pi.$

Similarly, we define _average spillover effect (ASE)_ from changing the
neighborhood score from $\pi$ to $\tilde{\pi}$ for each $d\in\\{0,1\\}$ as
follows:

$\displaystyle
ASE(\pi,\tilde{\pi},d)=\mathbf{E}[Y_{i}(d,\tilde{\pi})-Y_{i}(d,\pi)]=(\tilde{\pi}-\pi)\beta_{dm},$

which measures the effect of changing the neighborhood score from $\pi$ to
$\tilde{\pi}$ while fixing agent’s treatment status at $D_{i}=d$. Whether
$\beta_{0m}=0$ or $\beta_{1m}=0$ is of interest as it indicates whether there
are treatment spillovers at the outcome level.

###  Source of Endogeneity

In sum, our model of treatment choices and outcomes can be written as the
following semi-triangular system:

$\displaystyle
Y_{i}=Y_{i}(D_{i},\pi_{i}^{*})=\begin{cases}X_{i}^{\prime}\alpha_{1}+u_{1i}+\big{(}X_{i}^{\prime}\beta_{1}+e_{1i}\big{)}\pi_{i}^{*}&\textrm{if
$D_{i}=1$}\\\
X_{i}^{\prime}\alpha_{0}+u_{0i}+\big{(}X_{i}^{\prime}\beta_{0}+e_{0i}\big{)}\pi_{i}^{*}&\textrm{if
$D_{i}=0$}\end{cases}$ (19) $\displaystyle D_{i}=\mathbf{1}\\{v_{i}\leq
X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\pi_{i}^{*}\\}$ (20)
$\displaystyle\textrm{ s.t.
}\sigma_{i}^{*}=\Phi\Big{(}X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\pi_{i}^{*}\Big{)},\quad\forall
i\in\mathcal{N}_{n}.$ (21)

Using the formula
$Y_{i}=D_{i}Y_{i}(1,\pi_{i}^{*})+(1-D_{i})Y_{i}(0,\pi_{i}^{*})=Y_{i}(0,\pi_{i}^{*})+D_{i}(Y_{i}(1,\pi_{i}^{*})-Y_{i}(0,\pi_{i}^{*}))$,
19 can be written as follows:

$\displaystyle
Y_{i}=X_{i}^{\prime}\alpha_{0}+\pi_{i}^{*}X_{i}^{\prime}\beta_{0}+D_{i}X_{i}^{\prime}(\alpha_{1}-\alpha_{0})+D_{i}\pi_{i}^{*}X_{i}^{\prime}(\beta_{1}-\beta_{0})+\epsilon_{i}$
(22)

where

$\displaystyle\epsilon_{i}=u_{0i}+\pi_{i}^{*}e_{0i}+D_{i}\big{(}u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})\big{)}.$
(23)

Equation 22 gives the conventional linear regression model. Naturally, one may
consider estimating $(\alpha_{1},\alpha_{0},\beta_{1},\beta_{0})$ by the least
squares regression of $Y_{i}$ on
$(X_{i},\pi_{i}^{*}X_{i},D_{i}X_{i},D_{i}\pi_{i}^{*}X_{i})$. Resulting OLS
estimator is consistent only when $\epsilon_{i}$ is uncorrelated with the
regressors, i.e., $\mathbf{E}[\epsilon_{i}|D_{i},X_{i},\pi_{i}^{*}]=0$ which
requires that the following two conditions hold:

$\displaystyle\mathbf{E}[u_{0i}+\pi_{i}^{*}e_{0i}|D_{i}=0,X_{i},\pi_{i}^{*}]=^{(a)}\mathbf{E}[u_{0i}+\pi_{i}^{*}e_{0i}|X_{i},\pi_{i}^{*}]=^{(b)}0,$
$\displaystyle\mathbf{E}[u_{1i}+\pi_{i}^{*}e_{1i}|D_{i}=1,X_{i},\pi_{i}^{*}]=^{(a)^{\prime}}\mathbf{E}[u_{1i}+\pi_{i}^{*}e_{1i}|X_{i},\pi_{i}^{*}]=^{(b)^{\prime}}0.$

Since $\eta_{i}=(u_{1i},u_{0i},e_{1i},e_{0i})$ are uncorrelated with
$S=(G,X,Z)$ by construction, $(b)$ and $(b)^{\prime}$ are automatically
satisfied. Therefore, we only need to show that $(a)$ and $(a)^{\prime}$ are
satisfied. This is true only when $D_{i}$ is uncorrelated with $\eta_{i}$
conditional on $(X_{i},\pi_{i}^{*})$. This is the familiar selection-on-
observables assumption. Such assumption is unlikely to hold if the treatment
group and control group are systematically different in their unobserved
factors $\eta_{i}$ even after controlling for all relevant observables.
Indeed, the very fact that agents with the same observed characteristics
$(X_{i},\pi_{i}^{*})$ have made different treatment choices suggests that they
differ in their unobserved factors. Thus, the source of endogeneity comes from
the correlation between $v_{i}$ and $\eta_{i}$ even after conditional on $S$.

More specifically, note that the selection-on-observables assumption requires
that the following two conditions hold:

$\displaystyle Corr(Y_{i}(0,\pi_{i}^{*}),D_{i}|X_{i},\pi_{i}^{*})=0$ (24)

and

$\displaystyle
Corr(Y_{i}(1,\pi_{i}^{*})-Y_{i}(0,\pi_{i}^{*}),D_{i}|X_{i},\pi_{i}^{*})=0.$
(25)

Condition 24 requires that the idiosyncratic part of $Y_{i}(0,\pi_{i}^{*})$ is
uncorrelated with $D_{i}$, i.e., in the absence of the treatment, there should
be no difference in the mean potential outcomes across treatment group and
control group once we account for relevant observables $(X_{i},\pi_{i}^{*})$.
However, agents who take up the treatment may have unusual values of
$Y_{i}(0,\pi)$ even after controlling for $(X_{i},\pi_{i}^{*})$. If
individuals who take up the treatment tend to have higher values of
$Y_{i}(0,\pi)$ in terms of unobservables, then the naive least squares
regression would suffer from an upward bias since
$cov(D_{i},\epsilon_{i}|S)>0$. This is the case of classic selection problem.

The requirement 25 is also troublesome as the condition implies that the
unobserved gain from the treatment given $\pi_{i}^{*}$ should not vary across
treatment group and control group. This is not satisfied if the treatment
choice is correlated with unobserved gains from the treatment. It is plausible
that agents have some knowledge of likely idiosyncratic gains from the
treatment at the time they choose their treatment status. If agent’s treatment
choice is partially based on such knowledge, then 25 would not be satisfied.
This type of sorting on the unobserved gain, termed “ _essential
heterogeneity_ ” by Heckman et al. (2006), has been emphasized in the modern
program literature.

In conclusion, whenever selection problem or essential heterogeneity exists,
the naive OLS regression delivers inconsistent estimates of structural
parameters $(\alpha_{1},\alpha_{0},\beta_{1},\beta_{0})$.

##  Identification

In the previous section, we showed that the OLS regression of 22 suffers from
bias when $v_{i}$ is correlated with $\eta_{i}=(u_{1i},u_{0i},e_{1i},e_{0i})$
even when we control for $S$. In this section, we first show that the IV
methods do not identify the casual parameters of interest in the presence of
general heterogeneity. We then propose the alternative method known as control
function approach.

###  The Problem of Conventional IV Methods

Endogeneity is often addressed by IV methods such as two-stage least squares
(2SLS). In our setup, $Z_{i}$ is a valid IV for $D_{i}$ since (i) $D_{i}$ is
correlated with $Z_{i}$, and (ii) $Z_{i}$ is exogenous and is excluded from
the outcome equation. In fact, in the presence of spillovers in the first
stage, not only $Z_{i}$ but also $n$-dimensional vector $Z=(Z_{i},Z_{-i})$ is
a valid instrument for $D_{i}$ since in that case, $D_{i}$ is a function of
entire assignment vector $Z$.333Recall that when there exist spillovers in the
first stage choice model, not only $i$’s direct neighbor’s $Z$ but indirect
neighbors’ $Z$ also affect $D_{i}$. Therefore $Z_{j}$ for $j$ that are
eventually connected to $i$ is also relevant for $D_{i}$. However as the
network distance between $i$ and $j$ becomes greater, the dependence between
$Z_{j}$ and $D_{i}$ decays exponentially when $\lambda<1$. (See Xu (2018) and
Leung (2020b)). Therefore, using $Z_{j}$ that is too far from $i$ as an IV may
incur weak IV problem.. Therefore, we may run an IV regression to 22 where we
instrument $D_{i}$ by $Z_{i}$ or by $Z=(Z_{i},Z_{-i})$, depending on whether
spillovers exist in the first stage.

We argue that such strategy does not identify
$(\alpha_{0},\beta_{0},\alpha_{1},\beta_{1})$ in our setup. Suppose we
instrument $D_{i}$ by $Z_{i}$. The resulting IV estimator is consistent only
when the $\mathbf{E}[\epsilon_{i}|Z_{i},X_{i},\pi_{i}^{*}]=0$ where
$\epsilon_{i}=u_{0i}+\pi_{i}^{*}e_{0i}+D_{i}\big{(}u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})\big{)}$
as in 23. Note that,

$\displaystyle\mathbf{E}[\epsilon_{i}|Z_{i},X_{i},\pi_{i}^{*}]$
$\displaystyle=$
$\displaystyle\mathbf{E}[u_{0i}+\pi_{i}^{*}e_{0i}+D_{i}\big{(}u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})\big{)}|Z_{i},X_{i},\pi_{i}^{*}]$
$\displaystyle=$
$\displaystyle\underbrace{\mathbf{E}[u_{0i}+\pi_{i}^{*}e_{0i}|Z_{i},X_{i},\pi_{i}^{*}]}_{\textbf{A}}+\underbrace{\mathbf{E}[u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})|D_{i}=1,Z_{i},X_{i},\pi_{i}^{*}]}_{\textbf{B}}\underbrace{\Pr(D_{i}=1|Z_{i},X_{i},\pi_{i}^{*})}_{\textbf{C}}.$

$\textbf{A}=0$ since $\eta_{i}=(u_{1i},u_{0i},e_{1i},e_{0i})$ is uncorrelated
with $S$, and thereby with $(Z,X_{i},\pi_{i}^{*})$. C cannot be zero except
for trivial cases. Therefore $\mathbf{E}[\epsilon_{i}|Z,X_{i},\pi_{i}^{*}]=0$
only when $\textbf{B}=0$. This is satisfied when
$\mathbf{E}[u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})|D_{i}=1,Z_{i},X_{i},\pi_{i}^{*}]=\mathbf{E}[u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})|Z_{i},X_{i},\pi_{i}^{*}]$
as $\mathbf{E}[\eta_{i}|S]=0$ implies that the last term is zero. Note that
$u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})$ can be interpreted as an
idiosyncratic part of $Y_{i}(1,\pi_{i}^{*})-Y_{i}(0,\pi_{i}^{*})$. Therefore
we need to assume that $D_{i}$ is uncorrelated with the idiosyncratic gain
from taking the treatment once we condition on $(Z_{i},X_{i},\pi_{i}^{*})$.
Such requirement is unrealistic when agents have some knowledge on their
idiosyncratic gains and base their treatment decision on such knowledge, i.e.,
when there is sorting on unobserved gains.

Whether the $u_{1i}-u_{0i}+\pi_{i}^{*}(e_{1i}-e_{0i})$ is correlated with
$D_{i}$ is an empirical matter and should not be settled a priori. IV methods
rule out the possibility of such correlation and are subject to failure when
the correlation exists. This point has also been pointed out in the
traditional treatment effect literature which rules out spillover effects.
(See Hahn and Ridder (2011)). For instance, it is now well established in the
literature that IV/2SLS does not recover the average causal parameters such as
ATE under the heterogeneous responses model such as random coefficients models
(See Imbens and Angrist (1994)).

###  Control Function Approach

We now propose the alternative strategy known as the control function
approach. Control function approach addresses the endogeneity problem by
explicitly formulating the dependence between outcomes and treatments. To
apply this method, we first write the observed conditional means
$\mathbf{E}[Y_{i}|D_{i}=1,S]$ and $\mathbf{E}[Y_{i}|D_{i}=0,S]$ as follows:

$\displaystyle\mathbf{E}[Y_{i}|D_{i}=1,S]$ $\displaystyle=$
$\displaystyle\mathbf{E}[Y_{i}|D_{i}=1,\sigma_{i}^{*}(S,\theta),\pi_{i}^{*}(S,\theta),S]$
$\displaystyle=$
$\displaystyle\mathbf{E}[Y_{i}(1,\pi_{i}^{*}(S,\theta))|v_{i}\leq\Phi^{-1}(\sigma_{i}^{*}(S,\theta)),\sigma_{i}^{*}(S,\theta),\pi_{i}^{*}(S,\theta),S]$
$\displaystyle=$ $\displaystyle
X_{i}^{\prime}\alpha_{1}+\mathbf{E}[u_{1i}|v_{i}\leq\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S]+\pi_{i}^{*}(S,\theta)\Big{\\{}X_{i}^{\prime}\beta_{1}+\mathbf{E}[e_{1i}|v_{i}\leq\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S]\Big{\\}}$

since $D_{i}=1$ $\Longleftrightarrow$ $\Phi(v_{i})\leq\sigma_{i}^{*}$ (See
14). Similarly, the observed conditional mean for the control group is,

$\displaystyle\mathbf{E}[Y_{i}|D_{i}=0,S]$ $\displaystyle=$ $\displaystyle
X_{i}^{\prime}\alpha_{0}+\mathbf{E}[u_{0i}|v_{i}>\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S]+\pi_{i}^{*}(S,\theta)\Big{\\{}X_{i}^{\prime}\beta_{0}+\mathbf{E}[e_{0i}|v_{i}>\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S]\Big{\\}}.$

The terms
$\mathbf{E}[u_{1i}|v_{i}\leq\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S],\mathbf{E}[e_{1i}|v_{i}\leq\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S]$
and
$\mathbf{E}[u_{0i}|v_{i}>\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S],\mathbf{E}[e_{0i}|v_{i}>\Phi^{-1}(\sigma^{*}_{i}(S,\theta)),S]$
are “control functions” which account for the endogeneity of $D_{i}$.
Assumption 5 below restricts the form of these control functions.

######  Assumption 5.

For all $i\in\mathcal{N}_{n}$, $\eta_{i}=(u_{1i},u_{0i},e_{1i},e_{0i})$
satisfies the following conditions.

  1. (i)

$\eta_{i}$ is i.i.d. and is independent of $S$.

  2. (ii)

$\mathbf{E}[\eta_{i}|v_{i}]$ is a linear function of $v_{i}$.

Under these two conditions, we write

$\displaystyle\mathbf{E}[u_{1i}|v_{i},S]=\mathbf{E}[u_{1i}|v_{i}]=\rho_{u_{1}}v_{i},$
$\displaystyle\mathbf{E}[e_{1i}|v_{i},S]=\mathbf{E}[e_{1i}|v_{i}]=\rho_{e_{1}}v_{i},$
$\displaystyle\mathbf{E}[u_{0i}|v_{i},S]=\mathbf{E}[u_{0i}|v_{i}]=\rho_{u_{0}}v_{i},$
$\displaystyle\mathbf{E}[e_{0i}|v_{i},S]=\mathbf{E}[e_{0i}|v_{i}]=\rho_{e_{0}}v_{i}$

where $\rho=(\rho_{u_{1}},\rho_{e_{1}},\rho_{u_{0}},\rho_{e_{0}})$ captures
the covariances between each component of $\eta_{i}$ and $v_{i}$.

Assumption 5 (i) is often referred to as “separability” assumption and has
been utilized in literature as in Carneiro et al. (2011) and Brinch et al.
(2017). Under this assumption, the control functions depend only on the
individual propensity score $\sigma_{i}^{*}(S,\theta)$, e.g.,
$\mathbf{E}[u_{1i}|v_{i}\leq\Phi^{-1}(\sigma_{i}^{*}(S,\theta)),S]=\mathbf{E}[u_{1i}|v_{i}\leq\Phi^{-1}(\sigma_{i}^{*}(S,\theta))]$
so that the control functions are separated from $S$. As a result,
$\mathbf{E}[Y_{i}|D_{i}=1,S]$ and $\mathbf{E}[Y_{i}|D_{i}=0,S]$ depend on $S$
only though $(X_{i},\pi_{i}^{*},\sigma_{i}^{*})$. This step is necessary since
it is not possible to control for $S=(G,X,Z)$ itself as our data consist of
one large network.

Assumption 5 (ii) further allows us to write
$\mathbf{E}[u_{1i}|v_{i}\leq\Phi^{-1}(\sigma_{i}^{*})]$, for instance, as
$\rho_{u_{1}}\mathbf{E}[v_{i}|v_{i}\leq\Phi^{-1}(\sigma_{i}^{*})]$. Combined
with the normality assumption on $v_{i}$, we effectively assume that
$(\eta_{i},v_{i})$ are jointly normal. However, it can easily accommodate
alternative distributional assumptions on $v_{i}$ other than normality.

Under the joint normality assumption, control functions take a form of inverse
mills ratio. Define $\lambda_{1}(\cdot)$ and $\lambda_{0}(\cdot)$ as follows:
For $\sigma\in(0,1)$,

$\lambda_{1}(\sigma)=-\frac{\phi(\Phi^{-1}(\sigma))}{\sigma},\quad\lambda_{0}(\sigma)=\frac{\phi(\Phi^{-1}(\sigma))}{1-\sigma}.$

It follows that

$\displaystyle\mathbf{E}[Y_{i}|D_{i}=1,S]=X_{i}^{\prime}\alpha_{1}+\rho_{u_{1}}\lambda_{1}(\sigma_{i}^{*})+\pi_{i}^{*}\big{(}X_{i}^{\prime}\beta_{1}+\rho_{e_{1}}\lambda_{1}(\sigma_{i}^{*})\big{)},$
$\displaystyle\mathbf{E}[Y_{i}|D_{i}=0,S]=X_{i}^{\prime}\alpha_{0}+\rho_{u_{0}}\lambda_{0}(\sigma_{i}^{*})+\pi_{i}^{*}\big{(}X_{i}^{\prime}\beta_{0}+\rho_{e_{0}}\lambda_{0}(\sigma_{i}^{*})\big{)}.$

Let $\lambda_{i}=D_{i}\lambda_{1i}+(1-D_{i})\lambda_{0i}$. We see that
$(\alpha_{1},\beta_{1},\rho_{u_{1}},\rho_{e_{1}})$ is identified by regressing
$Y_{i}$ on
$(X_{i}^{\prime},\lambda_{i},\pi_{i}^{*}X_{i}^{\prime},\pi_{i}^{*}\lambda_{i})^{\prime}$
using the subsample of $D_{i}=1$. Similarly, we can identify
$(\alpha_{0},\beta_{0},\rho_{u_{0}},\rho_{e_{0}})$ by regressing $Y_{i}$ on
$X_{i},\lambda_{i}$ and their interactions with $\pi_{i}^{*}$ using the
subsample of $D_{i}=0$. The inclusion of $\lambda_{i}$ accounts for the
correlation between $\eta_{i}$ and $v_{i}$ so that we can test for the
endogeneity of $D_{i}$ by checking whether correlations are collectively zero
or not.

Our model achieves a point identification by exploiting a functional form
assumption between $\eta_{i}$ and $v_{i}$. We can relax the linearity
assumption and have more flexible parametric functional form by adding higher-
order terms. For instance, we may specify $\mathbf{E}[u_{1i}|v_{i}]$ as the
quadratic function of $v_{i}$ as follows:

$\mathbf{E}[u_{1i}|v_{i}]=\rho_{u_{1}}v_{i}+\tilde{\rho}_{u_{1}}v_{i}^{2}.$

Then it can be shown that

$\displaystyle\mathbf{E}[u_{1i}|v_{i}\leq\Phi^{-1}(\sigma_{i}^{*})]$
$\displaystyle=$
$\displaystyle-\rho_{u_{1}}\frac{\phi(\Phi^{-1}(\sigma_{i}^{*}))}{\sigma_{i}^{*}}+\tilde{\rho}_{u_{1}}\Big{[}\Phi^{-1}(\sigma_{i}^{*})\frac{\phi(\Phi^{-1}(\sigma_{i}^{*}))}{\sigma_{i}^{*}}+\Big{\\{}\frac{\phi(\Phi^{-1}(\sigma_{i}^{*}))}{\sigma_{i}^{*}}\Big{\\}}^{2}\Big{]}.$

This also offers a way to test for linearity assumption in a spirit of Lee
(1984).

##  Estimation

We propose a two-stage estimation procedure. In the first-stage, we estimate
the treatment choice games using a nested fixed point maximum likelihood
(NFXP-ML) method. In the second-stage, using first-stage estimates, we
estimate regression models of treatment outcomes with generated regressors.

###  First-Stage Estimation

Recall that the treatment choice models boil down to equation 20 subject to
the fixed-point requirement 28. Our sample log-likelihood function are defined
as follows:

$\displaystyle\widehat{\mathcal{L}}_{n}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Big{\\{}D_{i}\ln\sigma_{i}^{*}(S,\theta)+(1-D_{i})\ln(1-\sigma_{i}^{*}(S,\theta))\Big{\\}}$
(26)

Our estimator
$\hat{\theta}=(\hat{\theta}_{1},\hat{\theta}_{2},\hat{\theta}_{3})$ is defined
as the maximizer of $\widehat{\mathcal{L}}_{n}(\theta)$ subject to the
constraint that $\\{\sigma_{i}^{*}(S,\hat{\theta})\\}$ satisfies the fixed-
point requirement. Formally,

$\displaystyle\hat{\theta}=\arg\max_{\theta\in\Theta}\widehat{\mathcal{L}}_{n}(\theta)$
(27)

subject to

$\displaystyle\sigma_{i}^{*}(S,\hat{\theta})=\Phi\Big{(}X_{i}^{\prime}\hat{\theta}_{1}+\hat{\theta}_{2}Z_{i}+\hat{\theta}_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\hat{\theta})\Big{)},\quad\forall
i\in\mathcal{N}_{n}$ (28)

For computation, we use the nested fixed point (NFXP) algorithm. Specifically,
starting with an arbitrary initial guess for $\hat{\theta}$, we find the fixed
point of 28 via contraction iterations (it can be shown that 28 is a
contraction mapping when $\lambda<1$). We then compute the log-likelihood
function 26 using the obtained conditional choice probabilities. Update
$\hat{\theta}$ to $\hat{\theta}^{\prime}$ according to, say, Newton’s method.
Iterate the procedure until a sequence of estimates converges. Our NFXP-ML
estimator is taken as its limit.

###  Second-Stage Estimation

Let us define the set of regressors as

$W_{i}=[X_{i}^{\prime},\lambda_{i},\pi_{i}^{*}(S,\theta)X_{i}^{\prime},\pi_{i}^{*}(S,\theta)\lambda_{i}]^{\prime}$

where $\lambda_{i}=D_{i}\lambda_{1i}+(1-D_{i})\lambda_{0i}$ with
$\lambda_{1i}=\lambda_{1}(\sigma_{i}^{*}(S,\theta))$ and
$\lambda_{0i}=\lambda_{0}(\sigma_{i}^{*}(S,\theta))$.

Our estimators are based on the following moment conditions

$\displaystyle\mathbf{E}[Y_{i}|D_{i}=1,S]=W_{i}^{\prime}\gamma_{1},\quad\mathbf{E}[Y_{i}|D_{i}=0,S]=W_{i}^{\prime}\gamma_{0}$

where $\gamma_{1}=(\alpha_{1},\rho_{u_{1}},\beta_{1},\rho_{e_{1}})^{\prime}$
and $\gamma_{0}=(\alpha_{0},\rho_{u_{0}},\beta_{0},\rho_{e_{0}})^{\prime}$.

This suggests that $\gamma_{1}$ and $\gamma_{0}$ can be estimated by
regressing $Y_{i}$ on $W_{i}$, separately to the subsample with $D_{i}=1$ and
$D_{i}=0$, respectively. However, since $\lambda_{i}$ and $\pi_{i}^{*}$ are
functions of unknown first-stage parameters $\theta$, we need to replace
$\theta$ with $\hat{\theta}$. Define
$\hat{\lambda}_{1i}=\lambda_{1}(\sigma_{i}^{*}(S,\hat{\theta}))$ and
$\hat{\lambda}_{0i}=\lambda_{0}(\sigma_{i}^{*}(S,\hat{\theta}))$. Let
$\hat{\lambda}_{i}=D_{i}\hat{\lambda}_{1i}+(1-D_{i})\hat{\lambda}_{0i}$.
Similarly, we replace the unknown quantity
$\pi_{i}^{*}(S,\theta)\equiv\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta)$
with
$\hat{\pi}_{i}^{*}=\pi_{i}^{*}(S,\hat{\theta})=\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\hat{\theta})$.
Thus, our generated regressor $\hat{W}_{i}$ for $W_{i}$ is

$\hat{W}_{i}=[X_{i}^{\prime},\hat{\lambda}_{i},\hat{\pi}_{i}X_{i}^{\prime},\hat{\pi}_{i}\hat{\lambda}_{i}]^{\prime}.$

Estimator for $\gamma_{1}$ is then defined as

$\displaystyle\hat{\gamma}_{1}$ $\displaystyle=$
$\displaystyle\arg\min_{\gamma_{1}}\frac{1}{n}\sum_{i=1}^{n}D_{i}\big{(}Y_{i}-\hat{W}_{i}^{\prime}\gamma_{1}\big{)}^{2}$
$\displaystyle=$
$\displaystyle\Big{\\{}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}\Big{\\}}^{-1}\sum_{i=1}^{n}D_{i}\hat{W}_{i}Y_{i}.$

Similarly, estimator for $\gamma_{0}$ is

$\displaystyle\hat{\gamma}_{0}$ $\displaystyle=$
$\displaystyle\arg\min_{\gamma_{0}}\frac{1}{n}\sum_{i=1}^{n}(1-D_{i})\big{(}Y_{i}-\hat{W}_{i}^{\prime}\gamma_{0}\big{)}^{2}$
$\displaystyle=$
$\displaystyle\Big{\\{}\sum_{i=1}^{n}(1-D_{i})\hat{W}_{i}\hat{W}_{i}^{\prime}\Big{\\}}^{-1}\sum_{i=1}^{n}(1-D_{i})\hat{W}_{i}Y_{i}.$

###  Inference

For the asymptotic analysis, we consider large-network asymptotics in which a
number of individuals connected in a single network goes to infinity.
Moreover, for each $n$, we treat $S=(G,X,Z)$ as fixed. This is justified since
$S$ is an ancillary statistics, i.e., $S$ does not contain any information on
the parameters of interest.

####  Inference for the first-stage game

We first establish $\sqrt{n}$-consistency and asymptotic normality of the
first-stage estimator $\hat{\theta}$. The true parameter is denoted by
$\theta^{0}$. Therefore our data $\\{D_{i}\\}_{i=1}^{n}$ is assumed to be
generated from

$D_{i}=\mathbf{1}\\{v_{i}\leq
X_{i}^{\prime}\theta_{1}^{0}+\theta_{2}^{0}Z_{i}+\theta_{3}^{0}\pi_{i}^{*}(S,\theta^{0})\\}$

subject to
$\sigma_{i}^{*}(S,\theta^{0})=\Phi\big{(}X_{i}^{\prime}\theta_{1}^{0}+\theta_{2}^{0}Z_{i}+\theta_{3}^{0}\pi_{i}^{*}(S,\theta^{0})\big{)}$
for all $i\in\mathcal{N}_{n}$.

######  Theorem 2 (consistency of $\hat{\theta}$).

Under the following assumptions, $\hat{\theta}-\theta^{0}\xrightarrow{p}0$.

  1. (i)

The true parameter $\theta^{0}=(\theta_{1}^{0},\theta_{2}^{0},\theta_{3}^{0})$
lies in a compact set $\Theta\subseteq\mathbb{R}^{dim(\theta)}$ and
$|\theta_{3}^{0}|<\sqrt{2\pi}$. The support of $X_{i}$ is a bounded subset of
$\mathbb{R}^{k}$.

  2. (ii)

Let $R_{i}=(X_{i}^{\prime},Z_{i},\pi_{i}^{*}(S,\theta^{0}))^{\prime}$. For
large enough $n$, $\sum_{i=1}^{n}R_{i}R_{i}’$ is invertible, i.e.,

$\liminf_{n\to\infty}\det(\sum_{i=1}^{n}R_{i}R_{i}’)>0.$

See Appendix B.1 for the proof.

Assumption (i) ensures that there is unique equilibrium at the true parameter
(See Theorem 1) and that each equilibrium probability
$\sigma_{i}^{*}(S,\theta)\in(0,1)$ for all $i$. Assumption (ii) is the rank
condition for identification which requires that for all large enough $n$. the
moment matrix of regressors has full rank.

We now establish asymptotic normality of $\hat{\theta}$. Let us define the
information matrix as follows:

$\mathcal{I}_{n}(\theta)=\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\theta)\nabla_{\theta}l_{i}(\theta)^{\prime}\Big{|}S\Big{]}$

where
$l_{i}(\theta)=D_{i}\ln\sigma_{i}^{*}(S,\theta)+(1-D_{i})\ln(1-\sigma_{i}^{*}(S,\theta))$
is the individual log-likelihood function. Therefore
$\nabla_{\theta}l_{i}(\theta)$ is given by

$\displaystyle\nabla_{\theta}l_{i}(\theta)=D_{i}\frac{\nabla_{\theta}\sigma_{i}^{*}(S,\theta)}{\sigma_{i}^{*}(S,\theta)}+(1-D_{i})\frac{-\nabla_{\theta}\sigma_{i}^{*}(S,\theta)}{1-\sigma_{i}^{*}(S,\theta)}.$
(29)

######  Theorem 3 (asymptotic normality of $\hat{\theta}$).

In addition to the conditions for Theorem 2, assume

  1. (i)

The true parameter $\theta^{0}$ lies in the interior of the compact set
$\Theta\subseteq\mathbb{R}^{dim(\theta)}$.

  2. (ii)

For any $n$, $\mathcal{I}_{n}(\theta^{0})$ is nonsingluar.

Then

$\displaystyle(\mathcal{I}_{n}^{-1}(\theta^{0}))^{-1/2}\sqrt{n}(\hat{\theta}-\theta^{0})\xrightarrow{d}N(0,I_{dim(\theta)})$
(30)

where $I_{dim(\theta)}$ is the $dim(\theta)\times dim(\theta)$ identity
matrix.
See Appendix B.2 for proof.

##### Variance Estimation

The asymptotic variance of $\hat{\theta}$ can be estimated by
$\widehat{Var}(\hat{\theta})=\widehat{\mathcal{I}}_{n}^{-1}/n$ where

$\widehat{\mathcal{I}}_{n}\equiv\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\hat{\theta})\nabla_{\theta}l_{i}(\hat{\theta})^{\prime}.$

In order to compute $\nabla_{\theta}l_{i}(\hat{\theta})$ using equation 29, we
need to evaluate $\nabla_{\theta}\sigma_{i}^{*}(S,\hat{\theta})$. For this we
use the numerical approximation method: Take $\hat{\theta}+\epsilon$ for a
small perturbation $\epsilon$ (e.g., $\epsilon=10^{-5})$, then compute the new
equilibrium $\\{\sigma_{i}^{*}(S,\hat{\theta}+\epsilon)\\}_{i=1}^{n}$ by
solving the fixed point. $\nabla_{\theta}\sigma_{i}^{*}(S,\hat{\theta})$ is
then computed by
$(\sigma_{i}^{*}(S,\hat{\theta}+\epsilon)-\sigma_{i}^{*}(S,\hat{\theta}))/\epsilon$.

####  Inference for second-stage regression

Next, we establish $\sqrt{n}$-consistency and asymptotic normality of the
second-stage estimators $(\hat{\gamma}_{1},\hat{\gamma}_{0})$. Let us denote
the true parameters by $(\gamma_{1}^{0},\gamma_{0}^{0})$. We assume that our
model is correctly specified, i.e., $Y_{i}$ satisfies the following
conditional moment restrictions:

$\mathbf{E}[Y_{i}|S,D_{i}=1]=W_{i}^{\prime}\gamma_{1}^{0},\quad\mathbf{E}[Y_{i}|S,D_{i}=0]=W_{i}^{\prime}\gamma_{0}^{0}.$

We maintain the conditions for $\sqrt{n}$-consistency and asymptotic normality
of the first-stage estimator $\hat{\theta}$.

######  Theorem 4 (consistency of $(\hat{\gamma}_{1},\hat{\gamma}_{0})$).

Under the following assumptions,
$\hat{\gamma}^{0}_{1}-\gamma^{0}_{1}\xrightarrow{p}0$ and
$\hat{\gamma}^{0}_{0}-\gamma^{0}_{0}\xrightarrow{p}0$

  1. (i)

The true parameter $\gamma_{1}^{0}$ lies in a compact set
$\Gamma_{1}\subseteq\mathbb{R}^{dim(\gamma_{1})}$. Similarly, the true
parameter $\gamma_{0}^{0}$ lies in a compact set
$\Gamma_{0}\subseteq\mathbb{R}^{dim(\gamma_{0})}$.

  2. (ii)

Let

$\liminf_{n\to\infty}\det\Big{\\{}\sum_{i=1}^{n}\mathbf{E}[D_{i}W_{i}W_{i}^{\prime}|S]\Big{\\}}>0$

and

$\liminf_{n\to\infty}\det\Big{\\{}\sum_{i=1}^{n}\mathbf{E}[(1-D_{i})W_{i}W_{i}^{\prime}|S]\Big{\\}}>0.$

See Appendix B.3 for proof.

Next, we derive the asymptotic results for the second-step estimators. For
compactness, we only report results for $\hat{\gamma}_{1}$, as
$\hat{\gamma}_{0}$ case can be derived in an analogous way.

######  Theorem 5 (asymptotic normality of $\hat{\gamma}_{1}$).

Define

$\displaystyle\Upsilon_{n}$ $\displaystyle=$
$\displaystyle\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}|S]$
$\displaystyle\Psi_{n}$ $\displaystyle=$
$\displaystyle\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}\epsilon_{1i}^{2}|S]$
$\displaystyle+$
$\displaystyle\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\Big{|}S\Big{]}\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\theta^{0})\nabla_{\theta}l_{i}(\theta^{0})^{\prime}\Big{|}S\Big{]}^{-1}\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\Big{|}S\Big{]}^{\prime}$

In addition to the conditions for Theorem 4, assume

  1. (i)

The true parameter $\gamma_{1}^{0}$ lies in the interior of the compact set
$\Gamma_{1}\subseteq\mathbb{R}^{dim(\gamma_{1})}$.

  2. (ii)

For any $n$, $\Psi_{n}$ and $\Upsilon_{n}$ are nonsingular.

Then we have

$\Lambda_{n}^{-1/2}\sqrt{n}(\hat{\gamma}_{1}-\gamma_{1}^{0})\xrightarrow{d}N(0,I_{dim(\gamma_{1})})$

where $\Lambda_{n}=\Upsilon_{n}^{-1}\Psi_{n}\Upsilon_{n}^{-1}$. See Appendix
B.4 for proof.

If we ignore first-stage estimation, the asymptotic variance would be

$\Upsilon_{n}^{-1}\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}\epsilon_{1i}^{2}|S\Big{]}\Upsilon_{n}^{-1}$

which is smaller, in the positive semi-definite sense, than the correct
asymptotic variance $\Upsilon_{n}^{-1}\Psi_{n}\Upsilon_{n}^{-1}$.

##### Variance Estimation

The asymptotic variance $\Lambda_{n}$ can be estimated by replacing the
population means by sample counterparts. Specifically,

$\displaystyle\hat{\Upsilon}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}$
$\displaystyle\hat{\Psi}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}\hat{\epsilon}_{1i}^{2}+\Big{(}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\hat{\gamma}_{1}^{\prime}\nabla_{\gamma_{1}}W_{i}(\hat{\gamma}_{1})\Big{)}\Big{(}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\hat{\theta})\nabla_{\theta}l_{i}(\hat{\theta})\Big{)}\Big{(}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\hat{\gamma}_{1}^{\prime}\nabla_{\gamma_{1}}W_{i}(\hat{\gamma}_{1})\Big{)}^{\prime}$

where $\hat{\epsilon}_{1i}=D_{i}(Y_{i}-\hat{W}_{i}^{\prime}\hat{\gamma}_{1})$.

###  Monte Carlo Simulation

In this section, we illustrate the finite sample properties of our estimators
through simulation exercises.

##### Exogenous Variables

For simulation purpose, we imitate the environment of Dupas (2014). The
network $G$ is constructed from the GPS data of Dupas (2014). Specifically,
two households $i$ and $j$ are considered connected if they live within
500-meter radius. After removing isolated nodes, we have a sample size of 538.
The instrumental variable $Z$ is also taken from Dupas (2014) where the binary
$Z_{i}$ represents whether $i$ received a high level of subsidy or not.
Summary statistics of $(G,Z)$ can be found in the next section. Throughout the
simulation replications, $G$ and $Z$ are treated fixed. We do not consider
$X$.

##### Generating Endogenous Variables

Treatment choices are determined according to the following equation:

$D_{i}=\mathbf{1}\\{v_{i}\leq\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\pi_{i}^{*}\\}$

where $v_{i}\sim^{iid}N(0,1)$. We set
$\theta=(\theta_{1},\theta_{2},\theta_{3})=(-2,1,1.5)$ under which the
probability of $D=1$ is around 0.8. Since $|\theta_{3}|<2.5$, there exists a
unique equilibrium by the Theorem 1. Given our parameter values, we can
compute the unique equilibrium $\\{\sigma_{i}^{*}(G,Z,\theta)\\}_{i=1}^{n}$ by
calculating the fixed point to the following system:

$\sigma_{i}^{*}(G,Z,\theta)=\Phi\big{\\{}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(G,Z,\theta)\big{\\}},\quad\forall
i\in\mathcal{N}_{n}$

$\pi_{i}^{*}$ is then computed by
$\pi_{i}^{*}(G,Z,\theta)=\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(G,Z,\theta)/|\mathcal{N}_{i}|$.

Outcomes are realized according to the following rule:

$\displaystyle Y_{i}=\begin{cases}\alpha_{1i}+\beta_{1i}\pi_{i}^{*}&\textrm{if
$D_{i}=1$}\\\ \alpha_{0i}+\beta_{0i}\pi_{i}^{*}&\textrm{if
$D_{i}=0$}.\end{cases}$

We generate the random coefficients according to

$\displaystyle\alpha_{1i}|v_{i}\sim^{iid}N(2+0.3v_{i},1),\quad\beta_{1i}|v_{i}\sim^{iid}N(1+0.4v_{i},1),$
$\displaystyle\alpha_{0i}|v_{i}\sim^{iid}N(4+0.2v_{i},1),\quad\beta_{0i}|v_{i}\sim^{iid}N(3+0.2v_{i},1),$

so that
$(\mathbf{E}[\alpha_{1i}],\mathbf{E}[\beta_{1i}],\mathbf{E}[\alpha_{0i}],\mathbf{E}[\beta_{0i}])$
or $(\alpha_{1},\beta_{1},\alpha_{0},\beta_{0})$ is given as $(2,1,4,3)$.
Correlations between $(\alpha_{1i},\beta_{1i},\alpha_{0i},\beta_{0i})$ and
$v_{i}$ are given by
$(\rho_{\alpha_{1}},\rho_{\beta_{1}},\rho_{\alpha_{0}},\rho_{\beta_{0}})=(0.3,0.4,0.2,0.2)$
so that $D_{i}$ is endogenous with respect to all coefficients. ⁃

| coeff. | bias | se | cov.prob.
---|---|---|---|---
FS | $\theta_{1}$ | 0.007 | 0.276 | 0.948
| $\theta_{2}$ | -0.034 | 0.181 | 0.937
| $\theta_{3}$ | 0.026 | 0.231 | 0.942
SS | $\alpha_{1}$ | 0.004 | 0.277 | 0.964
| $\beta_{1}$ | -0.005 | 0.530 | 0.979
| $\alpha_{0}$ | -0.004 | 0.333 | 0.959
| $\beta_{0}$ | 0.004 | 0.783 | 0.972
Table 1: $n=538$ with 3000 simulations. Target coverage probability is 0.95.

Table 1 reports the results for the bias, standard errors, and coverage
probability for 3000 replications. The target coverage probability is 0.95. As
we observe from the first column, our estimators are unbiased. Our estimators
perform well in terms of coverage probabilities as well.

##  Application

###  Background and Data

Malaria is a life-threatening infectious disease responsible for approximately
1-3 million deaths per year. Most of these deaths are in children less than
five years of age in rural sub-Saharan Africa. The use of insecticide-treated
nets (ITNs) has been shown to be a cost-effective way to control malaria.
However, the rate of adoption remains low and many households exhibit low
willingness to pay (WTP) for ITNs. In addition, positive health externalities
generated from using ITNs render the private adoption level that is less than
the socially optimal one. For these reasons, public subsidy programs have been
proposed to achieve socially optimal coverage rate.

While it has been shown that distributing ITNs for free or at highly
subsidized prices is effective in increasing the adoption in the short run,
there have been concerns that the short-run, one-time subsidies would lower
household’s WTPs for the product later, and thus reduce the adoption rate in
the long-run. This could happen, for instance, when there exist reference
dependence effects in which households anchor their WTPs to previously paid
subsidized prices. Consequently, households may be unwilling to pay a higher
price for the product later once the subsidies end.

On the other hand, some argue that short-run subsidies would be beneficial for
the long-run adoption since households could learn the benefits of the product
better with prior experience. Such learning effects would increase consumer’s
future WTPs. Moreover, the adoption process can be facilitated with social
learning effects in which households learn benefits of the product from their
neighbors’ prior experiences. As a result, one-time subsidies would also be
beneficial for long-run adoption rate and household’s WTP.

Since ITNs need to be regularly replaced and re-purchased, understanding the
factors determining the short-run and long-run adoption decision is an
important task for sustainable public subsidy schemes. Depending on whether
reference dependence or learning effects exist, the subsidy schemes would lead
to different predictions on the short run and long run demand for ITNs. In
this application, therefore, we study the factors affecting the short-run and
long-run adoption (purchase) decision of ITNs. In doing so, we allow for
possible spillover effects in both short-run and long-run adoption decision.
As Dupas (2014) showed, social interactions seem to play an important role in
household’s bednet purchase decision. Depending on whether there exist
positive or negative peer effects in the short run and in the long run,
subsidy effectiveness may vary greatly.

variable | definition | mean | min | max
---|---|---|---|---
degree | number of neighbors | 16.41 | 1.00 | 38.00
$Z$ | 1(high subsidy) | 0.27 | 0.00 | 1.00
$D$ | 1(adoption at phase 1) | 0.47 | 0.00 | 1.00
$Y$ | 1(adoption at phase 2) | 0.16 | 0.00 | 1.00
female_educ | years of educ of female head | 5.37 | 0.00 | 22.00
wealth | wealth level | 20367.00 | 0.00 | 112273.00
Table 2: summary statistics ($n=583$)

##### Design of Experiment

We use data from a two-stage randomized pricing experiment conducted in Kenya
by Dupas (2014). In Phase 1, households within six villages were given a
voucher for the bednet at the randomly assigned subsidy level varying from
$100\%$ to $40\%$ with the corresponding prices varying from 0 to 250 Ksh. In
Phase 2, a year later, all study households in four villages were given a
second voucher for a bednet. This time, however, all households faced the same
subsidy level of $36\%$.

##### Data

Let $Z_{i}$ be a binary indicator representing that household $i$ received a
high subsidy (defined as the assigned price less than Ksh 50) in Phase 1.
Treatment variable $D_{i}$ equals to 1 if $i$ purchased a bednet in Phase 1.
$Y_{i}$ is also binary taking value 1 if $i$ purchased a bednet in Phase 2.
Following Dupas (2014), we may interpret $Y_{i}$ as a proxy for $i$’s WTP for
the future bednet.

Network Using GPS data, we construct the binarized spatial network. Two
households $i$ and $j$ are considered connected (i.e., $G_{ij}=1$) if they
live within 500-meter radius. We also consider 250-m, and 750-m radius. Since
the results do not differ much, we only report results for 500-m radius.

Other Covariates For household pre-treatment covariates, we consider wealth,
and the education level of the female head.

Summary statistics of the variables can be found on the Table 2. After
deleting $25$ isolated nodes, we have $n=538$ observations from four villages.

Figure 1: plot of estimated $(\sigma_{i}^{*},\pi_{i}^{*})$

###  Estimation Results

variable | estimates | marginal effects | p-value
---|---|---|---
spillover ($\pi$) | 2.308 | 0.661 | 0.000
subsidy | 0.694 | 0.199 | 0.000
female-educ | 0.223 | 0.064 | 0.026
wealth | 0.005 | 0.001 | 0.001
Table 3: estimation results for FS model ($n=583$)

##### Results on the short-run adoption

We first estimate the equation for the short-run adoption decision using our
game-theoretic model. Table 3 displays the estimates of coefficients, marginal
effects444Marginal effects are computed as the sample average of conditional
effects. For instance, the marginal effect of $Z_{i}$ is computed as
$\frac{1}{n}\sum_{i=1}^{n}\phi(X_{i}^{\prime}\hat{\theta}_{1}+\hat{\theta}_{2}Z_{i}+\hat{\theta}_{3}\pi_{i}^{*}(S,\hat{\theta}))\hat{\theta}_{2}$.,
as well as associated standard errors and p-values. As anticipated, high-
subsidy level is associated with higher adoption of the bednet. Education and
wealth are also positively associated with adoption decision in the short run.
These variables are all significant at 1 percent level. Figure 1 shows the
estimated plot of $(\hat{\sigma}_{i}^{*},\hat{\pi}_{i}^{*})$ by the value of
$Z_{i}$. The plot shows clearly that individual $Z_{i}$ is relevant for the
treatment choice.

Our results show strong evidence of the existence of positive spillover
effects in the short-run adoption decision. When the average adoption
probability of neighbors ($\pi_{i}^{*}$) increases by 10 percentage points,
$i$’s short-run adoption probability ($\sigma_{i}^{*}$) increases by 6.6
percentage points. The resulting conformity effects implies that if we ignore
spillover effects in the specification, we would underestimate the full effect
of the programs.

$D=1$ | estimates | p-value | $D=0$ | estimates | p-value
---|---|---|---|---|---
cons | 0.497 | 0.043 | cons | 0.128 | 0.174
female-educ | -0.094 | 0.530 | female-educ | -0.070 | 0.519
wealth | 0.003 | 0.388 | wealth | -0.002 | 0.325
lambda | 0.059 | 0.767 | lambda | 0.036 | 0.841
$\pi$ | -0.347 | 0.324 | $\pi$ | -0.021 | 0.940
$\pi$*female-educ | 0.031 | 0.906 |  $\pi$*female-educ | 0.176 | 0.513
$\pi$*wealth | -0.003 | 0.610 |  $\pi$*wealth | 0.013 | 0.098
$\pi$*lambda | 0.317 | 0.375 |  $\pi$*lambda | -0.063 | 0.832
Table 4: estimation results for SS model ($n=583$)

##### Results on the long-run adoption

Table 4 presents the estimates of own short-run adoption experience ($D_{i}$)
and average adoption probability of neighbors ($\pi_{i}^{*}$) on the long-run
adoption decision. Unfortunately, we have very limited statistical power
except for few constants due to small sample size. However, in terms of
magnitudes, estimated coefficients have implications on the spillover effects
in the long-run adoption decision.

Using the formula 16 and 18, we get the following estimated mean response
functions:

$\displaystyle\widehat{\mathbf{E}}[Y_{i}(1,\pi)]=0.497-0.347\pi,\quad\widehat{\mathbf{E}}[Y_{i}(0,\pi)]=0.128-0.02\pi$
(31)

First, let us consider $\widehat{\mathbf{E}}[Y_{i}(1,\pi)]$. Although the
coefficient on $\pi$ is not significant, we observe considerable negative
spillover effects in terms of magnitude: If $\pi$ increases by 10 percentage
points, the probability of the second-period adoption probability decreases by
3.4 percentage points. This is contrary to the positive spillovers observed in
the first period adoption decision. 555Dupas (2014) also report similar
results from their reduced-form regression models. Their results show that the
adoption in Phase 2 is negatively affected by the share of neighbors who
received a high subsidy in Phase 1. One possible explanation for such negative
spillovers in the treated response is that they result from positive health
spillovers occurring over time. For instance, household with higher value of
$\pi$ would anticipate higher coverage rate in their area, which would result
in lower malaria prevalence in the long run. This might make households less
likely to re-invest the product later. Such results highlight the importance
of distinguishing the mechanism of static spillovers from that of dynamic
spillovers.

Such effects do not seem to apply to the untreated households as
$\widehat{\mathbf{E}}[Y_{i}(0,\pi)]$ shows. However, the statistical power is
very limited.

##### Average Direct Effect

From 31, the average direct effect (ADE) of own short-run adoption on the
long-run adoption is computed as follows:

$\displaystyle\widehat{\mathbf{E}}[Y_{i}(1,\pi)-Y_{i}(0,\pi)]=0.369-0.326\pi$
(32)

The result suggests that the values of ADE vary greatly depending on the value
of $\pi$: when $\pi=0$, treated households are 36.9 percentage points more
likely to invest in the second bednet. However, such effect declines with the
neighborhood exposure rate $\pi$. When $\pi=1$, the effect is almost zero. The
fact that $ADE$ is positive for all possible values of $\pi$ points to the
existence of learning effects from prior experience, rather than reference
dependence effects.

##### Bias from ignoring spillovers

Suppose that we falsely ignore spillover effects in responses. Using the
conventional Heckit model, we obtain the following estimated average treatment
effect (ATE):

$\hat{\mathbf{E}}[Y_{i}(1)-Y_{i}(0)]=0.038.$

Above result suggests that the effect of $D$ on $Y$ is very limited. However
as equation 32 shows, there is substantial heterogeneity in the effect of $D$
on $Y$ depending on values of $\pi$: the effect of $D$ varies from almost 0
percent to 37 percent. Thus, by ignoring the spillover effects, we would draw
a misleading conclusion that there is no treatment effect.

##### Observed heterogeneity in effects

Let us turn to the effect heterogeneity due to observable covariates,
education and wealth. For the treated, the effect of education and wealth on
the adoption rate seems to be trivial in magnitude: coefficients are close to
zero and their associated p-values are large. We also compute the estimates
without covariates. The magnitude of the estimates resembles that with
covariates. Therefore we do not report the result here. This also suggests
that there seems to be little observed heterogeneity in
$\mathbf{E}[Y_{i}(1,\pi)]$ in terms of education and wealth.

On the other hand, for $D_{i}=0$ case, the magnitudes of the estimates on the
covariates are much higher than those for $D_{i}=1$ case. Consider education
first. The interaction between $\pi$ and education suggests that higher
education is associated with higher spillover effect — one more year of
education increases the effect of $\pi$ from $-0.02$ to $-0.02+0.17=0.15$.
Similarly if wealth level increases by 1000 units, the effect on $\pi$
increases by $1.2$ percentage point which is significant at 10 percent. Such
results suggest that control households with higher education and higher
wealth receive higher positive spillover effect.

###  Impact of Counterfactual Policies

One advantage of our structural approach is that it allows researchers to
simulate counterfactual policies. Suppose that a policy-maker is interested in
implementing means-tested subsidy schemes where $Z$ is determined according to
the following rule:

$\displaystyle Z_{i}=\mathbf{1}\\{wealth_{i}\leq\tau\\},\quad\forall
i\in\mathcal{N}_{n}\l$ (33)

i.e., household $i$ gets high subsidy only when their wealth level is below
some specified threshold $\tau$. The question is: what would be the expected
outcome under this new, counterfactual subsidy rule?

This problem is related to the literature on the policy-relevant treatment
effects (PRTE: Heckman and Vytlacil (2001)). In this framework, each
intervention or policy is defined by a manipulation on the exogenous variable
$S=(G,X,Z)$. In our setup, we assume that a policy maker has no means of
changing the underlying network structure $G$ or pre-treatment covariates $X$.
Thus, the only way to change $S$ is through changing $Z$. Let us denote the
new counterfactual policy as $S^{new}=(G,X,Z^{new})$ where we set the value of
$Z$ as $Z=Z^{new}$, which is not in the data. $i$’s expected outcome under the
new policy is given as $\mathbf{E}[Y_{i}|S=S^{new}]$. Note that for any $S$,

$\displaystyle\mathbf{E}[Y_{i}|S]=\mathbf{E}[Y_{i}|D_{i}=1,S]\Pr(D_{i}=1|S)+\mathbf{E}[Y_{i}|D_{i}=0,S]\Pr(D_{i}=0|S)$
(34)

Under our control function specification, $\mathbf{E}[Y_{i}|S]$ can be written
as follows::

$\displaystyle\mathbf{E}[Y_{i}|S]$ $\displaystyle=$
$\displaystyle\sigma_{i}^{*}(S)\Big{[}X_{i}^{\prime}\alpha_{1}+\lambda_{1}(\sigma_{i}^{*}(S))+\Big{\\{}X_{i}^{\prime}\beta_{1}+\lambda_{1}(\sigma_{i}^{*}(S))\Big{\\}}\pi_{i}^{*}(S)\Big{]}$
$\displaystyle+(1-\sigma_{i}^{*}(S))\Big{[}X_{i}^{\prime}\alpha_{0}+\lambda_{0}(\sigma_{i}^{*}(S))+\Big{\\{}X_{i}^{\prime}\beta_{0}+\lambda_{0}(\sigma_{i}^{*}(S))\Big{\\}}\pi_{i}^{*}(S)\Big{]}$
$\displaystyle=$
$\displaystyle\mathbf{E}[Y_{i}|X_{i},\sigma_{i}^{*}(S),\pi_{i}^{*}(S)]$

Note that $\mathbf{E}[Y_{i}|S]$ is a function of $S$ only through
$(X_{i},\sigma_{i}^{*}(S),\pi_{i}^{*}(S))$, thus we write
$\mathbf{E}[Y_{i}|X_{i},\sigma_{i}^{*}(S),\pi_{i}^{*}(S)]$. $i$’s expected
outcome under new policy is then given by
$\mathbf{E}[Y_{i}|X_{i},\sigma_{i}^{*}(S^{new}),\pi_{i}^{*}(S^{new})]$.

To estimate this, we first need to compute the new equilibrium choice
probabilities: $\\{\sigma_{i}^{*}(G,X,Z^{new})\\}_{i\in\mathcal{N}_{n}}$ where
$Z^{new}$ is determined according to 33. Under the identified first-stage
parameters, this is done by solving the new fixed point of the best-response
functions under the new data set $S^{new}=(G,X,Z^{new})$. We then estimate
$\hat{Y}_{i}\equiv\hat{\mathbf{E}}[Y_{i}|X_{i},\sigma_{i}^{*}(S^{new}),\pi_{i}^{*}(S^{new})]$
for each $i\in N_{n}$ using the formula above. Overall impact of policy
$S^{new}$ is computed by $\sum_{i=1}^{n}\hat{Y}_{i}/n$.

##### Results

Figure 2: counterfactual impact of means-tested subsidy on LR-adoption

See 2. The red line shows the effect of $\tau$ on the overall long-run
adoption level when we ignore interference effects. In such case, as $\tau$
increases, the long-run adoption level increases monotonically. This is
because as $\tau$ increases, more households get subsidy, and without
interference, treated agents are more likely to adopt in the long-run.

In the presence of spillovers, the effect of $\tau$ does not increase
monotonically anymore as the blue line shows. Higher $\tau$ also induces
higher $\pi_{i}^{*}$ which affect long-run adoption negatively. Therefore a
priori, we cannot expect that higher $\tau$ would give higher overall long-run
adoption rate in the population. In fact, as the blue line shows, the highest
long-run adoption rate is achieved under the subsidy scheme targeting the very
lowest percentile households.

The result also highlights complication involved in the use of subsidies to
increase long-run adoption rate. As the result shows, the highest expected
coverage is only 17 percent.

##  Concluding Remarks

In this paper, we propose a new methodological framework to analyze randomized
experiments with spillovers and noncompliance in a general network setup.
Using a game-theoretic framework, we allow for spillover effects to occur at
two stages: at the choice stage and outcome stage. Potential outcomes are
modeled as a random coefficient model to account for general unobserved
heterogeneity. We extend the traditional control function estimator of Heckman
(1979) to incorporate spillovers. Finally, we illustrate our methods using
Dupas (2014) data and show that our model can be used to evaluate the
counterfactual policies.

In our treatment choice games, we assumed that private information is
independently distributed across agents. Relaxing this assumption to allow for
network dependence in private information would be a rewarding task. Another
important issue is multiple equilibria – formalizing a problem of policy
evaluation and counterfactual prediction in the presence of multiple
equilibria is important for realistic policy design. Finally, we conclude by
noting that our model can be used to derive an ex ante optimal treatment
assignment rule under interference, especially in settings where a social
planner should take possible noncompliance and spillover into account.

## Appendix

##  Appendix A Proof of Theorem 1

Following Xu (2018), we show this by contradiction. Define
$\bar{\sigma}_{i}=\sum_{j\in\mathcal{N}_{i}}\sigma_{j}/|\mathcal{N}_{i}|$. Let
$\Gamma(X_{i},Z_{i},\bar{\sigma}_{i},\theta)=\Phi(X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\bar{\sigma}_{i})$
be $i$’s best-response function to inputs $(X_{i},Z_{i},\bar{\sigma}_{i})$,
and parameter value $\theta$. Suppose there are two non-identically equilibria
$\sigma^{*}=(\sigma_{i}^{*})_{i\in\mathcal{N}_{n}}$ and
$\sigma^{+}=(\sigma_{i}^{+})_{i\in\mathcal{N}_{n}}$. By definition, they
should satisfy

$\sigma_{i}^{*}=\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta),\quad\forall
i\in\mathcal{N}_{n}$

and

$\sigma_{i}^{+}=\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{+},\theta),\quad\forall
i\in\mathcal{N}_{n}.$

Taking difference and applying mean-value theorem, we have

$\displaystyle\sigma_{i}^{*}-\sigma_{i}^{+}$ $\displaystyle=$
$\displaystyle\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)-\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{+},\theta)$
$\displaystyle=$
$\displaystyle\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{m},\theta)}{\partial\bar{\sigma}_{i}}(\bar{\sigma}_{i}^{*}-\bar{\sigma}_{i}^{+})$

where $\bar{\sigma}_{i}^{m}$ is a mean value between $\bar{\sigma}_{i}^{*}$
and $\bar{\sigma}_{i}^{+}$. Taking an absolute value to the LHS,

$\displaystyle|\sigma_{i}^{*}-\sigma_{i}^{+}|$ $\displaystyle\leq$
$\displaystyle\Big{|}\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{m},\theta)}{\partial\bar{\sigma}_{i}}\Big{|}\cdot|\bar{\sigma}_{i}^{*}-\bar{\sigma}_{i}^{+}|$
(35) $\displaystyle\leq$
$\displaystyle\Big{|}\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{m},\theta)}{\partial\bar{\sigma}_{i}}\Big{|}\cdot\max_{j\in\mathcal{N}_{i}}|\sigma_{j}^{*}-\sigma_{j}^{+}|.$
(36)

From the definition of $\Gamma(\cdot)$, observe that

$\displaystyle\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i},\theta)}{\partial\bar{\sigma}_{i}}=\frac{\Phi(X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\bar{\sigma}_{i})}{\partial\bar{\sigma}_{i}}=\phi\big{(}X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\bar{\sigma}_{i}\big{)}\theta_{3}.$

Thus,

$\displaystyle\Big{|}\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{m},\theta)}{\partial\bar{\sigma}_{i}}\Big{|}\leq|\theta_{3}|\sup_{u}\phi(u)\equiv\lambda.$
(37)

Therefore we can write 36 as

$|\sigma_{i}^{*}-\sigma_{i}^{+}|\leq\lambda\max_{j\in\mathcal{N}_{i}}|\sigma_{j}^{*}-\sigma_{j}^{+}|.$

Taking $\max_{i\in N_{n}}$ to both sides gives,

$\max_{i\in\mathcal{N}_{n}}|\sigma_{i}^{*}-\sigma_{i}^{+}|\leq\lambda\max_{i\in\mathcal{N}_{n}}\max_{j\in\mathcal{N}_{i}}|\sigma_{j}^{*}-\sigma_{j}^{+}|\leq\lambda\max_{k\in\mathcal{N}_{n}}|\sigma_{k}^{*}-\sigma_{k}^{+}|$

which leads to contradiction when $\lambda<1$. $\blacksquare$

##  Appendix B Proofs for Asymptotic Results

###  Proof of consistency of first-stage estimators

Let $l_{i}(\theta)\equiv
D_{i}\ln\sigma_{i}^{*}(S,\theta)+(1-D_{i})\ln(1-\sigma_{i}^{*}(S,\theta))$ be
an individual log-likelihood function of $i$. Then
$\widehat{\mathcal{L}}_{n}(\theta)=\frac{1}{n}\sum_{i=1}^{n}l_{i}(\theta)$.

Define

$\mathcal{L}_{n}(\theta)=\mathbf{E}[\widehat{\mathcal{L}}_{n}(\theta)|S]$

where the population objective function, $\mathcal{L}_{n}(\theta)$, depends on
$n$ through the public state $S=(G,X,Z)$. Recall that the true parameter is
denoted by $\theta^{0}$. Following Gallant and White (1988) Theorem 3.3, we
establish consistency result by showing identifiable uniqueness and uniform
convergence result.

##### Identifiable Uniqueness

We show that
$\liminf_{n\to\infty}(\mathcal{L}_{n}(\theta^{0})-\mathcal{L}_{n}(\theta))>0$
for any $\theta$ such that $|\theta-\theta^{0}|\geq\epsilon>0$.

$\displaystyle-\liminf_{n\to\infty}(\mathcal{L}_{n}(\theta)-\mathcal{L}_{n}(\theta^{0}))$
$\displaystyle=$
$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\sum_{i=1}^{n}\mathbf{E}\Big{[}D_{i}\ln\frac{\sigma_{i}^{*}(S,\theta)}{\sigma_{i}^{*}(S,\theta^{0})}+(1-D_{i})\ln\frac{1-\sigma_{i}^{*}(S,\theta)}{1-\sigma_{i}^{*}(S,\theta^{0})}\Big{|}S\Big{]}$
$\displaystyle=$
$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\sum_{i=1}^{n}\Big{[}\sigma_{i}^{*}(S,\theta^{0})\ln\frac{\sigma_{i}^{*}(S,\theta)}{\sigma_{i}^{*}(S,\theta^{0})}+(1-\sigma_{i}^{*}(S,\theta^{0}))\ln\frac{1-\sigma_{i}^{*}(S,\theta)}{1-\sigma_{i}^{*}(S,\theta^{0})}\Big{|}S\Big{]}$
$\displaystyle\geq$
$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\sum_{i=1}^{n}\ln\big{(}\sigma_{i}^{*}(S,\theta)+1-\sigma_{i}^{*}(S,\theta)\big{)}=0.$

The second equality follows from
$\mathbf{E}[D_{i}|S]=\sigma_{i}^{*}(S,\theta^{0})$ and the last weak
inequality is due to Jensen’s inequality. To show that the inequality holds
strictly, we need to rule out the case of
$\liminf_{n\to\infty}(\mathcal{L}_{n}(\theta^{0})-\mathcal{L}_{n}(\theta))=0$.
This happens when for some large enough $n$,
$\sigma_{i}^{*}(S,\theta)=\sigma_{i}^{*}(S,\theta^{0})$ for all
$i\in\mathcal{N}_{n}=\\{1,2,\cdots,n\\}$, i.e., there exists $n$ that delivers
observationally equivalent choice probabilities.

Suppose this is the case. By the fixed point requirement, the following needs
to be satisfied for any arbitrary $\theta$, including the true parameter
$\theta^{0}$:

$\Phi^{-1}(\sigma_{i}^{*}(S,\theta))=X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta),\quad\forall
i\in\mathcal{N}_{n}$

and

$\Phi^{-1}(\sigma_{i}^{*}(S,\theta^{0}))=X_{i}^{\prime}\theta_{1}^{0}+\theta_{2}^{0}Z_{i}+\theta_{3}^{0}\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta^{0}),\quad\forall
i\in\mathcal{N}_{n}.$

If $\sigma_{i}^{*}(S,\theta)=\sigma_{i}^{*}(S,\theta^{0}),\ \forall
i\in\mathcal{N}_{n}$, we have,

$X_{i}^{\prime}(\theta_{1}-\theta_{1}^{0})+Z_{i}(\theta_{2}-\theta_{2}^{0})+(\theta_{3}-\theta_{3}^{0})\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\sigma_{j}^{*}(S,\theta^{0})=0,\quad\forall
i\in\mathcal{N}_{n}.$

Equivalently, $R_{i}^{\prime}(\theta-\theta^{0})=0,\ \forall
i\in\mathcal{N}_{n}$ where $R_{i}$ is defined as in Theorem 2. It follows that
$(\theta-\theta^{0})^{\prime}\sum_{i=1}^{n}R_{i}R_{i}^{\prime}(\theta-\theta^{0})=0$.
Given the assumption that $\sum_{i=1}^{n}R_{i}R_{i}^{\prime}$ is positive
definite for all large enough $n$, above equation holds only under
$\theta=\theta^{0}$ leading to contradiction. $\square$

Next, we verify that
$\sup_{\theta\in\Theta}|\widehat{\mathcal{L}}_{n}(\theta)-\mathcal{L}_{n}(\theta)|\xrightarrow{p}0.$
We first shows the pointwise convergence holds. Uniform convergence follows
then from Lipschitz conditions.

##### Pointwise Convergence

We first show that for any $\theta\in\Theta$,
$|\widehat{\mathcal{L}}_{n}(\theta)-\mathcal{L}_{n}(\theta)|\xrightarrow{p}0.$
It can be shown that

$\displaystyle\widehat{\mathcal{L}}_{n}(\theta)-\mathcal{L}_{n}(\theta)=\frac{1}{n}\sum_{i=1}^{n}\Big{\\{}\underbrace{(D_{i}-\sigma_{i}^{*}(S,\theta^{0}))\ln\frac{\sigma_{i}^{*}(S,\theta)}{1-\sigma_{i}^{*}(S,\theta)}}_{\zeta_{i}}\Big{\\}}.$

$\\{\zeta_{i}\\}_{i=1}^{n}$ is conditionally independent with mean zero given
$S$. It is also uniformly bounded due to Lemma 1. Therefore we can apply a LLN
for independent observations (e.g., Markov) and the result follows.

##### Uniform Convergence

Given pointwise convergence result, uniform convergence follows if we can
establish that
$\\{\widehat{\mathcal{L}}_{n}(\theta)-\mathcal{L}_{n}(\theta)\\}_{n}$ is
stochastically equicontinuous on $\Theta$ (theorem 1 in Andrews (1992)).
Sufficient condition for this is to show that the summand in the sample
objective function $\\{l_{i}(\theta)\\}$ is Lipschitz (Assumption W-LIP in
Andrews (1992)). Note that

$\displaystyle\nabla_{\theta}l_{i}(\theta)=D_{i}\frac{\nabla_{\theta}\sigma_{i}^{*}(S,\theta)}{\sigma_{i}^{*}(S,\theta)}+(1-D_{i})\frac{-\nabla_{\theta}\sigma_{i}^{*}(S,\theta)}{1-\sigma_{i}^{*}(S,\theta)}$

which is bounded by

$\displaystyle|\nabla_{\theta}l_{i}(\theta)|\leq\Big{|}\frac{\nabla_{\theta}\sigma_{i}^{*}(S,\theta)}{\sigma_{i}^{*}(S,\theta)}\Big{|}+\Big{|}\frac{\nabla_{\theta}\sigma_{i}^{*}(S,\theta)}{1-\sigma_{i}^{*}(S,\theta)}\Big{|}.$

By Lemma 1 and Lemma 2, $\sigma_{i}^{*}(S,\theta)$ and
$\nabla_{\theta}\sigma_{i}^{*}(S,\theta)$ are uniformly bounded. Therefore
$\\{l_{i}(\theta)\\}$ is Lipschitz-continuous and the result follows.
$\blacksquare$

###  Proof of asymptotic normality of first-stage estimators

$\hat{\theta}$ should satisfy the first-order condition for maximization:
$\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\hat{\theta})=0$. Given that
$\widehat{\mathcal{L}}_{n}(\hat{\theta})$ is smooth, we can apply the mean-
value theorem to the first-order condition around the true parameter
$\theta^{0}$:

$\displaystyle\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\hat{\theta})=\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\theta^{0})+\nabla_{\theta\theta}\widehat{\mathcal{L}}_{n}(\bar{\theta})(\hat{\theta}-\theta^{0})=0$
(38) $\displaystyle\Longleftrightarrow$
$\displaystyle\sqrt{n}(\hat{\theta}-\theta^{0})=-(\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\bar{\theta}))^{-1}\sqrt{n}\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\hat{\theta}^{0})$
(39)

where $\bar{\theta}$ is a mean value of the line joining $\hat{\theta}$ and
$\theta^{0}$. Define the Hessian matrix as

$\mathcal{H}_{n}(\theta)=\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta\theta}l_{i}(\theta)\Big{|}S\Big{]}$

and the information matrix as

$\mathcal{I}_{n}(\theta)=\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\theta)\nabla_{\theta}l_{i}(\theta)^{\prime}\Big{|}S\Big{]}.$

We first show that
$\nabla_{\theta\theta}\widehat{\mathcal{L}}_{n}(\bar{\theta})-\mathcal{H}_{n}(\theta^{0})\xrightarrow{p}0$
(ULLN of the Hessian matrix) and then
$\sqrt{n}\mathcal{I}_{n}^{-1}(\theta^{0})\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\hat{\theta}^{0})\xrightarrow{d}N(0,I_{dim(\theta)})$
(CLT on the score).

##### ULLN of the Hessian Matrix

We show that
$\nabla_{\theta\theta}\widehat{\mathcal{L}}_{n}(\bar{\theta})-\mathcal{H}_{n}(\theta^{0})\xrightarrow{p}0$.
Note that

$\displaystyle\nabla_{\theta\theta}\widehat{\mathcal{L}}_{n}(\bar{\theta})-\mathcal{H}_{n}(\theta^{0})$
$\displaystyle=\underbrace{\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta\theta}l_{i}(\bar{\theta})-\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta\theta}l_{i}(\theta^{0})}_{A}+\underbrace{\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta\theta}l_{i}(\theta^{0})-\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta\theta}l_{i}(\theta^{0})\Big{|}S\Big{]}}_{B}$

First, $A=o_{p}(1)$ since $\hat{\theta}-\theta^{0}\xrightarrow{p}0$ and
$\nabla_{\theta\theta}l_{i}(\cdot)$ is continuous as a result of Lemma 3.
Next, note that

$\displaystyle
B=\frac{1}{n}\sum_{i=1}^{n}\Big{\\{}\underbrace{\nabla_{\theta\theta}l_{i}(\theta^{0})-\mathbf{E}\big{[}\nabla_{\theta\theta}l_{i}(\theta^{0})\big{|}S\big{]}\Big{\\}}}_{\xi_{i}}$

$\\{\xi_{i}\\}$ is independent conditional on $S$ with mean zero. Also by
Lemma 3, it is uniformly bounded. Therefore by LLN for independent
observations, $B=o_{p}(1)$.

##### CLT on the Score

Note that
$\sqrt{n}\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\theta^{0})=\sqrt{n}\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\theta^{0})$
and that $\\{\nabla_{\theta}l_{i}(\theta^{0})\\}$ is independently distributed
conditional on $S$ with the uniformly bounded conditional variance
$\mathcal{I}_{n}(\theta^{0})$. Therefore we can apply Lyapunov’s CLT for
independent observations to get
$\sqrt{n}\mathcal{I}_{n}^{-1/2}(\theta^{0})\nabla_{\theta}\widehat{\mathcal{L}}_{n}(\theta^{0})\xrightarrow{d}N(0,I)$.

Combining all these results, we see that the equation 39 can be written as

$\displaystyle\sqrt{n}(\hat{\theta}-\theta^{0})=-(\mathcal{H}_{n}(\theta^{0})+o_{p}(1))^{-1}\mathcal{I}_{n}(\theta^{0})^{1/2}\sqrt{n}\mathcal{I}_{n}(\theta^{0})^{-1/2}\nabla\widehat{\mathcal{L}_{n}}(\theta^{0})$

By the information matrix inequality, when the model is correctly specified,
$\mathcal{H}_{n}(\theta^{0})=-\mathcal{I}_{n}(\theta^{0})$ so that we have

$\displaystyle\sqrt{n}(\hat{\theta}-\theta^{0})=(\mathcal{I}_{n}(\theta^{0})+o_{p}(1))^{-1}\mathcal{I}_{n}(\theta^{0})^{1/2}\sqrt{n}\mathcal{I}_{n}(\theta^{0})^{-1/2}\nabla\widehat{\mathcal{L}_{n}}(\theta^{0})$

Under the assumption that $\mathcal{I}_{n}(\theta^{0})$ is nonsingular, we get
the desired result:

$\sqrt{n}(\mathcal{I}_{n}^{-1}(\theta^{0}))^{-1/2}(\hat{\theta}-\theta^{0})\xrightarrow{d}N(0,I_{dim(\theta)}).$

$\blacksquare$

###  Proof of consistency of second-stage estimators

Our estimators are based on the following moment conditions

$\displaystyle\mathbf{E}[Y_{i}|D_{i}=1,S]=W_{i}^{\prime}\gamma_{1}^{0},\quad\mathbf{E}[Y_{i}|D_{i}=0,S]=W_{i}^{\prime}\gamma_{0}^{0}$

Let us focus on $\hat{\gamma}_{1}$ case as $\hat{\gamma}_{0}$ case can be
analyzed in an analogous way.

Given the moment condition
$\mathbf{E}[Y_{i}|D_{i}=1,S]=W_{i}^{\prime}\gamma_{1}^{0}$, we write the
equation in error form as

$Y_{i}=W_{i}^{\prime}\gamma_{1}^{0}+\epsilon_{1i},\quad\mathbf{E}[\epsilon_{1i}|D_{i}=1,S]=0.$

Estimator for $\gamma_{1}$ is defined as

$\displaystyle\hat{\gamma}_{1}$ $\displaystyle=$
$\displaystyle\arg\min_{\gamma_{1}}\frac{1}{n}\sum_{i=1}^{n}D_{i}\big{(}Y_{i}-\hat{W}_{i}^{\prime}\gamma_{1}\big{)}^{2}$
(40) $\displaystyle=$
$\displaystyle\arg\min_{\gamma_{1}}\frac{1}{n}\sum_{i=1}^{n}\big{(}D_{i}Y_{i}-D_{i}\hat{W}_{i}^{\prime}\gamma_{1}\big{)}^{2}$
(41) $\displaystyle=$
$\displaystyle\Big{\\{}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}\Big{\\}}^{-1}\sum_{i=1}^{n}D_{i}\hat{W}_{i}Y_{i}$
(42)

Note that
$D_{i}Y_{i}=D_{i}Y_{i}(1,\pi_{i}^{*}(S,\theta^{0}))=D_{i}(W_{i}^{\prime}\gamma_{1}^{0}+\epsilon_{1i})=D_{i}\big{(}\hat{W}_{i}^{\prime}\gamma_{1}^{0}+\epsilon_{1i}-(\hat{W}_{i}-W_{i})^{\prime}\gamma_{1}^{0}\big{)}$.
Plugging this into 42 gives that

$\displaystyle\hat{\gamma}_{1}$ $\displaystyle=$
$\displaystyle\big{(}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}\big{)}^{-1}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\big{(}\hat{W}_{i}^{\prime}\gamma_{1}^{0}+\epsilon_{1i}-(\hat{W}_{i}-W_{i})^{\prime}\gamma_{1}^{0}\big{)}$
$\displaystyle=$
$\displaystyle\gamma_{1}^{0}+\big{(}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}\big{)}^{-1}\sum_{i}D_{i}\hat{W}_{i}\big{(}\epsilon_{1i}-(\hat{W}_{i}-W_{i})^{\prime}\gamma_{1}^{0}\big{)}$

so that

$\displaystyle\hat{\gamma}_{1}-\gamma_{1}^{0}=\big{(}\underbrace{\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}}_{\textbf{A}}\big{)}^{-1}\underbrace{\frac{1}{n}\sum_{i}D_{i}\hat{W}_{i}\big{(}\epsilon_{1i}-(\hat{W}_{i}-W_{i})^{\prime}\gamma_{1}^{0}\big{)}}_{\textbf{B}}=\textbf{A}^{-1}\textbf{B}.$
(43)

##### Part $\bf{A}$

We show that
$\frac{1}{n}\sum_{i=1}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}-\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}|S]=o_{p}(1)$.
Decompose
$\frac{1}{n}\sum_{i=1}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}-\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}|S]$
into two parts as follows:

$\displaystyle\underbrace{\frac{1}{n}\sum_{i=1}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}-\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}}_{(a)}+\underbrace{\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}-\frac{1}{n}\sum_{i=1}^{n}\mathbf{E}[D_{i}W_{i}W_{i}^{\prime}|S]}_{(b)}.$

$(a)=o_{p}(1)$ since $\hat{\theta}-\theta^{0}\xrightarrow{p}0$ and
$W_{i}(\theta)$ is continuous in $\theta$. For $(b)$, note that the summand
$\\{D_{i}W_{i}W_{i}^{\prime}-\mathbf{E}[D_{i}W_{i}W_{i}^{\prime}|S]\\}$ is
conditionally independent given $S$ with mean zero. It is also uniformly
bounded. Therefore by LLN, $(b)=o_{p}(1)$. Finally, invertibility of
$\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}|S]$ follows from
the identification condition.

##### Part $\bf{B}$

Since $\hat{W}_{i}-W_{i}=o_{p}(1)$, we can write it $\bf{B}$ as

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}D_{i}(W_{i}+o_{p}(1))(\epsilon_{1i}-o_{p}(1))=\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\epsilon_{1i}$

Similar argument as above shows that

$\frac{1}{n}\sum_{i=1}^{n}\Big{(}D_{i}W_{i}\epsilon_{1i}-\mathbf{E}[D_{i}W_{i}\epsilon_{1i}|S]\Big{)}=o_{p}(1).$

It follows from the moment condition $\mathbf{E}[\epsilon_{1i}|D_{i}=1,S]=0$
that $\mathbf{E}[D_{i}W_{i}\epsilon_{1i}|S]=0$. Therefore we conclude that

$\textbf{B}=\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\epsilon_{1i}+o_{p}(1)=o_{p}(1).$

Combining with the result on part $\bf{A}$, we conclude that
$\hat{\gamma}_{1}-\gamma_{1}=o_{p}(1)$. $\blacksquare$

###  Proof of asymptotic normality of second-stage estimators

From 43,

$\displaystyle\sqrt{n}(\hat{\gamma}_{1}-\gamma_{1}^{0})=\Big{(}\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\hat{W}_{i}^{\prime}\Big{)}^{-1}\frac{1}{\sqrt{n}}\sum_{i}D_{i}\hat{W}_{i}\Big{(}\epsilon_{1i}-\gamma_{1}^{0\prime}(\hat{W}_{i}-W_{i})\Big{)}$
(44)
$\displaystyle=\Big{(}\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}W_{i}^{\prime}|S]+o_{p}(1)\Big{)}^{-1}\underbrace{\frac{1}{\sqrt{n}}\sum_{i}D_{i}\hat{W}_{i}\Big{(}\epsilon_{1i}-\gamma_{1}^{0\prime}(\hat{W}_{i}-W_{i})\Big{)}}_{\bf{C}}$
(45)

where the last step has been established in the previous section. Consider the
term $\hat{W}_{i}-W_{i}$ in $\bf{C}$. By mean-value theorem,

$\displaystyle\hat{W}_{i}-W_{i}=W_{i}(\hat{\gamma}_{1})-W_{i}(\gamma_{1}^{0})=\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})(\hat{\gamma}_{1}-\gamma_{1}^{0})$
$\displaystyle\Longrightarrow$
$\displaystyle\sqrt{n}(\hat{W}_{i}-W_{i})=\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})\sqrt{n}(\hat{\gamma}_{1}-\gamma_{1}^{0})$

where $\bar{\gamma}_{1}$ is a mean value of the line joining
$\hat{\gamma}_{1}$ and $\gamma_{1}^{0}$. By the asymptotic normality of the
first-step estimator $\hat{\theta}$ as in the equation 30, we can show that
$\sqrt{n}(\hat{\theta}-\theta^{0})$ is asymptotically linear. Specifically,
define the influence function as
$\eta_{i}=\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l_{i}(\theta^{0})\nabla_{\theta}l_{i}(\theta^{0})^{\prime}|S]\nabla_{\theta}l_{i}(\theta^{0})$,
then

$\sqrt{n}(\hat{\theta}-\theta^{0})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\eta_{i}+o_{p}(1).$

Therefore the term $\bf{C}$ in $\sqrt{n}(\hat{\gamma}_{1}-\gamma_{1}^{0})$ can
be written as

$\displaystyle\frac{1}{\sqrt{n}}\sum_{i}D_{i}\hat{W}_{i}(\epsilon_{1i}-\gamma_{1}^{0\prime}(\hat{W}_{i}-W_{i}))=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\epsilon_{1i}-\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\gamma_{1}^{0\prime}\sqrt{n}(\hat{W}_{i}-W_{i})$
$\displaystyle=\underbrace{\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\epsilon_{1i}}_{\textbf{C}(a)}-\underbrace{\Big{\\{}\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})\Big{\\}}}_{\textbf{C}(b)}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\eta_{i}+o_{p}(1)$

We first show that $\bf{C}(a)$ can be replaced by
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{i}W_{i}\epsilon_{1i}$ and that $\bf{C}(b)$
can be replaced by
$\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})]$.

#####  Part C(a)

We show that

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\Big{(}D_{i}\hat{W}_{i}\epsilon_{1i}-D_{i}W_{i}\epsilon_{1i}\Big{)}\xrightarrow{p}0$

Note htat

$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{i}(\hat{W}_{i}-W_{i})\epsilon_{1i}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{i}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})(\hat{\gamma}_{1}-\gamma_{1}^{0}\big{)}\epsilon_{1i}$
(46) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}D_{i}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})\sqrt{n}(\hat{\gamma}_{1}-\gamma_{1}^{0})\epsilon_{1i}$
(47) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}D_{i}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})\big{(}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\eta_{i}\big{)}\epsilon_{1i}$
(48) $\displaystyle=$
$\displaystyle\Big{(}\frac{1}{n}\sum_{i=1}^{n}D_{i}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})\epsilon_{1i}\Big{)}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\eta_{i}$
(49)

It can be shown easily that
$\frac{1}{n}\sum_{i=1}^{n}\Big{(}D_{i}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\epsilon_{1i}-\mathbf{E}[D_{i}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})\epsilon_{1i}|S]\Big{)}\xrightarrow{p}0$
where
$\mathbf{E}[D_{i}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\epsilon_{1i}|S]=0$
from the moment condition. Therefore equation 49 becomes $o_{p}(1)\times
O_{p}(1)$ and the result follows. $\blacksquare$

##### Part $\bf{C(b)}$

We show that

$\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})-\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})|S]=o_{p}(1).$

Decompose the LHS as

$\displaystyle\underbrace{\frac{1}{n}\sum_{i=1}^{n}D_{i}\hat{W}_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\bar{\gamma}_{1})-\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})}_{A}$
$\displaystyle+\underbrace{\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})-\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})|S]}_{B}.$

$A=o_{p}(1)$ since $\hat{\theta}-\theta^{0}\xrightarrow{p}0$. Also, since
$\\{D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\\}$
are conditionally independent given $S$ and uniformly bounded, we can apply
Markov LLN to show that $B=o_{p}(1)$. $\blacksquare$

##### Combining all the results,

term $\bf{C}$ can be written as

$\displaystyle\bf{C}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{i}W_{i}\epsilon_{1i}-\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\Big{|}S\Big{]}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\eta_{i}+o_{p}(1)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\underbrace{\Big{\\{}D_{i}W_{i}\epsilon_{1i}-\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\Big{|}S\Big{]}\eta_{i}\Big{\\}}}_{\zeta_{i}}.$

Since $\zeta_{i}|S$ has a mean zero and is independently distributed, we can
apply CLT for the independent observation and get
$\Psi_{n}^{-1/2}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\zeta_{i}\xrightarrow{d}N(0,I_{dim(\gamma_{1})})$
where
$\Psi_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{E}[\zeta_{i}\zeta_{i}^{\prime}|S]$
which can be simplified as

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\mathbf{E}[D_{i}W_{i}W_{i}^{\prime}\epsilon_{1i}^{2}]+\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\Big{|}S\Big{]}\frac{1}{n}\sum_{i=1}^{n}\mathbf{E}[\eta_{i}\eta_{i}^{\prime}|S]\mathbf{E}\Big{[}\frac{1}{n}\sum_{i=1}^{n}D_{i}W_{i}\gamma_{1}^{0\prime}\nabla_{\gamma_{1}}W_{i}(\gamma_{1}^{0})\Big{|}S\Big{]}^{\prime}$

as the cross-terms get crossed out due to
$\mathbf{E}[\epsilon_{1i}\eta_{i}^{\prime}|S]=0$, i.e., the first- and second-
stage moments are uncorrelated. Finally, from 45, and by defining
$\Upsilon_{n}=\mathbf{E}[\frac{1}{n}\sum_{i=1}^{n}W_{i}W_{i}^{\prime}|S]$, we
have

$\Lambda_{n}^{-1/2}\sqrt{n}(\hat{\gamma}_{1}-\gamma_{1}^{0})\xrightarrow{d}N(0,I_{dim(\gamma_{1})})$

for $\Lambda_{n}=\Upsilon_{n}^{-1}\Psi_{n}\Upsilon_{n}^{-1}$ as desired.
$\blacksquare$

##  Appendix C Auxiliary Lemmas

######  Lemma 1 (uniform boundedness of $\sigma_{i}^{*}(S,\theta)$).

There exists a constant $C\in(0,1)$ such that $\sigma_{i}^{*}(S,\theta)\geq C$
for any $i,S,\theta$ and $n$.

###### (Proof)

As in A, let us define agent’s best-response function as
$\Gamma(X_{i},Z_{i},\bar{\sigma}_{i},\theta)=\Phi(X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\bar{\sigma}_{i})$.
Recall that
$\sigma_{i}^{*}(S,\theta)=\Phi(X_{i}^{\prime}\theta_{1}+\theta_{2}Z_{i}+\theta_{3}\pi_{i}^{*}(S,\theta))$.The
result follows since $X_{i}$ is bounded, $Z_{i}$ is binary, and
$\pi_{i}^{*}(S,\theta)\leq 1$, $\blacksquare$

######  Lemma 2 (uniform boundedness of $\nabla\sigma_{i}$).

Suppose $\lambda<1$. There exists a finite constant $C_{1}$ such that

$\sup_{i,n,S,\theta,k}\big{|}\frac{\partial\sigma_{i}^{*}(S,\theta)}{\partial\theta_{k}}\big{|}<C_{1}<\infty.$

###### (Proof)

Recall that

$\sigma_{i}^{*}(S,\theta)=\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*}(S,\theta),\theta).$

Differentiating above equation with respect to $\theta_{k}$ gives

$\frac{\partial\sigma_{i}^{*}(S,\theta)}{\partial\theta_{k}}=\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\theta_{k}}+\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\bar{\sigma}_{i}^{*}}\frac{\partial\bar{\sigma}_{i}^{*}(S,\theta)}{\partial\theta_{k}}$

Equivalently,

$\displaystyle\frac{\partial\sigma_{i}^{*}(S,\theta)}{\partial\theta_{k}}=\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\theta_{k}}+\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\bar{\sigma}_{i}^{*}}\frac{\partial\sigma_{j}^{*}(S,\theta)}{\partial\theta_{k}}$
(50)

which gives the implicit function of
$[\partial\sigma_{i}^{*}(S,\theta)/\partial\theta_{k}]_{i\in\mathcal{N}_{n}}$.
Let us write 50 in matrix form by defining the following:

  * •

Let $\chi_{n}$ be $n\times 1$ vector with $i$th component
$\partial\sigma_{i}^{*}(S,\theta)/\partial\theta_{k}$.

  * •

Let $D_{n}$ be $n\times n$ matrix with $ij$th element

$\frac{1}{|\mathcal{N}_{i}|}\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\bar{\sigma}_{i}^{*}}$

if $G_{ij}=1$ and zero if $G_{ij}=0$.

  * •

Let $\tau_{n}$ be $n\times 1$ vector with $i$th component
$\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\theta_{k}}$.

Then we can write the system 50 as $\chi_{n}=D_{n}\chi_{n}+\tau_{n}$ or
equivalently,

$(I_{n}-D_{n})\chi_{n}=\tau_{n}$

which is invertible if $||D_{n}||_{\infty}<1$ where the induced matrix norm
$||D_{n}||_{\infty}$ is the maximum of the absolute values of row sums, i.e.,

$||D_{n}||_{\infty}=\max_{i\in\mathcal{N}_{n}}\Big{|}\frac{\partial\Gamma(X_{i},Z_{i},\bar{\sigma}_{i}^{*},\theta)}{\partial\bar{\sigma}_{i}^{*}}\Big{|}.$

37 implies that $||D_{n}||_{\infty}\leq\lambda$, thus $||D_{n}||_{\infty}<1$.
Therefore $D_{n}$ is invertible and
$(I_{n}-D_{n})^{-1}=\sum_{t=0}^{\infty}D_{n}^{t}$. It follows that
$\chi_{n}=(\sum_{t=0}^{\infty}D_{n}^{t})\tau_{n}$. Taking sup norm gives

$||\chi_{n}||_{\infty}\leq\sum_{t=0}^{\infty}||D_{n}^{t}||_{\infty}||\tau_{n}||_{\infty}=\frac{||\tau_{n}||_{\infty}}{1-\lambda}<\frac{C_{\tau}}{1-\lambda}$

since RHS does not depend on $(i,n,z_{n},\theta,k)$, we have the desired
result. $\blacksquare$

######  Lemma 3 (uniform boundedness of $\nabla^{2}\sigma_{i}$).

Suppose $\lambda<1$. There exists a finite constant $C_{2}$ such that

$|\frac{\partial^{2}\sigma_{i}^{*}(S,\theta)}{\partial\theta_{m}\partial\theta_{k}}|<C_{2}<\infty$

for any $i,n,S,\theta,k,m$ a.s.

###### (Proof)

Fix $m$. Differentiating the equation 50 w.r.t. $\theta_{m}$ gives

$\displaystyle\frac{\partial^{2}\sigma_{i}}{\partial\theta_{m}\partial\theta_{k}}=\frac{\partial^{2}\Gamma}{\partial\theta_{m}\partial\theta_{k}}+\frac{\partial^{2}\Gamma}{\partial\bar{\sigma}_{i}\partial\theta_{k}}\frac{\partial\bar{\sigma}_{i}}{\partial\theta_{m}}+\frac{\partial\Gamma}{\partial\bar{\sigma}_{i}}\frac{\partial^{2}\bar{\sigma}_{i}}{\partial\theta_{m}\partial\theta_{k}}+\frac{\partial\bar{\sigma}_{i}}{\partial\theta_{k}}\Big{\\{}\frac{\partial^{2}\Gamma}{\partial\bar{\sigma}_{i}^{2}}\frac{\partial\bar{\sigma}_{i}}{\partial\theta_{m}}+\frac{\partial^{2}\Gamma}{\partial\theta_{m}\partial\bar{\sigma}_{i}}\Big{\\}}.$

Let us write it compactly as follows:

$\displaystyle\framebox[1.1pt]{$\partial^{2}_{mk}\sigma_{i}$}=\Gamma_{mk}+\Gamma_{\bar{\sigma}k}\partial_{m}\bar{\sigma}_{i}+\Gamma_{\bar{\sigma}}\framebox[1.1pt]{$\partial^{2}_{mk}\bar{\sigma}_{i}$}+\Gamma_{\bar{\sigma}\bar{\sigma}}\partial_{k}\bar{\sigma}_{i}\partial_{m}\bar{\sigma}_{i}+\Gamma_{\bar{\sigma}m}\partial_{k}\bar{\sigma}_{i}.$
(51)

Write 51 in a matrix form by defining

  * •

Let $\tilde{\chi}_{n}$ be $n\times 1$ vector with $i$th component
$\partial^{2}_{mk}\sigma_{i}$.

  * •

Let $\tilde{\tau}_{n}$ be $n\times 1$ vector with $i$th component

$\Gamma_{mk}+\Gamma_{\bar{\sigma}k}\partial_{m}\bar{\sigma}_{i}+\Gamma_{\bar{\sigma}\bar{\sigma}}\partial_{k}\bar{\sigma}_{i}\partial_{m}\bar{\sigma}_{i}+\Gamma_{\bar{\sigma}m}\partial_{k}\bar{\sigma}_{i}.$

Then 51 can be written as

$(I_{n}-D_{n})\tilde{\chi}_{n}=\tilde{\tau}_{n}.$

As we have shown before, $D_{n}$ is invertible. For any $i\in\mathcal{N}_{n}$,
$|\tau_{i}|\leq
B_{\theta,\theta}+2B_{\bar{\sigma}\theta}C_{\partial\sigma}+B_{\bar{\sigma},\bar{\sigma}}C_{\partial\sigma}^{2}$,
so that $||\tau_{n}||_{\infty}=\max_{i}|\tau_{i}|$ is uniformly bounded.
Therefore,

$||\tilde{x}_{n}||_{\infty}\leq\frac{C_{\tau}}{1-\lambda}$

and the result follows. $\blacksquare$

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