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# Identification of Peer Effects with Miss-specified Peer Groups: Missing Data
and Group Uncertainty

Christiern Rose
University of Queensland

## 1\. Introduction

This paper addresses two challenges faced by empirical researchers studying
peer effects. First, most existing methods to identify/estimate peer effects
require that the researcher has access to group data: a sample of groups such
that within each group every member is observed (see Bramoullé et al. (2020)
for a recent review). Such data are often proprietary and/or restricted-access
(Breza et al., 2020), and widely available individual/household survey data
cannot be used because they are not designed to sample every group member
(e.g. neighbourhoods or industries in national household surveys/census
microsamples, grades in PISA sample of students etc.). The problem is
particularly acute in models of peer effects because, if even one peer is
entirely missing from the data (or has missing data for the variable of
interest), the peer average cannot be constructed. Standard empirical practice
is either to drop individuals with missing peer data from the estimation
sample, leading to sample selection and/or loss of information, or to compute
the average over the observed peers, leading to measurement error. Moreover,
the researcher may not even know that any peers are missing, hence might not
even be aware that their analysis may be compromised by these issues. We refer
to this as the missing data problem.

Second, the researcher often has to make a choice regarding which is the
relevant peer group for the particular outcome under study. Often there will
be multiple groups but the researcher will not know which is appropriate. For
example, studies based on the Dartmouth room-mate data have noted that it
isn’t clear whether peer effects ought to operate at the room level or the
floor level (Sacerdote, 2001; Glaeser et al., 2003; Angrist, 2014), and the
relevant group may be different for different outcomes, such as test-scores
and fraternity membership (Sacerdote, 2001).111These data are widely used
because of conditionally random allocation of students to dorm rooms. Standard
empirical practice would then be to re-estimate the peer effects for each
possible peer definition. This implicitly assumes that peer definition is the
same for all individuals (i.e. it is deterministic) and there is no reason for
this to be the case. In the context of the Dartmouth data, some dorms may
simply be more sociable than others, and the researcher is unlikely to know
which ones. We refer to this as the group uncertainty problem.

We consider a model in which individuals interact in groups and there are
endogenous (outcomes depend on peer outcomes), contextual (outcomes depend on
peer characteristics) and correlated (outcomes are correlated due to exposure
to a common environment) effects. This model is widely used in empirical
research, and has previously been studied by Lee (2007), Davezies et al.
(2009) and Bramoullé et al. (2009), among others. We begin by showing that
missing data and group uncertainty are both examples of a wider class of miss-
specification in which the members of the specified group are a subset of the
members of the (true) group. This is clear in the missing data case, and in
the group uncertainty case, it is applicable in the common empirical setting
in which the possible groups are nested. This is satisfied, for example, in
the Dartmouth data because rooms are nested in floors. Similarly, for peer
effects in education, interactions could take place at the class level or the
grade level. Finally, neighbourhood effects might operate at the street,
suburb or city levels. The members of the specified group can then be
guaranteed to be members of the group by specifying the smallest group.

Next we show that group miss-specification essentially reduces to knowledge of
the group size, provided that the members of the specified group are a subset
of the group members and specified-group fixed effects are used. Finally, we
show that peer effects can be identified through assumptions relating the
group size to the specified group size. We consider two such assumptions: one
for missing data and another for group uncertainty.

For missing data, if the group size is observed and the probability that an
individual is observed is uncorrelated with the error in the structural
equation (i.e. there is no sample selection problem), then peer effects are
identifiable. This result was first noted by Davezies et al.
(2009),222Davezies et al. (2009) do not study group uncertainty. but we extend
it to also consider the case in which the group size is unobserved. This is
important, because it allows for settings in which the researcher has no
information on missing individuals and need not even know whether they exist,
which may often be the case in applied work.

If the group size is unobserved, under a mild assumption on individuals’
probabilities of being observed,333Essentially requiring that they do not
depend on the error term in the structural equation (i.e. there is no sample
selection), that they are all equal in expectation, and that there is no
dependence across individuals. the observed group size conditional on the
group size follows a Binomial distribution, hence the specified group size
provides information on the group size, which can be used to identify the peer
effects. Viewed from the missing data perspective, our assumption is
essentially that some individuals are Missing Completely at Random, which is
widely used (often implicitly) in empirical work.444In principle we can relax
this assumption to Missing at Random by allowing the sampling probability to
depend on observables. We do not pursue this because the focus of this paper
is on replacing a group level sample with an individual levels sample, rather
than on sample selection of individuals. Hence we view the Missing Completely
at Random assumption as the benchmark case in which a well designed group
sample is replaced by a well designed individual sample. In this setting the
peer effects are identified if (i) the distribution of group sizes has at
least three points of support (i.e. there are at least three group sizes),
(ii) the endogenous and contextual effects are not such that they exactly
offset one another (yielding a zero reduced form peer effect) and (iii) the
distribution of group sizes is bounded. Conditions (i) and (ii) are necessary
even when there is no miss-specification (Lee, 2007; Davezies et al., 2009;
Bramoullé et al., 2009), whereas condition (iii) is a mild requirement used to
establish identification through the properties of an eigensystem. To estimate
the peer effects we propose a GMM estimator which combines information on the
observed group sizes with individual outcomes and exogenous characteristics of
the observed individuals. In practice, our method can be used when a
researcher has access only to an individual sample with group identifiers
(e.g. neighbourhoods or industries in national household surveys/census
microsamples, grades in PISA sample of students), which are far more widely
available than group data.

For group uncertainty, we consider a setting in which the group is one of two
types (e.g. dorm rooms and floors) at random. We can also allow the type
probability to depend on the exogenous characteristics, and an extension to
three or more groups is straightforward. We show that the parameters are
identified under conditions (i) and (ii) from the previous paragraph and a
condition on the support of the joint distribution of the sizes of the two
group types. This support condition is similar in spirit to that concerning
identifiability of bi-partite fixed effects models (e.g. the ‘connected set’
for the worker-firm fixed effect model of Abowd et al. (1999)). The basic idea
is that the reduced form peer effect is a weighted average of the reduced form
peer effects under each group type, each of which depends on the size of the
group type. Hence if there is sufficient joint variation in the group sizes
(e.g. room size and floor size), we can disentangle their reduced form peer
effects, and hence identify the peer effects.

### 1.1. Related literature

We now relate our work to the literature. The literature on sampled networks,
mismeasured networks and missing data has recently attracted much interest.
Lewbel et al. (2019) and Hardy et al. (2019) consider the implications of
measurement error in the network through which peer effects operate. Lewbel et
al. (2019) additionally consider the case in which only group identifiers are
available and no information on the network is available beyond group
identifiers. Boucher and Houndetoungan (2020) study peer effects when the
network is unknown but that its distribution is known or can be consistently
estimated.

Chandrasekhar and Lewis (2011) studies the implications of using sampled
network data in applied work, provide analytical corrections in some examples
of interest and develop a more general graphical reconstruction approach. In
the context of peer effects, the analytical correction can be applied if the
researcher has data on the network, outcomes and exogenous characteristics for
a subsample of individuals and knows the identities, outcomes and exogenous
characteristics of all of their peers. This means that data are only missing
for peers-of-peers. To apply graphical reconstruction, for every individual,
the researcher must observe the outcomes, exogenous characteristics and
variables which can be used to predict the propensity of individuals to form
links in a network formation model.

Breza et al. (2020) use a parametric network formation model to show that
network structure can be identified and used to construct network statistics
(e.g. centrality measures) if the researcher does not collect network data for
all individuals, but instead collects relational data for a sub-sample of
individuals and basic characteristics of all individuals.555Relational data
are collected by asking questions of the form ‘Consider all the individuals in
your group with whom you do activity X. How many of these have trait Y?’
(Breza et al., 2020) Sojourner (2013), Wang and Lee (2013) and Liu et al.
(2017) consider the case in which the network and all individuals are observed
without error but there are missing data in the outcomes and/or exogenous
characteristics.

All of these papers suppose that the researcher observes at least some data on
every individual.666Some additionally require specification and estimation of
one or more network formation model. In contrast, we allow for some
individuals to be entirely missing and do not require knowledge on whether
there even exist such individuals. These gains come from exploiting the
structure of the group interactions model we consider. Hence, though we
consider a less general network structure than other research, we also require
weaker assumptions on data availability.777We discuss this issue further in
Section 7.

Our work is also related to the literature on multiple peer effects.
Goldsmith-Pinkham and Imbens (2013), Arduini et al. (2020) and Reza et al.
(2021) study peer effects which operate through more than one network. These
papers suppose that every individual is subject to the same peer effects, but
that they operate through two or more networks. In contrast, we suppose that
individuals are exposed to peer effects operating through only one network,
but there is uncertainty as to which is the relevant network. Our model is
thus not a model of multiple peer effects, but one of peer effects with
uncertain peer groups.

## 2\. Model

We consider the group-interactions framework in which the data comprise
$N_{0}$ individuals from an i.i.d. sample of $G_{0}$ groups. Though our focus
is on identification, the asymptotic we have in mind is $G_{0}\to\infty$,
which is analagous to the large-$N$ fixed-$T$ asymptotic used for
microeconometric panel data. This model and asymptotic has been previously
studied by Lee (2007), Bramoullé et al. (2009) and Davezies et al. (2009),
among others, and can be microfounded based on a game with quadratic utility
in one’s own outcome and the average outcome of the others in the group (see
Blume et al. (2015)).

Individual $i\in\\{1,2,...,N_{0}\\}$ is in exactly one group denoted
$g_{0}(i)\in\\{1,2,...,G_{0}\\}$. The size of $i$’s group is $n_{g_{0}(i)}\geq
2$. When it is not required, we omit the dependence on $i$ and simply refer to
$g_{0}$ and $n_{g_{0}}$. The continuous outcome $y_{i}$ of individual $i$ is
determined by

$\displaystyle
y_{i}=\alpha_{g_{0}}+\left(\frac{1}{n_{g_{0}}-1}\sum_{\begin{subarray}{c}j:g_{0}(j)=g_{0}(i)\\\
j\neq
i\end{subarray}}y_{j}\right)\beta+\left(\frac{1}{n_{g_{0}}-1}\sum_{\begin{subarray}{c}j:g_{0}(j)=g_{0}(i)\\\
j\neq i\end{subarray}}x_{j}\right)\delta+x_{i}\gamma+\epsilon_{i},$ (2.1)

where $\alpha_{g_{0}(i)}$ is group unobserved heterogeneity, $x_{i}$ are
exogenous characteristics and $\epsilon_{i}$ is an error term satisfying

$\displaystyle\mathbb{E}[\epsilon_{i}|\alpha_{g_{0}},n_{g_{0}},\mathbf{x}_{g_{0}}]=0\quad\forall
i\in\\{1,2,...,N_{0}\\},$ (2.2)

where $\mathbf{x}_{g_{0}}=(x_{j})_{j:g_{0}(j)=g_{0}(i)}$ is $n_{g_{0}}\times
1$.888Since we only consider the within-transformation of the Fixed Effects
model below, (2.2) can be relaxed to
$\mathbb{E}[\epsilon_{i}|n_{g_{0}},\mathbf{x}_{g_{0}}]=0$. We maintain the
form of (2.2) to maintain comparability with the literature. The moment
condition supposes that group size and the characteristics of the group
members are strictly exogenous with respect to $\epsilon_{i}$. However,
$n_{g_{0}}$ and $\mathbf{x}_{g_{0}}$ can be arbitrarily dependent on
$\alpha_{g_{0}}$. The model includes correlated effects (captured by
$\alpha_{g_{0}}$), endogenous peer effects ($\beta$) and contextual peer
effects ($\delta$). For simplicity of exposition, following Bramoullé et al.
(2009) results are presented for the case where there is a single
characteristic $x_{i}$. All results continue to apply in the more general case
in which $x_{i}$, $\gamma$ and $\delta$ are K-dimensional vectors. To ensure
that the reduced form of (2.1) exists, we additionally assume that
$|\beta|<1$, in which case the reduced form is

$\displaystyle y_{i}$
$\displaystyle=\frac{\alpha_{g_{0}}}{1-\beta}+\left(\sum_{\begin{subarray}{c}j:g_{0}(j)=g_{0}(i)\\\
j\neq
i\end{subarray}}x_{j}\right)\pi_{1}(n_{g_{0}})+x_{i}\pi_{2}(n_{g_{0}})+u_{i},$
(2.3) $\displaystyle\pi_{1}(n)$
$\displaystyle=\left(\frac{(\delta+\beta\gamma)(n-1)^{-1}}{1-\beta(\beta+n-2)(n-1)^{-1}}\right),\quad\pi_{2}(n)=\left(\frac{\gamma+\beta(\delta-\gamma(n-2))(n-1)^{-1}}{1-\beta(\beta+n-2)(n-1)^{-1}}\right),$

where $u_{i}$ is the reduced form error. Applying the within-group
transformation

$\widetilde{y}_{i}=y_{i}-\frac{1}{n_{g_{0}}}\sum_{j:g_{0}(j)=g_{0}(i)}y_{j}$

to the reduced form yields,

$\displaystyle\widetilde{y}_{i}=\widetilde{x}_{i}\pi(n_{g_{0}})+\widetilde{u}_{i},,\quad\pi(n)=\left(\frac{\gamma-\delta(n-1)^{-1}}{1+\beta(n-1)^{-1}}\right)$
(2.4)

## 3\. (Possibly) Miss-specified Groups

We suppose now that the researcher specifies the group $g(i)\in\\{1,...,G\\}$
for individual $i$, with corresponding group size $n_{g(i)}$. These groups may
be miss-specified, and may also cover only a subset
$\mathcal{S}\subseteq\\{1,...,N_{0}\\}$ of individuals of cardinality $N\leq
N_{0}$. The latter could arise, for example, if the available data comprise a
random sample drawn from the $N_{0}$ individuals. Every individual in
$\mathcal{S}$ is in exactly one group. As with $g_{0}$, we omit the dependence
of $g(i)$ on $i$ unless it is required, writing simply $g$ instead.

Due to the group heterogeneity term ($\alpha_{g_{0}}$), it is immediately
clear that identification fails whenever the specified groups are not a subset
of the groups (i.e. $g(i)=g(j)$ does not imply $g_{0}(i)=g_{0}(j)$). This is
because the within-specified-group transformation

$\overline{y}_{i}=y_{i}-\frac{1}{n_{g}}\sum_{j:g(j)=g(i)}y_{j}$

does not eliminate $\alpha_{g_{0}}$. Moreover, identification fails even in
the absence of correlated effects (i.e. even if $\alpha_{g_{0}}=\alpha$) due
to an omitted variables problem. To see this, consider an example in which the
researcher has data on all individuals $N=N_{0}$ for some even number $N_{0}$
and specifies $G=1$ group containing all individuals, when in fact there are
$G_{0}=2$ groups of equal size. Solving for the reduced form and applying the
within-specified-group transformation yields, for individual $i$ in group
$g_{0}(i)=1$,

$\displaystyle\overline{y}_{i}$
$\displaystyle=\overline{x}_{i}\pi(N)+t_{1}+t_{2}+t_{3}+\overline{u}_{i}$
(3.1) $\displaystyle=\overline{x}_{i}\pi(N)+\overline{v}_{i}$ (3.2)

where

$\displaystyle t_{1}=\frac{\alpha_{1}-\alpha_{2}}{2(1-\beta)},\quad
t_{2}=\left(\frac{\gamma+\delta}{1-\beta}\right)\frac{1}{N}\left(\sum_{j:g_{0}(j)=1}x_{j}-\sum_{j:g_{0}(j)=2}x_{j}\right),$
$\displaystyle
t_{3}=-\frac{\overline{x}_{i}N(\gamma\beta+\delta)}{2(N-1+\beta)(N/2-1+\beta)}$

From the above, it is immediately clear that
$\mathbb{E}[\overline{v}_{i}|N,\mathbf{x}_{g}]\neq 0$. There are three reasons
for this, each corresponding to a term in equation (3.1). First, the
researcher specifies $\alpha_{1}=\alpha_{2}$, which may be violated. This
yields the term $t_{1}$. Second, the researcher includes the exogenous
characteristics and outcomes of those in group 2 on the right hand side of the
structural equations for the group 1 outcomes. This yields yields the term
$t_{2}$. Finally, the researcher miss-specifies the group size to be $N$
instead of $N/2$. This, leads the reduced form parameter to be incorrectly
specified as $\pi(N)$ rather than $\pi(N/2)$, yielding the term $t_{3}$. All
of these terms are correlated with $\overline{x}_{i}$, hence any estimator
(e.g. non-linear least squares) based on
$\mathbb{E}[\overline{v}_{i}|N,\mathbf{x}_{g}]=0$ is inconsistent for
$\gamma,\delta,\beta$.

The example above considered the case in which the specified group contained
the group. In contrast, if the groups contain the specified groups, the first
two sources of miss-specification are eliminated. This is because, if all
individuals in specified group $g$ are in group $g_{0}$ then they share the
same correlated effect, and, the sum of the outcomes and exogenous
characteristics over all those in group $g_{0}$ but not in group $g$ is common
to all those in group $g$. Hence, the terms $t_{1}$ and $t_{2}$ do not arise
if the specified groups are contained within the groups. Since it is unclear
how to otherwise mitigate these sources of miss-specification, we hence
proceed under the following assumption

######  Assumption 1 (Subset miss-specification).

For all $(i,j)\in\mathcal{S}^{2}$, $g(i)=g(j)\Rightarrow g_{0}(i)=g_{0}(j)$

and the remainder of the paper focuses on miss-specification arising due to
incorrect specification of the group size (i.e. due to $t_{3}$).

Assumption 1 covers at least two empirically important cases. First, it allows
for the case in which the researcher observes only a sample of $N<N_{0}$
individuals drawn from all $N_{0}$ individuals. This means that the researcher
may only have access to a sample of individuals with outcomes, characteristics
and group identifiers, but that the sample has not been designed to cover all
members of each group. For example, in the context of education, the
researcher might access a sample of students’ test-scores, characteristics and
classroom/school identifiers (Davezies et al., 2009) or have access to a large
educational survey such as the student level PISA survey, which contains
school and grade identifiers. In the context of risky behaviours, survey data
on smoking, drinking and illicit drug use among students can be incomplete
(Lundborg, 2006). In the context of neighbourhood effects, the neighbourhood
average outcome may be measured using survey data which includes geographic
identifiers (e.g. census tract). For example, Bertrand et al. (2000) use the
5% public use microsample of the 1990 Census to study welfare take-up. Glaeser
et al. (2003) use the same data for wages.

In all of these cases the researcher knows the groups but only observes a
subset of group members for each group. This implies that the group average
outcome/characteristic is measured with error due to a missing data problem.
As will be shown below, the form of the measurement error is such that the
structural parameters are identifiable when the reseacher has access to a
random sample of individuals and the group sizes are known, or, under some
mild additional restrictions on the sampling process, when the group sizes are
unknown. 999Identification under known group sizes was first established by
Davezies et al. (2009), whereas identification based on restricting the
missingness mechanism is novel.

Second, Assumption 1 covers the case in which there is group uncertainty but
the candidate peer groups are naturally nested. It is satisfied provided that
the researcher specifies the smallest of the groups. For example, Glaeser et
al. (2003) and Angrist (2014) estimate their model of peer effects for peer
groups based on dorm rooms and floors. As will be shown below, the structural
parameters are identifiable even if the relevant peer group is the room, the
floor, or sometimes the room and sometimes the floor.

## 4\. Identification

We now study identification under Assumption 1. Our identification analysis
follows Bramoullé et al. (2009), hence we say that the structural parameters
are (point) identified if and only if they can be uniquely recovered from the
reduced form parameters presented below (i.e. there is an injective
relationship). Our results are thus asymptotic in nature (see Manski (1995)),
and hence charaterise whether endogenous, contextual and correlated effects
can be distentangled if there is no limit to the number of groups $G_{0}$.

Under Assumption 1, the reduced form for individual $i\in\mathcal{S}$ is

$\displaystyle\overline{y}_{i}$
$\displaystyle=\overline{x}_{i}\pi(n_{g_{0}})+\overline{u}_{i}$ (4.1)

and taking conditional expectations yields101010To ease the notational burden
we do not make explicit that the expectation is also conditional on
$i\in\mathcal{S}$. Below, we consider only assumptions for which this omission
is innocuous and omit the condition $i\in\mathcal{S}$ for the remainder of the
paper.

$\displaystyle\mathbb{E}[\overline{y}_{i}|n_{g},\mathbf{x}_{g}]$
$\displaystyle=\overline{x}_{i}\mathbb{E}[\pi(n_{g_{0}})|n_{g},\mathbf{x}_{g}]+\mathbb{E}[\overline{u}_{i}|n_{g},\mathbf{x}_{g}]$
(4.2)

Equation (4.2) shows that identification prospects depend on whether there is
within-specified-group variation in the exogenous characteristic, on whether
$\mathbb{E}[\overline{u}_{i}|n_{g},\mathbf{x}_{g}]=0$, and if both of these
conditions are satisfied, on whether the structural parameters can be uniquely
recovered from the reduced form parameters
$\mathbb{E}[\pi(n_{g_{0}})|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]$ for all
$(\mathbf{x},n\geq 2)$ in the support of $(\mathbf{x}_{g},n_{g})$.111111We
require $n\geq 2$ because the term
$\mathbb{E}[\pi(n_{g_{0}})|n_{g}=1,\mathbf{x}_{g}]$ is not identifiable since
$n_{g}=1$ implies $\overline{y}_{i}=\overline{x}_{i}=0$.

In general, even under Assumption 1, the moment condition in (2.2) does not
imply $\mathbb{E}[\overline{u}_{i}|n_{g},\mathbf{x}_{g}]=0$. Suppose however
that we can make a restriction such that
$\mathbb{E}[\overline{u}_{i}|n_{g},\mathbf{x}_{g}]=0$, and also that there is
within-specified-group variation in $\mathbf{x}_{g}|n_{g}=n$ for all $n\geq 2$
in the support of $n_{g}$. Then equation (4.2) identifies the reduced form
parameter $\mathbb{E}[\pi(n_{g_{0}})|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]$, so
the structural parameters are point identified if they can be uniquely
recovered from

$\displaystyle\mathbb{E}[\pi(n_{g_{0}})|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]=\sum_{m=2}^{\infty}\mathbb{P}[n_{g_{0}}=m|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]\pi(n)$
(4.3)

for all $(\mathbf{x},n\geq 2)$ in the support of $(\mathbf{x}_{g},n_{g})$. The
distribution $n_{g_{0}}|n_{g},\mathbf{x}_{g}$ is not observed, hence the
structural parameters are not point identified unless it can be
restricted.121212Partial identification is possible because the probabilities
$\mathbb{P}[n_{g_{0}}=m|n_{g},\mathbf{x}_{g}]$ for $n=2,3....$ are non-
negative and sum to 1. We do not pursue partial identification, since, as
shown below, typical empirical settings faced by researchers can lead to point
identification.

We now consider three assumptions, each of which guarantees
$\mathbb{E}[\overline{u}_{i}|n_{g},\mathbf{x}_{g}]=0$ and restricts the
distribution of $n_{g_{0}}|n_{g},\mathbf{x}_{g}$. Each assumption includes the
standard model of peer effects with known peer groups as a special case,
respectively when $n_{g}=n_{g_{0}}$ (Assumption 2), when $\rho=1$ (Assumption
3) and when $\psi\in\\{0,1\\}$ (Assumption 4).

######  Assumption 2 (Missing data: Individial sample with known group size).

For each individual $i\in\\{1,2,...,N_{0}\\}$, the researcher observes
$(y_{i},x_{i},g_{0}(i))$ (i.e. $i\in\mathcal{S}$) with probability $p_{i}$,
$\mathbb{E}[\epsilon_{i}|\alpha_{g_{0}},n_{g_{0}},\mathbf{x}_{g_{0}},\mathbf{p}_{g_{0}}]=0$,
where $\mathbf{p}_{g_{0}}=(p_{j})_{j:g_{0}(j)=g_{0}(i)}$. The researcher
groups obseved individuals by $g_{0}$ and $n_{g_{0}}$ is observed or
$n_{g_{0}}=f(n_{g},\mathbf{x}_{g})$ for a known function $f$.

Assumption 2 covers the simplest case in which the group size is known but
individuals are sampled from the group. In practice, it means that the
researcher has access to a sample of individuals with outcomes,
characteristics, group membership indicators and group size. Identification
under Assumption 2 was first considered by Davezies et al. (2009), but we
include a brief discussion for comparison with our new results below. Under
Assumption 2, $\pi(m)$ is identified for all $m$ in the support of
$n_{g_{0}}$, such that the peer effects are identified when they can be
uniquely recovered from these parameters.

######  Assumption 3 (Missing data: Individial sample with known group size).

For each individual $i\in\\{1,2,...,N_{0}\\}$, the researcher observes
$(y_{i},x_{i},g_{0}(i))$ (i.e. $i\in\mathcal{S}$) with probability $p_{i}$,
$\mathbb{E}[p_{i}|\mathbf{x}_{g_{0}},n_{g_{0}}]=\rho\in(0,1]$,
$\mathbb{COV}[p_{i},p_{j}|\mathbf{x}_{g_{0}},n_{g_{0}}]=0$ for $j\in
g_{0}(i),j\neq i$, and
$\mathbb{E}[\epsilon_{i}|\alpha_{g_{0}},n_{g_{0}},\mathbf{x}_{g_{0}},\mathbf{p}_{g_{0}}]=0$,
where $\mathbf{p}_{g_{0}}=(p_{j})_{j:g_{0}(j)=g_{0}(i)}$. The researcher
groups obseved individuals by $g_{0}$.

Assumption 3 relaxes the requirement that the researcher knows the group size.
In practice, it allows for the case in which the researcher knows the number
of individuals sampled from each group ($n_{g}$) but does not know the group
size ($n_{g_{0}}$), nor whether there are any missing individuals in each
group.131313The group sizes would not be known even if $\rho$ were known.
Relative to Assumption 2, we additionally assume that the sample is unbiased,
in the sense that all individuals have the same ex-ante probability of being
sampled, and that the sampling probabilities are uncorrelated across
individuals. These are mild assumptions likely to be satisfied in well
designed samples.

Assumptions 2 and 3 both maintain that the sampling of individuals does not
induce sample selection bias (by restricting the conditional mean of the error
with respect to the sampling probabilities), whilst Assumption 3 additionally
maintains that the sampling probabilities do not depend on exogenous
covariates nor on the group size. Viewed from the missing data perspective,
Assumption 3 is essentially a type of Missing Completely at Random assumption,
which is widely used (often implicitly) in empirical work. In principle this
could be relaxed to Missing at Random, for example by allowing $\rho$ to
depend on observables. We do not pursue this because the focus of this paper
is not on sample selection at the individual level, but instead on replacing a
group level sample with an individual level sample. Hence we consider
Assumption 3 as the benchmark case in which the researcher has access to a
well designed sample of individuals rather than a well designed sample of
groups.

When conducting applied work, in an attempt to correct the measurement error
induced by possibly missing individuals, it might be tempting to proceed by
dropping any groups with missing members prior to estimation. There are four
problems with this. First, the researcher typically would not know which
groups having missing members and which do not (i.e. they may not even know if
$n_{g}<n_{g_{0}}$), so it might not be possible. Second, even if individuals
are missing completely at random, dropping every group with at least one
missing individual leads to an estimation sample of groups selected
systematically on group size.141414This is because larger groups are more
likely to have at least one missing member than smaller groups. Third, it is
not an efficient use of the data, because many of the groups that would be
dropped still provide information on the parameters. Moreover, if $\rho$ is
small and the group sizes are large, the vast majority of observed groups will
be missing at least one member. Fourth, with group fixed effects, the
researcher need only worry about the group size, rather than missing exogenous
characteristics and outcomes. The method we present below is an alternative to
dropping groups with missing members which corrects for the possibility of
missing individuals whilst making use of all observations.

We now discuss the informational content of Assumption 3, and how this can be
exploited for identification. Under Assumption 3 we have

$\displaystyle
n_{g}|n_{g_{0}},\mathbf{x}_{g}\sim\text{Binomial}(n_{g_{0}},\rho),$ (4.4)

which, together with the observed distribution of $n_{g}|n_{g}\geq
1,\mathbf{x}_{g}$, can lead to identification of $\rho$ and the peer
effects.151515Conditioning on $n_{g}\geq 1$ is important, because for some
groups the researcher may not observe any individuals. Hence it is the
distribution $n_{g}|n_{g}\geq 1,\mathbf{x}_{g}$ which is observed rather than
$n_{g}|\mathbf{x}_{g}$. As shown in the proof of Theorem 1 below, point
identification of $\rho$ comes from restrictions on the support of
$n_{g_{0}}$. The result uses that $n_{g_{0}}\geq 2$, which is maintained
throughout the literature (see e.g. Bramoullé et al. (2009), Lee (2007),
Davezies et al. (2009)).161616In practice, depending on the application one
may wish to impose a lower bound larger than two, which can improve the
performance of the estimator presented below.  Indeed, groups of size one
cannot provide information on peer effects and are not consistent with our
model. However, if individuals are sampled, then we may observe groups with
only one member even though all groups have at least two members (i.e.
$\mathbb{P}[n_{g}=1|\mathbf{x}_{g}]>0$ if $\rho<1$). Hence the distribution of
$n_{g}$ provides information on $\rho$, which combined with (4.4), leads to
its identification. Equation (4.3) can then be used to identify the peer
effects.

######  Assumption 4 (Group uncertainty).

For each individual $i\in\\{1,2,...,N_{0}\\}$ the researcher observes
$(y_{i},x_{i},g_{1}(i),g_{2}(i))$ (i.e. $\mathcal{S}=\\{1,...,N_{0}\\}$),
$2\leq n_{g_{1}}\leq n_{g_{2}}$ and $g_{1}(i)=g_{1}(j)\Rightarrow
g_{2}(i)=g_{2}(j)$ for all $(i,j)\in\mathcal{S}^{2}$. The researcher groups
individuals by $g_{1}$,
$\mathbb{E}[\epsilon_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g_{0}}]=0$ and
$\mathbb{P}[n_{g_{0}}=n_{g_{1}}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g_{0}}]=\psi\in[0,1]$
and
$\mathbb{P}[n_{g_{0}}=n_{g_{2}}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g_{0}}]=1-\psi$.

Assumption 4 covers the case in which there is group uncertainty but the
candidate groups are naturally nested. For simplicity of exposition we present
the case of two potential groups. Though an extension to three or more is
straightforward, we do not pursue it formally.

In the Dartmouth setting, Assumption 4 allows that peer effects operate at the
room level on some floors and at the floor level on others. We also require a
minor strengthening of the moment condition, replacing (2.2) with
$\mathbb{E}[\epsilon_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g_{0}}]=0$, and hence
$n_{g}$ with $n_{g_{1}},n_{g_{2}}$ in equations (4.2) and (4.3). The stronger
moment condition requires that both potential group sizes be strictly
exogenous with respect to the structural error. Under Assumption 4,
identification hinges on the support of the joint distribution of $n_{g_{1}}$
and $n_{g_{2}}$. Since the reduced form parameters are

$\displaystyle\mathbb{E}[\pi(n_{g_{0}})|n_{g_{1}}=n_{1},n_{g_{2}}=n_{2},\mathbf{x}_{g}=\mathbf{x}]$
$\displaystyle=\psi\pi(n_{1})+(1-\psi)\pi(n_{2})$ (4.5)

the first part of the identification argument identifies $\psi\pi(n_{1})$ and
$(1-\psi)\pi(n_{2})$ for all $(n_{1},n_{2})$ in a subset of the support of
$(n_{g_{1}},n_{g_{2}})$ using well known arguments for bi-partite fixed
effects models (e.g. (Abowd et al., 1999)). In particular, we use the notion
of the ‘connected-set’, which in our context corresponds to restrictions on
the bi-partite graph representing the support of $(n_{g_{1}},n_{g_{2}})$. We
thus use the following definition.

  * Definition

For two discrete random variables $X$ and $Y$ with supports $\mathcal{X}$ and
$\mathcal{Y}$, the bi-partite graph $\mathcal{G}(X,Y)$ has
$|\mathcal{X}|+|\mathcal{Y}|$ vertices, with each vertex corresponding to a
support point of either $X$ or $Y$. For two vertices, one corresponding to the
support point $x\in\mathcal{X}$ and another to the support point
$y\in\mathcal{Y}$, an undirected edge exists if $\mathbb{P}[X=x,Y=y]>0$ and
does not exist otherwise. There are no further edges in $\mathcal{G}(X,Y)$.

We now present the main identification result.

######  Theorem 1.

$(\gamma,\beta,\delta)$ are point identified if $\gamma\beta+\delta\neq 0$ and
any one of the following holds

  1. 1.

Assumption 2 holds and the support of the distribution of $n_{g_{0}}$ has at
least three elements (Davezies et al., 2009).

  2. 2.

Assumption 3 holds and the support of the distribution of $n_{g_{0}}$ is
bounded and has at least three elements.

  3. 3.

Assumption 4 holds, the support of the distribution of $n_{g_{1}}$ has at
least three elements, the support of the distribution of $n_{g_{1}}$ has at
least three elements and either exists a connected component of
$\mathcal{G}(n_{g_{1}},n_{g_{2}})$ with at least 3 vertices corresponding to
the support of $n_{g_{1}}$ or there exists a connected component of
$\mathcal{G}(n_{g_{1}},n_{g_{2}})$ with at least 3 vertices corresponding to
the support of $n_{g_{2}}$.

Theorem 1 shows that the structural parameters are identifiable if
$\gamma\beta+\delta\neq 0$ and the distribution of group sizes has sufficient
support. The former is a well known necessary identification condition in
models with correctly specified groups and both endogenous and contextual
effects (see for example Bramoullé et al. (2009) or Rose (2017)), which is
generically satisfied over the parameter space. If it is violated then the
endogenous and contextual effects exactly offset one another such that the
reduced form effect is $\pi(n)=\gamma$, which does not vary with $n$, and
hence no amount of variation in group sizes can identify $\beta$ and $\delta$.
All three results in Theorem 1 also require that the support of the
distribution of $n_{g_{0}}$ has at least three elements. This is also
necessary condition for identification when the groups are correctly specified
and there are endogenous, contextual and correlated effects (Lee, 2007;
Davezies et al., 2009; Bramoullé et al., 2009), hence it is also necessary if
the groups are (possibly) miss-specified.

Result 2 also uses the mild assumption that the distribution of $n_{g_{0}}$ be
bounded, which is needed to identify $\rho$ and the distribution of
$n_{g_{0}}|\mathbf{x}_{g}$ from the distribution of $n_{g}|n_{g}\geq
1,\mathbf{x}_{g}$ as a solution of an eigensystem.171717The lower bound on the
support of $n_{g_{0}}$ is 2 by assumption, and the upper bound need not be
known since it is identifiable from the distribution of $n_{g}|n_{g}\geq 1$.
This assumption is also used, for example, by Boucher and Houndetoungan
(2020). Result 3 uses a mild condition on the joint support of the candidate
group sizes $(n_{g_{1}},n_{g_{2}})$. It is satisfied, for example, whenever
there exists $n$ in the support of $n_{g_{1}}$ such that the support of
$n_{g_{2}}|n_{g_{1}}=n$ has at least three elements or the converse, reversing
the roles of $n_{g_{1}}$ and $n_{g_{2}}$, holds.

## 5\. Estimation

If the group sizes are known (Assumption 2), we estimate $\gamma,\delta,\beta$
by non-linear least squares based on

$\mathbb{E}[\overline{y}_{i}|n_{g_{0}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{0}})$

for $i\in\mathcal{S}$. Similarly, for group uncertainty (Assumption 4), we
apply non-linear least squares to based on

$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}[\psi\pi(n_{g_{1}})+(1-\psi)\pi(n_{g_{2}})]$

for $i\in\mathcal{S}$. If the researcher has a random sample of individuals
(Assumption 3), we apply maximum likelihood for $\rho$ and non-linear least
squares for $\gamma,\delta,\beta$. For maximum likelihood estimation of
$\rho$, we use the sample of $G$ specified group sizes with known distribution

$\displaystyle\mathbb{P}[n_{g}=n|n_{g}\geq
1]=\frac{\sum_{m=n}^{\overline{n}}\mathbb{P}[n_{g_{0}}=m]{m\choose
n}\rho^{n}(1-\rho)^{1-n}}{\sum_{l=1}^{\overline{n}}\sum_{m=l}^{\overline{n}}\mathbb{P}[n_{g_{0}}=m]{m\choose
l}\rho^{l}(1-\rho)^{1-l}}$ (5.1)

for $n=1,...,\overline{n}$, where $\overline{n}<\infty$ is the largest element
of the support of $n_{g_{0}}$.181818If $\overline{n}$ is unknown, it can be
consistently estimated using the largest observed group size. Thus, since
$\mathbb{P}[n_{g_{0}}=1]=0$ by assumption and
$\mathbb{P}[n_{g_{0}}=\overline{n}]=1-\sum_{m=2}^{\overline{n}-1}\mathbb{P}[n_{g_{0}}=m]$
we jointly estimate the $\overline{n}-1$ parameters
$\rho,\mathbb{P}[n_{g_{0}}=2],...,\mathbb{P}[n_{g_{0}}=\overline{n}-1]$. To
estimate $\gamma,\delta,\beta$, we apply non-linear least squares to

$\displaystyle\mathbb{E}[\overline{y}_{i}|n_{g},\mathbf{x}_{g}]=\overline{x}_{i}\sum_{n=2}^{\overline{n}}\mathbb{P}[n_{g_{0}}=n|n_{g}]\pi(n)$
(5.2)

for $i\in\mathcal{S}$, where $\mathbb{P}[n_{g_{0}}=n|n_{g}]$ is a known
function of the parameters $\rho$ and
$\mathbb{P}[n_{g_{0}}=2],...,\mathbb{P}[n_{g_{0}}=\overline{n}-1]$, which we
do not make explicit for simplicity of exposition.191919To control the number
of paramters to be estimated we make the additional restriction
$\mathbb{P}[n_{g_{0}}=n|n_{g},\mathbf{x}_{g}]=\mathbb{P}[n_{g_{0}}=n|n_{g}]$
for $n=1,..,\overline{n}$ to derive (5.2). This contrasts with our
identification results, for which this additional restriction is not made.

In practice we do not sequentially estimate the parameters by maximum
likelihood and then by non-linear least squares. Instead, we estimate the
parameters jointly by GMM based on the non-linear least squares moment
condition and equality to zero of the expectation of the score of the log-
likelihood function based on (5.1). This framework allows the researcher to
include additional information when it is available. For example, if the
researcher knows which observed groups have missing members (i.e. they observe
an indicator for $n_{g}=n_{g_{0}}$), they can also use
$\mathbb{P}[n_{g}=n_{g_{0}}|n_{g}]=\rho^{n_{g}}$. In the same way, it is also
straightforward to make a parametric assumption on the distribution of
$n_{g_{0}}$, which may be advantagenous if $\overline{n}$ is large.

## 6\. Monte-Carlo

We tailor the design to the well known data on roommates at Dartmouth college
studied by Sacerdote (2001), Glaeser et al. (2003) and Angrist (2014), among
others. The original data comprise 1589 freshman at Dartmouth college who were
randomly assigned to dorm-rooms. Fifty three percent of dorm-rooms were
doubles, 44 percent were triples and the remaining 3 percent were quads. The
first part of our Monte-Carlo experiment considers what would have happened
had only a random sample of these students been available. The second part
considers the issue of the definition of the relevant peer group, which could
be the room or the floor (Sacerdote, 2001; Glaeser et al., 2003; Angrist,
2014).

### 6.1. Missing data

We now suppose that only a random sample from the Dartmouth data were
available. The original Dartmouth data comprise a complete sample of freshman
(i.e. $\rho=1$). Our design varies $\rho\in\\{0.3,0.5,0.7,0.9,1\\}$. For
$\rho=1$ we set the sampling probability $p_{i}=1$ for all
$i\in\\{1,...,N_{0}\\}$ and for $\rho<1$ we set $p_{i}=\rho+q_{i}$, where
$q_{i}\overset{i.i.d.}{\sim}\text{Uniform}(-0.1,0.1)$. We set $G_{0}$ to be
the nearest integer to $M/(\rho\mathbb{E}[n_{g_{0}}])$, where $M$ is discussed
below. For each dataset, we draw $G_{0}$ dorm-rooms from the distribution of
dorm-room sizes, which we take to be $n_{g_{0}}-2\sim{\rm Binomial}(2,0.25)$
so as to match the sample mean and support of the Dartmouth data.202020In this
design, in expectation, 56.25% of rooms are doubles, 37.5% are triples and
6.25% are quads. We use a parametric distribution so as to evalute the
performance of the estimator both with and without a parametric restriction.
We use the Binomial distribution because it is bounded, and hence satisfies
the requirements of the identification in Theorem 1.

Since we draw a fixed number of dorm rooms, each with a random size, the
number of students $N_{0}$ varies from one dataset to the next. We then draw
an independent random sample of $N$ students, which includes each of the
$N_{0}$ students independently with probability $\rho$. Hence $N$ also varies
form one dataset to the next. Allowing $G_{0}$ to depend on $\rho$ as above
implies that the size of the observed sample is $N\approx M$ no matter the
value of $\rho$. Setting $M=1600$, our design matches the sample size of the
Dartmouth data. We thus consider the performance of our method had the
Dartmouth data been obtained from a random sample of freshman, rather than all
freshman, holding the number of observations constant as we vary $\rho$. We
also consider $M=8000$ to study the performance of the method under different
sample sizes.

Sacerdote (2001) exploits random assignment of freshman to dorms in the
Dartmouth data to identify peer effects in educational attainment, measured by
freshman GPA. It is argued that endogenous effects are difficult to identify
due to the reflection problem, though in principle the variation in dorm-room
sizes could be used to achieve this because there are three room-sizes (Lee,
2007; Davezies et al., 2009; Bramoullé et al., 2009). Sacerdote (2001) focuses
instead on credibly identifying the reduced form effect of average room-mate
high school attainment (Table 3, column 5) on freshman GPA. The estimated peer
effect is positive and statistically significant, though smaller in magnitude
than own high-school attainment. This reduced form regression has
$R^{2}=0.19$.

To match this setting as well as possible, in our design we set the own effect
to be $\gamma=1$, and the endogenous effect to be $\beta=0$, hence the reduced
form effect of room-mate’s high-school attainment is equal to the contextual
effect, which we set to be $\delta=0.5$. We set
$x_{i}\overset{i.i.d.}{\sim}\mathcal{N}(0,1)$,
$\epsilon_{i}\overset{i.i.d.}{\sim}\mathcal{N}(0,\sigma^{2})$ and
$\alpha_{g_{0}}\overset{i.i.d.}{\sim}\mathcal{N}(1,\sigma^{2})$ and choose
$\sigma^{2}=2(\gamma^{2}+\delta^{2}/\mathbb{E}[n_{g_{0}}-1])$. For the
expected dorm-room size, this choice corresponds to fraction 0.8 of
$\mathbb{V}[y_{i}]$ being due to $\alpha_{g_{0}(i)}+\epsilon_{i}$ and 0.2
being due to $x_{i}\gamma+(1-n_{g_{0}})^{-1}\sum_{j:g_{0}(j)=g_{0}(i),j\neq
i}x_{j}$, hence a population $R^{2}$ of 0.2 in a reduced form regression
without dorm nor dorm-room fixed effects, as estimated by Sacerdote (2001).
The specification that we estimate differs from that of Sacerdote (2001)
because we include dorm-room fixed effects, which are necessary so as to allow
for miss-specification.

####  6.1.1 Results

Table 1 presents the results. We consider four estimators; one for which the
observed groups are treated as if they are the groups (‘Miss-specified’), one
for which the group size is known (‘Known’), one for which the group size is
unknown but Assumption 3 holds (‘Unknown’) and a modification of Unknown under
which the parametric assumption $n_{g_{0}}-2\sim{\rm Binomial}(2,\omega)$ is
additionally imposed (‘Unknown-P’). The Unknown and Unknown-P specifications
take the upper bound on the support of the group sizes to be known as
$\overline{n}=4$, which, in the context of the Dartmouth data implies that the
researcher knows that there are no rooms of more than four students. The Miss-
specified, Unknown and Unknown-P estimators only use the outcome, exogenous
characteristic and group indicator for the $N$ sampled individuals. The Known
estimator additionally uses knowledge of the group size.

Beginning with the contextual effects only specifications ($\beta=0$ is
imposed) and $\rho=1$, we see that the results are very similar (identical to
three decimal places) for all four estimators. This suggests that there is
little cost in applying our method when there are no missing data. As $\rho$
decreases, the distribution of the Miss-specified estimator $\delta$ and
$\gamma$ both shift closer to zero. For $\rho\leq 0.5$, the bias becomes
sizeable. In contrast, the other estimators remain approximately unbiased for
all values of $\rho$, though their RMSE increases as $\rho$ decreases. This is
likely because, as $\rho$ decreases the number of groups with only one
observed member increases, and these groups provide no information on the
parameters due to the within-observed-group transformation.

The Known estimator performs at least as well as the Unknown estimator in
terms of RMSE. This is unsurprising since it also requires more information in
order to be implemented. For $\rho=0.9$ the difference in RMSE is small (0.086
vs 0.099 for $\delta$ when $N\approx M=8000$), though the gap widens as $\rho$
decreases. This is likely because, as $\rho$ decreases, there is less
variation in the distribution of observed group sizes, and hence, for a given
observed group size, more uncertainty as to the group size. The additional
parametric assumption on the group size distribution makes little difference
when $\rho\geq 0.7$, but when $\rho\leq 0.5$ the gap in RMSE between the
Unknown and Unknown-P estimators becomes non-neglible.

Moving on to the specifications with contextual and endogenous effects, it is
clear that, though identified due to there being at least three group sizes,
there is insufficient group size variation in this design to reliably separate
the two. In particular, the RMSE on $\beta$ is large and all estimators of
$\beta$ are biased upwards, whereas estimators of $\delta$ tend to be biased
downwards. Nevertheless, the qualiative conclusions regarding the relative
performance of the four estimators are the same as for the specifications with
contextual effects only.

### 6.2. Group uncertainty

Sacerdote (2001), Glaeser et al. (2003) and Angrist (2014) all discuss the
appropriate definition of the peer group, which may either be the room, the
floor or the entire dorm. It is also possible that the appropriate peer group
for academic attainment may be different to that of social outcomes such as
fraternity membership (Sacerdote, 2001). For this reason, these papers repeat
their empirical analysis for interactions at the room, floor and dorm levels.

We focus here on the distinction between the room and the floor. To match the
Dartmouth data, we first draw the rooms as described above. We then draw
$f_{1}\in\\{1,2,...,5\\}$ uniformly, and take rooms $\\{1,2,...,f_{1}\\}$ to
comprise the first floor. We then draw another integer
$f_{2}\in\\{1,2,...,5\\}$ and take rooms $\\{f_{1}+1,...,f_{1}+f_{2}\\}$ to
comprise the second floor. We proceed in this way until every room has been
allocated to a floor. In the Dartmouth data, the sample mean number of
students per floor is 8. This data generating process yields an expected
number of students per floor of 7.5. All other aspects of the data generating
process remain unchanged and we vary $\psi\in\\{0.2,0.4,0.6,0.8\\}$.

####  6.2.1 Results

Table 2 presents the results. We consider four estimators; one in which the
room is specified as the peer group (‘Room’), one in which the floor is
specified as the peer group (‘Floor’), one in which the researcher knows
whether the group is the room or the floor (‘Known’) and one in which the
group is unknown but Assumption 4 is assumed to hold (‘Uncertain’).

For brevity, the discussion focuses on the contextual effects only
specifications. Specifying the group to be the room leads to downwards bias of
$\delta$. The bias and RMSE grow as $\psi$ decreases. This is because the
proportion of incorrectly specified groups grows as $\psi$ decreases. For the
same reason the RMSE from specifying the group to be the dorm decreases as
$\rho$ decreases. The results suggest that the estimator based on specifying
the floor is unbiased. This is coincidental. It just so happens that in this
design the three sources of bias (the terms $t_{1},t_{2}$ and $t_{3}$
discussed in Section 3) offset one another. To demonstrate this, Table 3
modifies the design by setting
$\alpha_{g_{0}}\overset{i.i.d.}{\sim}\mathcal{N}(n_{g_{0}}^{-}1\sum_{j:j\in
g_{0}}x_{j},\sigma^{2})$ instead of
$\alpha_{g_{0}}\overset{i.i.d.}{\sim}\mathcal{N}(1,\sigma^{2})$. This has no
impact on the estimators Room, Known and Uncertain because all are based on
room fixed effects.212121For these estimators the results are numerically
identical to results reported in Table 2 because identical seeds were used to
draw the data for each replication. However it introduces substantial bias for
the Floor estimator which is based on floor fixed effects.

Comparing the Known estimator to the Uncertain estimator, as expected the RMSE
of the latter is larger since it requires less information to be implemented.
Nevertheless, depending on the sample size and value of $\psi$, the estimator
based on Assumption 4 has small bias and sufficiently small RMSE so as to be
distinguishable from zero. Considering now the specifications with endogenous
and contextual effects, though not well identified, the main qualiative
conclusions regarding the relative performance of the four estimators are the
same as for the specifications with contextual effects only. Notice also that
once endogenous effects are introduced the estimator based on specifying the
floor often has larger bias and RMSE than that based on Assumption 4. This is
particularly true for $\delta$.

## 7\. Discussion

Our identification and Monte-Carlo results demonstrate that it is possible to
conduct empirical analysis of peer effects despite missing data and group
uncertainty. Regarding missing data, we require only that the researcher has
access to a sample of individuals with outcome, exogenous characteristic and
group membership data. We propose a method which does not require information
on group size, nor on individuals which are not sampled, nor on whether such
individuals even exist. In principle, and subject to the limitations discussed
below, this opens up the possibility of peer effects studies based on widely
available individual level survey data (provided that it contains group
identifiers). Two examples of such data are the PISA sample of students, for
which school-grade identifiers are available, and household surveys with local
area identifiers (e.g. Census tracts or suburb identifiers). We also show that
peer effects can be identified under group uncertainty. Our results could also
be extended to incorporate both missing individuals and group uncertainty
simultaneously.

A limitation of this paper is that our results are specific to the group
interactions model we consider. Under more general network structures (e.g.
social networks), it is not the case the specified group fixed effects reduce
the problem of miss-specification to the problem of inferring the group size.
This is because the missing individuals do not all exert the same peer effect
on the sampled individuals.

The specificity of our results to the group interactions model also has
implications regarding applicability to survey data as discussed above. The
reason for this is that, even if the groups are correctly specified, there is
a weak identification problem when group sizes are large (Lee, 2007). Hence in
order to use our approach with survey data, the researcher would need to be
sure that groups are small. For example, neighbourhood effects based on
census-tract identifiers would likely be weakly identified, though the
prospects would be better for student survey data with classroom/grade
identifiers such as PISA. Future work may seek to resolve these issues, either
through appealing to additional data and/or through specification of a network
formation model. Another concern using survey data, particularly for studies
of neighbourhood effects, might be that there is selection of individuals into
groups. Though our focus is not on network endogeneity, our specification of
the correlated effect allows for arbitrary dependence of group members’
characteristics and the number of individuals in the group with unobserved
group heterogeneity, which considerably mitigates this issue.

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## Appendix

###### Proof.

of Theorem 1

The first result was established by (Davezies et al., 2009) and so we do not
repeat their proof here.

To establish the second result, first note that Assumption 3 implies
$\mathbb{E}[\overline{u}_{i}|n_{g},\mathbf{x}_{g}]=0$ for all
$i\in\mathcal{S}$. Let us now fix $\mathbf{x}_{g}=\mathbf{x}$ for some
$\mathbf{x}$ in its support. We begin by identifying $\rho$ and the
distribution of $n_{g_{0}}|\mathbf{x}_{g}=\mathbf{x}$ using the observable
distribution of $n_{g}|n_{g}\geq 1,\mathbf{x}_{g}=\mathbf{x}$ and the fact
that Assumption 3 implies

$n_{g}|n_{g_{0}},\mathbf{x}_{g}=\mathbf{x}\sim\text{Binomial}(n_{g_{0}},\rho)$

By the boundedness assumption, there exists
$\overline{n}<\infty:\mathbb{P}[n_{g_{0}}=\overline{n}]>0,\mathbb{P}[n_{g_{0}}\leq\overline{n}]=1$.
Note that, because $\rho>0$, $\overline{n}$ is identifiable as the integer $n$
such that $\mathbb{P}[n_{g}=n|n_{g}\geq 1]>0$. This implies that there exists
identifiable
$\overline{n}(\mathbf{x})<\infty:\mathbb{P}[n_{g_{0}}=\overline{n}(\mathbf{x})|\mathbf{x}_{g}=\mathbf{x}]>0,\mathbb{P}[n_{g_{0}}(\mathbf{x})\leq\overline{n}(\mathbf{x})|\mathbf{x}_{g}=\mathbf{x}]=1$.

Let $\mathbf{p}_{n}(\mathbf{x})=\mathbb{P}[n_{g}=n|n_{g}\geq
1,\mathbf{x}_{g}=\mathbf{x}]$,
$\mathbf{q}_{m}(\mathbf{x})=\mathbb{P}[n_{g_{0}}=m|\mathbf{x}_{g}=\mathbf{x}]$.
Under Assumption 3,
$\mathbf{p}(\mathbf{x})=(\mathbf{p}_{n}(\mathbf{x}))_{n=1,...,\overline{n}(\mathbf{x})}$
and
$\mathbf{q}(\mathbf{x})=(\mathbf{q}_{m}(\mathbf{x}))_{m=1,...,\overline{n}(\mathbf{x})}$
verify

$\displaystyle\mathbf{p}(\mathbf{x})=\frac{\mathbf{A}(\mathbf{x},\rho)\mathbf{q}(\mathbf{x})}{\iota_{\overline{n}(\mathbf{x})}^{\prime}\mathbf{A}(\mathbf{x},\rho)\mathbf{q}(\mathbf{x})}$
(7.1)

where
$\mathbf{A}(\mathbf{x},\rho)=(\mathbf{A}_{ij}(\mathbf{x},\rho))_{i=1,...,\overline{n}(\mathbf{x}),j=1,...,\overline{n}(\mathbf{x})}$,

$\mathbf{A}_{ij}(\mathbf{x},\rho)=\begin{cases}{j\choose
i}\rho^{i}(1-\rho)^{j-i}&i\leq j\\\ 0&\text{otherwise}\end{cases}$

and $\iota_{\overline{n}(\mathbf{x})}$ is the $\overline{n}(\mathbf{x})$
dimensional vector of ones. Rearranging (7.1) yields

$\displaystyle(\mathbf{p}(\mathbf{x})\iota_{\overline{n}(\mathbf{x})}^{\prime}\mathbf{A}(\mathbf{x},\rho)-\mathbf{A}(\mathbf{x},\rho))\mathbf{q}(\mathbf{x})=\mathbf{0}$
(7.2)

The inverse of $\mathbf{A}(\mathbf{x},\rho)$ exists since
$|\mathbf{A}(\mathbf{x},\rho)|=\rho^{T(\overline{n}(\mathbf{x}))}$ where
$T(\overline{n}(\mathbf{x}))$ is the $\overline{n}(\mathbf{x})^{th}$
triangular number and $\rho>0$, hence

$\displaystyle(\mathbf{A}(\mathbf{x},\rho)^{-1}\mathbf{p}(\mathbf{x})\iota_{\overline{n}(\mathbf{x})}^{\prime}\mathbf{A}(\mathbf{x},\rho)-\mathbf{I}_{\overline{n}(\mathbf{x})})\mathbf{q}(\mathbf{x})=\mathbf{0}$
(7.3)

Denote
$\mathbf{B}(\mathbf{x},\rho)=\mathbf{A}(\mathbf{x},\rho)^{-1}\mathbf{p}(\mathbf{x})\iota_{\overline{n}(\mathbf{x})}^{\prime}\mathbf{A}(\mathbf{x},\rho)$.
Then clearly the rank of $\mathbf{B}(\mathbf{x},\rho)$ cannot exceed 1.
Moreover, since $\rho>0$ we have $\mathbf{B}(\mathbf{x},\rho)\neq\mathbf{0}$,
hence $\mathbf{B}(\mathbf{x},\rho)$ has rank 1. From (7.3), we have that 1 is
an eigenvalue of $\mathbf{B}(\mathbf{x},\rho)$. Hence
$\mathbf{B}(\mathbf{x},\rho)$ has eigenvalues 1 of multiplicity 1 and 0 of
multiplicity $\overline{n}(\mathbf{x})-1$ and $\mathbf{q}(\mathbf{x})$ is an
eigenvector associated with the eigenvalue 1. Let
$\mathbf{v}(\mathbf{x},\rho)$ be an arbitrary eigenvector associated with the
eigenvalue 1. Then, since $\mathbf{q}(\mathbf{x})$ is a vector of
probabilities summing to one, we have
$\mathbf{q}(\mathbf{x})=\mathbf{v}(\mathbf{x},\rho)/(\iota_{\overline{n}(\mathbf{x})}^{\prime}\mathbf{v}(\mathbf{x},\rho))$.
Hence, given $\rho$ we can identify $\mathbf{q}(\mathbf{x})$.

To identify $\rho$, we use the restriction $\mathbf{q}_{1}(\mathbf{x})=0$,
which is known because $n_{g_{0}}\geq 2$. Since $\mathbf{B}(\mathbf{x},\rho)$
is a smooth function of $\rho$ for $\rho\in(0,1]$ and $1$ is always a simple
eigenvalue of $\mathbf{B}(\mathbf{x},\rho)$, the eigvenvector
$\mathbf{v}(\rho)$ is a smooth function of $\rho$ for $\rho\in(0,1]$, hence so
is $\mathbf{q}_{1}(\mathbf{x})$. Moreover, it is clear that
$\mathbf{q}_{1}(\mathbf{x})$ is strictly decreasing in $\rho$ if
$\mathbf{p}(\mathbf{x})$ is fixed. Hence the restriction
$\mathbf{q}_{1}(\mathbf{x})=0$ identifies $\rho$. By these arguments, $\rho$
and $\mathbf{q}(\mathbf{x})$ are identified.

Provided that there is variation in the elements of $\mathbf{x}$, the
conditional moment

$\displaystyle\mathbb{E}[\overline{y}_{i}|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]$
$\displaystyle=\overline{x}_{i}\mathbb{E}[\pi(n_{g_{0}})|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]$
(7.4)

identifies $\mathbb{E}[\pi(n_{g_{0}})|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}]$ for
all such $(\mathbf{x},n\geq 2)$ in the support of $(\mathbf{x}_{g},n_{g})$ (it
is not identified when $n=1$ nor when $\mathbf{x}=c\iota_{n}$ for some
constant $c$ because in these cases $\overline{y}_{i}=\overline{x}_{i}=0$).
The rest of the proof restricts attention to the case in which there is
variation in the elements of $\mathbf{x}$.

Define
$\mathbf{r}_{n}(\mathbf{x})=\mathbb{P}[n_{g}=n|\mathbf{x}_{g}=\mathbf{x}]$ and
$\mathbf{r}(\mathbf{x})=(\mathbf{r}_{n}(\mathbf{x}))_{n=1,...,\overline{n}(\mathbf{x})}$.
So

$\mathbf{r}(\mathbf{x})=(\iota_{\overline{n}(\mathbf{x})}^{\prime}\mathbf{A}(\mathbf{x},\rho)\mathbf{q}(\mathbf{x}))\mathbf{p}(\mathbf{x})$

is also identified. Now denote
$\mathcal{Q}(\mathbf{x})=\\{m\in\\{2,...,\overline{n}(\mathbf{x})\\}:\mathbf{q}_{m}(\mathbf{x})>0\\}$,
the largest element of which is $\overline{n}(\mathbf{x})$. Moreover, since
$\rho>0$, the support of $\mathbf{r}(\mathbf{x})$ is
$\mathcal{R}(\mathbf{x})=\\{1,...,\overline{n}(\mathbf{x})\\}$ if
$\rho\in(0,1)$ and $\mathcal{R}(\mathbf{x})=\mathcal{Q}(\mathbf{x})$ if
$\rho=1$. Let
$\widehat{\boldsymbol{\pi}}(\mathbf{x})=(\pi(m))_{m\in\mathcal{Q}(\mathbf{x})}$,
$\widehat{\mathbf{q}}(\mathbf{x})=(\mathbf{q}_{m}(\mathbf{x}))_{m\in\mathcal{Q}(\mathbf{x})}$,
$\widehat{\mathbf{r}}(\mathbf{x})=(\mathbf{r}_{n}(\mathbf{x}))_{n\in\mathcal{R}(\mathbf{x})\backslash
1}$ and $\widehat{\mathbf{A}}(\mathbf{x},\rho)$ be the sub-matrix of
$\mathbf{A}(\mathbf{x},\rho)$ formed from rows
$\mathcal{R}(\mathbf{x})\backslash 1$ and columns $\mathcal{Q}(\mathbf{x})$.
Then we have the equations

$\displaystyle(\mathbb{E}[\pi(n_{g_{0}})|n_{g}=n,\mathbf{x}_{g}=\mathbf{x}])_{n\in\mathcal{R}(\mathbf{x})\backslash
1}={\rm
diag}(\widehat{\mathbf{r}}(\mathbf{x}))^{-1}\widehat{\mathbf{A}}(\mathbf{x},\rho){\rm
diag}(\widehat{\mathbf{q}}(\mathbf{x}))\widehat{\boldsymbol{\pi}}(\mathbf{x}),$
(7.5)

for which the left hand side is identified. Since
$\mathbf{A}(\mathbf{x},\rho)$ has full rank,
$\widehat{\mathbf{A}}(\mathbf{x},\rho)$ has rank $|\mathcal{Q}(\mathbf{x})|$.
This is because, when $\rho=1$, we have
$\mathcal{R}(\mathbf{x})=\mathcal{Q}(\mathbf{x})$ and
$\widehat{\mathbf{A}}(\mathbf{x},\rho)=\mathbf{I}_{|\mathcal{Q}(\mathbf{x})|}$.
When $\rho\in(0,1)$, $\mathcal{R}=\\{1,...,\overline{n}(\mathbf{x})\\}$, hence
$\widehat{\mathbf{A}}(\mathbf{x},\rho)$ comprises rows
$\\{2,...,\overline{n}(\mathbf{x})\\}$ and a subset of columns
$\\{2,...,\overline{n}(\mathbf{x})\\}$ of $\mathbf{A}(\mathbf{x},\rho)$. The
sub-matrix of $\mathbf{A}(\mathbf{x},\rho)$ with rows
$\\{2,...,\overline{n}(\mathbf{x})\\}$ and columns
$\\{2,...,\overline{n}(\mathbf{x})\\}$ has determinant
$\rho^{T(\overline{n}(\mathbf{x})-1)}>0$, which implies that
$\widehat{\mathbf{A}}(\mathbf{x},\rho)$ has rank $|\mathcal{Q}(\mathbf{x})|$.
Moreover, ${\rm diag}(\widehat{\mathbf{q}}(\mathbf{x}))$ is invertible by
construction, hence (7.5) identifies $\widehat{\boldsymbol{\pi}}(\mathbf{x})$
for all $\mathbf{x}$ in the support of $\mathbf{x}_{g}$.

By assumption there are at least three different group sizes in the support of
$n_{g_{0}}$, so $|\cup_{\mathbf{x}}\mathcal{Q}(\mathbf{x})|\geq 3$. This
implies that $\pi(n_{1}),\pi(n_{2}),\pi(n_{3})$ are identified for at least
three different group sizes. We conclude by applying the well known result
established by Lee (2007), Davezies et al. (2009) and Bramoullé et al. (2009)
that $(\gamma,\delta,\beta)$ are identified when there are at least three
different group sizes and $\gamma\beta+\delta\neq 0$. Notice also that one can
replace $\rho$ with $\rho(\mathbf{x})$ everywhere in the above proof to allow
for a type of MAR.

To establish the third result, again fix $\mathbf{x}_{g}=\mathbf{x}$ where
$\mathbf{x}$ is some point in the support of $\mathbf{x}_{g}$. Provided that
there is variation in the elements of $\mathbf{x}$, the conditional moment

$\displaystyle\mathbb{E}[\overline{y}_{i}|n_{g_{1}}=n_{1},n_{g_{2}}=n_{2},\mathbf{x}_{g}=\mathbf{x}]$
$\displaystyle=\overline{x}_{i}\mathbb{E}[\pi(n_{g_{0}})|n_{g_{1}}=n_{1},n_{g_{2}}=n_{2},\mathbf{x}_{g}=\mathbf{x}]$
(7.6)

identifies
$\mathbb{E}[\pi(n_{g_{0}})|n_{g_{1}}=n_{1},n_{g_{2}}=n_{2},\mathbf{x}_{g}=\mathbf{x}]$
for all such $(\mathbf{x},n_{1}\geq 2,n_{2})$ in the support of
$(\mathbf{x}_{g},n_{g_{1}},n_{g_{2}})$ (it is not identified when $n_{1}=1$
nor when $\mathbf{x}=c\iota_{n_{1}}$ for some constant $c$ because in these
cases $\overline{y}_{i}=\overline{x}_{i}=0$). We obtain

$\displaystyle\mathbb{E}[\pi(n_{g_{0}})|n_{g_{1}}=n_{1},n_{g_{2}}=n_{2},\mathbf{x}_{g}=\mathbf{x}]=\psi\pi(n_{1})+(1-\psi)\pi(n_{2})$
(7.7)

We now consider the cases $\psi=0$, $\psi=1$ and $\psi\in(0,1)$ separately. If
$\psi=0$ then $(\gamma,\delta,\beta)$ are identified because
$\gamma\beta+\delta\neq 0$ and $n_{g_{2}}$ has at least three points in the
support of its distribution. If $\psi=1$ then $(\gamma,\delta,\beta)$ are
identified because $\gamma\beta+\delta\neq 0$ and $n_{g_{1}}$ has at least
three points in the support of its distribution. The rest of the proof now
considers $\psi\in(0,1)$. By standard arguments for two-way bi-partite fixed
effects models (e.g. Abowd et al. (1999)), equation (7.7) identifies
$\mu(n_{1})=\psi\pi(n_{1})+c$ and $\mu(n_{2})=(1-\psi)\pi(n_{2})-c$ for every
pair $(n_{1},n_{2})$ corresponding to two vertices in a connected component of
$\mathcal{G}(n_{g_{1}},n_{g_{2}})$, where $c$ is an unknown constant.

Consider a connected component of $\mathcal{G}(n_{g_{1}},n_{g_{2}})$ with at
least three vertices corresponding to the support of $n_{g_{1}}$. If such a
component does not exist we can instead consider a connected component of
$\mathcal{G}(n_{g_{1}},n_{g_{2}})$ with at least three vertices corresponding
to the support of $n_{g_{2}}$. One of these two exists by assumption, so
without loss of generality we present the first. Let the vertices in this
connected component respectively correspond to $\mathcal{N}_{1}$ and
$\mathcal{N}_{2}$, where $\mathcal{N}_{1}$ is a subset of the support of
$n_{g_{1}}$ and $\mathcal{N}_{2}$ is a subset of the support of $n_{g_{2}}$.
Then it must be that $|\mathcal{N}_{1}|\geq 3$ and $|\mathcal{N}_{2}|\geq 1$
(the latter is because $\mathcal{G}(n_{g_{1}},n_{g_{2}})$ is bi-partite). So
there exist $2\leq n_{1,1}<n_{1,2}<n_{1,3}$ and $2\leq n_{2}$ such that
$\\{n_{1,1},n_{1,2},n_{1,3}\\}\subseteq\mathcal{N}_{1}$ and
$\\{n_{2}\\}\subseteq\mathcal{N}_{2}$. Moreover, $\mu(n)$ is identified for
$n\in\\{n_{1,1},n_{1,2},n_{1,3},n_{2}\\}$. For $i\in\\{1,2,3\\}$, denote

$\displaystyle\Delta_{i}=\mu(n_{1,i})+\mu(n_{2})=\frac{\gamma(n_{2}-1)-\delta}{n_{2}-1+\beta}+\frac{\psi(\gamma\beta+\delta)(n_{1,i}-n_{2})}{(n_{1,i}-1+\beta)(n_{2}-1+\beta)}$
(7.8)

Then after some algebra, for $j\in\\{1,2,3\\}$ we have

$\displaystyle\Delta_{i}-\Delta_{j}=\frac{\psi(\gamma\beta+\delta)(n_{1,i}-n_{1,j})}{(n_{1,i}-1+\beta)(n_{1,j}-1+\beta)}$
(7.9)

Now since $\psi\neq 0$ and $\delta+\beta\gamma\neq 0$,

$\displaystyle\Delta\equiv\frac{\Delta_{1}-\Delta_{2}}{\Delta_{1}-\Delta_{3}}=\frac{(n_{1,1}-n_{1,2})(n_{1,3}-1+\beta)}{(n_{1,1}-n_{1,3})(n_{1,2}-1+\beta)}$
(7.10)

Since $\Delta$ is identified, $\beta$ is identified by (7.10) if
$\Delta(n_{1,1}-n_{1,3})\neq n_{1,1}-n_{1,2}$. Suppose that
$\Delta(n_{1,1}-n_{1,3})=n_{1,1}-n_{1,2}$. Then (7.10) implies
$n_{1,2}=n_{1,3}$, but here we have $n_{1,2}<n_{1,3}$, a contradiction. So
$\beta$ is identified. Since $\beta$ is identified, $\psi(\gamma\beta+\delta)$
is identified from (7.9), hence $\gamma$ and $\delta$ are identified from
(7.8). Finally, $\psi$ is identified because $\psi(\gamma\beta+\delta)$ and
$(\gamma,\delta,\beta)$ are identified and $\gamma\beta+\delta\neq 0$. Notice
also that one can replace $\psi$ with $\psi(\mathbf{x})$ and
$\mathcal{G}(n_{g_{1}},n_{g_{2}})$ with
$\mathcal{G}(n_{g_{1}},n_{g_{2}}|\mathbf{x}_{g}=\mathbf{x})$ everywhere in the
above proof, hence we can also allow for $\psi=\psi(\mathbf{x}_{g})$.

Table 1: Monte Carlo Results: Missing data

Contextual effect only ($\beta=0$ imposed), $N\approx M=8000$
---
|  | Miss-specified | Known | Unknown | Unknown-P
$\rho$ |  | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE
1 | $\delta$ | 0.502 | 0.084 | 0.502 | 0.084 | 0.502 | 0.084 | 0.502 | 0.084
$\gamma$ | 1.002 | 0.06 | 1.002 | 0.06 | 1.002 | 0.06 | 1.002 | 0.06
0.9 | $\delta$ | 0.426 | 0.112 | 0.501 | 0.086 | 0.499 | 0.099 | 0.499 | 0.099
$\gamma$ | 0.971 | 0.07 | 1.002 | 0.061 | 1.001 | 0.07 | 1.001 | 0.07
0.7 | $\delta$ | 0.328 | 0.198 | 0.504 | 0.097 | 0.502 | 0.153 | 0.5 | 0.151
$\gamma$ | 0.931 | 0.106 | 1.002 | 0.067 | 1.001 | 0.101 | 1 | 0.1
0.5 | $\delta$ | 0.264 | 0.266 | 0.498 | 0.112 | 0.518 | 0.257 | 0.504 | 0.238
$\gamma$ | 0.904 | 0.144 | 0.998 | 0.077 | 1.007 | 0.158 | 1 | 0.151
0.3 | $\delta$ | 0.225 | 0.332 | 0.505 | 0.146 | 0.896 | 1.602 | 0.54 | 0.468
$\gamma$ | 0.892 | 0.204 | 1.002 | 0.096 | 1.262 | 2.825 | 1.017 | 0.279
Contextual effect only ($\beta=0$ imposed), $N\approx M=1600$
1 | $\delta$ | 0.498 | 0.184 | 0.498 | 0.184 | 0.498 | 0.184 | 0.498 | 0.184
$\gamma$ | 0.999 | 0.132 | 0.999 | 0.132 | 0.999 | 0.132 | 0.999 | 0.132
0.9 | $\delta$ | 0.425 | 0.201 | 0.498 | 0.188 | 0.5 | 0.22 | 0.499 | 0.219
$\gamma$ | 0.973 | 0.142 | 1.003 | 0.134 | 1.003 | 0.153 | 1.003 | 0.152
0.7 | $\delta$ | 0.325 | 0.277 | 0.502 | 0.217 | 0.511 | 0.347 | 0.501 | 0.332
$\gamma$ | 0.926 | 0.189 | 1 | 0.148 | 1.002 | 0.224 | 0.997 | 0.219
0.5 | $\delta$ | 0.261 | 0.353 | 0.495 | 0.253 | 0.693 | 1.303 | 0.52 | 0.538
$\gamma$ | 0.903 | 0.251 | 0.998 | 0.175 | 1.083 | 0.649 | 1.006 | 0.336
Contextual and endogenous effects, $N\approx M=8000$
1 | $\beta$ | 0.095 | 0.772 | 0.095 | 0.772 | 0.095 | 0.772 | 0.095 | 0.772
$\delta$ | 0.471 | 0.245 | 0.471 | 0.245 | 0.471 | 0.245 | 0.471 | 0.245
$\gamma$ | 1.02 | 0.182 | 1.02 | 0.182 | 1.02 | 0.182 | 1.02 | 0.182
0.9 | $\beta$ | 0.092 | 0.822 | 0.076 | 0.788 | 0.04 | 0.81 | 0.04 | 0.81
$\delta$ | 0.394 | 0.325 | 0.479 | 0.246 | 0.485 | 0.263 | 0.485 | 0.263
$\gamma$ | 0.989 | 0.174 | 1.018 | 0.187 | 1.008 | 0.193 | 1.008 | 0.193
0.7 | $\beta$ | 0.133 | 0.899 | 0.069 | 0.795 | 0.048 | 0.875 | 0.052 | 0.876
$\delta$ | 0.269 | 0.48 | 0.483 | 0.253 | 0.481 | 0.322 | 0.478 | 0.322
$\gamma$ | 0.952 | 0.171 | 1.017 | 0.194 | 1.008 | 0.218 | 1.008 | 0.217
0.5 | $\beta$ | 0.122 | 0.928 | 0.052 | 0.825 | 0.067 | 0.914 | 0.07 | 0.919
$\delta$ | 0.205 | 0.577 | 0.484 | 0.272 | 0.485 | 0.413 | 0.475 | 0.404
$\gamma$ | 0.923 | 0.188 | 1.01 | 0.202 | 1.015 | 0.269 | 1.012 | 0.262
0.3 | $\beta$ | 0.077 | 0.952 | 0.059 | 0.851 | 0.066 | 0.91 | 0.098 | 0.926
$\delta$ | 0.19 | 0.647 | 0.489 | 0.301 | 0.734 | 1.463 | 0.5 | 0.651
$\gamma$ | 0.907 | 0.246 | 1.015 | 0.213 | 1.21 | 2.77 | 1.032 | 0.404
Contextual and endogenous effects, $N\approx M=1600$
1 | $\beta$ | 0.066 | 0.888 | 0.066 | 0.888 | 0.066 | 0.888 | 0.066 | 0.888
$\delta$ | 0.481 | 0.343 | 0.481 | 0.343 | 0.481 | 0.343 | 0.481 | 0.343
$\gamma$ | 1.015 | 0.245 | 1.015 | 0.245 | 1.015 | 0.245 | 1.015 | 0.245
0.9 | $\beta$ | 0.129 | 0.922 | 0.095 | 0.915 | 0.103 | 0.915 | 0.104 | 0.915
$\delta$ | 0.387 | 0.425 | 0.48 | 0.35 | 0.477 | 0.394 | 0.476 | 0.394
$\gamma$ | 1.004 | 0.246 | 1.029 | 0.26 | 1.031 | 0.276 | 1.031 | 0.275
0.7 | $\beta$ | 0.035 | 0.95 | 0.034 | 0.906 | 0.006 | 0.943 | 0.009 | 0.943
$\delta$ | 0.32 | 0.527 | 0.495 | 0.377 | 0.515 | 0.499 | 0.504 | 0.487
$\gamma$ | 0.939 | 0.255 | 1.009 | 0.263 | 1.008 | 0.335 | 1.004 | 0.326
0.5 | $\beta$ | 0.031 | 0.965 | 0.03 | 0.915 | 0.022 | 0.95 | 0.023 | 0.949
$\delta$ | 0.249 | 0.643 | 0.487 | 0.424 | 0.599 | 1.263 | 0.488 | 0.707
$\gamma$ | 0.911 | 0.305 | 1.007 | 0.284 | 1.051 | 0.658 | 1 | 0.432

Notes: Based on 1000 replications and reported to 3 decimal places. ‘Miss-
specified’ refers to the NLS estimator obtained by treating the observed
groups as if they were the groups (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g})$).
‘Known’ is the NLS estimator under Assumption 2 (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{0}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{0}})$).
‘Unknown’ is the GMM estimator under Assumption 3. ‘Unknown-P’ is the GMM
estimator under Assumption 3 imposing the parametric restriction
$n_{g_{0}}-2\sim{\rm Binomial}(2,\omega)$. Unknown and Unknown-P are based on
the upper bound on the support of $n_{g_{0}}$ being known to be
$\overline{n}=4$. Results are presented for $(\beta,\delta,\gamma)$ only.
Their true values are $(0,0.5,1)$. Results for $\rho$ and $\omega$ are
available on request.

Table 2: Monte Carlo Results: Uncertain groups

Contextual effect only ($\beta=0$ imposed), $N=8000$
---
|  | Room | Floor | Known | Uncertain
$\psi$ |  | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE
0.8 | $\delta$ | 0.41 | 0.123 | 0.501 | 0.135 | 0.496 | 0.066 | 0.516 | 0.131
$\gamma$ | 0.992 | 0.062 | 1.001 | 0.034 | 0.999 | 0.04 | 1.002 | 0.062
0.6 | $\delta$ | 0.321 | 0.198 | 0.498 | 0.135 | 0.5 | 0.061 | 0.504 | 0.142
$\gamma$ | 0.983 | 0.064 | 1 | 0.032 | 1 | 0.033 | 1.001 | 0.063
0.4 | $\delta$ | 0.232 | 0.281 | 0.497 | 0.133 | 0.5 | 0.064 | 0.501 | 0.145
$\gamma$ | 0.973 | 0.067 | 1 | 0.031 | 1 | 0.028 | 1.001 | 0.063
0.2 | $\delta$ | 0.144 | 0.366 | 0.499 | 0.129 | 0.497 | 0.08 | 0.503 | 0.145
$\gamma$ | 0.964 | 0.071 | 1 | 0.028 | 1 | 0.026 | 1.004 | 0.058
Contextual effect only ($\beta=0$ imposed), $N=1600$
0.8 | $\delta$ | 0.402 | 0.208 | 0.494 | 0.333 | 0.495 | 0.14 | 0.564 | 0.29
$\gamma$ | 0.983 | 0.132 | 0.997 | 0.077 | 0.997 | 0.088 | 1 | 0.135
0.6 | $\delta$ | 0.313 | 0.263 | 0.503 | 0.327 | 0.493 | 0.137 | 0.538 | 0.31
$\gamma$ | 0.975 | 0.134 | 1.001 | 0.075 | 0.996 | 0.073 | 0.999 | 0.135
0.4 | $\delta$ | 0.226 | 0.331 | 0.505 | 0.319 | 0.492 | 0.146 | 0.516 | 0.332
$\gamma$ | 0.966 | 0.136 | 1 | 0.071 | 0.997 | 0.063 | 0.999 | 0.134
0.2 | $\delta$ | 0.134 | 0.409 | 0.507 | 0.308 | 0.499 | 0.174 | 0.503 | 0.357
$\gamma$ | 0.955 | 0.139 | 1.001 | 0.065 | 0.999 | 0.058 | 1.006 | 0.13
Contextual and endogenous effects, $N=8000$
0.8 | $\beta$ | 0.041 | 0.814 | 0.127 | 0.683 | 0.054 | 0.46 | 0.055 | 0.799
$\delta$ | 0.392 | 0.36 | 0.429 | 0.384 | 0.466 | 0.214 | 0.492 | 0.302
$\gamma$ | 0.998 | 0.165 | 1.007 | 0.051 | 1.001 | 0.064 | 1.01 | 0.167
0.6 | $\beta$ | 0.039 | 0.848 | 0.112 | 0.663 | 0.07 | 0.461 | 0.099 | 0.8
$\delta$ | 0.301 | 0.495 | 0.431 | 0.385 | 0.462 | 0.227 | 0.458 | 0.354
$\gamma$ | 0.989 | 0.142 | 1.004 | 0.049 | 1.004 | 0.049 | 1.013 | 0.143
0.4 | $\beta$ | 0.042 | 0.873 | 0.115 | 0.679 | 0.072 | 0.491 | 0.106 | 0.769
$\delta$ | 0.205 | 0.64 | 0.427 | 0.394 | 0.458 | 0.256 | 0.446 | 0.379
$\gamma$ | 0.978 | 0.119 | 1.004 | 0.048 | 1.003 | 0.043 | 1.01 | 0.117
0.2 | $\beta$ | 0.032 | 0.881 | 0.119 | 0.653 | 0.106 | 0.561 | 0.123 | 0.724
$\delta$ | 0.123 | 0.772 | 0.422 | 0.391 | 0.436 | 0.303 | 0.433 | 0.393
$\gamma$ | 0.969 | 0.099 | 1.004 | 0.044 | 1.004 | 0.042 | 1.011 | 0.093
Contextual and endogenous effects, $N=1600$
0.8 | $\beta$ | 0.038 | 0.913 | 0.067 | 0.841 | 0.097 | 0.709 | 0.019 | 0.896
$\delta$ | 0.393 | 0.449 | 0.465 | 0.61 | 0.444 | 0.343 | 0.559 | 0.428
$\gamma$ | 0.995 | 0.228 | 1.001 | 0.098 | 1.002 | 0.123 | 1.005 | 0.239
0.6 | $\beta$ | 0.024 | 0.927 | 0.088 | 0.833 | 0.065 | 0.697 | 0.051 | 0.9
$\delta$ | 0.311 | 0.568 | 0.458 | 0.604 | 0.455 | 0.358 | 0.522 | 0.485
$\gamma$ | 0.986 | 0.209 | 1.006 | 0.096 | 0.998 | 0.096 | 1.009 | 0.22
0.4 | $\beta$ | 0.019 | 0.94 | 0.096 | 0.833 | 0.107 | 0.716 | 0.045 | 0.888
$\delta$ | 0.223 | 0.7 | 0.457 | 0.579 | 0.435 | 0.386 | 0.493 | 0.55
$\gamma$ | 0.974 | 0.192 | 1.006 | 0.091 | 1.001 | 0.081 | 1.005 | 0.198
0.2 | $\beta$ | -0.004 | 0.94 | 0.091 | 0.825 | 0.078 | 0.754 | 0.06 | 0.863
$\delta$ | 0.147 | 0.826 | 0.456 | 0.57 | 0.451 | 0.429 | 0.467 | 0.606
$\gamma$ | 0.962 | 0.177 | 1.004 | 0.085 | 1.001 | 0.075 | 1.011 | 0.176

Notes: Based on 1000 replications and reported to 3 decimal places. ‘Room’
refers to the NLS estimator obtained by treating the observed group 1 as if
they were the groups (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{1}})$).
‘Floor’ refers to the NLS estimator obtained by treating the observed group 2
as if they were the groups (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{2}})$).
‘Known’ is the NLS estimator under Assumption 2 (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{0}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{0}})$).
‘Uncertain’ is the NLS estimator based on
$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}[\psi\pi(n_{g_{1}})+(1-\psi)\pi(n_{g_{2}})]$.
Results are presented for $(\beta,\delta,\gamma)$ only. Their true values are
$(0,0.35,1)$. Results for $\psi$ are available on request.

Table 3: Monte Carlo Results: Uncertain groups (FE)

Contextual effect only ($\beta=0$ imposed), $N=8000$
---
|  | Room | Floor | Known | Uncertain
$\psi$ |  | Mean | RMSE | Mean | RMSE | Mean | RMSE | Mean | RMSE
0.8 | $\delta$ | 0.41 | 0.123 | 0.847 | 0.375 | 0.496 | 0.066 | 0.516 | 0.131
$\gamma$ | 0.992 | 0.062 | 1.3 | 0.303 | 0.999 | 0.04 | 1.002 | 0.062
0.6 | $\delta$ | 0.321 | 0.198 | 0.756 | 0.292 | 0.5 | 0.061 | 0.504 | 0.142
$\gamma$ | 0.983 | 0.064 | 1.224 | 0.227 | 1 | 0.033 | 1.001 | 0.063
0.4 | $\delta$ | 0.232 | 0.281 | 0.668 | 0.218 | 0.5 | 0.064 | 0.501 | 0.145
$\gamma$ | 0.973 | 0.067 | 1.149 | 0.153 | 1 | 0.028 | 1.001 | 0.063
0.2 | $\delta$ | 0.144 | 0.366 | 0.584 | 0.156 | 0.497 | 0.08 | 0.503 | 0.145
$\gamma$ | 0.964 | 0.071 | 1.075 | 0.081 | 1 | 0.026 | 1.004 | 0.058
Contextual effect only ($\beta=0$ imposed), $N=1600$
0.8 | $\delta$ | 0.402 | 0.208 | 0.841 | 0.488 | 0.495 | 0.14 | 0.564 | 0.29
$\gamma$ | 0.983 | 0.132 | 1.297 | 0.309 | 0.997 | 0.088 | 1 | 0.135
0.6 | $\delta$ | 0.313 | 0.263 | 0.762 | 0.43 | 0.493 | 0.137 | 0.538 | 0.31
$\gamma$ | 0.975 | 0.134 | 1.225 | 0.24 | 0.996 | 0.073 | 0.999 | 0.135
0.4 | $\delta$ | 0.226 | 0.331 | 0.679 | 0.375 | 0.492 | 0.146 | 0.516 | 0.332
$\gamma$ | 0.966 | 0.136 | 1.151 | 0.169 | 0.997 | 0.063 | 0.999 | 0.134
0.2 | $\delta$ | 0.134 | 0.409 | 0.596 | 0.328 | 0.499 | 0.174 | 0.503 | 0.357
$\gamma$ | 0.955 | 0.139 | 1.076 | 0.102 | 0.999 | 0.058 | 1.006 | 0.13
Contextual and endogenous effects, $N=8000$
0.8 | $\beta$ | 0.041 | 0.814 | 0.224 | 0.558 | 0.054 | 0.46 | 0.055 | 0.799
$\delta$ | 0.392 | 0.36 | 0.725 | 0.375 | 0.466 | 0.214 | 0.492 | 0.302
$\gamma$ | 0.998 | 0.165 | 1.318 | 0.324 | 1.001 | 0.064 | 1.01 | 0.167
0.6 | $\beta$ | 0.039 | 0.848 | 0.198 | 0.565 | 0.07 | 0.461 | 0.099 | 0.8
$\delta$ | 0.301 | 0.495 | 0.645 | 0.347 | 0.462 | 0.227 | 0.458 | 0.354
$\gamma$ | 0.989 | 0.142 | 1.237 | 0.244 | 1.004 | 0.049 | 1.013 | 0.143
0.4 | $\beta$ | 0.042 | 0.873 | 0.17 | 0.602 | 0.072 | 0.491 | 0.106 | 0.769
$\delta$ | 0.205 | 0.64 | 0.571 | 0.347 | 0.458 | 0.256 | 0.446 | 0.379
$\gamma$ | 0.978 | 0.119 | 1.159 | 0.168 | 1.003 | 0.043 | 1.01 | 0.117
0.2 | $\beta$ | 0.032 | 0.881 | 0.152 | 0.612 | 0.106 | 0.561 | 0.123 | 0.724
$\delta$ | 0.123 | 0.772 | 0.492 | 0.36 | 0.436 | 0.303 | 0.433 | 0.393
$\gamma$ | 0.969 | 0.099 | 1.082 | 0.094 | 1.004 | 0.042 | 1.011 | 0.093
Contextual and endogenous effects, $N=1600$
0.8 | $\beta$ | 0.038 | 0.913 | 0.138 | 0.747 | 0.097 | 0.709 | 0.019 | 0.896
$\delta$ | 0.393 | 0.449 | 0.785 | 0.62 | 0.444 | 0.343 | 0.559 | 0.428
$\gamma$ | 0.995 | 0.228 | 1.31 | 0.331 | 1.002 | 0.123 | 1.005 | 0.239
0.6 | $\beta$ | 0.024 | 0.927 | 0.129 | 0.772 | 0.065 | 0.697 | 0.051 | 0.9
$\delta$ | 0.311 | 0.568 | 0.715 | 0.582 | 0.455 | 0.358 | 0.522 | 0.485
$\gamma$ | 0.986 | 0.209 | 1.237 | 0.263 | 0.998 | 0.096 | 1.009 | 0.22
0.4 | $\beta$ | 0.019 | 0.94 | 0.138 | 0.789 | 0.107 | 0.716 | 0.045 | 0.888
$\delta$ | 0.223 | 0.7 | 0.617 | 0.565 | 0.435 | 0.386 | 0.493 | 0.55
$\gamma$ | 0.974 | 0.192 | 1.161 | 0.191 | 1.001 | 0.081 | 1.005 | 0.198
0.2 | $\beta$ | -0.004 | 0.94 | 0.136 | 0.804 | 0.078 | 0.754 | 0.06 | 0.863
$\delta$ | 0.147 | 0.826 | 0.521 | 0.563 | 0.451 | 0.429 | 0.467 | 0.606
$\gamma$ | 0.962 | 0.177 | 1.083 | 0.123 | 1.001 | 0.075 | 1.011 | 0.176

Notes: Based on 1000 replications and reported to 3 decimal places. ‘Room’
refers to the NLS estimator obtained by treating the observed group 1 as if
they were the groups (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{1}})$).
‘Floor’ refers to the NLS estimator obtained by treating the observed group 2
as if they were the groups (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{2}})$).
‘Known’ is the NLS estimator under Assumption 2 (i.e. based on the moment
$\mathbb{E}[\overline{y}_{i}|n_{g_{0}},\mathbf{x}_{g}]=\overline{x}_{i}\pi(n_{g_{0}})$).
‘Uncertain’ is the NLS estimator based on
$\mathbb{E}[\overline{y}_{i}|n_{g_{1}},n_{g_{2}},\mathbf{x}_{g}]=\overline{x}_{i}[\psi\pi(n_{g_{1}})+(1-\psi)\pi(n_{g_{2}})]$.
Results are presented for $(\beta,\delta,\gamma)$ only. Their true values are
$(0,0.35,1)$. Results for $\psi$ are available on request.