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# Identification of Peer Effects using Panel Data

Marisa Miraldo, Carol Propper, Christiern Rose Miraldo and Propper: Imperial
College Business School, Rose: University of Queensland. Corresponding author:
Christiern Rose, School of Economics, University of Queensland, 4072,
Australia<EMAIL_ADDRESS>This project is part of the Health
Foundation’s Efficiency Research programme. The Health Foundation is an
independent charity working to improve the quality of healthcare in the UK. We
are thankful for their financial support. Propper acknowledges financial
support from the ERC. Hospital Episode Statistics datasets were provided by
the Health and Social Care Information Centre (HSCIC, now National Health
Service Digital). The Hospital Episode Statistics are copyright
©2000/01-2013/14, re-used with the permission of The Health & Social Care
Information Centre. All rights reserved.

## 1\. Introduction

This paper provides new identification results for panel data models of peer
effects, through which outcomes depend on peers’ unobservable (to the
researcher) heterogeneity. Our framework also allows for correlated effects,
modelled as unobserved group heterogeneity which is permitted to be correlated
with individual heterogeneity in an unrestricted manner. We extend existing
identification results to apply both to a general network structure (e.g., a
social network) and to allow for correlated effects, and apply our results to
study innovation take-up among physicians.

Identification depends on a conditional mean restriction requiring exogenous
mobility of individuals between groups over time, as first formalised in Abowd
et al. (1999). That is, though our model of correlated effects allows, for
example, that high outcome individuals be systematically located in high
outcome groups, their mobility between groups ought not be determined by
transitory outcome shocks. Not all patterns of mobility suffice for
identification, and we provide identifying and non-identifying examples. We
also provide an extension of our identification results to allow for
endogenous peer effects, through which outcomes depend directly on peers’
outcomes. With endogenous effects, identification can also be attained using
the conditional mean restriction, however, for certain network structures,
additional conditional variance restrictions are necessary. We conduct a
Monte-Carlo with many individuals and few time periods and demonstrate that
the NLS estimator first proposed by Arcidiacono et al. (2012) works well in
practice. Increasing the number of time periods, the rate of mobility and the
richness of the network (e.g., social network data) improves the performance
of the estimator.

Our empirical work considers innovation take-up in cancer treatment. The
innovation we consider is keyhole surgery for colorectal cancer, and our data
are from the English National Health Service. Colorectal cancer is the third
most common cancer worldwide (Arnold et al., 2017). In England, it accounts
for 10% of cancer deaths and is the most expensive cancer to treat (Laudicella
et al., 2016). Keyhole surgery for colorectal cancer is an important
innovation. It has been shown to lead to better patient outcomes than the
alternative open procedure, particularly in the short term. Moreover, in the
National Health Service, keyhole surgery is less costly, primarily due to
shorter post-surgery hospital stays by patients (Lacy et al., 2002; COSTSG,
2004; Laudicella et al., 2016). Despite this, its take-up was slow, increasing
from 1% of eligible surgeries when it was first introduced in 2000 to 49% by
2014.

We use matched patient-surgeon-hospital-year data from 2000 to 2014 to
estimate peer effects in take-up of keyhole surgery, measured by the fraction
of colorectal cancer surgeries performed by keyhole. We find positive and
statistically significant peer effects. Our results suggest that a standard
deviation increase in the average latent take-up of other surgeons in the same
hospital leads to a 5 percentage point increase in take-up. We decompose this
effect by additionally estimating the effect of peer experience, which we find
accounts for some, but not all, of the peer effect.

### 1.1. Related Literature

Research has largely focussed on settings in which peer effects operate
through observable characteristics. That is, in addition to correlated
effects, outcomes depend on peers’ observables (Manski, 1993; Moffitt et al.,
2001; Lee, 2007; Bramoullé et al., 2009; Calvó-Armengol et al., 2009; Davezies
et al., 2009; De Giorgi et al., 2010; Goldsmith-Pinkham and Imbens, 2013;
Blume et al., 2015; De Paula, 2017; Cohen-Cole et al., 2018; Bramoullé et al.,
2019). These papers consider a sample comprising a cross-section of groups,
for which identification requires within-group variation in peers. Our panel
data allow us to relax these requirements. First, the researcher need not
observe individual characteristics, and if observed, need not know which ones
are exogenous nor which ones are appropriate to include. Second, variation
induced by mobility of individuals between groups over time means that there
is no need for within-group variation in peers. This implies that
identification is possible under the linear-in-means network structure, in
which peer effects operate through the group average, and which precludes
identification when only a cross-section of groups is available (Manski, 1993;
Bramoullé et al., 2009).

Another strand of the literature considers peer effects operating through
individual unobservables using a cross-section of groups (Graham, 2008; Rose,
2017). Relative to these papers, we allow for unobserved group level
heterogeneity to be arbitrarily correlated with individual heterogeneity, and,
though our results can be applied to any network, we show that panel data can
be used to identify peer effects for the linear-in-means network, which are
not identifiable with only a cross-section of networks when there are
correlated effects (Rose, 2017).

The most closely related work considers panel data models of peer effects. Key
contributions are Mas and Moretti (2009), Arcidiacono et al. (2012) and
Cornelissen et al. (2017), which, like us, study models of peer effects
operating through the unobserved heterogeneity of peers. We build on their
work by providing identification conditions which are straightforward to
verify in practice, and which can be applied to a general network structure
(e.g., social networks) in the presence of correlated effects. We also extend
our results to the canonical model of endogenous peer effects, in which
outcomes are simultaneously determined.111Arcidiacono et al. (2012) consider a
variant of endogenous effects through which outcomes depend on the expected
(as opposed to realized) outcomes of others, and study identification in an
example with 2 individuals. This model does not allow simulataneity in
outcomes.

Beyond the peer effects literature our work can be viewed as extending the
worker-firm fixed effects framework for wage decomposition used in labor
economics (e.g., Abowd et al. (1999)) to allow for within-firm interactions of
workers. That is, in addition to worker and firm heterogeneity, wages may
depend on the composition of other workers in the firm as well as their wages.
Such spillovers would be expected to operate in firms in which workers work in
teams (Mas and Moretti, 2009; Cornelissen et al., 2017).

Our empirical work constributes to the literature on innovation take-up,
particularly in the healthcare context. Related papers are Agha and Molitor
(2018) and Barrenho et al. (2020). Agha and Molitor (2018) study how the take-
up of new cancer drugs depends on the presence of local opinion leaders. To do
this, they compare diffusion patterns across regions, separating correlated
regional demand for new technology from information spillovers. They find that
take-up is fastest in the region in which the lead author on the clinical
trial practices. However, their work does not directly study peer effects in
innovation. Barrenho et al. (2020) study the take-up of keyhole surgery, and
estimate peer effects through surgeon observables. Our empirical contribution
is to additionally allow peer effects to operate through surgeons’ latent
propensity to innovate, which we show matters above and beyond the effect of
peer experience.

We proceed as follows. In Section 2, we present our baseline model. In Section
3 we provide identification results, apply them to two examples and discuss
estimation. In Sections 4 and 5 we conduct a Monte Carlo experiment and apply
our method to surgeons’ take-up of keyhole surgery for colorectal cancer. In
Section 6 we conclude. Proofs and extensions are in the Appendix.

### 1.2. Notation

If $M$ is a strictly positive integer, we denote $[M]=\\{1,2,...,M\\}$. If
$\mathbf{A}$ and $\mathbf{B}$ are $M\times P$ and $M\times Q$ matrices, we
denote the $M\times(P+Q)$ matrix obtained by concatenating $\mathbf{A}$ and
$\mathbf{B}$ by $[\mathbf{A},\mathbf{B}]$. If element $(i,j)$ of $\mathbf{A}$
is $\mathbf{A}_{ij}$, we write
$\mathbf{A}=(\mathbf{A}_{ij})_{i\in[M],j\in[P]}$, and if $\mathbf{a}$ is a
vector, $\mathbf{a}_{i}$ denotes entry $i$. We use $\mathbf{I}_{M}$ for the
$M$ dimensional identity, $\iota_{M}$ for the $M\times 1$ vector of ones and
$\mathbf{0}$ to denote a matrix of zeros. If its dimensions are ambiguous we
write $\mathbf{0}_{M,P}$ to denote a $M\times P$ matrix of zeros. We use
$\mathbf{1}(\cdot)$ to denote the indicator function.

## 2\. Model

We consider a pattern of mobility of $N$ workers between $M$ groups over $T$
periods. Our interest lies in the typical setting in which $T$ is small and
$N$ can be large. A mobility pattern is characterised by the $N\times M$
matrices of group membership indicators
$\mathbf{C}_{1},\mathbf{C}_{2},...,\mathbf{C}_{T}$ and the $N\times N$
interaction matrices $\mathbf{G}_{1},\mathbf{G}_{2},...,\mathbf{G}_{T}$, the
elements of which encode the peer effect exerted by one individual on another.
The groups are such that in each period, every individual is in exactly one
group and every group has at least one individual in at least one period. We
use $g(i,t)\in[M]$ to denote the group of individual $i\in[N]$ in year
$t\in[T]$ and $N_{g(i,t)}$ to denote its size.

The $N\times 1$ vector of continuous outcomes in period $t\in[T]$ is
$\mathbb{y}_{t}$, which is determined by

$\displaystyle\mathbf{y}_{t}$
$\displaystyle=\left(\mathbf{I}_{N}+\rho\mathbf{G}_{t}\right)\alpha+\mathbf{C}_{t}\gamma+\epsilon_{t}$
(2.1)

where $\alpha$ is an $N\times 1$ vector of time-invariant individual
unobserved heterogeneity, $\gamma$ an $M\times 1$ vector of unobserved group
heterogeneity, and $\epsilon_{t}$ an $N\times 1$ disturbance. In Section 3.1,
we consider an extension in which $\gamma$ is permitted to vary by period. The
parameter of interest is $\rho$, which captures the peer effect.

Unless otherwise stated $\mathbf{G}_{t}$ is unrestricted and can be
interpreted as the adjacency matrix of a weighted, directed network linking
$N$ individuals. In particular, denoting entry $(i,j)$ of $\mathbf{G}_{t}$ by
$\mathbf{G}_{ijt}$, it need not be the case that $\mathbf{G}_{ijt}=0$ when
$g(i,t)\neq g(j,t)$, nor when $i=j$, nor when $\mathbf{G}_{jit}=0$. From this
point onwards, we refer to $\mathbf{G}_{t}$ as the network. A typical example
of $\mathbf{G}_{t}$ is the linear-in-means network,

$\displaystyle\overline{\mathbf{G}}_{t}=\left(\mathbf{1}(g(i,t)=g(j,t))N_{g(i,t)}^{-1}\right)_{(i,j)\in[N]^{2}}.$
(2.2)

This implies that the peer effect operates through the group average of
$\alpha_{i}$, and is a natural choice when only group membership indicators
are available. If more detailed data on the network structure are available
(e.g., social network data), this information can be incorporated into
$\mathbf{G}_{t}$. Clearly, $\mathbf{G}_{t}$ can evolve over time as
individuals move between groups.

In our empirical application, $\mathbf{y}_{t}$ is surgeons’ take-up of keyhole
surgery for colorectal cancer, measured by the fraction of eligible surgeries
performed by keyhole in year $t$, $\mathbf{C}_{t}$ comprises indicators for
the hospital in which a surgeon practices, and we take $\mathbf{G}_{t}$ to be
linear-in-means, linear-in-others’-means (in which the focal surgeon is
excluded from the group average, see (3.14)) and a persistent version of
linear-in-others’-means, in which links persist when a surgeon moves to a new
hospital and are weighted by the number of years worked at the same hospital.
The latter captures cumulative peer exposure, which may be better suited to
innovation take-up. The vector $\alpha$ captures surgeons’ latent propensity
to take up keyhole surgery. It can account for heterogeneity in
education/training, ability (e.g., dexterity) and taste for innovation. The
vector $\gamma$ captures hospital level heterogeneity including resources
(e.g., equipment) and patient composition.

Stacking (2.1) by period yields
$\mathbf{y}=\left(\mathbf{J}+\rho\mathbf{G}\right)\alpha+\mathbf{C}\gamma+\epsilon$,
where $\mathbf{y}$ and $\epsilon$ are $NT\times 1$,
$\mathbf{C}=(\mathbf{C}_{1}^{\prime},\mathbf{C}_{2}^{\prime},\ldots,\mathbf{C}_{T}^{\prime})^{\prime}$,
$\mathbf{J}=(\mathbf{I}_{N},\mathbf{I}_{N},\ldots,\mathbf{I}_{N})^{\prime}$
and
$\mathbf{G}=(\mathbf{G}_{1}^{\prime},\mathbf{G}_{2}^{\prime},\ldots,\mathbf{G}_{T}^{\prime})^{\prime}$.
Since $\sum_{i=1^{N}}\mathbf{J}_{ki}=\sum_{f=1}^{M}\mathbf{C}_{kf}=1$ for all
$k\in[NT]$, we use the normalization $\gamma_{M}=0$ to obtain

$\displaystyle\mathbf{y}=\left(\mathbf{J}+\rho\mathbf{G}\right)\alpha+\mathbf{D}\gamma+\epsilon$
(2.3)

where $\mathbf{D}$ comprises the first $M-1$ columns of $\mathbf{C}$ and from
this point forwards
$\gamma=(\gamma_{1},\gamma_{2},...,\gamma_{M-1})^{\prime}$. This is without
consequence for identification of $\rho$.

Our identification results depend on variation in the network (both over
individuals and over time) and mobility of individuals between groups over
time. It is well known that mobility serves to separate the individual and
correlated effects (e.g., Abowd et al. (1999)). As we show below, mobility
also serves to separate individual and correlated effects from the peer
effect. This is because, in the typical setting in which there are no between
group links,222i.e., if $g(i,t)\neq g(j,t)$ then $\mathbf{G}_{ijt}=0$, though
our results do not require this. if an individual moves from one group to
another she ceases to interact with others in her previous group and begins to
interact with others in her new group.

## 3\. Identification

Our identification results treat the joint distribution of
$\mathbf{y},\mathbf{G},\mathbf{D}$ as observable. This is consistent with the
researcher accessing a sample from this distribution. For small $T$, our
approach is identical in spirit to that used throughout the peer effects
literature (i.e., with $T=1$), in which the researcher is assumed to observe a
cross-section of groups (see Bramoullé et al. (2019) for a review).

A more challenging but sometimes more realistic alternative is that the
researcher observes a sample of individuals from a single group, so that,
depending on the network structure, all individuals are potentially linked to
one another, at least indirectly. It is more challenging because additional
structure is required to deal with the dependence between individuals, though
this is primarily a concern for inference rather than identification.
Goldsmith-Pinkham and Imbens (2013) describe how asymptotic analysis could be
implemented based on a random variable measuring ‘distance’ between
individuals. Their argument is as follows. If distant individuals have low
probability of link formation (e.g., due to homophily), and distance has large
support, it may be possible to construct blocks of individuals such that each
pair of blocks could be treated as close to independent. This is in the same
spirit as certain types of asymptotic analysis for time-series, and allows the
researcher to view a sample of individuals from a single group similarly to a
sample of many groups.

Goldsmith-Pinkham and Imbens (2013) use the above arguments to justify
conducting identification analysis as if the joint distribuion of
$\mathbf{y},\mathbf{G},\mathbf{X}$ were observable, where $\mathbf{X}$ is a
matrix of exogenous characteristics through which peer effects may
operate.333Their model does not include correlated effects, hence the absence
of $\mathbf{D}$. Their arguments can be directly applied in our context to
justify analysis of a sample from the joint distribution of
$\mathbf{y},\mathbf{G},\mathbf{D}$ if $T$ is small. The only caveat is that
$\mathbf{y},\mathbf{G},\mathbf{D}$ must include all observed time periods for
the individuals and groups, so that temporal dependence is not an issue in
(hypothetically) constructing blocks of observations. In our application, such
blocks could be thought of as corresponding to regions of England because
mobility is primarily between hospitals in the same region (Goldacre et al.,
2013; Barrenho et al., 2020). In other applications such as wage decomposition
in labor economics, the relevant partition could be by industry and/or by
region.

We study identification of $\rho$ based on the conditional mean restriction

$\displaystyle\mathbb{E}[\mathbf{y}|\mathbf{G},\mathbf{D}]=\left(\mathbf{J}+\rho\mathbf{G}\right)\mathbb{E}[\alpha|\mathbf{G},\mathbf{D}]+\mathbf{D}\mathbb{E}[\gamma|\mathbf{G},\mathbf{D}],$
(3.1)

which implies exogeneity of the network and mobility of individuals between
groups over time with respect to the outcome shock $\epsilon$. The former is
typical in the peer effects literature and the latter is typical in the wage
decomposition literature. We do not restrict
$\mathbb{E}[\alpha|\mathbf{G},\mathbf{D}]$ nor
$\mathbb{E}[\gamma|\mathbf{G},\mathbf{D}]$, which allows for a limited form of
network endogeneity with respect to unobserved individual and group
heterogeneity.

We say that $\rho$ is identified when it can be uniquely recovered from the
right-hand side of (3.1). Our results are thus asymptotic in nature (see
Manski (1995)), and hence charaterize whether peer effects can be
distentangled from individual and group heterogeneity if there is no limit to
the number of mobility patterns observed. To simplify the exposition,
following Bramoullé et al. (2009) and Abowd et al. (1999), the remainder of
the paper presents the case in which $\mathbf{G}$ and $\mathbf{D}$ are treated
as fixed. To allow for the random case, we simply replace unconditional
expectations with expectations conditional on $\mathbf{G},\mathbf{D}$, and the
identification results below hold if there exists a realization in the support
of $\mathbf{G},\mathbf{D}$ which satisfies the relevant condition. In the same
way, it is straightforward to allow $N,M$ and $T$ to vary by mobility pattern.
Identical arguments are used throughout the peer effects literature (see,
e.g., Bramoullé et al. (2009)). Returning to the fixed case, (3.1) becomes

$\displaystyle\mathbb{E}[\mathbf{y}]=\left(\mathbf{J}+\rho\mathbf{G}\right)\mu^{\alpha}+\mathbf{D}\mu^{\gamma},$
(3.2)

where $\mu^{\alpha}=\mathbb{E}[\alpha]$ and $\mu^{\gamma}=\mathbb{E}[\gamma]$.
Since (3.2) is non-linear in the parameter $\rho$ and the (unknown)
$\mu^{\alpha}$, establishing identification is non-trivial, depending both on
the properties of the $NT\times 2N+M-1$ matrix
$[\mathbf{J},\mathbf{G},\mathbf{D}]$ and on the value of $\mu^{\alpha}$. For
example, it is clear that $\rho$ is not identified when
$\mu^{\alpha}=\mathbf{0}$.

Mobility of individuals between groups over time is necessary for
identification. In the absence of mobility,
$[\mathbf{J},\mathbf{G},\mathbf{D}]$ has rank at most $N$, so (3.2) yields $N$
equations in $N+M$ unknowns. For the same reason, we also require $T\geq 2$.
This is because the peer effect operates through time-invariant individual
unobserved heterogeneity, rather than through individual observables. However,
mobility alone is not sufficient for identification, as made clear in Example
2 below.

We now present our first identification result, which makes use of the within-
group annihilator for the correlated effects, given by
$\mathbf{W}=\mathbf{I}_{NT}-\mathbf{D}(\mathbf{D}^{\prime}\mathbf{D})^{-1}\mathbf{D}$,
and a decomposition of vectors
$\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime})^{\prime}$ which
lie in the null-space of $[\mathbf{WJ},\mathbf{WG}]$ such that
$\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are both $N\times 1$.

######  Proposition 1.

$\rho$ is identified if there does not exist a vector
$\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime})^{\prime}$ in the
null-space of $[\mathbf{WJ},\mathbf{WG}]$ and scalars $\delta_{1}$ and
$\delta_{2}\neq 0$ verifying
$\delta_{1}\mathbf{v}_{1}+\delta_{2}\mathbf{v}_{2}=\mu^{\alpha}$. Otherwise
$\rho$ is not identified.

Notice that full column rank of $[\mathbf{J},\mathbf{G},\mathbf{D}]$ is not
necessary because $\mathbf{v}=\mathbf{0}$ need not be the only vector in the
null-space of $[\mathbf{WJ},\mathbf{WG}]$. This is because
$[\mathbf{J},\mathbf{G},\mathbf{D}]$ has $2N+M-1$ columns but there are only
$N+M$ unknowns. Requiring $[\mathbf{J},\mathbf{G},\mathbf{D}]$ to have full
column rank is too strong because it rules out $T=2$.444This is because full
column rank requires $NT\geq 2N+M-1$, hence $T\geq 2+(M-1)/N$. $T=2$ is not
immediately ruled out when $M=1$, but $\rho$ is not identifiable in this case
because there can be no mobility if there is only 1 group. A common empirical
setting is when the rows of $\mathbf{G}$ sum to one (e.g., linear-in-means),
such that peer effects operate through a weighted average. If there are no
other collinearities among the columns of $[\mathbf{J},\mathbf{G},\mathbf{D}]$
then one can apply the following.

######  Corollary 1.1.

If $\mathbf{G}\iota_{N}=\iota_{NT}$, $\rho$ is identified if ${\rm
rank}([\mathbf{J},\mathbf{G},\mathbf{D}])=2N+M-2$ and there exists
$(i,j)\in[N]^{2}$ such that $\mu^{\alpha}_{i}\neq\mu^{\alpha}_{j}$.

Corollary 1.1 can be shown using the decomposition
$[\mathbf{WJ},\mathbf{WG}]=\mathbf{S}\mathbf{R}$ where $\mathbf{S}$ is the
$NT\times 2N-1$ full rank matrix formed by concatenating $\mathbf{WJ}$ and the
first $N-1$ columns of $\mathbf{WG}$ and

$\displaystyle\mathbf{R}=(\mathbf{S}^{\prime}\mathbf{S})^{-1}\mathbf{S}^{\prime}[\mathbf{WJ},\mathbf{WG}]=\begin{pmatrix}\mathbf{I}_{N}&\mathbf{0}_{N,N-1}&\iota_{N}\\\
\mathbf{0}_{N-1,N}&\mathbf{I}_{N-1}&-\iota_{N-1}\end{pmatrix}.$ (3.3)

From the structure of $\mathbf{R}$, it is immediate that vectors $\mathbf{v}$
in the null-space of $[\mathbf{WJ},\mathbf{WG}]$ are of the form
$\mathbf{v}_{1}=c\iota_{N}$, $\mathbf{v}_{2}=-\mathbf{v}_{1}$ for
$c\in\mathbb{R}$. Applying Proposition 1 yields identification of $\rho$ if
there exists $(i,j)\in[N]^{2}$ such that
$\mu^{\alpha}_{i}\neq\mu^{\alpha}_{j}$. If this condition is violated then
$\mu^{\alpha}=a\iota_{N}$ for some $a\in\mathbb{R}$, and
$(\mathbf{J}+\rho\mathbf{G})\mu^{\alpha}=(1+\rho)a\iota_{NT},$ so only
$(1+\rho)\mu^{\alpha}$ is identifiable. Intuitively, we require
$\mu^{\alpha}_{i}\neq\mu^{\alpha}_{j}$ because if individuals are homogeneous
then no amount of mobility can lead to changes in the average of $\alpha_{i}$
over peers, hence outcomes do not vary in response to changes in peer
composition over time.

The identification conditions above depend on $\mu^{\alpha}$, which is not
observed. We now ask how ‘large’ is the set of values of $\mu^{\alpha}$ for
which $\rho$ is identified. This is the notion of generic identification (see
Lewbel (2019)). If $\rho$ is generically identified, then it is identified for
all values of $\mu^{\alpha}$ with the exception of a few pathological cases,
which are ‘unlikely’ to arise in practice.

######  Corollary 1.2.

If ${\rm rank}([\mathbf{WJ},\mathbf{WG}])\geq N+1$ the set of $\mu^{\alpha}$
for which $\rho$ is not identified is a measure zero subset of
$\mathbb{R}^{N}$.

Corollary 1.2 means that $\rho$ is generically identified if ${\rm
rank}([\mathbf{WJ},\mathbf{WG}])\geq N+1$. This is because $\mathbf{v}$ lies
in a subspace of $\mathbb{R}^{2N}$ of dimension at most $N-1$, hence
$\delta_{1}\mathbf{v}_{1}+\delta_{2}\mathbf{v}_{2}$ lies in a subspace of
$\mathbb{R}^{N}$ of dimension at most $N-1$.

As with the well known rank condition in linear models (e.g., no perfect
multicollinearity among regressors for linear regression), the researcher can
check whether the rank requirements of Corollaries 1.1 and 1.2 hold in the
observed data. To do this, one interprets $N$ as the total number of observed
individuals, $M$ as the total number of observed firms and $T$ as the total
number of observed time periods, and constructs the observed values of
$\mathbf{J},\mathbf{G}$ and $\mathbf{D}$ accordingly.555In our identification
analysis, these quantities are the number of individuals, firms and time
periods in a single mobility pattern (i.e., a draw from the joint distribution
of $\mathbf{y},\mathbf{G},\mathbf{D}$). The observed data comprise the
realizations of many such patterns. If the rank requirement of either
Corollary holds then the researcher can be confident of identification.

Proposition 1 can be viewed as panel data analogue of the identification
results of Bramoullé et al. (2009), which imply that, for $T=1$ and exogenous
observed characteristic(s) $\mathbf{X}$ of dimension $N\times K$, the peer
effect $\dot{\rho}$ in the model

$\displaystyle\mathbb{E}[\mathbf{y}|\mathbf{X}]=\mathbf{X}\dot{\beta}+\mathbf{GX}\dot{\rho}+\mathbf{D}\dot{\mu}^{\gamma}$
(3.4)

is identified if and only if $\mathbf{WJ}$ and $\mathbf{WG}$ are linearly
independent (i.e., if there does not exist nonzero $\lambda\in\mathbb{R}^{2}$
such that $\lambda_{1}\mathbf{WJ}+\lambda_{2}\mathbf{WG}=\mathbf{0}$).666Since
$\alpha$ does not appear in (3.4), there is no need to impose $\gamma_{M}=0$,
which is no longer a normalization. In this case $\mathbf{D}$ ought to be
replaced by $\mathbf{C}$ and $\mathbf{W}$ by the annihilator for $\mathbf{C}$
when referring to the model in (3.4). We continue to use the notation
$\mathbf{D}$ and $\mathbf{W}$ to facilitate the comparison with our results.
This is a weaker rank requirement than that in Proposition 1, and can be
satisfied when $T=1$. This is because the peer effect is assumed to operate
through the observed $\mathbf{X}$ rather than through the unobserved $\alpha$.
In practice the researcher may not know what to include in $\mathbf{X}$ and/or
$\mathbf{X}$ may be of large dimension and/or endogenous. Proposition 1 shows
that identification can be attained with $T\geq 2$ without requiring any
knowledge on $\mathbf{X}$.777Of course this requires that $\mathbf{X}$ be
time-invariant. However, common choices of the components of $\mathbf{X}$ such
as gender and education are also time-invariant. An additional advantage of
panel data is that it facilitates identification for some network structures
for which peer effects are not otherwise identifiable. For example, if $T=1$
and the network is linear-in-means, then $\mathbf{WG}=\mathbf{0}$ and (3.4)
does not identify the peer effect. In contrast, as we show in Example 1 below,
peer effects are identifiable when $T=2$ provided that there is mobility of
individuals between groups over time.

It is straightforward to extend our model to include exogenous
characteristics, yielding

$\displaystyle\mathbb{E}[\mathbf{y}|\mathbf{X}]=\mathbf{X}\beta+\mathbf{F}\mathbf{X}\rho_{1}+\left(\mathbf{J}+\rho_{2}\mathbf{G}\right)\mu^{\alpha}(\mathbf{X})+\mathbf{D}\mu^{\gamma}(\mathbf{X})$
(3.5)

where $\mathbf{X}$ is $NT\times K$, $\mathbf{F}$ is a $NT\times NT$ block
diagonal matrix with blocks
$\mathbf{G}_{1},\mathbf{G}_{2},...,\mathbf{G}_{T}$,
$\mu^{\alpha}(\mathbf{X})=\mathbb{E}[\alpha|\mathbf{X}]$ and
$\mu^{\gamma}(\mathbf{X})=\mathbb{E}[\gamma|\mathbf{X}]$. Provided that the
entries of $\mathbf{X}$ vary over time, the parameters $\rho_{1}$ and
$\rho_{2}$ are separately identifiable under similar conditions to Proposition
1. For brevity, we do not pursue this formally, though we do estimate such a
specification in our empirical application.

In the Appendix we provide analagous idenification results which additionally
allow for endogenous peer effects, through which outcomes are simultaneously
determined. We also provide conditional variance restrictions in the spirit of
Graham (2008) and Rose (2017), which can be used when the conditional mean
restriction does not suffice for identification.

We now consider two examples of our baseline identification results with
$N=M=T=2$.

Example 1: An identifying mobility pattern. Consider the following mobility
pattern in which individuals one and two are respectively in groups 1 and 2 in
the first period. In the second period, individual one remains in group 1 and
individual two moves from group 2 to group 1. Under the linear-in-means
network, this mobility pattern yields

$\displaystyle[\mathbf{J},\mathbf{G},\mathbf{D}]=\begin{pmatrix}1&0&1&0&1\\\
0&1&0&1&0\\\ 1&0&1/2&1/2&1\\\ 0&1&1/2&1/2&1\\\
\end{pmatrix},\quad[\mathbf{WJ},\mathbf{W}\mathbf{G}]=\begin{pmatrix}1/3&-1/3&1/3&-1/3\\\
0&1&0&1\\\ 1/3&-1/3&-1/6&1/6\\\ -2/3&2/3&-1/6&1/6\end{pmatrix},$ (3.6)

which respectively have rank $4$ and $3$. Since
$[\mathbf{J},\mathbf{G},\mathbf{D}]$ has rank $2N+M-2$, by Corollary 1.1,
$\rho$ is identified if $\mu^{\alpha}_{1}\neq\mu^{\alpha}_{2}$. Since ${\rm
rank}[\mathbf{WJ},\mathbf{W}\mathbf{G}]=N+1$, Corollary 1.2 states that $\rho$
is generically identified. This is because the subset of $\mathbb{R}^{2}$ such
that $\mu^{\alpha}_{1}=\mu^{\alpha}_{2}$ has measure zero.

For the intuition, consider the underlying system of equations for the
outcomes $y_{it}$ of individual $i$ in period $t$,

$\displaystyle\begin{array}[]{ll}\mathbb{E}[y_{11}]=\mu^{\alpha}_{1}+\rho\mu^{\alpha}_{1}+\mu^{\gamma}_{1}&\mathbb{E}[y_{21}]=\mu^{\alpha}_{2}+\rho\mu^{\alpha}_{2}\\\
\mathbb{E}[y_{12}]=\mu^{\alpha}_{1}+\rho(\mu^{\alpha}_{1}+\mu^{\alpha}_{2})/2+\mu^{\gamma}_{1}&\mathbb{E}[y_{22}]=\mu^{\alpha}_{2}+\rho(\mu^{\alpha}_{1}+\mu^{\alpha}_{2})/2+\mu^{\gamma}_{1}\end{array}$
(3.9)

When individual two moves groups, individual one obtains a new peer, hence the
peer effect on individual one changes from $\rho\mu^{\alpha}_{1}$ in the first
period to $\rho(\mu^{\alpha}_{1}+\mu^{\alpha}_{2})/2$ in the second period,
whilst the individual and correlated effect are unchanged. The change due to
the peer effect is given by the expected change in the outcome of individual
one between the first and second period
$\mathbb{E}[y_{11}]-\mathbb{E}[y_{12}]=\rho(\mu^{\alpha}_{1}-\mu^{\alpha}_{2})/2$.
To identify $\rho$ we now need to identify
$\mu^{\alpha}_{1}-\mu^{\alpha}_{2}$. We can use the second period, in which
both individuals are in the same group, hence have the same correlated and
peer effects. This means that any difference in their expected outcomes is due
to differences in their individual effects, so
$\mathbb{E}[y_{12}]-\mathbb{E}[y_{22}]=\mu^{\alpha}_{1}-\mu^{\alpha}_{2}$. If
$\mu^{\alpha}_{1}\neq\mu^{\alpha}_{2}$ we obtain

$\displaystyle\rho=2(\mathbb{E}[y_{11}]-\mathbb{E}[y_{12}])/(\mathbb{E}[y_{12}]-\mathbb{E}[y_{22}]).$
(3.10)

If we were to observe this mobility pattern repeatedly, we could estimate the
expectations using sample means, yielding an estimator of $\rho$. Of course,
in practice we do not repeatedly observe the same mobility pattern, but a
variety of patterns, the information from which we combine through an
estimator based on the conditional mean restriction (3.1). Nevertheless, for
the purposes of identification we require only that there exists a single
identifying mobility pattern realized with non-zero probability. Finally,
notice that the above arguments are unchanged when the normalization
$\gamma_{M}=0$ is not used, in which case one has
$\mathbb{E}[y_{21}]=\mu^{\alpha}_{2}+\rho\mu^{\alpha}_{2}+\mu^{\gamma}_{2}$ in
(3.9).

Example 2: A non-identifying mobility pattern. Now modify Example 1 such that
both individuals move in the second period. This means that the individuals
are in different groups in the first period and swap groups in the second
period, implying that $\mathbf{G}=\mathbf{J}$. The null-space of
$[\mathbf{WJ},\mathbf{W}\mathbf{G}]$ comprises
$\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime})^{\prime}=(\mathbf{u}^{\prime},-\mathbf{u}^{\prime})^{\prime}$
for any $\mathbf{u}\in\mathbb{R}^{2}$. For any value of
$\mu^{\alpha}\in\mathbb{R}^{2}$, there clearly exists
$\mathbf{u}\in\mathbb{R}^{2}$ and scalars $\delta_{1}$ and $\delta_{2}\neq 0$
such that $(\delta_{1}-\delta_{2})\mathbf{u}=\mu^{\alpha}$, so by Proposition
1, $\rho$ is not identified. Note also that the identification condition in
Corollary 1.1 is violated since $[\mathbf{J},\mathbf{G},\mathbf{D}]$ has rank
$3<2N+M-2=4$ and the generic identification condition in Corollary 1.2 is
violated since $[\mathbf{WJ},\mathbf{W}\mathbf{G}]$ has rank $2<N+1=3$.

For the intuition, consider again the underlying system of equations

$\displaystyle\begin{array}[]{ll}\mathbb{E}[y_{11}]=\mu^{\alpha}_{1}+\rho\mu^{\alpha}_{1}+\mu^{\gamma}_{1}&\mathbb{E}[y_{21}]=\mu^{\alpha}_{2}+\rho\mu^{\alpha}_{2}\\\
\mathbb{E}[y_{12}]=\mu^{\alpha}_{1}+\rho\mu^{\alpha}_{1}&\mathbb{E}[y_{22}]=\mu^{\alpha}_{2}+\rho\mu^{\alpha}_{2}+\mu^{\gamma}_{1}\end{array}$
(3.13)

The correlated effect $\mu^{\gamma}_{1}$ is identified by individual one
moving groups ($\mu^{\gamma}_{1}=\mathbb{E}[y_{11}]-\mathbb{E}[y_{12}]$) but
the remaining equations are only sufficient to identify
$(1+\rho)\mu^{\alpha}$. The reason for this is that there is no variation in
the peers of either individual because the individuals are never in the same
group in the same period. Mobility of individuals between groups over time is
insufficient for identification of $\rho$ because it does not induce changes
in peer groups. This contrasts with the canonical wage decomposition model
imposing $\rho=0$, for which group-swapping would be sufficient to identify
$\mu^{\alpha}_{1},\mu^{\alpha}_{2},\mu^{\gamma}_{1}$.888Note that
identification of $\mu^{\alpha}$ and $\mu^{\gamma}$ is only up to the
normalization $\gamma_{M}=0$. Since it does not identify $\rho$, no matter how
many times we observe this mobility pattern in our data, it cannot be used to
construct an estimator of $\rho$.

### 3.1. Time-varying Correlated Effects

If correlated effects are time-varying, then $\gamma_{g(i,t)}$ is replaced by
$\gamma_{g(i,t)t}$, in which case $\mathbf{C}$ is a $NT\times MT$ block
diagonal matrix with blocks
$\mathbf{C}_{1},\mathbf{C}_{2},...,\mathbf{C}_{T}$, $\mathbf{D}$ comprises the
first $MT-1$ columns of $\mathbf{C}$ and $\gamma$ is $(MT-1)\times 1$. All
Propositions then apply as stated, provided that $\mathbf{W}$ is modified
accordingly. The parameters are not identifiable under the linear-in-means
network because the peer effect varies only at the group-period level, so
cannot be separated from the correlated effect.

### 3.2. Estimation

Arcidiacono et al. (2012) propose NLS estimation of (2.3), treating
$\alpha,\rho,\gamma$ as parameters to be estimated (i.e., a fixed effects
approach). The authors study its properties under the linear-in-others’-means
network,

$\displaystyle\widetilde{\mathbf{G}}_{t}=\left(\mathbf{1}(g(i,t)=g(j,t),i\neq
j)(N_{g(i,t)}-1)^{-1}\right)_{(i,j)\in[N]^{2}},$ (3.14)

and without correlated effects. Assuming that
$\mathbb{E}[\epsilon_{it}\epsilon_{js}]=0$ for all $i\neq j,t\neq s$,
$\mathbb{E}[\epsilon_{it}^{2}|g(i,t)]=\mathbb{E}[\epsilon_{jt}^{2}|g(j,t)]$
for all $i,j,t$ such that $g(i,t)=g(j,t)$,
$\mathbb{E}[\epsilon_{it}\alpha_{j}]=0$ for all $i,j,t$ and
$\rho<\min_{i,t}N_{g(i,t)}$, the authors show that the estimator of $\rho$ is
consistent and asymptotically normal in the number of individuals provided
that there are at least two time periods. The authors discuss a variant of
correlated effects appropriate to their application to peer effects in
education, which is allowed to vary over time but is restricted to be the same
across multiple groups, and argue that they expect similar behavior of the
estimator in this case. The proposed NLS estimator is based on the conditional
mean restriction (3.1), hence, subject to identification, could equally be
applied to other network structures and specifications of the correlated
effect. We do not formally establish its properties because our focus is on
identification and our empirical application. However we do explore this in
our Monte-Carlo experiment.

## 4\. Monte Carlo Experiment

The design is tailored to our empirical application, with 700 individuals, 140
groups, 15 time periods, mobility rate $p=0.03$ (this is the probability that
an individual moves groups from one period to the next, see summary statistics
in Table 2) and $\rho=0.5$. We also consider designs with 2 time periods and
mobility rate $p=0.1$.

The data generating process is as follows. In the first period, all
individuals are randomly assigned to groups of size five. In each subsequent
period, each individual moves group with probability $p$, in which case she
draws a new group with uniform probability over all groups. The expected group
size is 5 for all groups in all periods. The network structures we consider
are linear-in-means, linear-in-others’-means, and a variant of linear-in-
others’-means in which links persist when individuals move groups and are
weighted by the number of periods spent in the same group, and a social
network, in which there is within-group variation in peers.

For the persistent network, we let $\mathbf{A}_{s}$ be the $N\times N$ binary
adjacency matrix in period $s$, with element $(i,j)$ equal to 1 if
$g(i,s)=g(j,s)$ and $i\neq j$. Then we define $\mathbf{G}_{t}$ by taking
$\sum_{s=1}^{t}\mathbf{A}_{s}$ and rescaling its rows to sum to 1. This means
that $\mathbf{G}\alpha$ captures both contemporaneous and cumulative exposure
to others.

The social network is constructed as follows. In the first period, each
individual draws two links uniformly over other individuals in the same group.
In each subsequent period, links persist whilst individuals remain in the same
group. If an individual loses a link(s) due to mobility, a replacement link(s)
is drawn uniformly over the other individuals in the group with whom there is
not already a link. Links need not be reciprocal.999If an individual is in a
group of size 1 they have no link. If there exists a link between $i$ and $j$
in period $t$ then $\mathbf{G}_{ijt}$ is equal to the inverse of the number of
links that $i$ has in period $t$. Otherwise $\mathbf{G}_{ijt}=0$. If there are
three or fewer individuals in the group, each individual is linked to all
other individuals.

We take $\alpha_{i}=1+W_{i}+\eta_{i}$, where $W_{i}\in[0,1]$ is the number of
moves made by individual $i$ divided by $(T-1)$ and
$\eta_{i}\sim\mathcal{N}(0,1)$. We set $\epsilon_{it}\sim\mathcal{N}(0,1/2)$
and consider cases in which $\gamma=\mathbf{0}$ (which is also imposed on the
estimator) and in which $\gamma_{m}$ is the mean of $\alpha_{i}$ over all
members of group $m$ and all time periods. This design means that high
$\alpha_{i}$ individuals are more mobile and tend to be located in high
$\gamma_{m}$ groups. We choose the variances of $\alpha_{i}$ and
$\epsilon_{it}$ as above so that, on average, a standard deviation increase in
the peer effect on individual $i$ in period $t$ (i.e., in
$\sum_{j=1}^{N}\mathbf{G}_{ijt}\alpha_{j}$) leads to a 0.15-0.3 (depending on
the network structure) standard deviation increase in $y_{it}$. This matches
the effect sizes we find in our empirical application. We simulate 500
datasets for each experiment. Every dataset verifies the generic
identification condition in Corollary 1.2.

Table 1: Monte Carlo Results

|  Unobserved $\alpha$
---|---
LIM | 0.501 | 0.5 | 0.523 | 0.522 | 0.502 | 0.503 | 0.496 | 0.515
| (0.04) | (0.051) | (0.297) | (0.456) | (0.026) | (0.032) | (0.161) | (0.447)
LIOM | 0.5 | 0.5 | 0.505 | 0.454 | 0.501 | 0.502 | 0.494 | 0.437
| (0.029) | (0.035) | (0.216) | (0.355) | (0.019) | (0.022) | (0.121) | (0.386)
PLIOM | 0.501 | 0.507 | 0.505 | 0.394 | 0.503 | 0.503 | 0.489 | 0.455
| (0.068) | (0.085) | (0.414) | (0.641) | (0.046) | (0.052) | (0.231) | (0.361)
SOC | 0.501 | 0.5 | 0.5 | 0.462 | 0.5 | 0.5 | 0.503 | 0.496
| (0.019) | (0.023) | (0.185) | (0.361) | (0.013) | (0.014) | (0.084) | (0.248)
|  Observed $\alpha$
LIM | 0.5 | 0.5 | 0.498 | 0.493 | 0.501 | 0.502 | 0.499 | 0.491
| (0.009) | (0.037) | (0.027) | (0.175) | (0.009) | (0.024) | (0.026) | (0.12)
LIOM | 0.5 | 0.499 | 0.499 | 0.495 | 0.5 | 0.502 | 0.499 | 0.493
| (0.008) | (0.028) | (0.022) | (0.139) | (0.008) | (0.018) | (0.021) | (0.096)
PLIOM | 0.5 | 0.501 | 0.499 | 0.497 | 0.501 | 0.501 | 0.499 | 0.49
| (0.008) | (0.03) | (0.022) | (0.165) | (0.008) | (0.024) | (0.021) | (0.115)
SOC | 0.5 | 0.499 | 0.5 | 0.498 | 0.5 | 0.5 | 0.5 | 0.498
| (0.007) | (0.013) | (0.019) | (0.039) | (0.007) | (0.012) | (0.017) | (0.039)
Correlated Effects | No | Yes | No | Yes | No | Yes | No | Yes
Periods | 15 | 15 | 2 | 2 | 15 | 15 | 2 | 2
Mobility Rate ($p$) | 0.03 | 0.03 | 0.03 | 0.03 | 0.1 | 0.1 | 0.1 | 0.1
Individuals | 700 | 700 | 700 | 700 | 700 | 700 | 700 | 700
Groups | 140 | 140 | 140 | 140 | 140 | 140 | 140 | 140

Notes: We report the mean and standard deviation (in parentheses) estimate of
$\rho$ over 500 datasets, with true value $0.5$. ’LIM’ is the linear-in-means
network, ‘LIOM’ is linear-in-others’-means, ‘PLIOM’ is the persistent LIOM and
‘SOC’ is the social network. ‘Unobserved $\alpha$’ is the NLS estimator in
Section 3.2. ‘Observed $\alpha$’ is the OLS estimator taking
$\mathbf{X}=\mathbf{J}\alpha$ in (3.4). If there are no correlated effects the
estimator imposes $\gamma=\mathbf{0}$. Correlated effects are time-invariant.

### 4.1. Results

Table 1 reports the results. The top panel reports the NLS estimator of $\rho$
described in Section 3.2. To provide a benchmark for comparison, the bottom
panel reports the infeasible OLS estimator that would be used if $\alpha$ were
observable (i.e., for the model in (3.4) taking
$\mathbf{X}=\mathbf{J}\alpha$). Columns 1-2 of Table 1 show designs with 15
periods and mobility rate 0.03, which match our empirical application. The NLS
estimator of $\rho$ is centered on the true value over all networks and with
and without correlated effects. The variance of the NLS estimator is of a
similar order of magnitude to the infeasible OLS estimator, though of course
it is larger. Columns 3-4 reduce the number of periods to 2, which causes an
increase in the variance of the estimator. The final 4 columns increase the
rate of mobility to 0.1, which reduces the variance of the estimator.
Comparing the rows of Table 1, we find that peer effects are most precisely
estimated for the social network. This is likely because the social network
exhibits the most within-group variation in peers.

## 5\. Application

We study surgeons’ take-up of keyhole surgery for colorectal cancer in the
English National Health Service (henceforth, NHS). As discussed in the
introduction, colorectal cancer is prevalent, costly to treat, and accounts
for a high proportion of cancer deaths worldwide. An important innovation in
its treatment is keyhole surgery, which reduces costs and improves patient
outcomes relative to the older open procedure. Despite this, take-up of
keyhole surgery in England was slow, increasing from 1% of eligible surgeries
in 2000, the year it was introduced in England, to 49% by 2014 (see Table 2).
Our goal is to study the extent to which its diffusion was driven by peer
effects, hence by mobility of surgeons between hospitals over time.

The NHS is an ideal setting for our empirical work because it treats almost
all cancer patients in England,101010There is a small private sector in
England that mainly provides care for planned procedures for which there are
long waiting lists. Private sector provision for (any) cancers during the
period we examine was very limited and primarily focused on treatment of
overseas patients. it is a public system in which surgeons are salaried
employees of one hospital at any point in time (hence their renumeration does
not depend on the treatment provided), all hospitals operate under the same
financial rules set by central government and have the technology necessary
for keyhole surgery, and a two week waiting time guarantee for cancer referral
implies that allocation of patients to surgeons is close to random, based on
surgeon availability in a local hospital (Barrenho et al., 2020).

### 5.1. Data

Our data are from Barrenho et al. (2020), comprising a panel of NHS surgeons
and their take-up of keyhole surgery from its introduction in 2000 through to
2014. The data are obtained by merging Hospital Episode Statistics, which
provides treatment information for all patients in the English NHS with NHS
Workforce Statistics and the General Medical Council register. This provides
matched patient-surgeon-hospital-year data, which is collapsed into a surgeon-
hospital-year panel. To be included in the estimation sample, surgeons must be
observed at least twice, must perform more than 5 colorectal cancer
surgeries,111111This is to avoid including surgeons who do not routinely
perform the procedure. and hospital-year pairs must have at least two
surgeons. The resulting unbalanced panel comprises 11,923 observations of
1,363 surgeons over 15 years across 194 hospitals.

The dependent variable is surgeon take-up of keyhole surgery for colorectal
cancer, measured by the fraction of eligible colorectal cancer sugeries
performed by keyhole in the focal year.121212Keyhole surgery is suitable for
some, but not all, patients. The sample of patients considered is restricted
to those for which the surgeon has a choice between keyhole and the
alternative open surgery. A detailed description of patient eligibility can be
found in Barrenho et al. (2020). The surgeons we observe are senior
physicians, known as Consultants. They are entirely autonomous, and, for the
patients we consider, have discretion to choose either keyhole or open
surgery. We also observe surgeon demographics (gender and age, though we omit
gender from our models because it is time-invariant), and compute an
experience measure which is the cumulative number of eligible colorectal
cancer surgeries performed (both keyhole and open) by the beginning of the
focal year. To account for increasing average take-up over time, we also
include year fixed effects in all specifications.

We apply our model for surgeon $i$ in hospital $g(i,t)$ in year $t$,131313We
set a surgeon’s hospital to be that at which they practiced for the majority
of days of the year. and consider the linear-in-means and linear-in-
others’-means networks, as well as the persistent linear-in-others’-means
network used in our Monte-Carlo experiment. Recall that in the persistent
network, a surgeon is a peer if they have ever worked concurrently in the same
hospital, and peers are weighted by number of years worked together, with
weights summing to one. This network captures both contemporaneous and
cumulative exposure to others, hence measures a ‘stock’ of peer influence,
rather than the ‘flow’ measured by the other networks.

Table 2 summarises the data. In a typical hospital in a typical year there are
5 surgeons performing colorectal cancer surgery. The largest has 16 surgeons.
A typical surgeon is in their early 40s and has performed around 140
colorectal cancer surgeries by the beginning of a typical year. We only
observe surgeons beginning in 2000, hence experience is equal to zero in this
year for all surgeons. The minimum value of experience can be zero in any year
due to entry of newly qualified surgeons.141414For this reason, in addition to
taste and ability, $\alpha$ may be thought of as including experience in
colorectal cancer surgery prior to 2000.

Table 2: Summary Statistics

Year |  | 2000 | 2003 | 2006 | 2009 | 2012 | 2014 | All
---|---|---|---|---|---|---|---|---
Surgeons observed | 600 | 761 | 782 | 834 | 891 | 854 | 1363
Take-up | Mean | 0.008 | 0.022 | 0.104 | 0.293 | 0.421 | 0.485 | 0.218
S.D. | 0.031 | 0.072 | 0.173 | 0.273 | 0.28 | 0.291 | 0.28
Min | 0 | 0 | 0 | 0 | 0 | 0 | 0
Max | 0.256 | 1 | 1 | 1 | 1 | 1 | 1
Age $<40$ | Mean | 0.179 | 0.19 | 0.201 | 0.225 | 0.207 | 0.252 | 0.233
Age $40-44$ | Mean | 0.283 | 0.246 | 0.253 | 0.233 | 0.284 | 0.284 | 0.26
Age $45-49$ | Mean | 0.227 | 0.172 | 0.239 | 0.194 | 0.219 | 0.219 | 0.212
Age $50-54$ | Mean | 0.196 | 0.153 | 0.165 | 0.131 | 0.166 | 0.16 | 0.165
Experience (00’s) | Mean | 0 | 0.689 | 1.279 | 1.771 | 2.104 | 2.486 | 1.418
S.D. | 0 | 0.548 | 1.028 | 1.444 | 1.838 | 2.013 | 1.51
Min | 0 | 0 | 0 | 0 | 0 | 0.01 | 0
Max | 0 | 2.65 | 5.01 | 6.65 | 8.62 | 9.37 | 9.37
Moved | Mean | - | 0.035 | 0.028 | 0.038 | 0.035 | 0.023 | 0.029
Peers changed | Mean | - | 0.582 | 0.602 | 0.64 | 0.738 | 0.542 | 0.601
Hospitals observed | 151 | 151 | 149 | 148 | 143 | 139 | 194
No. of surgeons | Mean | 3.974 | 5.04 | 5.248 | 5.635 | 6.231 | 6.144 | 5.36
S.D. | 1.689 | 2.343 | 2.339 | 2.419 | 2.533 | 2.541 | 2.439
Min | 2 | 2 | 2 | 2 | 2 | 2 | 2
Max | 10 | 14 | 12 | 13 | 16 | 14 | 16

Notes: ‘Take-up’ is the fraction of colorectal cancer surgeries performed by
keyhole. ‘Age’ are indicators for the specified range. ‘Experience‘ is 00’s of
colorectal cancer surgery at the beginning of the year. ‘Moved’ is a binary
indicator for a surgeon being located in a different hospital in year $t$ than
in year $t-1$. ‘Peers changed’ is a binary indicator for a surgeons’ peers
being different in year $t$ than in $t-1$. ‘No. of surgeons’ is the number of
surgeons located in a hospital.

### 5.2. Identification

Our identification results are based on a sample from the joint distribution
of $\mathbf{y},\mathbf{G},\mathbf{D}$. As argued in Section 3, we also expect
them to be applicable if the sample can be partitioned into blocks based on
individuals’ ‘distance’ to one another, such that the dependence between each
pair of blocks is limited. Mobility of surgeons between hospitals is largely
within the local region (Goldacre et al., 2013; Barrenho et al., 2020), hence
such blocks could (hypothetically) be constructed by partitioning regions. Due
to this, we expect that our theory provides a reasonable approximation.

Identification is based on exogenous mobility of surgeons between hospitals
over time. We expect this to be the case because all hospitals have the
required technology for keyhole surgery, the NHS is a public system with
similar working conditions and renumeration nationwide, and colorectal cancer
forms only a small part of surgeons’ workloads. The leading reason for
mobility is to relocate closer to the pre-medical school family home (Goldacre
et al., 2013). Barrenho et al. (2020) provide empirical evidence of exogenous
mobility, showing that past take-up does not predict mobility, and that,
conditional on moving, there is no association between take-up of the surgeon
and take-up of the hospital moved to. These arguments support the conditional
mean restriction in (3.1). Since $\mathbb{E}[\alpha|\mathbf{G},\mathbf{D}]$
and $\mathbb{E}[\gamma|\mathbf{G},\mathbf{D}]$ are unrestricted, we allow for
dependence between a surgeon’s latent propensity to take-up keyhole surgery
and the nature of their network. Moreover, we do not rule out high take-up
surgeons being systematically located in high take-up hospitals, though we do
rule out their mobility between hospitals being driven by transitory take-up
shocks.

We now discuss the extent of identifying variation in the data. Though only 3%
of surgeons move hospital in a typical year (see Table 2), our panel is
relatively long, comprising 15 years in total. Moreover, each move changes the
peer groups of all those in the hospital left and all those in the hospital
joined. The median number of surgeons in a hospital-year pair is 5, hence a
surgeon moving from one median sized hospital to another changes the peer
groups of 10 surgeons. For this reason, 60% of surgeons’ peers change in a
typical year. These changes in peer groups can generate large fluctuations in
average peer take-up because peer groups are small. Our Monte-Carlo experiment
also demonstrates that the observed mobility rate is sufficient to accurately
estimate $\rho$.

All estimated models verify generic identification of $\rho$. In the baseline
model without surgeon demographics, the rank of $[\mathbf{WJ},\mathbf{WG}]$ is
2361 for the linear-in-means network, 2678 for the linear-in-others’-means
network and 2683 for the persistent network. Since there are $N=1363$
surgeons, the rank requirement of Corollary 1.2 is satisfied.151515For these
calculations, year dummies are included to construct the annihilator
$\mathbf{W}$, hence the partialling out is with respect to both hospital and
year dummies. For models with surgeon demographics, these are also included to
construct $\mathbf{W}$.

### 5.3. Results

Table 3: Peer Effects in Keyhole Surgery for Colorectal cancer

| Linear-in-means | Linear-in-others’-means | Persistent LIOM
---|---|---|---
$\mathbf{G}\alpha$ | 0.427 | 0.531 | 0.422 | 0.564 | 0.58 | 0.507 | 0.993 | 0.995 | 0.99
| (0.06) | (0.061) | (0.072) | (0.036) | (0.036) | (0.045) | (0.008) | (0.008) | (0.013)
Age$<40$ |  |  | -0.004 |  |  | 0.001 |  |  | 0.022
|  |  | (0.022) |  |  | (0.023) |  |  | (0.021)
Age$40-44$ |  |  | 0.03 |  |  | 0.035 |  |  | 0.047
|  |  | (0.017) |  |  | (0.018) |  |  | (0.017)
Age$45-49$ |  |  | 0.044 |  |  | 0.042 |  |  | 0.049
|  |  | (0.012) |  |  | (0.013) |  |  | (0.012)
Age$50-54$ |  |  | 0.03 |  |  | 0.029 |  |  | 0.032
|  |  | (0.008) |  |  | (0.008) |  |  | (0.008)
Exper. |  |  | 0.043 |  |  | 0.047 |  |  | 0.045
|  |  | (0.003) |  |  | (0.003) |  |  | (0.003)
$\mathbf{F}$Age $<40$ |  |  | -0.003 |  |  | -0.012 |  |  | 0.019
|  |  | (0.023) |  |  | (0.019) |  |  | (0.027)
$\mathbf{F}$Age $40-44$ |  |  | 0.01 |  |  | 0.007 |  |  | 0.022
|  |  | (0.019) |  |  | (0.016) |  |  | (0.02)
$\mathbf{F}$Age $45-49$ |  |  | -0.014 |  |  | -0.008 |  |  | 0.005
|  |  | (0.02) |  |  | (0.017) |  |  | (0.017)
$\mathbf{F}$Age $50-54$ |  |  | 0.002 |  |  | 0.003 |  |  | 0.013
|  |  | (0.017) |  |  | (0.013) |  |  | (0.011)
$\mathbf{F}$Exper. |  |  | 0.025 |  |  | 0.035 |  |  | 0.008
|  |  | (0.008) |  |  | (0.006) |  |  | (0.005)
$\rho\left(\sum_{i=1}^{N}\mathbf{G}_{ijt}^{2}\right)^{1/2}$ | 0.18 | 0.122 | 0.103 | 0.269 | 0.193 | 0.161 | 0.443 | 0.345 | 0.354
$\sigma_{\alpha}$ | 0.168 | 0.253 | 0.294 | 0.181 | 0.262 | 0.315 | 0.207 | 0.297 | 0.306
Surgeon FE | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes
Hospital FE | No | Yes | Yes | No | Yes | Yes | No | Yes | Yes
Year FE | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes | Yes
Observations | 11923 | 11887 | 11887 | 11923 | 11887 | 11887 | 11923 | 11887 | 11887

Notes: To allow for heteroskedasticity, standard errors are computed using a
wild bootstrap, as proposed by Arcidiacono et al. (2012).
$\sigma^{2}_{\alpha}$ is computed using the estimated $\alpha$, as proposed by
Arcidiacono et al. (2012).

Table 3 reports the results. The first row reports estimates of $\rho$. We
find positive peer effects in all specifications. The peer effect is precisely
estimated, and statistically distinguishable from zero in all specifications.
This is in line with our Monte-Carlo results, and, though their application to
peer effects in education is different to ours, with the results of
Arcidiacono et al. (2012).

The vector $\alpha$ combines both surgeons’ ability and taste for innovation,
but we cannot decompose the two with the available data. Clearly, the size of
the peer effect is interpretable only relative to the variance of $\alpha$. To
provide an interpretable benchmark, we treat $\mathbf{G}$ as fixed and
$\alpha_{i}$ as i.i.d., with variance $\sigma^{2}_{\alpha}$ estimated using
the estimated $\alpha$, as proposed by Arcidiacono et al. (2012). For surgeon
$i$ in year $t$, the effect on take-up of a standard deviation increase in
average peer latent propensity to innovate is
$\rho\sigma_{\alpha}\left(\sum_{i=1}^{N}\mathbf{G}_{ijt}^{2}\right)^{1/2}$.
For the linear-in-means network this is
$\rho\sigma_{\alpha}N_{g(i,t)}^{-1/2}$, and for linear-in-others’-means we
replace $N_{g(i,t)}$ by $N_{g(i,t)}-1$. Since the effect of a standard
deviation increase in own latent propensity to innovate is $\sigma_{\alpha}$,
the magnitude of the peer effect relative to the own effect is
$\rho\left(\sum_{i=1}^{N}\mathbf{G}_{ijt}^{2}\right)^{1/2}$. Table 3 reports
its average over all observations. With hospital fixed effects, it is around
0.1 for linear-in-means, 0.15-0.2 for linear-in-others’-means and 0.35 for the
persistent network. Our estimates imply that a standard deviation increase in
average peer latent propensity to innovate increases take-up by 3 percentage
points for linear-in-means, 5 percentage points for linear-in-others’-means
and 10 percentage points for the persistent network. Our finding that the
persistent network corresponds to the largest effect size suggests that both
contemporaneous and cumulative peer exposure play a role.

For each network, we also estimate a specifications which includes the age and
experience of a surgeon and their peers (i.e., the model in (3.5)). We find an
inverted-U profile in age, with take-up estimated to peak between 45 and 49.
The coefficient on experience is positive, suggesting that an additional
hundred career colorectal cancer surgeries leads to around a 5 percentage
point increase in take-up. The effect of peer age is close to zero, whilst
peer experience plays a similar role to own experience. The estimate of $\rho$
is smaller than in models in which peer effects operate through $\alpha$
alone, though it remains positive and statistically significant at
conventional levels. This suggests that some, but not all, of the peer effect
operates through experience.

## 6\. Conclusion

This paper provides and implements new identification results for panel data
models with peer effects. Our results suggest that these can typically be
separated from correlated effects provided either that there is sufficient
mobility in the data or that the network data are sufficiently detailed. We
find positive peer effects in surgeons’ take-up of keyhole surgery for
colorectal cancer, which operate through peer experience and latent propensity
to take-up the innovation. Our results imply that exposure to others with high
take-up and experience may be useful to increase innovation take-up. Based on
this, policymakers might consider programmes which expose low take-up surgeons
to high take-up and/or experienced surgeons. For example, one might conceive a
targeted secondment programme.

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## Appendix

Proof of Proposition 1

Denote $\theta=(\mu^{\alpha},\rho,\mu^{\gamma})$. Equation (3.2) yields
$\mathbb{E}[\mathbf{Wy}]=\mathbf{WJ}\mu^{\alpha}+\mathbf{WG}\rho\mu^{\alpha}$.
Suppose that there is $\overline{\theta}$ satisfying the conditional moment
restriction in (3.2). Then we have

$\displaystyle[\mathbf{WJ},\mathbf{WG}]\begin{pmatrix}\mu^{\alpha}-\overline{\mu}^{\alpha}\\\
\rho\mu^{\alpha}-\overline{\rho}\overline{\mu}^{\alpha}\end{pmatrix}=\mathbf{0}$
(A.1)

Let $\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime})$ be a vector
in the null-space of $[\mathbf{WJ},\mathbf{WG}]$. Then for some $\mathbf{v}$
we have

$\displaystyle\mathbf{v}_{1}=\mu^{\alpha}-\overline{\mu}^{\alpha},\quad\mathbf{v}_{2}=\rho\mu^{\alpha}-\overline{\rho}\overline{\mu}^{\alpha}$
(A.2)

which implies
$\overline{\rho}\mathbf{v}_{1}-\mathbf{v}_{2}=(\overline{\rho}-\rho)\mu^{\alpha}$.
If $\overline{\rho}\neq\rho$ then this is equivalently expressed as

$\displaystyle\frac{\overline{\rho}}{\overline{\rho}-\rho}\mathbf{v}_{1}-\frac{1}{\overline{\rho}-\rho}\mathbf{v}_{2}=\mu^{\alpha}$
(A.3)

If there does not exist $\mathbf{v}$ in the null-space of
$[\mathbf{WJ},\mathbf{WG}]$ and scalars $\delta_{1}$ and $\delta_{2}\neq 0$
verifying $\delta_{1}\mathbf{v}_{1}+\delta_{2}\mathbf{v}_{2}=\mu^{\alpha}$
then, by contradiction we have $\overline{\rho}=\rho$, so $\rho$ is
identified. Otherwise we can clearly have $\overline{\rho}\neq\rho$, so $\rho$
is not identified.

### Endogenous peer effects

We now consider an extension of our identification results to models with
endogenous peer effects. We consider the canonical model, in which outcomes
are simutaneously determined (see Bramoullé et al. (2019) for a review of
endogenous peer effects). The outcome equation is

$\displaystyle
y_{it}=\alpha_{i}+\psi\sum_{j=1}^{N}\mathbf{G}_{ijt}y_{jt}+\rho\sum_{j=1}^{N}\mathbf{G}_{ijt}\alpha_{j}+\gamma_{g(i,t)}+\epsilon_{it},$
(A.4)

or in stacked form,

$\displaystyle\mathbf{y}=\psi\mathbf{F}\mathbf{y}+(\mathbf{J}+\rho\mathbf{G})\alpha+\mathbf{D}\gamma+\epsilon,$
(A.5)

where $\psi$ is a scalar parameter capturing the endogenous peer effect and
$\mathbf{F}$ is a $NT\times NT$ block diagonal matrix with blocks
$\mathbf{G}_{1},\mathbf{G}_{2},...,\mathbf{G}_{T}$.

Throughout this section we suppose that the rows of $\mathbf{G}$ sum to one
($\mathbf{G}\iota_{N}=\iota_{NT}$), that there are no between-group
interactions ($\mathbf{G}_{ijt}=0$ if $g(i,t)\neq g(j,t)$) and $|\psi|<1$.
These are standard assumptions maintained in almost all papers concerning
identification of peer effects, and are made to ensure that the reduced form
exists and the reduced form correlated effect is proportional to the
structural correlated effect. Solving for the reduced form and taking
expectations yields

$\displaystyle\mathbb{E}[\mathbf{y}]=(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})\mu^{\alpha}+\mathbf{D}(1-\psi)^{-1}\mu^{\gamma}.$
(A.6)

We now modify Proposition 1 to allow for endogenous effects, making use of
$\mathbf{H}=((\mathbf{G}_{1}^{2})^{\prime},(\mathbf{G}_{2}^{2})^{\prime},...,(\mathbf{G}_{T}^{2})^{\prime})^{\prime}$
and vectors
$\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime},\mathbf{v}_{3}^{\prime})^{\prime}$
in the null-space of $[\mathbf{WJ},\mathbf{WG},\mathbf{WH}]$ such that
$\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3}$ are all $N\times 1$.

######  Proposition 2.

$\psi$ and $\rho$ are identified if $\psi+\rho\neq 0$ and there does not exist
a vector
$\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime},\mathbf{v}_{3}^{\prime})^{\prime}$
in the null-space of $[\mathbf{WJ},\mathbf{WG},\mathbf{WH}]$ and scalars
$\delta_{1},\delta_{2},\delta_{3},\delta_{4}$ satisfying
$\delta_{1}\mathbf{v}_{1}-\mathbf{v}_{2}=(\delta_{1}-\delta_{2})\mu^{\alpha}$,
$\delta_{3}\mathbf{v}_{1}+\mathbf{v}_{3}=(\delta_{3}-\delta_{4})\mu^{\alpha}$
and either $\delta_{1}\neq\delta_{2}$ or $\delta_{3}\neq\delta_{4}$.
Otherwise, $\psi$ and $\rho$ are not identified.

The identification conditions are similar in form to those of Proposition 1.
Similarly to Corollary 1.1, generic identification of $\rho$ and $\psi$ is
attained whenever $\mathbf{v}$ lies in a subspace of $\mathbb{R}^{N}$ of
dimension strictly less than $N$. A sufficient condition which can be used in
practice is that $[\mathbf{WJ},\mathbf{WG},\mathbf{WH}]$ has rank at least
equal to $2N+1$. The additional requirement that $\psi+\rho\neq 0$ is needed
because $\psi+\rho=0$ implies
$\mathbb{E}[\mathbf{y}]=\mathbf{J}\mu^{\alpha}+\mathbf{D}(1-\psi)^{-1}\mu^{\gamma},$
such that $\psi$ and $\rho$ are not identifiable. This is because the
endogenous effect exactly offsets the contextual effect, yielding no net peer
effect.

As with Proposition 1, Proposition 2 can also be related to the work of
(Bramoullé et al., 2009). The authors results imply that for $T=1$, the peer
effects $\dot{\psi}$ and $\dot{\rho}$ in the model

$\displaystyle\mathbb{E}[\mathbf{y}|\mathbf{X}]=(\mathbf{I}_{N}-\dot{\psi}\mathbf{F})^{-1}(\mathbf{X}\dot{\beta}+\mathbf{GX}\dot{\rho})+\mathbf{D}(1-\dot{\psi})^{-1}\dot{\mu}^{\gamma}$
(A.7)

are identified if and only if $\mathbf{WJ},\mathbf{WG}$ and $\mathbf{WH}$ are
linearly independent (i.e., if there does not exist nonzero
$\lambda\in\mathbb{R}^{3}$ such that
$\lambda_{1}\mathbf{WJ}+\lambda_{2}\mathbf{WG}+\lambda_{3}\mathbf{WG^{2}}=\mathbf{0}$)
and $\dot{\beta}\dot{\psi}+\dot{\rho}\neq 0$. Analagously to the discussion of
Proposition 1, the linear independence requirement is weaker and can be
satisfied with $T=1$ because $\mathbf{X}$ is assumed to be observable and
exogenous. These requirements can be relaxed when $T\geq 2$. The condition
$\dot{\beta}\dot{\psi}+\dot{\rho}\neq 0$ is analagous to $\psi+\rho\neq 0$ in
Proposition 2, and also rules out the case of no net peer effect.

Under the linear-in-means network, $\rho$ and $\psi$ are not separately
identifiable without additional restrictions. This is because
$\mathbf{H}=\mathbf{G}$, and we have

$\displaystyle\mathbb{E}[\mathbf{y}]=\left(\mathbf{J}+\frac{\psi+\rho}{1-\psi}\overline{\mathbf{G}}\right)\mu^{\alpha}+\mathbf{D}(1-\psi)^{-1}\mu^{\gamma}$
(A.8)

so that only the reduced form parameter $(\psi+\rho)(1-\psi)^{-1}$ is
identifiable. Since (A.8) takes the same form as (3.2), the results of Section
3 can be applied to establish identification of $(\psi+\rho)(1-\psi)^{-1}$.

To separately identify $\rho$ and $\psi$, one can use additional conditional
variance restrictions. To this end, we consider the variance

$\displaystyle\mathbb{V}[\mathbf{Wy}]=$
$\displaystyle\mathbf{W}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})\mathbb{V}[\alpha](\mathbf{J}+\rho\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}\mathbf{W}$
$\displaystyle+$
$\displaystyle\mathbf{W}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}\mathbb{V}[\epsilon](\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}\mathbf{W}$

and the restrictions

$\displaystyle\mathbb{COV}[\epsilon_{it},\epsilon_{js}]=\begin{cases}\sigma^{2}(g(i,t))&i=j,t=s\\\
0&\text{otherwise}\end{cases},\quad\mathbb{COV}[\alpha_{i},\alpha_{j}]=\begin{cases}\sigma^{2}_{\alpha}&i=j\\\
0&\text{otherwise}\end{cases}.$ (A.9)

As with the conditional mean restriction in (3.1) considered thus far, when
$\mathbf{G}$ and $\mathbf{D}$ are treated as random the variance and
restrictions above are conditional on $\mathbf{G},\mathbf{D}$. We continue to
present the fixed case for ease of exposition.

Our approach is similar in spirit to Graham (2008) and Rose (2017), which
consider a cross-section of networks. The restriction on the variance of
$\epsilon$ requires that the transitory shocks experienced by individuals in
the same group be uncorrelated and have equal variance. Uncorrelatedness
implies that outcomes are correlated only due to the correlated effect and
peer effects, whereas equality of variance implies that within a group, no
individuals are subject to systematically larger shocks than others.
Importantly, $\mathbb{COV}[\alpha_{i},\gamma_{g(j,t)}]$ is unrestricted for
all $i,j$ and $t$. This allows for sorting of individuals to groups based on
their time-invariant heterogeneity.

We first present the identification result for the linear-in-means network
structure. In the next section we derive an identification result which
applies to general network structures and under weaker conditional variance
restrictions. In general, we require only that
$\mathbb{COV}[\epsilon_{it},\epsilon_{js}]=0$ if $g(i,t)\neq g(j,t)$. The
stronger restriction in (A.9) is necessary for the linear-in-means network,
which is well known to be the most challenging case (Manski, 1993; Bramoullé
et al., 2009).

######  Proposition 3.

If $\mathbf{G}=\overline{\mathbf{G}}$, $\psi$ and $\rho$ are identified if
$\mathbf{W},\mathbf{WJJ^{\prime}W},\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W},\mathbf{WGG^{\prime}W}$
and $\mathbf{WFW}$ are linearly independent.

In contrast to Proposition 2 there is no requirement that $\psi+\rho\neq 0$.
This is because the endogenous effect operates through the outcomes, and hence
propagates variation both in $\alpha$ and $\epsilon$, whilst the contextual
effect operates through $\alpha$ alone. Since they operate through different
channels, the peer effects can be separated under restrictions on the within-
group variance of $\epsilon$ such as those in (A.9). The identification
condition in Proposition 3 fails if $T=1$ because
$\mathbf{F}=\mathbf{G}=\mathbf{G}^{\prime}=\mathbf{GG}^{\prime}$ and
$\mathbf{J}=\mathbf{I}_{N}$ so that
$\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}=2\mathbf{WGG^{\prime}W}=2\mathbf{WFW}$.
This is in agreement with Rose (2017), which shows that the conditional
variance restrictions in (A.9) are insufficient for identification with a
cross-section of networks.161616Graham (2008) shows that contextual effects
can be identified with a cross-section of networks under additional
restrictions, including on $\mathbb{COV}[\alpha_{i},\gamma_{g(j,t)}]$.

Due to its simplistic nature, the identification condition in Proposition 3
fails in Example 1. However, it typically holds in more realistic examples.
For example, identification is restored if a third individual is added and
individual three is in group 2 in both periods.

#### Conditional Variance Restrictions for a General Network Structure

In the general network case identification can be attained without making
restrictions on the within-group variance of $\epsilon$. This is in contrast
to the linear-in-means network.171717See the end of the proof of Proposition 3
for an explanation of why restrictions on the within-group variance are
necessary for the linear-in-means model. One can use

$\displaystyle\mathbb{COV}[\epsilon_{it},\epsilon_{js}]=0\quad g(i,t)\neq
g(j,s),\quad\mathbb{COV}[\alpha_{i},\alpha_{j}]=\begin{cases}\sigma^{2}_{\alpha}&i=j\\\
0&\text{otherwise}\end{cases}$ (A.10)

instead of (A.9), which allows for unrestricted within-group variance of
$\epsilon$. We make use of the following definition from Rose (2017),
reprinted here for convenience.

  * Definition

Consider $L$ matrices of the same dimension
$\mathbf{A}_{1},...,\mathbf{A}_{L}$. The matrix $\mathbf{A}_{l}$ ($l\in[L]$)
is maximally linearly independent of $\mathbf{A}_{1},...,\mathbf{A}_{L}$ if
$\lambda_{l}=0$ for all $\lambda\in\mathbb{R}^{L}$ such that
$\sum_{l=1}^{L}\lambda_{l}\mathbf{A}_{l}=\mathbf{0}$.

Note that linear independence of $\mathbf{A}_{1},...,\mathbf{A}_{L}$ is
equivalent to each of the $L$ matrices being maximally linearly independent.
In what follows we require only that a strict subset be maximially linearly
independent, which is weaker than linear independence. Identification hinges
on the covariance terms for outcomes of observations in different groups,
which are non-zero provided that there is mobility between groups. To extract
these covariance terms, we use $\mathcal{E}_{m}\subseteq[NT]$ to denote the
indices of the rows of $\mathbf{y}$ corresponding to group $m\in[M]$ and
define $\mathbf{E}_{m}$ as the matrix constructed from rows $\mathcal{E}_{m}$
of $\mathbf{I}_{NT}$. Pre-multiplying any conformable matrix by
$\mathbf{E}_{m}$ extracts rows $\mathcal{E}_{m}$, and post-multiplying by
$\mathbf{E}_{m}^{\prime}$ extracts columns $\mathcal{E}_{m}$. We also use
$\mathbf{E}_{-m}$ to extract the rows in the complement of $\mathcal{E}_{m}$.

######  Proposition 4.

$\psi,\rho,\sigma^{2}_{\alpha}$ are identified if $\psi+\rho\neq 0$ and there
exists $m\in[M]$ such that
$\mathbf{E}_{m}\mathbf{WJJ^{\prime}W}\mathbf{E}_{-m}^{\prime},\mathbf{E}_{m}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
and
$\mathbf{E}_{m}\mathbf{W}(\mathbf{JH}^{\prime}+\mathbf{HJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
are maximially linearly independent from
$\mathbf{E}_{m}\mathbf{WJJ^{\prime}W}\mathbf{E}_{-m}^{\prime},\mathbf{E}_{m}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime},\mathbf{E}_{m}\mathbf{W}(\mathbf{JH}^{\prime}+\mathbf{HJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime},\mathbf{E}_{m}\mathbf{W}(\mathbf{GH}^{\prime}+\mathbf{HG}^{\prime})\mathbf{W}\mathbf{E}_{-m},\mathbf{E}_{m}\mathbf{WGG^{\prime}W}\mathbf{E}_{-m}^{\prime},$
and $\mathbf{E}_{m}\mathbf{WHH^{\prime}W}\mathbf{E}_{-m}^{\prime}$.

Identically to Proposition 2, $\psi+\rho\neq 0$ is required so that the
endogenous and contextual effects do not exactly offset one another. This
contrasts with Proposition 3, which concerns the linear-in-means network, for
which $\psi+\rho\neq 0$ is not required. The reason for this is that
Proposition 3 additionally restricts the within-group variance of $\epsilon$
because restrictions on the between-group variance alone are insufficient for
identification. Since the endogenous effects operate both through $\epsilon$
and $\alpha$, whereas the contextual effects operate only through $\alpha$,
under restrictions on the within-group variance of $\epsilon$ there are no
values of $\psi$ and $\rho$ such that the two peer effects exactly offset one
another in the reduced form variance.

Proof of Proposition 2

Denote $\theta=(\psi,\rho,\alpha,\gamma)$. Under the conditional moment
restriction we have

$\displaystyle\mathbb{E}[\mathbf{y}_{t}]=(\mathbf{I}_{N}-\psi\mathbf{G}_{t})^{-1}(\mathbf{I}_{N}\mu^{\alpha}+\mathbf{G}_{t}\rho\mu^{\alpha}+\mathbf{D}_{t}\mu^{\gamma})$
(A.11)

for $t\in[T]$. The inverse of $(\mathbf{I}_{N}-\psi\mathbf{G}_{t})$ exists
because $\mathbf{G}_{t}\iota_{N}=\iota_{N}$ and $|\psi|<1$. Since
$\mathbf{G}_{t}\iota_{N}=\iota_{N}$ and $\mathbf{G}_{ijt}=0$ if $g(i,t)\neq
g(j,t)$ we have
$(\mathbf{I}_{N}-\psi\mathbf{G}_{t})^{-1}\mathbf{D}_{t}\mu^{\gamma}=\mathbf{D}_{t}(1-\psi)^{-1}\mu^{\gamma}$,
hence

$\displaystyle\mathbb{E}[\mathbf{y}_{t}]=(\mathbf{I}_{N}-\psi\mathbf{G}_{t})^{-1}(\mathbf{I}_{N}+\rho\mathbf{G}_{t})\mu^{\alpha}+\mathbf{D}_{t}(1-\psi)^{-1}\mu^{\gamma}$
(A.12)

Suppose that there is $\overline{\theta}$ satisfying the conditional moment
restriction. Then we have

$\displaystyle(\mathbf{I}_{N}-\psi\mathbf{G}_{t})^{-1}(\mathbf{I}_{N}+\rho\mathbf{G}_{t})\mu^{\alpha}+\mathbf{D}_{t}(1-\psi)^{-1}\mu^{\gamma}=(\mathbf{I}_{N}-\overline{\psi}\mathbf{G}_{t})^{-1}(\mathbf{I}_{N}+\overline{\rho}\mathbf{G}_{t})\overline{\mu}^{\alpha}+\mathbf{D}_{t}(1-\overline{\psi})^{-1}\overline{\mu}^{\gamma}$
(A.13)

Pre-multiplying both sides by
$(\mathbf{I}_{N}-\psi\mathbf{G}_{t})(\mathbf{I}_{N}-\overline{\psi}\mathbf{G}_{t})$
and rearranging yields

$\displaystyle\mathbf{I}_{N}(\mu^{\alpha}-\overline{\mu}^{\alpha})+\mathbf{G}_{t}\left((\rho-\overline{\psi})\mu^{\alpha}-(\overline{\rho}-\psi)\overline{\mu}^{\alpha}\right)+\mathbf{G}_{t}^{2}\left(-\overline{\psi}\rho\mu^{\alpha}+\psi\overline{\rho}\overline{\mu}^{\alpha}\right)$
$\displaystyle+\mathbf{D}_{t}\left((1-\overline{\psi})\mu^{\gamma}-(1-\psi)\overline{\mu}^{\gamma}\right)=\mathbf{0}$
(A.14)

Stacking for $t\in[T]$ yields

$\displaystyle\mathbf{J}(\mu^{\alpha}-\overline{\mu}^{\alpha})+\mathbf{G}\left((\rho-\overline{\psi})\mu^{\alpha}-(\overline{\rho}-\psi)\overline{\mu}^{\alpha}\right)+\mathbf{H}\left(-\overline{\psi}\rho\mu^{\alpha}+\psi\overline{\rho}\overline{\mu}^{\alpha}\right)+\mathbf{D}\left((1-\overline{\psi})\mu^{\gamma}-(1-\psi)\overline{\mu}^{\gamma}\right)=\mathbf{0}$
(A.15)

and applying $\mathbf{W}$ on the left yields

$\displaystyle[\mathbf{WJ},\mathbf{WG},\mathbf{WH}]\begin{pmatrix}\mu^{\alpha}-\overline{\mu}^{\alpha}\\\
(\rho-\overline{\psi})\mu^{\alpha}-(\overline{\rho}-\psi)\overline{\mu}^{\alpha}\\\
-\overline{\psi}\rho\mu^{\alpha}+\psi\overline{\rho}\overline{\mu}^{\alpha}\end{pmatrix}=\mathbf{0}$
(A.16)

Let
$\mathbf{v}=(\mathbf{v}_{1}^{\prime},\mathbf{v}_{2}^{\prime},\mathbf{v}_{3}^{\prime})$
be a vector in the null-space of $[\mathbf{WJ},\mathbf{WG},\mathbf{WH}]$. Then
for some $\mathbf{v}$,

$\displaystyle\mathbf{v}_{1}=\mu^{\alpha}-\overline{\mu}^{\alpha},\quad\mathbf{v}_{2}=(\rho-\overline{\psi})\mu^{\alpha}-(\overline{\rho}-\psi)\overline{\mu}^{\alpha},\quad\mathbf{v}_{3}=-\overline{\psi}\rho\mu^{\alpha}+\psi\overline{\rho}\overline{\mu}^{\alpha}$
(A.17)

Eliminating $\overline{\mu}^{\alpha}$ yields

$\displaystyle(\overline{\rho}-\psi)\mathbf{v}_{1}-\mathbf{v}_{2}=(\overline{\rho}-\psi+\overline{\psi}-\rho)\mu^{\alpha},\quad\psi\overline{\rho}\mathbf{v}_{1}+\mathbf{v}_{3}=(\psi\overline{\rho}-\overline{\psi}\rho)\mu^{\alpha}$
(A.18)

If there does not exist $\mathbf{v}$ in the null-space of
$[\mathbf{WJ},\mathbf{WG},\mathbf{WH}]$ and scalars
$\delta_{1},\delta_{2},\delta_{3},\delta_{4}$ satisfying
$\delta_{1}\mathbf{v}_{1}-\mathbf{v}_{2}=(\delta_{1}-\delta_{2})\mu^{\alpha}$,
$\delta_{3}\mathbf{v}_{1}+\mathbf{v}_{3}=(\delta_{3}-\delta_{4})\mu^{\alpha}$
and either $\delta_{1}\neq\delta_{2}$ or $\delta_{3}\neq\delta_{4}$ then we
have

$\displaystyle\overline{\rho}-\psi+\overline{\psi}-\rho=0,\quad\psi\overline{\rho}-\overline{\psi}\rho=0$
(A.19)

Solving the first equation for $\overline{\psi}$ and injecting into the second
yields $(\overline{\rho}-\rho)(\psi+\rho)=0$, hence $\overline{\rho}=\rho$
provided that $\psi+\rho\neq 0$. Injecting $\overline{\rho}=\rho$ into the
first equation yields $\overline{\psi}=\psi$.

Proof of Proposition 3

Denote $\theta=(\psi,\rho,\sigma^{2}_{\alpha})^{\prime}$. We have the reduced
form

$\displaystyle\mathbf{y}=(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})\alpha+\mathbf{D}(1-\psi)^{-1}\beta+(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}\epsilon$
(A.20)

Since $\mathbf{G}=\overline{\mathbf{G}}$, we have
$\mathbf{G}_{t}=\mathbf{G}_{t}^{2}$ for $t\in[T]$ and
$\mathbf{F}=\mathbf{F}^{2}$, hence
$(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}=\mathbf{I}_{NT}+\frac{\psi}{1-\psi}\mathbf{F}$
and

$\displaystyle\mathbf{W}\mathbf{y}=\mathbf{W}\left(\mathbf{J}+\frac{\psi+\rho}{1-\psi}\mathbf{G}\right)\alpha+\mathbf{W}\left(\mathbf{I}_{NT}+\frac{\psi}{1-\psi}\mathbf{F}\right)\epsilon$
(A.21)

with variance

$\displaystyle\mathbb{V}[\mathbf{Wy}]$
$\displaystyle=\sigma^{2}_{\alpha}\mathbf{W}\left(\mathbf{J}+\frac{\psi+\rho}{1-\psi}\mathbf{G}\right)\left(\mathbf{J}+\frac{\psi+\rho}{1-\psi}\mathbf{G}\right)\mathbf{W}$
(A.22)
$\displaystyle+\mathbf{W}\left(\mathbf{I}_{NT}+\frac{\psi}{1-\psi}\mathbf{F}\right)\Sigma\left(\mathbf{I}_{NT}+\frac{\psi}{1-\psi}\mathbf{F}\right)\mathbf{W}$
(A.23)
$\displaystyle=\sigma^{2}_{\alpha}\mathbf{WJJ^{\prime}W}+\frac{\sigma^{2}_{\alpha}(\psi+\rho)}{1-\psi}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}+\frac{\sigma^{2}_{\alpha}(\psi+\rho)^{2}}{(1-\psi)^{2}}\mathbf{W}\mathbf{GG}^{\prime}\mathbf{W}$
(A.24)
$\displaystyle+\mathbf{W}\Sigma\mathbf{W}+\frac{\psi(2-\psi)}{(1-\psi)^{2}}\mathbf{W}\mathbf{F}\Sigma\mathbf{F}\mathbf{W}$
(A.25)

where $\Sigma=\mathbb{V}[\epsilon]$. Consider group $m\in[M]$ and define
$\mathcal{E}_{m}\subseteq[NT]$ to be the indices of the rows of $\mathbf{y}$
which correspond to group $m$. Then we can define $\mathbf{E}_{m}$ as the
matrix constructed from rows $\mathcal{E}_{m}$ of $\mathbf{I}_{NT}$. Pre-
multiplying any conformable matrix by $\mathbf{E}_{m}$ extracts rows
$\mathcal{E}_{m}$, and post-multiplying by $\mathbf{E}_{m}^{\prime}$ extracts
columns $\mathcal{E}_{m}$. Consider first the within-group variance for group
$m$, given by

$\displaystyle\sigma^{2}_{\alpha}\mathbf{E}_{m}\mathbf{WJJ^{\prime}W}\mathbf{E}_{m}^{\prime}+\frac{\sigma^{2}_{\alpha}(\psi+\rho)}{1-\psi}\mathbf{E}_{m}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}\mathbf{E}_{m}^{\prime}$
(A.26)
$\displaystyle+\frac{\sigma^{2}_{\alpha}(\psi+\rho)^{2}}{(1-\psi)^{2}}\mathbf{E}_{m}\mathbf{W}\mathbf{GG}^{\prime}\mathbf{W}\mathbf{E}_{m}^{\prime}+\sigma^{2}(m)\mathbf{E}_{m}\mathbf{W}\mathbf{E}_{m}^{\prime}+\frac{\sigma^{2}(m)\psi(2-\psi)}{(1-\psi)^{2}}\mathbf{E}_{m}\mathbf{W}\mathbf{F}\mathbf{W}\mathbf{E}_{m}^{\prime}$
(A.27)

and the between group variance given by

$\displaystyle\sigma^{2}_{\alpha}\mathbf{E}_{m}\mathbf{WJJ^{\prime}W}\mathbf{E}_{-m}^{\prime}+\frac{\sigma^{2}_{\alpha}(\psi+\rho)}{1-\psi}\mathbf{E}_{m}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}+\frac{\sigma^{2}_{\alpha}(\psi+\rho)^{2}}{(1-\psi)^{2}}\mathbf{E}_{m}\mathbf{W}\mathbf{GG}^{\prime}\mathbf{W}\mathbf{E}_{-m}^{\prime}$
(A.28)

where $\mathbf{E}_{-m}$ extracts the rows $\mathcal{E}_{m}^{c}$, which denotes
the complement of $\mathcal{E}_{m}$. Notice first that the between-group
variance alone is insufficient to identify $\psi$ and $\rho$. Only
$(\psi+\rho)(1-\psi)^{-1}$ is identifiable based on
$\mathbb{COV}[\epsilon_{it},\epsilon_{jt}]=0$ for all $(i,j,t)\in[N]^{2}\times
T:g(i,t)\neq g(j,t)$ alone. However, under the additional restrictions on the
within-group variance in (A.9), we obtain two additional terms which allow
additionally for identification of $\psi(2-\psi)(1-\psi)^{-2}$. Now suppose
that there is $\overline{\theta}$ satisfying the conditional variance
restrictions in (A.9). Then, if
$\mathbf{W},\mathbf{WJJ^{\prime}W},\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W},\mathbf{WGG^{\prime}W}$
and $\mathbf{WFW}$ are linearly independent then there exists $m$ such that
$\overline{\sigma}^{2}(m)=\sigma^{2}(m)$,
$\overline{\sigma}^{2}_{\alpha}=\sigma^{2}_{\alpha}$ and

$\displaystyle\frac{\psi+\rho}{1-\psi}=\frac{\overline{\psi}+\overline{\rho}}{1-\overline{\psi}},\quad\frac{\psi(2-\psi)}{(1-\psi)^{2}}=\frac{\overline{\psi}(2-\overline{\psi})}{(1-\overline{\psi})^{2}}$
(A.29)

These two equations have two solutions given by
$\overline{\psi}=\psi,\overline{\rho}=\rho$ and
$\overline{\psi}=2-\psi,\overline{\rho}=(2\psi+3\rho-\psi\rho-2)(1+\psi)^{-1}$.
Since $|\psi|<1$ we have $|2-\psi|>1$, hence the second solution is
infeasible. Hence we obtain $\overline{\theta}=\theta$.

Proof of Proposition 4 Denote
$\theta=(\psi,\rho,\sigma^{2}_{\alpha})^{\prime}$. Under the variance
restriction (A.9) we have

$\displaystyle\mathbb{V}[\mathbf{y}]$
$\displaystyle=\sigma^{2}_{\alpha}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})(\mathbf{J}+\rho\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}+\frac{1}{(1-\psi)^{2}}\mathbf{D}\Sigma_{\gamma}\mathbf{D}^{\prime}$
(A.30)
$\displaystyle+(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}\Sigma(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}+\frac{1}{1-\psi}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})\Sigma_{\alpha\gamma}\mathbf{D}^{\prime}$
(A.31)
$\displaystyle+\frac{1}{1-\psi}\mathbf{D}\Sigma_{\alpha\gamma}^{\prime}(\mathbf{J}+\rho\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}+\frac{1}{1-\psi}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}\Sigma_{\epsilon\gamma}\mathbf{D}^{\prime}$
(A.32)
$\displaystyle+\frac{1}{1-\psi}\mathbf{D}\Sigma_{\epsilon\gamma}^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}$
(A.33)

where $\Sigma$ encodes the variance of $\epsilon$, $\Sigma_{\alpha\gamma}$
encodes the covariance terms for $\alpha,\gamma$ and similarly for
$\Sigma_{\epsilon\gamma}$. Suppose that there is $\overline{\theta}$
satisfying the variance restriction in (A.10). Then we have

$\displaystyle\sigma^{2}_{\alpha}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})(\mathbf{J}+\rho\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}+\frac{1}{(1-\psi)^{2}}\mathbf{D}\Sigma_{\gamma}\mathbf{D}^{\prime}$
(A.34)
$\displaystyle+(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}\Sigma(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}+\frac{1}{1-\psi}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}(\mathbf{J}+\rho\mathbf{G})\Sigma_{\alpha\gamma}\mathbf{D}^{\prime}$
(A.35)
$\displaystyle+\frac{1}{1-\psi}\mathbf{D}\Sigma_{\alpha\gamma}^{\prime}(\mathbf{J}+\rho\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}+\frac{1}{1-\psi}(\mathbf{I}_{NT}-\psi\mathbf{F})^{-1}\Sigma_{\epsilon\gamma}\mathbf{D}^{\prime}$
(A.36)
$\displaystyle+\frac{1}{1-\psi}\mathbf{D}\Sigma_{\epsilon\gamma}^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})^{-1}=$
(A.37)
$\displaystyle\overline{\sigma}^{2}_{\alpha}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})^{-1}(\mathbf{J}+\overline{\rho}\mathbf{G})(\mathbf{J}+\overline{\rho}\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})^{-1}+\frac{1}{(1-\overline{\psi})^{2}}\mathbf{D}\overline{\Sigma}_{\gamma}\mathbf{D}^{\prime}$
(A.38)
$\displaystyle+\epsilon(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})^{-1}\overline{\Sigma}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})^{-1}+\frac{1}{1-\overline{\psi}}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})^{-1}(\mathbf{J}+\overline{\rho}\mathbf{G})\overline{\Sigma}_{\alpha\gamma}\mathbf{D}^{\prime}$
(A.39)
$\displaystyle+\frac{1}{1-\overline{\psi}}\mathbf{D}\overline{\Sigma}_{\alpha\gamma}^{\prime}(\mathbf{J}+\overline{\rho}\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})^{-1}+\frac{1}{1-\overline{\psi}}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})^{-1}\overline{\Sigma}_{\epsilon\gamma}\mathbf{D}^{\prime}$
(A.40)
$\displaystyle+\frac{1}{1-\overline{\psi}}\mathbf{D}\overline{\Sigma}_{\epsilon\gamma}^{\prime}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})^{-1}$
(A.41)

Pre-multiplying both sides by
$\mathbf{W}(\mathbf{I}_{NT}-\psi\mathbf{F})(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})$
and post-multiplying by
$(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})\mathbf{W}$
yields

$\displaystyle\sigma^{2}_{\alpha}\mathbf{W}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})(\mathbf{J}+\rho\mathbf{G})(\mathbf{J}+\rho\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})\mathbf{W}+\mathbf{W}(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F})\Sigma(\mathbf{I}_{NT}-\overline{\psi}\mathbf{F}^{\prime})\mathbf{W}=$
$\displaystyle\overline{\sigma}^{2}_{\alpha}\mathbf{W}(\mathbf{I}_{NT}-\psi\mathbf{F})(\mathbf{J}+\overline{\rho}\mathbf{G})(\mathbf{J}+\overline{\rho}\mathbf{G})^{\prime}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})\mathbf{W}+\mathbf{W}(\mathbf{I}_{NT}-\psi\mathbf{F})\overline{\Sigma}(\mathbf{I}_{NT}-\psi\mathbf{F}^{\prime})\mathbf{W}$
(A.42)

Rearranging yields

$\displaystyle(\sigma^{2}_{\alpha}-\overline{\sigma}^{2}_{\alpha})\mathbf{WJJ^{\prime}W}+(\sigma^{2}_{\alpha}(\rho-\overline{\psi})-\overline{\sigma}^{2}_{\alpha}(\overline{\rho}-\psi))\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}$
(A.43)
$\displaystyle+(\overline{\sigma}^{2}_{\alpha}\psi\overline{\rho}-\sigma^{2}_{\alpha}\overline{\psi}\rho)\mathbf{W}(\mathbf{JH}^{\prime}+\mathbf{HJ}^{\prime})\mathbf{W}+(\overline{\sigma}^{2}_{\alpha}(\overline{\rho}-\psi)\psi\overline{\rho}-\sigma^{2}_{\alpha}(\rho-\overline{\psi})\overline{\psi}\rho)\mathbf{W}(\mathbf{GH}^{\prime}+\mathbf{HG}^{\prime})\mathbf{W}$
(A.44)
$\displaystyle+(\sigma^{2}_{\alpha}(\rho-\overline{\psi})^{2}-\overline{\sigma}^{2}_{\alpha}(\overline{\rho}-\psi)^{2})\mathbf{WGG^{\prime}W}+(\sigma^{2}_{\alpha}\overline{\psi}^{2}\rho^{2}-\overline{\sigma}^{2}_{\alpha}\psi^{2}\overline{\rho}^{2})\mathbf{WHH^{\prime}W}+\mathbf{W}(\Sigma-\overline{\Sigma})\mathbf{W}$
(A.45)
$\displaystyle+\mathbf{W}(\mathbf{F}(\psi\overline{\Sigma}-\overline{\psi}\Sigma)+(\psi\overline{\Sigma}-\overline{\psi}\Sigma)\mathbf{F}^{\prime})\mathbf{W}+\mathbf{WF}(\overline{\psi}^{2}\Sigma-\psi^{2}\overline{\Sigma})\mathbf{F^{\prime}W}=\mathbf{0}$
(A.46)

Since the within-group variance in $\epsilon$ is not restricted by (A.10),
identification hinges on between-group variance in the outcomes. Consider the
covariance terms for the outcomes of group $m$ with those in all other groups,
for which we have the equations

$\displaystyle(\sigma^{2}_{\alpha}-\overline{\sigma}^{2}_{\alpha})\mathbf{E}_{m}\mathbf{WJJ^{\prime}W}\mathbf{E}_{-m}^{\prime}+(\sigma^{2}_{\alpha}(\rho-\overline{\psi})-\overline{\sigma}^{2}_{\alpha}(\overline{\rho}-\psi))\mathbf{E}_{m}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
$\displaystyle+(\overline{\sigma}^{2}_{\alpha}\psi\overline{\rho}-\sigma^{2}_{\alpha}\overline{\psi}\rho)\mathbf{E}_{m}\mathbf{W}(\mathbf{JH}^{\prime}+\mathbf{HJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
(A.47)
$\displaystyle+(\overline{\sigma}^{2}_{\alpha}(\overline{\rho}-\psi)\psi\overline{\rho}-\sigma^{2}_{\alpha}(\rho-\overline{\psi})\overline{\psi}\rho)\mathbf{E}_{m}\mathbf{W}(\mathbf{GH}^{\prime}+\mathbf{HG}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
(A.48)
$\displaystyle+(\sigma^{2}_{\alpha}(\rho-\overline{\psi})^{2}-\overline{\sigma}^{2}_{\alpha}(\overline{\rho}-\psi)^{2})\mathbf{E}_{m}\mathbf{WGG^{\prime}W}\mathbf{E}_{-m}^{\prime}$
(A.49)
$\displaystyle+(\sigma^{2}_{\alpha}\overline{\psi}^{2}\rho^{2}-\overline{\sigma}^{2}_{\alpha}\psi^{2}\overline{\rho}^{2})\mathbf{E}_{m}\mathbf{WHH^{\prime}W}\mathbf{E}_{-m}^{\prime}=\mathbf{0}$
(A.50)

where $\mathbf{E}_{m}$ and $\mathbf{E}_{-m}^{\prime}$ are defined in the proof
of Proposition 3. If there exists $m\in[M]$ such that
$\mathbf{E}_{m}\mathbf{WJJ^{\prime}W}\mathbf{E}_{-m}^{\prime},\mathbf{E}_{m}\mathbf{W}(\mathbf{JG}^{\prime}+\mathbf{GJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
and
$\mathbf{E}_{m}\mathbf{W}(\mathbf{JH}^{\prime}+\mathbf{HJ}^{\prime})\mathbf{W}\mathbf{E}_{-m}^{\prime}$
are maximially linearly independent from the other matrices then we have
$\sigma^{2}_{\alpha}=\overline{\sigma}^{2}_{\alpha}$ and

$\displaystyle\overline{\rho}-\psi+\overline{\psi}-\rho=0,\quad\psi\overline{\rho}-\overline{\psi}\rho=0$
(A.51)

Solving the first equation for $\overline{\psi}$ and injecting into the second
yields $(\overline{\rho}-\rho)(\psi+\rho)=0$, hence $\overline{\rho}=\rho$
provided that $\psi+\rho\neq 0$. Injecting into the first equation yields
$\overline{\psi}=\psi$.