# Moment-Based Estimation of Diffusion and Adoption Parameters in Networks

L.S. Sanna Stephan 111I sincerely thank Prof. Matthew O. Jackson from Stanford
University, whose ideas and suggestions have been essential for the
development of the proposed estimator.

##  1 Introduction

According to standard econometric theory, Maximum Likelihood estimation (MLE)
is the efficient estimation choice, however, it is not always a feasible one.
In network diffusion models with unobserved signal propagation, MLE requires
integrating out a large number of latent variables, which quickly becomes
computationally infeasible even for moderate network sizes and time horizons.
Limiting the model time horizon on the other hand entails loss of important
information while approximation techniques entail a (small) error that.
Searching for a viable alternative is thus potentially highly beneficial. This
paper proposes two estimators specifically tailored to the network diffusion
model of partially observed adoption and unobserved network diffusion.
Akin to Generalized Method of Moments (GMM) estimators, they are constructed
using moment conditions and minimize the deviation of individual outcomes from
their unconditional expected value. They distinguish themselves from usual GMM
estimators in three important ways: first, moments are specific to
individuals. This also implies that the correlations in individual moment-
conditions are pair-specific. Second, these correlations potentially entail
small sample bias when individual moment conditions are aggregated. Third, for
computational reasons outlined below, each agent will contribute only to one
moment, resulting in moment-specific sample sizes whenever one chooses to
aggregate individual moment conditions.
Moment-based estimators have previously been derived for and applied to social
network models. Two strands of research can be distinguished. First, various
authors have applied moment-based estimates to network formation models where
the aim is to uncover the unobserved network structure (Snijders (2017),
Bickel et al. (2011)). Second, these techniques have been used for fitting
models of observed network interaction in general (Liu and fei Lee (2010)) and
spatial-auto-regressive models in particular (Su and Yang (2011)). However, to
the best of my knowledge, moment-based estimation has not been applied to the
“latent-diffusion-observed-adoption” model of Banerjee et al. (2013). Given
the widespread applicability of the latter, the contribution made in this
paper is thus important and relevant.
Network models entail the challenge that observations for individuals within
the same network are in general correlated. Correlation between the individual
observations that are aggregated into a moment is a challenge typically
encountered in Time Series analysis. Various papers have shown that the GMM
estimator is consistent if the data is mean stationary and ergodic (see Hansen
(2001)). The main difference between the application of GMM estimation to Time
Series and Network Models is that in the latter case, while correlations are
typically bounded and decay to zero asymptotically, yet no such notion as
covariance stationarity exists, making the estimation of the estimators’
covariance matrices computationally more challenging. Hansen and Lee (2019)
have derived conditions for asymptotic normality and consistent covariance
estimation in clustered samples, a theory that applies to network models. Yet
it is challenging at this point to state how covariance estimates based on
this theory perform in finite samples.
In the model at hand, as individuals exchange information in the network, the
network inter dependencies solely concern the information reception
probabilities, which are hard to evaluate due to the large number of
possibilities by which an agent may have been reached by the information.
Since any moment requires computation of these probabilities, moment-based
estimation in principle still suffers from the same problems as MLE.
Calculating the probability of any particular agent to receive the information
at any point in time $t$ requires integrating out the entire information
status matrix of all villagers from period one to period $t-1$. The remedy of
the dimensionality problem is not so much the idea to use moments itself, but
the fact that a moment-based approach enables me to focus on the set of moment
conditions that are easy to evaluate. As the system is over-identified, no
approximation technique is needed, exact estimation is feasible. Precisely, an
agent’s information reception probability can straightforwardly be computed in
the very first period in which she can be reached by the information. Here,
some shorthand formulas exist that do not require integrating out the
neighbours’ information scenarios. These formulas are correct in the one and
only period which is the first in which an agent can be reached by the signal.
What makes rates hard to compute in subsequent periods is the fact that the
information may stop or travel on when it reaches intermediate agents and that
the latter can thereafter exchange the information among themselves. This
increases the number of possible scenarios. When we consider an agent’s first
information reception opportunity, on the other hand, the information can only
possibly have travelled directly from its “injection point” (IP) to the agent.
Therefore, only the direct path from the agent to (an) IP(s) matter(s) and
linkages between intermediate agents on that path can be neglected. As a
consequence, the number of scenarios that are needed to be considered in order
to compute the (first-opportunity) reception probability are kept small and
easily tractable formulas can be used for most agents. Notably, these formulas
do get more and more sophisticated for agents further away from the
information source, but for an intermediate model time horizon (here $T=4$)
they remain tractable.
Due to the network dependencies, individual moments are in general correlated
within the village. If the sample size increases while village sizes are kept
constant (e.g. the increase in the sample size is achieved by adding more
villages), the variance of the estimator does converge to zero, albeit at a
rate smaller than the one observed for usual GMM estimators. This is
intuitive: if individuals contribute to only one moment condition, adding more
individuals will not decrease the variance proportionally.
The remainder is organized as follows. First, the model and assumptions are
recapitulated. Individual-specific moment conditions are the basis for any
moment-based estimation in this model. Section 3 introduces these conditions
and explains how they can be calculated and used for parameter identification.
This section includes the main novelty of the estimation approach, namely the
insight that individual means are in general hard, but under certain
conditions easy to calculate. It also discusses the challenges one faces when
wanting to set up an estimator based on the individual-specific moment
conditions. The following two sections propose two such estimators and
analyses their properties. Section 6 provides a comparison, listing
similarities and differences of the two estimators. Section 7 presents
evidence from a Monte Carlo study that compares the estimates and section 8 a
small application of the estimators to the real data. Section 9 concludes.
Proofs of the relevant theories are provided in the Appendix.

##  2 The Model

In this and the following sections, I use the terminology of the concrete
application despite the fact that the model and the estimation technique are
applicable to many research questions.
We dispose over a sample of villages, indexed by $v=1,...,V$.
$N_{v}$ individuals in village $v$ are linked in a network, represented by the
network adjacency matrix $G_{v}$. While villagers $i=1,...,N_{v}$ entertain
relationships among themselves, they are supposed not to interfere with agents
living in other villages. The symmetric matrix $G_{v}$ comprises binary
variables indicating the presence (absence) of a link. If the entry in row
$i$, column $j$ takes the value of one (i.e. $g_{v,ij}=g_{v,ji}=1$), we
denominate $i$ and $j$ as “neighbors” or “friends”. An organization enters the
community and starts providing a new technology to its members. In period
$t=0$, they advertise their innovation to a subset of them (referred to as the
“information injection points” or “IPs”) and thereafter rely on word-of-mouth
marketing.
In each subsequent period ($t=1,...,4$), two processes take place: first,
newly informed individuals face the choice of whether or not to adapt the
technology, second, informed individuals can instruct their neighbors about
the novel opportunity.
The process is modeled using the random matrices $Y_{v}$ and $S_{v}$,
containing dummy variables for respectively the participation and information
statuses of each of the villagers at each point in time. Let $Y$ and $S$ refer
to the sample outcome and information status matrices, respectively. Let
$Y_{it}=1$ ($S_{it}=1$) (i.e. the row $i$, column $t$ element of $Y$ ($S$)
displays the value one) implies that $i$ participates (is informed) in period
$t$. The distributions of the random matrices $Y$ and $S$ depend on $G_{v}$
and $s_{v,0}$.
For each village, we observe the network ($G_{v}$), the information
initialization $s_{v,0}$. and the individual participation decisions over time
(i.e. one realization of $Y_{v}$). The information statuses of all inhabitants
but the IPs are generally unobserved. In particular, there usually exist
various realizations of $S_{v}$ that are in accordance with the data at hand.
The following assumptions are made:

Assumption 1: Independence across villages.
Villages can be treated as independent entities. In particular, there is no
link between any two agents that reside in different communities.

Assumption 2: Exogenous network.
The network is exogenous, fixed and observed. Measurement error is negligible.

Assumption 3: Timing.
Each period consists of two processes: First, newly informed individuals
decide upon participation, second, information is exchanged. This implies that
$Y$ is of dimension $N_{v}\times 4$ while $S$ is of dimension $N_{v}\times 3$:
since information is exchanged after the participation decisions, modeling the
last period’s information exchange is redundant.

Assumption 4: Information is a pre-condition for participation.

Assumption 5: Participation is a one-time opportunity.
Each period only newly informed individuals face the participation decision.
Having opted in (out), the respective individual will thereafter stay in the
set of participants (non-participants) forever.

Assumption 6: Distributional assumption.
Conditional on being newly informed, the random variables $Y_{it}\hskip
4.26773pt\forall i=1,...,N_{v};\hskip 4.26773ptt=1,...,4$ are i.i.d. and
follow a Bernoulli distribution. The probability to participate conditional on
being newly informed is $p$.

Assumption 7: Information is never forgotten.
Individuals can only switch their information status once (from being
uninformed ($S_{i(t-1)}=0$) to being informed ($S_{it}=1$)). Any informed
individual remains potentially informing her neighbors every period.

Assumption 8: Information exchange.
In each period, informed individuals may transmit the information to their
neighbors. On average, they do so with probability $q$. Transmitting is
independent across individuals.
Let the random variable $I_{ij,t}$ denote the indicator that shows that
individual $i$ has sent the information to individual $j$ in period $t$.
Conditional on $i$ being a sender, the variables $I_{ij,t}\hskip
2.84544pt\forall i\neq j;t=1,...,3$ are i.i.d. and follow a Bernoulli
distribution with $E[I_{ij,t}]=q$.

The aim is to estimate
(i) the individual’s probability to participate if informed ($p$) and
(ii) individual’s probability to share information with acquaintances ($q$)

##  3 Unconditional Individual-Specific Moments and Moment Conditions

###  3.1 Individual-specific Mean-Conditions

From the above it is apparent that the outcomes follow a Bernoulli
distribution with individual and time specific means. Let

$\mu_{it}(p,q,S_{0},G)=E[Y_{it}|p,q,S_{0},G]$

be the unconditional expected value of individual $i^{\prime}s$ outcome in
period $t$. As the Bernoulli distribution is characterised in terms of its
mean, it is indispensable for any moment-based estimation to first derive a
closed-form expression for $\mu_{it}$. The data comprises $N\times T$
outcomes, for which $N\times T$ means can be derived. The mean depends on
whether and how the individual is connected to the information and hence it is
a function of the network, the information initiation and the parameters and
although its concrete functional form potentially differs across agents, it is
computed using the same (time period specific) formulas (outlined below) for
all agents. Once an expression for $\mu_{it}$ is available, it can be used to
construct individual-specific moment conditions that can be used to set up a
criterion function for estimation. The most simplistic moment-condition is
obtained by taking the individual’s outcome and subtracting its mean, e.g.

$g_{it}=Y_{it}-\mu_{it}(p,q,S_{0},G)$

Clearly,

$E[g_{it}]=0$

Due to network dependencies, the attempt to calculate $\mu_{it}$ can lead to
prohibitively complicated expressions. However, for the period that follows
the first period in which an agent can be reached by the information, the
unconditional mean can be straightforwardly derived. This is the aim of this
section.
As stated, for each agent, I wish to solely use the first period in which the
agent potentially faces a choice, e.g. the first period which they may enter
being informed. In the following, I use $N$ ($N_{v}$) to denote the number of
individuals in the sample (in village $v$) that can be reached by the
information within the modelled time horizon. Let $\tilde{N}_{t}$ be a vector
of indicator variables used to identify the set of individuals who enter the
information radius at time $t-1$ and who fulfill certain conditions outlined
below (e.g. $\tilde{N}_{it}=1$ implies that period $t-1$ is the first period
in which individual $i$ has a positive probability to be informed). Note that
these indicators solely depend on the network and are thus observable. Being
in reach at time $t-1$ implies that the individual faces a choice for the
first time at time $t$. Exact estimation is feasible if only the set of
individuals for which $\tilde{N}_{it}=1$ is used for estimation at each point
in time.
The indicators are used to select the correct period in which the individual-
specific moment condition shall be used for estimation. Since
$\tilde{N}_{it}=0$ in any period except the one to be employed, hence I may
simply multiply the individual- and time-specific moment conditions with the
individual- and time-specific indicator variables and sum up over time to
obtain the desired individual-specific moment condition, which is

$g_{i}=\sum_{t=1}^{T}\tilde{N}_{it}\big{(}Y_{it}-\mu_{it}(p,q,S_{0},G)\big{)}$

Since the term in brackets only becomes relevant if $\tilde{N}_{it}=1$, I need
to derive the mean of the outcome, conditional on $\tilde{N}_{it}=1$.

$\sum_{t=1}^{T}\tilde{N}_{it}\mu_{it}=\sum_{t=1}^{T}\tilde{N}_{it}E[Y_{it}|p,q,S_{0},G]=E[Y_{it}|\tilde{N}_{it}=1,p,q,S_{0},G]$

Since $Y_{it}$ is binary, thus

$E[Y_{it}|\tilde{N}_{it}=1,p,q,S_{0},G]=P(Y_{it}=1|\tilde{N}_{it}=1,p,q,S_{0},G)$

Observe that the individual must enter period $t-2$ as uninformed because
period $t-1$ is the first period in which she can be reached by the
information. Therefore, conditioning on $\tilde{N}_{it}=1$ incorporates the
information that $S_{i(t-2)}=0$.

$P(Y_{it}=1|\tilde{N}_{it}=1,p,q,S_{0},G)=P(Y_{it}=1|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)$

The second piece of information (besides not being informed in period $t-2$)
that is incorporated in $\tilde{N}_{it}=1$ is that the individual has a
positive probability to receive the information in period $t-1$, but a zero
probability to receive the information any earlier. Only if a switch in
information status has occurred in period $t-1$, a switch in the outcome is
possible in period $t$. Consequently, the probability of the outcome being one
can only be established conditioning on the agent’s information status in the
last two periods. Whether or not the agent received the information in the
exchange of period $t-1$ is not observable, hence I employ again the Law of
Total Probability to obtain

$P(Y_{it}=1|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)=$
$\sum_{s_{i,(t-1)}=0,1}\Big{\\{}P(Y_{it}=1|S_{i(t-1)}=s_{i(t-1)},S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)\times$
$P(S_{i(t-1)}=s_{i(t-1)}|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)\Big{\\}}$

Note that the outcome in period $t$ depends only on the parameter $p$ and on
whether or not the individual has been newly informed in the previous period.
As a consequence, it is sufficient to condition on $S_{i,(t-1)}$, which takes
the value one if the information has been received and zero if not,
$S_{i(t-2)}$, which is known to be zero, and the model parameter $p$.

$P(Y_{it}=1|S_{i(t-1)}=s_{i(t-1)},S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)=$
$P(Y_{it}=1|S_{i(t-1)}=s_{i(t-1)},S_{i(t-2)}=0,p)$ $\mbox{\it for
}s_{i(t-1)}=[0;1]$

and therefore

$P(Y_{it}=1|\tilde{N}_{it}=1,p,q,S_{0},G)=$
$P(Y_{it}=1|S_{i(t-1)}=0,S_{i(t-2)}=0,p)P(S_{i(t-1)}=0|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)+$
$P(Y_{it}=1|S_{i(t-1)}=1,S_{i(t-2)}=0,p)P(S_{i(t-1)}=1|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)$

Observe that the probability to participate in period $t$ is zero if the
individual did not receive the information, hence the first term cancels. Note
further that conditional on being newly informed, the individual’s probability
to participate is independent of any other agent’s information or
participation status and fixed at $p$. Thus

$P(Y_{it}=1|\tilde{N}_{it}=1,p,q,S_{0},G)=$
$P(Y_{it}=1|S_{i(t-2)}=0,S_{i(t-1)}=1,p)P(S_{i(t-1)}=1|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)$
$=pP(S_{i(t-1)}=1|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0})$

This shows that the unconditional expected value of the agent’s outcome in the
first period in which she potentially faces a choice is the parameter $p$
multiplied by the agent’s probability to receive the information at the first
possible occasion and I now seek to derive a closed-form expression for the
latter. The agent’s probability to receive the information in any period $t$
potentially depends on the information status of all other agents in period
$t-2$. This is because any agent $j$ who enters period $t-1$ as informed (e.g.
$S_{j(t-2)}=1)$ and who is linked to $i$ may send her the information.
Define the individual’s information reception probability at time $t-1$,
conditional on the information status vector of all villagers in period $t-2$
as

$r_{i(t-1)}(q,S_{t-2})=1-\prod_{j=1}^{N}(1-qg_{ij}S_{j(t-2)})$ (1)

Observe that $j$ can send to $i$ only if the two agents are linked $g_{ij}=1$
and further $j$ enters the period informed $S_{j(t-2)}=1$. Given both of these
conditions are fulfilled, then the probability that the information spreads
over a particular link is by assumption $q$. As a consequence, one minus the
product of all three terms results in the probability of $j$ not sending to
$i$. Intuitively, $i$ receives the information whenever somebody sends it,
thus the counterfactual (nobody sends it, the product of all relevant agents
not sending it) can be used to calculate $r_{it}$.
Information reception probabilities of the IPs are one, as they are known to
be informed. For all other agents

$E[r_{i(t-1)}(q)]=\sum_{s_{t-2}}r_{i(t-1)}(q,s_{t-2})P(S_{t-2}=s_{t-2},p,q,S_{0},G)$

Where I again used the Law of Total Probability: with $S_{t}$ (the information
status of all villagers at point in time $t$) being unobserved after period
$t=0$, thus calculating information reception probabilities hence implies
integrating out the information status variables of all individuals that are
linked to the agent. This technique is regularly employed in latent variable
models, what makes its application challenging for the model at hand is the
multitude of possible realizations $s_{t-2}$ of the random vector $S_{t-2}$.
Further, the information status of all villagers in period $t-2$ in turn
depends on the information status of their neighbours in the preceding period,
which again is a function of the precedent information status vector. This
argumentation can be applied until one ultimately ends up with the IPs, who
are known to be informed. Because in each period there are a multitude of
possibilities by which the information could have reached an agent, it is
necessary to integrate out the entire random matrix $S_{1:t-2}$, comprising
the information statuses of all villagers in all preceding time periods. Hence

$E[r_{i(t-1)}(q)]=\sum_{s_{1:t-2}}r_{i(t-1)}(q,s_{1:t-2})P(S_{1:t-2}=s_{1:t-2},p,q,S_{0},G)$

an expression that would have to be derived by iterated conditioning and then
applying yet again the law of total probability. With every agent having two
information statuses, the number of information scenarios (i.e. realizations
of $s_{1:t-1}$) grows exponentially in the number of agents. This demonstrates
that - as claimed in the introduction - calculating information reception
probabilities becomes prohibitively costly even for intermediate time horizons
and village sizes.
The probability of the agent entering period $t$ as informed in principle
depends on all information exchanges that have been taking place beforehand.
However, conditional on $\tilde{N}_{it}=1$, only the last information exchange
needs to be considered as it was agent $i^{\prime}s$ first opportunity to be
informed. Thus

$P(S_{i(t-1)}=1|S_{i(t-2)}=0,\tilde{N}_{it}=1,p,q,S_{0},G)=E[r_{i(t-1)}(q)]$

Then

$\mu_{it}(p,q,S_{0},G|\tilde{N}_{it}=1)=pE[r_{i(t-1)}(q)]$

And hence the moment conditions employed are

$g_{i}(p,q,S_{0},G)=\sum_{t=1}^{T}\tilde{N}_{it}(Y^{i}_{t}-pE[r_{i(t-1)}(q)])$
$E[g_{i}(p,q,S_{0},G)]=0$

Note that the mean formula $\mu_{it}$ was established conditioning on
$\tilde{N}_{it}=1$ and hence is correct in only that one precise period. Since
the indicator vector $\tilde{N}_{it}$ will be one in only one specific time
period (the first in which the individual can be reached by the information),
summing over time periods correctly selects the period for which the formula
is applicable.
Naturally, other moment conditions can be derived. However, for one-parameter
distributions, all higher order moments are functions of the mean and as such,
the above derivations remain relevant.
Having said that means are individual and time period specific is however
slightly sloppy: indeed they are time-period and link-portfolio specific,
since $r$ is pinned down by $G$ and $S_{0}$. Therefore, there are potentially
multiple individuals with the same mean. This will be important for
consistency of any estimator that is established on the basis of the vector
$g_{1},...,g_{N}$: as $N$ goes to infinity, every possible link portfolio will
be observed in the data sufficiently often for any function that is
established by summing up functions of the $g_{i}^{\prime}s$ to converge to
its expected value.

###  3.2 Information Reception Probabilities

Evaluating the individual-specific moment conditions above still requires an
expression for $\tilde{N}_{it}E[r_{i(t-1)}(q)]$. While (1) gives a formula how
to compute the reception rate $r_{i(t-1)}$ given a specific realisation of
$S_{t-2}$, the puzzle of how to circumvent the cumbersome computations of
integrating out the random matrix $S_{1:t-2}$ still remains to be solved. This
is the aim of this subsection.
For IP-neighbours, calculating the information reception rates is
straightforward. With $S_{0}$ being known, I can simply apply (1). Let
$g_{ij}=1$ if $i$ and $j$ are linked and zero otherwise, then

$\bar{r}_{i1}(q)=1-\prod_{\tilde{N}_{j1}=1}(1-qg_{ij})$

The information status vector of all individuals jointly fulfills the Markov
property: conditioning on the last period’s information status vector, I can
calculate the current period’s information reception probabilities. However,
in order to obtain the unconditional reception rates, the latent past
information status vector would have to be integrated out using once again the
law of total probability. The number of information scenarios to be considered
and hence the number of summands increases exponentially in the number of
agents and time periods, thus making the calculation prohibitively time
intense. The issue can be resolved in period two and three: given the choice
of moment conditions, I can replace the past latent variables by their
expectation (which is the respective individual’s reception rate). This leads
to the correct reception rate exclusively in the first period in which an
agent can be reached by the information. For the second and in some cases the
third information exchange, the same formula can be used as for the direct IP-
neighbours. The following Theorem, a proof of which can be found in the
Appendix, summarizes these findings.

######  Theorem 1.

For individuals that are two links away from the information injection, the
rates obtained from replacing the individuals’ neighbours’ latent variables
with their expected values will, in period 2 (the first period in which an
agent can be reached), coincide with the rates that would be obtained from
fully integrating out the random latent variables.
For individuals that are three links away from the information injection, the
rates obtained from replacing the individuals’ neighbours’ latent variables
with their expected values will, in period 3 (the first period in which an
agent can be reached), coincide with the rates that would be obtained from
fully integrating out the random latent variables if there are no circles
involving only non-IP non-participants on the way from the IP to the
individual.

I now apply Theorem 1 to the second information exchange. Precisely, the
random variables ($S_{j1}$) are replaced by the reception rates
($\bar{r}_{j1})$ in (1).

$\bar{r}_{i2}(q)=1-\prod_{\tilde{N}_{k2}=1}(1-\bar{r}_{k1}(q)qg_{ik})$

For the third information exchange (used for the moments in period four), it
is also not necessary to fully integrate out the past information status
matrix. Here, however, two different formulas are available, depending on the
final agent’s link portfolio. These formulas are applied in accordance with
the following Theorem.

######  Theorem 2.

For individuals that are three links away from the information injection and
for whom there are circles involving only non-IP non-participants on the way
from the IP to the individual, the rates obtained from replacing the
individual’s neighbours’ latent variables with their expected values and using
an adapted formula will, in period 3 (the first period in which an agent can
be reached), coincide with the rates that would be obtained from fully
integrating out the random latent variables if and only if their direct
neighbours do not have several friends in common that are directly linked to
an initially informed individual, i.e.

$\bar{r}_{i3}(q)=1-\prod_{\tilde{N}_{l3}=1}(1-\bar{r}_{l2}(q)qg_{il})\hskip
12.80365pt\mbox{if Theorem 1 applies}$
$\bar{r}_{i3}(q)=1-\prod_{\tilde{N}_{k2}=1}(1-\bar{r}_{1k}(q)(1-\prod_{\tilde{N}_{l3}=1}(1-q^{2}g_{kl}g_{li})))\hskip
12.80365pt\mbox{if Theorem 3 applies}$

The first formula again uses Theorem 1 and replaces the (scenario specific)
information status of $i^{\prime}s$ neighbours by their information reception
probabilities in (1).
The last formula merits some further explanation. Agent $K$, who is directly
linked to an IP is two links away from agent $i$ and there may be multiple
paths between them, each passing through a specific intermediate agent $l$
(such that $g_{kl}g_{li}=1$). Since these are lines, the possibility of the
information reaching $i$ through any one of them is simply $q^{2}$.
Consequently, the probability of the information not reaching $i$ through that
path is $1-q^{2}$ and if we take this to the power of the number of paths, I
obtain the probability of agent $i$ not being informed through any of them.
This conveniently facilitates the computation of agent $i$ being reached
through any (one or several) paths from agent $k$ to agent $i$ as one minus
the probability of her being reached through none. Since no path can frequent
an intermediate agent twice by the condition of Theorem 2, they can be treated
as independent. Finally, for agent $i$ to be informed through a path
originating at agent $k$, the latter must have received the information in the
first period.
It is crucial to take notice that all these rates are non-random, not scenario
specific, but deterministic functions, pinned down by the network and the
information initiation.
Recapitulate that $r_{i(t-1)}(S_{t-2},q)$ is individual $i^{\prime}s$
information reception probability in period $t-1$ given the previous
information scenario $S_{t-2}$. The crucial insight that drastically
facilitates computations and hence makes exact estimation possible is that

$E[r_{i(t-1)}(q)|\tilde{N}_{it}=1]=\sum_{t=1}^{T}\tilde{N}_{it}E[r_{i(t-1)}(q)]=\sum_{t=1}^{T}\tilde{N}_{it}\bar{r}_{i(t-1)}(q)$

where the expectation is taken by integrating out the information scenarios.
This is indeed what is stated in Theorems 1 and 2.
The aforementioned formulas are correct for the first period in which the
agent can be reached by the information. For period 1, this is trivial: the
IPs are known and there is no randomness. For period 2 and 3, a proof is
provided in the Appendix. The moment conditions become

$g_{i}(p,q,S_{0},G)=\sum_{t=1}^{T}\tilde{N}_{it}(Y_{it}-p\bar{r}_{i(t-1)})$
(2)

Naturally, in any period in which $\tilde{N}_{it}=0$, replacing the expected
value of the individual’s information reception probability with the estimate
computed on the basis of Theorem 1 or 2 is incorrect, but since
$\tilde{N}_{it}=1$ only for the one period in which using the simple formulas
leads to a correct calculation, hence this is irrelevant. Exact estimation is
thus feasible if, at each point in time, only a sub-sample of individuals is
chosen to contribute to the objective function, namely those for which the
rates can be straightforwardly obtained correctly. This implies that each
individual contributes to the objective function just once. Given that the
reception rates used are correct, there is no estimation bias.

###  3.3 Within-Village Correlation

This subsection explains how correlation in outcomes arises within the
village. The result of this correlation is that any moment-based estimator
exhibits a co-variance that is hard to evaluate or estimate. When villages are
numerous, this correlation has minor effects, yet given a small to moderate
number of villages, it impedes a fully efficient estimation.
Correlation arises through the information reception probabilities. For any
two agents residing in different villages, the moments are per definition
uncorrelated. Further, IP moments are uncorrelated within and across villages.
Assume that agent $i$ and agent $j$ are non-IP individuals residing in the
same village. Below I will use some very simple examples to illustrate how
correlation of individual moment conditions arise within the village. I derive
the covariation in individual moment conditions using formula (2).

$E[g_{i}(p,q,G,S_{0})g_{j}(p,q,G,S_{0})]=$
$E\Bigg{[}\Big{(}\sum_{t=1}^{T}(Y_{it}-p\bar{r}_{i(t-1)})\tilde{N}_{it}\Big{)}\Big{(}\sum_{t=1}^{T}(Y_{jt}-p\bar{r}_{j(t-1)})\tilde{N}_{jt}\Big{)}\Bigg{]}=$
$E\Bigg{[}\Big{(}\sum_{t=1}^{T}(Y_{it})\tilde{N}_{it}\Big{)}\Big{(}\sum_{t=1}^{T}(Y_{jt})\tilde{N}_{jt}\Big{)}\Bigg{]}-p^{2}\Big{(}\sum_{t=1}^{T}\bar{r}_{i(t-1)}\tilde{N}_{it}\Big{)}\Big{(}\sum_{t=1}^{T}\bar{r}_{j(t-1)}\tilde{N}_{jt}\Big{)}$

Assume that $\tilde{N}_{it}=\tilde{N}_{jt}\forall t$ e.g. both agents’ first
opportunity to receive the information is in the same period. Assume that
$\tilde{N}_{i3}=\tilde{N}_{j3}=1$, e.g. both are indirectly linked to the
information and that each disposes over exactly one IP connection, which
passes through one and the same intermediate agent $k$. This is the set-up
depicted in village graph 1.

Figure 1: Village Graph 1IP k i j

Then

$\sum_{t=1}^{T}\bar{r}_{i(t-1)}\tilde{N}_{it}=\sum_{t=1}^{T}\bar{r}_{j(t-1)}\tilde{N}_{jt}=\bar{r}_{k1}q$

Hence

$E[g_{i}(p,q,G,S_{0})g_{j}(p,q,G,S_{0})]=E[Y_{i3}Y_{j3}]-p^{2}(\bar{r}_{k1}q)^{2}$

Since $Y$ takes only the values 0 and 1 hence

$E[Y_{i3}Y_{j3}]=P[Y_{i3}=1\cap Y_{j3}=1]$

and since they only have one neighbour linked to the information (namely agent
$k$) this is

$P[Y_{i3}=1\cap Y_{j3}=1]=$ $P[Y_{i3}=1\cap
Y_{j3}=1|S_{k1}=1]P(S_{k1}=1)+P[Y_{i3}=1\cap Y_{j3}=1|S_{k1}=0]P(S_{k1}=0)$
$=P[Y_{i3}=1\cap Y_{j3}=1|S_{i2}=1\cap S_{j2}=1]P(S_{i2}=1\cap
S_{j2}=1|S_{k1}=1)P(S_{k1}=1)$

The outcomes can only both be one if both agents are informed.
Conditional on their common friend $k$ being informed, the probability of $k$
sending to $i$ is independent of the probability of $k$ sending to $j$, hence

$P(S_{i2}=1\cap S_{j2}=1|S_{k1}=1)=P(S_{i2}=1|S_{k1}=1)P(S_{j2}=1|S_{k1}=1)$

hence

$P[Y_{i3}=1\cap Y_{j3}=1|S_{i2}=1\cap
S_{j2}=1]P(S_{i2}=1|S_{k1}=1)P(S_{i2}=1|S_{k1}=1)P(S_{k1}=1)$
$=p^{2}q^{2}\bar{r}^{k}_{1}\neq p^{2}(\bar{r}^{k}_{1}q)^{2}$

and thus

$E[g_{i}(p,q,G,S_{0})g_{j}(p,q,G,S_{0})]\neq 0$

Correlation between moments of agents with the same set of indicators
$\tilde{N}$ arises through friends in common. This is because

$E[r_{i(t-1)}(q)]=\sum_{s_{0:t-2}}r_{i(t-1)}(q,S_{0:t-2})P(S_{1:t-2}|q,G,S_{0})$

hence friends in common show up in both agents’ expected value, creating
correlation in their outcomes. If on the other hand $i$ and $j$ were linked to
different intermediate agents (as depicted in village graph 2), then their
outcomes would be independent.

Figure 2: Village Graph 2IP k1 k2i j

Then

$E\Bigg{[}\sum_{t=1}^{T}(Y_{it})\tilde{N}_{it}\sum_{t=1}^{T}(Y_{jt})\tilde{N}_{jt}\Bigg{]}=$
$E\Bigg{[}\sum_{t=1}^{T}(Y_{it})\tilde{N}_{it}\Bigg{]}E\Bigg{[}\sum_{t=1}^{T}(Y_{jt})\tilde{N}_{jt}\Bigg{]}$
$E\Bigg{[}\sum_{t=1}^{T}(Y_{it})\tilde{N}_{it}\Bigg{]}=p\bar{r}_{i2}=p\bar{r}_{k1,1}(q)=\sum_{t=1}^{T}\bar{r}_{it}\tilde{N}_{it}$
$E\Bigg{[}\sum_{t=1}^{T}(Y_{jt})\tilde{N}_{jt}\Bigg{]}=p\bar{r}_{k2,1}(q)=\sum_{t=1}^{T}(\bar{r}_{jt})\tilde{N}_{jt}$

and thus

$E[g_{i}(p,q,G,S_{0})g_{j}(p,q,G,S_{0})]=0$

Assume now that $\exists t\hskip
2.84544pts.th.\tilde{N}_{it}\neq\tilde{N}_{jt}$ such that one agent has the
opportunity to receive the information earlier than the other. Again, I use a
very simple case for illustration. Assume that $i$ is linked directly to an IP
and that $j$ only disposes over one link to an IP, namely the one passing
through $i$. Then

$E\Bigg{[}\sum_{t=1}^{T}(Y_{it})\tilde{N}_{it}\Bigg{]}=p\bar{r}_{i1}$

and

$E\Bigg{[}\sum_{t=1}^{T}(Y_{jt})\tilde{N}_{jt}\Bigg{]}=p\bar{r}_{i1}q$

hence

$E[g_{i}(p,q,G,S_{0})g_{j}(p,q,G,S_{0})]=E[Y_{i2}Y_{j3}]-p^{2}(\bar{r}^{i}_{1})^{2}q$

But

$E[Y_{i2}Y_{j3}]=P[Y_{i2}=1\cap Y_{j3}=1|S_{i1}=1]P(S_{i1}=1)$
$+P[Y_{i2}=1\cap Y_{j3}=1|S_{i1}=0]P(S_{i1}=0)$ $=P[Y_{i2}=1\cap
Y_{j3}=1|S_{i1}=1]P(S_{i1}=1)$

since $i$ must be informed for her to have a non-zero probability to
participate and for her to pass on the information to $j$. Since conditional
on being informed, participation decisions are independent hence

$P[Y_{i2}=1\cap Y_{j3}=1|S_{i1}=1]P(S_{i1}=1)=$
$P[Y_{i2}=1|S_{i1}=1]P[Y_{j3}=1|S_{j2}=1]P(S_{j2}=1|S_{i1}=1)P(S_{i1}=1)=$
$p^{2}P(S_{j2}=1|S_{i1}=1)P(S_{i1}=1)$

but $i$ is $j^{\prime}s$ only source of information, so

$P(S_{j2}=1|S_{i1}=1)=q$

hence

$E[Y_{i2}Y_{j3}]=p^{2}q\bar{r}_{i1}\neq p^{2}(\bar{r}_{i1})^{2}q$

Implying that

$E[g_{i}(p,q,G,S_{0})g_{j}(p,q,G,S_{0})]\neq 0$

Correlation in this case arises due to $i$ and $j$ being connected.
Summing up, correlation arises through the information reception probabilities
hence the means of the outcomes. Since for IPs, reception probabilities are
non-random, thus per definition IP moments are uncorrelated with any other
agents’ moment, be it in the same or in a different village.

###  3.4 Parameter Identification from Individual Moment Conditions

As demonstrated above, the individual moment conditions are linear in $p$, but
nonlinear in $q$. A basic precondition for identification is that there is
some variation in the link portfolios, hence the information reception rates.
Without such a variation, only the product of $pr(q)$ was identified.
From the model on the other hand, it is always guaranteed that the sample
includes at least one IP. For this agent, the information reception rate is
one (she is sure to be informed). As a consequence

$E[Y_{i1}|\tilde{N}_{i1}=1]=p_{0}$

and hence the set of IPs can be used to identify $p$. With

$E\Bigg{[}\sum_{t=1}^{T}(Y_{it})|\tilde{N}_{i1}=0\Bigg{]}=E\Bigg{[}\sum_{t=2}^{T}(Y_{it})\tilde{N}_{it}\Bigg{]}=p_{0}\sum_{t=1}^{T}E[r_{i(t-1)}(q)]\tilde{N}_{it}=$
$p_{0}\sum_{t=1}^{T}\bar{r}_{i(t-1)}(q_{0})\tilde{N}_{it}$

obviously, the existence of non-IPs is necessary for the identification of
$q$. When all non-IPs are either participants or non-participants, $q$ will
not be identified in the interior of the parameter space and instead, since
$0<q<1$, the estimate will be set to respectively the minimal or maximal value
allowed. Disposing over at least one non-IP participant and one non-IP non-
participant are thus a necessary condition for identifying $q$ (in the sense
of not obtaining a corner solution).
In the following, I propose two estimators: one that does not aggregate
individual moment conditions at all and one that aggregates only IPs and non-
IPs.
The non-aggregated objective function is

$\hat{Q}_{na}=N^{-1}\sum_{i=1}^{N}g_{i}^{2}$

while the two-moment objective function is

$\hat{Q}_{2m}=(N_{1}^{-1}\sum_{i=1}^{N}g_{i}\tilde{N}_{1i})^{2}+(N_{2}^{-1}\sum_{i=1}^{N}g_{i}(1-\tilde{N}_{1i}))^{2}$

With $N_{1}$ and $N_{2}$ denoting the number of IPs and non-IPs, respectively.
The two-moment objective function has the advantage that the information on
the sign of the outcome’s deviation from its unconditional mean is preserved,
but the inconvenience that within-village correlation is amplified, since when
agents deviate in the same direction, the square of the sum of deviations
exceeds the sum of the squares of deviations.
The relationship between the two objective functions can be expressed as

$\hat{Q}_{2m}\approx\frac{1}{N}\hat{Q}_{na}+\frac{2}{N_{2}^{2}}\sum_{i=1}^{N}\sum_{j=1}^{N}g_{i}g_{j}$

If the co-variation in individual-specific moments converges to zero as $N$
goes to infinity (which is what one expects), the two objective functions
produce the same first-order-conditions, hence estimates. Since the latter
term converges to zero when more villages are added, the estimates resulting
from both functions are asymptotically equivalent but may exhibit different
finite sample properties.

##  4 The Non-aggregated Estimator

###  4.1 The Objective Function

If aggregation within the village potentially results in small-sample bias and
aggregation across villages is computationally costly, an idea would be to
simply not aggregate at all. In the following, I call this the non-aggregated
estimator. The sample moments are then

$\hat{g}_{i}=g_{i}$

Viewed as a GMM estimator, this would imply that we only have one observation
per moment condition. Stacking the individual moment conditions into a vector
$g_{s}$ and individual moment conditions in village $v$ into a vector $g_{v}$,
I obtain

$\hat{Q}_{na}=N^{-1}\hat{g}_{s}^{\prime}\mathbf{I}_{N}\hat{g}_{s}$

This is not a Method of Moments estimator in the sense it is generally defined
in econometrics as I do not use sample moments but individual moments and I
minimize the squared deviation of the observed variables from their
unconditional expected values. This corresponds to a GMM estimator that uses
agents of the same type (as defined by their link portfolio) to compose a
type-specific moment condition. Since

$\hat{Q}_{na}=N^{-1}\sum_{v=1}^{V}Q_{v,na}=N^{-1}\sum_{v=1}^{V}g_{v}^{\prime}\mathbf{I}_{N_{v}}g_{v}$

hence this objective function has the substantial advantage that villages can
be computed in parallel.
In the following sections, I check all requirements for parameter
identification and consistency.

####  4.1.1 The Limiting Function

Plugging in the expression for the individual-specific moments from above
leads to

$\hat{Q}_{na}=N^{-1}\sum_{i=1}^{N}\Big{(}\sum_{t=1}^{T}(Y_{it}-p\bar{r}_{i(t-1)}(q))\tilde{N}_{it}\Big{)}^{2}$
$Q_{0,na}=E[\hat{Q}_{na}]=E[g_{i}^{2}]=E\Big{[}\Big{(}\sum_{t=1}^{T}(Y_{it}-p\bar{r}_{i(t-1)}(q))\tilde{N}_{it}\Big{)}^{2}\Big{]}$

It appears surprising at first sight that the sample equivalent of $Q_{0,na}$
should converge if each individual moment condition has a sample size of one.
However, the objective function is established as a sum of the squared
deviations of individual outcomes from their unconditional expected value. The
latter is a function of the individual’s link portfolio i.e. the lengths, but
also concrete configuration of her link(s) to the IPs. There is a limited
albeit large number of possible link portfolios to be encountered in the
population. With several agents exhibiting the same link portfolio (thus
unconditional mean), the objective function can be split into partial sums by
aggregating agents of the same type. As the sample size increases, each link
portfolio occurs sufficiently often for each partial sum to converge to its
expected value and the fraction of agents of each type encountered in the
sample converges to the respective proportion in the underlying population.
Then $Q_{0,na}=E(Q_{na})$ can be seen as the squared difference between the
outcome and its unconditional expected value in the first period in which an
agent faces a choice for an agent who is representative for the underlying
population. Individuals can be segregated into $M$ types according to their
link portfolio such that $I_{im}=1$ if agent $i$ has link portfolio $m$.
Consequently

$Q_{0,na}=E[g_{i}^{2}]=\sum_{m=1}^{M}E[g_{i}^{2}|I_{im}=1]P(I_{im}=1)$

Let $\mu_{m}$ be the unconditional expected value of the outcome for an agent
with link portfolio $m$ in the first period in which she faces a choice and
let $tm$ denote this particular period, (e.g.
$\mu_{m}(p,q)=p\bar{r}_{m(tm-1)}$, then

$E\Big{[}\Big{(}\sum_{t=1}^{T}(Y_{it}-p\bar{r}_{i(t-1)}(q))\tilde{N}_{it}\Big{)}^{2}|I_{im}=1\Big{]}=$
$E\Big{[}\Big{(}Y_{i(tm)}-\mu_{m}\Big{)}^{2}|I_{im}=1\Big{]}=E\Big{[}\Big{(}Y_{i(tm)}-p\bar{r}_{m(tm-1)}(q)\Big{)}^{2}|I_{im}=1\Big{]}$

I can rewrite

$Q_{0,na}=\sum_{m=1}^{M}E\Big{[}\Big{(}Y_{i(tm)}-p\bar{r}_{m(tm-1)}(q)\Big{)}^{2}|I_{im}=1\Big{]}P(I_{im}=1)=$
$\sum_{m=1}^{M}\Bigg{(}E\big{[}Y_{i(tm)}^{2}|I_{im}=1\big{]}-2E\big{[}Y_{i(tm)}|I_{im}=1\big{]}p\bar{r}_{m(tm-1)}(q)$
$+\big{(}p\bar{r}_{m(tm-1)}(q)\big{)}^{2}\Big{)}P(I_{im}=1)$

Since $Y_{it}$ is binary, hence $E[Y_{it}^{2}]=E[Y_{it}]$. Further the
expected value of the outcome given that an agent is of type $m$ is the type
specific mean at the true parameters that generated the data, so

$E[Y_{i(tm)}|I_{im}=1]=\mu_{m}(p_{0},q_{0})=p_{0}r_{m(tm-1)}(q_{0})=p_{0}\bar{r}_{m(tm-1)}(q_{0})$

and consequently

$Q_{0,na}=\sum_{m=1}^{M}\Big{(}p_{0}\bar{r}_{m(tm-1)}(q_{0})-2p_{0}\bar{r}_{m(tm-1)}(q_{0})p\bar{r}_{m(tm-1)}(q)$
$+(p\bar{r}_{m(tm-1)}(q))^{2}\Big{)}P(I_{im}=1)$

This also shows that the limiting function is continuous in the parameters.

###  4.2 Identification

In order to demonstrate that $p_{0},q_{0}$ maximises $Q_{0,na}$ I take the
first order condition with respect to both parameters, which results in

$Q_{0,na}=E[g_{i}^{2}]=\sum_{m=1}^{M}E[g_{i}^{2}|I_{im}=1]P(I_{im}=1)$
$FOC_{p}:\sum_{m=1}^{M}E[g_{i}|I_{im}=1]\frac{\partial
E[g_{i}|I_{im}=1]}{\partial p}P(I_{im}=1)=0$
$FOC_{q}:\sum_{m=1}^{M}E[g_{i}|I_{im}=1]\frac{\partial
E[g_{i}|I_{im}=1]}{\partial q}P(I_{im}=1)=0$

But since it has been derived above that

$E[Y_{i(tm)}|I_{im}=1]=p_{0}r_{m(tm-1)}(q_{0})$

thus

$E[g_{i}(p_{0},q_{0})|I_{mi}=1]=p_{0}\bar{r}_{m(tm-1)}(q_{0})-p\bar{r}_{m(tm-1)}(q)=0$

hence $p_{0},q_{0}$ is a solution to the FOCs.

###  4.3 Consistency

What remains to be shown is that $\hat{Q}_{na}$ converges uniformly in
probability to $Q_{0,na}$ Recall that

$\hat{Q}_{na}=N^{-1}\sum_{i=1}^{N}\Big{(}\sum_{t=1}^{T}(Y_{it}-p\bar{r}_{i(t-1)}(q))\tilde{N}_{it}\Big{)}^{2}$

$\hat{Q}_{na}=N^{-1}\sum_{i=1}^{N}g_{i}^{2}$

If I sort the agents in my observed sample into the $M$ link portfolio-types
from the above and use $N_{m}$ to denote the number of agents with link
portfolio $m$ that are observed in the sample, this results in

$\hat{Q}_{na}=N^{-1}\sum_{i=1}^{N}\sum_{m=1}^{M}\big{(}g_{i}I_{mi})^{2}=N^{-1}\sum_{m=1}^{M}\sum_{i=1}^{N}\big{(}g_{i}I_{mi})^{2}$
$=N^{-1}\sum_{m=1}^{M}\frac{N_{m}}{N_{m}}\sum_{i=1}^{N}(g_{i}I_{im})^{2}=\frac{N_{m}}{N}\sum_{i=1}^{N}\frac{1}{N_{m}}(g_{i}I_{im})^{2}=$

Then by a Law of Large numbers

$\sum_{i=1}^{N}\frac{1}{N_{m}}(g_{i}I_{im})^{2}\overset{p}{\to}E[g_{i}^{2}|I_{im}=1]$

and

$\frac{N_{m}}{N}\overset{p}{\to}P(I_{im}=1)$

This is because the fraction of observed type-m agents converges to the
probability that an individual that is randomly drawn from the underlying
population is of type $m$. Further, since asymptotically every type is
observed infinitesimally often, hence the average of the squared deviations of
type $m$ agents from their common mean converges to its type-specific expected
value. As a consequence I can conclude that $\hat{Q}_{na}$ converges uniformly
to $Q_{0,na}$.
Note that convergence occurs even though individual outcomes are correlated in
the village. As long as variances and covariances are bounded, the LLN remains
applicable. The boundedness of the covariances is assured by the assumption
that an increase n the sample size is achieved by adding more villages,
guaranteeing that the vast majority of pair of agents are uncorrelated as they
reside in different villages.

###  4.4 Asymptotic Variance

Let $\theta=(p,q)$ be the vector of model parameters. Let $\hat{S}(\theta)$
denote the column vector of first derivatives of the observed objective
function with respect to $p$ and $q$ and $\hat{H}(\theta)$ the matrix of
second order derivatives, then a mean-value expansion of the vector $\hat{S}$
leads to

$\hat{S}(\hat{\theta})=\hat{S}(\theta_{0})+\hat{H}(\bar{\theta})(\hat{\theta}-\theta_{0})=0$

and thus

$(\hat{\theta}-\theta_{0})=\hat{H}(\bar{\theta})^{-1}\hat{S}(\theta_{0})$

such that the asymptotic variance is

$V_{na}=E[\hat{H}(\theta_{0})^{-1}\hat{S}(\theta_{0})\hat{S}(\theta_{0})^{\prime}\hat{H}(\theta_{0})^{-1}]$

Denote the matrix of partial derivatives of the moments conditions as

$D=\frac{\partial g_{s}}{\partial\theta}=\Big{(}\frac{\partial
g_{1}}{\partial\theta},...,\frac{\partial g_{N}}{\partial\theta}\Big{)}$

Observe that

$E[\hat{S}(\theta)\hat{S}(\theta)^{\prime}]=E\Bigg{[}\Big{(}\sum_{i=1}^{N}2\frac{\partial
g_{i}}{\partial\theta}g_{i}\Big{)}\Big{(}\sum_{i=1}^{N}2\frac{\partial
g_{i}}{\partial\theta}g_{i}\Big{)}^{\prime}\Bigg{]}=$
$4\begin{bmatrix}E\Big{[}\Big{(}\sum_{i=1}^{N}\frac{\partial g_{i}}{\partial
p}g_{i}\Big{)}^{2}\Big{]}&E\Big{[}\Big{(}\sum_{i=1}^{N}\frac{\partial
g_{i}}{\partial p}g_{i}\Big{)}\Big{(}\sum_{i=1}^{N}\frac{\partial
g_{i}}{\partial q}g_{i}\Big{)}\Big{]}\\\
E\Big{[}\Big{(}\sum_{i=1}^{N}\frac{\partial g_{i}}{\partial
p}g_{i}\Big{)}\Big{(}\sum_{i=1}^{N}\frac{\partial g_{i}}{\partial
q}g_{i}\Big{)}\Big{]}&E\Big{[}\Big{(}\sum_{i=1}^{N}2\frac{\partial
g_{i}}{\partial q}g_{i}\Big{)}^{2}\Big{]}\end{bmatrix}$
$=E[\hat{D}\hat{\Omega}\hat{D}]$

and

$E[\hat{H}(\theta)]=$
$2\begin{bmatrix}E\Big{[}\sum_{i=1}^{N}\Big{(}\frac{\partial g_{i}}{\partial
p}\Big{)}^{2}+\sum_{i=1}^{N}\Big{(}\frac{\partial^{2}g_{i}}{\partial^{2}p}g_{i}\Big{)}\Big{]}&E\Big{[}\sum_{i=1}^{N}\frac{\partial
g_{i}}{\partial p}\frac{\partial g_{i}}{\partial
q}+\sum_{i=1}^{N}\Big{(}\frac{\partial^{2}g_{i}}{\partial p\partial
q}g_{i}\Big{)}\Big{]}\\\ E\Big{[}\sum_{i=1}^{N}\frac{\partial g_{i}}{\partial
p}\frac{\partial g_{i}}{\partial
q}+\Big{(}\sum_{i=1}^{N}\frac{\partial^{2}g_{i}}{\partial p\partial
q}g_{i}\Big{)}\Big{]}&E\Big{[}\sum_{i=1}^{N}\Big{(}\frac{\partial
g_{i}}{\partial
q}\Big{)}^{2}+\sum_{i=1}^{N}\Big{(}\frac{\partial^{2}g_{i}}{\partial^{2}q}g_{i}\Big{)}\Big{]}\end{bmatrix}$

$E[\hat{H}(\theta)]=2\begin{bmatrix}E\Big{[}\sum_{i=1}^{N}\Big{(}\frac{\partial
g_{i}}{\partial p}\Big{)}^{2}&E\Big{[}\sum_{i=1}^{N}\frac{\partial
g_{i}}{\partial p}\frac{\partial g_{i}}{\partial q}\Big{]}\\\
E\Big{[}\sum_{i=1}^{N}\frac{\partial g_{i}}{\partial p}\frac{\partial
g_{i}}{\partial q}\Big{]}&E\Big{[}\sum_{i=1}^{N}\Big{(}\frac{\partial
g_{i}}{\partial q}\Big{)}^{2}\Big{]}\end{bmatrix}$

since $E[g_{i}]=0$. Hence

$E(H(\theta))=E[D^{\prime}\mathbf{I}_{N}D]$

then the results from the above show that $V$ can equivalently be represented
as

$V=(D^{\prime}\mathbf{I}_{N}D)^{-1}D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime}(D^{\prime}\mathbf{I}_{N}D)^{-1}$

which again underlines that the non-aggregated estimator can be perceived as a
GMM estimator. Note that since each moment condition has the same weight, the
weights just cancel out.
Because moments are correlated within a village, but not across villages,
hence $\Omega$ is block diagonal. One block comprises the village specific
covariance matrix $\Omega_{v}$. The latter has the individual variances on the
diagonal and covariances between individual moments on the off diagonal
elements. To construct a fully efficient estimator, all the off-diagonal
elements would have to be specified or estimated. If villages were identical,
$\Omega_{v}$ could be estimated and a fully efficient GMM estimate could be
constructed. In principle, correlations are specific to any particular pair of
agents and additional assumption would be needed to apply methods that
facilitate the calculation of covariances in the presence of clusters. The
alternative is to derive them in terms of the parameters. In the model at
hand, this is possible: with $G$ being observed, all the correlations and
hence all non-zero elements of $\Omega$ can be expressed in terms of $p$ and
$q$. However, this is computationally prohibitively expensive: for any two
agents in the same information radius, the number of friends in common as well
as the friends’ reception rates would have to be found while for any two
individuals at different IP-distance, it would have to be verified whether one
is on the path between the other and some IP. There would be
$(N_{v}-IP)(N_{v}-IP-1)$ of such checks per village and each would be
computationally cumbersome. As a consequence, specifying $\Omega$ correctly in
terms of $p$ and $q$ is computationally unrealistic.
Observe however that the sample variance is

$\hat{V}=(N^{-1}D^{\prime}\mathbf{I}_{N}D)^{-1}N^{-2}D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime}(N^{-1}D^{\prime}\mathbf{I}_{N}D)^{-1}$

The matrix

$D^{\prime}\mathbf{I}_{N}D$

does not involve covariances and an element of

$(D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{11}=\sum_{i=1}^{N}\big{(}\sum_{j=1}^{N}\frac{\partial
g_{j}}{\partial p}\rho_{ij}\big{)}\frac{\partial g_{i}}{\partial p}$
$(D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{12}=D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{21}=\sum_{i=1}^{N}\left(\sum_{j=1}^{N}\frac{\partial
g_{j}}{\partial p}\rho_{ij}\right)\frac{\partial g_{i}}{\partial q}$
$(D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{22}=\sum_{i=1}^{N}\left(\sum_{j=1}^{N}\frac{\partial
g_{j}}{\partial q}\rho_{ij}\right)\frac{\partial g_{i}}{\partial q}$

with $\rho_{ij}=E[g_{i}g_{j}]$ being the variance or co-variance respectively.
Correlations within the villages are always positive. The inner sum in the
expression above has at most $N_{v}$ non-zero terms. Assume that
$N\rightarrow\infty$ by means of $V\rightarrow\infty$, e.g. we obtain more
(not larger) villages. Observe that I can rewrite

$(D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{11}=\sum_{v=1}^{V}\sum_{i=1}^{N_{v}}\left(\sum_{j=1}^{N_{v}}\frac{\partial
g_{j}}{\partial p}\rho_{ij}\right)\frac{\partial g_{i}}{\partial p}$

and thus

$N^{-2}(D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{11}=\sum_{v=1}^{V}\left(\frac{N_{v}}{N}\right)^{2}\left(\frac{1}{N_{v}}\right)^{2}\sum_{i=1}^{N_{v}}\left(\sum_{j=1}^{N_{v}}\frac{\partial
g_{j}}{\partial p}\rho_{ij}\right)\frac{\partial g_{i}}{\partial p}$

Since there are at most $N_{v}$ positive correlations in village $v$ hence

$\left(\frac{1}{N_{v}}\right)^{2}\sum_{i=1}^{N_{v}}\left(\sum_{j=1}^{N_{v}}\frac{\partial
g_{j}}{\partial p}\rho_{ij}\right)\frac{\partial g_{i}}{\partial p}$

converges to its expected value. As such, with village sizes being fixed, the
variance of the estimator does converge to zero, albeit at a rate that is
slower than $N^{-1}$. To ease the exposition, assume villages have equal size
then

$(D^{\prime}\mathbf{I}_{N}\Omega\mathbf{I}_{N}D^{\prime})_{11}=V\left(\frac{N_{v}}{N}\right)^{2}\left(\frac{1}{N_{v}}\right)^{2}\sum_{i=1}^{N_{v}}\left(\sum_{j=1}^{N_{v}}\frac{\partial
g_{j}}{\partial p}\rho_{ij}\right)\frac{\partial g_{i}}{\partial p}$

As $N\rightarrow\infty$ with $N_{v}$ being fixed and $V$ increasing less than
proportional to $N$, thus

$\hat{V}\rightarrow 0$

albeit slower than $N^{-1}$. With fixed-$N_{v}$ asymptotics, e.g. an increase
in the sample size is achieved by adding more villages, the correlation in
individual moments play a minor role. This is also be the case if village
sizes are increased while holding the number of IPs constant and spreading
them out.
Having said that there are at most $N_{v}^{2}$ elements in the village-
specific sums (e.g. the correlation of each villagers’ moment condition with
herself and with all other villagers) does not mean that there are indeed this
many positive elements as villagers may well be uncorrelated if they are not
linked to the same IP and do not have friends in common. Presuming that
correlations among moments play a negligible role, one could presume that
$\Omega$ is approximately diagonal with individual-specific variances as
diagonal elements. The presumed diagonal matrix $\Omega$ can be used as a
weighting matrix in the moment estimation. Importantly, $E[\Omega]$ fails to
be diagonal if there is at least one pair of non-IP agents residing in the
same village with either one being on the path from the information to the
other or both being linked to the same direct source of information. However,
as the sample size increases, the error that is made by replacing with a
diagonal matrix becomes negligible and correlations within the village play a
minor role.

##  5 Two-moment GMM Estimator

###  5.1 The Objective Function

Since it potentially leads to bias to aggregate the individuals of one village
into one moment, a simple remedy is to aggregate all individuals into two
sample moments, one comprising the IPs and the other all remaining agents. The
moments for IPs are uncorrelated with the moments of any other individual,
even within the village. This is because correlation arises solely through
information reception probabilities. IPs already have the information and by
assumption conditional on being informed participation is an individual
choice. Consequently, the IP-moment is thus composed of i.i.d. data. For the
non-IP moment, individual moment conditions are positively correlated within
the village as derived above. One may aggregate non IPs in an attempt to
reduce small-sample bias, hoping that (correlated) positive deviations from
the individual-specific means in one village may cancel with (correlated)
negative deviations in another village.
Since I decide to use only the IPs for the first and the non-IPs for the
second moment, thus I choose the population moment conditions

$g_{1}=E[g_{i}|\tilde{N}_{1i}=1]$

and

$g_{2}=E[g_{i}|\tilde{N}_{1i}=0]$

Each individual moment condition can be split into

$g_{i}=(Y_{i1}-p)\tilde{N}_{i1}+\sum_{t=2}^{T}(Y_{it}-p\bar{r}_{i(t-1)})\tilde{N}_{it}$
(3)

where I made use of the fact that whenever $\tilde{N}_{i1}=1$ (an individual
faces a choice in period one, i.e. she is an IP), then $\bar{r}_{i0}=r_{i0}=1$
(she is known to be informed). This shows that each individual contributes to
one moment, i.e. the first moment is estimated from the sub-sample of IPs, the
second from the sub-sample of non-IPs. Let $N_{1}$ denote the number of IPS
and $N_{2}$ the number of non-IPs that are in the information radius. Then the
sample moments are

$\hat{g}_{1}=\frac{1}{N_{1}}\sum_{i=1}^{N}(Y_{1i}-p)\tilde{N}_{i1}$
$\hat{g}_{2}=\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}\big{(}Y_{it}-p\bar{r}_{i(t-1)}\big{)}\tilde{N}_{it}$

Note that summing up over all individuals is inconsequential as the index
vector ($\tilde{N}_{i}$) conveniently guarantees that individual $i$
contributes only to one of the two moments and hence the sum in $\hat{g}_{1}$
(respectively $\hat{g}_{2}$) will always have $N_{1}$ (respectively $N_{2}$)
terms.

$\hat{Q}_{a}=(\hat{g}_{1},\hat{g_{2}})^{\prime}\mathbf{I}_{2}(\hat{g}_{1},\hat{g_{2}})$

Note that this is but one weighting choice. However, as shall be demonstrated
below, as long as the first-order-condition (FOC) with respect to $q$ has a
solution, the choice of the weighting matrix has no impact. As a robustness
check, I computed the estimates using weights corresponding to the relative
sample sizes, i.e. $w_{1}=\frac{N_{1}}{N},w_{2}=\frac{N_{2}}{N}$. The
estimates were identical.

###  5.2 The limiting Function

As mentioned above, the first moment consists of the moment conditions for the
IPs, i.e. those individuals for which $\tilde{N}_{1i}=1$.

$E(g_{1})=E[g_{i}|\tilde{N}_{i1}=1]=E[(Y_{i1}-p)\tilde{N}_{i1}|\tilde{N}_{i1}=1)]$

because for IPs, the second term in (3) is zero.

$E(g_{1})=E[Y_{i1}|\tilde{N}_{i1}=1]-p$

By the assumptions stated above, these individuals are known to be informed
(i.e. $r_{i0}=1$), hence the expected value of their outcome is simply $p_{0}$
(i.e. the true probability that an informed individual participates). Thus

$E[g_{i}|\tilde{N}_{i1}=1]=P(Y_{i1}=1|S_{i0}=1)-p=p_{0}-p$

For all other individuals, the true unconditional expected value of the
outcome consists of the probability to participate multiplied by the
information reception probability. Since $\tilde{N}_{it}=1$ only in the period
that directly follows the first information exchange in which $i$ could have
been informed, hence

$E(g_{2})=E[g_{i}|\tilde{N}_{i1}=0]=E[\sum_{t=2}^{T}(Y_{it}-p\bar{r}_{i(t-1)}(q))\tilde{N}_{it}]$

because for non-IPs, the first part in (3) is zero. By the same argument as
above, I can segregate agents according to their link portfolio, i.e.
information reception probabilities. For simplicity, denote type $m=1$ as the
IPs, then

$E[g_{i}|\tilde{N}_{i1}=0]=\sum_{m=2}^{M}E[g_{i}|I_{im}=1]P(I_{im}=1|\tilde{N}_{i1}=0)$

i.e. the expected value of the individual moment condition of a non-IP is
calculated by the law of total probability taking into account how likely it
is that a randomly drawn non-IP exhibits link portfolio $m$ (which is
$P(I_{im}=1|\tilde{N}_{i1}=0)$).
Recalling that $tm$ is defined as the first period in which a type $m$ non-IP
agent faces a choice then

$E[g_{i}|I_{mi}=1]=p_{0}\bar{r}_{m,tm-1}(q_{0})-p\bar{r}_{m,tm-1}(q)$

Therefore

$E(g_{2})=\sum_{m=1}^{M}(p_{0}\bar{r}_{m,tm}(q_{0})-p\bar{r}_{m,tm}(q))P(I_{im}=1|\tilde{N}_{i1}=0)$
$E[Q_{a}]=Q_{0,a}=E(g_{1})^{2}+E(g_{2})^{2}$

###  5.3 Identification

Again it is necessary to show that the First-order Conditions (FOCs) of the
limiting function are zero at the true parameter value.

$FOC_{p}:-2E(g_{1})\frac{\partial E(g_{1})}{\partial
p}-2E(g_{2})\frac{\partial E(g_{2})}{\partial p}=0\Rightarrow$
$\big{(}p_{0}-p\big{)}+\left(\sum_{m=1}^{M}(p_{0}\bar{r}_{m,tm}(q_{0})-p\bar{r}_{m,tm}(q))P(I_{im}=1|\tilde{N}_{i1}=0)\right)$
$\left(\sum_{m=1}^{M}\bar{r}_{m,tm}(q)P(I_{im}=1|\tilde{N}_{i1}=0)\right)=0$
$FOC_{q}:-2E(g_{2})\frac{\partial E(g_{2})}{\partial q}=0\Rrightarrow$
$\Bigg{(}\sum_{m=1}^{M}(p_{0}\bar{r}_{m,tm}(q_{0})-p\bar{r}_{m,tm}(q))P(I_{im}=1|\tilde{N}_{i1})=0)\Bigg{)}$
$\left(\sum_{m=1}^{M}p\frac{\partial\bar{r}_{m,tm}(q)}{\partial
q}P(I_{im}=1|\tilde{N}_{i1}=0)\right)=0$

Thus apparently, $p=p_{0},q=q_{0}$ is a solution to the FOCs. Note firstly
that the estimate of $q$ is pinned down solely by means of the second moment.
Since the FOC of $q$ can be expressed as

$FOC_{q}:-2E(g_{1})E\frac{\partial g_{1}}{\partial q}=0$

Since the partial derivative is non-zero, if there is an exact solution to the
FOCs of $q$, then the latter implies that the $pq$ combination is chosen such
that $E(g_{1})=0$. This pins down $p$ as a function of $q$ such that this
condition is fulfilled. Substituting this into the FOCs for $p$ implies that
in the latter, the second term is zero and as such, the first moment
identifies $p$. This also highlights that is case of uniqueness, any weights
chosen are irrelevant to the resulting estimate, which was confirmed in the
Monte Carlo study.

###  5.4 Consistency

Recalling that

$\hat{g}_{1}=\frac{1}{N_{1}}\sum_{i=1}^{N}(Y_{1i}-p)\tilde{N}_{1i}$

As $N\rightarrow\infty$, also $N_{1}\rightarrow\infty$, albeit at a smaller
rate. I can express $N_{1}=N\frac{N_{1}}{N}=Nw_{1}$ where $w_{1}$ is the
fraction of $IPs$ in the underlying population. This implies that
$\hat{g}_{1}$ converges to $E(g_{1})$ by a Law of Large Numbers (LLN).
To see this note that

$\hat{g}_{1}=\frac{N}{N_{1}}\frac{1}{N}\sum_{i=1}^{N}(Y_{1i}-p)\tilde{N}_{1i}$

Then by a LLN

$\frac{1}{N}\sum_{i=1}^{N}(Y_{1i}-p)\tilde{N}_{1i}\overset{p}{\to}E[Y_{i1}\tilde{N}_{i1}]$

By the Law of iterated expectation

$E[(Y_{i1}-p)\tilde{N}_{i1}]=E[E[(Y_{i1}-p)\tilde{N}_{i1}]|\tilde{N}_{i1}]=E[\tilde{N}_{i1}E[Y_{i1}-p|\tilde{N}_{i1}]]$

Since $\tilde{N}_{i1}$ is binary hence

$E[\tilde{N}_{i1}E[Y_{i1}-p|\tilde{N}_{i1}]]=1\times
E[Y_{i1}-p|\tilde{N}_{i1}=1]P(\tilde{N}_{it}=1)+0\times
E[Y_{i1}-p|\tilde{N}_{i1}=0]P(\tilde{N}_{it}=0)$
$E[Y_{i1}-p|\tilde{N}_{i1}=1]P(\tilde{N}_{it}=1)=(p_{0}-p)P(\tilde{N}_{it}=1)$

Further by a LLN

$\frac{N}{N_{1}}\overset{p}{\to}P(\tilde{N}_{it}=1)^{-1}$

This is because the reciprocal of the fraction of IPs encountered in the
sample will converge to the probability that a randomly selected individual
from the underlying population is an IP. As a consequence

$\hat{g}_{1}\overset{p}{\to}P(\tilde{N}_{it}=1)^{-1}(p_{0}-p)P(\tilde{N}_{it}=1)=p_{0}-p=E(g_{1})$
$\hat{g}_{2}=\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}\big{(}Y_{it}-p\bar{r}_{i(t-1)}\big{)}\tilde{N}_{it}$

Which can be expressed as

$\frac{1}{N_{2}}\sum_{m=2}^{M}(Y_{it}-p\bar{r}_{m,tm}(q))I_{im}=\frac{N}{N_{2}}\sum_{m=2}^{M}\frac{1}{N}\sum_{i=1}^{N}(Y_{it}-p\bar{r}_{m,tm}(q))I_{im}$

Then we know that by a LLN

$\frac{1}{N}\sum_{i=1}^{N}(Y_{it}-p\bar{r}_{m,tm}(q))I_{im}\overset{p}{\to}E(Y_{it}-p\bar{r}_{m,tm-1}|I_{im}=1)P(I_{im}=1)=$
$(p_{0}\bar{r}_{m,tm-1}(q_{0})-p\bar{r}_{m,tm-1}(q))P(I_{im}=1)$

and

$\frac{N}{N_{2}}\overset{p}{\to}P(\tilde{N}_{i1}=0)^{-1}$

Finally since

$P(I_{im}=1|\tilde{N}_{i1}=0)=\frac{P(I_{im}=1)}{P(\tilde{N}_{i1}=0)}$

Hence

$\hat{g}_{2}\overset{p}{\to}(p_{0}\bar{r}_{m,tm-1}(q_{0})-p\bar{r}_{m,tm-1}(q_{0}))P(I_{im}=1|\tilde{N}_{i1}=0)=E(g_{2})$

From which I can deduct that $\hat{Q}_{a}$ converges uniformly to $Q_{0,a}$.
With the objective function converging to its expected value and the latter
being minimized at $p=p_{0},q=q_{0}$, we can thus conclude that the estimator
is consistent.

###  5.5 Asymptotic Variance

By the standard GMM results we have

$V=(D^{\prime}\mathbf{I}_{2}D)^{-1}(D^{\prime}\mathbf{I}_{2}\Omega\mathbf{I}_{2}D)(D^{\prime}\mathbf{I}_{2}D)^{-1}$

Now $\Omega$ is diagonal, since IPs and non-IPs are uncorrelated. Consequently
we have

$\Omega_{11}=\frac{p_{0}(1-p_{0})}{N_{1}}$
$\Omega_{22}=E\Big{[}\Big{(}\frac{1}{N_{2}}\sum_{i=1}^{N}g_{i}(1-\tilde{N}_{1i})\Big{)}\Big{(}\frac{1}{N_{2}}\sum_{i=1}^{N}g_{i}(1-\tilde{N}_{1i})\Big{)}\Big{]}=\sum_{i=1}^{N_{2}}\sum_{j=1}^{N_{2}}\frac{\rho_{ij}}{N^{2}}$

With $\rho_{ij}$ being the Variance of $g_{i}$ if $i=j$ or the covariance
between $i$ and $j$ if $i\neq j$. By the same argument as above, this can be
re-written as

$=\sum_{v=1}^{V}\frac{N_{2v}}{N_{2}}\sum_{i=1}^{N_{2v}}\sum_{j=1}^{N_{2v}}\frac{\rho_{ij}}{N_{v}^{2}}$

which shows that the estimator also converges at a rate slower than $N^{-1}$.

##  6 Comparison

From the first-order-condition of their respective limiting functions it is
apparent that whenever the solution is unique, both estimation methods
consistently identify it. In this case, the non-IPs identify $q$: $q$ is
chosen such that the derivative of the objective function with respect to $q$
(which only depends on non-IPs) is set to zero. Since choosing $q$ such that
the sum of deviations (hence the sum of individual non-IP moment conditions)
is zero is a solution to the foc for $q$, therefore in case of uniqueness, it
is the only solution in both cases. The IPs on the other hand then identify
$p$: since $q=q(p)$ is chosen such that the moment conditions for non-IPs are
zero for any $p$ satisfying the relationship that was derived from the foc for
$q$, hence non-IPs do not contribute any further information relevant to the
choice of $p$ beyond this relationship and it is the IPs that identify $p$.

###  6.1 Small Sample Properties

Though both estimators are consistent given uniqueness, yet the finite sample
estimates and also the estimates in case of multiplicity of solutions to the
FOCs vary. Multiplicity does not appear to be an issue for the problem at
hand, however, the difference in finite sample properties merits attention.
Recalling that agents can be segregated into $M$ link portfolios, the first
link portfolio denoting the IPs, then the finite sample first-order conditions
are:
Non-aggregated estimator:

$FOC_{p}:$
$\frac{N_{1}}{N}\frac{1}{N_{1}}\sum_{i=1}^{N}(Y_{i1}-p)\tilde{N}_{i1}+\frac{N_{2}}{N}\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}(Y_{it}-p\bar{r}_{i(t-1)})\bar{r}_{i(t-1)}\tilde{N}_{it}=0$

Two-moment estimator:

$FOC_{p}:$
$\frac{1}{N_{1}}\sum_{i=1}^{N}(Y_{i1}-p)\tilde{N}_{i1}+\Bigg{(}\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}(Y_{it}-p\bar{r}_{i(t-1)})\tilde{N}_{it}\Bigg{)}\Bigg{(}\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}\bar{r}_{i(t-1)}\tilde{N}_{it}\Bigg{)}=0$

Non-aggregated estimator:

$FOC_{q}:$
$\frac{N_{2}}{N}\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}(Y_{it}-p\bar{r}_{i(t-1)})p\frac{\partial\bar{r}_{i(t-1)}}{\partial
q}\tilde{N}_{it}=0$

Two-moment estimator:

$FOC_{q}:$
$\Bigg{(}\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}(Y_{it}-p\bar{r}_{i(t-1)})\tilde{N}_{it}\Bigg{)}\Bigg{(}\frac{1}{N_{2}}\sum_{i=1}^{N}\sum_{t=2}^{T}p\frac{\partial\bar{r}_{i(t-1)}}{\partial
q}\tilde{N}_{it}\Bigg{)}=0$

Note that the multiplication by $\frac{N_{1}}{N}$ and $\frac{N_{2}}{N}$ would
play a role only in case of non-uniqueness, which did not seem to be the case.
First, the difference concerns solely non-IPs. Second, for the FOC of $q$, for
the Two-Moment estimator, the term in the second bracket will naturally never
be zero, hence the Two-Moment-Estimator minimises the sum of deviations of all
non-IPs from their unconditional expected value, giving equal weight to each
agent (i.e. it minimises the term in the first brackets). The Non-aggregated
estimator in contrast minimises a weighted sum of deviations, the weights
being the partial derivative of the moment condition with respect to $q$. The
latter is a function of the derivative of agents’ information reception
probabilities ($\frac{\partial r_{i(t-1)}}{\partial q}$). The function is
decreasing and convex in the individual’s degree hence implying that less
weight is given to very well connected agents. For the foc of $p$, for the
two-moment estimator, $q=q(p)$ has been chosen such that the entire non-IP
term is as small as possible and it is thus predominantly the IP term that
pins down $p$. For the non-aggregated estimator on the other hand, individual
deviations from their expected value are weighted with the derivative of the
moment condition with respect to $p$, which is the information reception rate.
The latter is increasing in the individual degree with diminishing slope,
implying that more weight is allocated to well-connected agents.
The FOCs thus show that the non-aggregated estimator takes into account also
the marginal (as opposed to only the absolute) effect of a change in $p$ or
$q$ on the individual moment conditions.
Expressing

$\hat{\varepsilon}_{i}=\sum_{t=1}^{T}(Y_{it}-p\bar{r}_{i,t-1})\tilde{N}_{it}$

as the actual deviation from the mean observed for individual $i$. It is
apparent that

$Q_{na}=\frac{1}{N}\sum_{i=1}^{N}\hat{\varepsilon}_{i}^{2}$

and

$Q_{a}=(\frac{1}{N_{1}}\sum_{i=1}^{N}\hat{\varepsilon}_{i}\tilde{N}_{i1})^{2}+(\frac{1}{N_{2}}\sum_{i=1}^{N}\hat{\varepsilon}_{i}(1-\tilde{N}_{i1}))^{2}=$
$\frac{1}{N_{1}^{2}}\sum_{i=1}^{N}\hat{\varepsilon}_{i}^{2}\tilde{N}_{i1}+\frac{1}{N_{1}^{2}}\sum_{i=1}^{N}\sum_{j\neq
i}\hat{\varepsilon}_{i}\hat{\varepsilon}_{j}\tilde{N}_{i1}+\frac{1}{N_{2}^{2}}\sum_{i=1}^{N}\hat{\varepsilon}_{i}^{2}(1-\tilde{N}_{i1})+\frac{1}{N_{2}^{2}}\sum_{i=1}^{N}\sum_{j\neq
i}\hat{\varepsilon}_{i}\hat{\varepsilon}_{j}(1-\tilde{N}_{i1})(1-\tilde{N}_{j1})$

Again, whether or not the two moments are weighted does not impact the
solution in case of uniqueness. This shows that the Two-moment estimator has
the advantage and shortcoming that the information on the sign of individual
deviations from their unconditional expected values is preserved in the second
and forth term. The effect of a rare event combined with within-village
correlation can cause a extreme deviation of villagers from their mean that in
one direction. One would wish to employ the Two-moment estimator in the hope
that it can mitigate such an effect, because agents in other villages might
deviate in the opposite direction. One may advocate against the usage of the
Two moment estimator if one suspects that it will even more attenuate the
effect of such rare events.

###  6.2 Consistency requirements

Both estimates share the same consistency requirement. Convergence of both
$\hat{Q}_{na}$ and $\hat{Q}_{a}$ to their respective limiting distributions
require that

$\frac{1}{N_{m}}\sum_{i=1}^{N}Y_{itm}I_{im}\overset{p}{\to}p_{0}\bar{r}_{m,tm}(q_{0})\hskip
7.11317pt\forall m=1,...,M$

A sufficient condition is that

$\sum_{m=1}^{M}\frac{1}{N_{m}}\sum_{i=1}^{N}Y_{itm}I_{im}\overset{p}{\to}\sum_{m=1}^{M}w_{m}p_{0}\bar{r}_{m,tm}(q_{0})=E\Bigg{(}\sum_{m=1}^{M}\frac{1}{N_{m}}\sum_{i=1}^{N}Y_{itm}I_{im}\Bigg{)}$

or equivalently

$\sum_{m=1}^{M}\frac{1}{N_{m}}\sum_{i=1}^{N}(Y_{itm}-p_{0}\bar{r}_{m,tm}(q_{0}))I_{im}\overset{p}{\to}0$

which warrants the applicability of a Law of Large Numbers.
Showing that the sum converges to its expected value despite correlation among
outcomes requires showing that its variance shrinks to zero as $N$ goes to
infinity. A sufficient condition for this is that

$Var(\frac{1}{N}\sum_{i=1}^{N}\hat{g}_{i})\overset{p}{\to}0$

Define

$\bar{\sigma}^{2}=\frac{1}{N_{m}}\sum_{i=1}^{N}\hat{g}_{i}^{2}$

as the sample Variance moment conditions and

$\bar{\rho}_{v}=\frac{1}{N_{2,v}(N_{2,v}-1)}\sum_{i\in v}\sum_{j\in v,i\neq
j}\hat{g}_{i}\hat{g}_{j}(1-\tilde{N}_{i1})(1-\tilde{N}_{j1})$

the observed correlation between agents in village $v$. Note that correlation
within the villages only concerns non-IPs, hence the division by
$N_{2,v}(N_{2,v}-1)$ which is the largest possible pair of correlated agents
in the village and the multiplication by one minus the IP-index. Then

$Var(\frac{1}{N}\sum_{i=1}^{N}\hat{g}_{i})=\frac{1}{N^{2}}N\bar{\sigma}^{2}+\frac{1}{N^{2}}\sum_{i=1}^{N}g_{i}\sum_{j\neq
i}g_{j}(1-\tilde{N}_{i1})(1-\tilde{N}_{j1})=$
$\frac{1}{N}\Big{(}\bar{\sigma}^{2}+\frac{\sum_{v}N_{2,v}(N_{2,v}-1)\bar{\rho}_{v}}{N}\Big{)}$

which results from the fact that any correlation in individual outcomes (thus
moment conditions) is correlation within the village. Consistency thus
requires that

$\frac{1}{N}\Big{(}\bar{\sigma}^{2}+\frac{\sum_{v}N_{2,v}(N_{2,v}-1)\bar{\rho}_{v}}{N}\Big{)}\overset{p}{\to}0$

implying that the second term in brackets must converge to a constant or
decrease as the sample size increases to infinity. Intuitively, this implies
that new villages that are added to the sample are not larger or more densely
connected than the preceding ones. This condition is also sufficient to
guarantee convergence of the variances for both estimators, albeit potentially
at different rates.

###  6.3 Variance estimation and testing

Hansen and Lee (2019) derive conditions under which the Central Limit Theorem
(CLT) can be applied to the sample mean in presence of clusters, e.g. whenever
there are groups of observations that are correlated among themselves, but
uncorrelated with the observations from other groups. Because both moment-
based estimators are a function of the mean of the individual-specific moment
conditions, hence they are a sample mean, this theory can be applied for the
case at hand. Because individual-specific sample moment conditions are
strictly bounded between minus one and one, thus under quite mild conditions,
the sample moment covariance is a consistent estimator for the population
moment covariance and the estimators are asymptotically normally distributed.
The main insight here is that it is not necessary to know the limit of the
sample covariance matrix as long as one is certain that it does converge to
that limit. This is sufficient for standard testing purposes for which neither
heterogeneity nor correlation between moment conditions appears to be
problematic. Nonetheless, as always with testing based on asymptotic theory,
size distortions in small samples can be relevant and problematic.

##  7 Monte Carlo Study

Twelve real village networks are used for the simulation exercise. Following
the enumeration of villages as done by the authors that accumulated the data,
these are villages 1, 2, 4, 12, 23, 25, 31, 32, 45, 51, 57 and 73. The
villages are densely connected. As a result, a substantial fraction of agents
is directly linked to IPs and as such, they largely dominate the estimator. I
limit the number of direct IP-neighbours by reducing the number of IPs to half
of its original size. Accordingly, for each simulation run, the required
number of IPs was randomly drawn from the observed set of IPs. Using the
village network, the IP vector and the respective parameter combination, the
outcome data was simulated. This way, 96 samples were generated and for each
sample, the three estimation procedures were applied. The choice of the number
of villages and simulation runs was guided by computational aspects: disposing
over machines with 48 cores, three sample of villages could conveniently run
in parallel. This way, 24 jobs in total could generate the desired estimates.
To be able to run villages in parallel a common seed was first used to
generate the village specific seeds. For the data simulation, this common seed
equaled the simulation run (e.g. ranging from 1 to 96), for drawing the IPs,
the common seed equaled the simulation run plus one (e.g. ranging from 2 to
97). Using the common seed, I generated twelve village seeds and picked the
respective ones. Thereafter, the sample objective function is established by
aggregating the village objective functions and the peak is identified by grid
search.
Computational speed is decreasing in the number of IPs for the GMM estimator.
This is intuitive: remembering that for direct and indirect neighbours, the
reception probabilities are very easy to compute, a larger number of IPs
directly translate into a faster evaluation of the objective function. In
particular, for any agent three links away from the information, it is first
necessary to check whether they fulfill the requirements of theorem 1 or
theorem 2 in order to determine whether or not they can be used. It is these
checks that take up a long time hence speed is slowest when many villagers are
three links away from the IPs.
Three parameter configurations were analyzed:
$p_{0}=0.1,q_{0}=0.1,p_{0}=0.1,q_{0}=0.9$ and $p_{0}=0.5,q_{0}=0.5$. Beneath I
plot the 96 estimates (blue) together with the mean estimate (red) and
tabulate the bias as a percentage of the correct parameter value and the
observed standard deviation.

Figure 3: case 1: $p=0.1,q=0.1$

| p=0.1, q=0.1
---|---
| mean p | % bias | std. dev. | mean q | % bias | std. dev.
Non aggregated estimator | 0.1021 | 2.1 | 0.0237 | 0.123 | 23 | 0.0229
2 moments estimator | 0.103 | 3 | 0.0259 | 0.1217 | 21.7 | 0.028
Figure 4: case 2: $p=0.1,q=0.9$

| p=0.1,q=0.9
---|---
| mean p | % bias | std. dev. | mean q | % bias | std. dev.
Non aggregated estimator | 0.1003 | 0.3 | 0.0077 | 0.8886 | 1.266667 | 0.0901
2 moments estimator | 0.1058 | 5.8 | 0.0222 | 0.7682 | 14.64444 | 0.2249
Figure 5: case 3: $p=0.5,q=0.5$

| p=0.5, q=0.5
---|---
| mean p | % bias | std. dev. | mean q | % bias | std. dev.
Non aggregated estimator | 0.5039 | 0.78 | 0.0156 | 0.5286 | 5.72 | 0.0263
2 moments estimator | 0.5049 | 0.98 | 0.0441 | 0.5447 | 8.94 | 0.1065

Visibly, a substantial reduction in the spread of the estimates can be
achieved with the non-aggregated estimator. In line with this, standard
deviations are lowest for this estimator. In case 1, the non-aggregated
estimator for $q$ is biased, but the plot reveals that this is more the effect
of some outliers, rather than a general property. Interestingly, this is the
case for all three parameter configurations, but most pronounced in case 2.
This shows that first some gains can be made when later time periods are
considered and that within-village correlations increase the estimator
variance in finite samples, in line with the derivations above. It has to be
kept in mind that the superiority of the non-aggregated estimator comes with
the additional advantage of the possibility of parallelization over villages
which substantially increases speed.

##  8 Application

For the application, all 37 villages were used. A previous estimate had
generated $\hat{p}=0.16,\hat{q}=0.79$. Above are surface plots for both
estimation techniques. Unsurprisingly, the Non-aggregated estimator is very
close to the previous estimator. As mentioned above, the number of IPs is
large and as a consequence, many villagers are directly connected to them and
little can be gained by increasing the time horizon. The bad performance of
the Two-Moment Estimator is more surprising and highlighting once more that
within-village correlation is important, advocating that aggregating
individual moments in a GMM-style estimation can be problematic in the
“latent-diffusion-observed-adoption” model.

Figure 6: Real village estimation

##  9 Conclusion

Applying moment-based estimation to the network diffusion model with
unobserved information propagation is potentially beneficial, yet challenging.
The individuals’ information reception probabilities are in general hard to
evaluate. The problem can be tackled by using only one observation per
individual, namely the one of the period that immediately follows the first
information exchange in which the agent could have been informed. Here, some
short-hand formulas exist, making the computations tractable. This leads to
individual-specific moment conditions that can be used to set up an objective
function.
As mean and variances of the outcomes are link-portfolio specific and
correlation across individual means arises through the village network,
aggregating non-IP individuals into moment conditions can result in small
sample bias. Under regularity conditions outlined in Hansen and Lee (2019),
the researcher is able to obtain an unbiased estimate of the covariance of the
moment conditions and hence the parameters. Alternatively, when villages are
sufficiently small, the researcher may choose to compute each village
covariance matrix which is fully specified in terms of the model parameters
and the network. Finally, when villages become very numerous, the covariance
matrix of the moment conditions is sparse and the researcher may choose to
neglect its off diagonal non-zero entries. It remains to be investigated which
of these proceedings has the best performance.
The Monte Carlo results are in line with the theoretical derivations,
providing evidence that aggregating individual moments - as in a usual GMM
framework - can lead to a large spread of the estimates.
Given the promising results from using later time periods, further research in
finding shorthand formulas that enable a further extension of the modeled time
horizon has the potential to be highly beneficial.

##  10 Appendix - Proofs of Theorems 1 and 2

###  10.1 Proof of Theorem 1

###### Proof.

For the seek of clarity, I assume that all IP neighbours have the same number
of links to IPs, although the proof can also be established in general. Let

$E(r_{it}(q))=\sum_{S_{1:(t-1)}\in\mathbf{S}}E(S_{it}|S_{1:(t-1)},G,q)P(S_{(t-1)},G,q)$

the information reception rate achieved by integrating out the latent variable
and let $\omega_{i}$ be the receiver’s degree and let $\bar{\omega}$ be the
senders’ degree, same for all senders by assumption (i.e.
$\omega_{j}=\bar{\omega}\forall j=1,..,\omega_{i}$). I need to account for all
information scenarios that imply that $i$ has a nonzero probability to be
informed. First, she must have some informed neighbours. Given that she has
$\omega_{i}$ neighbours in total, the number of informed neighbours she has,
which I denote $x$, can vary from zero to $\omega_{i}$. Each time, there are
various (precisely $\binom{\omega_{i}}{x}$) possibilities to draw $x$ out of
$\omega_{i}$ individuals. For simplicity I have assumed that the $\omega_{i}$
individuals on the path between $i$ and the IP are homogeneous in the sense
that each has herself degree $\bar{\omega}$. Therefore, each of them has, by
formula (1), probability $(1-(1-q)^{\bar{\omega}})$ to be informed in the
first exchange and probability $(1-q)^{\bar{\omega}}$ to enter the second
period uninformed. Then
$\big{(}1-(1-q)^{\bar{\omega}}\big{)}^{x}\big{(}(1-q)^{\bar{\omega}}\big{)}^{\omega^{i}-x}$
is the probability that $x$ out of the $\omega_{i}$ friends are informed,
while the remaining $\omega_{i}-x$ stay uninformed. So far I have established
the probability that the second information exchange starts with $x$ out of
$\omega_{i}$ friends of $i$ being informed. I still need to evaluate what in
this case will be $i^{\prime}s$ probability to receive the information.
$(1-(1-q)^{x})$ is $i^{\prime}s$ chance to be informed, given that $x$ out of
her $\omega_{i}$ friends are informed. Letting $x$ vary from 0 to $\omega_{i}$
and summing up over all possibilities, this leads to

$E(r_{it}(q,S_{1:t-1}))=\sum_{x=0}^{\omega_{i}}\binom{\omega_{i}}{x}\big{(}1-(1-q)^{x}\big{)}\big{(}1-(1-q)^{\bar{\omega}}\big{)}^{x}\big{(}(1-q)^{\bar{\omega}}\big{)}^{\omega_{i}-x}$

To recapitulate: the binomial coefficient gives the number possibilities how
$x$ informed neighbours can be drawn out of the $\omega_{i}$ available ones,
$\big{(}1-(1-q)^{x}\big{)}$ gives $i^{\prime}s$ probability to receive the
information given she has $x$ informed neighbours and
$\big{(}1-(1-q)^{\bar{\omega}}\big{)}^{x}\big{(}(1-q)^{\bar{\omega}}\big{)}^{\omega_{i}-x}$
is the probability that indeed $x$ of her neighbours are informed and
$\omega_{i}-x$ stay uninformed. The total probability mass is then obtained by
summing over all possibilities for $x$. Evidently, if $x=0$, then $r_{it}=0$.

$r_{it}=\sum_{x=0}^{\omega_{i}}\binom{\omega_{i}}{x}\big{(}1-(1-q)^{\bar{\omega}})\big{)}^{x}\big{(}(1-q)^{\bar{\omega}}\big{)}^{\omega_{i}-x}$
$-\sum_{x=0}^{\omega_{i}}\binom{\omega^{i}}{x}\big{(}(1-q)^{x})\big{)}\big{(}1-(1-q)^{\bar{\omega}})\big{)}^{x}\big{(}(1-q)^{\bar{\omega}}\big{)}^{\omega_{i}-x}$

Using the formula for the binomial coefficient

$=\Big{(}1-(1-q)^{\bar{\omega}}+(1-q)^{\bar{\omega}}\Big{)}^{\omega_{i}}-\sum_{x=0}^{\omega_{i}}\binom{\omega_{i}}{x}\big{(}(1-q)(1-(1-q)^{\bar{\omega}})\big{)}^{x}\big{(}(1-q)^{\bar{\omega}}\big{)}^{\omega_{i}-x}$
$=1-\Big{(}(1-q)(1-(1-q)^{\bar{\omega}}+(1-q)^{\bar{\omega}}\Big{)}^{\omega_{i}}$
$=1-\Big{(}1-(1-q)^{\bar{\omega}}-q+q(1-q)^{\bar{\omega}}+(1-q)^{\bar{\omega}}\Big{)}^{\omega^{i}}$
$=1-\Big{(}1-q+q(1-q)^{\bar{\omega}}\Big{)}^{\omega_{i}}$
$=1-(1-q(\underbrace{1-(1-q)^{\bar{\omega}}}_{\bar{r}_{j(t-1)}})^{\omega_{i}}=\bar{r}_{it}$

∎

###  10.2 Proof of Theorem 2

###### Proof.

For clarity I assume that the IP neighbours linked to $i$ are homogeneous,
each having $J$ links to the final agent and the same number of links to IPs
and that there are in total $\omega_{i}$ such agents that are linked to the
IPs and indirectly linked to $i$. By the assumptions of theorem 2, none of the
agents on the (2 link) path between the IP-neighbours and the final agent is
frequented twice and as such, these paths can be treated as independent.

$E(r_{i3}(q)=\sum_{x=0}^{\omega_{i}}\binom{\omega_{i}}{x}(1-(1-q^{2})^{Jx})(\bar{r}_{k1})^{x}\big{(}1-\bar{r}_{k1}\big{)}^{\omega_{i}-x}$

The binomial coefficient again gives the number of possibilities to draw $x$
out of $\omega_{i}$ IP-neighbours to be informed. The probability that the
final agent receives the information through any one particular path is
$q^{2}$ and paths can be treated independently. If $x$ IP neighbours are
informed and each has $J$ paths to the final agent, then the total number of
paths is $Jx$ and consequently $i^{\prime}s$ probability to receive the
information through any path is $(1-(1-q^{2})^{Jx})$ e.g. one minus the
probability that she does not receive the information. Further
$(\bar{r}^{k}_{1})^{x}\big{(}1-\bar{r}^{k}_{1}\big{)}^{\omega_{i}-x}$ gives
the probability that $x$ out of the $\omega_{i}$ IP neighbours linked to $i$
are informed, while the remaining ones stay uninformed.

$=\sum_{x=0}^{\omega_{i}}\binom{\omega_{i}}{x}(\bar{r}_{k1})^{x}\big{(}1-\bar{r}_{k1}\big{)}^{\omega_{i}-x}$

$-\sum_{x=0}^{\omega^{i}}\binom{\omega^{i}}{x}((1-q^{2})^{Jx})(\bar{r}_{k1})^{x}\big{(}1-\bar{r}_{k1}\big{)}^{\omega_{i}-x}$

$=1-\sum_{x=0}^{\omega_{i}}\binom{\omega_{i}}{x}((1-q^{2})^{J}\bar{r}_{k1})^{x}\big{(}1-\bar{r}_{k1}\big{)}^{\omega_{i}-x}$

$1-\big{(}(1-q^{2})^{J}\bar{r}_{k1}+1-\bar{r}_{k1}\big{)}^{\omega_{i}}$

$1-\big{(}1-\bar{r}_{k1}(1-(1-q^{2})^{J})\big{)}^{\omega_{i}}=\bar{r}_{i3}$

∎

###  10.3 First Order Conditions (Non-aggregated Estimator)

$\frac{\partial(-Q)}{\partial
p}=2\sum_{t=2}^{T}\sum_{i\in\tilde{P}}r_{i(t-1)}-2\sum_{t=2}^{T}\sum_{i\in\tilde{N}}(r_{i(t-1)})^{2}p+2\\#IPP-2\\#IPp-=0$

$p=\frac{\sum_{t=2}^{T}\sum_{i\in\tilde{P}}(r_{i(t-1)})+\\#IPP}{\sum_{t=2}^{T}\sum_{i\in\tilde{N}}(r_{i(t-1)})^{2}+\\#IP}$
(4)

From 4 it is apparent that the optimal $p$ exceeds one if

$\\#IPP-\\#IP<\sum_{t=2}^{T}\sum_{i\in\tilde{N}}(r_{i(t-1)})^{2}-\sum_{t=2}^{T}\sum_{i\in\tilde{P}}(r_{i(t-1)})$
$\\#IPP-\\#IP<\sum_{t=2}^{T}\sum_{\begin{subarray}{c}i\in\tilde{N}\\\
i\notin\tilde{P}\end{subarray}}(r_{i(t-1)})^{2}+\sum_{t=2}^{T}\sum_{i\in\tilde{P}}\big{(}(r_{i(t-1)})^{2}-(r_{i(t-1)})\big{)}$

The left hand side (LHS) of this inequality is always non-positive. The first
term on the right hand side (RHS) is positive and increasing in the number of
paths leading to $i$ (her ”information degree”). The second term on the RHS is
negative but converges to zero for large enough information degrees.
This shows that the number of links (and thus the signal reception rates)
among non-ips in general and non-ip participants in particular must be
sufficiently large to prevent a corner solution.

$\frac{\partial(-Q)}{\partial
q}=2\sum_{t=2}^{T}\sum_{i\in\tilde{P}}p\frac{\partial r_{i(t-1)}}{\partial
q}-2\sum_{t=2}^{T}\sum_{i\in\tilde{N}}p^{2}r_{i(t-1)}\frac{\partial
r_{i(t-1)}}{\partial q}=0$
$p=\frac{\sum_{t=2}^{T}\sum_{i\in\tilde{P}}\frac{\partial r_{i(t-1)}}{\partial
q}}{\sum_{t=2}^{T}\sum_{i\in\tilde{N}}r_{i(t-1)}\frac{\partial
r_{i(t-1)}}{\partial q}}$ (5)

Intuitively, the RHS tends to zero when $q$ is large and there are few
participants that possess many links to the information.
Combining the FOCs leads to

$\frac{\sum_{t=2}^{T}\sum_{i\in\tilde{P}}(r_{i(t-1)})+\\#IPP}{\sum_{t=2}^{T}\sum_{i\in\tilde{N}}(r_{i(t-1)})^{2}+\\#IP}=\frac{\sum_{t=2}^{T}\sum_{i\in\tilde{P}}\frac{\partial
r_{i(t-1)}}{\partial
q}}{\sum_{t=2}^{T}\sum_{i\in\tilde{N}}r_{i(t-1)}\frac{\partial
r_{i(t-1)}}{\partial q}}$ (6)

6 pins down $q$ as a function of the number of IPs, the number of IP
participants as well as the agents in
$\tilde{N}_{2},..,\tilde{N}_{T},\tilde{P}_{2},..,\tilde{P}_{T},$ and their
respective link portfolios. Naturally, some variation in the outcomes for non-
ips is required, else the system of FOCs has no solution.

###  10.4 Convexity of the Objective Function (Non-aggregated Estimator)

The objective function $-Q$ is convex if and only if the Hessian is positive
semi definite. For the Hessian to be positive semi definite, the determinant
is required to be positive and which in turn requires that

$Det(\mathcal{H})=\frac{\partial^{2}Q}{\partial^{2}p}\frac{\partial^{2}Q}{\partial^{2}q}-\Big{(}\frac{\partial^{2}Q}{\partial
p\partial q}\Big{)}^{2}>0$

Since $Det(-Q)=Det(Q)$.

$Det(\mathcal{H})=$
$\underbrace{\left(2\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}(r_{i(t-1)})^{2}+\\#IP\right)}_{\frac{\partial^{2}Q}{\partial^{2}p}}\underbrace{\left(2\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}p^{2}\Bigg{(}\Bigg{[}\frac{\partial
r_{i(t-1)}}{\partial
q}\Bigg{]}^{2}+r_{i(t-1)}\frac{\partial^{2}r}{\partial^{2}q}\Bigg{)}-2\sum_{t=2}^{T}\sum_{i\in\tilde{P}_{t}}p\frac{\partial^{2}r_{i(t-1)}}{\partial^{2}q}\right)}_{\frac{\partial^{2}Q}{\partial^{2}q}}$
$-\underbrace{\left(4\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}pr_{i(t-1)}\frac{\partial
r_{i(t-1)}}{\partial q}-2\sum_{t=2}^{T}\sum_{i\in\tilde{P}_{t}}\frac{\partial
r_{i(t-1)}}{\partial q}\right)^{2}}_{\Big{(}\frac{\partial^{2}Q}{\partial
p\partial q}\Big{)}^{2}}$

A necessary condition is that

$\left(2\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}p^{2}\Bigg{(}\Bigg{[}\frac{\partial
r_{i(t-1)}}{\partial
q}\Bigg{]}^{2}+r_{i(t-1)}\frac{\partial^{2}r}{\partial^{2}q}\Bigg{)}-2\sum_{t=2}^{T}\sum_{i\in\tilde{P}_{t}}p\frac{\partial^{2}r_{i(t-1)}}{\partial^{2}q}\right)>0$
$\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}p\Bigg{(}\Bigg{[}\frac{\partial
r_{i(t-1)}}{\partial
q}\Bigg{]}^{2}+r_{i(t-1)}\frac{\partial^{2}r}{\partial^{2}q}\Bigg{)}>\sum_{t=2}^{T}\sum_{i\in\tilde{P}_{t}}\frac{\partial^{2}r_{i(t-1)}}{\partial^{2}q}$
(7)

The first derivative of $r_{i(t-1)}$ with respect to $q$ depends on $w^{i}$,
the number of paths to the information that individual $i$ possesses. It is
increasing in $w^{i}$ for small values of $q$, decreasing in $w^{i}$ in the
large-$q$ area and first increasing, then decreasing for intermediate values
of $q$. In any case, the slope decreases almost everywhere and hence the
second derivative is usually negative, implying that the RHS of 7 is negative.
The absolute value of the RHS increases in the number of non-ip participants
and in the number of ip connections they possess. As a consequence, it helps
fulfilling the convexity condition if non ip participants are numerous and
densely connected with the ips.
On the left-hand-side (LHS) the term

$\Bigg{[}\frac{\partial r_{i(t-1)}}{\partial
q}\Bigg{]}^{2}+r_{i(t-1)}\frac{\partial^{2}r}{\partial^{2}q}$

is positive for small and negative for larger values of $w^{i}$ and the
intersection with the horizontal axis occurs the earlier the higher $q$. The
function is first positive, crosses the axis and becomes negative and then
exhibits another turning point such that it finally converges to zero for very
large values of $w^{i}$. As a consequence, heterogeneity in $w^{i}$ helps
fulfilling the convexity condition as the function is positive for small
$w^{i}$ and close to zero for large $w^{i}$.
When all agents exhibit intermediate values of $w^{i}$ and/or participation
rates are low, on the other hand, the convexity condition may fail to hold.
Observe that a large number of non-ip participants also decreases the cross
partial derivative, and as such also that the determinant is positive. The
convexity condition is harder to fulfill in the high-$q$ area.
Given a particular $p,q$ point, a certain number of agents that contribute to
the objective function $\tilde{N}$ and a particular degree distribution among
them, equation 7 can be used to determine the minimal level of non-ip-
participation that guarantees convexity of the objective function.
Convexity of the objective function can also be investigated by means of a
Taylor series expansion, the aim being to demonstrate that higher order terms
vanish.
The third order terms are

$\frac{1}{2}\frac{\partial^{3}Q}{\partial^{2}p\partial
q}=2\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}r_{i(t-1)}\frac{\partial
r_{i(t-1)}}{\partial q}$ (8) $\frac{1}{2}\frac{\partial^{3}Q}{\partial
p\partial^{2}q}=2p\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}\Bigg{(}\Bigg{[}\frac{\partial
r^{i}_{(t-1)}}{\partial
q}\Bigg{]}^{2}+r_{i(t-1)}\frac{\partial^{2}r}{\partial^{2}q}\Bigg{)}-\sum_{t=2}^{T}\sum_{i\in\tilde{P}_{t}}\frac{\partial^{2}r_{i(t-1)}}{\partial^{2}q}$
(9) $\frac{1}{6}\frac{\partial^{3}Q}{\partial^{3}q}=$
$\frac{1}{3}p^{2}\sum_{t=2}^{T}\sum_{i\in\tilde{N}_{t}}\Bigg{(}3\frac{\partial
r_{i(t-1)}}{\partial
q}\frac{\partial^{2}r_{i(t-1)}}{\partial^{2}q}+r_{i(t-1)}\frac{\partial^{3}r_{i(t-1)}}{\partial^{3}q}\Bigg{)}-\frac{1}{3}\sum_{t=2}^{T}\sum_{i\in\tilde{P}_{t}}p\frac{\partial^{3}r_{i(t-1)}}{\partial^{3}q}$
(10) $\frac{1}{6}\frac{\partial^{3}Q}{\partial^{3}p}=0$ (11)

First, observe that the shape of $r_{i(t-1)}\frac{\partial
r_{i(t-1)}}{\partial q}$ and $\frac{\partial r_{i(t-1)}}{\partial
q}\frac{\partial^{2}r_{i(t-1)}}{\partial^{2}q}$ are reflections of one another
about the horizontal axis and hence mitigate one another. This is because the
first derivative is increasing and while the second derivative is decreasing.
Further, if $Det(\mathcal{H})>0$, then 9 is positive. The third derivative is
positive again. The forth term of 10 can mitigate the third term of 10 and 9
only if non-ip participants are sufficiently numerous.

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