Identification of a Rank-dependent Peer Effect Model

Identification of a rank-dependent peer effect model

Introduction

Economic agents are often connected. These connections raise important questions about how linked units influence one another through spillovers, known as peer effects. It has been well established in many empirical settings that peer effects are heterogeneous sacerdote2011peer,masten2018random,bramoulle2020peer. Despite this, the canonical models of peer effects assume homogeneous effects from peers manski1993identification,BDF. This paper develops an econometric framework that allows for flexible and heterogeneous patterns of peer effects. Specifically, we will focus on heterogeneous effects based on the ranking of each peer's outcome among the peer group: a rank-dependent peer effect model. As such, the effect of each peer of an individual is endogenously determined by the distribution of the outcomes of that individual's peers.

This framework provides empirical researchers with a new tool to uncover patterns in peer influence, particularly in settings where relative standing plays a crucial role, such as education, workplace productivity, and social interactions. For instance, when students in a school interact, the skills of your highest-performing friend may be the key determinant of how well you perform in school. On the other hand, in a manufacturing setting where workers operate on a conveyor-belt-style assembly line, the performance of the slowest worker might be the key factor determining the spillover effect. This motivates a model of peer effects that allows for differently ranked peers to have different effects on an individual's outcome.

The current literature on peer effects has primarily focused on estimating a scalar spillover parameter. The canonical model for peer effects has, following the seminal works of manski1993identification and BDF, been the linear-in-means (LIM) peer effect model. This model is defined as: for $i=1,\cdots,n$,

For example, consider spillovers in exam scores. Then $Y_i$ is the exam score for student $i$, $\bar{Y}_i$ is the mean of student $i$'s peers' exam scores and $x_i$ is the vector of student-level covariates for student $i$. $x_i$ could also include contextual effects: e.g., the mean of student $i$'s peers' covariates. In the LIM model, the total peer effect is the product of $\bar{Y}_i$ and a scalar peer effect parameter $\beta$.

However, the LIM model has several caveats, limiting the patterns of spillover it can analyse. For example, the LIM model assumes that inputs from different peers are perfectly substitutable. As such, there is no compositional effect from having perfectly homogeneous peers compared to having heterogeneous peers, as long as the mean is equal. This means that the LIM model assumes that the higher moments such as standard deviation of peer outcomes does not matter.

The limitations of the LIM model are also well established in the empirical literature sacerdote2011peer. To overcome them, empirical papers have investigated heterogenous patterns of peer effects by interacting average peer outcomes with exogenous variables, or by estimating non-linear functions of the average outcome of peers Foster,HL,GKN. More recently, there has been extensions to the LIM model by allowing for more general aggregators than the mean BRUZ. However, this analysis is still limited to looking at the effect of peer outcomes through a single aggregated measure. We further expand the scope of peer effects by developing a generalized econometric model that flexibly estimates heterogeneous spillovers from peers under an exchangeability assumption\footnote{In this context, exchangeability means that the spillover an individual experiences does not depend on the identity of their peers. This is implicitly assumed in existing models of peer effects, but needs to be made explicit in our model due to it's flexible nature. }. Specifically, we will let the inputs from peers affect an individual's outcome differently depending on the peer's relative position in the individual's peer group.

In our generalized model, the peer effect is a linear function of `ordered' peer outcomes. That is, for $i=1,\hdots,n$,

$d_i$ is the number of peers for individual $i$ and $\tilde{Y}_{i,k}$ is the outcome of the $k$-th lowest performing peer of individual $i$; $\tilde{Y}_{i,k}$ takes the $k$-th lowest value from the set $$ \{Y_j: \text{individual } j \text{ is a peer of individual } i. \}. $$ This model identifies the general peer effect under an assumption that peers are exchangeable and that the peer effect is linear in ordered peer outcomes. Note that this does not impose linear separability of peer outcomes. To see this, consider two connected individuals, $A$ and $B$. The impact of $A$'s outcome on $B$'s outcome depends on the outcomes of all the individuals that individual $B$ is connected to. This means that both the magnitude and sign of the peer effect coefficient from individual $A$ to $B$ is endogenous to all the peers of $B$, as their outcomes decide the ranking of $A$ in $B$'s peer group. In contrast, the canonical LIM model assumes this peer effect coefficient only depends on how many peers individual $B$ has.

The generality of this model allows us to capture many economically relevant features of peer effects. Firstly, as the model does not impose perfect substitution, it can analyse whether there are complementarities or substitutability in peer effects. Secondly, as the spillover from a peer can depend on the size of the peer group, the model can detect if individuals with more connections are systemically different from other individuals. For example, consider a simple model where individuals are affected differently by their relatively higher- and lower-performing peers:

This model is a simplification of the rank-dependent peer effect model, with $\beta_{1,d} = \ldots = \beta_{\lceil d/2 \rceil,d} = \beta_{\text{low}}$ and $\beta_{\lceil d/2 \rceil +1,d} = \ldots = \beta_{d,d} =\beta_{\text{high}}$ for each $d$. When $\beta_{\text{low}} = \beta_{\text{high}}$, all peer outcomes are perfectly substitutable with each other, as in the LIM model. When $\beta_{\text{low}} \neq \beta_{\text{high}}$, a sufficiently large change will move a peer from the lower-performing half to higher-performing half. This will change that peer's coefficient, and therefore also change their relative rate of substitution with the other peers.

Similarly, as the models' coefficients depend on $d_i$, it allows for returns to scale in the number of connections. As an illustrative example, consider a model where the median of peer outcomes is the sole determining factor in the peer effect, this becomes: $$ Y_i = \beta_{d_i} \tilde{Y}_{i,\lceil d_i/2 \rceil} + {x_i}^\intercal \gamma + \varepsilon_i. $$ If $\beta_{d} < \beta_{d'}$ when $d < d'$, we would conclude that peer effects exhibit returns to scale in the number of connections.

More generally, the rank-dependent peer effect model can be motivated as a general regression of an individual's outcome on the distribution of their peers' outcomes. As will be further discussed in Section (ref), ordered peer outcomes have one-to-one relationship with the (empirical) distribution of peer outcomes. As such, our model allows for a wide variety of heterogeneous patterns in the peer effects in terms of the distribution of peer outcomes. The primary contribution of this paper is the introduction and identification of the rank-dependent peer effect model, as shown in Equation (ref). Following the literature on peer effects, our identification result uses the conditional distribution of the outcomes given the network connections and the individual characteristics, with the network assumed to be exogenous. The steps of the identification argument are as follows. First, we show that the rank-dependent peer effect model admits a unique equilibrium when the magnitude of the total peer effect is bounded by one. Then, using the unique equilibrium result, we derive a reduced-form representation of the model. Given the reduced-form representation, we construct moment conditions to identify the peer effect coefficients $\{\beta_{k,d}\}_{k,d}$ and $\gamma$. This step relies on an instrument relevance condition closely related to the equivalent condition of the LIM model.

In implementation, we use Two-stage Least-squares (TSLS) to estimate the parameters. We show, through simulation exercises, how the finite-sample performance of the TSLS estimator depends on the complexity of the model, the sample size and the strength of our instruments. We also show how ex-ante knowledge about the structure of peer effects can aid in estimation in settings where the fully saturated model is challenging to estimate. We then show how our estimator can be applied to empirical settings by estimating spillovers in learning between students at two Norwegian middle schools. Our estimates show that the homogeneous estimates of the LIM model does not capture the heterogeneity of peer effects we find in the data. Instead, the highest-GPA peers are the driving factor in GPA spillovers. This supports the empirical relevance of the rank-dependent peer effect model in real world settings.

The paper most closely related to ours is BRUZ. They let peer outcomes be aggregated to a scalar summary measure by a broad class of aggregators, which determines the peer effect. This class of aggregators nests the mean used in the LIM model. Unlike our model, this aggregator depends on a single parameter, and does not allow for returns to centrality. In two extreme cases of their model, BRUZ construct the peer effect to be linear in the minimum peer outcome or the maximum peer outcome alone, both of which are allowed in our model as well. Similarly, tao2014social shows how to identify a model in which only the highest outcome friend matters for the peer effect. This is a special case of our model with $\beta_{d,d} = \beta_{max}$ for all d, and zero otherwise.

The empirical study of peer effects has been a large literature in economics bramoulle2020peer, with a seminal contribution by sacerdote2001peer. Some of the papers in this literature are close in spirit to our model, showcasing an interest in exploring these kinds of heterogeneities in the empirical literature. Existing work has, for example, let the peer effect be the mean of non-linear functions of peer outcomes, such as the proportion of peers above a cutoff Kang,GKN, through quantile regressions Kang, or looked at heterogeneity by interacting the peer effect with exogenous variables Foster,HL,alne2024peer,tincani2024heterogeneous. To the best of our knowledge, however, there is no previous work allowing for peer effects to depend on the distribution of peer outcomes, or a theoretical discussion of these methods.

The rest of this paper is structured as follows. Section (ref) discusses the model and its implications. Section (ref) formally identifies the parameters of the rank-dependent peer effect model. Section (ref) discusses finite-sample performances of a TSLS estimator in the rank-dependent peer effect model, using simulations. Finally, Section (ref) applies our methodology to educational peer effects in Norwegian schools.

Model

In our setting, a researcher observes individual-level outcomes and individual-level control covariates $\{Y_i, x_i\}_{i=1}^n$ and the network adjacency matrix $\{A_{i,j}\}_{1 \leq i,j \leq n}$. We let $A_{i,j}=1$ indicate that individual $i$ is linked to individual $j$, and $A_{i,j} = 0$ otherwise. We assume there are no self-loops in the network, so $A_{i,i} = 0$, and links may be directed, i.e., $A_{i,j}$ and $A_{j,i}$ are allowed to be different.

Consider a general model of nonparametric peer effects:

We call $h(\cdot) $ the peer effect function, or simply the peer effect, which takes a $n \times 1$ vector and produces a scalar peer effect.

Assumption (ref) assumes that the peer effect function $h$ is invariant to a permutation on its inputs. Then, we can construct a (derived) peer effect function $\tilde{h}$ that takes the ordered peer outcomes and the number of peers as its inputs, with which the nonlinear peer effect model (ref) holds. By additionally assuming linearity on $\tilde{h}$, i.e. $$ \tilde{h} \big( \{\tilde{Y}_{i,k}\}_{k=1}^{d_i}, d_i \big) = \sum_{k=1}^{d_i} \beta_{k,d} \tilde{Y}_{i,k}, $$ we obtain the rank-dependent peer effect model of our paper. In this sense, the key motivation for the rank-dependent peer effect model is Assumption (ref) where we assume that the names of individuals do not matter.

We define the rank-dependent peer effect model as:

for $i=1, \ldots, n$, where

Note that $\tilde{Y}_{i,1} \leq \ldots \leq \tilde{Y}_{i, d_i}$ for any $i$ and $\tilde{Y}_{i,k} = Y_j$ for some $j \neq i$. The individual-level control covariates $x_i$ may include functions of the network adjacency matrix $\{A_{i,j}\}_{1 \leq i,j \leq n}$ and the peer characteristics. For example, given some individual-specific characteristics $\{w_i\}_{i=1}^n$, the control covariate $x_i$ could be functions of $\{w_i\}_{i=1}^n$ and $\{A_{i,j}\}_{1 \leq i,j \leq n}$, allowing for contextual effects: \[ Y_{i} = \sum_{k=1}^{d_i} \tilde{Y}_{i,k} \beta_{k, d_i} +

^\intercal \gamma + \varepsilon_i =: \sum_{k=1}^{d_i} \tilde{Y}_{i,k} \beta_{k, d_i} + x_i \gamma + \varepsilon_i. \]

The rank-dependent peer effect model (ref) assumes that the outcomes of individual $i$'s peers affect individual $i$ in a way that the coefficient on a peer outcome depends on the peer's rank among individual $i$'s peer group. $\beta_{k,d}$ is the coefficient for the $k$-th lowest performing peer's outcome, when the given individual has $d$ peers in total. Note that the model allows a individual to affect different peers differently. Suppose that individual 1 is friends with both individual 2 and 3. If individual 1 is the lowest-performing peer of individual 2 while being the highest-performing peer of individual 3, the effect of individual 1's outcome will be $\beta_{1, d_2}$ for individual 2 and $\beta_{d_3, d_3}$ for individual 3.

The model nests both the linear-in-means (LIM) model and the linear-in-sums (LIS) model. By letting $\beta_{k,d} = \beta / d$, the peer effect term in (ref) becomes the LIM peer effect:

with $\bar{Y}_i = \frac{1}{d_i} \sum_{k=1}^{d_i} \tilde{Y}_{i,k} = \frac{\sum_{j=1}^n A_{i,j} Y_j}{\sum_{j=1}^n A_{i,j}}$. By letting $\beta_{k,d} = \beta$, the peer effect term becomes the LIS peer effect:

Though widely used, the LIM model and the LIS model have a caveat that using only a mean or a sum may ignore important information contained in the second or higher moments of the peer outcomes.\footnotemark \footnotetext{A notable exception is where the peer effect occurs through a binary variable, as in GPS}

The flexibility of our peer effect model (ref) allows us to address key questions in the peer effect literature. First, because the coefficients are rank-dependent, our model enables the analysis of complementarity versus substitutability, as has been discussed in Kang amongst others. For instance, when $\beta_{1,d} \neq 0$ and $\beta_{k,d} = 0$ for every $k \geq 2$ and $d$, the peer effect reflects perfect complementarity, meaning that only the lowest-performing peer influences the outcome. On the other hand, when $\beta_{k,d} = \beta_{k',d}$ for all $k, k'$, the peer effect exhibits global perfect substitutability, where each peer contributes equally. It is important to note that the linear structure of our model imposes local perfect substitutability. Specifically, all peer outcomes are perfectly substitutable with a constant marginal rate of substitution for small changes in $\tilde{Y}_{i,k}$ such that $\tilde{Y}_{i,k-1} \leq \tilde{Y}_{i,k} + \Delta \leq \tilde{Y}_{i,k+1}$. However, when the change in $\tilde{Y}_{i,k}$ is large enough to alter its rank, the slope of the isoquant curve adjusts accordingly.

Secondly, because the coefficients also depend on $d_i$, the size of the peer group, our peer effect model enables the exploration of returns to centrality. For simplicity, assume $Y_j = y$ for every $j$ in the peer group of individual $i$. In the LIM model, there is zero return to centrality, meaning that the peer effect remains unchanged when an additional peer is added. The LIS model imposes a constant return to centrality. In contrast, our model we can document the return to centrality in a flexible manner, by plotting $d \mapsto \sum_{k=1}^d \beta_{k,d} y$. These two features of peer effect heterogeneity are directly related to several empirically important questions: does a student benefit from a high-performing peer or get negatively influenced by a low-performing one? how substitutable are peer outcomes? does the number of friends impact the spillover effect?

While the rank-dependent peer effect model (ref) allows for nonlinearity in the peer effect, allowing for the peer effect of one peer to depend on the outcomes of all peers, it still admits a unique equilibrium under a natural extension of the assumption used in LIM models BDF.

Assumption (ref) assumes that the sum of the absolute values of peer effect coefficients for each individual is less than 1. Proposition (ref) states our unique equilibrium result.

Since the peer effect model admits a unique equilibrium, there exists a well-defined function from $\{\varepsilon_i\}_{i=1}^n$ to $\{Y_i\}_{i=1}^n$ given $\{x_i,A_{i,j}\}_{1 \leq i,j \leq n }$.

The unique equilibrium result from Proposition (ref) allows us to introduce a key ingredient in the identification of the peer effect parameters: the conditional distribution of $\{Y_i\}_{i=1}^n$ given $\{x_i, A_{i,j}\}_{1 \leq i,j \leq n}$. Without a unique equilibrium result as in Proposition (ref), we would have to assume an equilibrium selection mechanism to derive a conditional distribution of $\{Y_i\}_{i=1}^n$ given $\{x_i, A_{i,j}\}_{1 \leq i,j \leq n}$ from a conditional distribution of $\{\varepsilon_i\}_{i=1}^n$ given $\{x_i, A_{i,j}\}_{1 \leq i,j \leq n}$.\footnotemark \footnotetext{Identification does not necessarily require a unique equilibrium, see for example de2013econometric for examples from the literature on games with multiple equilibria. For example, an alternative identification strategy in our setting based on this literature would be to limit our attention to sets of $\{y_i\}_{i=1}^n$ that can only be an outcome of a single realization of $\{\epsilon_i\}_{i=1}^n$. Or, we may focus on extreme values of $\{x_i\}_{i=1}^n$ and discuss identification for samples with extreme values only. Though these approaches have been valuable in the contexts discussed in de2013econometric, they are not straightforward to implement in our setting. If there were multiple equilibria in the rank-dependent peer effect model, there would be $n!$, a massive number, different orderings to consider to find values of $\{y_i\}_{i=1}^n$ consistent with $\{\epsilon_i\}_{i=1}^n$.} However, thanks to Proposition (ref), we can assume Assumption (ref) and additional conditions on the distribution of $\{\varepsilon_i\}_{i=1}^n$, to construct moment conditions with $\{Y_i\}_{i=1}^n$.

Identification

To simplify notation, we use $\mathbf{E}_n$ to denote the conditional expectation of a random variable given the exogenous variables throughout the paper. That is for any random variable $W$, $$ \mathbf{E}_n \left[ W \right] := \mathbf{E} \left[ W | \{x_i, A_{i,j} \}_{1 \leq i,j \leq n} \right]. $$ We will follow the literature by assuming that the error term $\varepsilon_i$ is exogenous to all individual characteristics and the network.

Assumption (ref) rules out endogenous network formation and treats the network structure as fixed, meaning it does not depend on the randomness in $\{\varepsilon_i\}_{i=1}^n$. While there has been advances in analysing peer effect models with endogenous network formation goldsmith2013social,hsieh2016social,johnsson2021estimation,jochmans2023peer, we instead focus on endogeneity in peer effects conditional on the network.\footnote{To see this, note that we can view our model as a second step of a two-stage process where firstly, the network is formed and secondly, the peer effect is determined based on the connections established in the first step. Models of endogenous network formation address endogeneity in the first step, whereas our model addresses endogeneity in the second step by allowing peer effect coefficients to depend on $\{\varepsilon_i\}_{i=1}^n$.}

Assumption (ref) allows us to construct a reduced-form representation of $\mathbf{E}_n[Y_i]$, though unlike the LIM model this reduced form is not linear in $\{x_i\}_{i=1}^n$. Instead, the rank-dependence creates a nonlinear reduced-form representation as shown in Corollary (ref). To discuss the coefficients in the corollary, let us introduce the following notations: $\pi$ denotes an ordering on $\{1,\ldots, n\}$ in terms of $\{y_i\}_{i=1}^n$ and $\mathbb{B}(\pi)$ denotes a $n \times n$ peer effect coefficient matrix such that its $i$-th row $j$-th column component corresponds to the peer effect coefficient for individual $j$'s outcome on individual $i$'s outcome, given the ordering $\pi$. For any realization of $\{\epsilon_i\}_{i=1}^n$, Proposition (ref) gives us a unique equilibrium $\{y_i\}_{i=1}^n$ and thus a unique $\pi$ such that $$

= \mathbb{B} (\pi)

+

. $$ Corollary \ref{cor:reduced_form} takes expectations over the set of every possible ordering, $\Pi$, and gives us a reduced-form representation of $\mathbf{E}_n [Y_i]$.

Note that the residual terms $\eta_i$ and $\tilde{\eta}_{i,k}$ may not be linear in $\{x_i\}_{i=1}^n$.

To compare the reduced form from Corollary (ref) with more familiar models, such as the LIM model in (ref) and the LIS model, observe that for these models, Assumptions (ref) and (ref) directly yield a reduced-form representation of $\{Y_i\}_{i=1}^n$:

where the $n \times n$ matrix $\mathbb{B}$ is $\beta G := \beta \cdot \big( A_{i,j}/d_i \big)_{i,j}$ in the LIM model and $\beta A := \beta \cdot \big( A_{i,j} \big)_{i,j}$ in the LIS model. Importantly, this equation is linear in $\{x_i\}^n_{i=1}$.

However, this linearity does not hold in our model. Instead, the rank-dependent peer effect coefficients introduce nonlinearities in the conditional expectations due to the connection between ordering and $\{x_i\}^n_{i=1}$. Consider a simple case with three individuals where individual 1 is connected to both individuals 2 and 3, but individuals 2 and 3 are not connected to each other. From Assumption (ref), we have:

for some $\tilde{\beta}$, $\tilde{\gamma}_1$, $\tilde{\gamma}_2$, and $\tilde{\gamma}_3$. Without further assumptions on the distribution of $\{ \varepsilon_i \}_{i=1}^n$, the conditional expectations involve terms like $\mathbf{E}_n \left[ \varepsilon_2 \mathbf{1}\{ Y_2 \leq Y_3\}\right]$, which may not be zero or linear in $x_i$. An exception occurs when $\beta_{12} = \beta_{22}$, in which case the conditional expectations cancel out, resulting in a form similar to the LIM or LIS model.

These complications affect the relevance conditions we use to construct valid moment conditions. In the standard LIM and LIS models, the relevance condition on a set of instruments $\{z_i\}_{i=1}^n$ for identifying $\beta$ is given by ensuring that the matrix

has full rank. BDF discusses a similar condition in the LIM model when instruments are based on the mean covariates of peers or peers of peers.

Our conditions are different and will instead relate to the reduced forms presented in Corollary (ref). These conditions are given in Assumption (ref).

Part (ref) of Assumption (ref), together with Assumption (ref), provides the standard instrument exogeneity assumption commonly found in the literature. This includes, for example, instruments based on the average covariates of peers and peers-of-peers, as introduced by BDF for estimating the effects of peers' average outcomes. Given the increased number of endogenous variables in our model compared to the LIM model, a larger set of instruments is required. Beyond the instruments discussed in BDF, we also consider alternatives such as ordered peer covariates or higher moments of peer covariates. In the following discussion, we do not commit to a specific set of instruments, we focus on outlining the necessary conditions that any valid instruments must satisfy for identification.

Assumption (ref)-(ref) establishes a rank condition analogous to those found in BDF. The matrix in Assumption (ref)-(ref) decomposes into two components: $$ \sum_{i:d_i=d}

^\intercal and \sum_{i:d_i=d}

^\intercal. $$ Constructing instruments $\{z_{i,d}\}_{i=1}^n$ such that the first matrix is full rank is straightforward as long as there is sufficient non-transitivity in the network and variation in the covariates $\{x_i\}_{i=1}^n$. Specifically, we need the columns of $\mathbb{X} = \Big( x_1, \ldots, x_n \Big)^\intercal$ and $$

\cdots

$$ to be linearly independent. This condition is closely related to that found in \citet{BDF}, where there are no residual terms $\{\tilde{\eta}_{i,k}\}_{i,k}$. As such, when $\{\tilde{\eta}_{i,k}\}_{i,k}$ are close to zero, the sum of the two matrices will also have full rank, satisfying our relevance condition. In cases where $\{\tilde{\eta}_{i,k}\}_{i,k}$ are nonzero, it is challenging to interpret the rank condition, but we can empirically check whether or not it holds in a given sample.

To simplify the notation of our identification results, let

so that the rank-dependent peer effect model (ref) can be written as follows:

Let $W_i = \big( {\tilde{Y}_i}^\intercal, {x_i}^\intercal \big)^\intercal \in \mathbb{R}^{\frac{\bar{d}(\bar{d}+1)}{2}+l}$ and $z_i = \big( {z_{i,1}}^\intercal, \ldots, {z_{i,\bar{d}}}^\intercal, {x_i}^\intercal \big)^\intercal$. Lastly, let $\mathbb{Y}$, $\mathbb{W}$ and $\mathbb{Z}$ denote the row-stacked matrices of $Y_i$, $W_i$ and $z_i$. Theorem (ref) provides the identification result.

For identification, we consider a fixed $n$ and $\bar{d}$ environment. The minimal condition $n \geq \frac{\bar{d}(\bar{d}+1)}{2} + l$, along with Assumption (ref), guarantees that the matrix $\mathbf{E}_n \left[ \mathbb{Z}^\intercal \mathbb{W} \right]$ has a left inverse. However, in practice, when $\bar{d}$ is large relative to $n$, this can pose challenges. In such cases, it may be beneficial to consider an asymptotic framework where $\bar{d}$ grows with $n$ and thus the dimension of $\beta$ also grows with $n$. If the parameter $\beta$ satisfies some sparsity or smoothness restrictions, it may be possible to take advantage of recent developments for high-dimensional models with many endogenous variables. See for example belloni2022high. However, this literature primarily analyses cross-sectional data, and it is unclear if these results will extend to our setting due to the dependent nature of network data.

The moment condition (ref) consists of $n$ different equations, one for each student $i=1,\ldots,n$. Since $\tilde{Y}_i$ concatenates $\bar{d}$ different vectors of peer outcomes where only one out of the $\bar{d}$ vectors is nonzero for a given student, the $n \times \big(\frac{\bar{d}(\bar{d}+1)}{2} + l \big)$ matrix $\mathbf{E}_n[\mathbb{Z}^\intercal \mathbb{W}]$ can also be decomposed into $\bar{d}$ submatrices. Part (ref) of Assumption (ref) gives us each of the $\bar{d}$ submatrices having full rank, leading to $\mathbf{E}_n [\mathbb{Z}^\intercal \mathbb{W}]$ having full rank as a result. Given that $\mathbf{E}_n \left[ \mathbb{Z}^\intercal \mathbb{W} \right]$ has full rank, it is trivial to write $\beta$ and $\gamma$ as functions of the conditional moments of $\{Y_i\}_{i=1}^n$. In the case of the TSLS estimand, the first stage coefficient matrix from regressing $W_i$ on $z_i$ is used. Let $\Gamma$ denote the first stage coefficient matrix. Then, $$

:= \left( \mathbf{E}_n \left[ \left( \mathbb{Z} \Gamma \right)^\intercal \mathbb{W} \right] \right)^{-1} \mathbf{E}_n \left[ \left( \mathbb{Z} \Gamma \right)^\intercal \mathbb{Y} \right] =

. $$ For Sections (ref) and (ref), we use the sample analogue of the TSLS estimand as our estimator.

Comparison to other peer effect estimands

A natural question, given that our model generalizes the standard LIM model, is to what extent existing estimands recover the key parameters of interest from our model. A commonly accepted minimal standard for such estimands is that they represent weighted averages of the underlying heterogeneity, such as in the analysis of instrument variables mogstad2024instrumental or the analysis of Difference-in-Difference estimands de2020two,goodman2021difference. To explore this, we first assume instruments that satisfy the relevance condition used in the LIM model and define a LIM estimand as follows:

$$ \beta^{\text{LIM}} := \left( \mathbf{E}_n \left[ \mathbb{Z}^\intercal \mathbb{M}_X G \mathbb{Y} \right] \right)^{-1} \mathbf{E}_n \left[ \mathbb{Z}^\intercal \mathbb{M}_X \mathbb{Y} \right] $$ where $G = \big( A_{i,j} / d_i \big)_{i,j}$ and $\mathbb{M}_X = I_n - \mathbb{X} \left( \mathbb{X}^\intercal \mathbb{X} \right)^{-1} \mathbb{X}^\intercal$ with $\mathbb{X}$, a row-stacked matrix constructed with $\{x_i\}_{i=1}^n$. Similarly, for the LIS model, we define $$ \beta^{\text{LIS}} := \left( \mathbf{E}_n \left[ \mathbb{Z}^\intercal \mathbb{M}_X A \mathbb{Y} \right] \right)^{-1} \mathbf{E}_n \left[ \mathbb{Z}^\intercal \mathbb{M}_X \mathbb{Y} \right] $$ where $A = \big( A_{i,j} \big)_{i,j}$.

These conditions are the same instrument relevance conditions used in the LIM model and the LIS model, as in BDF.

Note that, in the case of LIM misspecification, the weights are applied not directly to $\beta_{k,d}$ but to $d \beta_{k,d}$. The rank-dependent peer effect coefficient $\beta_{k,d}$ represents the impact of a single peer outcome when there are $d$ peers in total, so it should be rescaled by $d$ to interpret it as a coefficient on a `representative' or`average' peer. For instance, when $\beta_{1,d} = \ldots = \beta_{d,d} = \beta$, we have: $$ \sum_{k=1}^d \beta_{k,d} \tilde{y}_{i,k} = d \beta \bar{y}_i, $$ where the coefficient on the average peer outcome is $d \beta_{k,d}$.

Proposition (ref) demonstrates that both the LIM estimand $\beta^{\textup{LIM}}$ and the LIS estimand $\beta^{\textup{LIS}}$ are weighted sums of the rank-dependent peer effect coefficients $\{\beta_{k,d}\}_{k,d}$. While the weights sum to one, which is reasonable, there is no guarantee that the weights have the same sign. As a result, it is possible for all $\beta_{k,d}$ to be positive, while the LIM estimand $\beta^{\textup{LIM}}$ is negative, and vice versa. The expressions for the weights are provided in the Appendix.

Simulation

To investigate the finite sample performance of the TSLS estimator for the rank-dependent peer effects, we simulate 1000 samples from the rank-dependent peer effect model while varying the model specifications. To investigate the small sample properties of the estimator, we consider sample sizes of either $n=100$ or $n=200$. The DGP we use is as follows: for $i = 1,\ldots, n$, $$ Y_i = \gamma_0 + \gamma_1 X_i + {\tilde{Y}_i}^\intercal \beta + \varepsilon_i $$ and

To simulate the network, we first generate the links based on a logit model, defining

with $V_{i,j} \overset{\text{iid}}{\sim} \text{logit}$ and $R_{i,j} \sim N(0,1)$. We then drop links from individuals with more links than $\bar{d}$ until all individuals have at most $\bar{d}$ links. This procedure generates a network with a roughly uniform degree distribution, with very few individuals having zero links. For the results in this section we will fix the network across simulations. Results when we do not fix the network across simulations can be seen in (ref). Finally, to avoid unnecessary repetition in our tables, we will only show results for a subset of $\beta$.

The instruments $\big\{Z_i = \big( Z_{i,1}, \ldots, {Z_{i,\bar{d}}}^\intercal \big)^\intercal\big\}_{i=1}^n$ used in the TSLS estimation are constructed by taking ordered statistics of peers' covariate $\{X_j\}_{j:A_{i,j}=1}$ for each student $i$. $Z_{i,d}$ is a $d$-dimensional vector that orders $\{X_j\}_{j:A_{i,j}=1}$ when $d_i=d$ and contains zeros when $d_i \neq d$. To avoid the simulations potentially suffering from the contamination bias discussed in goldsmith2024contamination, we stratify our estimator. We do this by splitting the dataset into $\bar{d}+1$ strata of $\{i:d_i=0\},\ldots, \{i:d_i=\bar{d}\}$ and the sets of the coefficients $\big(\gamma_0,\gamma_1 \big), \beta_{1,1}, \ldots, \big( \beta_{1,\bar{d}}, \ldots, \beta_{\bar{d},\bar{d}} \big)$ are separately estimated from each strata. A pooled version of our estimator generally performs similarly, but can sometimes be less robust. See (ref) for a discussion.

First, we set $\bar{d} = 2$. The bias and MSE of the TSLS estimator are compared to those of the OLS estimator in Table (ref). In this table, two key model parameters, $\gamma_1$ and $n$, vary across the columns. As expected, the OLS estimator shows a larger bias compared to the TSLS estimator. However, the TSLS estimator has a higher variance than the OLS estimator. Notably, as $|\gamma_1|$ increases, the variance of the TSLS estimator decreases. This indicates the finite sample performance of the estimator depends on the existence of strong covariates.

Additionally, as $|\gamma_1|$ increases, the performance of the OLS estimator also improves in terms of both bias and MSE. A larger $|\gamma_1|$ means that the ordering of $\{Y_i\}_{i=1}^n$ relies less on the variation in $\{\varepsilon_i\}_{i=1}^n$ and more on the variation in $\{\gamma_1 x_i\}_{i=1}^n$. If the endogeneity problem is exacerbated by the randomness in the mapping $\{Y_j\}_{j=1}^n \mapsto \tilde{Y}_{i,k}$, then a larger $|\gamma_1|$ may help reduce bias by decreasing this randomness.

Next, we increase the parameter space by setting $\bar{d} = 5$. Since the dimension of $\beta$ is proportional to $\bar{d}^2$, this introduces 15 different coefficients for the rank-dependent peer effect in our model. Panel A of Table (ref) presents the bias and MSE of the TSLS estimator under this setting. We observe that the TSLS estimator suffers from significant bias and highly volatile variance when $\gamma_1 = 1$ (columns 2 and 7). However, as $|\gamma_1|$ increases, the variance of the TSLS estimator decreases, as shown in columns 8 through 10.

Interestingly, the MSE is larger for the coefficients $\beta_{k,5}$ where $k = 2, 3, 4$, compared to $k = 1, 5$. This is related to our earlier observation that larger $|\gamma_1|$ improves the performance of the OLS estimator. The identity of the $k$-th lowest-performing peer tends to fluctuate more frequently for $k = 2, 3, 4$, leading to a more severe endogeneity problem from the randomness in the mapping $\{Y_j\}_{j=1}^n \mapsto \tilde{Y}_{i,k}$. In contrast, this issue is less pronounced for $k = 1, 5$. Therefore, it may be reasonable to impose smoothness restrictions on $\beta_{k,d}$ for values of $k$ that are not close to 1 or $d$.

To discuss the empirical advantage of a smoothness restriction, we additionally considered a restricted peer effect model with correct smoothness restrictions: for $d=1,\ldots,5$, $$ \beta_{k,d} = \frac{\beta_{-\max} }{d-1}\hspace{3mm} \forall k < d \hspace{5mm} \text{and} \hspace{5mm} \beta_{d,d} = \beta_{\max}. $$ With the restricted model, we estimated only two peer effect parameters: $\beta_{-\max}, \beta_{\max}$ . Panel B of Table (ref) contains the estimation results. Both in terms of bias and MSE, correct smoothness restriction greatly improves the performance of the TSLS estimator when $|\gamma_1|$ is small.

Finally, Table (ref) reports the average first-stage $R^2$ and the probability of the $R^2$ exceeding 0.7 for each of the endogenous variables $\tilde{Y}_{1,5}, \cdots, \tilde{Y}_{5,5}$. The results show that a weak first-stage regression exacerbates the performance of the TSLS estimator. This suggests that the potential costs of allowing for a more complex parameter space in terms of peer effect coefficients can be mitigated by carefully selecting the instruments $\{Z_i\}_{i=1}^n$ to ensure a sufficiently strong first-stage regression.

Empirical Illustration

To see how our model behaves with real data, we apply it to a dataset from alne2024peer, which is collected in two Norwegian middle schools and contains school grades, test scores for a nationwide test (“National Test”) taken prior to entering middle school, parental background variables, and friendship network of the students. We follow the regression specification of alne2024peer; the GPA is the outcome variable with possible spillovers and the National Test score and other socioeconomic and demographic variables are the control covariates. To simplify the presentation, we focus on a subset of the control covariates used in alne2024peer. Similarly, we run the estimation only for individuals with more than two friends to avoid any potential contamination bias. The instruments are constructed with the National test score from the test the students took before they started in middle school, as it is arguably exogenous to shocks to GPA in middle school.

In this paper, we expand the LIM model to allow for ordered peer outcomes to affect a student differently, as in (ref). Specifically, we focus on the highest-performing peer and the lowest-performing peer. Furthermore, whenever well-defined, we let

As $\tilde{Y}_{i,k}$ is the $k$-th lowest GPA among student $i$'s peers, $\bar{Y}_{i,-1-d_i}$ is the average GPA of student $i$'s peers minus the one with the lowest GPA and the one with the highest GPA, and so on. For the instruments, we construct $z_{i,1}$, $z_{i,d_i}$, $\bar{z}_{i,-1-d_i}$, $\bar{z}_{i,-1}$ and $\bar{z}_{i,-d_i}$ accordingly, using the National Test score. For each specification, we use the corresponding instruments to the $\tilde{Y}_i$ variables contained in the specification.

As controls for student background, we use the log of household income, whether the student was born in Norway, the average number of years the parents have been educated, the distance the students live from school as well as the student's gender.

Table (ref) shows the estimation results. There are five specifications we consider. The first is the classic LIM model. The second and third specifications separate out the lowest and highest friends GPA respectively. In the fourth we separate out both the highest and lowest GPA, while in the final specification we only include the highest and lowest GPA friends, not using the average GPA of the peers inbetween.

While the estimates are noisy, there are indications that the majority of estimated peer effect in the LIM model is driven by $\tilde{Y}_{d_i}$, the friend with the highest grade. We see that the coefficient on $\tilde{Y}_1$ is consistently smaller than the coefficient on $\tilde{Y}_{d_i}$. In our specification where we only separate out $\tilde{Y}_{d_i}$, the coefficient on the remaining LIM component is almost zero. However, the estimate of the coefficient on $\tilde{Y}_{d_i}$ is not significant in this specification. In total, this exercise shows that there may be large heterogeneity in peer spillovers in education masked by the estimated effect of the LIM model, though we do not have the power in this sample to conclude anything concrete.

Conclusion

This paper introduced the rank-dependent peer effect model. In this model, how you affect your peer depends not only on your own outcome level but also on the outcomes of the other peers that they have. The construction of the peer effect in the model is flexible enough to allow us to discuss many interesting questions regarding how the composition of a peer group affects the peer effect: complementarity v. substitutability, returns to centrality, etc. Traditional peer effect models, such as the linear-in-means (LIM) model, tend to oversimplify the dynamics of peer influence by focusing on a single scalar parameter and assuming perfect substitutability among peers. This oversimplification neglects the heterogeneity within peer effects, where individuals may experience varying degrees of influence depending on the structure and nature of their interactions, across different empirical contexts. By developing the rank-dependent peer effect model, this paper allows researchers to investigate richer patterns of peer interactions, with specific peers—such as the highest or lowest performers—having disproportionately larger effects on outcomes in certain contexts.

While we have focused on identification and estimation through TSLS, there are multiple avenues to establish estimators with better finite sample properties. For example, the model can benefit from having an estimator that has data-driven smoothness or sparsity property.

Finally, this paper has only considered settings with exogenous networks. As work on incorporating endogenous networks into peer effect models continue to develop, it would similarly be interesting to see if similar approaches can be applied to the rank-dependent peer effect model.