Testing for Peer Effects without Specifying the Network Structure

Testing for Peer Effects without Specifying the Network Structure

Introduction

A major stumbling block in the study of network effects is the need to specify the interaction structure. Most existing estimators and tests for network effects require a priori specification of the underlying network. Researchers often use data, if available, on geographical, economic, or social relationships between individuals (e.g., bilateral trade volume, friendship survey, etc.), along with a set of user-chosen rules (e.g., inverse distance, k-nearest neighbors, etc.), to determine the presence and strength of network connections in their models. However, the resulting network structure is subject to potential misspecification, and the misspecification risk is more severe when the specified network structure is purely based on theories or assumptions due to data limitations (e.g., linear-in-means).

A new body of literature on the identification and testing of network effects has emerged to tackle this issue. Blume2015 show that identification of network effects is possible even if the network structure is only partially known, as long as there are two individuals who are a priori known to be unconnected. Breza2020 propose a technique to estimate social links using aggregated relational data. BPR2021 introduce a new equilibrium concept for network formation models called “network competitive equilibrium," which allows recovery of unobserved social networks using only observable outcomes. LQT23 propose an identification strategy for cross-sectional social interaction models with many small networks, where unobserved network links are treated as random variables and network effects are identified from the “mean" relationship between the reduced form coefficients and structural parameters. For panel data models, several papers exploit the sparsity of network links commonly observed in social networks to directly estimate individual links using shrinkage estimation methods Bonaldi2015, Manresa2016, Rose2018. More recently, DRS24 consider a panel data model similar to those in the aforementioned papers, but their identification relies on differential popularity across individuals in a network, instead of the sparsity assumption.

To test for network effects, LP2018, LP2025 extend the Moran I test Moran to accommodate situations where the researcher faces multiple possible specifications of the underlying network structure. In the spatial econometric literature, some papers Ng2006, PUY2008, SYR2009, BFK2012, CGL2012, Pesaran2021 consider tests for cross-sectional correlation in panel data models with unspecified correlation structure. However, these tests are primarily designed for detecting cross-sectional correlation in the error term.

We contribute to this fast-growing literature by proposing an Anderson-Rubin (AR) test for the presence of peer effects that does not require specifying the network structure. The unrestricted model of our test is a linear panel data model of social interactions with dyad-specific peer effect coefficients for all potential peers. When no information on the network structure is available, all the other individuals in the network can be treated as potential peers. Our AR test evaluates if these peer effect coefficients are all zero.

Our test does not require the estimation of individual network links or peer effect coefficients, so it does not require restrictive regularity conditions and is much easier to implement than most existing methods in the literature. However, the merit comes at the cost of not being able to identify the strength of peer effects. Therefore, our test can be especially useful when the presence of peer effects is the primary concern or interest.\footnote{{Our AR test does not distinguish between endogenous and exogenous peer effects Manski93 because these two types of effects cannot be disentangled without any information or restrictions on the network Blume2015.} Therefore, the main objective of our test is to detect any forms of peer effects, rather than to identify the exact nature of the effects.} For instance, suppose a researcher conducts a causal analysis that relies on the Stable Unit Treatment Value Assumption, which requires no spillovers between treated and control units. To provide supportive evidence for this crucial assumption, the researcher may report some test statistics along with the estimation results. The proposed AR test is particularly advantageous in this context as it does not require costly and time-consuming data collection on network links or estimation of these links. On the other hand, if the presence of peer effects is the primary interest, our test can provide general evidence for peer effects that is not contingent on any specific assumption regarding the network structure, as demonstrated in our empirical applications.\footnote{A common strategy in applied research for identifying spatial or social effects is to use the spatial lags of covariates as instrumental variables (IVs) for endogenous spillover effects. However, as discussed in GO2012, this approach is vulnerable to weak identification. For instance, when the interaction structure is misspecified, the IVs derived based on the specified network may become invalid. Even when the network structure is correctly specified, the spatial lags of covariates are often highly correlated with each other or with other terms in the model, resulting in insufficient variation to identify the spillover effects. In such cases, our test can be an effective alternative to the existing identification strategy, as it does not rely on any parametric assumptions about the network structure.}

Our test is also closely related to the literature on inference with many instruments and/or many restrictions Bekker1994, Donald2003, AG2011, LO2012, Chao2014, Crudu2021, MS2022, AS2023. In our test, as the number of dyad-specific peer effect coefficients increases with the sample size, so do the number of restrictions under the null and the number of instrumental variables (IVs) employed to test the restrictions, leading to a testing problem with many restrictions and many IVs. To find a sufficient number of IVs to test the restrictions under the null, we exploit the exogenous characteristics of potential peers.\footnote{Our choice of IVs follows DRS24, but their approach assumes the number of agents in the network is fixed so that the number of IVs is fixed, which is different from our many-IV setting.} This is a unique many-IV scenario that arises naturally in the inference of network models without information on the network structure, and the two burgeoning research areas are nicely connected in this paper.

In this paper, we first illustrate the main idea of the proposed test using a simple panel data model without fixed effects. Then, we extend the test to a panel data model that includes both individual and time fixed effects. To the best of our knowledge, our paper is among the first to analyze Bekker1994's many-IV problem in a panel data model with two-way fixed effects. By adapting existing many-IV asymptotic results Hansen2008, AG2011, Chao2012, MS2022 to the panel data setting, we show that, under the null, our test statistic is asymptotically normal and has the correct size, allowing the number of agents in the network to increase to infinity at the same rate as the number of time periods.\footnote{The asymptotic validity of our test requires $nT \to \infty$ and $n < T$, where $n$ denotes the number of individuals and $T$ the number of time periods. So, our test requires a long panel. On the other hand, the power analysis in Remark (ref) suggests that when the number of null restrictions violated (i.e., the number of nonzero dyadic-specific peer effect coefficients) is fixed, consistency of our test requires that $T$ grows faster than $n$. Nevertheless, our Monte Carlo simulations indicate that the AR test performs reasonably well even when $n$ is comparable to $T$, such as $(n,T) = (30,50)$ or $(40,50)$. With the increasing availability of long panels in empirical research, our test is applicable in a wide range of settings.}

We conduct Monte Carlo simulations to investigate the finite sample performance of the proposed AR test. To study the power properties of the test, we consider various levels of network sparsity and interaction intensity. Furthermore, our simulations show the robustness and improved performance of the proposed AR test compared to some existing tests when the network is misspecified. We also provide two empirical applications to demonstrate how the proposed AR test can be applied in practice.

The remainder of the paper is organized as follows: Section (ref) and (ref) introduce the models and test statistics in the absence and presence of fixed effects; Section (ref) conducts Monte Carlo simulations examining empirical size and power of the test; Section (ref) applies the AR test to two empirical models: international growth spillover and National Basketball Association (NBA) player interaction models; and Section (ref) concludes. All the proofs of the theoretical results in this paper, the full version of the power analysis presented in Remark (ref), additional simulation results, and the estimator for the excess kurtosis of regression error discussed in Section (ref) are contained in Appendix.

Throughout this paper, we follow the convention of using boldface uppercase letters for matrices and row vectors, and boldface lowercase letters for column vectors.

AR Test for Peer Effects

In this section, we illustrate the main idea of the proposed test using a simple panel data model without fixed effects. We consider a more general panel data model with both individual and time fixed effects in Section (ref).

Consider a set of $n$ individuals $\mathcal{N}=\{1,2,\cdots ,n\}$. Let $ \mathcal{N}_{i}$ denote the set of potential peers of individual $i$ and $ n_{i}\equiv |\mathcal{N}_{i}|$ denote the cardinality of $\mathcal{N}_{i}$. When no information on $\mathcal{N}_{i}$ is available, all the other individuals in $\mathcal{N}$ can be treated as potential peers of individual $i$, i.e., $\mathcal{N}_{i}=\mathcal{N}/\{i\}$. In certain situations, researchers may know a priori that network links do not exist between certain pairs of individuals.\footnote{For instance, researchers may have prior knowledge that spillovers do not occur between some predetermined groups or clusters of individuals (see Remark (ref) for further discussion). We thank the Associate Editor for raising this point.} This information will reduce the number of potential peers of some individuals and thus the number of restrictions under the null hypothesis as we will see below. However, our test in general does not require any knowledge of $\mathcal{N}_{i}$, and this scenario is the primary focus of the paper. Suppose the outcome of individual $i$ in period $t$ is given by

for $i=1,\cdots ,n$ and $t=1,\cdots ,T$, where $\mathbf{X}_{it}$ is a $L$-dimensional row-vector of exogenous variables and $u_{it}$ is the error term. The coefficients $\alpha _{ij}$ represent dyad-specific endogenous peer effects Manski93.\footnote{In Remark (ref), we point out that a significant value of our test statistic indicates the presence of either endogenous or exogenous peer effects Manski93. See Remark (ref) for further discussion.} Our goal is to test for the presence of peer effects, i.e., $ H_{0}:\alpha _{ij}=0$ for all potential pairs of peers $(i,j)$.\footnote{An alternative approach to testing this joint null hypothesis is to test each restriction $\alpha _{ij}=0$ individually, using a Bonferroni-type procedure to control the familywise error rate (FWER). However, the Bonferroni procedure is known to be conservative, especially when the number of hypotheses is large, as it does not account for the dependence structure among the test statistics associated with each hypothesis. Therefore it may be desirable to test the restrictions jointly, as we propose in this paper. In addition, asymptotically controlling the FWER becomes nontrivial when the number of hypotheses increases with the sample size. For these reasons, we leave this direction for future research. We thank an anonymous referee for raising this point.} As the number of peer effect coefficients $\alpha _{ij}$ is proportional to the number of potential dyads in the network, the null hypothesis of our test imposes many restrictions AS2023.\footnote{The magnitude and density (i.e., the number of non-zero $\alpha_{ij}$) of the peer effect coefficients determine how strongly individuals are connected in the network and thus characterize the degree of deviation from the null hypothesis. A sparse network (or one with a small number of non-zero $\alpha_{ij}$) corresponds to a weak violation of the null. When the alternative is only weakly separated from the null, all tests inherently exhibit low power. In Remark (ref), we provide analytical results on how network density affects the power of our test.}

The peer effect term can be written more compactly as $\sum_{j\in \mathcal{N} _{i}}\alpha _{ij}y_{jt}=\mathbf{Y}_{it}\boldsymbol{\alpha} _{i}$, where $\mathbf{Y}_{it}$ is a row vector collecting the outcomes of individual $i$'s potential peers and $\boldsymbol{\alpha _{i}}$ is a column vector of corresponding coefficients. Let $\mathbf{e}_{i}$ denote the $i$th column of the identity matrix $\mathbf{I}_{n}$ and $\boldsymbol{\iota} _{n}$ denote an $n\times 1$ vector of ones. In matrix form, Equation ((ref)) can be written as

for $t=1,\cdots ,T$, where $\mathbf{y}_{t}=(y_{1t},\cdots ,y_{nt})^{\prime }$, $ \mathbf{Y}_{t}=(\mathbf{e}_{1}\mathbf{Y}_{1t},\cdots ,\mathbf{e}_{n}\mathbf{Y}_{nt})$, $\boldsymbol{\alpha} =(\boldsymbol{\alpha} _{1}^{\prime },\cdots ,\boldsymbol{\alpha} _{n}^{\prime })^{\prime }$, $\mathbf{X}_{t}=(\mathbf{X}_{1t}^{\prime },\cdots ,\mathbf{X}_{nt}^{\prime })^{\prime }$, and $\mathbf{u}_{t}=(u_{1t},\cdots ,u_{nt})^{\prime }$ . Stacking the observations over the $T$ periods together, we have

where $\mathbf{y}=(\mathbf{y}_{1}^{\prime },\cdots ,\mathbf{y}_{T}^{\prime })^{\prime }$, $ \mathbf{Y}=(\mathbf{Y}_{1}^{\prime },\cdots ,\mathbf{Y}_{T}^{\prime })^{\prime }$, $\mathbf{X}=(\mathbf{X}_{1}^{\prime },\cdots ,\mathbf{X}_{T}^{\prime })^{\prime }$, and $\mathbf{u}=(\mathbf{u}_{1}^{\prime },\cdots ,\mathbf{u}_{T}^{\prime })^{\prime }$.

The outcomes of individual $i$'s potential peers contained in $\mathbf{Y}_{it}$ are endogenous and natural instruments for $\mathbf{Y}_{it}$ are the exogenous characteristics of individual $i$'s potential peers denoted by $\mathbf{Z}_{it}$ DRS24. For instance, if $\mathcal{N}_{i}=\mathcal{N}/\{i\}$, then we could use $\mathbf{Z}_{it}=[\mathbf{X}_{jt}]_{j\in \mathcal{N}_{i}}=(\mathbf{X}_{1t},\cdots ,\mathbf{X}_{i-1,t},\mathbf{X}_{i+1,t},\cdots ,\mathbf{X}_{nt})$ as instruments for $ \mathbf{Y}_{it}=(y_{1t},\cdots ,y_{i-1,t},y_{i+1,t},\cdots ,y_{nt})$.\footnote{ In general, we could use linear combinations of the potential peers' exogenous characteristics as IVs for their outcomes. For instance, when $ \mathcal{N}_{i}=\mathcal{N}/\{i\}$, we could use $\mathbf{Z}_{it}=(\mathbf{X}_{1t}\mathbf{B},\cdots ,\mathbf{X}_{i-1,t}\mathbf{B},\mathbf{X}_{i+1,t}\mathbf{B},\cdots ,\mathbf{X}_{nt}\mathbf{B})$ as IVs for $ \mathbf{Y}_{it}=(y_{1t},\cdots ,y_{i-1,t},y_{i+1,t},\cdots ,y_{nt})$, where $\mathbf{B}$ is a $ L\times q$ matrix of known constants. If $\mathbf{B}=\mathbf{I}_{L}$, then $ \mathbf{Z}_{it}=(\mathbf{X}_{1t},\cdots ,\mathbf{X}_{i-1,t},\mathbf{X}_{i+1,t},\cdots ,\mathbf{X}_{nt})$. If $L$, the dimension of $\mathbf{X}_{it}$, is large, then we could choose a $\mathbf{B}$ with a small $q$ to reduce the number of IVs in $\mathbf{Z}$.} Let $\mathbf{Q}$ denote the IV matrix collecting linearly independent columns in $[\mathbf{X},\mathbf{Z}]$, where $\mathbf{Z}=(\mathbf{Z}_{1}^{\prime },\cdots ,\mathbf{Z}_{T}^{\prime })^{\prime }$ with $\mathbf{Z}_{t}=(\mathbf{e}_{1}\mathbf{Z}_{1t},\cdots ,\mathbf{e}_{n}\mathbf{Z}_{nt})$, and $K$ denote the number of columns in $\mathbf{Q}$. If the number of potential peers increases with $n$ such that $n_{i}=O(n)$ (e.g., when $\mathcal{N}_{i}=\mathcal{N}/\{i\}$), then the dimension of $\mathbf{Z}_{it}$ is $O(n)$ and, hence, $K=O(n^{2})$. Therefore, we are in the many-IV framework of Bekker1994.

Let $N=nT$, $\mathbf{P}=\mathbf{Q}(\mathbf{Q}^{\prime }\mathbf{Q})^{-1}\mathbf{Q}^{\prime }$ and $\mathbf{D}$ be a diagonal matrix containing the diagonal elements of $\mathbf{P}$. Using residuals computed from the restricted model $\widetilde{\mathbf{u}}=\mathbf{y}-\mathbf{X}\widetilde{ \boldsymbol{\beta} }$, where $\widetilde{\boldsymbol{\beta} }=(\mathbf{X}^{\prime }\mathbf{X})^{-1}\mathbf{X}^{\prime }\mathbf{y}$, we adopt the following jackknife AR test statistic MS2022 for $H_{0}:\boldsymbol{\alpha} =\mathbf{0}$

where $\widetilde{\Phi }$ is a consistent estimator of $\Phi =\mathrm{plim} _{N\rightarrow \infty }\frac{2}{K}\mathrm{tr}[(\mathbf{P}-\mathbf{D})\boldsymbol{\Omega} (\mathbf{P}-\mathbf{D})\boldsymbol{\Omega} ]$, with $\boldsymbol{\Omega} =\mathrm{E}(\mathbf{u}\mathbf{u}^{\prime }|\mathbf{X})$. The jackknife AR test statistic defined in Equation ((ref)) removes $\sum_{i}p_{ii}\widetilde{u}_{i}^2$ (= $\widetilde{\mathbf{u}}^{\prime }\mathbf{D}\widetilde{\mathbf{u}}$), where $p_{ii}$ denotes the $i$th diagonal element of $\mathbf{P}$, from the quadratic form $\widetilde{\mathbf{u}}^{\prime }\mathbf{P}\widetilde{\mathbf{u}}$ to re-center the test statistic to zero. This re-centering is referred to as leave-one-out or jackknife in the literature. The main advantage of the jackknife method is its robustness to heteroskedasticity of unknown form. This idea was introduced by Angrist1999 and has generated a rich literature in econometrics Hausman2012,Chao2012,Bekker2015,Crudu2021,MS2022.

Crudu2021 suggest estimating $\Phi $ by $\widetilde{\Phi }=\frac{2}{K}\mathrm{tr}[(\mathbf{P}-\mathbf{D}) \widetilde{\boldsymbol{\Omega} }(\mathbf{P}-\mathbf{D})\widetilde{\boldsymbol{\Omega} }]$, where $\widetilde{\boldsymbol{\Omega} }= \mathrm{diag}\{\widetilde{u}_{it}^{2}\}$. MS2022 point out that this variance estimator can lead to a loss of power and propose an alternative estimator for $ \Phi $. For brevity, we refer interested readers to thorough discussions on the consistent estimation of $\Phi $ in MS2022.

In our approach, for each endogenous regressor, $y_{jt}$, in Equation ((ref)) (i.e., the outcome of potential peer $j$), we use its corresponding exogenous characteristics, $\mathbf{X}_{jt}$, as the IVs. In other studies analyzing AR tests with many “weak” instruments MS2022, IVs are introduced through an auxiliary model (i.e., a first-stage regression) and are often weak in practice due to the difficulty of finding exogenous variables that affect the outcome $y$ only through the endogenous regressor. In contrast, in our setting, IVs are implicitly defined within the main regression model, i.e., Equation ((ref)), and thus are strong as long as some of the exogenous characteristics $\mathbf{X}$ are informative in predicting the outcomes. They remain strong even under the null hypothesis of no spillover effects. Since most applications include at least some exogenous variables that are predictive, we do not consider weak identification as a major concern for our test.

Another important feature of our testing environment, compared to existing studies on testing with many IVs, is that the number of parameters of interest or the number of restrictions under the null diverges with $n$ (i.e., testing with many IVs & many restrictions). The literature has typically studied the two testing problems separately -- testing with many IVs Bekker1994,Donald2003,AG2011,LO2012,Chao2014,Crudu2021,MS2022 and testing with many restrictions ANA2012,CAL2011,Catt2018a,Catt2018,AS2019,AS2023. However, as AS2019 notes, the asymptotic tools used for the two problems are often the same or similar, as they result in comparable test statistics (e.g., bilinear forms) and require similar econometric treatments to handle increasing dimensionality. Furthermore, our AR test statistic is evaluated under the null, where all dyad-specific peer effect coefficients are restricted to zero. As a result, the presence of many parameters in the unrestricted model does not materially affect the test statistic or its asymptotic behavior under the null. This is clearly an important merit of using the AR test in our setting, as it allows us to circumvent complications associated with estimating a large number of parameters AS2023.

To establish the asymptotic validity of the test statistic defined in Equation ((ref)), we maintain the following assumptions.

In the literature on inference with many instruments and/or many restrictions, it is common to assume that the error terms are independent Donald2003, AG2011, LO2012, Chao2014, Crudu2021, MS2022, AS2023. In the next section, we extend the model by introducing two-way fixed effects to partially account for dependence across individuals and over time. Assumption 2 implies that $\frac{1}{N}\mathrm{tr}(\mathbf{P})=\frac{K}{N}\leq C_{p}<1$. In our setting, as $K=O(n^{2})$ and $N=nT$, Assumption 2 allows $n$ to grow at the same rate as $T$, subject to the condition $n<T$. The main advantage of the AR test is that it only requires estimation of the restricted model, and Assumption 3 ensures that the exogenous variables $\mathbf{X}$ have enough variation and the OLS estimator for the restricted model is well behaved. The following proposition establishes the asymptotic normality of the proposed test statistic under the null hypothesis.\footnote{When $n$ is fixed, the number of IVs is fixed. Then, the asymptotic distribution of our test statistic reduces to a chi-square distribution with $K$ degrees of freedom (see the corollary to Proposition (ref) in Section (ref) for more details).}

AR Test in the Presence of Fixed Effects

To control for unobserved heterogeneity, we introduce individual and time fixed effects $\xi _{i}$ and $\eta _{t}$ to Equation ((ref)) so that the error term becomes

for $i=1,\cdots ,n$ and $t=1,\cdots ,T$, where $\epsilon _{it}$ are idiosyncratic random shocks. Then, in matrix form, Equation ((ref)) can be written as

where $\boldsymbol{\xi} =(\xi _{1},\cdots ,\xi _{n})^{\prime }$, $\boldsymbol{\eta} =(\eta _{1},\cdots ,\eta _{T})^{\prime }$, and $\boldsymbol{\epsilon} =(\boldsymbol{\epsilon} _{1}^{\prime },\cdots ,\boldsymbol{\epsilon} _{T}^{\prime })^{\prime }$ with $\boldsymbol{\epsilon} _{t}=(\epsilon _{1t},\cdots ,\epsilon _{nt})^{\prime }$.

To eliminate fixed effects, we apply a two-way within transformation by premultiplying Equation ((ref)) by $\mathbf{J}=(\mathbf{I}_{T}-T^{-1}\boldsymbol{\iota} _{T}\boldsymbol{\iota} _{T}^{\prime })\otimes (\mathbf{I}_{n}-n^{-1}\boldsymbol{\iota} _{n}\boldsymbol{\iota} _{n}^{\prime })$. The transformed model is

where $\mathbf{y}^{\ast }=\mathbf{Jy}$, $\mathbf{Y}^{\ast }=\mathbf{JY}$, $\mathbf{X}^{\ast }=\mathbf{JX}$, and $\boldsymbol{\epsilon} ^{\ast }=\mathbf{J}\boldsymbol{\epsilon} $. Moreover, let $\mathbf{Q}^{\ast }$ denote the IV matrix collecting linearly independent columns in $[\mathbf{X}^{\ast },\mathbf{Z}^{\ast }]$, where $\mathbf{Z}^{\ast }=\mathbf{JZ}$ , and $\mathbf{P}^{\ast }=\mathbf{Q}^{\ast }(\mathbf{Q}^{\ast \prime }\mathbf{Q}^{\ast })^{-1}\mathbf{Q}^{\ast \prime }$.

The jackknife AR test statistic defined in Equation ((ref)) has the advantage of being robust to heteroskedasticity of unknown form. However, when individual and time fixed effects exist and a data transformation is used to eliminate these effects, the standard jackknife method no longer re-center the test statistic properly. The within transformation introduces both cross-sectional and time-series dependence in the transformed errors and hence $\boldsymbol{\epsilon}^{\ast \prime} (\mathbf{P}^{\ast } - \mathbf{D}^{\ast }) \boldsymbol{\epsilon}^{\ast}$, where $\mathbf{D}^{\ast }$ is a diagonal matrix containing the diagonal elements of $\mathbf{P}^{\ast }$, does not have a zero mean. Other transformations such as the Helmert transformation also have the same issue in the presence of heteroskedasticity of unknown form. Hence, instead of using the jackknife method, we maintain the following assumption regarding the random shocks $\epsilon _{it}$ and re-center the quadratic form $\boldsymbol{\epsilon} ^{\ast \prime }\mathbf{P}^{\ast }\boldsymbol{\epsilon} ^{\ast }$ by subtracting out its mean as in AG2011 and AS2019.

The i.i.d. assumption for $\epsilon_{it}$ may seem restrictive. However, given that it is common practice among empirical researchers to use fixed effects to address potential heterogeneity and correlations in the error term, the i.i.d. assumption -- after accounting for the two-way fixed effects -- is not overly strong.\footnote{In the proof of Proposition (ref), we show that the dependence in the transformed errors induced by the within transformation asymptotically vanishes under homoskedasticity. Therefore, the homoskedasticity assumption plays an important role in establishing the asymptotic validity of our test statistic under two-way fixed effects.} As in the case without fixed effects considered in the previous section, we also impose the following assumptions.

Let $N^{\ast }=(n-1)(T-1)$ and $\widehat{\boldsymbol{\epsilon} }^{\ast }=\mathbf{y}^{\ast }-\mathbf{X}^{\ast }\widehat{\boldsymbol{\beta} }$ with $ \widehat{\boldsymbol{\beta} }=(\mathbf{X}^{\ast \prime }\mathbf{X}^{\ast })^{-1}\mathbf{X}^{\ast \prime }\mathbf{y}^{\ast }$. The test statistic for $H_{0}:\boldsymbol{\alpha} =\mathbf{0}$ in the presence of fixed effects is

where $\widehat{\Phi }^{\ast }$ is a consistent estimator of $\Phi ^{\ast }=(\mu _{4}-3\sigma ^{4})[\mathrm{plim}_{N\rightarrow \infty }\frac{1 }{K^{\ast }}\sum_{i=1}^N(p_{ii}^{\ast })^{2}-\bar{\lambda}]+2\sigma ^{4}(1-\bar{ \lambda})$, with $\mu _{4}=\mathrm{E}(\epsilon _{it}^{4}|\mathbf{X}_{it})$, $\bar{ \lambda}=\lim_{N\rightarrow \infty }K^{\ast }/N$, and $p_{ii}^{\ast }$ being the $i$th diagonal element of $\mathbf{P}^{\ast }$. In Appendix (ref), we provide a consistent estimator for the excess kurtosis, $\mu _{4}-3\sigma ^{4}$. It is worth pointing out that, when $\epsilon _{it}$ is mesokurtic (i.e., $\mu _{4}-3\sigma ^{4}=0$) or $\mathrm{plim}_{N\rightarrow \infty } \frac{1}{K^{\ast }}\sum_{i=1}^N(p_{ii}^{\ast })^{2}=\bar{\lambda}$, we have $ \Phi ^{\ast }=2\sigma ^{4}(1-\bar{\lambda})$.\footnote{The second case is satisfied when $p_{ii}^{\ast} \to \bar{\lambda}$ for all $i$, which is called an asymptotically balanced design of instruments/regressors in the many-IV literature. See AY2017 and AS2019 for related discussions.}

As discussed in AG2011 and Crudu2021, the normal approximation does not account for the number of instruments, which can be an issue in finite samples, particularly when the number of instruments is relatively small. Therefore, we consider the following chi-square approximation, valid when $K^{\ast}$ is either fixed or goes to infinity, and use it for our Monte Carlo simulations and empirical applications. Let $q_{f}(\tau)$ denote the $\tau$th quantile of the chi-square distribution with $f$ degrees of freedom.

Monte Carlo Simulations

In this section, we examine the empirical size and power of the AR test proposed in Equation ((ref)) (hereafter, denoted as $\texttt{T}_{\texttt{JL}}$) using simulations.\footnote{To save space, we do not report the results of the simulations without fixed effects. The results are available upon request.} The estimator proposed in Appendix (ref) is used to calculate the excess kurtosis in $\texttt{T}_{\texttt{JL}}$. All test statistics considered in this section use the same set of two-way within transformed residuals described in Section (ref). Appendix (ref) includes additional simulation results that are not reported in this section. The number of repetitions for each simulation specification is 5,000.

Size and Power

In the simulations, the data are generated from

where $x_{it} \sim \text{i.i.d.N}(0,1)$ and $u_{it} = \xi_i + \eta_t + \epsilon_{it}$ with $\xi_i \sim \text{i.i.d.U}(-1,1)$ and $\eta_t \sim \text{i.i.d.U}(-1,1)$. The random errors $\epsilon_{it}$ are generated from either normal: $\epsilon_{it} \sim \text{i.i.d.N}(0,1)$ (DGP1) or log-normal: $\epsilon_{it} \sim \text{i.i.d.}$ $[\exp(\zeta_{it})-\exp(0.5)]/[\exp(2)-\exp(1)]^{0.5}$ with $\zeta_{it} \sim \text{i.i.d.N}(0,1)$ (DGP2).

The true network is generated as a random graph er1959. Let $\texttt{ND}$ denote the proportion of dyads in the network with non-zero peer effect coefficients. Thus, $\texttt{ND}$ represents the density of the underlying network. We randomly select $\texttt{ND} \times 100\%$ of the dyads and set the corresponding peer effect coefficients $\alpha_{ij}=\rho$. \\

SIZE \, To demonstrate the robustness of the proposed AR test, $\texttt{T}_{\texttt{JL}}$, to many IVs, we compare $\texttt{T}_{\texttt{JL}}$ with two existing tests in the many-IV literature: Donald2003's J test ($\texttt{T}_{\texttt{DIN}}$), which is essentially $\texttt{T}_{\texttt{JL}}$ with the variance term $\Phi^{\ast}$ being $2\sigma^4$ due to the assumption of moderately many IVs such that $K^2/N \to 0$ AG2011; and AG2011's J test ($\texttt{T}_{\texttt{AG}}$), where $\Phi^{\ast} = 2\sigma^4(1 - \bar{\lambda})$ with $\bar{\lambda} = \lim_{N \to \infty} K^{\ast }/N$ due to the balanced covariate design assumption that $p_{ii}^{\ast} \to \bar{\lambda}$ for all $i$. All three tests use the same set of IVs described in Sections (ref). Chi-square approximation, described in Section (ref), is used to calculate the critical values.

Table (ref) reports the empirical size of the three tests for nominal 5% and 1% significance levels. When $n$ is large relative to $T$, which corresponds to the case where the number of IVs increases as fast as the sample size (i.e., $\bar{\lambda} \approx 1$),\footnote{In the simulations, $K^{\ast} = n(n-1) + 1$ and thus, $\bar{\lambda} \equiv K^{\ast}/N \approx 1$ when $n \approx T$, where $N = nT$.} $\texttt{T}_{\texttt{DIN}}$ exhibits significant under-rejections, while the rejection rates of the other tests are much closer to the nominal rates. This aligns with the theoretical prediction in AG2011 that $\texttt{T}_{\texttt{DIN}}$ under-rejects the null when $\bar{\lambda} > 0$ as the sample size increases. When $n$ is close to $T$, $\texttt{T}_{\texttt{JL}}$ slightly under-rejects the null, but the under-rejection vanishes as $T$ increases.

When the random error $\epsilon_{it}$ is normal, $\texttt{T}_{\texttt{AG}}$ and $\texttt{T}_{\texttt{JL}}$ show almost the same rejection rate, which is because the excess kurtosis in this case is zero and thus the variance term $\Phi^{\ast}$ in $\texttt{T}_{\texttt{JL}}$ reduces to the $\texttt{T}_{\texttt{AG}}$'s variance, $2\sigma^4(1 - \bar{\lambda})$. When the error is log-normal, however, the additional variance components in $\Phi^{\ast}$, that are associated with the excess kurtosis and the diagonal elements of the projection matrix, do not vanish, creating additional sampling variability that $\texttt{T}_{\texttt{AG}}$ does not account for. Thus, $\texttt{T}_{\texttt{AG}}$ exhibits significant over-rejections in this case. By contrast, the rejection rates of $\texttt{T}_{\texttt{JL}}$ are much closer to the nominal rates. These results also indicate that the estimator for the excess kurtosis proposed in Appendix (ref) performs well. \\

POWER \, Figure (ref) presents the power curves of the proposed AR test with varying $\rho$ and $\beta$, when $\texttt{ND} = 0.3$, $n=30$, and $T \in \{50,100\}$.

The rejection rates of the test quickly increase as the absolute values of $\rho$ and $\beta$ increase, confirming the analytic prediction in Remark (ref). Overall, it appears that the power is close to unity when $\rho = 0.3$, $\beta = 1$, and $T=50$.\footnote{Appendix (ref) compares the power of $\texttt{T}_{\texttt{AG}}$ and $\texttt{T}_{\texttt{JL}}$, where they exhibit almost the same rejection rates in many settings, indicating that there is little or no power loss when using $\texttt{T}_{\texttt{JL}}$, instead of $\texttt{T}_{\texttt{AG}}$, for testing the presence of peer effects without the balanced covariate design assumption.}

The next simulations analyze how network density ND affects the power of the test. The second column of Table (ref) reports the rejection rates of the AR test under different levels of network density ND. Overall, the power of the AR test increases as ND increases. However, when the network is sparse (e.g., ND $\leq 0.02$),\footnote{In these simulations, $n = 30$, so the number of possible links is $30 \times 29 = 870$. When ND = 0.02, approximately 17 nodes out of 870 are nonzero.} the power of the AR test is low.

Note that the peer effect coefficients $\alpha_{ij}$ are fixed to $\rho = 0.3$ in these simulations, so the overall level of peer interactions in the model is quite low when ND $\leq 0.02$. Since the proposed testing approach tests for all possible network connections without targeting any specific network structure, the low rejection rate in this scenario is expected (see Remark (ref) for more discussion). In this case, shrinkage methods that exploit the sparsity structure of the network may be preferable. Given that shrinkage methods may not perform well with dense networks, the two methods may complement each other in practice.\footnote{It is worth pointing out that the identification of both network links and peer effect parameters in DRS24 is considerably more challenging than testing for the existence of peer effects in our approach. Consequently, the identification strategy in DRS24 requires stronger regularity conditions. In particular, it requires that network effects do not cancel out such that $\beta_0\rho_0 + \gamma_0 \ne 0$ (Assumption A3), which excludes the null of no peer effects, i.e., $\rho_0=0$ and $\gamma_0=0$. As a result, their identification strategy and the adaptive elastic net GMM estimator developed under this assumption may not be suitable for testing the null of no peer effects directly.}

As discussed in Introduction, we envision the proposed AR test to be useful in scenarios where the primary parameter of interest is $\boldsymbol{\beta}$ and the researcher estimates $\boldsymbol{\beta}$ under the assumption of no network effects. The researcher can use our test to provide supportive evidence for the validity of the estimates. We would like to argue that the low rejection rates of the proposed AR test with a sparse network may not be a significant issue in this context. The third and fourth columns of Table (ref) report the bias and the coverage rate of 95 % confidence intervals (CI) of $\widehat{\beta }=(\mathbf{x}^{\ast \prime }\mathbf{x}^{\ast })^{-1}\mathbf{x}^{\ast \prime }\mathbf{y}^{\ast }$. As $\widehat{\beta}$ is a regression estimate of $\beta$ without accounting for peer effects, the bias and the distortion of the coverage rate of $\widehat{\beta}$ can represent the cost of type \expandafter\@slowromancap\romannumeral 2@ error. When $\texttt{ND}$ is less than 4%, the rejection rate is less than 50%, but the under-coverage of the 95% CI of $\widehat{\beta}$ is still less than 2%. If this level of bias and under-coverage is acceptable in practice, our test can still appeal to empirical researchers who prefer methods with less restrictive assumptions and simpler computations.

Next, we explore the idea of independent clusters discussed in Remark (ref). The simulations are based on the design used in Table (ref), with the modification that the $n$ individuals are divided into $R$ equal-sized groups. We then randomly select ND × 100% of the dyads within each group and set the corresponding $\alpha_{ij}=0.3$. As a result, the simulated network structure conforms to the assumption of independent clusters. We consider two tests, $\texttt{T}_{\texttt{JL}}$ and $\widetilde{\texttt{T}}_{\texttt{JL}}$. The former does not impose the assumption of independent clusters, whereas the latter is constructed under that assumption. In these simulations, we set $n = 30$. Consequently, the number of IVs in $\texttt{T}_{\texttt{JL}}$ is fixed at $K=n(n-1) = 870$ and that in $\widetilde{\texttt{T}}_{\texttt{JL}}$ varies with $R$ such that $K=n(n/R-1)$. For example, when $R = 2$, the number of IVs in $\widetilde{\texttt{T}}_{\texttt{JL}}$ is 420 and this number drops significantly to 60 when $R = 10$. The simulation results are presented in Table (ref).

Improvements in both size and power are observed for $\widetilde{\texttt{T}}_{\texttt{JL}}$. The improvement in size may be attributed to the chi-square approximation used to calculate the critical values. As shown in the corollary to Proposition 2, the approximation is valid when the number of IVs is either fixed or grows as fast as the sample size, but may yield better finite sample results when the number of IVs is relatively small.

Comparison against Tests with Misspecified Networks

An important merit of the proposed AR test is that it is not contingent on any specific form of network structure so it can be more robust compared to existing tests based on potentially misspecified networks. To show this merit, we compare the proposed AR test ($\texttt{T}_{\texttt{JL}}$) for $H_0:\alpha_{ij}=0$ for all $(i,j)$ in the model defined in Equation ((ref)) against a t-test ($\texttt{t-test}_{\texttt{TSLS}}$) for $H_0:\rho=0$ based on the two-stage least squares (TSLS) estimation of the model

where the adjacency matrix $\mathbf{W}$ is potentially misspecified. To carry out the TSLS estimation, we apply the two-way within transformation to obtain

and use $\mathbf{J}(\mathbf{I} _{T}\otimes\mathbf{W})\mathbf{X}$ as the IV for $\mathbf{J}(\mathbf{I} _{T}\otimes\mathbf{W})\mathbf{y}$.

We consider three cases of misspecification in the network structure that are commonly encountered in empirical research: size, location and direction. The first case misspecifies the sizes of the network links. In this case, the true network is generated as a random graph, where we randomly select $\texttt{ND} \times 100\%$ of the dyads and set the corresponding coefficients $\alpha_{ij} \sim \text{U}(0,1)$. To obtain the misspecified adjacency matrix $\mathbf{W}$ in Equation ((ref)), we define an indicator matrix $\mathbf{W}^{\ast}=[w_{ij}^{\ast}]$, where $w_{ij}^{\ast} = \mathds{1}\{ \alpha_{ij} > 0 \}$ for $i \ne j$, and then row-normalize $\mathbf{W}^{\ast}$ to get $\mathbf{W}$. This corresponds to the situation where a researcher knows the locations of the non-zero network links (i.e., who is connected with whom in the network) but does not know the size (or strength) of the links, so assumes each node is equally influenced by all its connections as an ad hoc solution.\footnote{An alternative interpretation is that the peer effects are heterogeneous in the data-generating process but the researcher (mistakenly) assumes that the peer effect is homogeneous.} We set $(n,T) = (30,50)$ and $\beta = 1$, and experiment with different values of $\texttt{ND}$. Note that, as $\texttt{ND}$ determines the number of non-zero links in the network, it also determines the number of misspecified entries in $\mathbf{W}$. Hence, $\texttt{ND}$ captures the degree of misspecification.

Table (ref) reports the rejection rates of $\texttt{T}_{\texttt{JL}}$ and $\texttt{t-test}_{\texttt{TSLS}}$ under varying $\texttt{ND}$. The power of $\texttt{t-test}_{\texttt{TSLS}}$ is higher than the power of $\texttt{T}_{\texttt{JL}}$ when $\texttt{ND}$ is small, but as $\texttt{ND}$ increases, the power of $\texttt{T}_{\texttt{JL}}$ improves, whereas the power of $\texttt{t-test}_{\texttt{TSLS}}$ deteriorates. This is because, as $\texttt{ND}$ increases, the overall peer effect in the model increases but the number of misspecified links in $\mathbf{W}$ also increases -- the former increases the power of $\texttt{T}_{\texttt{JL}}$ while the latter decreases the power of $\texttt{t-test}_{\texttt{TSLS}}$. When $\texttt{ND}$ is close to one, $\texttt{t-test}_{\texttt{TSLS}}$ has little power and eventually is not applicable. This is because, as $\texttt{ND}$ gets close to one, the misspecified network $\mathbf{W}$ converges to a complete and homogeneous network, and the model defined in Equation ((ref)) becomes the linear-in-means model. As discussed in Remark (ref), in the presence of fixed effects, the peer effect coefficient $\rho$ cannot be identified in the linear-in-means model. In contrast, the proposed AR test performs well even if the true network is complete as long as the true $\alpha_{ij}$'s are heterogeneous.

The second case pertains to misspecification in the locations of non-zero network links. In this experiment, the true network is a circular network where all individuals are equally spaced around a circle and are friends only with their two nearest neighbors. We assume individuals are equally influenced by their friends. Specifically, in Equation ((ref)), we set $\alpha_{ij} = 0.3$ if $j$ is $i$'s nearest neighbor (i.e., right next to $i$) and $\alpha_{ij} = 0$ otherwise. To obtain the misspecified adjacency matrix $\mathbf{W}=[w_{ij}]$ in Equation ((ref)), we assume the researcher mistakenly takes the $\texttt{m}^{\text{th}}$ nearest neighbors as one's friends. Specifically, $w_{ij} = 0.5$ if $j$ is $i$'s $\texttt{m}^{\text{th}}$ nearest neighbor and $w_{ij} = 0$ otherwise.\footnote{Note that each individual always has two friends, so $w_{ij} = 0.5$ is obtained when the adjacency matrix is row-normalized.} Therefore, as $\texttt{m}$ increases, the degree of misspecification increases. We set $(n,T) = (30,50)$ and $\beta=1$, and experiment with different values of $\texttt{m}$.

The top panel of Figure (ref) displays the power of $\texttt{T}_{\texttt{JL}}$ and $\texttt{t-test}_{\texttt{TSLS}}$, and the bottom panel displays the average estimate of the peer effect coefficient $\rho$ in TSLS. The power of $\texttt{t-test}_{\texttt{TSLS}}$ diminishes rapidly as $\texttt{m}$ (i.e., the degree of misspecification) increases, whereas the proposed AR test is robust to the misspecification. The power of $\texttt{t-test}_{\texttt{TSLS}}$ exhibits a nonlinear relationship with the degree of misspecification: it drops below 0.1 at m = 3 and then increases up to 0.4. The average estimate of $\rho$ in TSLS on the bottom panel indicates that the increase in the power is due to the sizable “negative" estimates after m = 3. It signifies that misspecification can result in not only power loss but also misleading results.

Lastly, we perform two sets of simulations to examine misspecification in the direction of peer effects.\footnote{We thank an anonymous referee for suggesting this exercise.} Practitioners typically assume homogeneity in the direction of peer effects, but in the following exercises, we allow for the coexistence of positive and negative peer effects. The first exercise adopts the simulation design used for Table (ref), but sets $\alpha_{it} = 0.3$ or $-0.3$, varying the proportion of negative coefficients ($\texttt{prop}_{\texttt{neg}}$). The misspecified adjacency matrix $\mathbf{W}$ for $\texttt{t-t}_{\texttt{TSLS}}$ is obtained by $\mathbf{W}^{\ast}=[w_{ij}^{\ast}]$, where $w_{ij}^{\ast} = \mathbf{1}\{ \alpha_{ij} \ne 0 \}$ for $i \ne j$, and then row-normalize $\mathbf{W}^{\ast}$ to get $\mathbf{W}$. Thus, the size and location of the network links are correctly specified, but the direction of peer interactions is misspecified. We set $(n,T) = (30,50)$, $\beta = 1$, and $\texttt{ND} = 0.3$. The simulation results are reported in Table (ref).

It is clear from the results that the power of $\texttt{T}_{\texttt{JL}}$ is robust to the proportion of negative links, whereas the rejection rate of $\texttt{t-t}_{\texttt{TSLS}}$ significantly decreases as $\texttt{prop}_{\texttt{neg}}$ increases. The TSLS estimate of $\rho$ when $\texttt{prop}_{\texttt{neg}} = 0.5$ is almost zero under normal errors, which explains the deterioration in the power of $\texttt{t-t}_{\texttt{TSLS}}$.

The next simulation uses the design used for Figure (ref), but assigns a positive weight for one neighbor and a negative weight for the other. Specifically, we consider $n=30$ and divide them into two groups such that the first group includes individuals from 1 to 15 and the other group includes the rest. Each group forms a circular network, and within groups, we set $\alpha_{ij} = 0.3$ if $j$ is $i$'s nearest right-side neighbor. Across groups, we set $\alpha_{ij} = -0.3$ if $j = i + n/2$ for $i \leq n/2$ and $j = i - n/2$ for $i > n/2$. All other coefficients are set to zero. As a result, the peer effects within groups are positive and those across groups are negative. The misspecified adjacency matrix $\mathbf{W}$ for $\texttt{t-t}_{\texttt{TSLS}}$ is obtained from $\mathbf{W}^{\ast}=[w_{ij}^{\ast}]$, where $w_{ij}^{\ast} = \mathbf{1}\{ \alpha_{ij} \ne 0 \}$ for $i \ne j$, and then row-normalize $\mathbf{W}^{\ast}$ to get $\mathbf{W}$. We set $(n,T) = (30,50)$, and $\beta = 1$. For comparison, we compute the results where the peer effects are positive both within and across groups. The simulation results are reported in Table (ref).

Similar to the previous results, the power of $\texttt{T}_{\texttt{JL}}$ is robust to the direction of peer effects, whereas the rejection rate of $\texttt{t-t}_{\texttt{TSLS}}$ falls below 30% when both negative and positive peer interactions exist. Overall, these exercises demonstrate that the proposed AR test can be an effective and reliable alternative when the misspecification of the network structure is a concern.

Empirical Applications

Growth Spillovers among OECD Countries

We apply our AR test to the international growth spillover model considered in EK2007 and HO2013 among others. The papers introduce spatial externalities to the classical Solow growth model by augmenting the model with spatial lags to account for spatial interdependence between countries due to knowledge transfer and technological spillover. The spatially augmented Solow model requires specification of the dependence structure to identify the spatial effects, for which the papers use geographic distance or bilateral trade volume. Since our test does not rely on a particular network structure, our result can be interpreted as more general evidence for global interdependence.

We use a balanced panel of 28 OECD member countries over the period 1975 - 2015 and specify our (unrestricted) model as follows:\footnote{The specification ((ref)) is a simplified version of the real income model used in EK2007. As there is no information about potential peers in the data, we set $ \mathcal{N}_{i}=\mathcal{N}/\{i\}$. The 28 OECD countries are the countries that joined the OECD by 2010 and have data for the entire period of analysis: Australia, France, Republic of Korea, Sweden, Austria, Greece, Mexico, Switzerland, Belgium, Iceland, Netherlands, Turkey, Canada, Ireland, New Zealand, United Kingdom, Norway, United States, Denmark, Italy, Portugal, Finland, Japan, Spain, Germany, Hungary, Luxembourg, and Poland.}

The outcome variable $y_{it}$ is the real GDP per worker. The exogenous variables $p_{it}$ and $s_{it}$ are the average annual working-age population growth and average saving rate, respectively, over the last five years. More specifically, $s_{it}$ is measured by the average investment share in GDP.\footnote{As is common in the literature, we suppose the sum of exogenous technical progress rate and capital depreciation rate in the model is 0.05.} We compiled the panel of our analysis from the OECD database (https://data.oecd.org) for the working-age population data and the PennWorld Tables, version 10.0, for the rest of the data.

As discussed in Footnote (ref) of Section (ref), the IV matrix $\mathbf{Q}$ collects linearly independent columns in $[\mathbf{X},\mathbf{Z}]$, where $\mathbf{Z}=(\mathbf{Z}_{1}^{\prime },\cdots ,\mathbf{Z}_{T}^{\prime })^{\prime }$ with $\mathbf{Z}_{t}=(\mathbf{e}_{1}\mathbf{Z}_{1t},\cdots ,\mathbf{e}_{n}\mathbf{Z}_{nt})$. If $\mathbf{Z}_{it}=(\mathbf{X}_{1t},\cdots ,\mathbf{X}_{i-1,t},\mathbf{X}_{i+1,t},\cdots ,\mathbf{X}_{nt})$, then the total number of IVs is $L+n(n-1)L$, where $L$ is the number of exogenous regressors in $\mathbf{X}_{it}$. If $L$ is large, the number of IVs could be larger than the sample size and violate the regularity condition that the IV matrix has the full column rank. In this application, to reduce the number of IVs, we use $\mathbf{Z}_{it}=(\mathbf{X}_{1t}\boldsymbol{\iota} _{L},\cdots ,\mathbf{X}_{i-1,t}\boldsymbol{\iota} _{L},\mathbf{X}_{i+1,t}\boldsymbol{\iota} _{L},\cdots ,\mathbf{X}_{nt}\boldsymbol{\iota} _{L})$, where $\boldsymbol{\iota} _{L}$ is a $L\times 1$ vector of ones. With IVs constructed in this way, the number of IVs is less than the sample size as long as $n<T$.

The spatial lag is the source of growth spillovers in this model. Therefore, we test for the presence of growth spillovers by testing $H_0: \alpha_{ij} = 0$ for all $(i,j)$. Our AR test strongly supports the presence of global spillovers with a near-zero p-value.\footnote{The test statistic for the chi-square approximation is 1070.70, while the critical value for the nominal 1% test is 810.22.} Our result echoes the significant spillover effects that have been identified in EK2007 and HO2013. However, compared to the existing studies, our result does not rely on any specification assumption for the underlying network structure and is thus more robust.

Player Interaction in the NBA

Our second application examines player interactions in the National Basketball Association (NBA) games. We use the NBA 2015-16 season data used in HJS20,\footnote{They estimate peer effects among NBA players but their empirical model imposes a particular network structure such that players are affected only by the same type of players, where “types" are the player positions: $Guards$ or $Forwards$.} and follow the paper to create the outcome and exogenous variables for our empirical model. The data include player-period level offensive and defensive statistics such as points, fouls, steals and etc, where a period represents any contiguous game period in which the same ten players are on the court. In this case, the player networks are time-varying, so the data are conceptualized for repeated cross-sections.

As our model requires a panel, we focus on the most frequently used lineups of players in the eastern and western conference winners of the season, Cleveland (CLE) and Golden State Warriors (GSW), respectively, and construct panels for the two lineups for the season. The lineup for CLE includes L. James, K. Love, J. Smith, T. Thompson, and K. Irving, and the lineup for GSW includes H. Barnes, D. Green, A. Bogut, K. Thompson, and S. Curry. The panel for CLE (GSW) includes 140 (106) time periods and spans 306.1 (307) minutes in total. Hereafter, we call the two lineups the best lineups.

The empirical model is similar to Equation ((ref)) and uses the Wins Produced for the outcome variable, which is a leading measure of NBA player production based on the work of sports economist Berri99:

where $3PT_{it}$, $2PT_{it}$, $FT_{it}$, $REB_{it}$, $STL_{it}$, $BLK_{it}$, $MFG_{it}$, $TO_{it}$, and $Mins_{it}$ are 3-point field goals made, 2-point field goals made, free throws made, rebounds, steals, blocks, missed field goals, missed free throws, turnovers, and minutes played, respectively, by player $i$ in period $t$. Wins Produced per minute (or wins per minute) estimates a player's marginal win productivity based upon player-level variables related to team-winning.

Our exogenous variables include two player-level exogenous variables, $Experience_{it}$ and $Fatigue_{it}$.\footnote{ HJS20 also include three “team-level" exogenous variables, which are controlled for by the time fixed effect in our model.} The $Experience_{it}$ is minutes played from the start of the game to the end of period $t-1$, and $Fatigue_{it}$ is minutes continuously played until the end of period $t-1$.

Table (ref) includes AR test statistics for player interaction in the best lineups over the season, where the statistics are computed for the chi-square distribution approximation. All test statistics fail to reject the null at 5% significance level, which implies that interactions among players in the two lineups were not significant. Overall, it appears that the data contains little signal after the two-way transformation, rendering very small estimated coefficients for the exogenous variables under the null. According to the simulation results, this is the case where the power of our test can be weak.

However, interactions between players may vary over time and the test statistics aggregated for the entire season may mask the time-varying interaction effects. Therefore, we also examine the changes in the test statistic over the season using a rolling window analysis, where we repeatedly compute the p-value of the AR test statistic with a rolling window of fifty time periods. Figure (ref) plots the p-values for the best lineups over the season.\footnote{The splines are computed using the function “fit" with “smoothingspline" option in MatLab. The smoothing parameter was automatically determined by the function.}

In the case of GSW, p-values are still quite large. However, in the case of CLE, the p-value falls significantly below 0.05 between the 5th and 30th time periods, indicating that the player performances in the lineup were significantly interdependent during that time period. Overall, we observe substantial heterogeneity in the test statistics for the two lineups over the season, which strongly suggests that player interactions are not fixed, but change over time to adapt to the game environment.

Conclusion

This paper proposes an AR test for the statistical significance of dyad-specific peer effects in a linear panel data model of social interactions. The main advantage of the proposed test is that it does not require specifying the interaction structure. In our test, both the number of null restrictions and the number of IVs employed to test them increase with the sample size, bridging the literature on network models with unknown interaction structures and the literature on inference with many restrictions and/or IVs.

An important assumption of the proposed test is that network effects are invariant over time. When the true network effects are time-varying, the proposed AR test may yield misleading results. A partial solution is to conduct a rolling window analysis as illustrated in Section (ref). Another restrictive assumption is that the random shocks $\epsilon_{it}$ need to be i.i.d. in the presence of individual and time fixed effects. The jackknife AR test statistic presented in Section (ref) is robust to heteroskedasticity of unknown form, but the standard jackknife method does not properly re-center the test statistic in the presence of two-way fixed effects. Although including fixed effects can alleviate potential heteroskedasticity to a certain extent, developing a heteroskedasticity-robust test for network effects in the presence of two-way fixed effects remains an important direction for future research.\\