2.1 The Linear Model

2.1.1 Setup and Identification

The cross-sectional model of interest takes the form of the linear network-regression model: for any ,

(1) $$ \begin{align} y_{M,m} = x_{M,m}' \beta + \varepsilon_{M,m} \ \ \ \ \ \forall m\in\mathcal{M}_{N}^{}, \end{align} $$

where

(2) $$ \begin{align} Cov(\varepsilon_{M, m}, \varepsilon_{M, m'}\mid X_{M}) = 0 \quad \text{unless } m \text{ and } m' \text{ are connected}, \end{align} $$

and $\beta $ is a $K\times 1$ vector of the regression coefficients and $X_{M}$ denotes the $M\times K$ matrix that records the observed dyad-specific characteristics, that is, .

In this paper, we assume that $\beta $ is identified, which follows from standard assumptions on strict exogeneity, lack of multicollinearity and the existence of finite second moments of $y_{M,m}$ and $x_{M,m}$ . (For completeness, see Assumption B.1 and Proposition B.1 in the Supplementary Material.)

We note that equation (2) allows for there to be spillovers across the error terms even when dyads m and $m'$ are not adjacent, as long as they are connected through indirect links. By comparison, applied researchers such as Harmon et al. (Reference Harmon, Fisman and Kamenica2019) and Lustig and Richmond (Reference Lustig and Richmond2020) (and the estimators of Aronow et al. (Reference Aronow, Samii and Assenova2015) and Tabord-Meehan (Reference Tabord-Meehan2019)) consider a variant of the linear regression (1) under the assumption

(3) $$ \begin{align} Cov( \varepsilon_{M, d(i,j)}, \varepsilon_{M, d(k,l)}\mid x_{M, d(i,j)}, x_{M, d(k,l)} ) = 0 \hspace{5mm} \text{unless } \quad \{i,j\}\cap \{k,l\} \neq \emptyset, \end{align} $$

with $m = d(i,j)$ representing the dyad between i and j. This specific assumption would be equivalent to setting

(4) $$ \begin{align} Cov( \varepsilon_{M, m}, \varepsilon_{M, m'}\mid X_{M} ) = 0 \hspace{5mm} \text{unless } m \text{ and } m' \text{ are adjacent}. \end{align} $$

2.1.2 Examples

Whether to allow indirect spillovers (as in (2)) or not (as in (4)) depends on the researchers’ applications. We now present examples where our approach may be preferable.

Example 2.1 Gravity Model of Bilateral Trade Flow

A researcher is studying the trade flow from country i to j, with (log) exports from i to j denoted $y_{ij}$ . Following the literature, (s)he assumes $y_{ij}$ follows the structural gravity equation (e.g., Anderson and van Wincoop Reference Anderson and van Wincoop2003; Eaton and Kortum Reference Eaton and Kortum2002; Helpman, Melitz, and Rubinstein Reference Helpman, Melitz and Rubinstein2008; Melitz Reference Melitz2003):

(5) $$ \begin{align} y_{ij} = \alpha+\beta z_{ij} + \gamma \sum_{k\neq i}g_{ki} y_{ki} + \eta_{ij}, \end{align} $$

where $z_{ij}$ represents a dyadic characteristic of i and j, such as the shipping cost, whether both countries are democratic (e.g., Mansfield, Milner, and Rosendorff Reference Mansfield, Milner and Rosendorff2000), or whether both participate in WTO/GATT (e.g., Gowa and Kim Reference Gowa and Kim2005); $\sum _{k}g_{ki} y_{ki}$ is the amount i spends on imports ( $g_{ki}$ equals one if country i purchases goods from country k and zero otherwise), and $\eta _{ij}$ captures unobserved heterogeneity pertaining to the trade flow between countries i and j.

To see our main point, suppose there are only four countries ( $1$ , $2$ , $3$ , and $4$ ) which trade, where country $1$ exports to country $2$ , which in turn exports to country $3$ , and country $3$ exports to country $4$ . Equation (5) then simplifies to $y_{12} = \alpha + \beta z_{12} + \eta _{12}$ , $y_{23} = \alpha + \beta z_{23} + \gamma y_{12} + \eta _{23}$ , and henceforth.

Rearranging these equations implies that the trade flow from country $3$ to country $4$ can be written as

$$ \begin{align*} y_{34} &= \alpha + \alpha \gamma + \alpha \gamma^2 + \gamma^{2} \beta z_{12} + \gamma \beta z_{23} + \beta z_{34} + \gamma^{2} \eta_{12} + \gamma \eta_{23} + \eta_{34}. \end{align*} $$

Therefore, $Cov(y_{12}, y_{34}\mid z) = \gamma ^{2} Var(\eta _{12}\mid z) \neq 0$ , where $z \equiv \{z_{12}, z_{23}, z_{34}\}$ . Hence, there can be nonzero correlation between trade flows $y_{12}$ and $y_{34}$ even if they do not have a country in common. This is because an idiosyncratic shock to an upstream country can propagate through the trade network.

Example 2.2 Legislative Voting

A researcher is interested in whether seating arrangements in legislatures can affect a politician’s behavior, $y_i$ (e.g., propensity to vote “Yes” on a roll call, as Harmon et al. Reference Harmon, Fisman and Kamenica2019, or the amount of co-sponsoring, as Lowe and Jo Reference Lowe and Jo2021; Saia Reference Saia2018, among others). For concreteness, suppose there are four politicians with the seating arrangements given by Figure 1.

The researchers posit that i’s behavior can be influenced by the (average) of its seating neighbors’ own voting behavior through a parameter $\gamma $ as follows:

(6) $$ \begin{align} y_{A} = \alpha + \gamma y_{B} + \eta_{A}, \qquad\qquad &y_{B}= \alpha + \gamma \frac{y_{A}+y_{C}}{2} + \eta_{B}, \end{align} $$

(7) $$ \begin{align} y_{C} = \alpha + \gamma \frac{y_{B}+y_{D}}{2} + \eta_{C}, \qquad y_{D} = \alpha + \gamma y_{C} + \eta_{D}. \end{align} $$

If $\gamma \neq 0$ , A is affected by their neighbor B, while B is affected by both of its neighbors (A and C) and so forth. The researcher is interested in whether neighbors’ decisions are more highly correlated than the decisions among non-neighbors.

Denote $y_{ij}$ as the dyadic outcome of interest (e.g., a measure of correlation between i and j’s decisions). Both $y_{AB}$ and $y_{CD}$ involve $y_{B}$ and $y_{C}$ , which are themselves a function of $\eta _{B}$ and $\eta _{C}$ . Hence, $Cov(y_{AB}, y_{CD}) \neq 0$ , even if the two pairs of legislators do not share a common member.

2.1.3 Estimation

Throughout this paper, we focus on the OLS estimator of $\beta $ , denoted by $\hat {\beta }$ . Under the assumptions above, we can write

(8) $$ \begin{align} \hat{\beta} - \beta &= \Bigg( \sum_{j\in\mathcal{M}_{N}^{}} x_{M, j}x_{M, j}' \Bigg)^{-1} \sum_{m\in\mathcal{M}_{N}^{}} x_{M, m} \varepsilon_{M, m}. \end{align} $$

It is straightforward to verify that $\hat {\beta }$ is unbiased for $\beta $ under our identification conditions (Assumption B.1 in the Supplementary Material). However, a consistency result is by no means trivial due to the dependence along the network which induces a complex form of cross-sectional dependence, hindering a naïve application of the standard theory for independently and identically distributed ( $i.i.d.$ ) random vectors.

2.2 Outline of Our Procedure

2.2.1 Inference

Inference about $\beta $ is based on a normal approximation of the distribution of $\hat {\beta }$ around $\beta $ . We focus on hypothesis testing conducted using the expression

(9) $$ \begin{align} \big(\widehat{Var}(\hat{\beta})\big)^{-\frac{1}{2}} (\hat{\beta}-\beta), \end{align} $$

where $\widehat {Var}(\hat {\beta })$ is a consistent estimator of the asymptotic variance of $\hat {\beta }$ . Our main result in Section 3.4 is providing such an appropriate estimator, which takes the form

(10)

where $\kappa _{m,m'}$ is an appropriate kernel function that will formally be defined in Section 3.4; $h_{m,m'}$ represents an indicator function that takes one if dyads m and $m'$ are connected and zero otherwise; and .

This paper derives conditions under which $\widehat {Var}(\hat {\beta })$ is consistent for the asymptotic variance of $\hat {\beta }$ . Before doing so, let us compare the variance estimator (10) with an often used estimator based on one-way clustering of dyad groupings.

Remark 2.1 Dyadic-Robust Variance Estimator

An increasing number of applied researchers, such as Harmon et al. (Reference Harmon, Fisman and Kamenica2019) and Lustig and Richmond (Reference Lustig and Richmond2020), estimate model (1) and conduct inference using the following dyadic-robust variance estimators proposed by Aronow et al. (Reference Aronow, Samii and Assenova2015) and Tabord-Meehan (Reference Tabord-Meehan2019):

(11)

where equals one if dyads m and $m'$ are adjacent and zero otherwise.

Note that the use of the dyadic-robust variance estimator sets cases in which two dyads are not adjacent, but connected, to zero. Meanwhile, our estimator (10) accounts for network spillovers by accommodating the correlation across both adjacent and connected dyads.Footnote ⁶ As our examples above suggest, the structure of the variance estimator (11) may not be compatible with indirect spillovers in some settings, which should assume the specification (2) instead. This suggests that the dyadic-robust variance estimator may be inconsistent when non-adjacent dyads can still affect the correlation structure and outcomes of dyad m.Footnote ⁷ This conjecture is formally proven in Corollary 3.1 and illustrated in Monte Carlo simulations in Section 4. We note that this is a feature of applying such dyadic-robust variance estimators to network spillovers, and not a feature of those estimators per se.

3.1 Network-Dependent Processes

Let $Y_{M,m}$ be a random vector defined as

(12)

andFootnote ⁸ denote .Footnote ⁹ From equations (8) and (12), we can write

(13) $$ \begin{align} \hat{\beta}-\beta = \Bigg( \frac{1}{M} \sum_{j\in\mathcal{M}_{N}^{}} x_{M, j}x_{M, j}^{\prime} \Bigg)^{-1} \frac{1}{M} \sum_{m\in\mathcal{M}_{N}} Y_{M,m}.\end{align} $$

Our interest lies in proving the asymptotic properties of $\hat {\beta }$ taking advantage of the expression (13). However, the presence of $\varepsilon _{M,m}$ in $Y_{M,m}$ , which is allowed to be correlated along the network over active dyads, renders our approach nonstandard and unsuitable for applications of canonical results, such as those for $i.i.d.$ random variables or even other variants, including spatially correlated and time-series data.

The main insight of this paper is that the spillovers across connected—even if not adjacent—dyads can be rewritten as the dependence of $Y_{M,m}$ ’s along the network of active dyads (hereby, referred to as the “network”). This allows us to embrace such complex cross-sectional dependence and appropriately rewrite the problem so that recent results on network dependent random variables (Kojevnikov et al. Reference Kojevnikov, Marmer and Song2021) can be applied. To do so, the dependence between random variables for any two sets of dyads A and B, $Y_{M,A}$ and $Y_{M,B}$ , which are at a distance s from one another, is assumed to be controlled by a sequence of bounded (random) coefficients $\theta _{M,s}$ . As the minimum distance, s, between A and B grows, the dependence $\{\theta _{M,s}\}$ between $Y_{M,A}$ and $Y_{M,B}$ , is assumed to go to zero. A formal description is provided in Appendix A of the Supplementary Material.

3.2 Definitions

As will become transparent shortly, asymptotic theories for $\hat {\beta }$ rest on tradeoffs between the correlation of the network-dependent random vectors (i.e., the dependence coefficients) and the denseness of the underlying network. To measure the denseness, we first define two concepts of neighborhoods: for each $m\in \mathcal {M}_{N}$ and ,

where $\rho _{M}(m,m')$ denotes the geodesic distance between dyads m and $m'$ .Footnote ¹⁰ The former set collects all the m’s neighbors whose distance from m is no more than s (which we call a neighborhood), while the latter registers all the m’s neighbors whose distance from m is exactly s (which we call a neighborhood shell).

Next, we define two types of density measures of a network: for $k, r>0$ ,

(14)

where it is assumed that $\mathcal {M}_{N}(m'; -1) = \emptyset $ . The former measure gauges the denseness of a network in terms of the average size of a version of the neighborhood. Kojevnikov et al. (Reference Kojevnikov, Marmer and Song2021) show that controlling the asymptotic behavior of an appropriate composite of these two measures (denoted by $c_M$ and defined in Assumption 3.6) is sufficient for the Law of Large Numbers (LLN) and Central Limit Theorem (CLT) of the network dependent random variables (Condition ND).

Note that there are two different units at play here: the number of sampling units (i.e., individuals), N, and the number of dyads, M. We now assume that $M \to \infty $ as $N \to \infty $ , eliminating the possibility of extremely sparse networks among individuals. This is empirically relevant and consistent with both applied and theoretical literatures (see Appendix A.1 of the Supplementary Material for a discussion).

3.3 Asymptotic Properties of $\hat {\beta }$

We make use of the following two regularity assumptions for the proof of consistency of $\hat \beta $ for $\beta $ (Theorem 3.1) and to derive its asymptotic distribution (Theorem 3.2).Footnote ¹¹ All proofs can be found in Appendix B of the Supplementary Material.

Assumption 3.3 Kojevnikov et al. Reference Kojevnikov, Marmer and Song2021, Assumption 3.4

There exists a positive sequence $r_{M}\rightarrow \infty $ such that for $k=1,2$ ,

$$ \begin{align*} & \frac{M^{2}\theta_{M,r_{M}}^{1-1/p}}{\sigma_{M}} \stackrel{a.s.}{\rightarrow} 0, \quad \text{and} \quad \frac{M}{\sigma_{M}^{2+k}} \sum_{s\geq 0} c_{n}(s,r_{M};k)\theta_{M,s}^{1-\frac{2+k}{p}} \stackrel{a.s.}{\rightarrow} 0, \end{align*} $$

as $M\rightarrow \infty $ , where $p>4$ is the same as in Assumption 3.2.

Assumption 3.2 requires that the errors are not too large once conditioned on common shocks. Together with the standard full-rank assumption for identification of $\beta $ , this implies Assumption 3.1 of Kojevnikov et al. (Reference Kojevnikov, Marmer and Song2021) for each uth element of $Y_{M,m}^{}$ , denoted by $Y_{M,m}^{u}$ with $u\in \{1,\ldots , K\}$ .

Assumption 3.3 is a condition that controls the tradeoff between the denseness of the underlying network and the covariability of the random vectors. If the network becomes dense, then the dependence of the associated random variables has to decay much faster. This embodies the idea that spillovers decay as they propagate farther (see, e.g., Kelejian and Prucha (Reference Kelejian and Prucha2010)), which is consistent with the applications described above. For instance, Acemoglu, García-Jimeno, and Robinson (Reference Acemoglu, García-Jimeno and Robinson2015) assume that network spillovers are zero if agents are sufficiently distantly connected on a geographical network. This assumption may be violated for very dense networks with low decay of spillovers.

3.3.2 Asymptotic Normality

Let , which is present in $\hat \beta $ in equation (13). Let $S_{M}^{u}$ be the uth entry of $S_{M}$ for $u\in \{1,\ldots ,K\}$ and denote the unconditional variance of $S_{M}^{u}$ by . Since $S_{M}^{u}$ is not a sum of independent variables, its variance cannot be simply expressed as a sum of the variances of $Y_{M,m}$ . We thus need to explicitly take into account covariance between the random variables $\{Y_{M,m}^{u}\}_{m\in \mathcal {M}_{N}}$ . We study the CLT for the normalized sum of $Y_{M,m}^{u}$ , which is given by $\frac {S_{M}^{u}}{\tau _{M}}$ .

Assumption 3.4 bridges the conditional variance (assumed in Assumptions 3.2 and 3.3) and the unconditional variance of $\frac {S_{M}}{\tau _{M}}$ , which we are interested in. The final assumption for the asymptotic normality result is a standard regularity condition guaranteeing that the asymptotic variance is well defined,Footnote ¹² which follows from both matrices in the expression being well-defined.

Assumption 3.4 Growth Rates of Variances

$\frac {\sigma _{M}^{2}}{\tau _{M}^{2}} \stackrel {a.s.}{\rightarrow } 1$ as $N\rightarrow \infty $ .

Assumption 3.5. (a) For all $N\geq 1$ , $\{ x_{M, m} \}_{m\in \mathcal {M}_{N}}$ have uniformly bounded support.

(b) $\lim _{N\rightarrow \infty } \Big (\frac {1}{M} \sum _{k\in \mathcal {M}} E \big [ x_{M,k} x_{M,k}^{\prime } \big ]\Big )$ is positive definite.

(c) $\lim _{N\rightarrow \infty } \frac {N}{M^{2}} \sum _{m\in \mathcal {M}_{N}^{}} \sum _{m'\in \mathcal {M}_{N}^{}} E\big [ \varepsilon _{M, m}\varepsilon _{M, m'} x_{M, m}x_{M, m'}^{\prime } \big ]$ exists with finite elements.

Under these assumptions, the asymptotic distribution of $\hat {\beta }$ is given by the following.

Theorem 3.2 Asymptotic Normality of $\hat \beta $

Under Assumptions 3.1–3.5, $\sqrt {N} \big ( \hat {\beta }-\beta \big ) \stackrel {d}{\rightarrow } \mathcal {N}(0, AVar(\hat {\beta }))$ as $N\rightarrow \infty $ , where

(15) $$ \begin{align} AVar(\hat{\beta}) = \lim_{N\rightarrow\infty} N \Bigg( \sum_{k\in\mathcal{M}_{N}^{}} E\big[ x_{M, k}x_{M, k}^{\prime} \big]\Bigg)^{-1} \Bigg( \sum_{m\in\mathcal{M}_{N}^{}} \sum_{m'\in\mathcal{M}_{N}^{}} E\big[ \varepsilon_{M, m}\varepsilon_{M, m'} x_{M, m}x_{M, m'}^{\prime} \big] \Bigg) \Bigg( \sum_{k\in\mathcal{M}_{N}^{}} E\big[ x_{M, k}x_{M, k}^{\prime} \big] \Bigg)^{-1}, \end{align} $$

which is positive semidefinite with finite elements.

3.4 Consistent Estimation of the Asymptotic Variance of $\hat {\beta }$ under Network Spillovers

Our objective is to consistently estimate $AVar(\hat {\beta })$ defined in Theorem 3.2. As errors are mean zero, $Y_{M,m}$ is centered, that is, $E\big [ Y_{M,m} \big ]=0$ for each $m\in \mathcal {M}_{N}$ .

3.4.1 The Estimator

The proposed estimator is a type of kernel estimator. Let $b_{M}$ denote the bandwidth, or the lag truncation (its choice is described in Section 3.4.2) and a kernel function such that $\omega (0)=1$ , $\omega (z)=0$ whenever $|z|>1$ , and $\omega (z)=\omega (-z)$ for all . The feasible variance estimator of interest is

(16) $$ \begin{align} \widehat{Var}(\hat{\beta}) = \Bigg( \frac{1}{M}\sum_{k\in\mathcal{M}_{N}} x_{M,k}x_{M,k}^{\prime} \Bigg)^{-1} \Bigg( \frac{1}{M^{2}} \sum_{s\geq 0}\sum_{m\in\mathcal{M}_{N}} \sum_{m'\in\mathcal{M}_{N}^{\partial}(m;s)} \omega_{M}(s) \hat{Y}_{M,m}\hat{Y}_{M,m'}^{\prime} \Bigg) \Bigg( \frac{1}{M}\sum_{k\in\mathcal{M}_{N}} x_{M,k}x_{M,k}^{\prime} \Bigg)^{-1}, \end{align} $$

with for all $s \geq 0$ and , where .

3.5 Consistency of the Proposed Estimator

The consistency of the variance estimator requires two sets of additional assumptions. The first set is Assumption 4.1 of Kojevnikov et al. (Reference Kojevnikov, Marmer and Song2021), but stated here in terms of the network over dyads.

Assumption 3.6 Kojevnikov et al. Reference Kojevnikov, Marmer and Song2021, Assumption 4.1

There exists $p>4$ such that

(a) $\sup _{N\geq 1} \max _{m\in \mathcal {M}_{N}} E\big [ | \varepsilon _{m} |^{p} \mid \mathcal {C}_{M} \big ] < \infty $ $a.s.$ ;

(b) $\lim _{M\rightarrow \infty } \sum _{s\geq 1} |\omega _{M}(s)-1| \delta _{M}^{\partial }(s)\theta _{M,s}^{1-\frac {2}{p}} = 0$ $a.s.$ ;

(c) $\lim _{M\rightarrow \infty }\frac {1}{M} \sum _{s\geq 0} c_{M}(s,b_{M};2) \theta _{M,s}^{1-\frac {4}{p}} = 0$ $a.s.$ , where

Assumption 3.6(a) is a stronger counterpart to Assumption 3.2, as it requires that a higher-order (i.e., higher than fourth order) conditional moment be well-defined. Assumption (b) posits a tradeoff between the kernel function, the denseness of a network, and the dependence coefficients. Specifically, the kernel function $\omega _{M}$ is required to converge to one sufficiently fast. Kojevnikov et al. (Reference Kojevnikov, Marmer and Song2021) demonstrate primitive conditions under which this requirement is fulfilled (Proposition 4.2). Assumption (c) requires that the correlation coefficients decay much faster relative to the denseness of the network. This is satisfied in the suggested choice for $b_M$ above.

Another set of conditions restricts the denseness of the network, ruling out the situation where the network becomes progressively dense: most notably, the case where every single individual unit is directly linked to every other individual.

The following theorem is the main theoretical contribution of this paper.

Theorem 3.3 establishes the consistency of our proposed variance estimator accounting for network spillovers across dyads in the sense of the Frobenius norm.

3.6 When to Use the Proposed Estimator and the Role of Decaying Spillover Effects?

It follows from Theorem 3.3 that the dyadic-robust variance estimator (11) is inconsistent for the true variance when the underlying network involves a non-negligible degree of far-away correlations, as suggested in the examples of the previous section.Footnote ¹³ Specifically, the following corollary states that the dyadic-robust variance estimator of Aronow et al. (Reference Aronow, Samii and Assenova2015) may not necessarily be consistent when it is naïvely applied to the network-regression model with nonzero correlations beyond direct neighbors.

Corollary 3.1 Inconsistency of Dyadic-Robust Estimators with Network Spillovers

Suppose that the assumptions required in Theorem 3.3 hold. Assume, in addition, that

(17) $$ \begin{align} \inf_{N\geq 1} \frac{1}{M} \bigg\| \sum_{s\geq 2} \sum_{m\in\mathcal{M}_{N}} \sum_{m'\in\mathcal{M}_{N}^{\partial}(m;s)} E\big[ \varepsilon_{M, m} \varepsilon_{M, m'} x_{M, m} x_{M, m'}^{\prime} \big] \bigg\|_{F}> 0. \end{align} $$

Then, the dyadic-robust estimator (11) applied to the network-regression models (1) and (2) is inconsistent.

The added condition (17) in Corollary 3.1 pertains to both the network configuration of active dyads and the regression variables. It represents a setting where the spillovers from far-away neighbors are non-negligible even when N is large. For instance, (17) can hold even if there are not many neighbors, as long as the covariances between the error terms are sufficiently large. This builds on Erikson et al. (Reference Erikson, Pinto and Rader2014) that inference with dyadic data may be biased if one only partially accounts for such spillovers. On the other hand, if far-away neighbors in the network have only a negligible effect on the cross-sectional dependence, the dyadic-robust variance estimator remains a good approximation for the asymptotic variance of linear dyadic data models with network spillovers across dyads. These insights are investigated further in Section 4 using numerical simulations.

These observations extend to settings where the spillovers decay along the network (i.e., when the correlation along unobservables decreases with the geodesic distance among dyads). Indeed, our estimator already accounts for such decay through the indirect covariances in its expression (16). When such spillovers propagate and decay is not too high, then condition (17) is satisfied, as such spillovers are non-negligible. On the other hand, if they decay at a very high rate (in the limit, a $100\%$ decay from adjacent to connected dyads), then our estimator will become very similar to the dyadic-robust variance estimator.

However, some researchers may be willing to tolerate some asymptotic bias to still implement the dyadic-robust estimator. Then, when should they prefer our proposed estimator? While a general answer is complex because the bias depends on both the strength of indirect spillovers and the network configuration, the example below, together with the subsequent simulations, provides useful directions for salient settings.

Example 3.1 Maximum Admissible Bias in the Dyadic-Robust Variance Estimator

Suppose that spillovers decay exponentially with distance along the network: i.e., $E\big [ \varepsilon _{M, m} \varepsilon _{M, m'} x_{M, m} x_{M, m'} \big ] = \gamma ^{s}$ , where s is the geodesic distance between dyads m and $m'$ , $\gamma \in (0,1)$ and S the longest path in the network.

Let $B>0$ denote the maximum tolerance for condition (17) that the researcher is willing to allow when using the dyadic-robust variance estimator (11). Then, a sufficient condition for the researcher to prefer the proposed estimator (16) over (11) is that the decay rate $\gamma $ is higher than a threshold $\bar {\gamma }$ , where

$$ \begin{align*} \ln \bar{\gamma} = \frac{2}{S+2} \bigg\{ \ln B - \ln (S-1) - \frac{1}{S-1} \sum_{s\geq 2} \ln \delta_{M}^{\partial}(s) \bigg\}. \end{align*} $$

When the tolerated bias is small ( $B \to 0$ ), dependence is large enough and does not decay too fast (large $\gamma $ ), or the network is more dense, our approach is preferable because it provides a consistent estimator even with non-negligible spillovers. Since the network is observed, S and the last term are estimable and can be used for such diagnostics. Appendix C.5 of the Supplementary Material provides further discussions, while the next section presents for our baseline results and discusses simulation exercises.

4.1 Simulation Design

We compare three types of variance estimators across different specifications and network configurations. We use the Eicker–Huber–White estimator as a benchmark,Footnote ¹⁴ the dyadic-robust estimator of Tabord-Meehan (Reference Tabord-Meehan2019) as a comparison accounting for the dyadic nature of the data (when inappropriately used in the presence of network spillovers), and our proposed estimator which is robust to network spillovers across dyads.

We first generate networks on which random variables are assigned by following Canen, Schwartz, and Song (Reference Canen, Schwartz and Song2020), among others, by employing two models of random graph formations. They are referred to as Specifications 1 and 2. Specification 1 uses the Barabási and Albert (Reference Barabási and Albert1999) model of preferential attachment, with the fixed number of edges $\nu \in \{1,2,3\}$ being established by each new node.Footnote ¹⁵ Specification 2 is based on the Erdös–Renyi random graph (Erdös and Rényi Reference Erdös and Rényi1959, Reference Erdös and Rényi1960) with probability $p= \frac {\lambda }{N}$ for N denoting the number of nodes and $\lambda \in \{1,2,3\}$ being a parameter that governs the probability relative to the node size. The summary statistics for the networks generated by Specifications 1 and 2 are given in Appendix C.1 of the Supplementary Material.

For each of the randomly generated networks, the simulation data are generated from the following simple network-linear regression:

$$ \begin{align*} y_{m} = x_{m} \beta + \varepsilon_{m}, \end{align*} $$

with Footnote ¹⁶ representing the dyad between agent i and j. The dyad-specific regressor $x_{m}$ is defined as , where both $z_{i}$ and $z_{j}$ are drawn independently from $\mathcal {N}(0,1)$ . The regression coefficient is fixed to $\beta =1$ across specifications.

The dyad-specific error term $\varepsilon _{m}$ is constructed to exhibit nonzero correlation with $\varepsilon _{m'}$ as long as dyads m and $m'$ are connected (i.e., in the network terminology, there exists a path in the simulated network), while the strength of the correlation is assumed to decay as they are more distant. This decay is parametrized by $\gamma $ (see Appendix C of the Supplementary Material for details). If $\gamma =1$ , then spillover effects are the same no matter how far the agents are apart, that is, the spillover effects do not decay. If $\gamma = 0$ , there are no spillover effects, so the dyadic-robust variance estimator should be consistent. The case of $S=2$ corresponds to a situation where up to friends of friends may matter for spillovers.

We consider three scenarios for each type of network. In the main text, we set $S=2$ and $\gamma =0.8$ . The results for $S=2$ with $\gamma =0.2$ are given in Appendix C.5 of the Supplementary Material, and the ones for $S=1$ with $\gamma =0.8$ are in Appendix C.6 of the Supplementary Material. For comparison purposes, we employ the mean-shifted (by one) rectangular kernel with the lag truncation equal to two throughout the experiments.

5.1 Data

We adopt the dataset used in Harmon et al. (Reference Harmon, Fisman and Kamenica2019), which collects the MEP-level data on votes cast in the EP. The dataset records what each MEP voted for (Yes or No), where she was seated, and a number of individual characteristics (e.g., country, age, education, gender, and tenure). We restrict the sample to the policies voted in Strasbourg during the seventh term, and we focus on the seating pattern between July 14 and July 16, 2009 (which involved 116 different proposals being voted on). The resulting sample has 2,431,261 observations, which are split into 422 politicians forming 26,099 pairs (i.e., dyads) of MEPs over 116 proposals.Footnote ¹⁸ Further information on the construction of our sample is detailed in Appendix D.3 of the Supplementary Material.

5.2 Empirical Setup

We follow Harmon et al. (Reference Harmon, Fisman and Kamenica2019) in assuming that two MEPs who are seated next to each other within the same political group are treated as an active dyad and that such relations are exogenously determined. Their main specification is a linear model:

(18) $$ \begin{align} Agree_{d(i,j), t} = \beta_{0} + \beta_{1} SeatNeighbors_{d(i,j), t} + \varepsilon_{d(i,j), t}, \end{align} $$

where $Agree_{d(i,j), t}$ is an indicator that takes one if MEP i and j cast the same vote on proposal t, and zero otherwise, $SeatNeighbors_{d(i,j), t}$ is a binary variable that equals one if MEP i and j are seated next to each other when the vote for proposal t is taken place, and zero otherwise. The authors originally conducted inference using the estimator in Aronow et al. (Reference Aronow, Samii and Assenova2015), assuming that dyads $m = d(i,j)$ cannot be correlated with $m' = d(k,l)$ unless they share a common member.

We compare this approach to using the variance estimator introduced in Section 3.4, which allows the error terms to exhibit nonzero correlations as long as they are connected on the network over dyads represented by the adjacency relation of seating arrangements in Parliament. We use the mean-shifted rectangular kernel with the lag truncation equal the longest path in the constructed network, which accommodates all the possible correlations across connected dyads (i.e., pairs of MEPs), placing equal weight on each of them.Footnote ¹⁹

Inspired by Harmon et al. (Reference Harmon, Fisman and Kamenica2019), we consider three specifications: (I) a simple linear regression model as given in (18); (II) the model (18) augmented with a flexible set of other demographic variables;Footnote ²⁰ and (III) the model (18) with both a flexible set of other demographic variables and day-specific fixed effects. When fixed effects are present in their original estimation, we estimate a within-difference model via OLS.