Linear Regression with Centrality Measures Job Market Paper Yong Cai

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Linear Regression with Centrality Measures

Job Market Paper

Yong Cai ∗

Department of Economics

Northwestern University

yongcai2023@u.northwestern.edu

This version: October 19, 2022.

Newest version here.

Abstract

This paper studies the properties of linear regression on centrality measures when net-

work data is sparse – that is, when there are many more agents than links per agent

– and when they are measured with error. We make three contributions in this set-

ting: (1) We show that OLS estimators can become inconsistent under sparsity and

characterize the threshold at which this occurs, with and without measurement er-

ror. This threshold depends on the centrality measure used. Speciﬁcally, regression

on eigenvector is less robust to sparsity than on degree and diﬀusion. (2) We develop

distributional theory for OLS estimators under measurement error and sparsity, ﬁnd-

ing that OLS estimators are subject to asymptotic bias even when they are consistent.

Moreover, bias can be large relative to their variances, so that bias correction is nec-

essary for inference. (3) We propose novel bias correction and inference methods for

OLS with sparse noisy networks. Simulation evidence suggests that our theory and

methods perform well, particularly in settings where the usual OLS estimators and

heteroskedasticity-consistent/robust t-tests are deﬁcient. Finally, we demonstrate the

utility of our results in an application inspired by De Weerdt and Dercon (2006), in

which we consider consumption smoothing and social insurance in Nyakatoke, Tanza-

nia.

Keywords: networks, diﬀusion centrality, eigenvector centrality.

JEL Classiﬁcation Codes: C18, C21, C81.

∗I am grateful to Ivan Canay, Eric Auerbach and Joel Horowitz for guidance on this project. I have

also beneﬁted from the comments and suggestions of Eduardo Campillo-Betancourt, Piotr Dworczak, Grant

Goehring and Deborah Kim.

arXiv:2210.10024v1 [econ.EM] 18 Oct 2022

1 Introduction

A large and rapidly growing body of work documents the inﬂuence of networks in a wide

range of economic outcomes: peer eﬀects drive academic achievement, production networks

shape shock propagation in the macroeconomy, social networks aﬀect information- and risk-

sharing with important implications for development (see Sacerdote 2011,Carvalho and

Tahbaz-Salehi 2019 and Breza et al. 2019 for recent reviews). Many other examples abound.

One particular strand of research has fruitfully explored the relationship between an

agent’s network position and their economic outcomes. For example, Hochberg et al. (2007)

considers the network of venture capital ﬁrms and ﬁnds that better-networked ﬁrms success-

fully exit a greater proportion of their investments. Meanwhile, Cruz et al. (2017) examines

the social networks in the Philippines and shows that more central families are dispropor-

tionately represented in political oﬃces. Similarly, Banerjee et al. (2013) studies the problem

of diﬀusing microﬁnance in India and establishes that seeding information to more central

agents led to greater participation in the program.

In these papers, researchers often estimate linear models by ordinary least squares (OLS),

using centrality measures as explanatory variables. Centrality measures are node-level statis-

tics that capture notions of importance in a network. Since nodes can be important for many

reasons, a variety of centrality measures exist, each capturing a particular aspect of network

position. For example, the degree centrality of an agent reﬂects the number or intensity

of their direct links, while eigenvector centrality is designed so that inﬂuence of agents is

proportional to that of their connections. The correlation between an outcome variable and

a particular centrality measure may be revealing about the types of interactions that drive

a given economic phenomenon: an outcome that is well-predicted solely by degree is likely

to be determined in an extremely local manner, whereas one that is more strongly asso-

ciated with eigenvector centrality may involve non-linear interactions between agents that

are further apart. As such, when researchers estimate these correlations and test their sta-

tistical signiﬁcance, they frequently do so with the goal of drawing conclusions about the

economic signiﬁcance of various centrality measures and the implied mechanisms for out-

come determination. Such an exercise is credible only if the OLS estimator is close to the

estimand, and if the chosen test statistic (typically the heteroskedasticity-consistent/robust

t-statistics) is well described by its asymptotic distribution (standard normal for t-statistics)

in ﬁnite sample.

However, network data have two features that may threaten the statistical validity of

OLS. Firstly, networks may be sparse, with many more agents than links per agent. This

could happen because interactions are observed with low frequency, or because the interac-

tions in question are rare. Chandrasekhar (2016) argues that many economic networks are

sparse, providing evidence from commonly used social network data (e.g. AddHealth; Kar-

nataka Villages (Banerjee et al. 2013); Harvard social network (Leider et al. 2009)). Sparsity

poses a challenge to estimation and inference: if networks are largely empty, there might not

be enough variation in centrality measures to identify the parameters of interest. Despite

its importance, sparsity has received relatively little attention in the network econometrics

literature.

Secondly, the observed network may diﬀer from the true network of interest. Centrality

measures are often calculated on data which are obtained by survey or constructed using

some proxy for interaction between agents, though subsequent analysis would frequently treat

the true network as known. Ignoring measurement error may thus lead to estimates that

perform poorly. A growing literature works with networks that are assumed to be measured

with error. However, they generally do not consider sparse settings. This is important

since sparsity and measurement error are mutually reinforcing: sparser networks contain

weaker signals, which are in turn more diﬃcult to pick out from noisy measurements. The

upshot is that OLS estimators computed on sparse, noisy networks may have particularly

poor properties. Asymptotic theory that ignore these features will provide similarly poor

approximations to their ﬁnite sample behavior. Consequently, estimation and inference

procedures based on these theories may lead to invalid conclusions about the economic

signiﬁcance of centrality measures.

This paper studies the statistical properties of OLS on centrality measures in an asymp-

totic framework which features both measurement error and sparsity. Our analysis is centered

on degree, diﬀusion and eigenvector centralities, which are among the most popular mea-

sures. Our contribution is threefold: (1) We characterize the amount of sparsity at which

OLS estimators become inconsistent with and without measurement error, ﬁnding that this

threshold varies depending on the centrality measure used. Speciﬁcally, regression on eigen-

vector centrality is less robust to sparsity than that on degree and diﬀusion. This suggests

that researchers should be cautious about comparing regressions on diﬀerent centrality mea-

sures, since they may diﬀer in statistical properties in addition to economic signiﬁcance. (2)

We develop distributional theory for OLS estimators under measurement error and sparsity.

We restrict ourselves to sparsity ranges under which OLS is consistent, but we ﬁnd that

asymptotic bias can be large even in this case. Furthermore, the bias may be of larger order

than variance, in which case bias correction would be necessary for obtaining non-degenerate

asymptotic distributions. Additionally, we ﬁnd that under sparsity, the estimator converges

at a slower rate than is reﬂected by the usual heteroskedasticity-consistent(hc)/robust stan-

dard errors, requiring a diﬀerent estimator. (3) In view of the distributional theory, we

propose novel bias-corrected estimators and inference methods for OLS with sparse, noisy

networks. We also clarify the settings under which hc/robust t-statistics are appropriate for

testing.

Our theoretical results are derived in an asymptotic framework where networks are mod-

eled as realizations of sparse random graphs. As n→ ∞, the expected number of links per

agent grows much more slowly than n. Because our statistical model captures important

features of real world data, we expect our methods to be reliable for estimation and infer-

ence with sparse, noisy networks. We provide simulation evidence supporting this view. The

utility of our results is also evident from an application inspired by De Weerdt and Dercon

(2006), where we conduct a stylized study of consumption smoothing and social insurance

in Nyakatoke, Tanzania.

Our choice of asymptotic framework poses technical challenges. Firstly, the eigenvectors

and eigenvalues of sparse random graphs are diﬃcult to characterize. We draw on recent

advances in random matrix theory (Alt et al. 2021a;b;Benaych-Georges et al. 2019;2020)

to overcome this challenge. Secondly, spectral norms of random matrices concentrate slowly

in sparse regimes. Instead, we develop bounds for moments of noisy adjacency matrices by

relating them to counts of particular graphs, in the spirit of Wigner (1957) (see Chapter 2

of Tao 2012 more generally). Finally, in order for bias correction to improve mean-squared

error, the bias needs to be estimated at a suﬃciently fast rate. Because variance is of lower

order than bias, a naive plug-in approach does not work for estimating higher order bias

terms, although it is suﬃcient for the ﬁrst order term. We leverage this fact to recursively

construct good estimators for higher order terms.

Related Literature

Our work is most closely related to papers that study linear regression with centrality statis-

tics. To our best knowledge, we are the ﬁrst to study linear regression with diﬀusion cen-

trality, though there exist prior work on eigenvector centrality. Le and Li (2020) studies

linear regression on multiple eigenvectors of a network assuming the same type of measure-

ment error as this paper. They focus on denser settings than we do and provide inference

method only for the null hypothesis that the slope coeﬃcient is 0. We are concerned only

with eigenvector centrality, which is the leading eigenvector, but our results cover the sparse

case as well as tests of non-zero null hypotheses (more details in Remark 5). Our paper is

also related to Cai et al. (2021), which proposes penalized regressions on the leading left and

right singular vectors of a network. They consider networks that are as sparse as the ones we

study, but their networks are observed with an additive, normally distributed error (more

details in Remark 4). Outside of the linear regression setting, Cheng et al. (2021) considers

inference on deterministic linear functionals of eigenvectors. They study symmetric matrices

with asymmetric noise, proposing novel estimators that leverage asymmetry to improve per-

formance when eigengaps are small. We focus on symmetric matrices with symmetric noise

and study the plug-in estimator in which eigenvector is estimated using the noisy adjacency

matrix in place of the true matrix.

Our paper also relates to a growing literature that considers sampling and measurement

error in networks. Chandrasekhar and Lewis (2016) examines settings in which researchers

have access to a panel of networks, but which are constructed using only a partial sample

of nodes or edges. Thirkettle (2019) studies a similar missing data problem, but in a cross-

sectional setting with only one network. It is concerned with forming bounds on centrality

statistics and does not consider subsequent linear regression. Griﬃth (2022) considers the

censoring in network data, which arises when agents are only allowed to list a ﬁxed number

of relationships during the sampling process. The above papers study missing data problems

under the assumption that the observed network is without error. We assume that the

entirety of one network is observed but with error. Lewbel et al. (2021) studies measurement

error in peer eﬀects regression, ﬁnding that 2SLS with friends-of-friends instruments is valid

as long as measurement error is small. They do not discuss centrality regressions.

This paper is also connected to the nascent literature on the statistical properties of

sparse networks. A strand of this literature is concerned with network formation models

that can give rise to sparsity in the observed data. Dong et al. (2020) and Motalebi et al.

(2021) consider modiﬁcations to the stochastic block model. A more general model takes

the form of inhomogeneous Erdos-Renyi graph, which are generated by a graphon with a

sparsity parameter that tends to zero in the limit (see for instacne Bollob´as et al. 2007

and Bickel and Chen 2009). Our paper takes this approach. Yet another model for sparse

graphs is based on graphex processes, which generalizes graphons by generating vertices

through Poisson point processes (see Borgs et al. 2018,Veitch and Roy 2019 and references

therein). Our choice of inhomogeneous Erdos-Renyi graphs is motivated by their prevalence

in econometrics (Section 3 of De Paula 2017 and Section 6 of Graham 2020a provide many

examples), as well as tractability considerations. To our best knowledge, few papers have

tackled the challenges that sparse networks pose for regression. Two notable exceptions study

network formation models, which take the form of edge-level logistic regressions (Jochmans

2018;Graham 2020b). A separate literature considers estimation of peer eﬀects regressions

involving sparse networks using panel data (Manresa 2016;Rose 2016;De Paula et al. 2020).

Here, sparsity is an assumption used to justify regularization methods. We consider a node-

level regression in a cross-sectional setting with one large network.

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LinearRegressionwithCentralityMeasuresJobMarketPaperYongCai*DepartmentofEconomicsNorthwesternUniversityyongcai2023@u.northwestern.eduThisversion:October19,2022.Newestversionhere.AbstractThispaperstudiesthepropertiesoflinearregressiononcentralitymeasureswhennet-workdataissparse{thatis,whenthereareman...

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