Dependence matters Statistical models to identify the drivers of tie formation in economic networks Giacomo De Nicola1 Cornelius Fritz2 Marius Mehrl3 Göran Kauermann1_2

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Dependence matters: Statistical models to identify the

drivers of tie formation in economic networks

Giacomo De Nicola1,∗, Cornelius Fritz2, Marius Mehrl3, Göran Kauermann1

Department of Statistics, LMU Munich, Ludwigstr. 33, 80539 Munich, Germany1

Department of Statistics, Pennsylvania State University, 100 Thomas Building,

16802 State College, PA, USA2

School of Politics and International Studies, University of Leeds, LS2 9JT Leeds,

United Kingdom3

Abstract

Networks are ubiquitous in economic research on organizations, trade, and many

other areas. However, while economic theory extensively considers networks, no gen-

eral framework for their empirical modeling has yet emerged. We thus introduce two

diﬀerent statistical models for this purpose – the Exponential Random Graph Model

(ERGM) and the Additive and Multiplicative Eﬀects network model (AME). Both

model classes can account for network interdependencies between observations, but

diﬀer in how they do so. The ERGM allows one to explicitly specify and test the

inﬂuence of particular network structures, making it a natural choice if one is sub-

stantively interested in estimating endogenous network eﬀects. In contrast, AME

captures these eﬀects by introducing actor-speciﬁc latent variables aﬀecting their

propensity to form ties. This makes the latter a good choice if the researcher is

interested in capturing the eﬀect of exogenous covariates on tie formation without

having a speciﬁc theory on the endogenous dependence structures at play. After

introducing the two model classes, we showcase them through real-world applica-

tions to networks stemming from international arms trade and foreign exchange

activity. We further provide full replication materials to facilitate the adoption of

these methods in empirical economic research.

Keywords – Inferential Network Analysis, Network Data, Endogeneity, Arms Trade,

Foreign Exchange Networks, Statistical Modeling

JEL classiﬁcation – C20, C49, F14, F31, L14

Declaration of interest: none

1 Introduction

The study of networks has established itself as a central topic in economic research (Jack-

son, 2008). Within the broader context of the study of complex and interdependent sys-

tems (see e.g. Flaschel et al., 1997, 2007, 2018), networks can be deﬁned as interconnected

structures which can naturally be represented through graphs. In the economic litera-

ture, networks have been extensively considered from a theoretical perspective, with the

primary goal of understanding how economic behavior is shaped by interaction patterns

(Jackson and Rogers, 2007). Indeed, the adequate modelling of such interactions has been

described as one of the main empirical challenges in economic network analysis (Jackson

et al., 2017). Research in this direction on, e.g., organizations as networks, diﬀusion in

networks, network experiments, or network games, is surveyed in Bramoullé et al. (2016),

Jackson (2014), and Jackson et al. (2017). These theoretical advances ﬁnd application in

∗Corresponding author: giacomo.denicola@stat.uni-muenchen.de

arXiv:2210.14860v3 [stat.ME] 7 Jul 2023

many diﬀerent ﬁelds in which network structures naturally arise, such as national and in-

ternational trade, commercial agreements, ﬁrms’ organization, and collaboration activity.

However, such advances have not yet been accompanied by a corresponding shift in the

standard methods used to empirically validate them. Some recent contributions (see e.g.

Atalay et al., 2011; Chaney, 2014; Morales et al., 2019) develop estimators tailored specif-

ically to their network-based theoretical models, but more generally applicable modeling

frameworks for the analysis of real-world network data have not yet emerged. Statistical

methods speciﬁcally designed to empirically test theories where interdependencies arise

from network structures, such as the Exponential Random Graph Model (ERGM), exist

but are not yet widely used by economists. Jackson (2014), for instance, discusses ERGMs

but argues that they “suﬀer from proven computational problems” (2014, p.76). Jackson

et al. (2017) further explain that “it is practically impossible to estimate the likelihood of

a given network at even a moderately large scale”, concluding that with ERGMs, “there is

an important computational hurdle that must be overcome in working with data” (2017,

p.85).

Contrasting this assessment, we argue that recent work in the realm of empirical

network analysis provides robust and scalable methods with readily available implemen-

tations in the Rstatistical software (R Core Team, 2021). Computational issues thus

do not represent an insurmountable barrier to employ robust inferential network meth-

ods anymore. In this paper, we demonstrate the eﬀectiveness and usability of some of

those methods by applying them to real economic data. We speciﬁcally focus on models

which aim to capture the mechanisms leading to network formation, i.e. to measure how

the probability of forming a tie is inﬂuenced by (a) nodal characteristics, (b) pairwise

covariates, and (c) the rest of the network. In particular, our focus is on Exponential

Random Graph Models (ERGM) (Robins et al., 2007) and Additive and Multiplicative

Eﬀect (AME) network models (Hoﬀ, 2021), respectively implemented in the Rpackages

statnet (Handcock et al., 2008) and amen (Hoﬀ, 2015). We ﬁnd these two model classes

to be among the most promising ones for applications in the economic sciences, as they

are well suited for answering two broad categories of research questions. The ERGM is

an ideal ﬁt if, based on economic theory, the researcher envisages a particular dependence

structure for the existence of ties in the network at hand and wants to test whether their

theory is corroborated by empirical data. On the other hand, AME, and more generally

continuous latent variable models, are a good choice when the researcher is interested in

capturing the eﬀect of exogenous variables on tie formation without having prior knowl-

edge on which endogenous network dependence mechanisms are at play. In this case,

AME oﬀers the possibility to estimate the eﬀect of both nodal and pairwise covariates

while simultaneously controlling for network eﬀects, which may induce bias if ignored (see

Lee and Ogburn, 2021). In addition, the estimated latent structure can provide insight

on the underlying network mechanisms for which they are controlling.

The principal aim of this paper is to showcase ERGM and AME by focusing on

their value for economic research. After introducing each model class, we demonstrate

their empirical usage by respectively applying them to two relevant economic questions

stemming from real-world networks. We ﬁrst use the ERGM to model the international

trade of major conventional weapons, where a directed tie exists if one country transfers

arms to another. In line with Chaney (2014), network eﬀects such as directed triadic

closure (e.g. the positive impact of an increase in the volume of trade between countries

A and B on the probability that country C, that already exports to A, starts exporting

to B) are of explicit theoretical interest in this application, and the ERGM allows for

their proper speciﬁcation and testing. We then make use of the AME model to study a

historical network of global foreign exchange activity, where a directed edge is present if

one country’s national currency is actively traded within the other country. AME allows us

to estimate how relevant country features, such as per-capita gdp and the gold standard,

and pairwise covariates, such as the distance between two countries and their reciprocal

trade volume, inﬂuence tie formation, while controlling for network eﬀects to provide

unbiased estimates. We further compare the two model classes, weighing pros and cons

of each approach and providing guidance on which tool is appropriate for applications to

diﬀerent empirical settings and research questions. Finally, in addition to a step-by-step

analysis and interpretation of these application cases, we provide full replication code in

our GitHub repository1, allowing for seamless reproducibility. We, therefore, demonstrate

the “oﬀ-the-shelf” applicability of these methods, and oﬀer applied researchers a head-

start in employing them to study substantive economic problems.

Our contribution is related to various strands of the growing literature on economic

networks (e.g. Jackson and Rogers, 2007; Jackson, 2008; Bramoullé et al., 2016). Due

to its focus on economic questions, our work diﬀers from surveys in physics (Newman,

2003), statistics (Goldenberg et al., 2010), or political science (Cranmer et al., 2017).

Several articles provide overviews and surveys of existing economic network models from

a theoretical perspective (Jackson, 2014; Graham, 2015; Jackson et al., 2017; De Paula,

2020). None of these articles concentrates on discussing broadly applicable statistical

modeling frameworks, such as ERGM and AME, from an empirical perspective. In this

sense our paper is similar in spirit to van der Pol (2019) who, however, only focuses on

ERGM, without comparing alternative approaches. Indeed, one of the goals of this paper

is to shed light on the emerging AME model class (and, more generally, on latent variable

network models) for future applications in the economic literature.

The remainder of the paper is structured as follows. Section 2 discusses existing

literature and presents the mathematical and notational framework used to deﬁne and

discuss networks throughout the paper. Section 3 introduces the ERGM and applies it to

the international arms trade network. Section 4 is dedicated to AME and its application to

the global foreign exchange network. Section 5 concludes the paper with a brief discussion

on the two model classes, contrasting their diﬀerent uses and highlighting pros and cons

of each approach.

2 Economic Networks

2.1 Related literature

Even though network structures naturally arise in many aspects of economics and are

subject of prominent research in the ﬁeld, much of the previous literature has ignored

the implied interdependencies, instead opting for regression models assuming ties to be

independent conditional on the covariates (e.g. Anderson and Van Wincoop, 2003, Rose,

2004, Lewer and Van den Berg, 2008). This assumption is often unreasonable in practice.

It would, for example, imply that Germany imposing economic sanctions on Russia is

independent of Italy imposing sanctions on Russia, and, in the directed case, even of

Russia imposing them on Germany itself. While no standard framework for the modeling

of empirical network data has emerged in economics so far, a number of contributions in

1https://github.com/gdenicola/statistical-network-analysis-in-economics

– or adjacent to – the ﬁeld do make use of statistical network models. We shortly survey

these works here to show that the models we present are indeed suitable for the analysis of

economic data. Possibly the most obvious kind of economic network is the international

trade network (see Chaney, 2014) and many of these studies accordingly seek to model the

formation of trade ties. In this vein, two early studies (Ward and Hoﬀ, 2007; Ward et al.,

2013) apply latent position models to show that trade exhibits a latent network structure

beyond what a standard gravity model can capture (see also Fagiolo, 2010; Dueñas and

Fagiolo, 2013). More recently, numerous contributions have used the ERGM to explicitly

theorize and understand network interdependence in the general trade (Herman, 2022;

Liu et al., 2022; Smith and Sarabi, 2022) as well as the trade in arms (Thurner et al.,

2019; Lebacher et al., 2021), patents (He et al., 2019), and services (Feng et al., 2021).

That being said, empirical research on economic networks is not limited to trade.

Smith et al. (2019) use multilevel ERGMs to study a production network consisting of

ownership ties between ﬁrms at the micro-level and trade ties between countries at the

macro-level, while Mundt (2021) explores the European Union’s sector-level production

network via ERGMs as well as an alternative methodology, the stochastic actor-oriented

model (SAOM). The latter is another prominent tool in the realm of network analysis,

which is suitable for modeling longitudinal network data. As we, in the interest of brevity,

focus on models for static networks (i.e. networks that are observed only at one point

in time), we do not treat the SAOM, and instead refer to Snijders (1996, 2017) for an

introduction to the model class. Going back to empirical research on economic networks

in the literature, Fritz et al. (2023) deploy ERGMs to investigate patent collaboration

networks. Studies on foreign direct investments document network inﬂuences using latent

position models (Cao and Ward, 2014), or seek to model them via extensions of the ERGM

(Schoeneman et al., 2022). Finally, economists also study networks of interstate alliances

and armed conﬂict (see e.g. Jackson and Nei, 2015; König et al., 2017), both of which

have been modeled via ERGMs (Cranmer et al., 2012; Campbell et al., 2018) and AME

(Dorﬀ et al., 2020; Minhas et al., 2022). This short survey indicates that both ERGM and

AME can be used to answer questions which are of substantive interest to economists.

2.2 Setup

Before introducing models for networks in which dependencies between ties are expected,

we brieﬂy introduce the mathematical framework for networks, as well as the necessary

notation. Let y

y= (yi j)i,j=1,...,nbe the adjacency matrix representing the observed binary

network, comprising nﬁxed and known agents (nodes). In this context, yi j = 1 indicates

an edge from agent ito agent j, while yi j = 0 translates to no edge between the two. Since

self-loops are not admitted for most studied networks, the diagonal of y

yis left unspeciﬁed

or set to zero. Depending on the application, the direction of an edge can carry additional

information. If it does, we call the network directed. In this article, we mainly focus

on this type of networks. Also note that all matrix-valued objects are written in bold

font for consistency. In addition to the network connections, we often observe covariate

information on the agents, which can be at the level of single agents (e.g. the gdp of a

country) or at the pairwise level (e.g. the distance between two countries). We denote

covariates by x

x1, ..., x

xp, and our goal is to specify a statistical model for Y

Y, that is the

random variable corresponding to y

y, conditional on x

x1, ..., x

xp. A natural way to do this

is to specify a probability distribution over the space of all possible networks, which we

deﬁne by the set Y. Two main characteristics diﬀerentiate our modeling endeavor from

classical regression techniques, such as Probit or logistic regression models. First, for

most applications, we only observe one realization y

yfrom Y

Y, rendering the estimation

of the parameters to characterize this distribution particularly challenging. Second, the

entries of Y

Yare generally co-dependent; thus, most conditional dependence assumptions

inherent to common regression models are violated. Generally, we term mechanisms that

induce direct dependence between edges to be endogenous, while all eﬀects external to

the modeled network, such as covariates, are called exogenous.

3 The Exponential Random Graph Model

The ERGM is one of the most popular models for analyzing network data. First intro-

duced by Holland and Leinhardt (1981) as a model class that builds on the platform of

exponential families, it was later extended with respect to ﬁtting algorithms and more

complex dependence structures (Lusher et al., 2012; Robins et al., 2007). We next intro-

duce the model step-by-step to highlight its ability to progressively generalize by building

on conditional dependence assumptions.

3.1 Accounting for dependence in networks

We begin with the simplest possible stochastic network model, the Erdös-Rényi-Gilbert

model (Erdös and Rényi, 1959; Gilbert, 1959), where all edges are assumed to be inde-

pendent and to have the same probability of being observed. In stochastic terms, each

observed tie is then a realization of a binomial random variable with success probability

π, which yields

Pπ(Y

Y=y

y) =

i=1 Y

j̸=i

πyi j (1 −π)1−yi j (1)

for the probability to observe y

y. Evidently, model (1), which implies equal probability

for all possible ties, is too restrictive to be applied to real world problems. In the next

step, we, therefore, additionally incorporate covariates xi j by letting πvary depending on

those covariates, leading to edge-speciﬁc probabilities πi j. Following the common practice

in logistic regression, we parameterize the log-odds by log πi j

1−πi j =θ⊤xi j, where xi j is a

vector of exogenous statistics with the ﬁrst entry set to 1 to incorporate an intercept, and

get

Pθ(Y

Y=y

y) =

i=1 Y

j̸=i exp{θ⊤xi j}

1 + exp{θ⊤xi j}!yi j 1

1 + exp{θ⊤xi j}!1−yi j

.(2)

From (2), the analogy to standard logistic regression being a special case of generalized

linear models (Nelder and Wedderburn, 1972) becomes apparent. The joint distribution

of Y

Ycan be formulated in exponential family form, yielding

Pθ(Y

Y=y

y|x) = exp{θ⊤s(y

y)}

κ(θ),(3)

where s(y

y)=(s1(y

y), ..., sp(y

y)),sq(y

y) = Pn

i=1 Pj̸=iyi jxi j,q∀q= 1, ..., p, with xi j,qas q−th

entry in xi j and κ(θ) = Qn

i=1 Qj̸=i(1 + exp{θ⊤xi j}). In the jargon of exponential families,

we term s(y

y)suﬃcient statistics.

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Dependencematters:StatisticalmodelstoidentifythedriversoftieformationineconomicnetworksGiacomoDeNicola1,∗,CorneliusFritz2,MariusMehrl3,GöranKauermann1DepartmentofStatistics,LMUMunich,Ludwigstr.33,80539Munich,Germany1DepartmentofStatistics,PennsylvaniaStateUniversity,100ThomasBuilding,16802StateColle...

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Dependence matters Statistical models to identify the drivers of tie formation in economic networks Giacomo De Nicola1 Cornelius Fritz2 Marius Mehrl3 Göran Kauermann1_2

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