Network Synthetic Interventions A Causal Framework for Panel Data Under Network Interference Anish Agarwal1 Sarah H. Cen2 Devavrat Shah2 and Christina Lee Yu3

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Network Synthetic Interventions: A Causal Framework

for Panel Data Under Network Interference

Anish Agarwal1, Sarah H. Cen∗2, Devavrat Shah2, and Christina Lee Yu3

1Industrial Engineering and Operations Research, Columbia University

2Electrical Engineering and Computer Science, Massachusetts Institute of Technology

3Operations Research and Information Engineering, Cornell University

Abstract

We propose a generalization of the synthetic controls and synthetic interventions methodol-

ogy to incorporate network interference. We consider the estimation of unit-speciﬁc potential

outcomes from panel data in the presence of spillover across units and unobserved confounding.

Key to our approach is a novel latent factor model that takes into account network interference

and generalizes the factor models typically used in panel data settings. We propose an estima-

tor, Network Synthetic Interventions (NSI), and show that it consistently estimates the mean

outcomes for a unit under an arbitrary set of counterfactual treatments for the network. We

further establish that the estimator is asymptotically normal. We furnish two validity tests for

whether the NSI estimator reliably generalizes to produce accurate counterfactual estimates. We

provide a novel graph-based experiment design that guarantees the NSI estimator produces ac-

curate counterfactual estimates, and also analyze the sample complexity of the proposed design.

We conclude with simulations that corroborate our theoretical ﬁndings.

1 Introduction

There is growing interest in the identiﬁcation and estimation of causal eﬀects in the context of

spillover on networks, in which the outcomes of a unit are aﬀected by the treatments assigned to

other units, known as the unit’s “neighbors.” Here, a unit could be an individual, customer cohort,

or region, and correspondingly, treatments could be recommendations, discounts, or legislation.

For example, whether an individual gets COVID-19 is a function of not only the individual’s

vaccination status but also the vaccination status of that individual’s social network. In the setting

of e-commerce, the number of goods sold of a particular product is a function of not only whether

that product gets a discount, but the discount level of other products that are substitutes or

complements of it. That is, there is network interference.

In this work, we focus on network inference with panel data, a ubiquitous manner in which

data is structured, where we collect multiple measurements of diﬀerent units, and each unit can

undergo a diﬀerent sequence of treatments. See Figure 1for an example of panel data and the

type of causal question we are interested in. Causal inference with panel data has recently received

signiﬁcant attention, and a popular class of estimators in such settings are known as matching

estimators, where one represents the outcomes of one unit as some combination of other units to

answer counterfactual questions. Such estimators have been very popular in practice due to their

∗Correspondence to Sarah H. Cen at shcen@mit.edu.

arXiv:2210.11355v2 [econ.EM] 12 Oct 2023

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0% 10% 0% ... 25% 25% 25%

Product #1

! = 1

Product #2

! = 2 ... ! = % − 1 ! = %! = 3 ! = % − 2

!(: 0 !(: 1 !(: 1 !(: 0 ... !(: 0 !(: 0 !(: 0 !(: 0 ...

!): 1 !): 1 !): 1 !): 1 ... !): 1 !): 0 !): 0 !): 0 ...

!*: 0 !*: 0 !*: 1 !*: 1 ... !*: 1 !*: 0 !*: 0 !*: 1 ...

!+: 1 !+: 1 !+: 1 !+: 1 ... !+: 1 !+: 1 !+: 0 !+: 1 ...

Unit 1

Unit 2

Unit 3

Unit 4

prediction period

Product #N

units

Example question of interest: Given each product’s sales numbers under the discounts above,

how would Product #2 have sold across # = % − 2, … , % if different discounts had been applied instead?

discounts applied to products across time

Figure 1: Panel data setting illustrated via an online retail example. Each row corresponds to

a product (or unit). Each column corresponds to a week (or measurement). A discount (the

treatment) is applied to each product each week. In this work, we ask questions of the form:

Given every product’s sales numbers across time and under various discounts, what would the sales

numbers (i.e., potential outcomes) have been for a speciﬁc unit, like Product #2, under a diﬀerent

set of end-of-year discounts?

ﬂexibility and simplicity in addition to the fact that they provide valid causal estimates under

unobserved confounding with appropriate assumptions. Some examples of matching estimators

with panel data include Diﬀerence-in-Diﬀerences (DiD) (Bertrand et al. 2004), Synthetic Controls

(SC) (Abadie 2021), and variants thereof. However, such matching estimators rely on the Stable

Unit Treatment Value Assumption (SUTVA), which implies that there is no spillover across units,

i.e., the treatment applied to one unit does not aﬀect the outcomes of other units. Failing to

account for spillovers can lead to biased estimates.

We propose a novel latent factor model—which is a generalization of models studied in the

panel data literature—that accounts for network interference. Given this model, we establish an

identiﬁcation result where the counterfactual potential outcome for a given unit and its neighbors

can be written as a linear combination of the observed outcomes of a carefully selected set of

other units. This identiﬁcation result leads to a natural estimator, which we call Network Synthetic

Interventions (NSI), a simple two-step procedure, that estimates the mean counterfactual potential

outcome for a given unit. We then show that, given our latent factor model, the NSI estimator is

ﬁnite-sample consistent and asymptotically normal under suitable conditions. NSI and our analysis

of it can be viewed as a generalization of the Synthetic Interventions (Agarwal et al. 2020b) and,

in turn, Synthetic Controls frameworks to account for network interference.

We furnish two validity tests that verify whether the treatment assignment pattern and the

observed data have enough variation such that valid counterfactual estimates can be produced.

Motivated by these tests, we provide a novel graph-based experiment design.

To explain the eﬃcacy of the experiment design and the NSI estimator, we consider the setting

of a regular network graph with degree d≥2. We show that the proposed experiment design

requires only O(d3) training samples in order to guarantee that the training data has enough

variation such that it is possible to generalize to a given target counterfactual treatment. Further,

NSI obtains an estimate within error εwith high probability under the proposed experiment design

when Opoly(d)/ε4training samples per unit are available. This is a signiﬁcant improvement over

the O(exp(d)/ε2) training samples that a naive procedure would require.

We conclude with simulations showing that NSI is robust to spillovers under which existing

estimators are biased.

1.1 Related work

The literature on causal inference with network interference or spillover eﬀect has mostly considered

the setting of a single measurement per unit, whether in the setting of a randomized experiment or

an observational study. Under fully arbitrary interference, it has been shown that it is impossible

to estimate any desired causal estimands as the model is not identiﬁable (Manski 2013,Aronow

et al. 2017,Basse and Airoldi 2018a,Karwa and Airoldi 2018). Subsequently, various models

have been proposed in the literature that impose restrictions on the exposure functions (Manski

2013,Aronow et al. 2017,Viviano 2020,Auerbach and Tabord-Meehan 2021,Li et al. 2021),

interference neighborhoods (Ugander et al. 2013,Bargagli-Stoﬃ et al. 2020,Sussman and Airoldi

2017a,Bhattacharya et al. 2020), parametric structure (Toulis and Kao 2013,Basse and Airoldi

2018b,Cai et al. 2015,Gui et al. 2015,Eckles et al. 2017), two-sided platforms (Johari et al.

2022,Bajari et al. 2021) or a combination of these, each leading to a diﬀerent solution concept. A

comprehensive review on network interference models is given by De Paula (2017). In this work,

we focus on network interference that is additive across the neighbors, referred to in the literature

as the joint assumptions of neighborhood interference, additivity of main eﬀects, or additivity of

interference eﬀects (Sussman and Airoldi 2017a,Yu et al. 2022,Cortez et al. 2022a,b).

Distinct to our work is that we consider a panel data setting in which there are multiple mea-

surements (e.g., a time series) for each unit. The potential outcomes function is thus also dependent

on both the unit and the measurement. Additionally, we allow for the estimation of unit-speciﬁc

counterfactuals under multiple treatments, whereas the existing literature has largely focused on

binary treatments. Key to our approach is a novel latent factor model that takes into account net-

work interference and is a generalization of the factor models typically used in panel data settings.

Previous work has focused on causal estimands that capture population-level eﬀects, such as the

average direct treatment eﬀect (the average diﬀerence in outcomes if only one unit and none of its

neighbors get treated (Basse and Airoldi 2018b,Jagadeesan et al. 2020,S¨avje et al. 2021,Sussman

and Airoldi 2017a,Leung 2019,Ma and Tresp 2021)) and the average total treatment eﬀect (the

average diﬀerence in outcomes if all units get treated versus if they do not (Ugander et al. 2013,

Eckles et al. 2017,Chin 2019,Yu et al. 2022,Cortez et al. 2022a,b)). Alternately there has been

some literature that focuses on hypothesis testing for the presence of network interference (Aronow

2012,Bowers et al. 2013,Athey et al. 2018,Pouget-Abadie et al. 2017,Saveski et al. 2017); these

results do not immediately extend to estimation as they are based on randomization inference with

a ﬁxed network size, and focus on testing the sharp null hypotheses.

While a majority of the literature focuses on randomized experiments, there is a growing in-

terest in the literature to account for network interference when analyzing observational studies.

The existing literature generally assumes partial interference, where the network consists of many

disconnected sub-communities (Tchetgen and VanderWeele 2012,Perez-Heydrich et al. 2014,Liu

et al. 2016,DiTraglia et al. 2020,Vazquez-Bare 2022). Without this strong clustering condition,

other works impose strong parametric assumptions on the potential outcomes function, assuming

that the potential outcomes only depend on a known statistic of the neighborhood treatment, e.g.

the number or fraction of treated (Verbitsky-Savitz and Raudenbush 2012,Chin 2019,Ogburn et al.

search results #1

search results #2

Spillover

: How do the discounts (treatments) applied to

similar products (units) affect the sale of ?

50% off!

25% off!

neighbors

Spillover

: Discounts (treatment) of )’s neighbors

* ) can affect sales (potential outcome) of ).

Network Graph +

Figure 2: Example of spillover eﬀects and how they can be captured via a graph network. On the

left, suppose that an online retailer presents similar products alongside one another. Then, the sale

of one product (e.g., orange sunglasses) is aﬀected by the discounts applied to similar products; in

this case, other sunglasses. On the right, spillover is often modeled via a network graph G, in which

the treatments applied to the neighbors N(n) of a unit nmay aﬀect the potential outcomes of n.

2017). This reduces estimation to a regression task under requirements of suﬃcient diversity in the

treatments. Belloni et al. (2022) also consider a setting in which the exposure mapping is known

but allow the “radius” of interference to vary across units, then learn this radius from data to de-

vise a doubly robust estimator. Forastiere et al. (2021) consider a general exposure mapping model

alongside an inverse propensity weighted estimator, but the estimator has high variance when the

exposure mapping is complex. De Paula et al. (2018) and De Paula et al. (2019) derive identiﬁ-

cation conditions when the observational panel data contains no information about the social ties

(i.e., network). Further, building on recent works in panel data (Agarwal et al. 2020b,2021a), we

allow for unobserved confounding in treatment assignment as long as there exist low-rank latent

factors that mediate the unobserved confounding, i.e., there is “selection on latent factors”.

2 Setup & Model

We begin with some notation. Let [X]:={1, . . . , X}for any positive integer X. For vector

a∈[D]Nand set S⊆[N], let aS∈[D]|S|denote the vector containing the elements of aindexed

by Sand ai∈[D] denote the i-th element of a. Let Ixdenote the x×xidentity matrix and

⊗denote the Kronecker product. Let Ind(·) denote the indicator function. Let ∥·∥ψ2denote the

Orlicz norm. Let Opdenote a probabilistic version of big-Onotation and ˜

Ω denote the variation

on big-Ω notation that ignores logarithmic terms (see Appendix Afor precise deﬁnitions). For sets

of indices S1⊆[m1] and S2⊆[m2] and a matrix Π ∈Rm1×m2, let Π[S1, S2]∈R|S1|×|S2|denote the

submatrix corresponding to the rows indexed by S1and columns index by S2. We use “ : ” as a

shorthand for all indices such that Π[ : , S2]∈Rm1×|S2|and Π[S1,: ] ∈R|S1|×m2. Let X∗denote the

∗-product space, where its length is not pre-determined. Let Π+denote the pseudo-inverse of Π.

2.1 Setup

Consider N≥1 units, D≥1 treatments, and T≥1 measurements of interest. We denote the

potential outcome for a given unit nand measurement tby the real-valued random variable Y(a)

t,n ,

where a∈[D]Ndenotes the vector of treatments over all Nunits. This deﬁnition allows for spillover

eﬀects because the potential outcome for a given unit is a function of the treatment assignment of

all units. To model spillover across units, we use a network graph. Let G= ([N],E) denote a graph

over the Nunits, where E ⊆ [N]×[N] denotes the edges of the graph. Throughout, we assume that

Gis ﬁxed and known. Let N(n) denote the neighbors of unit n∈[N] with respect to Gsuch that

j∈ N(n)⇐⇒ (j, n)∈ E. For simplicity of notation, let self-edges be included, i.e., (n, n)∈ E for

all n∈[N]. We assume that the network graph Gcaptures spillover eﬀects in the following way.

Assumption 1 (Stable Neighborhood Treatment Value Assumption (SNTVA)).The potential

outcome of measurement t∈[T]for unit n∈[N]under treatments a∈[D]Nis given by

Y(a)

t,n =Y(aN(n))

t,n ,

where aN(n)∈[D]|N(n)|denotes the treatments assigned to the units in n’s neighborhood N(n)for

measurement t. That is, the potential outcome of unit ndepends on its neighbors’ treatments but

does not depend the treatment of any other unit j∈[N]\ N(n).

See Figure 2for an example of spillover and its network representation. Several prior works

on network interference also assume SNTVA, e.g., as the Neighborhood Interference Assumption

(NIA) (Sussman and Airoldi 2017b). It can be viewed as a particular instantiation of exposure

mappings, as deﬁned by Aronow and Samii (2017), and eﬀective treatment functions (e.g., under

the constant treatment response assumption) (Manski 2013).

Remark 1. SNTVA only captures ﬁrst-order spillover eﬀects, i.e., assumes that the potential

outcome of unit nis only aﬀected by the treatments of its immediate neighbors. One could capture

higher-order spillover eﬀects by adding edges to G. The trade-oﬀ is that, as the number of edges in

Gincreases, the estimation bounds for the NSI estimator in Section 4get correspondingly weaker.

Remark 2. Although we assume Gis an undirected graph, our results can be adapted for directed

graphs by changing the deﬁnition of N(i). When Gis directed, j∈ N(i)if and only if (j, i)∈ E.

2.2 Network latent-factor model

In this section, we introduce the model that we use to develop our estimator and formal results.

Assumption 2. Let the potential outcome of measurement t∈[T]for unit n∈[N]under graph G

and treatments a∈[D]Nbe given by:

Y(aN(n))

t,n =⟨un,n,wt,an⟩+X

j∈N(n)\nuj,n,wt,aj+ϵ(aN(n))

t,n ,(1)

where u·,·∈Rrand w·,·∈Rrrepresent latent (unobserved) factors; ϵ(aN(n))

t,n represents addi-

tive, idiosyncratic shocks, and ris the “rank” or model complexity. Further, we assume that

Eϵ(aN(i))

t,i |LF = 0, where LF :=uj,i,wt,a :i, j ∈[N], t ∈[T],and a∈[D].

We make several remarks. First, we note that Assumption 2automatically satisﬁes Assumption

1. Second, the latent factor uj,n captures the eﬀect in the potential outcome Y(aN(n))

t,n due to the

interaction between node nand its neighbour j; analogously wt,ajcaptures the eﬀect due to the

treatment that neighbor jreceives (i.e., aj) for measurement t. Speciﬁcally, their eﬀect is captured

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NetworkSyntheticInterventions:ACausalFrameworkforPanelDataUnderNetworkInterferenceAnishAgarwal1,SarahH.Cen∗2,DevavratShah2,andChristinaLeeYu31IndustrialEngineeringandOperationsResearch,ColumbiaUniversity2ElectricalEngineeringandComputerScience,MassachusettsInstituteofTechnology3OperationsResearchand...

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Network Synthetic Interventions A Causal Framework for Panel Data Under Network Interference Anish Agarwal1 Sarah H. Cen2 Devavrat Shah2 and Christina Lee Yu3

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