Econometrics of Network Models

Áureo de Paula1

1UCL, São Paulo School of Economics, CeMMAP, IFS and CEPR

ESWC (August 2015)

de Paula Econometrics of Networks

Networks are . . .

. . . vulgar!

of or relating to the common people; generallycurrent; of the usual, typical, or ordinarykind. (“Vulgar” in Merriam-Webster.com, 2011)

de Paula Econometrics of Networks

. . . everywhere.

Many social and economic activities are not directly mediatedby prices (e.g., spillovers in education, labor market search,non-cognitive outcomes).

Many other behaviors are, but it also matters how agents are incontact (e.g., production and financial networks).

“Connections” (direct and indirect) define (and are possiblydefined by) how information, prices and quantities reverberate.

de Paula Econometrics of Networks

Look around!

de Paula Econometrics of Networks

de Paula Econometrics of Networks

de Paula Econometrics of Networks

What this review does and does not cover.

- This is a selective overview of recent advances in theapplied econometric literature.My focus is on econometric models of network formationand econometric models where outcome determination ismediated by networks.

- It does not elaborate on the measurement ofnetwork-related phenomena in macroeconomics, industrialorganisation, finance, and trade. Nor does it aspires tocover econometric issues in subclasses of network models(e.g., barganing and matching in graphs).

de Paula Econometrics of Networks

Some Basic TerminologyI Networks ≡ graphs: g = (Ng , Eg).

(Ng : nodes, vertices); (Eg : edges, links, ties)

- Eg = unordered (ordered) node pairs⇒ undirected(directed) network.(e.g., Fafchamps-Lund [2003]) (e.g., Atalay et al. [2011])

- Connections can also be “weighted.”(e.g., Diebold-Yilmaz [2015])

I Ni(g): set of neighbours incident with node i in g.(degree of node i = |Ni(g)|)

I Adjancency matrix: W|Ng |×|Ng |.(Wij represents ij edge)

I walks, paths, distance, cycles, clustering . . .

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I Centrality measures register the “importance” of a node.- degree centrality: how many neighbours;- closeness centrality: how far from any other node;- betweeness centrality: how crucial in connecting other

nodes;- “centrality-referencing” measures.

eigenvector centrality (Gould [1967], Bonacich [1972]);Katz centrality (Katz [1953]): ascribe βk to connections of length k .

Bonacich centrality (Bonacich [1987]): α(I− βW )−1W1;. . .diffusion centrality (Banerjee et al. [2014]);Google’s PageRank index (Brin and Page [1998]).

I Other measures registering different features (e.g.,clustering, etc.)(see, e.g., Jackson [2009])

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I We can also define probabilistic graph models:

(G, σ(G),P).

- Example: i.i.d. link formation on n nodes (Gilbert [1959],Erdös-Rényi [1960]). (N ↑ ∞ ⇒ Poi degree distr.)

- Example: “iterative” models like preferential attachment(Barabási-Albert [1999]).

- Bollobás [2001], Jackson [2009], Kolaczyk [2009]

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Outcomes on Networks

I Many interdependent outcomes are mediated byconnections (“networks”).

I A popular representation follows the linear-in-meansspecification suggested in Manski [1993]. For example,

yi = α+ βN∑

j=1

Wijyj + ηxi + γ

N∑j=1

Wijxj + εi ,

with E(εi |x,W ) = 0.

I In matrix form, we have

yN×1 = α1N×1 + βWN×NyN×1 + ηxN×1 + γWN×NxN×1 + εN×1

⇔y = α(I− βW )−11 + (I− βW )−1(ηI + γW )x + (I− βW )−1ε

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I This system can be obtained from interaction models withmaximizing agents with quadratic payoffs.

- Example: Blume, Brock, Durlauf and Jayaraman [2015]. Bayes-Nashequilibrium with

Ui(y;W ) =

α+ ηxi + γ∑j 6=i

Wijxj + zi

yi + β∑j 6=i

Wijyiyj −12

y2i .

- Example: Calvo-Armengól, Patacchini and Zenou [2009]. Nashequilibrium with yi = ei + εi and

Ui(ei , ε;W ) =

ηxi + γ∑j 6=i

Wijxj

ei−12

e2i +(αWi1+νi)εi−

12ε2

i +βN∑

j=1

Wijεiεj

⇒ y =α

β(I− βW )−1βW1 + (ηI + γW )x + (I− βW )−1ν.

(e.g., Denbee, Julliard, Li and Yuan [2014] and other studies.)

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I Manski [1993] categorises “social effects” as:- Endogenous effect: group outcomes on individual outcome;- Exogenous or contextual effect: group characteristics on

individual outcome;- Correlated effects.

. . . and the “reflection problem”.

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If |β| < 1, ηβ + γ 6= 0, Wij = (N − 1)−1 if i 6= j and Wii = 0,(α, β, η, γ) is not point-identified.Corollary to Proposition 1 in Bramoullé et al. [2009], also in Manski[1993], Kelejian et al. [2006] and others.

I Outlook improves with further restrictions on the modeland/or data.

- Example. Take the related representation originally considered inManski [1993]:

yi = α+ βE(yj |w) + ηxi + γE(xj |w) + εi , E(εi |x,w) = δw .

Manski [1993] (Prop 2)⇒ (α, β, η) are point-identified when δ = γ = 0and 1,E(xj |w), xi are “linearly independent in the population”.(A similar result appears in Angrist [2014].)

I This identification argument uses between-group variationin E(xj |w), not used in the proposition.

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I Alternative strategies explore restrictions to highermoments.

If |β| < 1, Wij = (N − 1)−1 if i 6= j ,Wii = 0, and V(ε|x) = σ2Ithen (α, β, η, γ) is point-identified.Moffitt [2001] (N = 2) and reminiscent of results like Fisher [1966].

- Additive group effect or shock⇒ identification with cov restrictions andat least two groups of different size. (Davezies, d’Haultfoeuille andFougére [2009])

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- Graham [2008] also uses higher moments to identify

yl Nl×1 = γWl Nl×Nl εl Nl×1 + αl1Nl×1 + εl Nl×1,

(see also Glaeser, Sacerdote and Scheinkman [2003]).γ is identified if there are two groups under randomassignment and additional distributional restrictions.

- Blume, Brock, Durlauf and Jayaraman [2015] exploresimilar ideas for the more general model.

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I Another avenue: “exclusion restrictions” in W .

If ηβ + γ 6= 0 and I,W ,W 2 are linearly independent,(α, β, η, γ) is point-identified.(Bramoullé, Djebbari and Fortin [2009])

- Wij = (N − 1)−1, i 6= j;Wii = 0⇒ W 2 = (N − 1)−1I+(N − 2)/(N − 1)W

- W block diagonal and two blocks of different sizes⇒

yi =α

1− β +

[η +

β(ηβ + γ)

(1− β)(Nl − 1 + β)

]xi +

ηβ + γ

(1− β)(1 + βNl−1 )

x i + νi .

(Lee [2007], Davezies, d’Haultfoeuille and Fougére [2009])

I Linear independence valid more generally. In fact,∑Nj=1 Wij = 1 and I,W ,W 2 linearly dependent⇒W block

diagonal with blocks of the same size and nonzero entriesare (Nl − 1)−1.(Blume, Brock, Durlauf and Jayaraman [2015])

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I What if W is unknown?“If researchers do not know how individuals form referencegroups and perceive reference-group outcomes, then it isreasonable to ask whether observed behavior can be usedto infer these unknowns” (Manski [1993])

I In fact . . .

If W is diagonalizable such that∑N

j=1 Wij = 1 and Wii = 0for any i ∈ {1, . . . ,N}, I,W and W 2 are linearlyindependent and βη + γ 6= 0, then α, β, η, δ and W areuniquely determined by the reduced-form equation system.

(de Paula, Rasul and Souza [2015])

de Paula Econometrics of Networks

I The reduced form coefficient matrix

Π = (I− βW )−1(ηI + γW )

can be estimated by OLS if repeated observations (T ) areavailable and T > N.

I T > N not necessarily attainable . . . but notice that(observed) Ws are “sparse” in many cases.(e.g., Atalay et al. [2011] < 1%; Carvalho [2014] ≈ 3%; AddHealth≈ 2%).

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I If Π is itself sparse, one can then estimate

πi = argminπi

1T

∑t

(yit − π>i xt )2 + λ

∑j

pT (πij).

(e.g., Manresa [2014] (β = 0, LASSO), Bonaldi, Hortacsu and Kastl[2014] (elastic net).)

I If β 6= 0, Π will not necessarily be sparse. In this case, onecan focus on

min(W ,β,δ,γ)1T∑

t ‖yt − Πxt‖22 + λ∑

i 6=j pT (Wij)

s.t. (I− βW )Π− (ηI + γW ) = 0

I Monte Carlo results in de Paula, Rasul and Souza [2015]are encouraging.

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I Nonlinearities:- “social effects might be transmitted by distributional features other

than the mean” Manski [1993], and/or- in the “link” function (i.e., yi = f

(∑Nj=1 Wijyj , xi ,

∑Nj=1 Wijxj , εi

)).

- Example: Tao and Lee [2007], Tincani [2015].

- Example: Brock and Durlauf [2001, 2007], Xu and Lee [forthcoming]←Bramoullé et al. [2014]; Blume, Brock, Durlauf and Ioannides [2011].

I Multiplicity. (de Paula [2013])

I Manski [2013]: potential outcomes with social interactions.

yi(d) = f (Wi , y−i(d),d, εi)

(Consumption in PROGRESA, Angelucci and De Giorgi [2009];spillovers in scholarship program, Dieye et al. [2014]; epidemiology)

- W also possibly affected by the treatment (Comola and Prina [2014]).

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Other Considerations

I Spillovers mediated through networks: myriad of economicand social circumstances.

- Example: Input-output networks. Suppliers and clients.(Carvalho et al. [2014], Bernard et al. [2014])

- Example: Taxation networks. VAT “binds” compliancethrough network.(de Paula and Scheinkman [2010], Pomeranz [forthcoming])

- Example: Propagation of micro shocks in production andfinancial systems.(Carvalho [2014], Acemoglu et al. [forthcoming])

. . . econometric insights may be useful in these and othercontexts.

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Network Formation

I In some cases, peer structure plausibly (econometrically)exogenous or predetermined . . .. . . but many times network formed in articulation withoutcomes or incentives determined on those verynetworks.(Instruments (when available) can possibly be used in the previousmodels (see, e.g., Qu and Lee [2015]).)

I Models for network formation are of interest per se and fortheir articulation with the determination of outcomes.

I Useful (though possibly imperfect) categorization:- Statistical Models- Strategic Models

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Statistical Models

I Statistical model: (G, σ(G),P), where P is a class ofprobability distributions on (G, σ(G)).

I Data is one or more networks.

- Example: Erdös-Rényi. G is the set of 2N(N−1)/2 graphs on N nodes, Pis indexed by p.(Zheng, Salganik and Gelman [2006] study a heterogeneous version,see also Hong and Xu [2014])

- Example: A generalization is given by the ERGM:

P(G = g) = exp

( p∑k=1

αk Sk (g)− A(α1, . . . , αp)

),

where Sk (g), k = 1, . . . , p enumerate features of the graph g andA(α1, . . . , αp) ensures that probabilities integrate to one.

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I ERGM ∈ exponential family.- (Sk (g))p

k=1 is a sufficient statistic for (αk )pk=1 (natural

parameter);- A(α1, . . . , αp) = ln

[∑g∈G exp

(∑pk=1 αk Sk (g)

)]is its

cumulant generating function;- . . .

I In principle, we can use MLE . . . but A(α1, . . . , αp) involvesa sum over 2N(N−1)/2 graphs.

- N = 24⇒ |G| > # atoms in universe!

- One strategy: (log) pseudo-likelihood∑{i,j} lnP(Wij = 1|W−ij = w−ij ;α) (Besag [1975], Strauss

and Ikeda [1990]). Unreliable if not close to indep links.

- Two main alternative avenues:> Variational principles ( Jordan and Wainwright [2008]);> MCMC (Kolaczyk [2009], recent articles).

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I P(Wij = 1|W−ij = w−ij ;α) = P(Wij = 1;α)⇒ focus ondyads.

- Example: Holland and Leinhardt [1981] (directed network).

P(Wij = Wji = 1) ∝ exp(αrec + 2α+ αouti + αin

i + αoutj + αin

j )

andP(Wij = 1,Wji = 0) ∝ exp(α+ αout

i + αinj ).

Dzemski [2015] takes αs to be “fixed effects.”

- Example: Chatterjee et al. [2011], Yan and Xu [2013] (undirectednetwork, β-model). Graham [2014] characterizes MLE (with covar) andstudies a conditional ML (using sufficient stats for αi ).

I Chandrasekhar and Jackson [2014]. Use additionalsubgraphs (Gl)

Kl=1 (beyond pairs): SUGM.

(e.g., K = 2, G1 = pair, G2 = triangle)

de Paula Econometrics of Networks

Strategic FormationI Statistical framework “indexed” by economic models.

(Payoff structure and equilibrium notion)

I Typically, ui(g) (in undirected network) is a variation of∑j 6=i

Wij ×(u + εil

)+∣∣∪j:Wij=1Nj(g)− Ni(g)− {i}

∣∣ ν +∑j

∑k>j

WijWik Wjkω

I Similar specifications for directed networks.

I Transferable or non-transferable utility.

I Network formation:- iterative;

(Blume [1993], Watts [2001], Jackson and Watts [2002])- static.

(Jackson and Wolinsky [1996], Bala and Goyal [2000])

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I Iterative network formation: sequential meeting protocoland individuals add or subtract links at each iteration

- Example: Christakis, Fowler, Imbens and Kalyanaraman[2010], undirected.(formation ≈ stochastic stability analysis in Jackson and Watts [2002])

- Example: Mele [2013], Badev [2013], directed.(Potential function⇒ NE or k-Nash stable equilibria w/o unobservables)(Meeting protocol + myopic updating⇒ unique invariant distr on graphs)

> i.i.d. EV unobservables⇒ ERGM. . .(Mele [2013] suggests MC scheme to improve on performance)

- Models are fitted to AddHealth data on friendships and outcomes(smoking, Badev [2013]) using Bayesian methods or ML.

de Paula Econometrics of Networks

I “Static” network formation: e.g., pairwise stability (Jacksonand Wolinsky [1996]).

I Example: N = 3 with payoffs∑

j∈1,...,n,j 6=i δd(i,j;g)−1 (1 + εij)− |Ni(g)|.

For εij = εji , 0 < ε23 < δ/(1− δ):

ε12

ε13

-

6

HHHH

H

AAA

{12, 13}{12, 13}

{12, 23}

{12, 13}{12, 23}{13, 23}

{12, 13} {13, 23}

{12, 23}{12}

{12}{23}�{23}

{13, 23}

{23} - {23}

δ/(1− δ)

-1-1

−1− ε23δ

−1− ε23δ

δ/(1− δ)

-{13}{23} {13}

de Paula Econometrics of Networks

I Usual approach (e.g., Berry and Tamer [2006])⇒ boundson δ.Issue: explore equilibrium networks in the space of unobservables fordifferent δ, but N = 24⇒ |G| > # atoms in universe!

I Sheng [2014]. Use small size subnetworks consistent withPS + additional payoff structures⇒ bounds. (Maybe tooconservative if N � subnetwork size.)

I Miyauchi [2014]. Payoff restrictions⇒ complementarity(supermodularity). Use lattice structure of equilibrium setto improve computation.

I Other examples: Boucher and Mourifié [2013], Leung [2015] . . .

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I de Paula, Richards-Shubik and Tamer [2015]: pairwisestability in (non-traferable utility) large networks.

- Large networks: N is continuous (see Lovasz [2012] on cont graphs).

- Payoffs: depend on characteristics (not identity), finite links and finitedepth⇒ sparse, bounded degree graph (graphing).

> Focus on network types: characteristics of local payoff-relevantnetworks. Covariates with finite support⇒ # network types is finite.

Given parameters, proportion of network types in possible equilibria canbe matched to data.

? Verifying whether parameter is consistent with (necessary, sometimessufficient) conditions for pairwise stability is a quadratic programme!

N = 500⇒ 30secs. per parameter (on average).

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de Paula Econometrics of Networks

I Incomplete information: e.g., Gilleskie and Zheng [2009],Leung [2015].

I Dynamic (farsighted) network formation: e.g., Lee andFong [2011] (bipartite), Johnson [2012].(≈ empirical dynamic games)

I Network formation and outcomes: Gilleskie and Zheng[2009], Badev [2013]. Goldsmith-Pinkham and Imbens[2013] (dyadic formation + linear-in-means), Hsieh and Lee[2013] (ERGM + linear-in-means).(Partial identification in formation model⇒ partial identification in

outcome model parameters. E.g., Ciliberto, Murry and Tamer [2015],Chesher and Rosen [2014].)

I Surveys: Kolaczyk [2009], Goldenberg et al. [2009],Hunter et al. [2012] (statistics); Graham [2015],Chandrasekhar [2015] (econometrics).

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Measurement

I Measurement is (obviously) essential!

- Complete networks are not always and costly to observe.(e.g., ERGM not projective (Rinaldo and Shalizi [2013]); Handcock andGile [2010], Koskinen et al. [2010])

(Sampling and inference schemes for network features fromincompletely observed networks (see Kolaczyk [2009] could be used incertain models of interest (e.g., Chandrasekhar and Jackson [2014], dePaula, Richards-Shubik and Tamer [2015])

I Measurement error in outcomes and/or covariatescomplicate inference strategies in interaction systems.(e.g., Moffitt [2001], Ammermueller and Pischke [2009], Angrist [2014]).

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- Some results are still possible:

Suppose there are two groups g = 1, 2 such that N1 = 2 and N2 = 3. If|β| < 1, Wij,g = (Ng − 1)−1 if i 6= j,Wii,g = 0, V(εg |xg) = σ2Ig andyi,g = yi,g + vi,g where vi,g ⊥⊥ vj,h and vi,g ⊥⊥ yj,h for any i, j, g and h,then β is identified.

- This would not work if measurement error is in thecovariates (as highlighted by the papers above).

. . . but nevertheless demonstrate that the networkinteraction structure may itself be used to handle themeasurement error.

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Looking ahead

I Networks are everywhere and research has tackled manyquestions and contexts.

I More is needed.

- Heterogeneity (e.g., Masten [2015]);- Nonlinearities (e.g., Tincani [2015]);- Dynamics (forward-looking);- Measurement.

I Many more details in the paper!

de Paula Econometrics of Networks