Description and Examples

13 The model consists of two layers: a model of the network, and a model of agents’ outcomes on the network.

14 The network is represented by a graph, or more generally a hypergraph, featuring nodes and edges. Nodes in the network correspond to economic agents, and edges represent their links or collaborations. The network can be static or dynamic, and we will see that, under the assumption that the network is exogenous, a specification of the network formation model is not needed.

15 A key feature of the framework is the presence of agent-specific types, which govern sorting patterns and affect outcomes, yet are latent to the econometrician.

16 Throughout the paper, we will refer to economic agents as workers and firms as a leading example. In settings with workers and firms, we model the network as bipartite, links are employment relationships, and both workers and firms are heterogeneous. We show an example in Figure 1. Employment, job mobility and wages all depend on the workers’ and firms’ latent types. Prominent models of this kind were proposed in Becker [1973], Shimer and Smith [2000], Postel-Vinay and Robin [2002], and Lentz, Piyapromdee and Robin [2022], among many others.

Figure 1

A worker-firm network

A worker-firm network

Note: Nodes are workers (numbered from 1 to 5) and firms (numbered from 1 to 3). Edges are lines linking workers to firms, representing employment relationships. Workers 1, 4, 5 stay in a single firm, while workers 2 and 3 move between two firms.

17 Outcomes in the network depend on the agents’ latent types, and on covariates that are observed by the econometrician. Outcomes also depend on idiosyncratic errors, or “shocks.”

18 A key assumption that we will maintain in this paper is that the network is exogenous, in the sense that network links are independent of the shocks conditional on agents’ types and covariates. In models with workers and firms, Bonhomme, Lamadon and Manresa [2019] show that this assumption is satisfied in the models of Shimer and Smith [2000] and Lentz, Piyapromdee and Robin [2022] but that it fails in the model with sequential bargaining of Postel-Vinay and Robin [2002], for example.

19 Our main focus will be on estimating parameters governing the model of outcomes while conditioning on the network. This approach only restricts the network insofar as exogeneity with respect to idiosyncratic shocks is required. However, it allows for general forms of matching and sorting patterns. Beyond the example of workers and firms, this setup is relevant to other bipartite network settings appearing in the economics of education, empirical finance, and trade (Bonhomme [2020]). Non-bipartite settings, such as models of team production, are also encompassed by this framework (Ahmadpoor and Jones [2019]; Bonhomme [2021]).

20 In applications, the network formation model may also be of interest. Since the framework accounts for unobserved heterogeneity, it can be used to study network formation models with additive or non-additive heterogeneity and independent link-specific shocks (Graham [2017]; Bickel and Chen [2009]). However, since our approach relies on a parametric likelihood for the distribution of outcomes, it is not well-suited to study the determinants of link formation in models with strategic interactions (De Paula, Richards-Shubik and Tamer [2018]; Gualdani [2021]; Sheng [2020]).

Probabilistic Framework

21 Let Y denote a vector of outcomes, one for every link (or edge) in the network. Let A denote the vector of latent types, one for every agent (or node). Finally, let X denote a matrix indicating which agents are linked together. In models with covariates, which may be agent-specific or link-specific, we include those in X.

22 We postulate a parametric model for outcomes conditional on the latent types and the network links (and possibly covariates),

23 where f_θ is a parametric distribution indexed by a finite-dimensional parameter vector θ.

24 We leave the joint distribution of latent types and network links (and possibly covariates) fully unrestricted. Formally,

25 where π is an unknown (i.e., nonparametric) distribution.

26 The combination of a parametric outcome distribution and a nonparametric distribution of heterogeneity and covariates is common in the panel data literature. Indeed, Equations 1 and 2 nests the standard panel data case, where X simply indicates which observations correspond to the same worker. Nevertheless, the current framework is not limited to panel data.

27 A key assumption implied by the specification of a likelihood conditional on X (and A) is that the network is assumed exogenous. In panel data models, this corresponds to the assumption of strict exogeneity of covariates. This assumption is commonly relaxed in linear panel models, for example using the sequential moment restrictions introduced by Arellano and Bond [1991]. A subsequent literature allows for sequentially exogenous network links in linear models (e.g., Kuersteiner and Prucha [2020]). Allowing for sequential exogeneity of covariates in nonlinear panel data models is still a frontier research area (see Bonhomme, Dano and Graham [2023]).

28 In a setting with workers and firms, a leading example of Equation 1 is a model of wage determination. Following the pioneering approach of Abowd, Kramarz and Margolis [1999], a linear specification for log-wages is

29 where i ∈ {1,…, N} are workers, t ∈ {1,…, T} are time periods, j(i, t) ∈ {1,…, J} denotes the firm where i is employed at t, ε_it is an idiosyncratic shock, and we are abstracting from exogenous covariates for simplicity. In this model, network exogeneity is often referred to as “exogenous mobility,” meaning that job transitions (represented by the firm indicators j(i, t)) are assumed independent of the ε_it’s conditional on the α_i’s and the ψ_j’s.

30 In order to allow for complementarity patterns, one may be interested in a nonlinear extension of Equation 3, such as a Constant Elasticity of Substitution (CES) specification in logarithms,

31 Nonlinear log wage specifications have been proposed and estimated by Bonhomme, Lamadon and Manresa [2019] and Lentz, Piyapromdee and Robin [2022]. However, a difference between their approaches and the one we propose here is that those authors restrict the distribution of heterogeneity and how it relates to employment relationships and job transitions. That is, in both papers, π in Equation 2 is restricted, while it is fully unrestricted in the current framework. In addition, both Bonhomme, Lamadon and Manresa [2019] and Lentz, Piyapromdee and Robin [2022] rely on large firms to recover firm heterogeneity, whereas our approach will yield valid moment restrictions irrespective of the degree of sparsity of the network.

32 In models 3 and 4 one can form an NT × (N + J) matrix X, by having each row denoting an observation (or edge) (i, t), and having both sets of agents (or nodes) i and j in the columns of X. In this case, one can equivalently write Equation 3 and 4 as

33 where the vector A contains the α’s and ψ’s, and the function g takes a known (CES) form. Since we assume that the model is parametric, we specify the distribution of ε conditional on X and A, for example as a normal distribution with zero mean and diagonal covariance matrix with variance σ². In this case, θ = σ² in the linear model, and θ = (λ, ρ, γ, σ²) in the CES model.

34 The framework in Equations 1 and 2 also nests certain models of network formation. As an example, consider the model of link formation in Graham [2017]. Undirected binary links Y_ij ∈ {0, 1} between agents i and j are determined based on covariates X_ij and agent-specific types A_i and A_j as

35 where the ε_it are standard logistic, independent of A and X, and i.i.d. across pairs of agents (i, j) ∈ {1,…, N}², i ≠ j Graham [2017] leaves the joint distribution of X and A unrestricted, and his model is thus a special case of the framework in Equations 1 and 2.

Parameter θ

37 To answer the first question, the following proposition shows how to characterize moment restrictions on θ that are valid irrespective of the heterogeneity distribution.

Proposition 1. Let ϕ_θ(y, x) be a function of outcomes and network links (and/or covariates). Suppose that

is bounded. The following three statements are equivalent:

• i. For any joint distribution of (A, X) we have

• ii. Almost surely in A, X,

• iii. For almost all a and x,

38 It is immediate that (ii) implies (i), and that (ii) and (iii) are equivalent. The implication from (i) to (ii) comes from the fact that we require the moment restriction to hold irrespective of the distribution of A and X. A formal argument is provided in Appendix I. Note also that (ii) implies moment restrictions conditional on X,

39 Proposition 1 is the main insight of the functional differencing approach (Bonhomme [2012]). Indeed, finding a ϕ satisfying Equation 6 amounts to finding an element in the null space of the conditional expectation operator associated with the parametric conditional model of Y given (A, X). To this end, Bonhomme [2012] proposed numerical projection methods, while Honoré and Weidner [2020] relied on symbolic computing to find analytical ϕ functions in dynamic discrete choice settings. In some models, however, one can show that no non-trivial solution ϕ exists, which implies the absence of informative moment equality restrictions on θ. To deal with such cases, Dhaene and Weidner [2023] propose an approximate functional differencing approach. In dynamic panel data logit models, Dobronyi, Gu and Kim [2021] derive the identified set on the parameters, which combines moment equality and inequality restrictions.

40 While most applications of the functional differencing approach so far are confined to panel data settings without interactions between agents, Proposition 1 makes clear that the scope of the approach is not limited to those settings. A conventional panel data model consists of a collection of individual-specific submodels indexed by the individual heterogeneity A_i, the vector of individual covariates (X^ʹ_i1,…, X^ʹ_iT)ʹ, and the parameter θ that is common across individuals. In contrast, in network settings all units may potentially be related, and both the network matrix X and the vector of heterogeneity A may affect all outcomes in the network. As Proposition 1 illustrates, this difference between panel data and network settings is immaterial from the point of view of the applicability of functional differencing. In the next sections we will provide examples to illustrate the usefulness of the functional differencing approach when applied to networks.

Average Effects

41 We now turn our attention to linear functionals of the distribution of A and X, and answer the second question. Using functional differencing, we show how to obtain, for given θ, moment restrictions on an average effect of the form

42 where m_θ(·) is known given θ, and the expectation is taken with respect to the joint distribution π of (A, X). As an example, in the CES model of wage determination 4, one may be interested in estimating average marginal effects of worker or firm heterogeneity on log wages, while accounting for the presence of complementarity between worker and firm effects. The following proposition provides a counterpart to Proposition 1 for such target parameters.

Proposition 2. Let ψ_θ(y, x) be a function of outcomes and network links (and/or covariates). Suppose that

is bounded. The following three statements are equivalent:

• i. For any joint distribution of (A, X) we have

• ii. Almost surely in A, X,

• iii. For almost all a and x,

43 The equivalence between the three parts is again easy to see (see Appendix I), and the usefulness of the result comes from the fact that Equation 7 is a linear functional equation. In this case as well, the linear operator in Equation 7 is known given θ. Proposition 2 shows that the functional differencing approach initially applied in Bonhomme [2012] to obtain restrictions on model parameters in panel data models can be applied to derive restrictions on average effects in other models with latent variables, including network settings.

44 Finding a moment representation for µ amounts to solving a linear functional system, which is a Fredholm integral equation of the first kind. Numerical and analytical methods can be used to construct ψ functions that satisfy Equation 7. In dynamic panel logit models, Aguirregabiria and Carro [2021], Dobronyi, Gu and Kim [2021] and Dano [2023] show that the computation of average marginal effects is analytically straightforward. However, in models with continuous outcomes the inverse problem in Equation 7 is generally ill-posed (Carrasco, Florens and Renault [2007]; Engl, Hanke and Neubauer [1996]). Given a solution ψ to the functional system, using it for estimation in a finite sample thus typically requires regularization. In this paper we focus on finding moment functions ϕ and ψ, and leave a detailed study of estimators and their properties to future work. See the eighth section for further discussion.

Parameters β and σ²

48 In this subsection we derive moment restrictions on β and σ². For this purpose we rely on Proposition 1. We start by noting that Equation 6 can be equivalently written as

49 It is useful to introduce the Moore-Penrose pseudo-inverse x^†_i of x₁ and to write the two orthogonal projectors associated with x₁ as x₁x^†_i = u₁u^ʹ₁ and I_n – x₁x^†_i = u₂u^ʹ₂, where u = (u₁, u₂) is orthogonal. Let ν = y − x₂β, ν₁ = u^ʹ₁ν, and ν₂ = u^ʹ₂ν.

50 We then note that Equation 6 can equivalently be written as

51 Since u = (u₁, u₂) is orthogonal, we equivalently obtain

52 In Proposition 1 we are looking for restrictions holding for all real vectors a. Since the rows of u^ʹ₁x₁ are linearly independent, we equivalently search for the following equation being satisfied for all real vectors b: [1]

53 This convolution equation has the unique solution:

54 We have thus shown the following.

55 By Proposition 1, Proposition 3 characterizes all the available moment restrictions on the parameter θ = (βʹ, σ²) in model 8. Now, there are many possible choices for φ_θ, leading to many choices of moment functions in this model.

56 As a first example, let us take

57 We obtain

58 This implies the following conditional moment restrictions on β:

59 In a panel data setting, Chamberlain [1992] shows that the efficiency bound for β based on the quasi-differencing restrictions 9 coincides with the bound based on a semiparametric model where ε has mean zero but is otherwise not restricted. In our setup, Equation 9 remains valid in general non-Gaussian linear network regression models where

60 As a second example, let n₂ = dimν₂ = T race(I_n – x₁x^†₁), and take

61 We obtain

62 This implies the following conditional moment restrictions on β and σ²:

63 Note that Equation 10 remains valid under non-Gaussianity, provided

Quadratic Forms

64 In this subsection we derive moment restrictions on a quadratic form

65 Using the above reparameterization in terms of (v₁, v₂), we equivalently have

66 We then have the following result, shown in Appendix I.

67 By Proposition 2, Proposition 4 characterizes all the available moment restrictions on

68 The trace correction in Equation 12 is a well-known formula to obtain unbiased estimators of quadratic forms. Andrews et al. [2008], and Kline, Saggio and Sølvsten [2020] in a heteroskedastic context, apply such corrections to estimate variance components in log wage variance decompositions.

69 Moreover, given the particular solution 12, any other solution is of the form

70 where, as in Proposition 3, the function φ_θ satisfies

Remark 1. Proposition 2 can be applied to find estimators of other average effects beyond quadratic forms. Consider the quantity

, for some function m_θ(·). Examples are higher-order moments of A, its distribution function, or some nonlinear moments of A. Using similar arguments to the ones leading to Proposition 4, one can derive a formula for all moment restrictions on μ,

However, the expression of ψ_θ, which we derive in Appendix II, depends on the inverse of the Fourier transform operator, and it does not generally admit a closed-form expression when m_θ(a, x) is not polynomial in a. In addition, estimating μ based on Equation 13 generally requires regularizing ψ_θ (as shown in a panel data setting in the working paper version of Bonhomme [2012]).

Characterization

78 We start by noting that, in model 14, Equation 6 can be equivalently written as

79 where we have denoted as y_i the ith element of y, x^ʹ_i1 the ith row of x₁, and x^ʹ_i2 the ith row of x₂, for i ∈ {1,…, n}.

80 Equivalently, we have

81 that is,

82 Letting, for all

83 Finally, since exp(sʹa), for

84 Proposition 5 provides an exhaustive characterization of available moment restrictions in the logit model 14. For a non-zero ϕ to exist, it is necessary that, for some

Panel Data: Conditional Logit

85 Consider first the panel data model

86 Let i ∈ {1,…, N}, y_i = (y_i1,…,y_iT)ʹ and x_i = (x^ʹ_i1,…,x^ʹ_iT)ʹ. For simplicity we search for functions ϕ_θ(y_i, x_i) that only depend on (y, x) through (y_i, x_i). We thus look for ϕ_θ(y_i, x_i) such that

87 Consider the T = 2 case, and take s = 1. We obtain

88 which coincides with the moment function of conditional logit (Rasch [1960]; Andersen [1970]).

Network Formation: Tetrad Logit

89 Consider next the logistic network formation model 5 introduced in Graham [2017]. Links are undirected, [4] and there are n = N(N – 1)/2 observations (one for each dyad), where N is the number of agents. In this model, the sufficient statistic s = x^ʹ₁y in Proposition 5 is the vector of degrees, i.e., the degree sequence of the network.

90 For simplicity we focus on functions of tetrads formed by four agents

91 Taking first s₁ = s₂ = s₃ = s₄ = 1, we obtain

92 Considering next s₁ = s₂ = s₃ = s₄ = 2, we obtain

93 Finally, for s₁ = s₂ = 2, s₃ = s₄ = 1, we obtain

94 and there will be analogous restrictions associated with permutations of the degree sequence (2, 2, 1, 1). All other possible degree sequences have no identifying content for θ.

95 Together, taking ϕ as in Equations 18, 19 and 20 (alongside its permutations), implies the moment restrictions underpinning the “tetrad logit” estimator in Graham [2017]. However, Proposition 5 clarifies that these restrictions may not be unique, and it provides all available moment restrictions in the logistic network formation model.

Binary Choice on a Network: AKM Logit

96 In this subsection we derive moment restrictions on θ in model 15. We focus on the case where X_it does not vary within job spells. For example, when controlling for the worker’s age, job seniority only varies between spells. [5] We focus the analysis on the T = 2 case, and we consider several subnetwork configurations of the data displayed in Figure 2. Given a subnetwork configuration, we verify if moment conditions on θ exist, and what form they take.

97 Configuration A: one worker in the same firm. Suppose worker i stays in the same firm j in both periods. We look for ϕ_θ(y_it, y_i,t+1, x) such that, for s ∈ {0, 1, 2},

98 Given that x_{2it =} x_{2i, t + 1} (since the covariate does not vary within spell), this implies

99 Hence θ drops out from the equation, and there is no information to estimate θ in this configuration.

Figure 2

Configurations of worker-firm subnetworks in the logit model with worker and firm heterogeneity

Configurations of worker-firm subnetworks in the logit model with worker and firm heterogeneity

Note: Worker nodes are in the left columns, indicated in grey. Firm nodes are in the right columns, indicated in black. In panels C and F we derive non-trivial moment restrictions on the parameter θ in model 15, while no such restrictions exist in panels A, B, D, E.

100 Configuration B: one worker moving between two firms. Suppose worker i moves between firms j and j ′. We look for ϕ_θ(y_it, y_i,t+1, x) such that

101 However, in this case, (s₁, s₂) fully determines (y_it, y_{i t+1}). Hence, for each (s₁, s₂) we obtain ϕ_θ(y_it, y_i,t+1, x) = 0, which shows there is no information about θ in this configuration.

102 Configuration C: two workers moving between the same two firms. Suppose workers i and i′ both move between the same firms j and j′. We look for a function ϕ_θ(y_it, y_i,t+1, y_iʹt, y_iʹ,t+1, x) such that

103 It turns out that, in this subnetwork configuration, there exist non-trivial moment restrictions on θ. To see this, take s₁ = s₂ = s₃ = s₄ = 1. We obtain

104 This implies the conditional moment restriction

105 Configuration D: two workers moving between different firms. Suppose worker i moves between j and j′, and worker i′ moves between different firms j″ and j″′. We look for ϕ_θ(y_it, y_i,t+1, y_iʹt, y_iʹ,t+1, x) such that

106 It is easy to see there is no non-trivial ϕ function in this case. Intuitively, since workers never share a firm, it is not possible to “difference out” the firm component of heterogeneity.

107 Configuration E: two workers moving to different firms from the same firm. Suppose worker i moves between j and j′, and worker i′ moves from the same firm j to a different firm j″. We look for ϕ_θ(y_it, y_i,t+1, y_iʹt, y_iʹ,t+1, x) such that

108 It is easy to see there is no information about θ in this configuration.

109 Configuration F: three workers in a loop. There are many other subnetwork configurations providing information beyond configuration C. Indeed, consider three workers who move as follows: i moves between firms j and j′, i′ moves between j′ and j″, and i″ moves between j″ and j. We look for ϕ_θ(y_it, y_i,t+1, y_iʹt, y_iʹ,t+1, y_iʺt, y_iʺ,t+1 x) such that

110 Taking s₁ = s₂ = s₃ = s₄ = s₅ = s₆ =,1 one obtains

111 This implies the conditional moment restriction