Handbook of Social Economics (eBook)

vii Social networks with unknown network structure 42

All the results in this section so far have taken the social network matrix A as known. This severely restricts the domain of applicability of existing identification results on social networks. We finish this section by considering how identification may proceed when this matrix is unknown. In order to do this, we believe it is necessary to consider the full implications of the interpretation of linear social interactions models as simultaneous equations systems. While this interpretation is given in studies like Bramoullé et al., the full implications of this equivalence have not been explored. This is evident if one observes that the matrix form of the general social networks model may be written as

I−JA )ω=( cI+dA )x+ε

where for expositional purposes, the constant term is ignored. From this vantage point, it is evident that social networks models are special cases of the general linear simultaneous equations system of the form

ω=Βx+ε.

Systems of this type, of course, are the focus of the classical identification in econometrics, epitomized in Fisher (1966) and comprehensively summarized in Hsiao (1983). One can go further and observe that the assumption that the same network weights apply to both contextual and endogenous social interactions is not well motivated by theory, and regard equation (55) as the general specification of a linear social networks model where the normalization Γii = 1 for all i is imposed. From this vantage point it is evident that the distinction between J and A is of interest only when A is known a priori, as is the case both for the linear in means model and the more general social networks framework.

Following the classical literature, one can then think of the presence or absence of identification in terms of whether particular sets of restrictions on (55) produce identification. All previous results in this section are examples of this perspective but rely on the very strong assumption of a particular way of imposing these restrictions, i.e., Γ = I − JA and B = cI + dA for known A. Note that the results we have described do not employ information on the variance covariance matrix of the reduced form error structure, which is one source of identifying information and the basis for Graham’s (2008) results. The simultaneous equations perspective makes clear that the existing results on identification in linear social networks models can be extended to much richer frameworks. We consider two classes of models in which we interpret all agents i = 1,…, nV as arrayed on a circle. We do this so that agents 1 and nV are immediate neighbors of one another, thereby allowing us to work with symmetric interaction structures.

First, assume that each agent only reacts to the average behaviors and characteristics of his two nearest neighbors, but is unaffected by anyone else. This is a linear variation of the model studied in Blume (1993). In terms of the matrices Γ and B, one way to model this is to assume that, preserving our earlier normalization, Γii = 1 and Γii−1 Γii+1 = γ1 for all i, Γij = 0 otherwise; Bii = b0, Bii − 1 = Bii+1 = b1 for all i, and Bij = 0 otherwise, where here (and for the remainder of this discussion), all indices are mod nV. The model is identified under theorem 5 since the nearest neighbor model may be interpreted via the original social networks model via restrictions on A. For our purposes, what is of interest is that identification will still hold if one relaxes the symmetry assumptions so that Γii−1 = γi−1, Γii+1 = γi1, Bii−1 = bi−1 and Bii+1 = bi1. If these coefficients are nonzero, then the matrices Γ and B fulfill the classical rank conditions for identification, cf. Hsiao (1983, theorem 3.3.1) and one does not need to invoke theorem 4 at all. Notice that it is not necessary for the interactions parameters to be the same across agents in different positions in the network. Relative to Bramoullé et al., what this example indicates is that prior knowledge of A can take the form of the classical exclusion restrictions of simultaneous equations theory. From the vantage point of the classical theory, there is no need to impose equal coefficients across interactions as those authors do. Imposition of assumptions such as equal coefficients may be needed to account for aspects of the data, e.g., an absence of repeated observations of individuals. But if so, then the specification of the available data moments should be explicitly integrated into the identification analysis, something which has yet to be done. Further, data sets such as Add Health, which produce answers to binary questions concerning friends, are best interpreted as providing 0 values for a general A matrix, but nothing more in terms of substantive information.

This example may be extended as follows. Suppose that one is not sure whether or not the social network structure involves connections between agents that are displaced by 2 on the circle, i.e., one wishes to relax the assumption that interactions between agents who are not nearest neighbors are 0. In other words, we modify the example so that for all i, Γii = 1, Γii−1 = Γii−1, Γii−2 = γi−2, Γii+2 = γi2, Γij = 0 otherwise, Bii= bi0, Bii−1 = bi−1, Bii+1 = bi1, Bii−2 = bi−2, Bii+2 = bi2, and Bij = 0 otherwise. If the nearest neighbor coefficients are nonzero, then by Hsiao’s theorem 3.3.1 the coefficients in this model are also identified regardless of the values of the coefficients that link non-nearest neighbors. This is an example in which aspects of the network structure are testable, so that relative to Bramoullé et al. one does need to exactly know A in advance in order to estimate social structure. The intuition is straightforward, the presence of overlapping network structures between nearest neighbors renders the system overidentified: so that the presence of some other forms of social network structure can be evaluated relative to it. This form of argument seems important as it suggests ways of uncovering social network structure when individual data are available, and again has yet to be explored. Of course, not all social network structures are identified for the same reason that without restrictions, the general linear simultaneous equations model is unidentified. What our argument here suggests is that there is much to do in terms of uncovering classes of identified social networks models that are more general than those that have so far been studied.

For a second example, we consider a variation of the model studied by Bramoullé et al., which involves geometric weighting of all individuals according to their distance; as before we drop the constant term for expositional purposes. Specifically, we consider a social networks model

i =c x i +d ∑ j≠i a ij( γ ) x j +J ∑ j≠i a ij ( γ ) ω j + ε i .

The idea is that the weights assigned to the behaviors of others are functions of an underlying parameter γ. In vector form, the model is

=cx+dA( γ )x+JA( γ )ω+ε,

where

( γ )=( 0 γ γ2 … γ0γγ γ 2… γ k γ k γ k−1…… γ k γ k γ k−1⋮ γ 2…γγ γ 20 )

Following Bramoullé et al., x is a scalar characteristic. The parameter space for this model is P = {(c, d, J, γ) ∈ R2 × R+ × [0, 1)}. The reduced form for this model is

= ( I−JA(γ) ) −1 ( cI+dA(γ ) )x+ ( I−JA(γ) ) −1 ε.

Denote by F : P → nv2 the map

( c,d,J,γ )= ( I−JA(γ) ) −1 ( cI+dA(γ ) ).

The function F characterizes the mapping of structural model parameters (c, d, J, γ) to reduced form parameters. We will establish what Fisher (1959) calls complete identifia- bility of the structural...