Identification in Dynamic Dyadic Network Formation Models with Fixed Effects

Abstract: This paper establishes (set) identification results in a dynamic dyadic network formation model with time-varying observed covariates, lagged local network statistics, and unobserved heterogeneity in the form of fixed effects. Our framework accommodates observed-covariate homophily, transitivity through common friends, second-order or indirect-friend effects, and more general local subgraph statistics within a single dynamic index model. The analysis combines two complementary ways of handling fixed effects: inequalities that integrate out time-invariant dyad heterogeneity by treating each dyad as a short panel, and signed-subgraph comparisons that difference out fixed effects algebraically through intertemporal variation within each dyad. We show that the semiparametric identifying restrictions can be sharpened using either or both of the following assumptions: (i) error distribution is serially independent with a known distribution, (ii) pairwise fixed effect takes the form of additive individual fixed effects. Combining (i) and (ii) under i.i.d. logit shocks, we obtain an exact conditional logit representation and provide sufficient conditions for point identification.

• The paper develops semiparametric identification methods that separate structural state dependence, observed homophily, and unrestricted time-invariant dyad effects using panel variation and bounding or signed-subgraph comparisons.
• The framework remains valid under arbitrary within-dyad serial correlation and accommodates lagged links, common friends, friends-of-friends, and other local network statistics, though the resulting identified sets are not shown to be sharp.
• Under additive node effects and i.i.d. logistic shocks, node-balanced intertemporal configurations yield an exact conditional-logit likelihood and can achieve point identification when transformed covariates have full support.

Overview

This paper, by Wayne Yuan Gao and Yi Niu (2604.07488), studies identification of dynamic dyadic network formation models with fixed effects. The model of interest is

$$D_{ijt} = 1\{Z_{ijt}'\alpha_0 + X_{ij,t-1}'\lambda_0 + A_{ij} - U_{ijt} \ge 0\},$$

where $D_{ijt}$ is an undirected link indicator, $Z_{ijt}=|Z_{it}-Z_{jt}|$ captures time-varying observed homophily, $X_{ij,t-1}$ is a fixed-dimensional vector of lagged local network statistics (own lagged link status, common friends, friends-of-friends, or arbitrary subgraph counts), $A_{ij}$ is a time-invariant dyad fixed effect, and $U_{ijt}$ is an idiosyncratic shock. The framework nests Graham's (2016) dynamic transitivity model and contains the static tetrad-logit setup of Graham (2017) as effectively nested special cases. The central econometric problem is that observed link dynamics conflate structural state dependence, homophily, and time-invariant unobserved heterogeneity; the paper's contribution is to separate these components using panel variation across both nodes and time.

The paper's positioning relative to prior work is precise. Relative to Graham (2016), whose stable-neighborhood argument becomes unwieldy once explicit time-varying covariates are introduced—by analogy to Honoré–Kyriazidou-type issues in nonlinear dynamic panels—the paper develops tools that remain applicable under observed-covariate homophily. Relative to Gao–Li–Xu on static strategic formation, it exploits intertemporal variation in a sequentially exogenous setting, which eliminates the need for subnetwork-CCP identifiability conditions and yields results that hold even without additive node effects.

Semiparametric identification under unrestricted dyad effects

Under Assumption 1—shocks i.i.d. across dyads, jointly independent of $(A_{ij})$ and the full covariate array, with homogeneous marginals over time but possibly serially correlated within dyads—the paper derives two complementary families of restrictions.

Dyad-panel route. Applying the "bounding-by-$c$" technique of Gao and Wang to handle endogeneity from lagged outcomes, Proposition 1 constructs intertemporally aggregated bounds $$\overline L(c\mid h;\theta)\le \underline U(c\mid h;\theta)$$ for all thresholds $c$ and covariate histories $D_{ijt}$0, defining an identified set $D_{ijt}$1. The construction integrates out the fixed effect against its unknown distribution while exploiting time-homogeneity of the shock marginals. The paper is careful to state that this set is not claimed to be sharp; sharpness "appears substantially harder" and is left open.

Signed-subgraph route. Propositions 2 and 3 difference out $D_{ijt}$2 algebraically via balanced signed comparisons over edge-time cells $D_{ijt}$3: collections $D_{ijt}$4 and $D_{ijt}$5 such that each dyad appears equally often on each side. On such events the dyad effect cancels exactly, yielding sup/inf envelope inequalities conditional only on exogenous histories. These inequalities use only exogeneity—not homogeneous marginals—and therefore remain valid under arbitrary serial correlation within dyads. The two identified sets are not nested in general.

Unified perspective. Proposition 4 organizes both routes as endpoints of a spectrum of partial differencing/partial integration designs. Comparison objects sharing a residual-load vector $D_{ijt}$6 yield envelope inequalities in which zero-load dyads are differenced out and nonzero-load dyads are absorbed into a latent CDF $D_{ijt}$7 held fixed while profiling over nuisance histories. This taxonomy clarifies why the two approaches are complementary rather than redundant, though pooling occurs only within a fixed residual-load class.

Sharpening under additional structure

Known marginal CDF plus serial independence. Proposition 5 shows that when $D_{ijt}$8 is known, continuous, and shocks are serially independent, every fully differenced comparison has a known composite-error CDF given by convolutions of $D_{ijt}$9 and its reflection, converting the sandwich bounds into explicit ones. A max-score-type corollary follows because the difference of two i.i.d. continuous variables has CDF satisfying $Z_{ijt}=|Z_{it}-Z_{jt}|$0, giving a dynamic analog of maximum-score-type inequalities at threshold zero.

Additive node effects with unknown CDF. Under $Z_{ijt}=|Z_{it}-Z_{jt}|$1 with i.i.d. exchangeable node types (Assumption 2), Proposition 6 enlarges the admissible weighted-differencing class substantially: complete elimination now requires only that weighted node incidence sums $Z_{ijt}=|Z_{it}-Z_{jt}|$2 vanish—a weaker condition than dyad balancing—and partial elimination is admissible via conditioning on retained-node histories and profiling over eliminated-node histories. Dynamic triads, weighted stars, tetrads, and longer cycles all contribute valid semiparametric restrictions, valid under arbitrary serial correlation since homogeneous marginals are not used.

Exact conditional logit. Theorem 1 combines both strengthenings under i.i.d. standard logistic shocks (Assumption 3). For any completely node-balanced configuration of edge-time cells,

$Z_{ijt}=|Z_{it}-Z_{jt}|$3

with point identification if the support of $Z_{ijt}=|Z_{it}-Z_{jt}|$4 spans $Z_{ijt}=|Z_{it}-Z_{jt}|$5. The logit case is special precisely because $Z_{ijt}=|Z_{it}-Z_{jt}|$6 is affine; for probit or other nonlogit $Z_{ijt}=|Z_{it}-Z_{jt}|$7, node effects do not cancel algebraically and no exact conditional likelihood exists. The theorem strictly generalizes the per-period tetrad logit—which the authors candidly note is not new, following directly from Graham (2017)—to intertemporal tetrads, triadic cycles on three nodes, even-length cycles, and weighted stars. Two practical gains are highlighted: triadic cycles expand available comparisons when $Z_{ijt}=|Z_{it}-Z_{jt}|$8 is small, and intertemporal configurations can restore full rank when count-valued network statistics (e.g., common friends) have limited within-date integer-valued variation. Under the logit assumption, the moment-inequality results remain simultaneously applicable, providing overidentifying restrictions for specification testing and additional identifying power when the support condition fails.

Limitations and open questions

The paper states several concessions explicitly. Sharpness of none of the proposed identified sets is established; the term "identified set" is used without sharpness claims, and establishing sharpness in this dynamic-network environment is deferred. All semiparametric results require $Z_{ijt}=|Z_{it}-Z_{jt}|$9, and larger $X_{ij,t-1}$0 expands the class of comparisons without guaranteeing monotone shrinkage of the identified set, since conditioning objects also grow. The i.i.d.-across-dyads assumption rules out community-level unobserved shocks affecting multiple dyads at a date—an extension left to future work. The exact conditional logit requires fully i.i.d. logistic shocks, a substantial strengthening relative to the semiparametric results that allow arbitrary serial correlation. Finally, an appendix on latent-distance heterogeneity $X_{ij,t-1}$1 shows that the dyad-panel, signed-subgraph, and known-CDF results survive unchanged, but weighted node-differencing and the node-balanced conditional logit generally fail, since latent-distance terms do not cancel under node-balanced configurations. Inference and estimation are not developed here.

Conclusion

The paper delivers a unified semiparametric identification architecture for dynamic dyadic network formation with fixed effects, spanning a difference-out/integrate-out spectrum, and shows how known error distributions, additive node effects, and logit errors sharpen it progressively—culminating in an exact conditional logit for any completely node-balanced configuration of edge-time cells. The main open questions left by the paper are sharpness of the identified sets, inference, robustness to cross-dyad dependence, and identification under non-additive dyad heterogeneity such as latent distances.