Identification in Dynamic Dyadic Network Formation Models with Fixed Effects

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abstractThis paper establishes 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. Keywords: network formation, dynamic, dyadic, fixed effects, homophily, transitivity, subgraph, identification, semiparametric, conditional moment inequalities, logit JEL Classification: C14, C23, C31

Introduction

Dynamic dyadic network formation models with state dependence, homophily, and local network spillovers create a natural tension between substantive realism and econometric tractability. On the one hand, lagged local network covariates such as common friends, friends-of-friends, and related subgraph counts are central for describing persistence, transitivity, and other forms of local clustering in network formation. On the other hand, once fixed effects are introduced, identification becomes difficult since observed link dynamics mix together structural state dependence, observed homophily, and time-invariant unobserved heterogeneity. An important econometric question in this context is whether these components can be separated using a panel of network data.

The paper studies a dynamic dyadic network formation model in which current link surplus depends on time-varying observed dyadic covariates, a vector of lagged local network statistics, and time-invariant unobserved heterogeneity (fixed effects). The key insight is that, once the network statistics are lagged and observed, the model can be studied as a dynamic panel with lagged endogenous network covariates.

Under unrestricted time-invariant dyad effects and an unknown error distribution, we propose two complementary semiparametric identification routes. The first route treats each dyad as a short panel and integrates out the fixed effect with respect to an unknown distribution, while the second route uses dynamic signed-subgraph comparisons to difference out fixed effects algebraically through intertemporal variation within each dyad. Both routes rely on a “bounding-by-$c$” technique proposed in GaoWang26 to handle the endogeneity issue arising from the lagged outcome variables. In addition, we synthesize the two routes into an umbrella framework that produces a general class of identifying restrictions along the “difference-out” / “integrate-out” spectrum.

The paper then shows how additional structure can be exploited to further sharpen the identification results. First, the assumption that errors are serially independent with a known distribution induces additional identifying restrictions on subgraphs with differenced-out fixed effects. Second, when pairwise fixed effect takes an additive form of individual-level fixed effects, we can obtain additional identifying restrictions using a weighted differencing argument. Third, we combine the two additional structures above with a logit error specification and show that the resulting model admits an exact conditional logit representation for any completely node-balanced configuration of edge-time cells---a class that includes within-date tetrads (the per-period analogue of graham_2017) but also intertemporal tetrads, triadic cycles, and other configurations that exploit both cross-node and cross-period variation. We provide sufficient conditions for point identification based on this enlarged class.

Related Literature

Our paper builds upon and contributes to the econometric literature on network formation models. See, e.g., DEPAULA2020a,DEPAULA2020b and GRAHAM2020, for general surveys on this topic.

More specifically, our paper belongs to the line of econometric work on dyadic network formation models with homophily effects and individual unobserved heterogeneity (fixed effects), as pioneered by graham_2017. graham_2017 provides the canonical dyadic setup under the logit error specification. Candelaria2017, toth2017, Jochmans2018, GAO2020, and \citet*{GAO2023} consider various generalizations and adaptations of graham_2017, but all focus on the static setting where the network is observed only once.

In contrast, this paper considers a dynamic environment in which current link formation depends on lagged local network statistics, following the conceptual framework of graham_2016. Specifically, graham_2016 considers a dyadic network formation model with lagged common-friends transitivity, unrestricted dyad heterogeneity, and a fully i.i.d. logit shock specification. Its main identification device is the stable-neighborhood argument, designed to separate transitivity from unrestricted time-invariant dyad heterogeneity. However, graham_2016 does not incorporate observed covariates; in fact, once one introduces explicit time-varying observed homophily, the stable-neighborhood approach becomes less convenient, analogously to a similar issue in nonlinear dynamic panel models in honore2000panel: one would need to compare dyads whose local network environments are sufficiently stable while simultaneously matching on time-varying covariate histories. Relative to graham_2016, our contribution is to develop identification tools that remain applicable once explicit time-varying observed homophily is brought into the model. Furthermore, we provide results not only for the parametric setting with i.i.d. logit setup, but also for a semiparametric setting where errors are allowed to be serially correlated with unknown distributions.

The paper also draws directly on GaoWang26, which develops panel-style “bounding-by-$c$” arguments for nonlinear dynamic models with fixed effects. The model setup in this paper is analogous to a nonlinear panel model with lagged endogenous regressors. However, in the current paper, at each time point, the "cross-sectional" data structure is given by a network of individuals along with their covariates, which is different from the “purely individual” data structure in GaoWang26. Hence, while the core idea of GaoWang26 continues to be useful, our paper considers a data structure not covered in GaoWang26, exploits nontrivial adaptations of the “bounding-by-$c$” technique, and obtains identifying restrictions that have no direct analog in the standard panel data setting.

The paper is also related to and different from \citet*{GaoLiXu26}, which studies static strategic network formation models. First, \citet*{GaoLiXu26} considers a data structure where a single large network is observed once, while our current paper focuses on the alternative “panel” data structure where we have network data over multiple time periods. The time dimension in our current paper allows us to carry out intertemporal comparisons that have no direct analog in \citet*{GaoLiXu26}. Second, both \citet*{GaoLiXu26} and this paper provide econometric methods to study how local network structure affects the linking decision between two individuals, but the two papers approach this issue from two very different, and likely complementary, perspectives: \citet*{GaoLiXu26} considers strategic interactions and simultaneity issues in a static setting, while the current paper considers a sequentially exogenous setup based on lagged networks. One implication is that, in our current paper, there is no need to impose separate subnetwork-CCP identifiability conditions as required in \citet*[Assumption 4 and Section 4]{GaoLiXu26}. Third, while both papers exploit signed-subgraph and weighted-differencing techniques to eliminate fixed effects, the current paper features results with no analogues in \citet*{GaoLiXu26}, since here we can exploit intertemporal variations and the “bounding-by-$c$” technique from GaoWang26, and obtain results even without the additive fixed effect structure, which is always assumed in \citet*{GaoLiXu26}.

The rest of the paper proceeds as follows. Section (ref) introduces the model setup. Section (ref) develops the paper's main semiparametric identification architecture under arbitrary dyad effects, including both dyad-panel and dynamic signed-subgraph arguments and the unified partial-differencing perspective linking them. Section (ref) studies how additional structure sharpens those results through known composite-error distributions and additive-node restrictions. Section (ref) concludes.

Model Setup

This section introduces the paper's baseline dynamic dyadic network formation model and the notation used throughout the identification analysis.

Consider a set of nodes (representing individuals or other types of economic agents) indexed by $i$ with dyads, i.e., pairs of nodes, indexed by $ij$. Throughout this paper, we focus on undirected and unweighted networks. Writing $D_{ijt}\in\{0,1\}$ as the link indicator for dyad $ij$ at time $t$, we consider the following dynamic network formation model

equation[equation omitted — 175 chars of source]

with $Z_{ijt} := |Z_{it}-Z_{jt}| \in \mathbb{R}^{d_h}$ denoting the observed time-varying dyadic covariates at time $t$, where the node-level covariate $Z_{it}$ may be vector-valued and $|\cdot|$ denotes coordinate-wise absolute value. Here $X_{ij,t-1}\in\mathbb{R}^{d_x}$ is a vector of observed lagged network covariates of fixed dimension, $A_{ij}$ is a time-invariant unobserved dyad fixed effect, and $U_{ijt}$ are idiosyncratic time-varying dyadic shocks. The unknown parameter vector $\theta_0 := (\alpha_0',\lambda_0')'$ consists of the coefficient vector on observed homophily $\alpha_0\in\mathbb{R}^{d_h}$ and that on lagged network covariates $\lambda_0\in\mathbb{R}^{d_x}$.\footnote{Because the error distribution is left unspecified in the semiparametric analysis, the model is invariant to a common positive rescaling of $(\theta,A_{ij},U_{ijt})$. The semiparametric identified sets derived below fully reflect this scale indeterminacy. Scale is pinned once the error distribution is specified, as in the logit specification of Section (ref).}

Note that any time-invariant dyadic observable is absorbed by the fixed effect $A_{ij}$ in the unrestricted-dyad-effects baseline; the semiparametric identification arguments therefore exploit variation in the time-varying covariates $Z_{ijt}$ and the lagged network statistics $X_{ij,t-1}$.

The framework incorporates several familiar ingredients in network formation models. Since $Z_{ijt}$ is constructed as distances between node-level observed characteristics, the model captures homophily with respect to observed characteristics. If $X_{ij,t-1}$ includes lagged common friends, the model captures potential preference for transitivity. If $X_{ij,t-1}$ includes lagged friends-of-friends or other second-order reachability measures, it captures indirect-friend effects. More generally, $X_{ij,t-1}$ may collect any fixed-dimensional vector of lagged local subgraph statistics that a researcher deems relevant for the network formation problem.

It is also useful to explicitly relate our model to the setup in graham_2016, whose baseline dynamic specification is

equation[equation omitted — 159 chars of source]

where $R_{ij,t-1} := \sum_{k \neq i,j} D_{ik,t-1}D_{jk,t-1}$ is the lagged number of common friends. Note that equation (ref) is a special case of (ref), obtained by omitting the $Z_{ijt}'\alpha_0$ term and setting \[ X_{ij,t-1} := \bigl(D_{ij,t-1},R_{ij,t-1}\bigr)', \quad \lambda_0 := (\beta_0,\gamma_0)'. \] Our framework is therefore broader in two directions at once: it allows explicit observed-covariate homophily through time-varying $Z_{ijt}$ and it allows a general fixed-dimensional vector of lagged local network covariates rather than only lagged own-link status and common friends. The current model also contains the static formation model of graham_2017 as an effectively nested special case, which can be obtained by suppressing the lagged-network vector $X_{ij,t-1}$, restricting the fixed effect to take the additive-node form $A_{ij}=\nu_i+\nu_j$, and interpreting the resulting model at a single time point. Nothing in the semiparametric arguments below uses the special two-regressor form $(D_{ij,t-1},R_{ij,t-1})$ beyond the fact that it is an observed lagged vector that satisfies certain exogeneity conditions, and the proofs go through unchanged for any fixed-dimensional $X_{ij,t-1}$.

\paragraph{Observed data.} The econometrician observes the node-level covariates $(Z_{it})_{t=1}^{T}$ for each node $i$ and the network $(D_{ijt})_{t=0}^{T}$ for all dyads $ij$. Because $X_{ij,t-1}$ is computed from the lagged network, its construction at $t=1$ requires the initial network $(D_{ij0})_{ij}$, which is treated as given. No distributional assumption is placed on the initial network.

In the following, it would be convenient to write $\theta := (\alpha',\lambda')'$ and \[ W_{ijt}(\theta) := Z_{ijt}'\alpha + X_{ij,t-1}'\lambda, \quad V_{ijt} := U_{ijt} - A_{ij}, \] so that model (ref) becomes \[ D_{ijt} = \mathbf{1}\{V_{ijt} \le W_{ijt}(\theta_0)\}. \] From the viewpoint of dyad $ij$, the model is therefore a dynamic binary panel with one time-invariant dyad effect and lagged endogenous network covariates. Below we explain how to exploit the intertemporal variations of the panel structure, as well as the additional two-dimensional network structure at each fixed time point, to obtain identifying restrictions.

Semiparametric Identification

This section develops the paper's semiparametric identification approach under unrestricted form of dyad fixed effects. The first subsection integrates the fixed effect out by treating each pair as a short panel. The second subsection differences the dyad effect out directly through dynamic signed-subgraph comparisons. The third shows that these are two endpoints of a broader spectrum that combines differencing and integration.

assumption[Idiosyncratic Dyadic Shocks] Write $U_{ij}^{1:T}: = (U_{ij1},\dots,U_{ijT})'$ and $Z_i^{1:T}: = (Z_{i1}',\dots,Z_{iT}')'$. The dyad-level shock vectors $\{U_{ij}^{1:T}: i<j\}$ are i.i.d. across dyads and are jointly independent of $(A_{ij})_{ij}$ and $(Z_i)_i$, i.e., \[ \{U_{ij}^{1:T}: i<j\} \perp \left( \{A_{ij}: i<j \in\{1,...,n\}\}, \{Z_i^{1:T}: i\in\{1,\ldots,n\}\} \right). \] Moreover, the distribution of $U_{ijt}$ is homogeneous across time $t$, i.e., for each dyad $ij$ and each pair of dates $t,s \in \{1,\ldots,T\}$, $U_{ijt} \sim U_{ijs}.$

Throughout, all conditional distributions are assumed to admit regular versions, so that conditioning on exact realizations of covariate histories and taking suprema or infima over their supports are well-defined operations.\footnote{Equivalently, the reader may interpret all sup/inf operations as essential suprema/infima with respect to the relevant marginal measures.}

Assumption (ref) is standard in the dyadic network formation literature. It says that the dyad-level shock process is i.i.d.\ across dyads, exogenous relative to both the time-invariant latent heterogeneity and the entire observed exogenous covariate array, has homogeneous marginals over time, and may nevertheless be serially correlated within a dyad. Arbitrary dependence between $A_{ij}$ and the covariate histories is still allowed. The i.i.d.\ assumption across dyads rules out unobserved community-level shocks that simultaneously affect multiple dyads at the same date; such extensions are left to future work. The i.i.d.\ logit assumption in graham_2016 can be viewed as a strengthening of Assumption (ref).

Dyadic Panel Identification

We apply the “bounding-by-$c$” technique in GaoWang26 and obtain bounds free of lagged outcome variables, which allows us to exploit the independence and time-homogeneity assumption on idiosyncratic dyadic shocks $U_{ijt}$. Specifically, fix $h\in\operatorname{Supp}(Z_{ij}^{1:T})$, $c\in\mathbb{R}$, and pair of dates $(t,s)$. If $D_{ijt}=1$ and $W_{ijt}(\theta_0)\le c$, then by (ref), \[ V_{ijt}\le W_{ijt}(\theta_0)\le c \implies D_{ijt}\mathbf{1}\{W_{ijt}(\theta_0)\le c\} \le \mathbf{1}\{V_{ijt}\le c\}. \] Taking expectations conditional on $Z_{ij}^{1:T}=h$ gives \[ L_t(c \mid h; \theta) :=\mathbb{E}\left[D_{ijt}\mathbf{1}\{ W_{ijt}(\theta)\le c\} \mid Z_{ij}^{1:T}=h \right] \le \mathbb{P}(V_{ijt}\le c \mid Z_{ij}^{1:T}=h). \] Similarly, if $D_{ijs}=0$ and $W_{ijs}(\theta_0)\ge c$, one can get \[ U_s(c \mid h; \theta) :=1-\mathbb{E}\!\left[ (1 - D_{ijs})\mathbf{1}\{W_{ijs}(\theta)\ge c\} \mid Z_{ij}^{1:T}=h \right] \ge \mathbb{P}(V_{ijs}\le c \mid Z_{ij}^{1:T}=h). \] By the joint independence and the homogeneous-marginal parts of Assumption (ref), $\mathbb{P}(V_{ijt}\le c \mid Z_{ij}^{1:T}=h)$ is common across dates. After taking supremum over $t$ and infimum over $s$, we obtain an identified set for $\theta$. We summarize the results in the following proposition.

proposition[Dyadic Panel Identifying Restrictions] For any $\theta=(\alpha',\lambda')'$, define the intertemporally aggregated bounds \[ \overline L(c \mid h; \theta) := \max_{t=1,\ldots,T} L_t(c \mid h; \theta), \quad \underline U(c \mid h; \theta) := \min_{t=1,\ldots,T} U_t(c \mid h; \theta). \] Then under (ref) and Assumption (ref), we have $\theta_0 \in \Theta_I^{\mathrm{dyad}}$, where \[ \Theta_I^{\mathrm{dyad}} := \left\{ \theta: \overline L(c \mid h; \theta) \le \underline U(c \mid h; \theta) \text{ for all } c\in\mathbb{R} \text{ and all } h\in\operatorname{Supp}(Z_{ij}^{1:T}) \right\}. \]

remark[About Sharpness] Proposition (ref) shows that $\theta_0$ belongs to the displayed restriction set, but it does not claim that the set is sharp. Throughout this paper, we use “identified set” in this standard sense without claiming sharpness. Establishing sharpness in the present dynamic-network environment appears substantially harder and is left to future work.

remark[Role of the time dimension] All semiparametric results in this section require at least $T\ge 2$ time periods, since the identifying restrictions compare outcomes across distinct dates. A larger $T$ enlarges the class of available comparisons: additional dates contribute to the maximum over $t$ and minimum over $s$ in Proposition (ref), and enlarge the class of admissible balanced signed subgraphs in Propositions (ref)--(ref). This does not automatically imply monotone shrinkage of the identified set, since the conditioning objects also grow with $T$, but it does expand the set of identifying restrictions that can be brought to bear.

Signed Subgraph Identification

The signed-subgraph approach is closer to GaoLiXu26. It uses time as an additional differencing dimension and constructs events over edge-time cells so that fixed effects cancel algebraically. Because the network regressors are lagged, one can compare edge-time cells without confronting contemporaneous simultaneity. The key point is that the propositions below use only the exogeneity part of Assumption (ref); they do not use homogeneous marginals and therefore remain valid under arbitrary serial correlation. We begin with the smallest nontrivial case, a two-period transition for one dyad, and then state the general signed-subgraph version.

proposition[Dyad-transition inequalities] Fix two dates $t \neq s$ and define \[ \Delta_{ts}W_{ij}(\theta) := W_{ijt}(\theta)-W_{ijs}(\theta), \quad \Delta_{ts}U_{ij} := U_{ijt}-U_{ijs}, \quad \mathcal Z_{ij}^{1:T} := \bigl({Z_i^{1:T}}',{Z_j^{1:T}}'\bigr)'. \] Under (ref) and Assumption (ref), for every $c\in\mathbb{R}$ and every $z$ in the support of $\mathcal Z_{ij}^{1:T}$, \[ \mathbb{E}\left[ D_{ijt}(1-D_{ijs}) \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\le c\} \mid \mathcal Z_{ij}^{1:T}=z \right] \le \mathbb{P}\bigl(\Delta_{ts}U_{ij}<c\bigr), \] and \[ \mathbb{E}\left[ (1-D_{ijt})D_{ijs} \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\ge c\} \mid \mathcal Z_{ij}^{1:T}=z \right] \le \mathbb{P}\bigl(\Delta_{ts}U_{ij}>c\bigr). \] Consequently, \[ \begin{aligned} \sup_z\, \mathbb{E}\left[ D_{ijt}(1-D_{ijs}) \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\le c\} \mid \mathcal Z_{ij}^{1:T}=z \right] \le\;& \inf_z \Bigl[ 1- \mathbb{E}\left[ (1-D_{ijt})D_{ijs} \right.\\ &\left. \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\ge c\} \mid \mathcal Z_{ij}^{1:T}=z \right] \Bigr]. \end{aligned} \]

Proposition (ref) is the simplest dynamic analog of the Gao-Li-Xu differencing logic. The dyad effect $A_{ij}$ appears once with a positive sign and once with a negative sign, so it cancels exactly. The conditioning is only on exogenous $Z$ histories; the lagged network vector $X_{ij,t-1}$ remains inside the random index difference and need not be conditioned on. Call a triple $(i,j,t)$ with $i<j$ and $t\in\{1,\ldots,T\}$ an edge-time cell. For any finite collection $\mathcal C$ of edge-time cells, let $N(\mathcal C)$ denote the set of nodes appearing in $\mathcal C$, and define the corresponding exogenous history vector by \[ \mathcal Z_{\mathcal C}^{1:T} := \bigl({Z_m^{1:T}}'\bigr)_{m\in N(\mathcal C)}'. \]

proposition[Dynamic signed-subgraph inequalities] Let $\mathcal C^{+}$ and $\mathcal C^{-}$ be nonempty finite collections of edge-time cells. Suppose they are balanced in the sense that for every dyad $(i,j)$, \[ \#\{t:(i,j,t)\in\mathcal C^{+}\} = \#\{t:(i,j,t)\in\mathcal C^{-}\}. \] Define $$Y_{\mathcal C}^+ : = \prod_{e\in\mathcal C^+} D_e \prod_{e\in\mathcal C^-} (1-D_e),\quad Y_{\mathcal C}^- : =\prod_{e\in\mathcal C^+} (1-D_e) \prod_{e\in\mathcal C^-} D_e,$$ where $D_e$ denotes the link indicator attached to cell $e$. Also define \[ \Delta_{\mathcal C}W(\theta) := \sum_{(i,j,t)\in\mathcal C^{+}} W_{ijt}(\theta) - \sum_{(i,j,t)\in\mathcal C^{-}} W_{ijt}(\theta),\quad \Delta_{\mathcal C}U := \sum_{(i,j,t)\in\mathcal C^{+}} U_{ijt} - \sum_{(i,j,t)\in\mathcal C^{-}} U_{ijt}. \] Under (ref) and Assumption (ref), for every $c\in\mathbb{R}$, \[ \sup_z \mathbb{E}\left[ Y_{\mathcal C}^+ \mathbf{1}\{\Delta_{\mathcal C}W(\theta_0)\le c\} \mid \mathcal Z_{\mathcal C}^{1:T}=z \right] \le \inf_z \left[ 1- \mathbb{E}\left[ Y_{\mathcal C}^- \mathbf{1}\{\Delta_{\mathcal C}W(\theta_0)\ge c\} \mid \mathcal Z_{\mathcal C}^{1:T}=z \right] \right]. \]

remarkProposition (ref) is the direct dynamic analog of the Gao-Li-Xu subgraph argument. Proposition (ref) is its two-cell special case, obtained by taking $\mathcal C^+=\{(i,j,t)\}$ and $\mathcal C^-=\{(i,j,s)\}$. Dyad transitions are the smallest balanced signed subgraphs, but richer dynamic objects are possible. One can mix cross-sectional and intertemporal differencing in the same construction, provided each dyad appears with net sign zero. Because the network covariates are lagged, the isolation arguments that are required in simultaneous static strategic models are not necessary here. Unlike Proposition (ref), these signed-subgraph inequalities do not use homogeneous marginals over time. Like Proposition (ref), they continue to hold under arbitrary serial correlation of the pairwise shock process. Analogously to Proposition (ref), define \[ \Theta_I^{\mathrm{subgraph}} := \left\{ \theta: \text{the inequality in Proposition \ref{prop:signed-subgraph} holds for all balanced } (\mathcal C^+,\mathcal C^-) \text{ and all } c\in\mathbb{R} \right\}. \] Then $\theta_0\in\Theta_I^{\mathrm{dyad}}\cap\Theta_I^{\mathrm{subgraph}}$, and the two identified sets are in general not nested.

Unified Partial-Differencing Perspective

The two semiparametric approaches above can be viewed as extreme points of a broader spectrum: \[

tikzpicture[tikzpicture omitted — 448 chars of source]

\] The basic object is a signed comparison over edge-time cells in which some fixed-effect components cancel algebraically, while the remaining components are absorbed into a common latent CDF.

The clean economic interpretation is exactly a split between two roles: First, the differenced-out parts. These are the dyad components whose fixed effects cancel algebraically. For them, one only needs the exogeneity part of Assumption (ref). Their exogenous histories may therefore be conditioned on freely and then profiled out through sup/inf operations. Second, the integrated-out parts. These are the dyad components whose fixed effects do not cancel. For them, one relies on the homogeneity/common-law part of Assumption (ref). Their residual contribution is absorbed into a latent CDF that is held fixed while one takes envelopes over admissible comparison objects.

Define a comparison object as an ordered pair $g: = (\mathcal C_g^+,\mathcal C_g^-)$, where $\mathcal C_g^+$ and $\mathcal C_g^-$ are finite collections of edge-time cells indexed by $g$. Define \[ Y_g^+ := \prod_{e\in\mathcal C_g^+} D_e \prod_{e\in\mathcal C_g^-} (1-D_e), \quad Y_g^- := \prod_{e\in\mathcal C_g^+} (1-D_e) \prod_{e\in\mathcal C_g^-} D_e, \] \[ \Delta_gW(\theta) := \sum_{e\in\mathcal C_g^+} W_e(\theta) - \sum_{e\in\mathcal C_g^-} W_e(\theta), \quad \Delta_gU := \sum_{e\in\mathcal C_g^+} U_e - \sum_{e\in\mathcal C_g^-} U_e. \] For each dyad $(i,j)$, let \[ \rho_g(i,j) := \#\{t:(i,j,t)\in\mathcal C_g^+\} - \#\{t:(i,j,t)\in\mathcal C_g^-\}. \] Also, define the residual-dyad set and the vector of dyadic-covariate histories for the uncanceled dyads \[ \mathcal R_g := \{(i,j):\rho_g(i,j)\neq 0\},\quad Z_{\mathcal R_g}^{1:T} := \bigl({Z_{ij}^{1:T}}'\bigr)'_{(i,j)\in\mathcal R_g}. \]

Assume that for each $g\in\mathcal G$, both $\mathcal C_g^+$ and $\mathcal C_g^-$ are nonempty. On the event $Y_g^+=1$, \[ \Delta_gU < \Delta_gW(\theta_0) + \sum_{(i,j)\in\mathcal R_g}\rho_g(i,j)A_{ij}, \] and on $Y_g^-=1$ the reverse strict inequality holds. Thus, if one defines \[ M_g := \Delta_gU - \sum_{(i,j)\in\mathcal R_g}\rho_g(i,j)A_{ij}, \] then $Y_g^+=1$ implies $M_g<\Delta_gW(\theta_0)$ and $Y_g^-=1$ implies $M_g>\Delta_gW(\theta_0)$.

proposition[Partial-differencing envelope within a fixed residual-load class] Let $\mathcal G$ be a family of comparison objects $g$ such that: \begin{enumerate} • all $g\in\mathcal G$ have the same residual-load vector $\rho_g=\rho$ and hence the same residual-dyad set $\mathcal R$; • for each $g\in\mathcal G$, one can partition the observable exogenous histories entering $g$ into a retained component $S_g$ and a nuisance component $T_g$; • the retained component is common across $g\in\mathcal G$, in the sense that $S_g=S_{\mathcal R}$ for all $g$, where $S_{\mathcal R}$ is built from the exogenous histories of the uncanceled dyads in $\mathcal R$; • conditional on $S_{\mathcal R}=s$, the distribution of $M_g$ is common across $g\in\mathcal G$ and does not depend on $T_g$. \end{enumerate} Then for every $c\in\mathbb{R}$ and every $s$ in the support of $S_{\mathcal R}$, there exists a CDF $F(c\mid s)$ such that \[ \sup_{g\in\mathcal G} \sup_{t\in\operatorname{Supp}(T_g\mid S_{\mathcal R}=s)} \mathbb{E}\left[ Y_g^+ \mathbf{1}\{\Delta_gW(\theta_0)\le c\} \mid S_{\mathcal R}=s,\; T_g=t \right] \le F(c\mid s), \] and \[ F(c\mid s) \le \inf_{g\in\mathcal G} \inf_{t\in\operatorname{Supp}(T_g\mid S_{\mathcal R}=s)} \left[ 1- \mathbb{E}\left[ Y_g^- \mathbf{1}\{\Delta_gW(\theta_0)\ge c\} \mid S_{\mathcal R}=s,\; T_g=t \right] \right]. \]

remarkProposition (ref) provides a taxonomic framework that nests the two semiparametric approaches developed above, but it does so within a fixed residual-load class. That is, the proposition pools only over comparison objects that leave the same uncanceled dyads with the same residual coefficients. Different residual-load vectors generate different retained conditioning objects and therefore different envelope inequalities; the overall identified set is obtained by intersecting the restrictions from those separate classes. First, the complete integration out class. Take $\mathcal G=\{g_t:t=1,\ldots,T\}$, where $g_t$ is the one-cell comparison object built from edge-time cell $(i,j,t)$, so that $\mathcal C_{g_t}^+=\{(i,j,t)\}$ and $\mathcal C_{g_t}^-=\varnothing$. Then $\rho(i,j)=1$, so no dyad effect is canceled. Set $S_{\mathcal R}=Z_{ij}^{1:T}$ and let $T_g$ be empty. The common conditional law of $M_{g_t}=U_{ijt}-A_{ij}$ given $Z_{ij}^{1:T}=h$ follows from the joint independence and the homogeneous-marginal part of Assumption (ref). Although these one-cell comparison objects have $\mathcal C_{g_t}^-=\varnothing$ and hence fall outside the strict-inequality setting of the proposition, the resulting weak-inequality bounds are still valid and recover Proposition (ref). Second, the complete differencing out class. Take $\mathcal G=\{g\}$ with $\mathcal C_g^+=\{(i,j,t)\}$ and $\mathcal C_g^-=\{(i,j,s)\}$. Then $\rho\equiv 0$, so the dyad effect is canceled completely and $M_g=\Delta_{ts}U_{ij}$. Set $S_{\mathcal R}$ to be degenerate and let $T_g=\mathcal Z_{ij}^{1:T}$. This gives Proposition (ref). More generally, any balanced signed subgraph has $\rho\equiv 0$ and falls under Proposition (ref). Lastly, the partial differencing / partial integration class. For any fixed class of intermediate comparison objects with the same residual-load vector $\rho$, the zero-load dyads are differenced out, while the nonzero-load dyads are integrated out through the unknown CDF $F(c\mid s)$. This provides an organizing perspective under arbitrary dyad effects. In the notation of Proposition (ref), $T_g$ should be read as the exogenous histories attached to the differenced-out pieces, while $S_{\mathcal R}$ should be read as the exogenous histories attached to the absorbed pieces. Exogeneity lets one condition on $T_g$ and then profile over it, whereas homogeneity/common-law restrictions are used to compare the latent CDF indexed by $S_{\mathcal R}$ across comparison objects. This unified view explains why the dyad-panel and signed-subgraph approaches are complementary rather than redundant. The dyad-panel approach sits at the “fully integrated” end of the spectrum, while the signed-subgraph approach sits at the “fully differenced” end. Intermediate partial-differencing designs lie between those extremes, but each residual-load class contributes its own envelope inequality rather than all classes pooling into a single common CDF. The later strengthenings below move along the same spectrum by making some composite-error CDFs explicit or by enlarging the class of admissible partial-differencing designs.

Sharper Identification under Additional Structures

Section (ref) imposed neither parametric knowledge of the shock process nor additional structure on $A_{ij}$. Two strengthenings are especially useful. First, if the common marginal CDF of $U_{ijt}$ is known and the shock process is serially independent, then every fully differenced comparison has a known composite-error CDF and the bounding inequalities become explicit. Second, the additive-node structure $A_{ij}=\nu_i+\nu_j$ enlarges the class of valid weighted-differencing arguments even when the CDF is unknown.

Known Marginal CDF and Serial Independence

Suppose now that the common marginal CDF of $U_{ijt}$ is known, continuous, and denoted by $F_U$. Suppose in addition that the shock process $U_{ijt}$ is serially independent within each dyad $(i,j)$. Because Assumption (ref) already gives i.i.d.\ shock vectors across dyads, this strengthening implies independence across all distinct edge-time cells. Therefore every differencing design that fully removes the relevant fixed effects produces a composite error with known CDF.

proposition[Explicit bounds under a known marginal CDF and serial independence] Suppose Assumption (ref) holds, the common marginal CDF $F_U$ of $U_{ijt}$ is known and continuous, and $U_{ij1},\ldots,U_{ijT}$ are independent for every dyad $(i,j)$. \begin{enumerate} • For any pair of dates $t\neq s$, define \[ \Delta_{ts}U_{ij} := U_{ijt}-U_{ijs}, \quad F_{\Delta}(c) := \mathbb{P}(\Delta_{ts}U_{ij}\le c) = \int F_U(c+u)\, dF_U(u). \] Then \[ \sup_z\, \mathbb{E}\left[ D_{ijt}(1-D_{ijs}) \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\le c\} \mid \mathcal Z_{ij}^{1:T}=z \right] \le F_{\Delta}(c) \] \[ \le \inf_z \left[ 1- \mathbb{E}\left[ (1-D_{ijt})D_{ijs} \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\ge c\} \mid \mathcal Z_{ij}^{1:T}=z \right] \right]. \] • For any balanced signed subgraph $(\mathcal C^+,\mathcal C^-)$ as in Proposition (ref), define \[ \Delta_{\mathcal C}U := \sum_{e\in\mathcal C^+} U_e - \sum_{e\in\mathcal C^-} U_e, \quad F_{\mathcal C}(c) := \mathbb{P}(\Delta_{\mathcal C}U\le c). \] Then $F_{\mathcal C}$ is known from $F_U$; specifically, it is the convolution of $|\mathcal C^+|$ copies of $F_U$ and $|\mathcal C^-|$ copies of the reflected CDF $F_{-U}(x):=\mathbb{P}(-U_{ijt}\le x)$. Moreover, \[ \sup_z \mathbb{E}\left[ Y_{\mathcal C}^+ \mathbf{1}\{\Delta_{\mathcal C}W(\theta_0)\le c\} \mid \mathcal Z_{\mathcal C}^{1:T}=z \right] \le F_{\mathcal C}(c) \] \[ \le \inf_z \left[ 1- \mathbb{E}\left[ Y_{\mathcal C}^- \mathbf{1}\{\Delta_{\mathcal C}W(\theta_0)\ge c\} \mid \mathcal Z_{\mathcal C}^{1:T}=z \right] \right]. \] \end{enumerate}

remarkProposition (ref) is the most direct nonlogit sharpening available in the present setting. The gain comes from combining a known marginal CDF with serial independence, not from the marginal CDF alone. Once fixed effects are fully differenced out, the middle term in the lower/upper sandwich is pinned down by a known composite-error distribution. Logit remains distinct for a different reason: under additive node effects it yields an exact conditional-logit representation with algebraic cancellation of the fixed effects. There is also a useful max-score-type special case. Because $\Delta_{ts}U_{ij}=U_{ijt}-U_{ijs}$ is the difference of two i.i.d.\ continuous variables, its CDF is symmetric around zero and satisfies \[ F_{\Delta}(0)=\frac{1}{2}. \] Hence Proposition (ref) implies \[ \sup_z\, \mathbb{E}\left[ D_{ijt}(1-D_{ijs}) \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\le 0\} \mid \mathcal Z_{ij}^{1:T}=z \right] \le \frac{1}{2}, \] \[ \sup_z\, \mathbb{E}\left[ (1-D_{ijt})D_{ijs} \mathbf{1}\{\Delta_{ts}W_{ij}(\theta_0)\ge 0\} \mid \mathcal Z_{ij}^{1:T}=z \right] \le \frac{1}{2}. \] This is the appropriate dynamic analog of the maximum-score-type special case in GaoWang26, so it is best viewed as a sharpening of the dyad-transition bounds under serial independence, in the spirit of GaoWang26, rather than as a separate identification result.

Additive Node Effects with Unknown CDF

Now suppose instead that the dyad effect is additive in nodes: \[ A_{ij}=\nu_i+\nu_j. \] This assumption alone sharpens the semiparametric analysis because weighted differencing can now be organized around nodes rather than dyads. The admissible class of weighted configurations is therefore much larger than the dyad-balanced signed subgraphs used under unrestricted dyad effects.

assumption[Additive node effects and exchangeable node types] Assume that (i) $A_{ij}=\nu_i+\nu_j$, (ii) the node histories $\{(\nu_i,Z_i^{1:T}): i=1,2,\ldots\}$ are i.i.d.\ across $i$, and (iii) the dyad-level shock process is independent of the full node history $\{(\nu_i,Z_i^{1:T}): i=1,\ldots,n\}$.

Let $\mathcal C$ be a finite nonempty collection of edge-time cells $e=(i,j,t)$, let $\omega_e\neq 0$ be an associated real weight, and let $\dot{e} = \{i,j\}$ be the set of nodes appearing in the dyad component of cell $e$. Define the positive and negative cells \[ \mathcal C^+ := \{e\in\mathcal C:\omega_e>0\}, \quad \mathcal C^- := \{e\in\mathcal C:\omega_e<0\}, \] and, for each node $m$ appearing in $\mathcal C$, define its weighted incidence sum $\sigma_m := \sum_{e\in\mathcal C:\, m\in \dot{e}} \omega_e.$ Let $S_0 := \{m:\sigma_m=0\}$ be the set of nodes whose fixed effects are eliminated by the weighted configuration, and let $S_R := \{m:\sigma_m\neq 0\}$ be the set of retained nodes. Also let \[ \mathcal Z_{S_0}^{1:T} := \bigl({Z_m^{1:T}}'\bigr)'_{m\in S_0},\quad \mathcal Z_{S_R}^{1:T} := \bigl({Z_m^{1:T}}'\bigr)'_{m\in S_R}. \] Assume throughout that both $\mathcal C^+$ and $\mathcal C^-$ are nonempty. Define \[ Y_{\mathcal C}^+ := \prod_{e\in\mathcal C^+} D_e \prod_{e\in\mathcal C^-} (1-D_e), \quad Y_{\mathcal C}^- := \prod_{e\in\mathcal C^+} (1-D_e) \prod_{e\in\mathcal C^-} D_e, \] and write \[ \Delta_{\mathcal C,\omega}W(\theta) := \sum_{e\in\mathcal C}\omega_e W_e(\theta), \quad \widetilde U_{\mathcal C,\omega} := \sum_{e\in\mathcal C}\omega_e U_e - \sum_{m\in S_R}\sigma_m \nu_m. \]

proposition[Weighted node-differencing under additive fixed effects] Under (ref) and Assumptions (ref)-(ref), for every $c\in\mathbb{R}$ and every realization $z_R$ of $\mathcal Z_{S_R}^{1:T}$, there exists a CDF $F_{\mathcal C,\omega}(\cdot\mid z_R)$ such that \[ \sup_{z_0\in\operatorname{Supp}(\mathcal Z_{S_0}^{1:T}\mid \mathcal Z_{S_R}^{1:T}=z_R)} \mathbb{E}\left[ Y_{\mathcal C}^+ \mathbf{1}\{\Delta_{\mathcal C,\omega}W(\theta_0)\le c\} \mid \mathcal Z_{S_R}^{1:T}=z_R,\; \mathcal Z_{S_0}^{1:T}=z_0 \right] \le F_{\mathcal C,\omega}(c\mid z_R), \] and \[ \inf_{z_0\in\operatorname{Supp}(\mathcal Z_{S_0}^{1:T}\mid \mathcal Z_{S_R}^{1:T}=z_R)} \left[ 1- \mathbb{E}\left[ Y_{\mathcal C}^- \mathbf{1}\{\Delta_{\mathcal C,\omega}W(\theta_0)\ge c\} \mid \mathcal Z_{S_R}^{1:T}=z_R,\; \mathcal Z_{S_0}^{1:T}=z_0 \right] \right] \ge F_{\mathcal C,\omega}(c\mid z_R). \]

remarkProposition (ref) is the lagged-dynamic analog of the triad, weighted-star, and general cycle arguments in \citep*{GaoLiXu26}. It sharpens the unknown-CDF analysis even before specifying any parametric CDF. The key gain is combinatorial. First, complete elimination now requires only that weighted node incidences sum to zero, which is weaker than dyad balancing. Second, partial elimination is also admissible, because one conditions on the retained-node histories $\mathcal Z_{S_R}^{1:T}$ and profiles over the eliminated-node histories $\mathcal Z_{S_0}^{1:T}$, leaving the residual node effects inside the latent CDF $F_{\mathcal C,\omega}(\cdot\mid z_R)$. Additionally, dynamic versions of triads, weighted stars, tetrads, and longer cycles can all be used, and they can all contribute valid semiparametric restrictions. In particular, if one defines $\Theta_I^{\mathrm{add}}$ as the set of $\theta$ satisfying the envelope implications from Proposition (ref) for all admissible weighted configurations $(\mathcal C,\omega)$, all thresholds $c$, and all retained conditioning values $z_R$, then \[ \theta_0 \in \Theta_I^{\mathrm{dyad}} \cap \Theta_I^{\mathrm{add}}. \] Thus additive node effects sharpen the semiparametric analysis even when the marginal CDF of $U_{ijt}$ is left unknown, so this sharpening is complementary to Proposition (ref). It is worth noting precisely which components of Assumption (ref) drive the result. The additive representation $A_{ij}=\nu_i+\nu_j$ enables the combinatorial gain of organizing weighted differencing around nodes rather than dyads. The i.i.d.\ node-history condition (Assumption (ref)(ii)) and the stronger shock independence (Assumption (ref)(iv)) are used in the proof to ensure that the retained node effects are conditionally independent of the eliminated-node histories given the retained histories, which is what allows profiling over $\mathcal Z_{S_0}^{1:T}$ while holding the latent CDF fixed. The proof does not use homogeneous marginals across dates, so this sharpening remains valid under arbitrary serial correlation within dyads. If, in addition, the assumptions of Proposition (ref) hold and the weighted configuration achieves complete node balance $\sigma_m=0$ for every node in $\mathcal C$, then $S_R$ is empty, $\widetilde U_{\mathcal C,\omega}=\sum_{e\in\mathcal C}\omega_e U_e$, and the now-unconditional CDF $F_{\mathcal C,\omega}$ is the known convolution of the scaled shock marginals.

Additive Node Fixed Effects with IID Logit Specification

The previous two subsections sharpened identification in two complementary directions: Section (ref).1 used a known marginal CDF with serial independence to make composite-error distributions explicit, while Section (ref).2 used additive node effects to enlarge the class of admissible differencing designs. This subsection combines the two strengthenings under a logit specification and shows that the combination yields an exact conditional logit representation that goes well beyond the per-period analogue of graham_2017's static tetrad logit. The key gain is that cross-node differencing (from Section (ref).2) and cross-period differencing (from Section (ref)) can be combined freely: any configuration of edge-time cells that achieves complete node balance produces an exact conditional logit, whether or not the cells share a common date. This yields a much larger class of identifying restrictions and a correspondingly weaker sufficient condition for point identification.

We begin by motivating why logit is special. Suppose additive node effects are combined with a known conditional CDF $F$ for the current shock. At each date $t$, the model is \[ D_{ijt} = \mathbf{1}\left\{ Z_{ijt}'\alpha_0 + X_{ij,t-1}'\lambda_0 + \nu_i + \nu_j - U_{ijt} \ge 0 \right\}. \] Unlike the static strategic model studied in \citep*{GaoLiXu26}, there is no contemporaneous endogenous network statistic here, so the isolation machinery from that paper is not needed. If, conditional on the node effects and the lagged observables, the current shock on edge $(i,j)$ at date $t$ has CDF $F$, then \[ p_{ij,t} := \mathbb{P}\!\left( D_{ijt}=1 \mid Z_{ijt},X_{ij,t-1},\nu \right) = F\!\left(Z_{ijt}'\alpha_0 + X_{ij,t-1}'\lambda_0 + \nu_i + \nu_j\right). \] For any configuration $\mathcal C=(\mathcal C^+,\mathcal C^-)$ of edge-time cells, the ratio $\mathbb{P}(Y_{\mathcal C}^+=1\mid \mathcal Z_{\mathcal C},\nu)/\mathbb{P}(Y_{\mathcal C}^-=1\mid \mathcal Z_{\mathcal C},\nu)$ involves terms of the form $F(\eta_e)/(1-F(\eta_e))$. The additive node effects cancel from the exponent $\sum_{e\in\mathcal C^+}\eta_e - \sum_{e\in\mathcal C^-}\eta_e$ if and only if $\sigma_m=0$ for every node $m$. But the multiplicative product of odds ratios reduces to an exponential of this sum if and only if $\log[F(\cdot)/(1-F(\cdot))]$ is affine---that is, up to location-scale normalization, exactly the logit case. For nonlogit $F$ (such as normal/probit), $\log[F(\cdot)/(1-F(\cdot))]$ is nonlinear and the node effects do not cancel algebraically from the product, so there is no exact conditional likelihood of the graham_2017 type.

The semiparametric results above allow arbitrary serial correlation. The logit result below is sharper, but it does require a fully i.i.d.\ logistic shock structure.

assumption[IID logistic shocks with additive node effects] Assume that (i) $A_{ij}=\nu_i+\nu_j$, and (ii) the shocks $\{U_{ijt}: i<j,\ t=1,\ldots,T\}$ are i.i.d.\ across dyads and dates with standard logistic CDF, and are jointly independent of the full latent-heterogeneity array and the full exogenous covariate array.

The standard logistic specification in Assumption (ref)(ii) fixes both the location and scale of the error distribution, thereby resolving the scale indeterminacy present in the semiparametric analysis.

Let $\mathcal C=(\mathcal C^+,\mathcal C^-)$ be a configuration of edge-time cells $e=(i,j,t)$, and recall the notation $\sigma_m=\#\{e\in\mathcal C^+: m\in\dot{e}\}-\#\{e\in\mathcal C^-: m\in\dot{e}\}$ for the signed incidence of node $m$. Say $\mathcal C$ is completely node-balanced if $\sigma_m=0$ for every node $m$ appearing in $\mathcal C$. Define \[ \Delta_{\mathcal C}W(\theta) := \sum_{e\in\mathcal C^+}W_e(\theta) - \sum_{e\in\mathcal C^-}W_e(\theta), \] and let $\mathcal Z_{\mathcal C}$ denote the collection of observed exogenous histories for all edges appearing in $\mathcal C$.

theorem[Conditional logit under node-balanced comparisons] Under Assumption (ref), let $\mathcal C=(\mathcal C^+,\mathcal C^-)$ be any completely node-balanced configuration. Then \[ \log \frac{ \mathbb{P}(Y_{\mathcal C}^+ = 1 \mid \mathcal Z_{\mathcal C}) }{ \mathbb{P}(Y_{\mathcal C}^- = 1 \mid \mathcal Z_{\mathcal C}) } = \Delta_{\mathcal C}W(\theta_0). \] Equivalently, \[ \mathbb{P}(Y_{\mathcal C}^+ =1 \mid Y_{\mathcal C}^+ + Y_{\mathcal C}^- = 1,\;\mathcal Z_{\mathcal C}) = \frac{\exp(\Delta_{\mathcal C}W(\theta_0))}{1+\exp(\Delta_{\mathcal C}W(\theta_0))}. \] If the support of $\{\Delta_{\mathcal C}W(\theta_0):\mathcal C\ \text{completely node-balanced}\}$ spans $\mathbb{R}^{d_h+d_x}$, then $\theta_0=(\alpha_0',\lambda_0')'$ is point identified.

remark[Tetrad logit as a special case] The within-date tetrad is the simplest completely node-balanced configuration: for four distinct nodes $(i,j,h,k)$ and a single date $t$, set $\mathcal C^+=\{(i,j,t),(h,k,t)\}$ and $\mathcal C^-=\{(i,k,t),(j,h,t)\}$. Each node appears once in $\mathcal C^+$ and once in $\mathcal C^-$, so $\sigma_m=0$ for $m\in\{i,j,h,k\}$. In this case Theorem (ref) reduces to a per-period application of graham_2017's static tetrad logit with lagged regressors entering the index. That per-period result is not new: it follows directly from graham_2017 once the lagged network covariates are treated as predetermined. The contribution of Theorem (ref) is that it generalizes the tetrad logit by exploiting both the cross-node differencing of Section (ref).2 and the cross-period differencing of Section (ref), thereby producing a substantially richer class of exact conditional logit restrictions.

remark[Examples of new configurations] The following completely node-balanced configurations go beyond the within-date tetrad and are specific to the dynamic setting of this paper. Intertemporal tetrads. Take four distinct nodes $(i,j,h,k)$ and (possibly distinct) dates $t_1,t_2,t_3,t_4$: set $\mathcal C^+=\{(i,j,t_1),(h,k,t_2)\}$ and $\mathcal C^-=\{(i,k,t_3),(j,h,t_4)\}$. Each node still appears once in $\mathcal C^+$ and once in $\mathcal C^-$. The covariate contrast $\Delta_{\mathcal C}W(\theta_0)$ now mixes cross-sectional and temporal variation, providing directions in $\mathbb{R}^{d_h+d_x}$ not available from any single-date tetrad. Triadic cycles. Take three distinct nodes $\{i,j,k\}$ and six dates: set $\mathcal C^+=\{(i,j,t_1),(j,k,t_2),(i,k,t_3)\}$ and $\mathcal C^-=\{(i,j,t_4),(j,k,t_5),(i,k,t_6)\}$. Each node appears in exactly two edges of $\mathcal C^+$ and two edges of $\mathcal C^-$, so $\sigma_m=0$ for $m\in\{i,j,k\}$. This uses only three nodes rather than four, so triadic comparisons are available even in smaller networks. Longer cycles and weighted stars. More generally, any node-balanced cycle of length $2k$ or any star configuration in which the hub's positive and negative incidences cancel produces an exact conditional logit. The class of such configurations grows combinatorially with the number of nodes and dates.

remark[Point identification: weakened support condition] The support condition in Theorem (ref) is stated over all completely node-balanced configurations, not just within-date tetrads. This is strictly weaker than requiring the within-date tetrad-differenced covariate vector to span $\mathbb{R}^{d_h+d_x}$, because intertemporal tetrads and triadic cycles contribute additional directions. The gain is substantive in at least two settings. First, when $n$ is small (so few tetrads exist at any given date), triadic cycles on three nodes expand the set of available comparisons. Second, when $X_{ij,t-1}$ contains count-valued network statistics such as common friends, its within-date tetrad difference takes integer values with limited variation, but intertemporal configurations pool across dates and can restore full rank. As a concrete illustration, consider $d_h=1$ and $X_{ij,t-1}=(D_{ij,t-1},R_{ij,t-1})'$ with $d_x=2$. Within-date tetrads generate covariate contrasts in $\mathbb{R}\times\mathbb{Z}^2$. The continuous covariate $Z_{it}$ provides variation along the first coordinate (though with possible point masses from the tetrad combination), and the integer-valued lagged-network differences provide the remaining two dimensions whenever the network is sufficiently heterogeneous. If the within-date tetrad support alone does not span $\mathbb{R}^3$, one can supplement it with intertemporal tetrads: at distinct dates $t_1\neq t_3$ or $t_2\neq t_4$, the lagged-network differences $\Delta X$ draw from different network configurations, and the resulting covariate contrasts can fill out missing directions.

remark[Moment inequality restrictions] Beyond the exact conditional logit of Theorem (ref), the semiparametric results of Sections (ref)--(ref) also continue to apply under Assumption (ref). In particular, Assumption (ref) implies both a known marginal CDF (standard logistic) and serial independence, so Proposition (ref) gives explicit sandwich bounds for every dyad-balanced signed subgraph with the composite-error CDF computed as a known convolution of logistic distributions. Simultaneously, Proposition (ref) applies, and for any completely node-balanced configuration the composite error has a known CDF. These moment inequality restrictions supplement the conditional logit of Theorem (ref) in two ways: they provide overidentifying restrictions useful for specification testing, and they supply additional identifying power through the “bounding-by-$c$” technique when the conditional logit support condition for point identification fails.

Conclusion

This paper studies a broad class of dynamic dyadic network formation models with time-varying observed covariates, lagged local network statistics, and unobserved heterogeneity. The framework nests observed-covariate homophily, transitivity, second-order or indirect-friend effects, and more general local subgraph statistics within a single dynamic index model. The main message is that, once these network covariates are lagged and observable, the model can be studied through a unified difference-out / integrate-out perspective rather than only through exact logit likelihood methods. Three principal strengthenings then sharpen that semiparametric analysis: a known marginal CDF combined with serial independence, additive node effects, and the special affine-log-odds structure of logit. Combining all three under i.i.d.\ logit with additive node effects yields an exact conditional logit representation for any completely node-balanced configuration of edge-time cells, generalizing the per-period analogue of graham_2017's tetrad logit by exploiting both cross-node and cross-period variation. Sharpness, inference, and further econometric development are left to future work.