C31 | Macro Paper Warehouse

Normal Approximation in Large Network Models

Mon, 01 Jan 0001 00:00:00 +0000

This paper proves a central limit theorem (CLT) for network formation models with strategic interactions and homophilous agents, addressing a foundational inferential gap in the econometrics of large networks. The setting is one where the econometrician observes a single large network — the asymptotic framework sends network size n to infinity — which is the empirically relevant case for most network datasets. The network moments of interest are averages of node-level statistics (1/n) Σ ψ_i, where ψ_i can capture degree, clustering coefficients, or subnetwork counts (triangles, k-stars) that have been used for structural inference in network formation games.

The model is a pairwise-stability network formation game augmented onto a latent-space/geometric-graph structure. Each node i has an i.i.d. type (X_i, Z_i), where X_i is a continuously distributed position vector capturing homophilous attributes. Two nodes i and j form a link if a joint-surplus function V(·) exceeds zero, where V depends on the scaled distance r_n^{-1}‖X_i − X_j‖ between positions, a vector of strategic interaction statistics S_{ij} (functions of neighboring links), node attributes Z_i, Z_j, and an i.i.d. utility shock ζ_{ij}. Homophily enters as a monotonicity requirement: V is decreasing in the distance component, so dissimilar nodes are less likely to link. Sparsity is ensured by setting r_n = (κ/n)^{1/d}, which keeps expected degree asymptotically bounded.

Strategic interactions enter through S_{ij}, which depends on links involving neighbors of i or j (local externalities), generating chains of cross-sectional dependence that are the central obstacle to the CLT. The paper identifies two distinct sources of dependence: (1) link interdependencies from best-response chains, where the realization of one link influences neighboring links; and (2) global coordination in equilibrium selection, where agents may condition on a common signal.

The main technical contribution is adapting “stabilization” conditions from the literature on geometric graphs (Penrose and Yukich 2003, 2008) to the strategic setting. Exponential stabilization (Assumption 5) requires that the radius of stabilization R_i — the smallest neighborhood of i such that ψ_i depends only on nodes within that neighborhood — has a distribution with exponential tails. This bounds the effective dependence neighborhood and provides the weak dependence structure needed for the CLT.

To verify stabilization from primitive conditions, the paper employs branching process theory. The key construct is the “strategic neighborhood” C_i^+, the component of i in the network of non-robust links D (pairs where strategic interactions can change the link outcome). The paper bounds |C_i^+| by a subcritical Galton-Watson branching process: if the mean offspring is below 1 (subcriticality, Assumption 7, stated as ‖h*‖_m < 1), the process is non-explosive and its size has exponential tails, yielding the required stabilization. The subcriticality condition directly restricts the strength of strategic interactions and is the network analog of the condition ‖β‖ < 1 in linear autoregressive models. A second condition (Assumption 8, decentralized selection) requires that equilibrium selection operates independently across disjoint strategic neighborhoods, ruling out global coordination; this holds under myopic best-response dynamics.

For inference, the paper proposes a network HAC variance estimator hat_Σ_n = (1/n) Σ_i Σ_j k(d_{ij}/b_n) hat_ψ_i hat_ψ_j^T, where k(·) is a kernel, d_{ij} is the path distance in A, and b_n is a bandwidth, and a network bootstrap that resamples nodes with replacement. Both are shown to be consistent (Theorem 3). Simulation results with n up to 500, varying strategic interaction strength θ_2 from 0 to 0.5, show that the network HAC estimator achieves nominal 5% rejection rates and 95% coverage for n ≥ 500, while the bootstrap slightly over-rejects in small samples and performance degrades as θ_2 increases.

The scope conditions are explicit: the CLT applies to sparse networks (expected degree bounded), undirected networks with local externalities, models admitting a pairwise-stability equilibrium, and equilibrium selection satisfying decentralization. Extensions to directed or denser networks are left for future work.

Q: What is the primary research question and why does it require new theory? A: The paper asks when sample averages of network statistics — degree, clustering, subnetwork counts — satisfy a CLT in strategic network formation models observed as a single large network. Standard CLT proofs require weakly dependent observations, but strategic interactions generate chains of link dependence of a priori unbounded length, and multiple equilibria allow global coordination, both of which can destroy asymptotic normality. Prior work (Leung 2019b; Menzel 2024) established laws of large numbers but not CLTs, which require stronger conditions.

Q: What is the stabilization condition and why is it the right formulation of weak dependence? A: Exponential stabilization (Assumption 5) requires that the radius of stabilization R_i — the smallest K such that ψ_i depends only on the K-neighborhood of i in the network — has a distribution with exponential tails: lim sup_{w→∞} w^{-η} max{log τ_{b,ε}(w), log τ_p(w)} < 0 for some η ∈ (0,1]. This implies that each node’s statistic depends effectively only on a bounded fraction of the network, making {ψ_i} weakly dependent. The condition is a modification of stabilization conditions from the geometric graph literature (Penrose and Yukich 2003, 2008) adapted to allow strategic interactions.

Q: How does the paper connect the abstract stabilization condition to primitive model conditions? A: The paper defines the strategic neighborhood C_i^+ as the union of one-step network neighborhoods of nodes in i’s component in the non-robust link network D (where D_{ij} = 1 iff the link A_{ij} can be switched by strategic interactions). The size |C_i^+| controls the radius of stabilization. By mapping exploration of C_i via breadth-first search onto a Galton-Watson branching process, subcriticality (mean offspring < 1, i.e., ‖h*‖_m < 1) implies that |C_i^+| has exponential tails, which yields exponential stabilization with η = 1 (Theorem 2).

Q: What is the subcriticality condition and what does it restrict? A: Subcriticality (Assumption 7) requires that the mean interaction-strength measure satisfies ‖h*‖_m < 1, where h* bounds the probability that a given link is non-robust as a function of node attributes. This restricts how strongly the existence of one link influences the probability of neighboring links. The authors explicitly analogize this to the condition ‖β‖ < 1 in linear autoregressive models: both bound the magnitude of “autoregressive” dependence below one to prevent explosive propagation of dependence.

Q: What is the decentralized selection condition and what does it rule out? A: Assumption 8 (decentralized selection) requires that the equilibrium selection mechanism operates independently across disjoint strategic neighborhoods: A_{H_l} = λ_{|H_l|}(r^{-1}T_{H_l}, ζ_{H_l}) for each disjoint strategic neighborhood H_l. This rules out global coordination where agents condition on a common signal (such as the type of a particular node) to jointly select an equilibrium. The condition is satisfied by myopic best-response dynamics and is described as the single-network analog of requiring equilibrium selection to be independent across networks under many-network asymptotics.

Q: What is the structure of the CLT proof? A: The proof has two steps. Step 1 proves a CLT for the Poissonized model where the number of nodes N_n ~ Poisson(n), leveraging results from Penrose and Yukich (2008) for geometric graphs extended to the strategic setting. Step 2 is a de-Poissonization argument that transfers the Poissonized CLT back to the fixed-n model. The abstract CLT (Theorem 1) requires Assumptions 5 and 6, and Theorem 2 establishes that Assumptions 1–8 imply Assumption 5 with η = 1.

Q: How does the network HAC estimator work and what are its consistency conditions? A: The estimator is hat_Σ_n = (1/n) Σ_i Σ_j k(d_{ij}/b_n) hat_ψ_i hat_ψ_j^T, where d_{ij} is the path distance between i and j in the observed network A, k(·) is a kernel function, b_n is a bandwidth, and hat_ψ_i = ψ_i(N_n) − (1/n) Σ_j ψ_j(N_n) is the demeaned statistic. Consistency (hat_Σ_n →^p Σ_n) is established under appropriate conditions on the bandwidth b_n (Theorem 3). The bandwidth plays the same role as in time-series HAC estimation, controlling the window over which covariances are summed.

Q: What do the simulations show about finite-sample performance? A: Using a DGP with X_i ~ U([0,1]^2), ζ_{ij} ~ N(0,1), and θ_2 varying from 0 to 0.5 to control strategic interaction strength, the network HAC estimator achieves nominal 5% rejection rates and 95% coverage at n ≥ 500 across all settings. The bootstrap slightly over-rejects in small samples. Performance of all procedures degrades as θ_2 increases (stronger strategic interactions), consistent with the theoretical condition that subcriticality must hold. These results support practical use of the inference procedures based on Theorem 1.

Q: How does this paper relate to prior work on CLTs for network data? A: Kojevnikov et al. (2021) prove a CLT for node-level data conditional on the network, but this does not apply to network formation because the network is the outcome, not a conditioning variable. Leung (2019b) and Menzel (2024) prove laws of large numbers for strategic network formation but not CLTs. Kuersteiner (2019) takes a different approach using a conditional mixingale assumption. The paper’s abstract CLT extends Penrose and Yukich (2008) by modifying the stabilization condition to accommodate strategic interactions; the primitive conditions are new and use branching process tools that build on Leung (2019b).

Q: What network moments can the CLT be applied to? A: The CLT applies to any average of node statistics ψ_i that depends only on the K-neighborhood of i in the network (Assumption 4 with finite K). Explicit examples include average degree (ψ_i = Σ_j A_{ij}), average clustering coefficient, and counts of connected subnetworks such as triangles and k-stars. Subnetwork counts have been used as the basis for structural identification and estimation of network formation games (Sheng 2020), making the CLT directly applicable to inference in those models.

Q: What are the scope limitations and directions for future work? A: The CLT applies to sparse undirected networks with local externalities (Assumption 2), homophily in positions (Assumption 1), and equilibrium selection satisfying decentralization (Assumption 8). It does not cover directed networks, denser networks where expected degree grows with n, or models with global link externalities. The authors identify extending results to directed and denser networks and developing more powerful inference procedures exploiting network structure as priorities for future work.

Stabilization (exponential): The condition that the radius of stabilization R_i — the smallest neighborhood of i beyond which ψ_i does not depend on further nodes — has a distribution with exponential tails (lim sup_{w→∞} w^{-η} log τ(w) < 0 for η ∈ (0,1]). This is the paper’s operative formulation of weak dependence for network statistics and is adapted from geometric graph theory to the strategic setting.

Strategic neighborhood (C_i^+): The union of one-step neighborhoods of nodes in i’s component in the non-robust link network D. A link (i,j) is non-robust (D_{ij} = 1) if strategic interactions can change its realization — i.e., the surplus V can be positive under some interaction configurations and non-positive under others. The size of C_i^+ governs the radius of stabilization and hence the degree of cross-sectional dependence.

Subcriticality (‖h*‖_m < 1): The condition that the mean-field interaction strength measure satisfies ‖h*‖_m < 1, where h* bounds the conditional probability that a link is non-robust. Subcriticality ensures that breadth-first search of the strategic neighborhood is dominated by a subcritical Galton-Watson process (mean offspring < 1), preventing explosive growth of the dependence neighborhood. The paper explicitly frames this as the network analog of ‖β‖ < 1 in autoregressive models.

Decentralized selection (Assumption 8): The requirement that the equilibrium selection mechanism assigns outcomes independently across disjoint strategic neighborhoods: A_{H_l} = λ_{|H_l|}(r^{-1}T_{H_l}, ζ_{H_l}) for each disjoint H_l. This rules out global coordination — agents conditioning on a common signal to select among equilibria — while permitting local coordination within strategic neighborhoods. Satisfied by myopic best-response dynamics.

Pairwise stability: The solution concept underlying the model. A network A satisfies pairwise stability under transferable utility if A_{ij} = 1{V_{ij} > 0}, meaning a link forms exactly when the joint surplus is positive. This is the equilibrium condition from which the strategic interaction statistics S_{ij} and non-robustness indicators D_{ij} are derived.

Network HAC estimator: The variance estimator hat_Σ_n = (1/n) Σ_i Σ_j k(d_{ij}/b_n) hat_ψ_i hat_ψ_j^T, where d_{ij} is the path distance in the observed network, k(·) is a kernel, and b_n is a bandwidth. It is the network analog of heteroskedasticity- and autocorrelation-consistent (HAC) estimators in time series, using path distance in place of temporal lag distance.

Homophily (in this paper’s sense): The property that the joint-surplus function V is decreasing in the first argument r_n^{-1}‖X_i − X_j‖ (scaled positional distance), so nodes that are more dissimilar in position are strictly less likely to form links. Combined with the sparsity scaling r_n = (κ/n)^{1/d}, this ensures that links decay with distance in social space and that the network remains sparse as n grows.

Peer Effects in Consideration and Preferences

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a general nonparametric model of discrete choice in which peers influence agents through two distinct channels: (1) the set of alternatives an agent considers (consideration set effects) and (2) the agent’s preferences over those alternatives (preference effects). The framework embeds these peer mechanisms in a continuous-time Markov process where agents revise choices at Poisson alarm-clock rates. A peer is classified as a consideration peer, a preference peer, or both, and the network is encoded as two directed edge sets rather than one.

The central identification challenge is recovering network structure, consideration probabilities, and preferences simultaneously, without relying on exogenous variation in covariates or the menu of available options. The paper shows this is achievable using time-series variation in the choices made by connected agents. The key insight is that consideration peers who adopt alternative v change the probability that the focal agent considers v — entering only the “consideration” term of the conditional choice probability (CCP) — while preference peers who adopt alternatives other than v change only the “conditional-on-consideration” selection probability. These cross-alternative patterns in the CCPs allow the researcher to distinguish the two channels. Once consideration-only peers are isolated, their choices serve as exclusion restrictions that mimic artificial menu variation, enabling nonparametric recovery of preferences.

Identification proceeds in stages: (i) recover the full reference group of each agent from changes in CCPs; (ii) separate consideration-only peers from preference-affecting peers using cross-order effects across alternatives; (iii) distinguish preference-only peers from consideration-and-preference peers under an exclusion restriction (Assumption 4) requiring that an agent with a dual-channel peer also has at least one single-channel peer; (iv) recover consideration ratios Q(v|n+1)/Q(v|n) and then the full choice rule. The results allow arbitrary heterogeneity across agents and do not require exogenous menu variation or covariate shifters.

For continuous-time data (Dataset 1), the CCPs and Poisson rates are exactly identified from the observed revision history. For discrete-time panel data (Dataset 2), identification is generic under a mild eigenvalue condition on the transition rate matrix.

The empirical application studies store-opening decisions by China’s two dominant high-end tea chains — Heytea and Nayuki — across prefecture-level cities from their founding through end-2020. By that date, Nayuki had 485 stores in 57 cities and Heytea had 729 stores in 46 cities, in an industry whose total revenue grew from 42.2 to 83.1 billion yuan between 2017 and 2020. Each firm-market pair is modeled as an agent deciding whether to open a new store. The key exclusion restriction is that the cumulative store count of either firm in geographically neighboring markets shifts consideration probabilities but does not enter marginal profitability directly.

Estimation via maximum likelihood yields four substantive findings: (1) Firms exhibit limited consideration — consideration probabilities for markets with no prior presence by either firm are substantially below one. (2) Stores in neighboring markets significantly raise consideration probabilities for a given market, for both own-firm and rival stores; this peer effect in consideration is described as economically large. (3) Own-market store density raises marginal profitability (density economies) while rival presence lowers it (competitive effects). (4) A full-consideration model that omits the attention stage overestimates the negative competitive effect and underestimates positive density effects.

Counterfactual simulations show that removing attention constraints (full consideration) accelerates market penetration substantially: firms enter new markets earlier and achieve broader geographic coverage. Removing peer effects in consideration only — while retaining attention constraints — slows the diffusion of store openings across neighboring markets, because peer effects in consideration function as an informational cascade. Limited consideration also reduces competition by delaying rival entry into high-profitability markets, explaining a significant share of the geographic concentration in first- and second-tier cities during the early expansion phase. The paper’s scope is limited to settings with repeated, non-durable choices; it does not model forward-looking behavior or multiple equilibria, which the authors note as directions for future research.

Q: What are the two peer-effect channels in the model, and how do they differ structurally? A: A consideration peer influences whether an alternative enters the agent’s consideration set — specifically, the probability Q_a(v | n) that alternative v is considered is a function of the number n of consideration peers currently adopting v. A preference peer influences the choice rule R_a(v | y, C) — the probability that v is selected conditional on it being in the consideration set. Importantly, the paper models the two channels as affecting logically separate stages of the decision process, so the observed CCP factors into a consideration term and a conditional-selection term that respond to distinct sets of peers.

Q: Why does the standard identification approach of varying menus fail here, and how does the paper substitute for it? A: Menu variation requires the researcher to observe the same agent facing different sets of available alternatives, which is unavailable in many empirical settings. The paper replaces exogenous menu variation with endogenous variation generated by consideration-only peers: when a consideration-only peer adopts alternative v, the focal agent’s probability of considering v rises, effectively mimicking the removal of other alternatives from her consideration set. This peer-induced variation in consideration is then used to trace out the choice rule R_a over counterfactual menus without any actual menu changes.

Q: How does the paper separate consideration peers from preference peers in the data? A: The decomposition exploits an asymmetry in how the two peer types appear in the log-CCP. When a consideration peer switches to alternative v, the term ln Q_a(v | .) changes but the conditional-selection term ln D_a(v | .) remains unchanged, because the agent already considers v. Conversely, when a preference peer adopts an alternative other than v, only the conditional-selection term shifts. The paper formalizes this via cross-order effects of peers across alternatives in the CCPs (Propositions 3.1–3.3) and invokes Assumption 4 — requiring at least one single-channel peer when a dual-channel peer exists — to complete the separation.

Q: What is Assumption 4 and why is it necessary? A: Assumption 4 states that if agent a has a peer in N_CR_a (a peer affecting both consideration and preferences), then a also has at least one additional peer affecting only consideration or only preferences. Without this exclusion restriction, the consideration and preference effects of a dual-channel peer are not separately identified from each other; the single-channel peer provides the variation needed to pin down each component separately.

Q: What does Proposition 2.1 establish and what does it require? A: Proposition 2.1 establishes existence and uniqueness of an invariant equilibrium distribution mu over choice configurations, with full support. It requires Assumptions 1 (independent consideration), 2(i) (strictly positive consideration probability for every alternative), and 3(i) (strictly positive probability of selecting any non-default alternative from some reachable consideration set). The continuous-time Poisson structure ensures zero probability of simultaneous revisions, which rules out multiple equilibria in the data-generating process.

Q: How does the paper handle discrete-time panel data, where only periodic snapshots of choices are observed? A: The paper invokes results from Blevins (2017, 2026) to show that the transition rate matrix W of the continuous-time process is generically identified from the discrete-time transition matrix observed at interval Delta, provided the eigenvalues of W do not differ by integer multiples of 2pii/Delta. Once W is identified, the CCPs P and Poisson rates lambda_a are recovered. This result is described as generic, meaning it holds except on a measure-zero set of parameter values.

Q: What data does the empirical application use, and what are the key sample statistics? A: The application uses city-level store registration data sourced from the National Enterprise Credit Information Publicity System (via CnOpenData, 2021), supplemented by regional statistics from the China City Statistical Yearbook (2016–2021). The sample ends in 2020 to avoid COVID-19 demand shifts. By end-2020, Nayuki had 485 stores across 57 cities and Heytea had 729 stores across 46 cities. The high-end tea industry’s total revenue grew from 42.2 to 83.1 billion yuan between 2017 and 2020.

Q: What is the key exclusion restriction in the empirical specification, and why is it plausible? A: Stores in geographically neighboring markets (parameterized by distance bins d(m,m’)) enter the attention index pi_tilde but are excluded from the marginal profit index pi_bar. The rationale is that nearby store counts are informative signals that draw managerial attention to a market (an informational spillover) but do not directly alter the profitability of operating in that market — profitability depends on local demand, competition within the market, and own firm density, not on activity in adjacent markets. This restriction identifies the consideration-only peer channel.

Q: What does the paper find about biases from ignoring limited consideration? A: When the two-stage model (consideration + choice) is replaced by a single-stage full-consideration model, the estimated payoff parameters differ substantially. Specifically, the full-consideration model overestimates the negative effect of competition (rival presence in the same market) and underestimates the positive effect of own-store density. The intuition is that correlated entry patterns driven by shared consideration spillovers are misattributed to payoff interactions when the consideration stage is omitted.

Q: What do the counterfactual simulations show about the role of limited consideration in market dynamics? A: Three counterfactuals are compared against the baseline. Under full consideration (no attention constraints), market penetration is substantially faster — firms enter new markets earlier and achieve broader geographic coverage. Removing peer effects in consideration while retaining attention constraints slows geographic diffusion because the informational cascade that propagates entry to neighboring markets is eliminated. Limited consideration also reduces competition by delaying rival entry into high-profitability markets; markets with high potential demand remain underserved for longer. Collectively, limited consideration explains a significant portion of the geographic concentration of tea chain stores in first- and second-tier cities during the early expansion period.

Q: What forms of heterogeneity does the identification allow, and what does it not require? A: The nonparametric identification results accommodate arbitrary heterogeneity across agents in consideration mechanisms Q_a, choice rules R_a, Poisson revision rates lambda_a, and network positions. The identification requires neither exogenous covariates that shift preferences or consideration, nor variation in the set of available alternatives across observations. It relies solely on time-series variation in the choices made by connected agents, which are endogenous to the model and are themselves identified in the first stage.

Q: How does the paper model history dependence, and does it change the main identification results? A: Section 4.1 extends the model to allow consideration probabilities and choice rules to depend on the agent’s own choice history h_t in addition to the current configuration y. Proposition 4.1 states that under Assumptions 1–4 applied conditional on both y_{at} and h_t, all identification propositions from Section 3.1 remain valid. The extension also allows consideration probabilities to equal one, enabling nontrivial dynamics in consideration sets driven by past choices.

Q: How is the unobservable default handled in the empirical application? A: When the default alternative (e.g., “do not open a store”) is unobserved, the Poisson revision rate lambda_a cannot be separately identified from the CCPs without normalization. The paper normalizes lambda_a = 1 for each agent in the empirical application, treating the revision opportunity rate as fixed and recovering all remaining primitives under this normalization.

Consideration set: The subset C of the full menu Y that agent a actually attends to at the moment of revision; formed before the choice rule is applied. Alternative v enters C independently with probability Q_a(v | n), where n is the number of consideration peers currently adopting v. The default alternative is always in the consideration set.

Conditional choice probability (CCP): P_a(v | y), the ex-ante probability that agent a selects alternative v given choice configuration y; equal to the product of the consideration probability Q_a(v | .) and the conditional-selection probability D_a(v | .), integrated over all possible consideration sets.

Choice configuration: The vector y = (y_a)_{a in A} recording the current alternative selected by every agent in the network simultaneously; the state variable of the continuous-time Markov process.

Consideration-only peer: A peer a’ in N_C_a \ N_R_a whose choices enter the consideration probability Q_a but not the choice rule R_a. Variation in the choices of consideration-only peers serves as an exclusion restriction that mimics artificial menu variation for identifying preferences.

Preference-only peer: A peer a’ in N_R_a \ N_C_a whose choices enter the choice rule R_a but not the consideration probability Q_a.

Cross-order peer effect: The pattern in the CCP by which a consideration peer’s adoption of alternative v changes ln P_a(v | .) but not the conditional-selection component, while a preference peer’s adoption of a different alternative v’ changes the conditional-selection component but not the consideration component; this asymmetry is the key to separating the two channels.

Limited consideration: The situation in which Q_a(v | n) is strictly less than one for at least some alternatives v and peer counts n, so that the agent does not evaluate all available options before choosing; distinct from full rationality in which all alternatives are always considered.

Mean attention index (pi_tilde): The latent index governing the consideration probability in the empirical specification; it depends on own and rival store counts in the same and neighboring markets and on firm fixed effects, but is excluded from the marginal profit index — constituting the empirical exclusion restriction that separates the consideration and payoff channels.