📈 economics

Heterogeneous Peer Effects with Endogenous Network Formation

This paper proposes a novel Bayesian Selection-corrected Heterogeneous Spatial Autoregressive (SCHSAR) framework that jointly models endogenous network formation and heterogeneous peer effects, revealing significant but varied impacts of peer interactions on U.S. firms' R&D investments to inform targeted policy design.

Original authors: Duong Trinh, Santiago Montoya-Blandón

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Duong Trinh, Santiago Montoya-Blandón

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out why some students get better grades than others. You might guess it's because their friends study hard. But here's the tricky part: do the friends make the student smarter, or did the student just choose friends who were already smart?

This is the core puzzle of "peer effects" in economics. For a long time, economists had a hard time solving two big problems at once:

The "Who is who?" problem: People react differently to their friends. Some are easily influenced (like a sponge), while others are stubborn and do their own thing (like a rock). Old models assumed everyone was a sponge or everyone was a rock, which isn't true.
The "Chicken or Egg" problem: People choose their friends based on hidden traits (like personality or ambition) that also affect their success. If you don't account for this, you get the math wrong.

This paper introduces a new tool called SCHSAR (Selection-corrected Heterogeneous Spatial Autoregressive model) to solve both problems at the same time. Here is how it works, using simple analogies:

1. The "Magic Grouping" (Heterogeneity)

Imagine a classroom where the teacher assumes everyone learns the same way. The teacher says, "If the class average goes up, everyone's grade goes up by 10%."

But in reality, the class is a mix of Sponges (who soak up the class energy) and Rocks (who stay the same regardless of the class).

The old models treated everyone as a "Medium Sponge."
This new paper says: "Let's use a magic lens to secretly sort the students into Sponge Groups and Rock Groups without us knowing who is in which group beforehand."
It finds that about 34% of the companies in their study are "Sponges" (very influenced by peers), while 66% are "Rocks" (mostly influenced by their own costs).

2. The "Hidden Magnet" (Endogenous Network Formation)

Now, imagine these students are choosing who to sit next to.

The "Sponges" might secretly prefer sitting near other "Sponges" because they are naturally chatty.
The "Rocks" might prefer sitting near other "Rocks" because they like quiet.
The Trap: If you just look at the grades, you might think the friends caused the grades. But really, the hidden "chattiness" or "quietness" (the hidden magnet) caused them to sit together and determined their grades.
The paper's model acts like an X-ray machine. It looks at the friendship choices and the grades simultaneously to figure out the hidden magnet. It corrects the math so we don't blame the friends for what was actually the student's own hidden personality.

3. The "Detective's Toolkit" (Bayesian Estimation)

Solving this math puzzle is incredibly hard because there are so many hidden variables (who is a sponge? who is a rock? what is the hidden magnet?).

Instead of trying to solve it with one giant, impossible equation, the authors use a Bayesian Detective approach.
Think of it like a game of "Guess Who?" played thousands of times. The computer makes a guess about who is in which group, checks if it fits the data, makes a better guess, and repeats this millions of times.
Eventually, the guesses settle into a clear picture of reality. This allows them to say, "We are 95% sure that Group A is influenced by peers, while Group B is not."

The Real-World Test: U.S. Companies and R&D

The authors tested this tool on 1,150 U.S. companies that trade technology and patents. They wanted to see how government tax breaks for Research & Development (R&D) actually work.

What they found:

The "Peer-Driven" Group (The Sponges): About 34% of companies are highly influenced by what their partners do. If their partners spend more on R&D, these companies follow suit. However, they are less sensitive to their own tax breaks. They are "herd followers."
The "Self-Driven" Group (The Rocks): The other 66% are mostly influenced by their own costs and tax breaks. They don't care as much about what their partners are doing.
The "Hidden Magnet": Companies that are naturally good at making connections (high "social capital") tend to be the ones forming the networks and investing in R&D. If you ignore this, you overestimate how much friends influence each other.

Why This Matters for Policy

The paper shows that a "one-size-fits-all" policy doesn't work.

If the government wants to boost innovation by giving tax breaks, they should target the "Self-Driven" companies because those companies will react directly to the money.
However, if the government wants to spread a new idea through the whole network, they should target the "Peer-Driven" companies (or the central hubs in the network). These companies act like amplifiers; if you give them a nudge, they pass it on to everyone else.

In short: This paper gives economists a better way to separate "influence" from "selection." It proves that people (and companies) aren't all the same, and they choose their friends for hidden reasons. By fixing the math to account for this, we can design better policies that actually work.

Technical Summary: Heterogeneous Peer Effects with Endogenous Network Formation

Problem Statement
The paper addresses two persistent methodological challenges in modeling social and economic interactions: heterogeneous peer effects and endogenous network formation. Existing literature often assumes that peer effects are uniform across all agents or that the network structure is exogenously given. However, in reality, agents differ in how they respond to peers (some amplify, others absorb spillovers), and links are often formed strategically based on unobserved traits that also influence outcomes. Failing to account for these factors leads to severe selection bias and incorrect inferences regarding the magnitude and direction of peer effects. The authors note that while weighted networks can capture interaction intensity, they cannot explain the variation in how agents translate peer behavior into their own decisions. Furthermore, standard spatial autoregressive (SAR) models typically treat the spatial weights matrix as fixed, ignoring the endogeneity arising from unobserved individual characteristics that drive both link formation and outcomes.

Methodology: The SCHSAR Framework
The authors propose a unified econometric framework called the Selection-corrected Heterogeneous Spatial Autoregressive (SCHSAR) model. This approach jointly models network formation and outcome determination through two stages:

Outcome Equation (Heterogeneity): The model extends the standard SAR specification by incorporating a finite mixture structure. Individuals are assigned to one of $G$ latent types, each governed by a distinct set of parameters ( $\lambda_g, \beta_g, \delta_g, \sigma^2_{ug}$ ). This allows peer effects ( $\lambda_g$ ) and other coefficients to vary across unobserved clusters, capturing rich heterogeneity where some agents may have positive, negative, or zero peer responses.
Network Formation Equation (Endogeneity): The adjacency matrix $W$ is modeled as the realization of a strategic decision-making process. A latent utility function determines whether a link $w_{ij}$ exists, driven by observed dyadic characteristics (e.g., homophily) and unobserved individual-specific factors ( $a_i$ ). The function $f(a_i, a_j)$ captures either unobserved degree heterogeneity (additive effects) or unobserved homophily (distance effects).
Joint Modeling and Correction: The core innovation is the inclusion of the same unobserved factor $a_i$ in both the network formation and outcome equations. By explicitly modeling $a_i$ as a latent variable that influences both the probability of forming a link and the individual's outcome, the framework corrects for the selection bias (omitted variable bias) that arises when $W$ is endogenous.

Estimation Strategy
Given the high-dimensional integration required for maximum likelihood estimation (due to latent variables $w^*$ , $z$ , and $a$ ), the authors develop a fully Bayesian data augmentation approach using Markov Chain Monte Carlo (MCMC).

Data Augmentation: The algorithm samples latent variables (individual random effects $a$ , latent type indicators $z$ , and latent link utilities $w^*$ ) alongside model parameters from the joint posterior distribution.
Sampling Scheme: The procedure utilizes Gibbs sampling for most parameters due to conjugate priors. For the peer effect parameters $\lambda_g$ , which do not have a standard posterior form, a Metropolis-within-Gibbs step with Adaptive Scaling Metropolis (ASM) is employed.
Flexibility: The framework accommodates both binary networks (probit link) and weighted networks (Tobit specification) and allows for different specifications of unobserved heterogeneity (degree heterogeneity vs. homophily).

Key Contributions

Unified Framework: The paper bridges the literature on heterogeneous spatial models and network formation econometrics. It nests existing models (e.g., Goldsmith-Pinkham and Imbens 2013; Cornwall and Parent 2017) as special cases (e.g., when $G=1$ or when the network is exogenous) while relaxing their restrictive assumptions.
Addressing Endogeneity without Instruments: Unlike traditional instrumental variable approaches, the SCHSAR model corrects for network endogeneity by modeling the structural source of the bias (unobserved heterogeneity) directly, avoiding the need for external instruments which are often weak or unavailable.
Data-Driven Heterogeneity: Instead of relying on pre-defined observable groups (e.g., gender, industry) to define heterogeneity, the model uses a finite mixture approach to identify latent types based on the data, allowing for more flexible and potentially more accurate grouping of agents.

Empirical Results
The authors apply the SCHSAR model to a dataset of 1,150 U.S. firms involved in a technology collaboration network (1980–2014), analyzing corporate R&D investments.

Network Formation: Results confirm significant homophily in link formation based on industry (SIC) and technology class. Unobserved degree heterogeneity is also a key driver of network structure.
Peer Effects: After correcting for selection bias, the model reveals significant positive but heterogeneous peer effects on R&D.
Latent Types: The data supports a two-type mixture:
- Peer-driven firms (~34%): Exhibit a larger network effect ( $\lambda \approx 0.215$ ) but weaker own-price elasticity ( $\approx -2.2$ ). These firms are more responsive to peers' R&D and less sensitive to their own tax costs.
- Self-driven firms (~66%): Exhibit a smaller network effect ( $\lambda \approx 0.127$ ) but stronger own-price elasticity ( $\approx -9.5$ ). These firms are more sensitive to their own cost structures and less influenced by peers.
Policy Implications: The framework allows for the quantification of firm-level direct, spillin (receiving), and spillout (transmitting) effects. The analysis identifies distinct roles: "responsive absorbers" (high spillin, peer-driven) and "influential transmitters" (high spillout, self-driven but central). Targeting firms with high total spillout effects (often large, central firms in high-tech sectors) is shown to be a cost-effective strategy for maximizing network-wide innovation diffusion.

Significance and Claims
The paper claims that the SCHSAR framework provides a flexible and robust approach for drawing credible conclusions about peer effects in settings where both individual heterogeneity and network endogeneity are present.

Simulation Evidence: Monte Carlo experiments demonstrate that the proposed Bayesian estimators are computationally tractable and yield near-unbiased estimates with nominal coverage, whereas naive approaches (ignoring endogeneity or homogeneity) produce severe bias and unreliable inference.
Policy Design: By disentangling heterogeneous responses and network positions, the model offers valuable insights for evidence-based policy. It suggests that uniform policies may be inefficient and that targeted interventions (e.g., focusing on "transmitters" vs. "absorbers") can optimize innovation diffusion under budget constraints.
Methodological Advancement: The paper positions itself as a necessary evolution in econometric modeling, moving beyond the assumption of exogenous networks and homogeneous agents to better reflect the complex realities of economic and social interactions.