When economists or social scientists try to measure how individuals are influenced by the behaviors or outcomes of their peers, they often turn to linear-in-means models. These models are straightforward in concept: an individual's outcome is modeled as a function of their own characteristics and the average characteristics or outcomes of their peer group. But what happens when we try to push this analysis one step further and use "peers-of-peers" as instruments—that is, the characteristics or behaviors of the friends of someone's friends—to isolate causal peer effects? While this approach sounds promising, it introduces a set of intricate challenges, both in theory and in practice. Understanding these obstacles is crucial for researchers aiming to draw reliable conclusions about social influence.
Short answer: Using peers-of-peers as instruments in linear-in-means models is fraught with challenges, primarily because such instruments can violate key econometric assumptions, lead to reflection problems, and suffer from weak identification due to the underlying network structure and potential loss of crucial information. These issues can undermine the validity of causal inference about peer effects.
Why Do Researchers Use Peers-of-Peers Instruments?
Peer effects are notoriously difficult to identify. If you simply regress an individual's outcome on the average outcome of their peers, you run into the classic "reflection problem": it’s hard to tell whether the peer group influences the individual, the individual influences the peer group, or both are responding to shared external factors. One way around this is to find an instrumental variable—something that affects the peers but not the individual directly.
The idea behind using peers-of-peers as instruments is that the characteristics or actions of a person’s friends’ friends might influence the friend group but not the individual directly. For example, in a classroom, a student’s behavior might be influenced by their immediate classmates, and those classmates could in turn be influenced by their own other friends in the school. If the network is large and well-mapped, it seems plausible to use these second-degree connections as a source of exogenous variation.
The Reflection Problem and Endogeneity
However, as highlighted in the theoretical work discussed by the National Bureau of Economic Research (nber.org), there is a fundamental problem with separating out causal peer effects from correlated group behaviors. The “reflection problem” means that in linear-in-means models, the average outcome of peers is mechanically linked to the individual’s own outcome, creating simultaneity. When you introduce peers-of-peers as instruments, you risk inheriting the same simultaneity or endogeneity that plagues the original peer variable.
The NBER working paper explains that when organizational forms change—such as when a market is replaced by a firm—there is a loss of information from market prices, which cannot be fully replicated by internal transfer prices. This analogy carries over to peer effects: the information structure of the network is crucial. Using peers-of-peers may not provide truly independent variation if the network is dense or highly interconnected, because the "information from market prices is lost" as you move to more indirect connections (nber.org). This weakens the power of peers-of-peers as valid instruments.
Instrument Validity: Exclusion Restriction and Weak Instruments
For an instrument to be valid, it must satisfy the "exclusion restriction": it should affect the outcome only through its effect on the endogenous peer variable, not directly. In practice, peers-of-peers may violate this. For example, in a school setting, a friend’s friend might also be a direct classmate or interact with the individual in other ways, contaminating the source of variation. If the network is not perfectly mapped or contains overlapping connections, the "peers-of-peers" variable may not be sufficiently independent, undermining the exclusion restriction.
Moreover, as discussed in the context of market versus firm organization by NBER, when a system centralizes authority (akin to centralizing information in a network), "information from market prices is lost... [and] cannot be replicated by internally generated transfer prices." This is similar to the problem of "weak instruments" in peer effect estimation: as you move to more remote network links, the signal provided by peers-of-peers becomes weaker, reducing the statistical power to identify causal effects.
The structure of the social network is critical. In sparse networks, a person’s friends may have few connections that are not shared with the individual, making it difficult to find suitable peers-of-peers who are not also direct peers. In dense networks, almost everyone is connected, so the distinction between peers and peers-of-peers blurs. According to nber.org, when markets (or information sources) are "closed," there is a "loss of information... that can be used to reduce the cost of contracting." In the context of networks, this means that using more distant network members as instruments can strip away useful variation, especially if the network is not well understood or is highly clustered.
Agency Issues and Coordination
The NBER paper also touches on agency issues: when principals (decision makers) cannot directly observe or contract with all agents (participants), they must rely on intermediaries, which introduces coordination challenges. In network models, using peers-of-peers as instruments is analogous to relying on indirect information to coordinate or infer effects. If the structure is poorly specified, or if there is hidden influence running through other channels, the estimated peer effect may be biased or misleading.
Practical Data Challenges
Empirically, gathering accurate data on peers-of-peers is much more demanding than mapping immediate peer groups. If the network is large or not fully observed, any error in identifying connections can further degrade the quality of the instrument. This is compounded by potential measurement error in defining who counts as a peer or a peer-of-peer, which can introduce additional bias.
Summary and Broader Implications
To wrap up, the use of peers-of-peers as instruments in linear-in-means models is theoretically appealing but riddled with practical and conceptual difficulties. The reflection problem, weak instrument issues, violation of the exclusion restriction, and loss of information due to network structure all conspire to make causal inference challenging. As emphasized in the NBER working paper, the choice of information structure—whether in markets, firms, or social networks—has profound implications for the reliability of inference. The analogy that "information from market prices is lost" when moving to internal systems holds true for social networks as well: moving to more indirect measures (like peers-of-peers) can mean losing the very variation needed for robust identification.
While sciencedirect.com and aeaweb.org did not provide additional relevant detail for this specific question, the insights from nber.org are clear: researchers must be wary of over-relying on peers-of-peers instruments without careful attention to network structure, the validity of the exclusion restriction, and the potential loss of information. Otherwise, the estimated peer effects may tell us less about actual social influence and more about the limitations of our econometric tools.
