Why Link Formation Matters for Peer Effects
A peer-effect estimate depends not only on outcomes and characteristics, but also on which individuals are treated as one another’s peers. If people choose their friends, coworkers, or discussion groups partly on the basis of traits that also affect outcomes, the observed network is not an innocuous sampling arrangement. The estimated relationship between peer exposure and an outcome can then combine a causal peer effect with selection into links.
This issue is distinct from the simultaneous feedback problem in linear-in-means models. Reflection concerns how outcomes can be jointly determined within a given network. Endogenous network formation concerns how that network came to be observed, and whether link formation is related to unobserved outcome determinants. Both problems can arise at once.
A Model That Separates the Main Sources of Dependence
Let \(g_{ij}\) indicate whether individual \(i\) is linked to individual \(j\), with \(g_{ii}=0\). Let \(G\) denote the matrix of links; when rows are normalized, \(Gy\) is the vector of average linked-peer outcomes. A common linear-in-means outcome equation is
Here \(y\) is the vector of outcomes, \(X\) contains observed individual or contextual characteristics, and \(u\) collects unobserved outcome determinants. The coefficient \(\beta\) represents a structural response to peers’ outcomes under the model’s assumptions. The equation alone does not ensure that \(\beta\) is identified: \(Gy\) is generally jointly determined with \(y\), and the links in \(G\) may themselves be selected.
Three mechanisms are especially important. Homophily is the tendency for individuals with similar characteristics to connect. If an unobserved trait such as motivation affects both link choices and outcomes, linked peers may have similar outcomes even if one person’s outcome does not causally affect another’s. Shared shocks arise when connected individuals face the same unobserved environment, such as a workplace policy change or a local disruption. Endogenous link formation is the broader selection process in which link choices depend on observed or unobserved factors that also predict outcomes. These mechanisms overlap, but they are not identical.
The key concern can be written as a conditional moment condition. For a peer exposure \(q_i\), such as a peer characteristic or peer treatment, a regression interpretation requires a condition like
If link formation makes \(q_i\) related to \(u_i\), this condition fails. Adding observed controls can address selection on those measured characteristics, but it does not automatically remove selection on latent traits or unmeasured shocks. Nor does a small standard error repair a failure of this condition: standard errors quantify sampling uncertainty around an estimator, not whether its identifying assumptions hold.
How Homophily Enters the Estimate
A simple omitted-variable calculation isolates the selection component. Suppose an outcome satisfies \(y_i=\beta q_i+\gamma z_i+e_i\), where \(z_i\) is an unobserved trait and \(E[e_i\mid q_i,z_i]=0\). If \(y_i\) is regressed on \(q_i\) alone, the population slope is
The second term is the bias relative to \(\beta\). Homophilous link formation can produce \(\operatorname{Cov}(q_i,z_i)\ne 0\): for example, individuals with high motivation may connect to similarly motivated peers, making their peer exposure predictive of their own motivation. The sign of the bias depends on both the relationship between \(z_i\) and outcomes, represented by \(\gamma\), and the relationship between \(q_i\) and \(z_i\). It need not always be positive.
Worked Example: Homophily and an Inflated Peer-Exposure Slope
Consider four hypothetical individuals. Let \(q_i\) be an observed peer exposure and \(z_i\) an unobserved high-motivation indicator. Suppose outcomes follow \(y_i=2q_i+3z_i\), so the causal slope on exposure is 2. The data are:
| Individual | \(q_i\) | \(z_i\) | \(y_i=2q_i+3z_i\) |
|---|---|---|---|
| A | 0 | 0 | 0 |
| B | 1 | 0 | 2 |
| C | 2 | 1 | 7 |
| D | 3 | 1 | 9 |
Suppose, for illustration, that the observed exposure is associated with motivation because individuals tend to form links with others like themselves. Regressing \(y_i\) on \(q_i\) alone gives \(\bar q=1.5\) and \(\bar y=4.5\). The ordinary least-squares slope is
The regression slope is 3.2, not the structural effect 2. The omitted-variable expression gives the same result: \(\sum_i(q_i-\bar q)(z_i-\bar z)=2\), so the extra component is \(3(2/5)=1.2\), and \(2+1.2=3.2\). This small example isolates homophily-related confounding; it is not a complete simultaneous network-lag model.
Shared Shocks and Changing Networks
A common shock can make linked peers’ outcomes move together without either peer causing the other’s outcome. If a group is exposed to an unobserved local event, its members’ outcomes may share a component \(c\). When peer exposure also varies with that event, a regression that omits \(c\) attributes some of the shared movement to peer influence.
Group fixed effects can remove a time-invariant group component under the relevant panel assumptions, but they do not eliminate time-varying group shocks. They also do not automatically solve selection into links that changes over time. For example, individuals may form new links in response to changing health, employment, or expectations, all of which may also predict later outcomes. Controlling for observed history helps only if the remaining conditional independence assumptions are credible.
Worked Example: A Shared Shock Masquerading as Peer Influence
Suppose four hypothetical community-period observations have peer exposure \(q=(0,1,1,2)\). A common local shock is \(c=(0,1,0,1)\), and the outcome is \(y=2q+3c\). The true coefficient on peer exposure is 2. The resulting outcomes are \(y=(0,5,2,7)\).
If the regression omits the shock, then \(\bar q=1\) and \(\bar y=3.5\). Its slope is
The slope exceeds 2 because the omitted shock is associated with peer exposure in these observations. This is a demonstration of confounding, not evidence that every common shock biases an estimate in the same direction. If the shock were conditionally unrelated to exposure, this particular omitted-variable component would be zero.
The general diagnostic question is whether the source of variation in peer exposure is independent of unobserved outcome determinants. A network can be observed without error and still be endogenous. Conversely, link formation need not invalidate an estimate if the relevant selection is fully accounted for by credible controls or design assumptions.
What Can Support Credible Identification?
A causal interpretation requires a source of variation in peer exposure that is not itself generated by unobserved outcome determinants. The appropriate argument depends on the setting; no estimator can substitute for stating and defending its identifying assumptions.
- Conditional exogeneity of the network. One may assume that, conditional on observed covariates and design variables, link formation is unrelated to unobserved potential outcomes. This is a substantive assumption, not a consequence of including many controls. Its credibility depends on what determined links and outcomes.
- Randomized link or peer assignment. Randomly assigning peers, groups, or opportunities to connect can generate exogenous variation. The assignment mechanism, the relevant exposure mapping, and the possible spillovers must be specified. Randomizing an invitation is not equivalent to randomizing realized links if individuals choose whether to accept.
- Instrumental-variable variation. An instrument can shift link formation or peer exposure while satisfying relevance, exclusion, and independence conditions. Exclusion requires that the instrument affect the outcome only through the defined exposure; independence requires that it be unrelated to unobserved outcome determinants. These conditions need substantive justification.
- Panel or natural-experiment variation. Repeated observations or policy-driven network changes may help separate persistent individual differences from exposure changes. This strategy still requires that time-varying unobservables do not jointly determine the change in links and outcomes, or that the design otherwise addresses them.
The earlier material on identification with endogenous spatial regressors and identifying peer effects from network variation provides related identification tools. For this topic, the essential additional question is whether the network variation those tools use can be treated as exogenous to the outcome disturbances. A strong first stage or substantial network variation is not enough if the variation is produced by unobserved selection.
Worked Example: A Randomized Link Encouragement as an Instrument
Suppose a hypothetical program randomly assigns an encouragement \(Z_i\) to connect with a study partner. Let \(D_i\) indicate whether the individual actually receives the specified peer exposure, and let \(Y_i\) be a test score in points. The group means are:
| Encouragement assignment | Mean exposure \(D_i\) | Mean score \(Y_i\) |
|---|---|---|
| \(Z_i=1\) | 0.70 | 5.6 |
| \(Z_i=0\) | 0.30 | 4.4 |
Before interpreting a Wald estimate, require that assignment is randomized, that it changes the probability of the defined exposure, that it has no outcome effect except through that exposure, and that there is no defier behavior (monotonicity). In a network setting, also define the exposure mapping and assume it captures all outcome-relevant spillovers: the encouragement must not change other relevant links or exposures in an unaccounted-for way.
The first-stage difference is \(0.70-0.30=0.40\), and the reduced-form difference in scores is \(5.6-4.4=1.2\) points. The Wald ratio is
Under the stated assumptions and a binary exposure, this is interpreted as an estimated local average causal effect of the defined exposure for individuals whose exposure status is changed by the encouragement: 3.0 score points. It is not automatically the average effect for everyone, nor does randomization of encouragement alone guarantee the exclusion and interference assumptions.
Common Mistakes and Practical Checks
- Calling observed similarity a peer effect. Similar outcomes among linked people may reflect homophily or shared shocks. Similarity alone does not distinguish causation from selection.
- Assuming controls solve all selection. Controls address only the variation they measure and model appropriately. An unobserved trait that affects both links and outcomes can remain a source of bias.
- Confusing link exogeneity with outcome-model identification. Even an exogenous network does not, by itself, resolve simultaneous determination of peer and own outcomes. Conversely, an approach to simultaneous feedback does not establish that link formation is exogenous.
- Treating random encouragement as random realized exposure. Noncompliance can be handled only under additional assumptions, including exclusion and monotonicity for the stated instrumental-variable interpretation.
- Using robust standard errors as a bias correction. An appropriate variance estimator can affect uncertainty estimates, but it does not make an endogenous peer exposure exogenous. Inference under network dependence is a separate issue.
A useful analysis therefore reports the formation process, the proposed source of identifying variation, and the assumptions connecting that variation to the causal effect. Sensitivity to alternative network definitions may be informative, but stability across definitions does not prove exogeneity. The central question remains whether, under the design and controls used, the variation in peer exposure is plausibly unrelated to unobserved outcome determinants except through the exposure of interest.
Check Your Understanding
Use the distinctions and assumptions in this tutorial to answer the following questions.
- How can homophily create bias in a regression of outcomes on peer exposure even when there is no causal peer effect?
- In the omitted-variable calculation, what determines the sign of the bias term?
- Why might group fixed effects fail to remove confounding from a time-varying shared shock?
- In the encouragement example, which assumptions are needed to interpret the Wald ratio as a local average causal effect?
- Explain why randomizing an invitation to form a link does not necessarily randomize the realized network.
- Why can network-robust standard errors not correct bias caused by endogenous link formation?