Identifying Peer Effects from Network Variation

Why Network Variation Matters

In a peer-effects model, an individual's outcome may depend on both the outcomes and the observed characteristics of connected peers. These two sources of dependence pose different identification problems. Peer characteristics can help identify a peer-characteristic effect only if their network averages vary independently of the other included regressors. Peer outcomes are different: because outcomes are jointly determined, a peer's outcome is generally correlated with the structural disturbance. Identifying that effect requires valid excluded instruments as well as sufficient instrument relevance.

The network describes who is connected to whom, but its existence alone does not guarantee identification. A network can produce peer averages that are constant, or that duplicate variation already captured by own characteristics. Likewise, a plausible network-based instrument can fail to distinguish a peer-outcome regressor from the included controls. The key questions are therefore about variation, exclusion, and rank.

A Linear Network Model and Its Conditions

Let \(n\) individuals be connected by a known \(n\times n\) matrix \(W\). Assume \(W\) has zero diagonal, so individuals are not their own peers, \(W_{ij}\geq 0\) for all \(i,j\), and each row is normalized to sum to one. Thus, \((Wy)_i\) is the average outcome among individual \(i\)'s peers, and \((Wx)_i\) is the average observed characteristic among those peers. Consider the model

y=\rho Wy+\theta Wx+X\beta+\varepsilon.

Here \(y\) is the vector of outcomes, \(x\) is one observed individual characteristic, \(X\) contains the included individual characteristics and any other included controls, \(\rho\) is the peer-outcome coefficient, and \(\theta\) is the peer-characteristic coefficient. The vector \(\varepsilon\) contains structural disturbances. The \(Wx\) term is displayed separately to make its role clear; if \(x\) is also a column of \(X\), it is included there as well.

Identification setup: Treat \(W\) as known and fixed for the analysis. Assume \(I-\rho W\) is nonsingular, the included regressors have the required variation, and \(E[\varepsilon\mid W,X]=0\). To use network transformations of \(x\) as excluded instruments, also assume they have no direct effect on \(y\) beyond the terms included in the structural equation. These are substantive assumptions; network algebra alone cannot establish them.

The nonsingularity condition ensures that the simultaneous system has a well-defined reduced form. It is distinct from the instrument conditions: a stable, well-defined model can still be unidentified if its regressors do not have the necessary independent variation or if its instruments are invalid.

The coefficient \(\theta\) is identified by variation in \(Wx\) that remains after accounting for the other included regressors, including own characteristics. If \(Wx\) is a constant or is an exact linear combination of the intercept and other included regressors, the data cannot separate its coefficient from theirs. In the simplest case, this requires that \(Wx\) not lie in the span of the intercept and \(X\).

The coefficient \(\rho\) creates a different difficulty. Since \(Wy\) is built from outcomes generated in the same system, it is generally correlated with \(\varepsilon\). Ordinary least squares does not resolve that simultaneity. Instrumental variables can identify \(\rho\) only if the excluded instruments both satisfy an exclusion restriction and predict \(Wy\) after the included regressors have been accounted for.

Why Network Transformations Can Be Instruments

A common candidate instrument is \(W^2x\), the average of peers' peer-characteristics under the two-step network weighting. More generally, one may consider \(W^k x\) for \(k\geq 2\), or transformations of other exogenous characteristics. Such variables are not automatically valid instruments. Their proposed validity rests on two claims: they are excluded from the structural equation except through the included regressors, and they are uncorrelated with the structural disturbance. If a second-order peer characteristic directly affects outcomes, for example, excluding \(W^2x\) would be inappropriate unless that direct pathway is modeled.

The relevance of these candidates is also not automatic. A network can make \(W^2x\) constant or nearly predictable from the included regressors. It can also leave \(W^2x\) with little association with the part of \(Wy\) that is not explained by those regressors. The relevant association is conditional on the included regressors, not merely the raw correlation between two network variables.

Rank condition: Let \(Q\) denote the matrix of included exogenous regressors, and let \(Z\) collect excluded instruments. For a single endogenous regressor \(Wy\), identification requires that at least one excluded-instrument direction have a nonzero population cross-moment with \(Wy\) after partialling out \(Q\). For \(m\) endogenous regressors, the matrix of these partialled-out cross-moments must have rank \(m\).

To state the condition precisely, write \(\tilde Z\) and \(\tilde D\) for the excluded instruments and endogenous regressors after removing their linear projections on \(Q\). The population rank condition is

\operatorname{rank}\left(E[\tilde Z^{\mathsf T}\tilde D]/n\right)=m,

where \(m\) is the number of endogenous regressors. With one endogenous regressor, this condition says that the excluded instruments, taken together, must have some direction with a nonzero population cross-moment with that regressor. Redundant instruments do not invalidate identification: they add no new information, but a nonzero relevant direction can still identify the single coefficient.

Worked Examples

Worked Example: Peer-Characteristic Variation Can Disappear

Consider four individuals with characteristic values \(x=(0,0,1,1)^{\mathsf T}\). In one hypothetical network, the links are \(1\!-\!2\), \(1\!-\!3\), \(2\!-\!4\), and \(3\!-\!4\). Each individual has two peers. The resulting peer-characteristic averages are

Wx=(0.5,0.5,0.5,0.5)^{\mathsf T}.

Because \(Wx=0.5\mathbf{1}\), it is exactly a multiple of the intercept. After removing the intercept, the residualized peer-characteristic average is zero. Thus the data cannot identify a separate \(\theta\) in this network, even though individuals have different \(x\) values and every person has peers.

Now consider a line network with links \(1\!-\!2\), \(2\!-\!3\), and \(3\!-\!4\), with the same \(x\). Each person's peer average is the average over their adjacent neighbors, so

Wx=(0,0.5,0.5,1)^{\mathsf T}.

To check whether this provides variation beyond an intercept and \(x\), first subtract the overall mean of \(Wx\), which is \(0.5\). Within the \(x=0\) group, the mean of \(Wx\) is \(0.25\); within the \(x=1\) group, it is \(0.75\). The residuals after projecting on the intercept and \(x\) are therefore

\widetilde{Wx}=(-0.25,0.25,-0.25,0.25)^{\mathsf T}.

These residuals are not all zero, so the peer-characteristic average is not collinear with the intercept and \(x\). This is a necessary variation condition for identifying \(\theta\) in this simplified design. It does not, by itself, establish that the full model is identified or that any instrument for \(Wy\) is valid.

Worked Example: Checking Relevance and Computing an IV Estimate

Suppose a hypothetical analysis includes an intercept, \(x\), and \(Wx\) as included regressors, and proposes \(W^2x\) as an excluded instrument for \(Wy\). After projecting all variables on the included regressors, suppose the residualized instrument, endogenous regressor, and outcome are as follows. These are illustrative residualized values, not observations from a real study.

Individual	\(\tilde z\)	\(\widetilde{Wy}\)	\(\tilde y\)
1	-1	-2	-3
2	0	1	1.5
3	1	2	3
4	0	-1	-1.5

The sample cross-product for relevance is

\sum_{i=1}^4 \tilde z_i\widetilde{(Wy)}_i =(-1)(-2)+(0)(1)+(1)(2)+(0)(-1)=4.

The nonzero cross-product shows that this instrument is relevant in this residualized sample. With one excluded instrument and one endogenous regressor, the just-identified IV estimate for \(\rho\), after partialling out the included regressors, is the ratio of the instrument-outcome cross-product to the instrument-regressor cross-product:

\hat{\rho}_{IV} =\frac{\sum_i\tilde z_i\tilde y_i} {\sum_i\tilde z_i\widetilde{(Wy)}_i} =\frac{(-1)(-3)+(0)(1.5)+(1)(3)+(0)(-1.5)}{4} =\frac{6}{4}=1.5.

The arithmetic gives an IV coefficient of \(1.5\) in the outcome units per unit of peer outcome. The calculation does not prove that \(W^2x\) is exogenous or excluded. Those claims require the model assumptions and substantive knowledge of the setting; nor does this small illustration provide a basis for inference.

Worked Example: Redundant Instruments and the Rank Condition

Suppose there is one endogenous regressor, and after partialling out included regressors its values are \(\tilde d=(-1,0,1)^{\mathsf T}\). Two proposed excluded instruments happen to be identical: \(\tilde z_1=\tilde z_2=(-1,0,1)^{\mathsf T}\). Their cross-moments with the endogenous regressor are

\frac{1}{3} \begin{bmatrix} \tilde z_1^{\mathsf T}\tilde d & \tilde z_2^{\mathsf T}\tilde d \end{bmatrix} = \frac{1}{3} \begin{bmatrix} 2 & 2 \end{bmatrix}.

This row matrix has rank one, which is the required rank for one endogenous regressor. The instruments are redundant, but identification does not fail: either identical instrument supplies a nonzero relevant direction. By contrast, if both residualized instruments were orthogonal to \(\tilde d\), the cross-moment row would be \([0\;\;0]\), of rank zero, and the endogenous coefficient would not be identified by them.

For two endogenous regressors, the requirement is stronger: the cross-moment matrix must have rank two. A matrix such as

\begin{bmatrix} 2&2\\ 2&2 \end{bmatrix}

has rank one, so it cannot identify two separate endogenous coefficients. The issue is insufficient independent instrument information for the endogenous regressors, not merely that some instrument columns are linearly dependent.

A Short Justification of the Identification Conditions

The rank condition follows from the IV moment equations. After partialling out the included regressors, write the relevant structural equation as \(\tilde y=\tilde D\gamma+\tilde\varepsilon\), where \(\gamma\) collects the coefficients on the endogenous regressors. If the instruments are valid, their population moments satisfy \(E[\tilde Z^{\mathsf T}\tilde\varepsilon]/n=0\). Consequently,

E[\tilde Z^{\mathsf T}\tilde y]/n = E[\tilde Z^{\mathsf T}\tilde D]/n\;\gamma.

If the cross-moment matrix has rank \(m\), it has enough independent directions to determine the \(m\) entries of \(\gamma\), subject to the stated model and moment conditions. If its rank is less than \(m\), there is a nonzero vector \(a\) such that \(E[\tilde Z^{\mathsf T}\tilde D]a/n=0\). Then changing the coefficient vector in direction \(a\) does not change these instrument moments, so those moments cannot distinguish all the coefficients. This establishes why rank is necessary and, for the linear moment system, why full column rank gives unique identification from those moments.

In the model above, \(\rho\) is the coefficient on the endogenous regressor \(Wy\). Network transformations of exogenous characteristics can supply the excluded-instrument directions, but only if their partialled-out cross-moments with \(Wy\) satisfy the rank condition. The exogeneity moment condition and the rank condition do different jobs: the first rules out systematic correlation with the structural error, while the second ensures enough predictive variation to identify the coefficient.

Common Pitfalls and Practical Interpretation

A frequent error is to conclude that a peer effect is identified because the network is irregular, because \(Wx\) varies, or because an instrument is labeled “network-based.” None of these facts alone is enough. For \(\theta\), check whether \(Wx\) retains variation after accounting for included regressors. For \(\rho\), assess both instrument validity and the rank condition after partialling out those regressors. A visually complex network can still yield a constant peer average, and a variable \(W^2x\) can still be irrelevant to the residualized \(Wy\).

Another mistake is to treat any linear dependence among excluded instruments as an identification failure. Redundant instruments do not add independent information, but they do not erase a relevant direction already present. With one endogenous regressor, identification fails for this reason only if every excluded-instrument direction has zero population cross-moment with it after residualization. With several endogenous regressors, the relevant question is whether the full cross-moment matrix has enough rank for all of them.

Finally, the exclusion restriction deserves substantive scrutiny. If a second-order network characteristic affects outcomes directly, \(W^2x\) may not be excluded. Including additional network terms may address that direct pathway, but it can also change which network transformations remain available as excluded instruments. Identification is therefore a property of the specified structural model, the network, and the assumed moment conditions together—not of a network matrix in isolation.

Key takeaway: Peer-characteristic effects need independent residual variation in peer characteristics. Peer-outcome effects need valid excluded instruments and a full-rank relationship between those instruments and the endogenous peer outcomes. Redundant instruments are not by themselves fatal; insufficient rank is.

Check Your Understanding

Use the model and conditions developed above to answer the following questions.

Why does a constant \(Wx\) fail to identify a separate peer-characteristic coefficient when the model includes an intercept?
What two distinct conditions must a proposed \(W^2x\) instrument satisfy to help identify the coefficient on \(Wy\)?
With one endogenous regressor, why can two identical but relevant excluded instruments still satisfy the rank condition?
What does a rank-one cross-moment matrix imply when there are two endogenous regressors?
Why does finding a nonzero sample cross-product between a residualized instrument and \(Wy\) not establish the instrument's exclusion restriction?