Nested and Crossed Cluster Structures

Why the Clustering Structure Matters

Cluster-robust inference allows observations within a cluster to have dependent errors. But “cluster” is not just a label to attach to a data set: the covariance estimator must reflect which observations can share shocks. Students in the same school may share school-level conditions, while students taught by the same teacher may share instructional conditions. If teachers each work at one school, teacher membership is nested within school. If teachers work across multiple schools, school and teacher memberships are crossed.

These distinctions determine which score contributions are grouped together in a covariance estimator. Clustering at too narrow a level can omit dependence shared by a larger unit. Conversely, treating crossed dimensions as if one were nested within the other can omit dependence along the dimension that was not used. The regression coefficients need not change, but their estimated uncertainty can.

Regression Scores and One-Way Clustering

Consider the linear regression model \(y=X\beta+u\), estimated by ordinary least squares (OLS), with \(n\) observations and \(k\) regressors. Write \(x_i\) for the \(k\)-vector of regressors for observation \(i\), and \(\hat{u}_i=y_i-x_i'\hat{\beta}\) for its OLS residual. The observation’s score contribution for the full coefficient vector is the \(k\)-vector \(s_i=x_i\hat{u}_i\). For a cluster \(g\), its score sum is \(S_g=\sum_{i\in g}s_i\).

The one-way cluster-robust covariance estimator groups the score contributions by cluster and takes an outer product of each cluster sum. Let \(G\) denote the number of clusters. Its uncorrected form, often called CR0, is:

The sandwich’s middle term allows arbitrary covariance among observations in the same cluster; the cluster sums retain products between all their score contributions. The estimator relies on a suitable asymptotic argument with many independent clusters and regularity conditions, including that no cluster dominates the information. Finite-sample corrections and degrees-of-freedom choices can matter, particularly when the number of clusters is small; the formula above is used here to show the covariance structure and arithmetic.

Crossed Dimensions and Multiway Clustering

Suppose each observation belongs to one cluster in dimension A and one in dimension B, and neither dimension is nested within the other. A standard two-way clustering assumption allows error dependence between observations that share an A cluster or share a B cluster, with dependence restricted across pairs that share neither. As with one-way clustering, consistent inference needs suitable asymptotics and regularity conditions; in a common two-way framework, both dimensions must have enough clusters.

For each A cluster \(a\), define \(S_a=\sum_{i\in a}s_i\); define \(S_b\) for each B cluster \(b\). An intersection \(a\cap b\) contains observations belonging to both clusters. The two-way CR0 covariance estimator uses inclusion–exclusion:

The A-cluster term counts score products for pairs sharing A; the B-cluster term counts pairs sharing B. Pairs sharing both dimensions appear in both sums, so the intersection term subtracts their duplicate count. The formula includes diagonal products as well: an observation shares both of its own cluster memberships with itself.

Choosing the Dimensions: A Decision Procedure

Nested Clusters and Intersections

If teachers are nested within schools, two-way clustering by school and teacher does not generally add a new dependence dimension beyond school clustering. Each teacher cluster is contained in a school cluster, and the school-teacher intersections are exactly the teacher clusters. Thus the two-way meat simplifies algebraically:

This identity does not imply that teacher-level clustering is adequate when school-wide dependence is plausible. It shows that adding a nested dimension to school clustering does not create a separate crossed dimension in the two-way inclusion–exclusion estimator. With a hierarchy of dependence, choose a level that permits the dependence of concern; clustering at an upper level allows arbitrary dependence among its lower-level units.

Common Mistake and Practical Interpretation

Mistake: “Teachers are the smallest groups, so I should cluster by teacher.” That reasoning assumes errors are independent across teachers. If students taught by different teachers still share school shocks, teacher-only clustering excludes those cross-teacher covariances. Correction: identify the shocks that can jointly affect errors, then choose a covariance estimator whose grouping structure retains the corresponding score cross-products.

A second error is to use a two-way estimator merely because two group identifiers are present. Two-way clustering represents dependence along two crossed dimensions; it is not a general remedy for every hierarchical or complex dependence pattern. State the proposed dependence assumptions, check whether the dimensions are nested or crossed, and then check the cluster counts. For a coefficient contrast, use the relevant quadratic form of the estimated covariance matrix, \(c'\widehat{\mathrm{Var}}(\hat{\beta})c\); the observation score \(x_i\hat{u}_i\) itself is a vector contribution for the full coefficient vector.

The central practical point is that clustering is a statement about the error structure, not just about the sampling frame. One-way clustering is appropriate when independent top-level clusters contain the possible dependence. Crossed clustering is appropriate when more than one nonnested grouping can link observations. In both cases, a covariance estimate can be computed even when the number of clusters is too small for its usual approximation to be dependable.