Clusterrobust

Clusterrobust standard errors, also known as cluster-robust covariance estimators, are used in regression analysis to obtain valid standard errors when observations are grouped into clusters and errors may be correlated within those clusters. The approach preserves the assumption that observations from different clusters are independent, while allowing arbitrary forms of dependence inside each cluster. This makes clusterrobust methods particularly suitable for data with hierarchical or panel structures, such as students within schools, employees within firms, or repeated measurements within individuals.

The core idea is a sandwich-type variance estimator that aggregates the within-cluster information to adjust the estimated coefficient variability. In ordinary least squares, the cluster-robust covariance matrix can be written as V_hat_cr = (X'X)^{-1} [sum over clusters of X_c' u_c u_c' X_c] (X'X)^{-1}, where X_c and u_c are the design matrix and residual vector for cluster c. This formulation allows residuals to be heteroskedastic and correlated within clusters, while treating clusters as independent units. The resulting standard errors are then used for confidence intervals and hypothesis tests.

Applications of clusterrobust standard errors are widespread in econometrics and social sciences, especially with panel data, clustered sampling, or any data with natural groupings. A key consideration is the number of clusters: with a small number of clusters, the standard errors can be biased downward, leading to overconfident inferences. Practitioners may employ finite-sample corrections, bootstrap methods, or alternative estimators such as multiway clustering or Driscoll-Kraay corrections when there is cross-sectional dependence.

Software implementations are common in statistical packages, enabling users to request cluster-robust standard errors by specifying the clustering variable. The concept remains a practical default when intra-cluster correlation is plausible and the clustering structure is well-defined.