TLDR: Clustered standard errors are a way to estimate the uncertainty in statistical models when observations are grouped into clusters and there is correlation within each cluster. They are commonly used in econometrics and other fields to account for clustering effects.

Clustered standard errors are used when observations are grouped into clusters, such as classrooms or geographic regions, and there is correlation within each cluster. This can happen when, for example, a new teaching technique is implemented in some classrooms but not others, and student test scores within each classroom are not independently distributed. In such cases, traditional standard errors may not be appropriate because they assume independence between observations. Clustered standard errors take into account the correlation within clusters and provide more accurate estimates of the uncertainty in statistical models.

These standard errors are similar to other types of standard errors, such as Huber-White standard errors for heteroscedasticity and Newey-West standard errors for autocorrelation. They are consistent in the presence of cluster-based sampling or treatment assignment. Clustered standard errors are often used in applied econometric settings, including difference-in-differences or experiments.

Mathematically, clustered standard errors can be derived using a "sandwich" estimator. This estimator takes into account the block-diagonal structure of the covariance matrix, where each block corresponds to a cluster. By constructing plug-in matrices for the within-cluster analogues of the covariates and residuals, an estimator for the clustered standard errors can be obtained. The number of clusters needed for reliable estimation depends on the specific context, but a common rule of thumb is around 30-50 clusters.

While the precise justification for clustering is still a topic of debate, clustered standard errors are widely used in practice to address the correlation within clusters and provide more accurate estimates of the uncertainty in statistical models.

In summary, clustered standard errors are a way to estimate the uncertainty in statistical models when observations are grouped into clusters and there is correlation within each cluster. They are commonly used in econometrics and other fields to account for clustering effects and provide more accurate estimates of the uncertainty in statistical models.