Linear regression models, estimated by ordinary least squares or weighted least squares, are a ubiquitous tool across a diverse range of fields. Classically, inference in such models is based on the assumption that the model's errors are independent and homoskedastic (or more generally, that the errors follow a known structure of a low-dimensional parameter). However, such assumptions will be unreasonable in many applications. Cluster-robust variance estimation methods provide a basis for inference under the weaker assumption that the observations can be grouped into clusters, where the errors in different clusters are independent but errors within a common cluster may be correlated.

Let us consider the linear regression model

$$ y_{ij} = \beta_0 + \beta_1 x_{1ij} + \beta_2 x_{2ij} + \cdots + \beta_{p-1} x_{p-1,ij} + e_{ij}, $$

where

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