16.5 Cluster-robust standard errors

Whatever the estimator, the disturbances of the same unit are usually correlated over time, through \(\alpha _i\) if it is left in the error, or through serial correlation of \(\varepsilon _{it}\). Treating the \(NT\) observations as independent then gives standard errors that are too small. The remedy generalises White’s estimator to clusters: with independence across units but arbitrary correlation within them, \[ \widehat \Var _{\text {CL}}(b) = (X'X)^{-1}\Bigl (\sum _{i=1}^NX_i'\hat u_i\hat u_i'X_i\Bigr )(X'X)^{-1}, \] where \(X_i\) and \(\hat u_i\) are the rows and residuals of unit \(i\). It is consistent as \(N \to \infty \) with \(T\) fixed, by the argument of Theorem 10.2.1 applied to the \(N\) independent cluster sums \(X_i'u_i\). It requires many clusters: with few, it is unreliable and usually too small. How much the conventional formula understates the variance depends on the correlation of the regressor within clusters as well as that of the disturbance. With \(T\) observations per cluster, intraclass correlation \(\rho _u\) of the disturbances and \(\rho _x\) of the regressor, the variance ratio is approximately the Moulton factor \(1 + (T - 1)\rho _x\rho _u\) (Moulton 1990), which can be large when the regressor is constant within clusters, as for a policy that applies to a whole state or school.

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