Clustering, Spatial Correlations and Randomization Inference

It is standard practice in empirical work to allow for clustering in the error covariance matrix if the explanatory variables of interest vary at a more aggregate level than the units of observation. Often, however, the structure of the error covariance matrix is more complex, with correlations varying in magnitude within clusters, and not vanishing between clusters. Here we explore the implications of such correlations for the actual and estimated precision of least squares estimators. We show that with equal sized clusters, if the covariate of interest is randomly assigned at the cluster level, only accounting for non-zero covariances at the cluster level, and ignoring correlations between clusters, leads to valid standard errors and confidence intervals. However, in many cases this may not suffice. For example, state policies exhibit substantial spatial correlations. As a result, ignoring spatial correlations in outcomes beyond that accounted for by the clustering at the state level, may well bias standard errors. We illustrate our findings using the 5% public use census data. Based on these results we recommend researchers assess the extent of spatial correlations in explanatory variables beyond state level clustering, and if such correlations are present, take into account spatial correlations beyond the clustering correlations typically accounted for.