Refined Cluster Robust Inference

It has become standard for empirical studies to conduct inference robust to cluster dependence and heterogeneity. With a small number of clusters, the normal approximation for the $t$-statistics of regression coefficients may be poor. This paper tackles this problem using a critical value based on the conditional Cram\'er-Edgeworth expansion for the $t$-statistics. Our approach guarantees third-order refinement, regardless of whether a regressor is discrete or not. The critical value is a closed-form function of the estimated score skewness and kurtosis. Simulations show that our proposal can make a difference in size control with as few as 10 clusters. Keywords: Cluster robust inference, Cram\'er-Edgeworth expansion, Asymptotic refinement