Cross Sectional Regression with Cluster Dependence: Inference based on Averaging

We re-investigate the asymptotic properties of the traditional OLS (pooled) estimator, $\hat{\beta} _P$, in the context of cluster dependence. The present study considers various scenarios under various restrictions on the cluster sizes and number of clusters. It is shown that $\hat{\beta}_P$ could be inconsistent in many realistic situations. We propose a simple estimator, $\hat{\beta}_A$ based on data averaging. The asymptotic properties of $\hat{\beta}_A$ are studied. It is shown that $\hat{\beta}_A$ is consistent even when $\hat{\beta}_P$ is inconsistent. It is further shown that the proposed estimator $\hat{\beta}_A$ is more efficient than $\hat{\beta}_P$ in many practical scenarios. As a consequence of averaging, we show that $\hat{\beta}_A$ retains consistency, asymptotic normality under classical measurement error problem circumventing the use of Instrumental Variables (IV). A detailed simulation study shows the efficacy of $\hat{\beta}_A$. It is also seen that $\hat{\beta}_A$ yields better goodness of fit.