Inference about clustering and parametric assumptions in covariance matrix estimation

Selecting an estimator for the covariance matrix of a regression's parameter estimates is an important step in hypothesis testing. From less to more robust estimators, the choices available to researchers include Eicker/White heteroskedasticity-robust estimator, cluster-robust estimator, and multi-way cluster-robust estimator. The rationale for choosing a less robust covariance matrix estimator is that tests conducted using this estimator can have better power properties. This motivates tests that examine the appropriate level of robustness in covariance matrix estimation. In this paper, we propose a new robustness testing strategy, and show that it can dramatically improve inference about the proper level of robustness in covariance matrix estimation. In an empirically relevant example, namely the placebo treatment application of Bertrand, Duflo and Mullainathan (2004), the power of the proposed robustness testing strategy against the null hypothesis "no clustering" is 0.82 while the power of the existing robustness testing approach against the same null is only 0.04. We also show why the existing clustering test and other applications of the White (1980) robustness testing approach often perform poorly, which to our knowledge has not been well understood. The insight into why this existing testing approach performs poorly is also the basis for the proposed robustness testing strategy.