Fixed-b asymptotics for panel models with two-way clustering

This paper studies a cluster robust variance estimator proposed by Chiang, Hansen and Sasaki (2024) for linear panels. First, we show algebraically that this variance estimator (CHS estimator, hereafter) is a linear combination of three common variance estimators: the one-way unit cluster estimator, the “HAC of averages” estimator, and the “average of HACs” estimator. Based on this finding, we obtain a fixed-$b$ asymptotic result for the CHS estimator and corresponding test statistics as the cross-section and time sample sizes jointly go to infinity. Furthermore, we propose two simple bias-corrected versions of the variance estimator and derive the fixed-$b$ limits. In a simulation study, we find that the two bias-corrected variance estimators along with fixed-$b$ critical values provide improvements in finite sample coverage probabilities. We illustrate the impact of bias-correction and use of the fixed-$b$ critical values on inference in an empirical example on the relationship between industry profitability and market concentration.