  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 from Thompson
(2011) on the relationship between industry profitability and market
concentration.
