  While batching methods have been widely used in simulation and statistics, it
is open regarding their higher-order coverage behaviors and whether one variant
is better than the others in this regard. We develop techniques to obtain
higher-order coverage errors for batching methods by building Edgeworth-type
expansions on $t$-statistics. The coefficients in these expansions are
intricate analytically, but we provide algorithms to estimate the coefficients
of the $n^{-1}$ error term via Monte Carlo simulation. We provide insights on
the effect of the number of batches on the coverage error where we demonstrate
generally non-monotonic relations. We also compare different batching methods
both theoretically and numerically, and argue that none of the methods is
uniformly better than the others in terms of coverage. However, when the number
of batches is large, sectioned jackknife has the best coverage among all.
