  Bias correction can often improve the finite sample performance of
estimators. We show that the choice of bias correction method has no effect on
the higher-order variance of semiparametrically efficient parametric
estimators, so long as the estimate of the bias is asymptotically linear. It is
also shown that bootstrap, jackknife, and analytical bias estimates are
asymptotically linear for estimators with higher-order expansions of a standard
form. In particular, we find that for a variety of estimators the
straightforward bootstrap bias correction gives the same higher-order variance
as more complicated analytical or jackknife bias corrections. In contrast, bias
corrections that do not estimate the bias at the parametric rate, such as the
split-sample jackknife, result in larger higher-order variances in the i.i.d.
setting we focus on. For both a cross-sectional MLE and a panel model with
individual fixed effects, we show that the split-sample jackknife has a
higher-order variance term that is twice as large as that of the
`leave-one-out' jackknife.
