We show that the blockwise bootstrap approximation for the distribution of a studentized statistic computed from dependent data is second-order correct provided we choose an appropriate variance estimator. We also show how to adapt the $BC_a$ confidence interval of Efron to the a dependent case. For the proofs we extend the results of Götze and Hipp on the validity of the formal Edgeworth expansion for a sum to the studentized mean.
"Second-order correctness of the blockwise bootstrap for stationary observations." Ann. Statist. 24 (5) 1914 - 1933, October 1996. https://doi.org/10.1214/aos/1069362303