The Stationary Bootstrap

A plain block bootstrap resamples a return series by cutting it into fixed-length chunks — say, blocks of 20 days — and gluing randomly chosen blocks back together to build a fake history that still has the right day-to-day dependence within each block. The awkward part is that a fixed block length creates a seam: the resampled series isn't really a stationary process, because the statistical behavior right at a block boundary differs subtly from the behavior in the middle of a block, and results can be sensitive to exactly which length you picked.

The stationary bootstrap fixes this by making the block length itself random. Instead of always cutting 20-day blocks, each block's length is drawn from a geometric distribution with some average length, so blocks vary — some short, some long — and the resulting resampled series is provably stationary (its statistical properties don't depend on where you start looking), which the fixed-block version isn't. In practice you pick only the average block length, not an exact one, which turns out to matter less for the final answer than the exact length choice does in the plain block bootstrap.

The average block length still needs to be chosen sensibly — too short and you destroy the real serial dependence you're trying to preserve (like volatility clustering); too long and you don't get enough independent resampled blocks to estimate anything precisely. A common rule of thumb ties the average length to how quickly the series' autocorrelation decays.