Stationary & Block Bootstrap for Dependent Data
How can a bootstrap preserve dependence between adjacent market observations?
A stationary bootstrap draws blocks of random geometric length, retaining short-run dependence that single-observation resampling destroys. Its interval is useful only when the series, statistic and chosen block length satisfy the needed assumptions.
Block bootstrap intervals still require suitable stationarity, dependence, statistic and block-length assumptions.
Evidence map
| Aspect | Finding |
|---|---|
| What it is | Resampling schemes that preserve serial dependence by drawing blocks of consecutive observations instead of single points. |
| Key result / formula | For a series x_1..x_T, pick a starting index uniformly at random and, at each step, continue to the next observation with probability (1 − p) or restart at a new random index with probability p. |
| Why it matters for backtesting | Returns and strategy P&L are serially dependent — volatility clusters, positions persist, labels overlap — so an i.i.d. bootstrap of daily results manufactures confidence intervals that are far too narrow. |
What it is
The moving-block bootstrap draws fixed-length blocks; the stationary bootstrap (Politis & Romano 1994) draws blocks of geometrically distributed random length, which makes the resampled series stationary — a property the fixed-block version lacks.
Key result / formula
Block lengths are then geometric with mean 1/p, and the resampled series is stationary. The tuning parameter is the expected block length: too short destroys the dependence the procedure exists to preserve, too long leaves few effectively independent blocks and inflates the variance of the bootstrap estimate. Politis & White (2004), with the Patton–Politis–White (2009) correction, give a data-driven rule for choosing it from the estimated autocovariance structure.
Why it matters for backtesting
The block bootstrap is the standard machinery underneath resampling-based backtest tests (reality checks, stepwise multiple-testing procedures) and one way to build confidence bands for suitable statistics when stationarity and dependence assumptions are defensible. A block-based null can preserve within-series dependence while breaking a predeclared alignment with a signal; the null construction, not the resampling label, determines whether it is valid.
Worked resampling
A plain bootstrap draws individual daily returns and destroys a three-day volatility cluster. A moving-block bootstrap can instead draw blocks of length five, preserving short local dependence within each block. A stationary bootstrap varies block length geometrically, avoiding fixed block boundaries. Run the full estimator on each replicate and compare intervals across plausible mean block lengths, such as 5, 10 and 20 days. The block length is a modeling choice tied to dependence scale, not a knob for obtaining significance. Neither method manufactures unseen crises; if the original record contains no liquidity freeze, its resamples cannot validate behavior in one. Record the random seed, number of replicates and resampled unit.
Source
Politis & Romano, "The Stationary Bootstrap", JASA 89(428), 1994, 1303-1313; Künsch, "The Jackknife and the Bootstrap for General Stationary Observations", Annals of Statistics 17(3), 1989; Politis & White, "Automatic Block-Length Selection for the Dependent Bootstrap", Econometric Reviews 23(1), 2004, with correction in Patton, Politis & White, Econometric Reviews 28(4), 2009. Primary source
Related in the library
The Virtue of Complexity Debate: More Parameters Than Observations
Efron's Bootstrap: Resampling the Sample to Get a Standard Error
Guide on this topic: Monte Carlo simulation for trading
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