Block bootstrap: Definition & Examples

A block bootstrap estimates sampling uncertainty for dependent observations by resampling contiguous stretches of a time series, with replacement, and recomputing a statistic. For a quant analysing returns, the purpose is to retain local dependence that an ordinary observation-by-observation bootstrap would break. It creates alternative samples from the observed history; it does not create new market information.

Why the order can change the error bar

Consider the mean of $n$ observations from a covariance-stationary process. Write $\gamma_k=\text{Cov}(X_t,X_{t+k})$ . Expanding the variance of the sum gives

\text{Var}(\bar X_n)=\frac{1}{n}\left[\gamma_0+2\sum_{k=1}^{n-1}\left(1-\frac{k}{n}\right)\gamma_k\right].

Independent resampling makes distinct bootstrap observations conditionally independent, so it loses the serial-covariance contribution. Whether that matters depends on the statistic and dependence.

For a concrete model comparison, take a stationary AR(1) process with coefficient $0.5$ , marginal variance $1$ , and $n=100$ . Its covariance is $\gamma_k=0.5^k$ . The formula gives variance of the mean approximately $0.0296$ , or standard error $0.1720$ . An independence calculation using the same marginal variance gives $0.01$ and $0.1000$ . These are model quantities, not fitted bootstrap results. Positive serial covariance makes the independence calculation too small here; negative covariance can change the direction.

Weak correlation in raw returns also does not establish independence: dependence in squared returns can matter for volatility statistics. Choose a resampling method for the quantity whose uncertainty you need. Lahiri's dependent-data example, §2.3, shows why an iid bootstrap can converge to the wrong variance.

Build one moving-block sample

For observations

x_1,\ldots,x_n

and fixed block length

\ell

, the moving-block bootstrap uses the

n-\ell+1

overlapping blocks

B_i=(x_i,\ldots,x_{i+\ell-1}),\qquad i=1,\ldots,n-\ell+1.

Draw block starts independently and uniformly with replacement. Concatenate the selected blocks in draw order. One common convention draws

\lceil n/\ell\rceil

blocks and truncates the last block to obtain exactly

n

observations. Recompute the statistic for each replicate; the resulting collection approximates its sampling distribution under the method's assumptions. Künsch's original construction resamples whole blocks rather than reducing the data to independent observations. Künsch (1989), abstract.

Use the fictional sequence

(1,2,3,4,5,6)

with

\ell=2

. The eligible blocks are

(1,2)

(2,3)

(3,4)

(4,5)

and

(5,6)

. Choosing zero-based starts

0,3,1

gives

(1,2)\;|\;(4,5)\;|\;(2,3),\qquad \bar x^*=\frac{17}{6}\approx2.833.

The original mean is $3.5$ . A replicate need not match it: variation across replicates is the point. The increasing sequence is a construction aid, not evidence of stationarity or a suitable dataset for inference.

Five overlapping blocks of length two from fictional values 1 to 6. Drawing zero-based starts 0, 3 and 1 produces 1, 2, 4, 5, 2, 3, with artificial joins 2 to 4 and 5 to 2; its mean is 17/6.

The copied pairs retain their original order. Dashed dividers identify the new joins between independently selected blocks.

Within a copied block, observed neighbours travel together. At the joins, the new neighbours

2\to4

and

5\to2

were not adjacent in the original series. Thus “preserves dependence” means preserving stretches of it, with boundary distortion. Statistics built from lagged pairs need particular care over these joins; blocking vectors of lagged observations is one construction discussed in Lahiri, §2.5.

The block pool defines the variant

For the same six observations and length two:

Variant	Eligible blocks	Distinguishing feature
Moving blocks	Five: $(1,2)$ through $(5,6)$	All starts that fit inside the observed history
Nonoverlapping blocks	Three: $(1,2)$ , $(3,4)$ , $(5,6)$	A fixed partition gives fewer eligible starts
Circular blocks	Six: the five moving blocks plus $(6,1)$	Wrapping permits a start at every observation

All three draw blocks with replacement. State how the nonoverlapping method handles a remainder when

n

is not divisible by

\ell

. Circular wrapping equalises observations' representation in the pool, but the end-to-start adjacency is synthetic. These are different resampling distributions. Lahiri, §2.6; arch time-series bootstrap documentation.

The stationary bootstrap uses random block lengths and circular indexing. Its construction and stationarity property deserve their own treatment. Lahiri, §2.7.2.

Block length is an inference choice

Short blocks cut dependence frequently; long blocks retain longer runs but leave fewer effectively distinct stretches to estimate uncertainty. The resulting bias–variance trade-off concerns the bootstrap estimate of a distributional feature, such as a variance or a tail probability. More bootstrap repetitions reduce simulation noise conditional on the data; they cannot repair the wrong block length.

Under regularity conditions for certain smooth statistics, the leading mean squared error of a long-run variance estimate has the form

\text{MSE}(\ell)\approx \frac{A}{\ell^2}+C\frac{\ell}{n},

with positive, process-dependent constants. This motivates an

n^{1/3}

block-length scale in that setting. It is not a universal prescription: the target functional, dependence, bootstrap variant and unknown constants affect the choice. A selector justified for variance estimation is not automatically optimal for confidence-interval coverage. Nordman and Lahiri (2014), introduction and §2.1.

The classical justification starts from a stationary, weakly dependent process with suitable moment and statistic regularity conditions; commonly

\ell\to\infty

while

\ell/n\to0

. Stationarity alone is insufficient. An abrupt regime break, long-range dependence, or a statistic without the required regularity can invalidate the standard argument. Report the blocking convention and examine sensitivity to plausible lengths rather than treating one selection as proof of validity. Künsch (1989), abstract; Nordman and Lahiri, §3.1.

Resampling blocks also serves a different purpose from chronological holdout: rearranged history does not establish performance on unseen future data. And sequential bootstrap, which changes selection probabilities to address overlapping financial labels, is a different algorithm.

For the underlying resample–recompute–collect workflow, Simulation and Resampling develops bootstrap standard errors and percentile intervals. The public page is a preview; the full lesson requires membership.

Sources

Künsch (1989), The Jackknife and the Bootstrap for General Stationary Observations, original abstract in the author's uploaded record: stationary-data construction and consistency conditions.
Lahiri (2003), Resampling Methods for Dependent Data, chapter excerpts, §§2.3, 2.5–2.7: dependence, block pools and boundary effects.
Nordman and Lahiri (2014), Convergence Rates of Empirical Block Length Selectors for Block Bootstrap, introduction, §§2.1 and 3.1: target-specific block selection and assumptions.
arch 8.0.0, Time-series Bootstraps: fixed-length moving and circular block conventions.

Why the order can change the error bar

Build one moving-block sample

The block pool defines the variant

Block length is an inference choice

Sources

Related terms

Stationary bootstrap