Resample centred innovations and regenerate a dependent series through an increasing-order AR approximation; distinguish the recursion from statistic-specific validity.
Sieve bootstrap
The sieve bootstrap estimates sampling uncertainty in a time series by fitting an increasingly flexible model, resampling its estimated innovations, and regenerating dependent observations. In the usual autoregressive version, a quant can assess uncertainty in a mean or fitted persistence statistic while retaining dependence through an autoregressive model, rather than independently shuffling observed levels.
Why it is a sieve
The approximating AR order pn grows with sample size, slowly enough to control estimation error. This distinguishes an AR sieve from a bootstrap that permanently assumes an AR(1) model. The fixed-order recursion below is its building block. Bühlmann's original report, §§1–2.
For a series already expressed relative to its estimated mean, one construction is
Yt∗=j=1∑pnϕ^jYt−j∗+εt∗.
Fit the AR approximation, calculate residuals, subtract their sample mean, then draw innovations independently with replacement from that centred residual pool. Feed each innovation through the fitted recursion. Restore the estimated mean when working with the original level convention. Unlike the block bootstrap, this generates new observations through a fitted filter rather than copying stretches of observed levels. Bühlmann, §2.
Three steps through one fitted filter
Take the stipulated fitted coefficient ϕ^=0.5, no intercept, a centred residual pool (−1,0,1) with equal selection probabilities, and initial state Y0∗=2. These inputs are supplied for arithmetic; no original dataset or estimated fit is being claimed.
Select the innovation sequence (1,−1,0). The recursion gives:
Step
Previous level
Fitted contribution
Innovation
New level
1
2
0.5(2)=1
+1
2
2
2
0.5(2)=1
−1
0
3
0
0.5(0)=0
0
0
The complete path, including the initial state, is (2,2,0,0). Each draw is independent conditional on the fitted model and pool; the generated levels depend on previous generated levels. The displayed innovation sequence is one possible outcome of independent sampling, not an instruction to copy the innovations' original time order.
Stipulated fitted AR(1) with coefficient 0.5 and initial value 2: innovation draws 1, minus 1 and 0 create the level path 2, 2, 0, 0. No intercept or burn-in is used.
Independent innovation draws enter a recursive filter; the resulting levels retain the filter's dependence. The initial point is a fixed starting condition.
Centring the residual pool sets its empirical mean to zero. It does not set the starting state to zero, demean each generated path, or make the fitted coefficient known without error. Here the start is fixed and all three generated steps are retained: this is a finite conditional illustration with no burn-in. A stationary sampling target needs an appropriate initial distribution or a burn-in that makes the chosen initial state negligible. For this stable coefficient, changing only the start changes step t by 0.5t times the initial difference.
What gets recomputed
Across replicates, regenerate a full sample and recompute the statistic. If it is an estimated AR coefficient, refit the AR model on each bootstrap sample; retaining ϕ^ as the reported statistic would create no coefficient-estimation distribution. A sample mean instead needs its mean recomputed. State whether order selection and other preprocessing are repeated, according to the inferential procedure being justified.
The generating coefficients are held at their original fitted values within this plug-in bootstrap. Sampling variation in the refitted statistic approximates estimation uncertainty under that fitted model; it does not establish that the original estimate is the true parameter. Conditional prediction from a fixed observed state and stationary-sample inference are different targets.
Matching dependence is not enough
Classical justification uses suitable causal, invertible linear processes, stable AR approximations, controlled lag growth, moment conditions and regular statistics. Merely requiring pn/n→0 is insufficient. More bootstrap repetitions cannot repair a poor approximation.
Validity is statistic-specific. Kreiss, Paparoditis and Politis show extensions beyond linear processes for some statistics, including the sample mean under their assumptions. They also show failure for sample autocovariances in some linear processes: matching second-order dependence need not reproduce the higher-order structure that governs a statistic's uncertainty. Uncorrelated residuals need not be independent. The ordinary AR sieve is therefore no universal solution for nonlinear dependence, nonstationary series or every statistic. Kreiss, Paparoditis and Politis (2011), §§1–3, Theorem 3.1 and Examples 3.1–3.2.
For the underlying bootstrap workflow, Simulation and Resampling develops resample–recompute distributions and distinguishes sampling uncertainty from simulation noise. The public page is a preview; the full lesson requires membership.
