This example from Efron and Tibshirani’s Introduction to the Bootstrap.
Given a zero-centered time series , a first order autoregressive scheme is as follows:
Or in words: the observed value at time t is a multiple of the time before plus some random error.
Using least squares or maximum likelihood, one can find an estimate for .
Then a bootstrap accuracy of can be carried out by drawing bootstrap replicates from the empirical distribution of . These ‘approximate distrubances’ can be calculated as:
Then a new bootstrap time series is given by:
(t = 2, \ldots, N)
Where is drawn from .
The bootstrapped time series using the first order autoregressive scheme resembles the original time series much more than simply bootstrapping the original .