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 .

R script for this simulation

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