Bootstrapping time series

This example from Efron and Tibshirani’s Introduction to the Bootstrap.

Given a zero-centered time series z_1, \ldots, z_N , a first order autoregressive scheme is as follows:

z_t = \beta z_{t-1} + \epsilon_t

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 \hat{\beta} for \beta.

Then a bootstrap accuracy of \hat{\beta} can be carried out by drawing bootstrap replicates from the empirical distribution of \hat{F} = \{ \hat{\epsilon}_2, \ldots, \hat{\epsilon}_N \}.  These ‘approximate distrubances’ can be calculated as:

\hat{\epsilon}_t  =  z_t - \hat{\beta}^* z_{t-1}

Then a new bootstrap time series z^* is given by:

z^*_1 = z_1

z^*_t = \hat{\beta}^* z^*_{t-1} + \epsilon^*   (t = 2, \ldots, N)

Where \epsilon^* is drawn from \hat{F}.

The bootstrapped time series using the first order autoregressive scheme resembles the original time series much more than simply bootstrapping the original z_t.

R script for this simulation

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