Block Bootstrap for Time Series

A nonparametric block bootstrap series for a simulated time series is generated and the sample autocorrelations at lags 1, …, 10 for the and series are compared. The series is simulated as an ARMA(1,1), , where the are independent normal random variables with mean 0 and variance 1. The theoretical autocorrelation for the ARMA(1,1) series is shown by the light gray lines and the sample autocorrelations of the series by the orange lines. The blue points show the sample autocorrelations of the simulated bootstrap series. The block size parameter may be fixed or in the stationary case it is randomly distributed with a mean of from a truncated geometric distribution.


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In applications, it is of interest to see what block size is needed to preserve autocorrelations in the bootstrap series. To see the effect of randomness, try automatic animation with the random seed. The low-order autocorrelations at lags 1 and 2 are of special interest.
Time series bootstrap methods are reviewed in the recent paper [1]. [2] uses a nonparametric bootstrap to preserve temporal correlations in a nonparametric weather simulator.
[1] D. M. Politis, "The Impact of Bootstrap Methods on Time Series Analysis," Statistical Science, 18(2), 2003 pp. 219-230.
[2] L. M. King, A. I. McLeod, and S. P. Simonovic, "A Multisite, Multivariate K-Nearest Neighbour Weather Generator for Simulation of Historical and Future Climate Data," to appear.
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