Block Bootstrap for Time Series

ϕ

0.5

θ

0.

data series length, n

100

500

1000

5000

bootstrap series length, m

100

500

1000

5000

block size, b

3

random seed

39145

stationary

A nonparametric block bootstrap series

y

1

,…,

y

m

for a simulated time series

z

1

,…,

z

n

is generated and the sample autocorrelations at lags 1, …, 10 for the

y

and

z

series are compared. The

z

series is simulated as an ARMA(1,1),

z

t

=ϕ

z

t-1

+

a

t

-θ

a

t-1

, where the

a

t

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

z

series by the orange lines. The blue points show the sample autocorrelations of the simulated bootstrap series. The block size parameter

b

may be fixed or in the stationary case it is randomly distributed with a mean of

b

from a truncated geometric distribution.