The wild bootstrap residuals are computed as \(residuals*V\), where \(V\) is a sampling from a random variable (see details section).
rwild(residuals, type = "golden")residuals
Type of distribution of V.
The wild bootstrap residuals computed using a sample of the random variable \(V\).
For the construction of wild bootstrap residuals, sampling from a random variable \(V\) such that \(E[V^2]=0\) and \(E[V]=0\) is needed. A simple and suitable \(V\) is obtained with a discrete variable of the form:
``golden'', Sampling from golden section bootstrap values suggested by Mammen (1993). $$P\Bigg\{ V=\frac{1-\sqrt{5}}{2} \Bigg\} = \frac{5+\sqrt{5}}{10} \, and \, P\Bigg\{ V=\frac{1+\sqrt{5}}{2} \Bigg\} = \frac{5-\sqrt{5}}{10},$$ which leads to the golden section bootstrap.
``Rademacher'', Sampling from Rademacher distribution values \(\big\{-1,\,1\big\}\) with probabilities \(\big\{\frac{1}{2},\,\frac{1}{2}\big\}\), respectively.
``normal'', Sampling from a standard normal distribution.
Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255 285. Davidson, R. and E. Flachaire (2001). The wild bootstrap, tamed at last. working paper IER1000, Queens University.
# NOT RUN {
n<-100
# For golden wild bootstrap variable
e.boot0=rwild(rep(1,len=n),"golden")
# Construction of wild bootstrap residuals
e=rnorm(n)
e.boot1=rwild(e,"golden")
e.boot2=rwild(e,"Rademacher")
e.boot3=rwild(e,"normal")
summary(e.boot1)
summary(e.boot2)
summary(e.boot3)
# }