Demonstration of bootstrapping the residuals of an AR(1) time series model

ar1-residuals.R

#
# Demonstration of bootstrapping AR(1) residuals,
# using the AirPassergers data.
#
# This code performs a simple bootstrap, not a
# block bootstrap, because it assumes the model's residuals
# are independent. If your residuals exhibit autocorrelation,
# then a block bootstrap would be more appropriate.
# (However, if your residuals *do* show autocorrelation,
# then your time series model needs improvement.)
#

# y is the time series, T is the series' length
y <- AirPassengers
T <- length(y)

# Fit the time series to an AR(1) model
model <- arima(y, order = c(1,0,0))

# This function computes one bootstrap replication.
# Using the resampled residuals, it simulates one
# AR(1) time series and returns the statistic for
# that series.
statistic <- function(data, ind) {
  resids <- data[ind]
  ts <- arima.sim(model = model$model, n = T,
                  innov = resids)

  # For this demonstration, we compute the
  # Augmented Dickey-Fuller statistic.
  # You can change this to compute your statistic of interest.
  htest <- tseries::adf.test(ts)

  # Return just the statistic
  htest$statistic
}

# Perform the bootstrap
b <- boot::boot(data = resid(model),
                statistic = statistic,
                R = 999 )

# Plot the bootstrap replications
plot(b)

# Show the bootstrap confidence intervals
boot::boot.ci(b, type = c("norm", "basic"))

# Note: The AR(1) model of AirPassengers is a very poor model,
# so the bootstrap confidence intervals will be wide.