Wild bootstrap heteroscedasticity test

Calibrates the Breusch-Pagan Lagrange multiplier statistic with a null-imposed wild bootstrap, providing a p-value that is accurate in small samples and under non-normal errors without relying on the asymptotic \(\chi^2\) approximation.

Usage

performWildBootstrapTest(
  model,
  data,
  B = 499,
  distribution = c("rademacher", "mammen"),
  progress = interactive()
)

Arguments

model

A fitted stats::lm object describing the mean structure whose residual variance is to be assessed.

data

A base::data.frame (or compatible object) containing the variables referenced in model. The data must include all observations used to fit model and should not contain unresolved missing values.

B

Integer number of bootstrap replications. Defaults to 499.

distribution

Multiplier distribution used for the wild perturbation. Supported options are "rademacher" and "mammen".

progress

Logical flag controlling whether progress should be reported when running the bootstrap loop.

Value

An object of class htest containing the observed Breusch-Pagan statistic, bootstrap p-value, and additional details under the bootstrap element.

Details

The observed statistic is the usual \(n R^2\) from the auxiliary regression of the squared residuals on the regressors. The bootstrap reference distribution, however, must be generated under the homoscedastic null — otherwise it inherits the heteroscedasticity the test is looking for and the procedure loses all power. Each bootstrap sample is therefore built from leverage-standardised, mean-centred residuals \(r_i = e_i/\sqrt{1-h_{ii}}\) (which have a common variance under \(H_0\)) resampled i.i.d. and perturbed by a wild multiplier \(v_i\), so that \(y_i^* = \hat y_i + r^*_i v_i\). The statistic is recomputed on each refit and the p-value is \((1 + \#\{T_b^* \ge T\})/(B + 1)\). Under \(H_0\) this controls size even for heavy-tailed errors; under the alternative the observed statistic is extreme relative to the homoscedastic reference, giving power.

References

Davidson, R., & Flachaire, E. (2008). The wild bootstrap, tamed at last. Journal of Econometrics, 146(1), 162–169.

Godfrey, L. G. (2006). Tests for regression models with heteroskedasticity of unknown form. Computational Statistics & Data Analysis, 50(10), 2715–2733.

Wu, C. F. J. (1986). Jackknife, bootstrap and other resampling methods in regression analysis. The Annals of Statistics, 14(4), 1261–1295.

Examples

data(mtcars)
model <- lm(mpg ~ wt + qsec, data = mtcars)
if (interactive()) {
  set.seed(123)
  performWildBootstrapTest(model, mtcars, B = 199, progress = FALSE)
}