Explaining bootstraps and robustness

In this note we consider several versions of the bootstrap and argue that it can be helpful in explaining and thinking about such procedures to use an explicit representation of the random resampling process. To illustrate the point we give such explicit representations and use them to produce some results about bootstrapping linear models that are, apparently, not widely known, at least in the econometric literature. Among these are a demonstration of the equivalence, to order n-1 of the covariance matrix of the bootstrap distribution of the least squares estimator and the Eicker(1967)/White(1980) heteroscedasticity robust covariance matrix estimate. The method also shows the precise relations between an Efron(1979) bootstrap procedure and the Bayesian bootstrap of Rubin(1981)