# Ecplaining Bootstraps and Robustness

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Explaining Bootstraps and Robustness Tony Lancaster, Brown University, Providence RI 029121 Abstract In this note we consider several versions of the bootstrap and ar- gue that it can be helpful in explaining and thinking about such pro- cedures 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 mod- els 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) het- eroscedasticity robust covariance matrix estimate. The method also shows the precise relations between an Efron(1979) bootstrap proce- dure and the Bayesian bootstrap of Rubin(1981) KeyWords heteroscedasticity; Bayes; Least Squares 1 INTRODUCTION The bootstrap is usually explained algorithmically, as a set of computational instructions. (This description seems to apply to books, e.g. Efron and Tibshi- rani(1993); survey articles, e.g. Horowitz(2001); and to textbooks, e.g. Wooldridge(2002).) In the case of the Efron nonparametric bootstrap the algorithm would be some- thing like 1. Randomly sample your data, with replacement, n times. 2. Compute the statistic of interest using your new data 3. Repeat steps 1 and 2 B times 4. Calculate the standard deviation of the B values of the statistic. Justi cation for the algorithm would then be provided by explaining that the empirical distribution of the data, say Fn is an approximation to the true but unknown distribution F and that repeated sampling from Fn is approximately the same as repeated sampling from F which, in turn, is what is required to 1Tony Lancaster is Professor of Economics at Brown University, Providence RI 02912 (email: [email protected]) 1 calculate a repeated sampling distribution. Further justi c

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