Bootstrapping Regression Models With BLUS Residuals∗

and FCAR for their financial support. We would also like to thank a referee and an Associate Editor for their comments which led to an improved presentation of the simulation results. †Département de mathématiques et de statistique, Universite ́ de Montréal To bootstrap a regression problem, pairs of response and explanatory variables or residuals can be resampled, according to whether we believe that the explanatory variables are random or fixed. In the latter case, different residuals have been proposed in the literature, including the ordinary residuals (Efron, 1979), standardized residuals (Bickel and Freedman, 1983) and studentized residuals (Weber, 1984). Freedman (1981) has shown that the bootstrap from ordinary residuals is asymptotically valid when the number of cases increases and the number of variables is fixed while Weber (1984) has done the same for studentized residuals. Bickel and Freedman (1983) have shown the asymptotic validity for ordinary residuals when the number of variables as well as the number of cases increase provided that the ratio of the two converges to 0 at an appropriate rate. In this paper, we introduce the use of Best Linear Unbiased Scaled (BLUS) residuals in bootstrapping regression models. The main advantage of the BLUS residuals, introduced in Theil (1965), is that they are uncorrelated. The main disadvantage is that only n−p residuals can be computed for a regression problem with n cases and p variables. The asymptotic results of Freedman (1981) and Bickel and Freedman (1983) for the ordinary (and standardized) residuals are generalized to the BLUS residuals. A small simulation study shows that even though only n − p residuals are available, bootstrapping BLUS residuals is as good in small samples, and sometimes better, than bootstrapping from standardized or studentized residuals. Key words and phrases: BLUS residuals, standardized residuals, studentized residuals, bootstrap, regressio