INTRODUCTION

The bootstrap is a method for estimating the distribution of an estimator or test statistic by resampling the data. It amounts to treating the data as if they were the population for the purpose of evaluating the distribution of interest. Under mild regularity conditions, the bootstrap yields an approximation to the distribution of an estimator or test statistic that is at least as accurate as the approximation obtained from first-order asymptotic theory. Thus, the bootstrap provides a way to substitute computation for mathematical analysis if calculating the asymptotic distribution of an estimator or statistic is difficult. The maximum score estimator (Manski (1975, 1985)), the statistic developed by Härdle et al. (1991) for testing positive-definiteness of income-effect matrices, and certain functions of time-series data (Blanchard and Quah (1989), Runkle (1987), West (1990)) are examples in which evaluating the asymptotic distribution is difficult and bootstrapping has been used as an alternative.

In fact, the bootstrap is often more accurate in finite samples than first-order asymptotic approximations but does not entail the algebraic complexity of higher-order expansions. Thus, it can provide a practical method for improving upon first-order approximations. First-order asymptotic theory often gives a poor approximation to the distributions of test statistics with the sample sizes available in applications. As a result, the nominal levels of tests based on asymptotic critical values can be very different from the true levels.