Handout 1 . The Bootstrap Econ 379

The bootstrap is a powerful tool for inference in statistical models. This note describes briefly what the bootstrap is and how to use it. Suppose X1, . . . , Xn is an i.i.d. sample drawn from an unknown distribution P . Consider testing the null hypothesis H0 : θ = θ0 versus the alternative H1 : θ 6= θ0 where θ is a parameter related to the distribution P . The first step is to construct a test statistic Tn, which is some function of the data. Let Fn(t) := Pr(Tn ≤ t) denote the distribution of the test statistic under the null hypothesis. Often we do not know Fn because we do not know P . The exception to this rule is if Tn is what is called a pivotal statistic. If Fn is unknown we instead must approximate it. The asymptotic approach relies on the limiting distribution, F∞(t) = limn→∞ Fn(t) which can usually be derived analytically. The limiting distribution is used to obtain a critical value, cn, such that F∞(cn) = 1−α/2. The size of the test is then Pr(|Tn| > cn) = 2(1−Fn(cn)) ≈ α. If F∞ is a good approximation of Fn, then the critical value cn provides a test procedure with accurate size control. The bootstrap is an alternative way of approximating Fn(t) for the purposes of constructing a critical value. There are two reasons why we might need a different approximation. First, in some cases the bootstrap will provide a better approximation. Second, the asymptotic approach may be difficult computationally. The bootstrap uses the actual sample to estimate the distribution of the test