BOOTSTRAP TESTS: HOW MANY BOOTSTRAPS?
Russell Davidson James G. MacKinnon

ABSTRACT
In practice, bootstrap tests must use a nite number of bootstrap samples.
This means that the outcome of the test will depend on the sequence of
random numbers used to generate the bootstrap samples, and it necessarily
results in some loss of power. We examine the extent of this power loss and
propose a simple pretest procedure for choosing the number of bootstrap
samples so as to minimize experimental randomness. Simulation experiments
suggest that this procedure will work very well in practice.

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Abstract
We rst propose a procedure for estimating the rejection probabilities for bootstrap
tests in Monte Carlo experiments without actually computing a bootstrap
test for each replication. This procedure is only about twice as expensive (per
replication) as estimating rejection probabilities for asymptotic tests. We then
propose a procedure for computing modi ed bootstrap P values that will often be
more accurate than ordinary ones. This procedure is closely related to the double
bootstrap, but it is far less computationally demanding. Simulations show that
both procedures often work very well.

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Abstract
Although it is common to refer to “the bootstrap,” there are actually a great many
different bootstrap methods that can be used in econometrics. We emphasize the
use of bootstrap methods for inference, particularly hypothesis testing, and we also
discuss bootstrap confidence intervals. There are important cases in which bootstrap
inference tends to be more accurate than asymptotic inference. However, it is not
always easy to generate bootstrap samples in a way that makes bootstrap inference
even asymptotically valid.

Bootstrap Inference in a Linear Equation
Estimated by Instrumental Variables
Russell Davidson

Abstract
We study several tests for the coefficient of the single right-hand-side endogenous variable
in a linear equation estimated by instrumental variables. We show that all the
test statistics—Student’s t, Anderson-Rubin, Kleibergen’s K, and likelihood ratio
(LR)—can be written as functions of six random quantities. This leads to a number
of interesting results about the properties of the tests under weak-instrument asymptotics.
We then propose several new procedures for bootstrapping the three non-exact
test statistics and a conditional version of the LR test. These use more efficient estimates
of the parameters of the reduced-form equation than existing procedures.
When the best of these new procedures is used, K and conditional LR have excellent
performance under the null, and LR also performs very well. However, power considerations
suggest that the conditional LR test, bootstrapped using this new procedure
when the sample size is not large, is probably the method of choice.

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