750 Microeconometrics: Methods and Developments
t 1 ∗,...,tB∗. For a symmetric test rejectH 0 if the original samplet-statistic falls
outside theαquantile of|t 1 ∗|,...,|t∗B|.Note thatt∗in step 1 is centered on̂θas the bootstrap views the original sample,
withθ=̂θ, as the DGP. For equal-tailed two-sided tests (or for one-sided tests)
this procedure leads to asymptotic refinement with true sizeα+O(N−^1 ), rather
thanα+O(N−0.5), using bootstrap standard errors (or standard errors obtained
using equation (14.21)). For a two-sided symmetrical test (or a chi-squared test)
the corresponding rates are instead, respectively,α+O(N−1.5)andα+O(N−^1 ).
There are as many ways to bootstrap as there are different ways to obtain
resamples, and there are many ways to use these resamples.
The resampling method used above is called a paired bootstrap, as oftenwi=
(yi,xi)and here bothyiandxiare being resampled. By contrast, a residual boot-
strap, for a model with additive error, holdsxifixed and resamples over residuals
̂u 1 ,...,̂uNto yield resampled valueswi∗=(y∗i,xi), wherey∗i =xi′̂β+̂u∗i. A para-
metric bootstrap uses distributional knowledge, such as a specified distribution for
yi|xi, to resampleyigivenxi. For clustered data any resampling is over clusters.
For hypothesis tests it is best, if possible, to imposeH 0 in drawing the bootstrap
sample. More bootstrap replications are needed when the goal of the bootstrap is
asymptotic refinement for a test statistic.
Much development of the bootstrap has been done in the statistics literature. The
econometrics literature is surveyed in Horowitz (2001), and MacKinnon (2002) pro-
vides much useful practical advice. Econometric studies have focused on bootstraps
for estimation methods used mainly by econometricians. For overidentified GMM
models one should recenter so that the population moment condition is imposed
in the sample; see Hall and Horowitz (1996).
The bootstrap needs to be used with caution, as standard bootstraps can provide
inconsistent standard error estimates for nonsmooth estimators and for less than√
N-consistent estimators. This has led to a currently active literature. Abrevaya
and Huang (2005) consider the maximum score estimator, Abadie and Imbens
(2008) consider matching treatment effects estimators, and Moreira, Porter and
Suarez (2004) consider IV with weak instruments. Sub-sampling, due to Politis and
Romano (1994), works in a wider range of settings than the bootstrap.
In applied microeconometrics the main use of the bootstrap is to obtain standard
errors. Bootstraps with asymptotic refinement are rarely done, as sample sizes are
felt to be fairly large. But a bootstrap with asymptotic refinement can correct for
many well-documented problems associated with standard tests, including the lack
of invariance to parameterization for the Wald test and the poor finite-sample
performance of auxiliary regressions used in computing LM tests and conditional
moment tests.
14.5 Causation
The preceding sections presented estimation and inference methods for quite gen-
eral regression models. Econometrics is distinguished by a desire to go beyond