Palgrave Handbook of Econometrics: Applied Econometrics

A. Colin Cameron 749

This method is convenient whenever standard errors are difficult to obtain
by conventional methods. Leading examples are (i) two-step estimators when
estimation at the first step complicates inference at the second step; (ii) Haus-
man tests that require computation of the variance of the difference between
two estimators when neither estimator is efficient under the null hypothesis; and
(iii) estimation with clustered errors when a package does not compute cluster-
robust standard errors (in this case a cluster bootstrap that resamples over clusters
is used). Given bootstrap standard errors, a standard Wald test ofH 0 :θ=θ 0 uses
t=(̂θ−θ 0 )/ŝθ,Bootand asymptotic normal critical values.
The preceding bootstrap is theoretically no better than usual first-order asymp-
totic theory. The attraction is the practical one of convenience.
Some bootstraps, however, provide a better asymptotic approximation, called an
asymptotic refinement. The econometrics literature focuses on asymptotic refine-
ment for test statistics. Consider a test ofH 0 :θ=θ 0 with nominal significance
level or nominal sizeα. An asymptotic approximation yields an actual rejection
rate or true sizeα+O(N−j), whereO(N−j)means is of orderN−jandj>0, with
oftenj= 1 /2orj=1. Then the true size goes toαasN→∞. Largerjis preferred,
however, as then convergence toαis faster. A method with asymptotic refinement
(or higher-order asymptotics) is one that yieldsjlarger than that obtained using
conventional asymptotics. The hope is that such asymptotic refinement will lead
to tests with true size closer toαfor moderate sample sizes, though this is not
guaranteed. Asymptotic refinement may be possible if the bootstrap is applied to
an asymptotically pivotal statistic, meaning one with asymptotic distribution that
does not depend on unknown parameters.
The bootstrap standard error procedure does not lead to asymptotic refinement
for the Wald test. Nor does the percentile method which rejectsH 0 :θ=θ 0 if
θ 0 falls outside the lowerα/2 and upperα/2 quantiles of the bootstrap estimates
̂θ∗ 1 ,...,̂θB∗. The problem is that̂θis bootstrapped and̂θis not asymptotically pivotal,

since even underH 0 its asymptotic normal distribution depends on an unknown

parameter (the variance).

Instead, the Wald statistic itself should be bootstrapped, ast=(̂θ−θ 0 )/ŝθis

asymptotically pivotal, since it is asymptoticallyN[0, 1]underH 0 , an asymp-

totic distribution with no unknown parameters. The bootstrap-tor percentile-t

procedure for a two-sided test ofH 0 :θ=θ 0 at levelαis:

1. Do the followingBtimes:
   
   
   • Draw a bootstrap resamplew
     ∗
     1 ,...,w
   
   
   
   

∗

Nby sampling with replacement from

the original data (called a paired bootstrap).

• Obtain an estimatêθ∗, standard errorŝθ∗andt-statistict∗=(̂θ∗−̂θ)/ŝθ∗.

2. Use theBstatisticst 1 ∗,...,tB∗to approximate the distribution oft=(̂θ−θ 0 )/ŝθ. For
   an equal-tailed (or nonsymmetric) test rejectH 0 if the original samplet-statistic
   falls outside the lowerα/2 and upperα/2 quantiles of the bootstrap estimates