  We present a new robust bootstrap method for a test when there is a nuisance
parameter under the alternative, and some parameters are possibly weakly or
non-identified. We focus on a Bierens (1990)-type conditional moment test of
omitted nonlinearity for convenience, and because of difficulties that have
been ignored to date. Existing methods include the supremum p-value which
promotes a conservative test that is generally not consistent, and test
statistic transforms like the supremum and average for which bootstrap methods
are not valid under weak identification. We propose a new wild bootstrap method
for p-value computation by targeting specific identification cases. We then
combine bootstrapped p-values across polar identification cases to form an
asymptotically valid p-value approximation that is robust to any identification
case. The wild bootstrap does not require knowledge of the covariance structure
of the bootstrapped processes, whereas Andrews and Cheng's (2012, 2013, 2014)
simulation approach generally does. Our method allows for robust bootstrap
critical value computation as well. Our bootstrap method (like conventional
ones) does not lead to a consistent p-value approximation for test statistic
functions like the supremum and average. We therefore smooth over the robust
bootstrapped p-value as the basis for several tests which achieve the correct
asymptotic level, and are consistent, for any degree of identification. They
also achieve uniform size control. A simulation study reveals possibly large
empirical size distortions in non-robust tests when weak or non-identification
arises. One of our smoothed p-value tests, however, dominates all other tests
by delivering accurate empirical size and comparatively high power.
