  Dovonon and Hall (Journal of Econometrics, 2018) proposed a limiting
distribution theory for GMM estimators for a p - dimensional globally
identified parameter vector {\phi} when local identification conditions fail at
first-order but hold at second-order. They assumed that the first-order
underidentification is due to the expected Jacobian having rank p-1 at the true
value {\phi}_{0}, i.e., having a rank deficiency of one. After reparametrizing
the model such that the last column of the Jacobian vanishes, they showed that
the GMM estimator of the first p-1 parameters converges at rate T^{-1/2} and
the GMM estimator of the remaining parameter, {\phi}_{p}, converges at rate
T^{-1/4}. They also provided a limiting distribution of
T^{1/4}({\phi}_{p}-{\phi}_{0,p}) subject to a (non-transparent) condition which
they claimed to be not restrictive in general. However, as we show in this
paper, their condition is in fact only satisfied when {\phi} is overidentified
and the limiting distribution of T^{1/4}({\phi}_{p}-{\phi}_{0,p}), which is
non-standard, depends on whether {\phi} is exactly identified or
overidentified. In particular, the limiting distributions of the sign of
T^{1/4}({\phi}_{p}-{\phi}_{0,p}) for the cases of exact and overidentification,
respectively, are different and are obtained by using expansions of the GMM
objective function of different orders. Unsurprisingly, we find that the
limiting distribution theories of Dovonon and Hall (2018) for Indirect
Inference (II) estimation under two different scenarios with second-order
identification where the target function is a GMM estimator of the auxiliary
parameter vector, are incomplete for similar reasons. We discuss how our
results for GMM estimation can be used to complete both theories and how they
can be used to obtain the limiting distributions of the II estimators in the
case of exact identification under either scenario.
