In this paper we derive an alternative asymptotic
approximation to the sampling distribution of the
limited information maximum likelihood estimator and
a bias-corrected version of the two-stage least
squares estimator. The approximation is obtained by
allowing the number of instruments and the
concentration parameter to grow at the same rate as
the sample size. More specifically, we allow for
potentially nonnormal error distributions and obtain
the conventional asymptotic distribution and the
results of Bekker (1994,
Econometrica 62, 657–681) and
Bekker and Van der Ploeg (2005, Statistica
Neerlandica 59, 139–267) as special
cases. The results show that when the error
distribution is not normal, in general both the
properties of the instruments and the third and
fourth moments of the errors affect the asymptotic
variance. We compare our findings with those in the
recent literature on many and weak instruments.