Abstract :
In this paper we derive an alternative asymptotic approximation to the sampling
distribution of the limited information maximum likelihood estimator and a biascorrected
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.
