We consider the estimation of coefficients of a structural equation with many instrumental
variables in a simultaneous equation system. It is mathematically equivalent
to an estimating equation estimation or a reduced rank regression in the statistical
linear models when the number of restrictions or the dimension increases with
the sample size. As a semi-parametric method, we propose a class of modifications
of the limited information maximum likelihood (LIML) estimator to improve its
asymptotic properties as well as the small sample properties for many instruments
and persistent heteroscedasticity. We show that an asymptotically optimal modification
of the LIML estimator, which is called AOM-LIML, improves the LIML
estimator and other estimation methods. We give a set of sufficient conditions for
an asymptotic optimality when the number of instruments or the dimension is large
with persistent heteroscedasticity including a case of many weak instruments.
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