The paper considers the OLS, the IV, and two method-of-moments estimat
ors, MM and MMK, of the coefficients of a single equation, where the e
xplanatory variables are correlated with the disturbance term. The MM
and MMK estimators are generalizations of the LIML and LIMLK estimator
s, respectively. Multivariate first-order approximations to the distri
butions are derived under normality, using a parameter sequence where
the number of instruments increases as the number of observations incr
eases. Numerical results show these approximations are more accurate,
compared to large-sample approximations, even if the number of instrum
ents is small. The moments of the multivariate limit distributions of
the MM and MMK estimators can be consistently estimated under a variet
y of parameter sequences, including the large-sample sequence. The new
approximate confidence regions perform well in terms of exact levels,
compared to traditional ones. The IV estimator of the coefficient of
a single explanatory endogenous variable is interpreted as a shrinkage
estimator, which is dominated, in practical cases, by the MM and MMK
estimators in terms of nearness to the true value in the sense of Pitm
an.