ALTERNATIVE APPROXIMATIONS TO THE DISTRIBUTIONS OF INSTRUMENTAL VARIABLE ESTIMATORS

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.