ALTERNATIVE APPROXIMATIONS TO THE DISTRIBUTIONS OF INSTRUMENTAL VARIABLE ESTIMATORS: 1 INTRODUCTION 2 THE MODEL AND THE ESTIMATORS 3. THE CHOICE OF PARAMETER SEQUENCE 4. ASYMPTOTIC RESULTS A. Asymptotic Distributions B. A Skewed Approximation C. Estimation 5. A SINGLE EXPLANATORY ENDOGENOUS VARIABLE A. Reduction to a Canonical Form B. Approximations C. Pitman Nearness 6. NUMERICAL EVALUATIONS A. Approximations to Tabulated Distributions B. Simulations 7 EFFECT OF NORMALITY 8 CONCLUSION APPENDIX A. OLS and IV B. A Transformation C. MM and MMK REFERENCES

The paper considers the OLS, the IV, and two method-of-moments estimators, MM and MMK, of the coefficients of a single equation, where the explanatory variables are correlated with the disturbance term. The MM and MMK estimators are generalizations of the LIML and LIMLK estimators, respectively.Multivariate first-order approximations to the distributions are derived under normality, using a parameter sequence where the number of instruments increases as the number of observations increases. Numerical results show these approximations are more accurate, compared to large-sample approximations, even if the number of instruments is smallThe moments of the multivariate limit distributions of the MM and MMK estimators can be consistently estimated under a variety 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 Pitman.KEYWORDS' Instrumental variables, method of moments, limited information maximum likelihood, asymptotic distributions, parameter sequence, shrinkage estimator, Pitman nearness, Monte Carlo experiments