A Note on the Exact Finite Sample Frequency Functions of Generalized Classical Linear Estimators in Two Leading Over-Identified Cases

Abstract

Abstract The asymptotic unbiasedness and normality of alternative statistical estimators θ, θ,1 ··· of a given parameter θ* are generally proved without reference to any explicit knowledge of the exact finite sample distribution functions, F n (x), G n (x) ···. (Here n denotes sample size.) Within the class of asymptotically unbiased and normally distributed estimators of a given parameter it is sometimes possible to demonstrate that one estimator possesses a smaller asymptotic variance than another, or that one estimator possesses the smallest asymptotic variance within a particular subclass. Asymptotic theory obviously does not predict anything about finite sample distribution functions Fn(x), Gn(x) ···. In particular we cannot deduce from asymptotic theorems that the estimator with the smallest asymptotic variance will continue to exhibit the smallest dispersion in finite samples. Consequently it remains an essential task in positive estimation theory to derive the exact finite sample distribution functions of the alternative estimators that appear to be promising on the basis of asymptotic considerations. In econometric statistics the alternative estimators of structural coefficients in systems of simultaneous equations appear to possess rather complicated finite sample distribution functions. Monte Carlo experiments are resorted to for the purpose of providing useful leads in mathematical research into the nature of these distributions. In this article are presented some leading results of a mathematical investigation of the exact finite sample distribution functions of generalized classical linear estimators. The results presented here exhibit some particularly important implications for the conduct and evaluation of Monte Carlo experiments in econometric statistics.