A General Theorem in the Theory of Asymptotic Expansions as Approximations to the Finite Sample Distributions of Econometric Estimators

THERE HAS RECENTLY BEEN A GROWING INTEREST in the use of asymptotic series expansions of the Edgeworth type to approximate finite sample distributions in econometrics. Working in the framework of a conventional simultaneous equations model, a number of authors [1, 2, 6, 7, and 13] have derived such expansions for various single-equation estimators and Sargan [10] has considered the problem of developing an expansion of the distribution of the full information maximum likelihood estimator (FIML). In addition, Sargan [11] has recently established an important general theorem on the validity of Edgeworth expansions for sample distributions of statistics which can be represented as very general functions of sample data, imposing only weak conditions on the class of functions. This result covers a wide variety of econometric estimators and test statistics. Nevertheless, work in this field to date has been based on two limiting assumptions: normally distributed structural disturbances and nonrandom exogenous variables. The latter is particularly unfortunate since models in practice usually involve lagged variables in the regressor set. On the other hand, there is no reason in principle, at least, why valid expansions cannot be obtained in more general models. The present paper, therefore, is concerned with extending Sargan's approximation theorem in [11] to include such cases. The central result of the paper is stated and proved in Section 2. In Section 3 we provide some discussion of the theorem and its conditions and attempt to relate them to the contemporaneous work of Sargan in [12].