Stability Comparisons of Estimators

A property of estimators called stability is investigated in this paper. The stability of an estimator is a measure of the magnitude of the affect of any single observation in the sample on the realized value of the estimator. High stability often is desirable for robustness against misspecification and against highly variable observations.

Stabilities are determined and compared for a wide variety of estimators and econometric models. Estimators considered include: least squares, maximum likelihood (including both LIML and FIML), instrumental variables, M-, and multi-stage estimators such as tow and three stage least squares, Zellner’s feasible Aikten estimator of the multivariate regression model, and Heckman’s estimator of censored regression and self-selection models. The general results of the paper apply to numerous additional estimators of various and sundry models.

The stability of an estimator is found to depend on the number of finite moments of its influence curve (evaluated at a random observation in the sample). An estimator’s stability increases strictly and continuously from zero to one as the number of finite moments of its influence curve increases from one to infinity. The more moments, the higher the stability. Since it often is possible to construct estimators with a specified influence function, estimators with different stabilities can be constructed. For example, one can attain the maximum stability possible by formulating a bounded influence estimator, since they have an infinite number of finite moments.