On the arbitrariness of some asymptotic test statistics based on generalized inverses

Under appropriate conditions, some asymptotic test statistics based on generalized inverses (g-inverses) are shown to be arbitrary in the sense that any desired numerical value for a statistic can be obtained by appropriately choosing a g-inverse of an estimator. Examples of statistics based on g-inverses include score-type and Hausman-type statistics. Some versions of these statistics considered in the literature can be viewed as polar cases in the sense that their weighting matrices have either minimum or maximum rank. By associating a statistic with an estimator of a variance-covariance matrix and by appropriately choosing the estimator, it is possible to construct a statistic that is invariant with respect to the choice of a g-inverse of the estimator. Copyright 2005 Royal Economic Society