COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 761R

"Asymptotic Results for Generalized Wald Tests"

Donald W.K. Andrews

September 1985
Revised April 1986

This note presents conditions under which a quadratic form based on a g-inverted weighting matrix converges to a chi-square distribution as the sample size goes to infinity. Subject to fairly weak underlying conditions, a necessary and sufficient condition is given for this result. The result is of interest, because it is needed to establish asymptotic significance levels and local power properties of generalized Wald tests (i.e., Wald tests with singular limiting covariance matrices). Included in this class of tests are Hausman specification tests and various goodness of fit tests, among others. The necessary and sufficient condition is relevant to procedures currently in the econometrics literature, because it illustrates that some results stated in the literature only hold under more restrictive assumptions than those given.

JEL Classification: 211

Keywords: Generalized inverse, Wald test, asymptotics, chi-square