Econometrica: Jul, 1986, Volume 54, Issue 4

The Exact Distribution of the Wald Statistic

https://doi.org/0012-9682(198607)54:4<881:TEDOTW>2.0.CO;2-W
p. 881-896

P. C. B. Phillips

This paper derives the exact finite sample distribution of the Wald statistic for testing general linear restrictions on the coefficients in the multivariate linear model. This generalizes all previously known results, including those for the standard F statistic in linear regression, for Hotelling's generalized T^2"0 test. The results presented here encompass both the null and the non-null distributions. They also yield in a simple and elegantway the asymptotic distribution theory and related higher order asymptotic expansions. Various specializations of our general result are presented, including a computable formula for the null distribution in the case of a test of single restriction. Conventional classical assumptions of normally distributed errors and nonrandom exogenous variables are employed.