In a recent article (1984a) Phillips showed that the distribution of the limited information maximum likelihood (LIML) estimator of the coeﬀicients of the endogenous variables in a single structural equation is multivariate Cauchy in the leading (totally unidentiﬁed) case. The purpose of the present note is to show that the same result holds for the full information maximum likelihood (FIML) estimator. Our proof relies on the theory of invariant measures on a Stiefel manifold. This approach provides a major simpliﬁcation of the derivation of the LIML result given in the earlier article and extends to the FIML case without diﬀiculty. We start by illustrating its use for LIML.
Phillips, Peter C.B., "The Distribution of FIML in the Leading Case" (1985). Cowles Foundation Discussion Papers. 979.