This paper derives the exact probability density function of the limited information maximum likelihood (LIML) estimator of the coeﬀicient vector of the endogenous variables in a structural equation containing n + 1 endogenous variables and L ≥ 1 degrees of overidentiﬁcation. This generalizes the presently known results for the two endogenous variable case (n + 1 = 2) and the leading case analyses in the author’s earlier paper (1982). Upon appropriate symbolic translation the results may be applied directly to the maximum likelihood estimator in the multivariate linear functional relationship.
Phillips, Peter C.B., "The Exact Distribution of LIML: II" (1983). Cowles Foundation Discussion Papers. 896.