It is shown that the exact ﬁnite sample distribution of the limited information maximum likelihood (LIML) estimator in a general and leading single equation case is multivariate Cauchy. When the LIML estimator utilizes a known error covariance matrix (LIMLK) it is proved that the same Cauchy distribution still applies. The corresponding result for the instrumental variable (IV) estimator is a form of multivariate t density where the degrees of freedom depend on the number of instruments.
Phillips, Peter C.B., "The Exact Distribution of LIML: I" (1982). Cowles Foundation Discussion Papers. 891.