In the context of a single linear structural equation under classical assumptions, we derive the joint conditional density of the LIML endogenous coeﬀicient estimator, and the usual characteristic root arising from the LIML procedure, given the OLS estimates of the reduced form coeﬀicients for the excluded exogenous variables. This provides the joint distributions for various combinations of the statistics commonly used for inference in this model, and is hence an important stepping stone in the analysis of these procedures. The main result also leads to a new derivation of the density of the LIML estimator itself, and provides a result which is directly comparable to earlier results for IV estimators, including OLS and TSLS. We also consider briefly the density of the LIML structural variance estimator, and the joint density of the LIML and TSLS estimators for the endogenous coeﬀicients.
Hillier, Grant H., "Joint Distribution Theory for Some Statistics Based on LIML and TSLS" (1987). Cowles Foundation Discussion Papers. 1083.