General formula for the ﬁnite sample and asymptotic distributions of the instrumental variable estimators and the Wald statistics in a simultaneous equation model are derived. It is assumed that the coeﬀicient vectors of both endogenous and exogenous variables are only partially identiﬁed, even though the order condition for identiﬁcation is satisﬁed. This work extends previous results in Phillips (1989) where the coeﬀicient vector of the exogenous variables is partially identiﬁed and that of the endogenous variables is totally unidentiﬁed. The eﬀect of partial identiﬁcation on the ﬁnite sample and asymptotic distributions of the estimators and the Wald statistics is analyzed by isolating identiﬁable parts of the coeﬀicient vectors using a rotation of the coordinate system developed in Phillips (1989). The pdf’s of the estimators and the Wald statistics are illustrated using simulation and compared with their respective asymptotic distributions.
Choi, In and Phillips, Peter C.B., "Asymptotic and Finite Sample Distribution Theory for IV Estimators and Tests in Partially Identified Structural Equations" (1989). Cowles Foundation Discussion Papers. 1172.