Optimal significance tests in simultaneous equation models

Consider testing the null hypothesis that a single structural equation has specified coefficients. The alternative hypothesis is that the relevant part of the reduced form matrix has proper rank, that is, that the equation is identified. The usual linear model with normal disturbances is invariant with respect to linear transformations of the endogenous and of the exogenous variables. When the disturbance covariance matrix is known, it can be set to the identity, and the invariance of the endogenous variables is with respect to orthogonal transformations. The likelihood ratio test is invariant with respect to these transformations and is the best invariant test. Furthermore it is admissible in the class of all tests. Any other test has lower power and/or higher significance level.