In the context of the single structural equation model, we derive a number of exact results that extend and/or simplify results hitherto available. First, we obtain expressions for both the conditional and unconditional densities of the limited information maximum likelihood estimator for the coefficients of the endogenous variables. The unconditional result is considerably simpler than the corresponding result obtained earlier by Phillips (1985), and we indicate how this result can be used to obtain distribution results for the coefficients of the exogenous variables in exactly the manner used in Phillips (1984a) for the ordinary least squares and two-stage least squares estimators. Next, we obtain expressions for the mean square error of the ordinary least squares/two-stage least squares estimators for the coefficients of the exogenous variables. Finally, a number of generalizations of these results are indicated, and we explain briefly how these results can contribute to further attempts to understand the general problems of inference in this model.