In this paper we investigate the distribution of the score statistics for testing hypothesis about the slope parameter in a simple structural regression model. It is shown that for two of the most common ways of making the model identifiable, the distribution of the score statistics under the null hypothesis can be found exactly as an increasing function of an F statistics, providing thus exact test statistics for testing hypothesis about the slope parameter. It is unknown if such results hold in general for the likelihood ratio statistics. Use is made of orthogonal parameterizations obtained in the literature. Generalizations to an elliptical structural model are also investigated.