In the context of the linear regression model, this paper considers testing for a single Hildreth-Houck random coefficient against the alternative that the coefficient follows the return to normalcy model. We attempt to construct a point-optimal invariant test but find that we have to resort to the class of approximate point-optimal invariant (APOI) tests introduced by King (1987). Empirical power calculations show that these tests have good small-sample properties compared to the likelihood ratio and Wald tests. A particular APOI test is recommended and is found to be remarkably robust to nonnormality.