Inference on the autocorrelation coefficient p of a linear regression model with first-order autoregressive normal disturbances is studied. Both stationary and nonstationary processes are considered. Locally best and point-optimal invariant tests for any given value of p are derived. Special cases of these tests include tests for independence and tests for unit root hypotheses. The powers of alternative tests are compared numerically for a number of selected testing problems and for a range of design matrices. The results suggest that point-optimal tests are usually preferable to locally best tests, especially for testing values of p greater than or equal to one.