We consider the power functions of five popular tests for AR(1) errors in a linear regression model from which relevant regressors have inadvertently been omitted. These functions are derived by numerically evaluating the finite-sample distributions of the test statistics. With this form of model mis-specification, it is found that the performances of the tests are not independent of the scale of the errors' distribution. The omission of seasonal effects or a linear trend component can have serious implications, especially if testing against positive autocorrelation, and some of the well known advantages of the "Alternative Durbin Watson test" (King (1981)) are found to still apply when the model is underspecified.