This paper investigates the construction and application of point-optimal invariant (POI) tests of joint AR(1)-AR(4) disturbances against jointMA(1)-MA(4) disturbances in the linear regression model. A Monte Carlo experiment is conducted to assess and compare the small sample performances of two asymptotic tests and two POI tests. Of the asymptotic tests, the Lagrange multiplier test is found to have distinctly better small sample properties than the prediction test. We also find that the extra computation required to perform a POI test is rewarded by a clear improvement in size and power properties in comparison to the asymptotic tests. The use of a POI test is illustrated with an application to a quarterly model of price inflation in the United Kingdom during 1947-1970. The paper concludes with some discussion of the problem of testing general AR(p) disturbances against MA(q) disturbances.