A complete class of tests when the likelihood is locally asymptotically quadratic

We construct an asymptotically complete class of tests (i.e. for a test not within this class one can find a test within the class that has a better or equal asymptotical power function) for testing one parametric restriction, when the likelihood function can be approximated by a quadratic function asymptotically: The coefficients of this quadratic form may be random variables, so that our results apply to the problem of unit-root testing: We show that the LR test (in the case of unit roots the DF-test) is not within this class.