Theory often specifies a particular cointegrating vector among integrated variables, and testing for a unit root in the known cointegrating vector is often required. Although it is common to simply use a univariate test for a unit root for this test, it is known that this does not take into account all available information. We show here that in such testing situations, a family of tests with optimality properties exists. We use this to characterize the extent of the loss in power from using popular methods, as well as to derive a test that works well in practice. We also characterize the extent of the losses of not imposing the cointegrating vector in the testing procedure. We apply various tests to the hypothesis positing that price forecasts from the Livingston data survey are cointegrated with prices, and find that although most tests fail to reject the presence of a unit root in forecast errors, the tests presented here strongly reject this (implausible) hypothesis.