Optimal Power for Testing Potential Cointegrating Vectors with Known

Theory often specifies a particular cointegrating vector amongst integrated variables and it is often required that one test for a unit root in the known cointegrating vector. It is common to simply employ a univariate test for a unit root, but this does not take into account all available information. We show 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 characterize the extent of the losses of not imposing the cointegrating vector in the testing procedure. We apply various tests to the hypothesis 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.