The paper introduces a test for detecting bilinearity in a stochastic unit root process. It is shown that, under the null of no bilinearity, the tests statistics are asymptotically normally distributed. Finite sample results conclude that the suggested two-step testing procedure is consistent in the sense that the size of step one test is not affected by the possible detection of bilinearity at step two. The empirical part describes testing the unit root bilinearity for 64 world-wide stock market indices. It is shown that for at least 70% cases the hypothesis of no unit root bilinearity has to be rejected.