This paper introduces a t-ratio type test for detecting bilinearity in a stochastic unit root process. It appears that such a process is a realistic approximation for many economic and financial time series. It is shown that, under the null of no bilinearity, the test statistics are asymptotically normally distributed. Proofs of these asymptotic normality results require the Gihman and Skorohod theory for multivariate diffusion processes. Finite sample results describe speed of convergence, power of the tests and possible distortions to unit root testing which might appear due to the presence of bilinearity. It is concluded that the two-step testing procedure suggested here (the first step for the linear unit root and the second step for its bilinearity) is consistent in the sense that the size of the step one test is not affected by the possible detection of bilinearity at step two. The empirical part of the paper describes testing unit root bilinearity for 65 GARCH-adjusted stock market indices from mature and emerging markets. It is shown that for at least 70% of these markets, the hypothesis of no unit root bilinearity has to be rejected.