The paper considers the question of dominance, in the context of financial markets, of the deterministic unit root processes with a structural break by the bilinear unit root model without such break or vice versa. In the deterministic unit root process breaks are usually interpreted as exogenous, while the unit root bilinearity is mostly attributed to speculation. A series of Monte Carlo experiments show substantial size distortions in testing for the deterministic unit root process in the presence of unit root bilinearity and vice versa. To eliminate this problem, two additional tests are proposed here: one for the joint testing of the process with a structural break and unit root bilinearity, and the other for testing the unit root bilinearity conditional on the break. The asymptotic properties of these tests have been analysed. The tests are applied for the daily stock price indices for 63 countries, for the period 1992-2005. It has been found out that in 34 cases the bilinearity is present in the series, and in only two cases a structural break was discovered without the presence of bilinearity. Since for most of the series a possible break occurs either in 2000 or 2001, it sheds some new light on the reasons for the stock market breakdown at the beginning of the 21st century.
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