Abstract: To enrich the theory of asset price bubble detection and use the testing model in the empirical analysis correctly, based on the fixed rolling window mode, the statistic distribution of unit root and intercept are studied according to whether the data generation process contains drift terms.The theoretical results show that the related test statistics converge to the functional of Wiener process under large sample size. To facilitate the use of these test statistics, Monte Carlo simulation is used to obtain critic values for several significant levels commonly used in finite sample size. Simulation results show that these critic values increase with sample size and decrease with window parameters. Empirical analysis suggests that, based on the classical model, the intercept should be checked to determine whether it is zero or not to select the appropriate test model as well as to improve the test power.