Abstract: Bubble refers to the explosive growth of asset prices, and the crash is rapidly decline of stock or stock index after the bubble bursts. A series of asset prices plummet in a short period of time causing a sharp oscillation in the stock market, which in turn damage the asset interests of investorsand disrupt the orderly development of the capital market.In this paper, a new method is developed through the Log-Periodic Power Law Model (LPPL) to help investors identify bubbles and track the risks of bubbles in financial activities effectively. The LPPL model is ln［p(t)］=A+(tc－t)α{B+Ccos［ωln(tc－t)+φ］}, where ln［p(t)］ is the logarithm of assets price, α is the exponent of the power law growth and tc is the critical point of bubble burst.The two big bull market crashes of the Shanghai and Shenzhen 300 Index (CSI 300) in 2007 and 2015 are studied. Based on the LPPL model, two methods including “rolling window” and “fixing the initial point and moving the end point” are introduced for resampling, then out-of-sample prediction daily is made to calculate the critical point of the bubble bursts and construct the dynamic confidence intervals for the bubble critical times respectively. Theempirical results show that the confidence interval of the critical times can basically cover the time when the bubbles crash actually occurs as time goes by. Compared with the simple method that use the LPPL model to predict the single critical time, the confidence interval method overcomes the randomness of the critical time prediction well, and also well display the change trajectory of the stock market bubble critical interval. The innovative method of this paper provides an effective tool for the risk management, which is a beneficial expansion for the researches of the crash early warning.

Key words: log-periodic power lawmodel；stock crash；financial bubbles；confidence interval