A statistical model of speculative bubbles, with applications to the stock markets of the United States, Japan, and China

It is common knowledge that the more prices deviate from fundamentals, the more likely it is for prices to reverse. Taking this into account, we propose a simple statistical model to identify speculative bubbles in financial markets. Through the estimates of the time varying parameters, including transition probabilities, we can identify when and how newly born bubbles grow and burst over time. The model can be estimated by recursive computations, which require a huge storage capacity for standard computers. For this reason, we introduce an approximation in the computation, maintaining the recursive nature of our estimation technique. We then apply this model to the stock markets of the United States, Japan, and China, estimate its parameters and the probabilities of a bubble crash, and obtain several interesting results: the time series data of the stock price bubble show an inherently non-stationary development and the probability of a bubble crash indeed increases as the stock price becomes too high or too low.