By combining (i) the economic theory of rational expectation bubbles, (ii) behavioral finance on imitation and herding of investors and traders and (iii) the mathematical and statistical physics of bifurcations and phase transitions, the logperiodic power law (LPPL) model has been developed as a flexible tool to detect bubbles. The LPPL model considers the faster-than-exponential (power law with finite-time singularity) increase in asset prices decorated by accelerating oscillations as the main diagnostic of bubbles. It embodies a positive feedback loop of higher return anticipations competing with negative feedback spirals of crash expectations. We use the LPPL model in one of its incarnations to analyze two bubbles and subsequent market crashes in two important indexes in the Chinese stock markets between May 2005 and July 2009. Both the Shanghai Stock Exchange Composite index (US ticker symbol SSEC) and Shenzhen Stock Exchange Component index (SZSC) exhibited such behavior in two distinct time periods: 1) from mid-2005, bursting in October 2007 and 2) from November 2008, bursting in the beginning of August 2009. We successfully predicted time windows for both crashes in advance [24, 1] with the same methods used to successfully predict the peak in mid-2006 of the US housing bubble [37] and the peak in July 2008 of the global oil bubble [26]. The more recent bubble in the Chinese indexes was detected and its end or change of regime was predicted independently by two groups with similar results, showing that the model has been well-documented and can be replicated by industrial practitioners. Here we present more detailed analysis of the individual Chinese index predictions and of the methods used to make and test them. We complement the detection of log-periodic behavior with Lomb spectral analysis of detrended residuals and (H, q)-derivative of logarithmic indexes for both bubbles. We perform unit-root tests on the residuals from the log-periodic power law model to confirm the Ornstein-Uhlenbeck property of bounded residuals, in agreement with the consistent model of ‘explosive’ financial bubbles [16].

According to the theory of unit root test, Lee-Carter model and generalized linear model, which are widely used in mortality projection, impose key implicit assumptions respectively which are inconsistent with each other. Log mortality rate (the force of mortality or the central mortality rate) is described as a unit root process in Lee-Carter model, while it is modeled as a deterministic trend process in generalized linear model. We use panel LM unit-root tests with level shifts to test the assumptions in above models, based on mortality data of the 7 most developed countries(G7) and Nordic countries(Denmark, Finland, Norway, Sweden). The test results show that a mortality projection model, whatever it is Lee-Carter model or generalized linear model, is not always appropriate to predict dynamic mortality rates of different countries. Further, we explain period effect and cohort effect of dynamic mortality according to the results of structural break test. Based on the empirical results, we extend Lee-Carter model, which includes a special case of generalized linear model. To check the performance of the extended model, we use it to forecast USA and Sweden mortality and we find that the extended Lee-Carter model works better than the original Lee-Carter model.