Research on Regime-Switching Between Different Stochastic Dynamical Systems

Regime-Switching (RS) is an important phenomenon and modeling technique in time series analysis that the observed process follows different patterns during different time periods. To the best of our knowledge, there is no study on the RS phenomenon between different dynamical systems. In this dissertation, we finished 3 projects on this topic. In Project I, we studied the RS modeling between different stochastic dynamical systems with known parameter forms. We proposed a heteroskedasticity-based E-M algorithm to infer this model since it cannot be estimated under the likelihood framework. We also proposed a hypothesis testing procedure, named as RS testing, to test whether the RS phenomenon exists or not through testing whether the state prediction agrees with the observation or not. We demonstrated its power by comprehensive simulations and proved that VIX has the RS phenomenon. In Project II, we extended this method to the Realized Variance (RV) processes of the stock market. To calculate RV, we proposed a novel data cleaning method for the TAQ transaction-level dataset to achieve a better trade-off between the data quality and data size. We proved that the RS phenomenon is universal in the stock market's volatility. In Project III, we studied the RS phenomenon in scientific processes. The state-of-the-art forecasting method for scientific processes is time-invariant Empirical Dynamic Modeling (EDM). In this project, we proposed a time-dependent EDM framework and proposed a Periodically-Regime-Switching (PRS) model to combine the strong periodicity and Markovian property of the latent state process. We proved our method's performance on the chlorophyll forecasting problem. At the end, we made a comprehensive discussion on these 3 projects and summarized the application scenarios.