Markov switching quantile regression models with time-varying transition probabilities

Markov switching models are widely used in the time series field for their ability to describe the impact of latent regimes on the behaviour of response variables. Meanwhile, Markov switching quantile regression models with fixed transition probabilities (MSQR-FTP) also provide rich dynamics to modeling financial data, however, it is not always clear how to describe the dynamics on the transition probabilities. This paper extends the transition probabilities to be the time-varying case by allowing them to include information from related variables. By establishing a connection between a quantile regression and an asymmetric Laplace distribution, this paper proposes a maximum likelihood estimation (MLE) method for MSQR-TVTP, and shows the consistency of the MLE. Finally, the performance of the proposed method is illustrated through a simulation study. As an empirical application, we further apply the method to the S&P 500 weekly percentage returns.