Estimation and testing nonhomogeneity of Hidden Markov model with application in financial time series

Both homogeneous and nonhomogeneous Hidden Markov models (HMM) have been gaining increased attention in financial time series modeling. The homogeneous HMM assumes constant transition probabilities, while nonhomogeneous HMM assumes varying transition matrix depended on some covariates. While both assumptions may seem plausible in different applications, there is a lack of studies from a statistical inference aspect. In this paper, we study the nonhomogeneous hidden Markov model, and propose an estimation via a modified EM algorithm, the kernel regression and local likelihood techniques. The motivation for this new procedure is that it enables us to employ a generalized likelihood ratio test procedure to test whether the transition matrix actually depends on a specific covariate. We propose the CV method to select bandwidth and the BIC method to select number of states, and further propose conditional boot-strap method to assess the standard errors of the estimates. We conduct a simulation study to demonstrate our procedure, and show that the Wilk's type of phenomenon holds for the proposed model. Furthermore, we analyze S&P 500 Index return data. Our analysis reveals different patterns in bull and bear markets, and show that the time varying transitions are statistically significant.