Shrinkage Estimation for Multivariate Hidden Markov Models

Motivated from a changing market environment over time, we consider high-dimensional data such as financial returns, generated by a hidden Markov model that allows for switching between different regimes or states. To get more stable estimates of the covariance matrices of the different states, potentially driven by a number of observations that are small compared to the dimension, we modify the expectation–maximization (EM) algorithm so that it yields the shrinkage estimators for the covariance matrices. The final algorithm turns out to reproduce better estimates not only for the covariance matrices but also for the transition matrix. It results into a more stable and reliable filter that allows for reconstructing the values of the hidden Markov chain. In addition to a simulation study performed in this article, we also present a series of theoretical results that include dimensionality asymptotics and provide the motivation for certain techniques used in the algorithm. Supplementary materials for this article are available online.