Locally adaptive Bayesian covariance regression

Multivariate time series data arise in many applied domains, and it is often crucial to obtain a good characterization of how the covariance among the different variables changes over time. Certainly this is the case in financial applications in which covariance can change dramatically during times of financial crisis, revealing different associations among assets and countries than occur in a healthier economic climate. Our focus is on developing models that allow the covariance to vary flexibly over continuous time, and additionally accommodate locally adaptive smoothing of the covariance. Locally adaptive smoothing to accommodate varying smoothness in a trajectory over time has been well studied, but such approaches have not yet been developed for time-varying covariance matrices to our knowledge. To address this gap, we generalize recently develop methods for Bayesian covariance regression to incorporate random dictionary elements with locally varying smoothness. Using a differential equation representation, we additionally develop a fast computational approach via MCMC, with online algorithms also considered. The performance of the models is assessed through simulation studies and the methods are applied to financial time series.