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 dierent variables changes over time. Certainly this is the case in nancial applications in which co- variance can change dramatically during times of nancial crisis, revealing dierent associations among assets and countries than occur in a healthier economic climate. Our focus is on developing models that allow the covariance to vary exibly over con- tinuous time, and additionally accommodate locally adaptive smoothing of the covari- ance. 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 dierential equation rep- resentation, 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 nancial time series.