Conditionally linear, matrix normal state space models
We develop a class of linear state space models for matrix-valued time series data where the state is a latent matrix normal process. We derive matrix versions of the Kalman filter, log-likelihood, and smoother enabling estimation of the latent state matrix as well as the model's parameters. To conduct Bayesian inference, we provide algorithms that draw from the joint posterior distribution of the latent state matrices conditional on the observed data and parameters. We apply these methods to a large panel of U.S. macroeconomic time series across the 50 U.S. states. The proposed framework accommodates mixed-frequency data, heteroskedasticity, and outliers within a unified matrix-valued structure. Empirically, we find that a small number of latent factors captures the joint dynamics across states and variables, providing a parsimonious and scalable approach to modeling high-dimensional macroeconomic systems.