Bayesian Copula Factor Autoregressive Models for Time Series Mixed Data

In this paper, we propose a Bayesian copula factor autoregressive (BCFAR) model for analyzing time series mixed data, accommodating main effects and interactions. Our main motivation lies in inferring dynamic interactions between macroeconomic variables and stock market indices. The BCFAR model assumes conditional independence and applies latent factors in both response time series and high-dimensional mixed-type covariates in quadratic time series regression using copula functions. To complement this, we introduce a simpler time series Bayesian factor regression (TS-BFR) model, designed specifically for continuous Gaussian multivariate time series data. Both models leverage the quadratic autoregression (QAR) framework, employ latent factors for efficient dimension reduction, and characterize the main effects and interactions of the covariates by integrating the latent variables into the response time series. To enhance computational efficiency, we employ a semiparametric time series extended rank likelihood for the margins of explanatory variables in the BCFAR model, reducing the number of estimated parameters and ensuring a fast computational algorithm. To compute the posterior distribution of latent factors and model parameters, we introduce flexible Bayesian algorithms, employing Metropolis-Hastings (MH) and Forward Filtering Backward Sampling (FFBS) algorithms within Gibbs sampling. The effectiveness of these algorithms is demonstrated through rigorous simulation studies, and we further validate our approach with the analysis of a macroeconomic dataset.