Bayesian prior modeling in vector autoregressions via the Yule-Walker equations

Abstract

In multivariate time series analysis, the Yule-Walker method refers to a system of equations relating the cross-covariances of a stationary vector autoregressive (VAR) model with the matrices of the autoregressive coefficients and the covariance matrix of the noise, both of which are unknown to be estimated. In Bayesian inference of VAR models, one of the key problems is the setting of the prior distributions on these unknown parameters. The Yule-Walker equations are used here to develop a novel prior specification that exploits the reparameterization of the unknowns in terms of the mean, the cross-covariances, and the covariance of the process. Further, the cross-covariance matrices are separated out in terms of the standard deviations and the correlations. All these new quantities are easier to handle because it is more common to have prior information on the mean and the correlation structure of a multiple time series rather than the underlying autoregressive coefficients and the white noise process. The proposed prior specification is suitable for both non informative and informative settings. Through the Yule-Walker based prior, parameter estimation and structure learning of the stationary VAR models are performed via Markov chain Monte Carlo methods. The methodology is illustrated via some synthetic data sets, a benchmark example, and an environmental time series.