Bayes shrinkage estimation for high-dimensional VAR models with scale mixture of normal distributions for noise

We propose Bayesian shrinkage methods for coefficient estimation for high-dimensional vector autoregressive (VAR) models using scale mixtures of multivariate normal distributions for independently sampled additive noises. We also suggest an efficient selection procedure for the shrinkage parameter as a computationally feasible alternative to the traditional MCMC sampling methods for high-dimensional data. A shrinkage parameter is selected at the minimum point of a newly proposed score function which is asymptotically equivalent to the mean squared error of the model coefficients. The selected shrinkage parameter is presented in a closed form as a function of sample size, level of noise, and non-normality in data, and it can be efficiently estimated by using a suggested variation of cross validation. Consistency of both of the cross validation estimator and proposed shrinkage estimator is proved. The competitiveness of the proposed methods is demonstrated based on comprehensive experimental results using simulated data and high-dimensional plant gene expression data in the context of coefficient estimation and structural inference for VAR models. The proposed methods are applicable to high-dimensional stationary time series with or without near unit roots.