Multivariate stochastic regression in time series modeling

This paper begins with a brief review of multivariate time series analysis, covering canonical correlation analysis and scalar components of vector ARMA models, pioneered by Tiao and his collaborators, and vector ARMAX models in linear systems theory. It then presents a fast stepwise regression procedure that includes parsimonious variable selection followed by rank selection in stochastic regression models. The procedure overcomes a long-standing difficulty with parameter estimation in these models, the dauntingly large number of parameters in the matrix of regression coefficients relative to the sample size n. Recent attempts to address this difficulty have used group lasso and hard thresholding of small singular values to take advantage of coefficient and rank sparsity. However, the underlying theory is based on non-random or independent regressors, whereas the procedure and its underlying theory developed herein are applicable to stochastic regressors in multivariate time series models.