On the Detection of Contemporaneous Relationships among Multiple Time Series

The Box and Tiao (1977) canonical time series approach considers linear combinations of multiple time series and ranks them according to their predictability. When dealing with individually non stationary sequences, it is the least predictable components that are of interest as they suggest cointegrated stationary relationships. In this article, we review the Box and Tiao (1977) approach and point out a feature that escaped notice for more than 30 years. Their measure of the predictability of a linear combination assumes that the history on all individual series is available, and not just the past realizations of that particular linear combination. As a consequence, their canonical analysis leads to only a lower bound for the number of unpredictable components.

In this article, we investigate the sampling properties of the canonical analysis approach and study through simulations whether this method can detect the number of cointegrated relationships and can estimate their coefficients. On the basis of our simulation results, we propose to supplement the canonical analysis with a regression approach that constructs the cointegrated relationships from regressions on a smaller subset of the series. Two examples are considered for illustration: the annual Chinese real income by sector, and the Quenouille hog data considered by Box and Tiao in their original article.