Cross-Correlation Matrices for Tests of Independence and Causality Between Two Multivariate Time Series

Abstract

An often-studied problem in time series analysis is that of testing for the independence of two (potentially multivariate) time series. Toeplitz matrices have demonstrated utility for the related setting of time series goodness-of-fit testing—ergo, herein, we extend those concepts by defining a nontrivial block Toeplitz matrix for use in the setting of independence testing. We propose test statistics based on the trace of the square of the matrix and determinant of the matrix; these statistics are connected to one another as well as known statistics previously proposed in the literature. Furthermore, the log of the determinant is argued to relate to a likelihood ratio test and is proven to be more powerful than other tests that are asymptotically equivalent under the null hypothesis. Additionally, matrix-based tests are presented for the purpose of inferring the location or direction of the causality existing between the two series. A simulation study is provided to explore the efficacy of the proposed methodology—the methods are shown to offer improvement over existing techniques, which include the famous Granger causality test. Finally, data examples involving U.S. inflation, trade volume, and exchange rates are given. Supplementary materials for this article are available online.

KEY WORDS:

• Granger causality
• Residual diagnostics
• Vector ARMA