On testing for causality in variance between two multivariate time series

Verifying the existence of a relationship between two multivariate time series represents an important consideration. In this article, the procedure developed by Cheung and Ng [A causality-in-variance test and its application to financial market prices, J. Econom. 72 (1996), pp. 33-48] designed to test causality in variance for univariate time series is generalized in several directions. A first approach proposes test statistics based on residual cross-covariance matrices of squared (standardized) residuals and cross products of (standardized) residuals. In a second approach, transformed residuals are defined for each residual vector time series, and test statistics are constructed based on the cross-correlations of these transformed residuals. Test statistics at individual lags and portmanteau-type test statistics are developed. Conditions are given under which the new test statistics converge in distribution towards chi-square distributions. The proposed methodology can be used to determine the directions of causality in variance, and appropriate test statistics are presented. Monte Carlo simulation results show that the new test statistics offer satisfactory empirical properties. An application with two bivariate financial time series illustrates the methods.