On Testing for Independence between Several Time Series

Abstract

Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more that two time series, checking pairwise independence does not lead to consistent procedures. Thus a finite family of empirical processes relying on multivariate lagged residuals are constructed, and we derive their asymptotic distributions.In order to obtain simple asymptotic covariance structures, Mobius transformations of the empirical processes are studied, and simplifications occur. Under the null hypothesis of independence, we show that these transformed processes are asymptotically Gaussian, independent, and with tractable covariance functions not depending on the estimated parameters. Various procedures are discussed, including Cramer-von Mises test statistics and tests based on non-parametric measures. The ranks of the residuals are considered in the new methods, giving tests statistics which are asymptotically margin-free. Generalized cross-correlations are introduced, generalizing the concept of cross-correlation to an arbitrarily number of time series; portmanteau procedures based on them are discussed. In order to detect the dependence visually, graphical devices are proposed. Simulations are conducted to explore the finite sample properties of the methodology, which is found to be powerful against various types of alternatives when the independence is tested between two and three time series. An application is considered, using the daily log-returns of Apple, Intel and Hewlett-Packard traded on the Nasdaq financial market.