We study the pairwise and mutual independence testing problem for multivariate functional data. Using a basis representation of functional data, we reduce this problem to testing the independence of multivariate data, which may be high-dimensional. For pairwise independence, we apply tests based on distance and Hilbert-Schmidt covariances as well as their marginal versions, which aggregate these covariances for coordinates of random processes. In the case of mutual independence, we study asymmetric and symmetric aggregating measures of pairwise dependence. A theoretical justification of the test procedures is established. In extensive simulation studies and examples based on a real economic data set, we investigate and compare the performance of the tests in terms of size control and power. An important finding is that tests based on distance and Hilbert-Schmidt covariances are usually more powerful than their marginal versions under linear dependence, while the reverse is true under non-linear dependence.

Discrimination among multivariate autoregressive processes by Bayes' method is described. The determination of the true order of the multivariate autoregressive equation is also investigated; the estimator of the true order is obtained by minimizing the posterior risk.