This paper develops a general nonparametric test for the null hypothesis that the vector of time series under scrutiny is temporally and cross sectionally independent. The null class of the models can be extended to include weak dependence. This test can be used to test the adequacy of a fitted model. As an application of the test we show how we test diagnostically a vector auto regressive model fitted to the given data. This procedure is legitimate because the first order asymptotic distribution of the test statistic is robust to the estimated residual vector.
This paper is published in Baek, Ehung G., and William A. Brock. "A nonparametric test for independence of a multivariate time series." Growth Theory, Nonlinear Dynamics, and Economic Modelling: Scientific Essays of William Allen Brock (2001): 304.
Baek, Khung G. and Brock, William A., "A Nonparametric Test For Independence Of A Multivariate Time Series" (1989). Economic Staff Paper Series. 197.