This paper is concerned about testing whether a cross-covariance matrix deviates from a pre-assigned one or not. For this purpose, a new test statistic is constructed based on the Frobenius norm of the difference between the sample cross-covariance matrix and the pre-assigned matrix. The test is implemented by applying the parametric bootstrap scheme. We conduct a simulation study to examine the performance of the test and compare it with other competitive tests. As multiple simulation examples show, our empirical powers are clearly superior to others in detecting any deviation of the cross-covariance from the pre-assigned matrix. In addition, the proposed test is insensitive to non-cross-covariance elements in the covariance matrix. As an illustration, we also investigate its performance in testing pairwise time-reversibility.
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