The comparison of two time series often arises in climatology, environmental science, and econometrics. Through natural and physical circumstances these series are often dependent. We develop a hypothesis test for the equality of autocovariance functions for two linearly dependent multivariate time series. Previous tests for two independent series are reviewed and extended to the dependent case. A univariate bootstrapped statistic that automatically selects the order of the test is extended to the multivariate setting as well. The performance of the tests are compared through simulation and the methods are applied to univariate temperature and multivariate air quality series. Empirical results show that by accounting for the correlation between series substantial improvements in power can be made in the detection of differences in the autocovariance.