The bootstrap for testing the equality of two multivariate time series with an application to financial markets

The problem of testing the equality of the generating processes of two multivariate time series is addressed in this work. To this aim, we construct four tests based on a distance measure between stochastic processes. The metric is defined in terms of the quantile cross-spectral densities of both processes. A proper estimate of this dissimilarity is the cornerstone of the proposed tests. The first test employs the asymptotic distribution of the estimate, which we derive from some standard results on complex random variables and which is useful in its own right. The bad behaviour of this test when compared with alternative ones is shown. The three remaining techniques are based on the bootstrap. Specifically, a particular bootstrap method for spectral densities and extensions of the moving blocks bootstrap and the stationary bootstrap are used for their construction. The approaches are assessed in a broad range of scenarios under the null and the alternative hypothesis. The results from the analyses show that the procedure based on the stationary bootstrap exhibits the best overall performance in terms of both size and power. The proposed techniques are used to answer the question about whether or not the dotcom bubble crash of 2000s permanently impacted the global market behavior.