Testing for the equality of integration orders is an important topic in time series analysis because it constitutes an essential step in testing for (fractional) cointegration in the bivariate case. For the multivariate case, there are several versions of cointegration, and the version given in Robinson and Yajima (2002) has received much attention. In this definition, a time series vector is partitioned into several sub-vectors, and the elements in each sub-vector have the same integration order. Furthermore, this time series vector is said to be cointegrated if there exists a cointegration in any of the sub-vectors. Under such a circumstance, testing for the equality of integration orders constitutes an important problem. However, for multivariate fractionally integrated series, most tests focus on stationary and invertible series and become invalid under the presence of cointegration. Hualde (2013) overcomes these difficulties with a residual-based test for a bivariate time series. For the multivariate case, one possible extension of this test involves testing for an array of bivariate series, which becomes computationally challenging as the dimension of the time series increases. In this paper, a one-step residual-based test is proposed to deal with the multivariate case that overcomes the computational issue. Under certain regularity conditions, the test statistic has an asymptotic standard normal distribution under the null hypothesis of equal integration orders and diverges to infinity under the alternative. As reported in a Monte Carlo experiment, the proposed test possesses satisfactory sizes and powers.
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