SummaryWe investigate the collinearity of vector time series in the frequency domain, by examining the rank of the spectral density matrix at a given frequency of interest. Rank reduction corresponds to collinearity at the given frequency. When the time series is nonstationary and has been differenced to stationarity, collinearity corresponds to co-integration at a particular frequency. We examine rank through the Schur complements of the spectral density matrix, testing for rank reduction via assessing the positivity of these Schur complements, which are obtained from a nonparametric estimator of the spectral density. New asymptotic results for the test statistics are derived under the fixed bandwidth ratio paradigm; they diverge under the alternative, but under the null hypothesis of collinearity the test statistics converge to a non-standard limiting distribution. Subsampling is used to obtain the limiting null quantiles. A simulation study and an empirical illustration for 6-variate time series data are provided.

中文翻译：

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测试向量时间序列的共线性

总结我们通过检查给定感兴趣频率下频谱密度矩阵的秩，研究了频域中向量时间序列的共线性。等级降低对应于给定频率下的共线性。当时间序列是非平稳的并且已经变得平稳时，共线性对应于特定频率下的协整。我们通过频谱密度矩阵的Schur补码检查等级，通过评估这些Schur补码的阳性来测试等级降低，这是从光谱密度的非参数估计器获得的。在固定带宽比率范式下得出了用于测试统计量的新渐近结果；他们在另一种情况下有所分歧，但是在共线性的零假设下，检验统计量收敛到非标准的极限分布。二次采样用于获得极限零分位数。提供了六变量时间序列数据的仿真研究和经验例证。