For vector Itô semimartingale dynamics, we derive the asymptotic distributions of likelihood-ratio-type test statistics for the purpose of identifying the eigenvalue structure of both integrated and spot covariance matrices estimated using high-frequency data. Unlike the existing approaches where the cross-section dimension grows to infinity, our tests do not necessarily require large cross-section and thus allow for a wide range of applications. The tests, however, are based on non-standard asymptotic distributions with many nuisance parameters. Another contribution of this paper consists in proposing a bootstrap method to approximate these asymptotic distributions. While standard bootstrap methods focus on sampling point-wise returns, the proposed method replicates features of the asymptotic approximation of the statistics of interest that guarantee its validity. A Monte Carlo simulation study shows that the bootstrap-based test controls size and has power for even moderate size samples.

对于向量 Itô semimartingale 动力学，我们推导出似然比型检验统计量的渐近分布，以识别使用高频数据估计的积分和点协方差矩阵的特征值结构。与横截面尺寸增长到无穷大的现有方法不同，我们的测试不一定需要大横截面，因此允许广泛的应用。然而，这些测试基于具有许多有害参数的非标准渐近分布。本文的另一个贡献在于提出了一种引导方法来近似这些渐近分布。虽然标准的引导方法专注于采样逐点返回，但 所提出的方法复制了保证其有效性的感兴趣统计数据的渐近近似的特征。Monte Carlo 模拟研究表明，基于 bootstrap 的测试控制大小，并且即使是中等大小的样本也具有效力。