Rank Tests at Jump Events^*

We propose a test for the rank of a cross-section of processes at a set of jump events. The jump events are either specific known times or are random and associated with jumps of some process. The test is formed from discretely sampled data on a fixed time interval with asymptotically shrinking mesh. In the first step, we form nonparametric estimates of the jump events via thresholding techniques. We then compute the eigenvalues of the outer product of the cross-section of increments at the identified jump events. The test for rank <i>r</i> is based on the asymptotic behavior of the sum of the squared eigenvalues excluding the largest <i>r</i>. A simple resampling method is proposed for feasible testing. The test is applied to financial data spanning the period 2007–2015 at the times of stock market jumps. We find support for a one-factor model of both industry portfolio and Dow 30 stock returns at market jump times. This stands in contrast with earlier evidence for higher-dimensional factor structure of stock returns during “normal” (non-jump) times. We identify the latent factor driving the stocks and portfolios as the size of the market jump.本文提出一种针对一组跳跃事件（jump events）下多过程截面秩的检验方法。该跳跃事件既可指代特定已知时点，也可指代与某一过程跳跃行为相关的随机时点。该检验基于固定时间区间内的离散采样数据构建，且采样网格具备渐近收缩特性。第一步，我们通过阈值化技术（thresholding techniques）完成跳跃事件的非参数估计（nonparametric estimates）。随后，我们计算已识别跳跃事件处各过程增量截面的外积（outer product）矩阵的特征值（eigenvalues）。针对秩小于<em>r</em>的检验，其构建基于排除前<em>r</em>个最大特征值后的平方特征值之和的渐近行为。本文提出一种简易重采样方法（resampling method）以实现可行的检验。我们将该检验方法应用于2007至2015年间覆盖股票市场跳跃时点的金融数据集。研究结果表明，在市场跳跃时点，行业投资组合与道琼斯30（Dow 30）成分股的收益率均符合单因子模型（one-factor model）。这与此前“常态”（非跳跃时期）下股票收益率存在高维因子结构的研究结论相悖。我们将驱动个股与投资组合收益的潜在因子识别为市场跳跃的幅度。