Robust Jump Regressions

We develop robust inference methods for studying linear dependence between the jumps of discretely observed processes at high frequency. Unlike classical linear regressions, jump regressions are determined by a small number of jumps occurring over a fixed time interval and the rest of the components of the processes around the jump times. The latter are the continuous martingale parts of the processes as well as observation noise. By sampling more frequently the role of these components, which are hidden in the observed price, shrinks asymptotically. The robustness of our inference procedure is with respect to outliers, which are of particular importance in the current setting of relatively small number of jump observations. This is achieved by using nonsmooth loss functions (like L1) in the estimation. Unlike classical robust methods, the limit of the objective function here remains nonsmooth. The proposed method is also robust to measurement error in the observed processes, which is achieved by locally smoothing the high-frequency increments. In an empirical application to financial data, we illustrate the usefulness of the robust techniques by contrasting the behavior of robust and ordinary least regression (OLS)-type jump regressions in periods including disruptions of the financial markets such as so-called “flash crashes.” Supplementary materials for this article are available online.

本研究致力于开发稳健的推断方法，以研究高频离散观测过程中跳跃的线性相关性。与经典的线性回归不同，跳跃回归由固定时间间隔内发生的一小部分跳跃所决定，而跳跃时间周围的其余过程成分则构成跳跃回归的其余部分。后者包括过程的连续鞅部分以及观测噪声。通过更频繁地采样，这些隐藏在观测价格中的成分的作用逐渐减小。我们的推断程序在异常值方面的稳健性，在当前跳跃观测数量相对较少的设置中尤为重要。这是通过在估计中使用非光滑损失函数（如L1）来实现的。与经典的稳健方法不同，此处目标函数的极限仍然是非光滑的。所提出的方法对观测过程中的测量误差也具有稳健性，这是通过局部平滑高频增量来实现的。在金融数据的实证应用中，我们通过对比包括金融市场中断（如所谓的“闪崩”）在内的时期的稳健回归和普通最小二乘（OLS）型跳跃回归的行为，展示了稳健技术在金融数据中的应用价值。本文的补充材料可在网上获得。